# Publications

Note that the pdf files archived here are the papers submitted by the authors as allowed by the journals' copyright policies. Please see the links for the copy of record, which may have different formatting and slightly different content. Also note that code implementing many of the semidefinite optimization techniques proposed in the following papers is available in the package SDP_PF in the latest release of MATPOWER.

[ Preprints ] [ 2024 ] [ 2023 ] [ 2022 ] [ 2021 ] [ 2020 ] [ 2019 ] [ 2018 ] [ 2017 ] [ 2016 ] [ 2015 ] [ 2014 ] [ 2013 ] [ 2012 ] [ 2011 ] [ 2010 ]

[ Monographs ] [ Journal Papers ] [ Conference Proceedings ] [ Ph.D. Dissertation ] [ Master's Thesis ] [ Technical Reports ]

M. Pollack, R. Piansky, S. Gupta, A. Kody, and

**D.K. Molzahn**, "Equitably Allocating Wildfire Resilience Investments for Power Grids – The Curse of Aggregation and Vulnerability Indices," submitted.

Wildfires ignited by power systems infrastructure are among the most destructive wildfires; hence some utility companies in wildfire-prone regions have pursued a proactive policy of emergency power shutoffs. These shutoffs, while mitigating the risk of disastrous ignition events, result in power outages that could negatively impacts vulnerable communities. In this paper, we consider how to equitably allocate funds to underground and effectively de-risk power lines in transmission networks. We explore the impact of the 2021 White House resource allocation policy called the Justice40 initiative, which states that 40% of the benefits of federally-funded climate-related investments should go to socially vulnerable communities. The definition of what constitutes a vulnerable community varies by organization, and we consider two major recently proposed vulnerability indices: the Justice40 index created under the 2021 White House and the Social Vulnerability Index (SVI) developed by the Center for Disease Control and Prevention (CDC). We show that allocating budget according to these two indices fails to reduce power outages for indigenous communities and those subject to high wildfire ignition risk using a high-fidelity synthetic power grid dataset that matches the key features of the Texas transmission system. We discuss how aggregation of communities and "one size fits all" vulnerability indices might be the reasons for the misalignment between the goals of vulnerability indices and their realized impact in this particular case study. We provide a method of achieving an equitable investment plan by adding group-level protections on percentage of load that is shed across each population group of interest. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri, R. Gupta and

**D.K. Molzahn**, "Optimized LinDistFlow for High-Fidelity Power Flow Modeling of Distribution Networks," submitted.

The DistFlow model accurately represents power flows in distribution systems, but the model's nonlinearities result in computational challenges for many optimization applications. Accordingly, a linear approximation known as LinDistFlow is commonly employed. This paper introduces an algorithm for enhancing the accuracy of the LinDistFlow approximation, with the goal of aligning the outputs more closely with those from the nonlinear DistFlow model. Using sensitivity information, our algorithm optimizes the LinDistFlow approximation's coefficient and bias parameters to minimize discrepancies in predictions of voltage magnitudes relative to the nonlinear DistFlow model. The algorithm employs the Truncated Newton Conjugate-Gradient (TNC) optimization method to fine-tune coefficients and bias parameters during an offline training phase in order to improve the LinDistFlow approximation's accuracy in optimization applications. Numerical results underscore the algorithm's efficacy, showcasing accuracy improvements in L1-norm and L∞-norm losses of up to 92% and 88%, respectively, relative to the traditional LinDistFlow model. We assess how the optimized parameters perform under changes in the network topology and also validate the optimized LinDistFlow approximation's efficacy in a hosting capacity optimization problem. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Gupta and

**D.K. Molzahn**, "Improving Fairness in Photovoltaic Curtailments via Daily Topology Reconfiguration for Voltage Control in Power Distribution Networks," submitted.

In PV-rich power distribution systems, over-voltage issues are often addressed by curtailing excess generation from PV plants (in addition to reactive power control), raising fairness concerns. Existing fairness-aware control schemes tackle this problem by incorporating fairness objectives into the cost function. However, such schemes result in increased overall curtailments. This paper proposes a solution through daily topology reconfiguration, ensuring that different PV plants face varying grid conditions each day, leading to different curtailment levels and enhancing fairness. We illustrate that implementing this approach enhances overall fairness without significantly increasing overall curtailments. The optimization problem involves two stages. The day-ahead stage optimizes the network topology using day-ahead forecasts of PV generation and demand, minimizing net curtailment and accounting for fairness based on curtailments from prior days. The real-time stage implements the optimized topology and computes active and reactive power setpoints for the PV plants. Day-ahead grid constraints are modeled using LinDistFlow, and real-time control employs a linearized model with a first-order Taylor approximation. The proposed scheme is numerically validated on several benchmark test cases. Results are compared using the Jain Fairness Index, considering fairness and reconfiguration scenarios. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah and

**D.K. Molzahn**, "A Fault-Tolerant Distributed Termination Method for Distributed Optimization Algorithms," submitted.

This paper proposes a fully distributed termination method for distributed optimization algorithms solved by multiple agents. The proposed method guarantees terminating a distributed optimization algorithm after satisfying the global termination criterion using information from local computations and neighboring agents. The proposed method requires additional iterations after satisfying the global terminating criterion to communicate the termination status. The number of additional iterations is bounded by the diameter of the communication network. This paper also proposes a fault-tolerant extension of this termination method that prevents early termination due to faulty agents or communication errors. We provide a proof of the method's correctness and demonstrate the proposed method by solving the optimal power flow problem for electric power grids using the alternating direction method of multipliers. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, K.R. Davis, and M.L. Crow, "Tightening QC Relaxations of AC Optimal Power Flow through Improved Linear Convex Envelopes," submitted.

AC optimal power flow (AC OPF) is a fundamental problem in power system operations. Accurately modeling the network physics via the AC power flow equations makes AC OPF a challenging nonconvex problem. To search for global optima, recent research has developed various convex relaxations that bound the optimal objective values of AC OPF problems. The QC relaxation convexifies the AC OPF problem by enclosing the non-convex terms within convex envelopes. The QC relaxation's accuracy strongly depends on the tightness of these envelopes. This paper proposes two improvements for tightening QC relaxations of OPF problems. We first consider a particular nonlinear function whose projections are the nonlinear expressions appearing in the polar representation of the power flow equations. We construct a polytope-shaped convex envelope around this nonlinear function and derive convex expressions for the nonlinear terms using its projections. Second, we use sine and cosine expression properties, along with changes in their curvature, to tightness this convex envelope. We also propose a coordinate transformation to tighten the envelope by rotating power flow equations based on individual bus-specific angles. We compare these enhancements to a state-of-the-art QC relaxation method using PGLib-OPF test cases, revealing improved optimality gaps in 68% of the cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, R. Harris, S. Litchfield, D. Huggins, and

**D.K. Molzahn**, "Detecting Shared Data Manipulation in Distributed Optimization Algorithms," submitted.

This paper investigates the vulnerability of the Alternating Direction Method of Multipliers (ADMM) algorithm to shared data manipulation, with a focus on solving optimal power flow (OPF) problems. Deliberate data manipulation may cause the ADMM algorithm to converge to suboptimal solutions. We derive two sufficient conditions for detecting data manipulation based on the theoretical convergence trajectory of the ADMM algorithm. We evaluate the detection conditions' performance on three data manipulation strategies we previously proposed: simple, feedback, and bilevel optimization attacks. We then extend these three data manipulation strategies to avoid detection by considering both the detection conditions and a neural network (NN) detection model in the attacks. We also propose an adversarial NN training framework to detect shared data manipulation. We illustrate the performance of our data manipulation strategy and detection framework on OPF problems. The results show that the proposed detection conditions successfully detect most of the data manipulation attacks. However, a bilevel optimization attack strategy that incorporates the detection methods may avoid being detected. Countering this, our proposed adversarial training framework detects all the instances of the bilevel optimization attack. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Harris, M. Alkhraijah, and

**D.K. Molzahn**, "Optimally Managing the Impacts of Convergence Tolerance for Distributed Optimal Power Flow," submitted.

The future power grid may rely on distributed optimization to determine the set-points for huge numbers of distributed energy resources. There has been significant work on applying distributed algorithms to optimal power flow (OPF) problems, which require separate controllers to agree on shared boundary variable values. Looser tolerances for the mismatches in these shared variables generally yield faster convergence at the expense of exacerbating constraint violations, but there is little quantitative understanding of how the convergence tolerance affects solution quality. To address this gap, we first quantify how convergence tolerance impacts constraint violations when the distributed OPF solution is applied to the power system. Using insights from this analysis, we then develop a bound tightening algorithm which guarantees that operating points from distributed OPF algorithms will not result in violations despite the possibility of shared variable mismatches within the convergence tolerance. We also explore how bounding the cumulative shared variable mismatches can prevent unnecessary conservativeness in the bound tightening. While ensuring feasibility, the proposed approach enables control of the trade-off between computational speed, which improves with looser convergence tolerance, and distributed OPF solution cost, which degrades with looser convergence tolerance due to tightened constraints. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason, S. Misra, J.P. Watson, and

**D.K. Molzahn**, "Adaptive Power Flow Approximations with Second-Order Sensitivity Insights," submitted.

The power flow equations are fundamental to power system planning, analysis, and control. However, their inherent non-linearity and non-convexity can pose significant challenges when solving problems that involve these equations. To address these challenges, linear approximations are frequently used to simplify computations at the potential cost of compromising accuracy and feasibility. Typical power flow approximations rely on general assumptions that are not always valid. Additionally, many existing linear approximations, such as the first-order Taylor series, overlook important information contained within higher-order terms. To gain a deeper understanding of the relationships among variables within a specific operational range and to achieve more accurate approximations, this paper considers a second-order sensitivity analysis of the power flow equations. By modeling the curvature of the power flow manifold, this paper uses the second-order sensitivities in an importance sampling method to improve the accuracy of adaptive power flow linearizations. Furthermore, this paper explores approximations formulated as rational functions with linear numerators and denominators that can be rewritten as linear constraints in power system optimization problems. These approximations, inspired by the Pad\'e approximant, utilize second-order power flow sensitivities. This approach is extended to enhance accuracy beyond what linear functions can achieve across various operational scenarios, and can also be constructed to be conservative (over- or under-estimate a quantity of interest). [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]C. Hettle, S. Gupta, and

**D.K. Molzahn**, "Fair and Reliable Reconnections for Temporary Disruptions in Electric Distribution Networks using Submodularity," submitted.

The economic cost of power outages has been estimated to be between $35 billion and $50 billion annually in the United States. We analyze and advance methodologies for electrical distribution network reconfiguration in order to quickly restore power, an important lever for improving reliability. For short-term network disruptions, we give a novel model for reconfiguration by considering switches that can be sequentially closed upon detection of a network anomaly upstream. The order in which switches close has an impact on reliability metrics such as SAIDI, which are the basis for regulator-imposed financial performance incentives. We introduce the "Minimum Reconnection Time (MRT" problem, which aims to find an optimal switch ordering that minimizes the cost of outages and show that it generalizes metrics like SAIDI. We first show that MRT is a special case of the well-known minimum linear ordering problem from submodular optimization literature and that MRT is also NP-hard. We show how to approximate MRT using recent kernel-based randomized rounding approaches, and in turn improve the state-of-the-art for a broad class of MLOP instances. Finally, we note that the choice of reliability metric can result in significant differences for low and medium voltage buses. We therefore consider optimizing multiple metrics simultaneously using local search methods that reconfigure the system's base tree to optimize service disruptions, reconnection times, and energy losses. We computationally validate our reconfiguration methods on the NREL SMART-DS Greensboro synthetic urban-suburban network and show significant improvement in reliability metrics. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Gupta and

**D.K. Molzahn**, "Analysis of Fairness-promoting Optimization Schemes of Photovoltaic Curtailments for Voltage Regulation in Power Distribution Networks," submitted.

Active power curtailment of photovoltaic (PV) generation is commonly exercised to mitigate over-voltage issues in power distribution networks. However, fairness concerns arise as certain PV plants may experience more significant curtailments than others depending on their locations within the network. Existing literature tackles this issue through fairness-promoting/aware optimization schemes. These schemes can be broadly categorized into two types. The first type maximizes an additional fairness objective along with the main objective of curtailment minimization. The second type is formulated as a feedback controller, where fairness is accounted for by assigning different weights (as feedback) in the curtailment minimization objective for each PV plant based on previous curtailment actions. In this work, we combine these two schemes and provide extensive analyses and comparisons of these two fairness schemes. We compare the performance in terms of fairness and net curtailments for several benchmark test networks. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Talkington, R. Gupta, R. Asiamah, P. Buason, and

**D.K. Molzahn**, "Strategic Electric Distribution Network Sensing via Spectral Bandits," submitted.

Despite their wide-scale deployment and ability to make accurate, high-frequency voltage measurements, communication network limitations have largely precluded the use of smart meters for real-time monitoring purposes in electric distribution systems. While smart meter communication networks have very limited bandwidth available per meter, they also have the ability to dedicate higher bandwidth to varying subsets of meters. Leveraging this capability in order to enable real-time monitoring from smart meters, this paper proposes an online bandwidth-constrained sensor sampling algorithm that exploits the graphical structure inherent in the power flow equations. The key idea is to use a spectral bandit framework where the estimated parameters are the graph Fourier transform coefficients of the nodal voltages. The structure provided by this framework promotes a sampling policy that strategically accounts for electrical distance. Maxima of sub-Gaussian random variables model the policy rewards, which relaxes distributional assumptions common in prior work. The scheme is implemented on realistic electrical networks to dynamically identify meters exposing violations of voltage magnitude limits and illustrating the effectiveness of the proposed method. [ abstract ] [ pdf ] [ Ref: bib txt ]

## Journal Papers

## Conference Proceedings

A. Jalilian, B. Taheri, and

**D.K. Molzahn**, "Co-Optimization of Damage Assessment and Restoration: A Resilience-Driven Dynamic Crew Allocation for Power Distribution Systems," to appear in*IEEE Transactions on Power Systems*.

This study introduces a mixed-integer linear programming (MILP) model, effectively co-optimizing patrolling, damage assessment, fault isolation, repair, and load re-energization processes. The model is designed to solve a vital operational conundrum: deciding between further network exploration to obtain more comprehensive data or addressing the repair of already identified faults. As information on the fault location and repair timelines becomes available, the model allows for dynamic adaptation of crew dispatch decisions. In addition, this study proposes a conservative power flow constraint set that considers two network loading scenarios within the final network configuration. This approach results in the determination of an upper and a lower bound for node voltage levels and an upper bound for power line flows. To underscore the practicality and scalability of the proposed model, we have demonstrated its application using IEEE 123-node and 8500-node test systems, where it delivered promising results. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason, S. Misra, and

**D.K. Molzahn**, "A Data-Driven Sensor Placement Approach for Detecting Voltage Violations in Distribution Systems,"*Electric Power Systems Research*, vol. 232, no. 110387, July 2024.

Stochastic fluctuations in power injections from distributed energy resources (DERs) combined with load variability can cause constraint violations (e.g., exceeded voltage limits) in electric distribution systems. To monitor grid operations, sensors are placed to measure important quantities such as the voltage magnitudes. In this paper, we consider a sensor placement problem which seeks to identify locations for installing sensors that can capture all possible violations of voltage magnitude limits. We formulate a bilevel optimization problem that minimizes the number of sensors and avoids false sensor alarms in the upper level while ensuring detection of any voltage violations in the lower level. This problem is challenging due to the nonlinearity of the power flow equations and the presence of binary variables. Accordingly, we employ recently developed conservative linear approximations of the power flow equations that overestimate or underestimate the voltage magnitudes. By replacing the nonlinear power flow equations with conservative linear approximations, we can ensure that the resulting sensor locations and thresholds are sufficient to identify any constraint violations. Additionally, we apply various problem reformulations to significantly improve computational tractability while simultaneously ensuring an appropriate placement of sensors. Lastly, we improve the quality of the results via an approximate gradient descent method that adjusts the sensor thresholds. We demonstrate the effectiveness of our proposed method for several test cases, including a system with multiple switching configurations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Piansky, G. Stinchfield, A. Kody,

**D.K. Molzahn**, and J.P. Watson, "Long Duration Battery Sizing, Siting, and Operation Under Wildfire Risk Using Progressive Hedging," to appear in*Electric Power Systems Research*, to be presented at*23rd Power Systems Computation Conference (PSCC)*, June 4-7, 2024.

Battery sizing and siting problems are computationally challenging due to the need to make long-term planning decisions that are cognizant of short-term operational decisions. This paper considers sizing, siting, and operating batteries in a power grid to maximize their benefits, including price arbitrage and load shed mitigation, during both normal operations and periods with high wildfire ignition risk. We formulate a multi-scenario optimization problem for long duration battery storage while considering the possibility of load shedding during Public Safety Power Shutoff (PSPS) events that de-energize lines to mitigate severe wildfire ignition risk. To enable a computationally scalable solution of this problem with many scenarios of wildfire risk and power injection variability, we develop a customized temporal decomposition method based on a progressive hedging framework. Extending traditional progressive hedging techniques, we consider coupling in both placement variables across all scenarios and state-of-charge variables at temporal boundaries. This enforces consistency across scenarios while enabling parallel computations despite both spatial and temporal coupling. The proposed decomposition facilitates efficient and scalable modeling of a full year of hourly operational decisions to inform the sizing and siting of batteries. With this decomposition, we model a year of hourly operational decisions to inform optimal battery placement for a 240-bus WECC model in under 70 minutes of wall-clock time. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri and

**D.K. Molzahn**, "Optimizing Parameters of the DC Power Flow," to appear in*Electric Power Systems Research*, to be presented at*23rd Power Systems Computation Conference (PSCC)*, June 4-7, 2024.

Many power system operation and planning problems use the DC power flow approximation to address computational challenges from the nonlinearity of the AC power flow equations. The DC power flow simplifies the AC power flow equations to a linear form that relates active power flows to phase angle differences across branches, parameterized by coefficients based on the branches' susceptances. Inspired by techniques for training machine learning models, this paper proposes an algorithm that seeks optimal coefficient and bias parameters to improve the DC power flow approximation's accuracy. Specifically, the proposed algorithm selects the coefficient and bias parameter values that minimize the discrepancy, across a specified set of operational scenarios, between the power flows given by the DC approximation and the power flows from the AC equations. Gradient-based optimization methods like Broyden-Fletcher-Goldfarb-Shanno (BFGS), Limited-Memory BFGS (L-BFGS), and Truncated Newton Conjugate-Gradient (TNC) enable solution of the proposed algorithm for large systems. After an off-line training phase, the optimized parameters are used to improve the accuracy of the DC power flow during on-line computations. Numerical results show several orders of magnitude improvements in accuracy relative to a hot-start DC power flow approximation across a range of test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri and

**D.K. Molzahn**, "AC Power Flow Feasibility Restoration via a State Estimation-Based Post-Processing Algorithm," to appear in*Electric Power Systems Research*, to be presented at*23rd Power Systems Computation Conference (PSCC)*, June 4-7, 2024.

This paper presents an algorithm for restoring AC power flow feasibility from solutions to simplified optimal power flow (OPF) problems, including convex relaxations, power flow approximations, and machine learning (ML) models. The proposed algorithm employs a state estimation-based post-processing technique in which voltage phasors, power injections, and line flows from solutions to relaxed, approximated, or ML-based OPF problems are treated similarly to noisy measurements in a state estimation algorithm. The algorithm leverages information from various quantities to obtain feasible voltage phasors and power injections that satisfy the AC power flow equations. Weight and bias parameters are computed offline using an adaptive stochastic gradient descent method. By automatically learning the trustworthiness of various outputs from simplified OPF problems, these parameters inform the online computations of the state estimation-based algorithm to both recover feasible solutions and characterize the performance of power flow approximations, relaxations, and ML models. Furthermore, the proposed algorithm can simultaneously utilize combined solutions from different relaxations, approximations, and ML models to enhance performance. Case studies demonstrate the effectiveness and scalability of the proposed algorithm, with solutions that are both AC power flow feasible and much closer to the true AC OPF solutions than alternative methods, often by several orders of magnitude in the squared two-norm loss function. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]G. Nilsson, A.D. Owen Aquino, S. Coogan, and

**D.K. Molzahn**, "GreenEVT: Greensboro Electric Vehicle Testbed,"*IEEE Systems Journal*, vol. 18, no. 1, pp. 600-611, March 2024.

The ongoing electrification of the transportation fleet will increase the load on the electric power grid. Since both the transportation network and the power grid already experience periods of significant stress, joint analyses of both infrastructures will most likely be necessary to ensure acceptable operation in the future. To enable such analyses, this paper presents an open-source testbed that jointly simulates high-fidelity models of both the electric distribution system and the transportation network. The testbed utilizes two open-source simulators, OpenDSS to simulate the electric distribution system and the microscopic traffic simulator SUMO to simulate the traffic dynamics. Electric vehicle charging links the electric distribution system and the transportation network models at vehicle locations determined using publicly available parcel data. Leveraging high-fidelity synthetic electric distribution system data from the SMART-DS project and transportation system data from OpenStreetMap, this testbed models the city of Greensboro, NC down to the household level. Moreover, the methodology and the supporting scripts released with the testbed allow adaption to other areas where high-fidelity geolocated OpenDSS datasets are available. After describing the components and usage of the testbed, we exemplify applications enabled by the testbed via two scenarios modeling the extreme stresses encountered during evacuations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ GitHub Repository ]M. Alkhraijah, R. Harris, C. Coffrin, and

**D.K. Molzahn**, "PowerModelsADA: A Framework for Solving Optimal Power Flow using Distributed Algorithms,"*IEEE Transactions on Power Systems (Letters)*, vol. 39, no. 1, pp. 2357-2360, January 2024.

This letter presents PowerModelsADA, an open-source framework for solving Optimal Power Flow (OPF) problems using Alternating Distributed Algorithms (ADA). PowerModelsADA provides a framework to test, verify, and benchmark both existing and new ADAs. This paper demonstrates use cases for PowerModelsADA and validates its implementation with multiple OPF formulations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ GitHub Repository ]S. Talkington, D. Turizo, S. Grijalva, J. Fernandez, and

**D.K. Molzahn**, "Conditions for Estimation of Sensitivities of Voltage Magnitudes to Complex Power Injections,"*IEEE Transactions on Power Systems*, vol. 39, no. 1, pp. 478-491, January 2024.

Voltage phase angle measurements are often unavailable from sensors in distribution networks and transmission network boundaries. Therefore, this paper addresses the conditions for estimating sensitivities of voltage magnitudes with respect to complex (active and reactive) electric power injections based on sensor measurements. These sensitivities represent submatrices of the inverse power flow Jacobian. We extend previous results to show that the sensitivities of a bus voltage magnitude with respect to active power injections are unique and different from those with respect to reactive power. The classical Newton-Raphson power flow model is used to derive a novel representation of bus voltage magnitudes as an underdetermined linear operator of the active and reactive power injections—parameterized by the bus power factors. Two conditions that ensure the existence of unique complex power injections given voltage magnitudes are established for this underdetermined linear system, thereby compressing the solution space. The first is a sufficient condition based on the bus power factors. The second is a necessary and sufficient condition based on the system eigenvalues. We use matrix completion theory to develop estimation methods for recovering sensitivity matrices with varying levels of sensor availability. Simulations verify the results and demonstrate engineering use of the proposed methods. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason, S. Misra, and

**D.K. Molzahn**, "Sample-Based Conservative Bias Linear Power Flow Approximations," to appear in*IEEE IAS Industrial and Commercial Power System Asia (IEEE I&CPS Asia)*, July 9-12, 2024.

The power flow equations are central to many problems in power system planning, analysis, and control. However, their inherent non-linearity and non-convexity present substantial challenges during problem-solving processes, especially for optimization problems. Accordingly, linear approximations are commonly employed to streamline computations, although this can often entail compromises in accuracy and feasibility. This paper proposes an approach termed*Conservative Bias Linear Approximations (CBLA)*for addressing these limitations. By minimizing approximation errors across a specified operating range while incorporating conservativeness (over- or under-estimating quantities of interest), CBLA strikes a balance between accuracy and tractability by maintaining linear constraints. By allowing users to design loss functions tailored to the specific approximated function, the bias approximation approach significantly enhances approximation accuracy. We illustrate the effectiveness of our proposed approach through several test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Gupta, P. Buason, and

**D.K. Molzahn**, "Fairness-aware Photovoltaic Generation Limits for Voltage Regulation in Power Distribution Networks using Conservative Linear Approximations,"*8th Texas Power and Energy Conference (TPEC)*, February 12-13, 2024.

This paper proposes a framework for fairly curtailing photovoltaic (PV) plants in response to the over-voltage problem in PV-rich distribution networks. The framework imposes PV generation limits to avoid overvoltages. These limits are computed a day ahead of real-time operations by solving an offline stochastic optimization problem using forecasted scenarios for PV generation and load demand. The framework minimizes the overall curtailment while considering fairness by reducing disparities in curtailments among different PV owners. We model the distribution grid constraints using a conservative linear approximation (CLA) of the AC power flow equations which is computed using a set of sampled power injections from the day-ahead predicted scenarios. The proposed framework is numerically validated on a CIGRE benchmark network interfaced with a large number of PV plants. We compare the performance of the proposed framework versus an alternative formulation that does not incorporate fairness considerations. To this end, we assess tradeoffs between fairness, as quantified with the Jain Fairness Index (JFI), and the total curtailed energy. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A.D. Owen Aquino, S. Talkington, and

**D.K. Molzahn**, "Managing Vehicle Charging during Emergencies via Conservative Distribution System Modeling,"*8th Texas Power and Energy Conference (TPEC)*, February 12-13, 2024.

Combinatorial distribution system optimization problems, such as scheduling electric vehicle (EV) charging during evacuations, present significant computational challenges. These challenges stem from the large numbers of constraints, continuous variables, and discrete variables, coupled with the unbalanced nature of distribution systems. In response to the escalating frequency of extreme events impacting electric power systems, this paper introduces a method that integrates sample-based conservative linear power flow approximations (CLAs) into an optimization framework. In particular, this integration aims to ameliorate the aforementioned challenges of distribution system optimization in the context of efficiently minimizing the charging time required for EVs in urban evacuation scenarios. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri and

**D.K. Molzahn**, "AC Power Flow Informed Parameter Learning for DC Power Flow Network Equivalents,"*8th Texas Power and Energy Conference (TPEC)*, February 12-13, 2024.

This paper presents an algorithm to optimize the parameters of power systems equivalents to enhance the accuracy of the DC power flow approximation in reduced networks. Based on a zonal division of the network, the algorithm produces a reduced power system equivalent that captures inter-zonal flows with aggregated buses and equivalent transmission lines. The algorithm refines coefficient and bias parameters for the DC power flow model of the reduced network, aiming to minimize discrepancies between inter-zonal flows in DC and AC power flow results. Using optimization methods like BFGS, L-BFGS, and TNC in an offline training phase, these parameters boost the accuracy of online DC power flow computations. In contrast to existing network equivalencing methods, the proposed algorithm optimizes accuracy over a specified range of operation as opposed to only considering a single nominal point. Numerical tests demonstrate substantial accuracy improvements over traditional equivalencing and approximation methods. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Harris and

**D.K. Molzahn**, "Detecting and Mitigating Data Integrity Attacks on Distributed Algorithms for Optimal Power Flow using Machine Learning,"*57th Hawaii International Conference on System Sciences (HICSS)*, January 3-6, 2024.

Using distributed algorithms, multiple computing agents can coordinate their operations by jointly solving optimal power flow problems. However, cyberattacks on the data communicated among agents may maliciously alter the behavior of a distributed algorithm. To improve cybersecurity, this paper proposes a machine learning method for detecting and mitigating data integrity attacks on distributed algorithms for solving optimal power flow problems. In an offline stage with trustworthy data, agents train and share machine learning models of their local subproblems. During online execution, each agent uses the trained models from neighboring agents to detect cyberattacks using a reputation system and then mitigate their impacts. Numerical results show that this method reliably, accurately, and quickly detects data integrity attacks and effectively mitigates their impacts to achieve near-feasible and near-optimal operating points. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]O. Kuryatnikova, B. Ghaddar, and

**D.K. Molzahn**, "Two-Stage Robust Quadratic Optimization with Equalities and Its Application to Optimal Power Flow,"*SIAM Journal on Optimization*, vol. 33, no. 4, pp. 2830-2857, December 2023.

In this work, we consider two-stage quadratic optimization problems under ellipsoidal uncertainty. In the first stage, one needs to decide upon the values of a subset of optimization variables (control variables). In the second stage, the uncertainty is revealed, and the rest of the optimization variables (state variables) are set up as a solution to a known system of possibly nonlinear equations. This type of problem occurs, for instance, in optimization for dynamical systems, such as electric power systems as well as gas and water networks. We propose a convergent iterative algorithm to build a sequence of approximately robustly feasible solutions with an improving objective value. At each iteration, the algorithm optimizes over a subset of the feasible set and uses affine approximations of the second-stage equations while preserving the nonlinearity of other constraints. We implement our approach and demonstrate its performance on Matpower instances of AC optimal power flow. Although this paper focuses on quadratic problems, the approach is suitable for more general setups. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]I. Aravena,

**D.K. Molzahn**, S. Zhang, F.E. Curtis, S. Tu, A. Waechter, E. Wei, E. Wong, A. Gholami, K. Sun, X.A. Sun, S.T. Elbert, J.T. Holzer, and A. Veeramany, "Recent Developments in Security-Constrained AC Optimal Power Flow: Overview of Challenge 1 in the ARPA-E Grid Optimization Competition,"*Operations Research*, special issue on*Computational Advances in Short-Term Power System Operations*, vol. 71, no. 6, pp. 1997-2014, November-December 2023.

The optimal power flow problem is central to many tasks in the design and operation of electric power grids. This problem seeks the minimum cost operating point for an electric power grid while satisfying both engineering requirements and physical laws describing how power flows through the electric network. By additionally considering the possibility of component failures and using an accurate AC power flow model of the electric network, the security-constrained AC optimal power flow (SC-AC-OPF) problem is of paramount practical relevance. To assess recent progress in solution algorithms for SC-AC-OPF problems and spur new innovations, the U.S. Department of Energy's Advanced Research Projects Agency-Energy (ARPA-E) organized Challenge 1 of the Grid Optimization (GO) competition. This paper describes the SC-AC-OPF problem formulation used in the competition, overviews historical developments and the state of the art in SC-AC-OPF algorithms, discusses the competition, and summarizes the algorithms used by the top three teams in Challenge 1 of the GO Competition (Teams gollnlp, GO-SNIP, and GMI-GO). [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]F.E. Curtis,

**D.K. Molzahn**, S. Tu, A. Wächter, E. Wei, and E. Wong, "A Decomposition Algorithm for Large-Scale Security-Constrained AC Optimal Power Flow,"*Operations Research*, special issue on*Computational Advances in Short-Term Power System Operations*, vol. 71, no. 6, pp. 2031-2044, November-December 2023.

**Description of Team GO-SNIP's algorithm which finished second overall in the Department of Energy's Grid Optimization Competition Challenge 1.**

A decomposition algorithm for solving large-scale security-constrained AC optimal power flow problems is presented. The formulation considered is the one used in the ARPA-E Grid Optimization (GO) Competition, Challenge 1, held from November 2018 through October 2019. The techniques found to be most effective in terms of performance in the challenge are presented, including strategies for contingency selection, fast contingency evaluation, handling complementarity constraints, avoiding issues related to degeneracy, and exploiting parallelism. The results of numerical experiments are provided to demonstrate the effectiveness of the proposed techniques as compared to alternative strategies. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Turizo and

**D.K. Molzahn**, "Invertibility Conditions for the Admittance Matrices of Balanced Power Systems,"*IEEE Transactions on Power Systems*, vol. 38, no. 4, pp. 3841-3853, July 2023.

The admittance matrix encodes the network topology and electrical parameters of a power system in order to relate the current injection and voltage phasors. Since admittance matrices are central to many power engineering analyses, their characteristics are important subjects of theoretical studies. This paper focuses on the key characteristic of*invertibility*. Previous literature has presented an invertibility condition for admittance matrices. This paper first identifies and fixes a technical issue in the proof of this previously presented invertibility condition. This paper then extends this previous work by deriving new conditions that are applicable to a broader class of systems with lossless branches and transformers with off-nominal tap ratios. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]F. Milano and

**D.K. Molzahn**, "Forward to the EPSR Special Issue for PSCC 2022,"*Electric Power Systems Research*, vol. 217, no. 109148, April 2023.

[ pdf ] [ link ] [ Ref: bib txt ]L.A. Roald, D. Pozo, A. Papavasiliou,

**D.K. Molzahn**, J. Kazempour, and A. Conejo, "Power Systems Optimization under Uncertainty: A Review of Methods and Applications,"*Electric Power Systems Research*, vol. 214, Part A, no. 108725, January 2023. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

Electric power systems and the companies and customers that interact with them are experiencing increasing levels of uncertainty due to factors such as renewable energy generation, market liberalization, and climate change. This raises the important question of how to make optimal decisions under uncertainty. This paper aims to provide an overview of existing methods for modeling and optimization of problems affected by uncertainty, targeted at researchers with a familiarity with power systems and optimization. We also review some important applications of optimization under uncertainty in power systems and provide an outlook to future directions of research. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri,

**D.K. Molzahn**, and S. Grijalva, "Improving Distribution System Resilience by Undergrounding Lines and Deploying Mobile Generators,"*Electric Power Systems Research*, vol. 214, Part A, no. 108804, January 2023.

To improve the resilience of electric distribution systems, this paper proposes a stochastic multi-period mixed-integer linear programming model that determines where to underground new distribution lines and how to coordinate mobile generators in order to serve critical loads during extreme events. The proposed model represents the service restoration process using the linearized DistFlow approximation of the AC power flow equations as well as binary variables for the undergrounding decisions of the lines, the configurations of switches, and the locations of mobile generators during each time period. The model also enforces a radial configuration of the distribution network and considers the transportation times needed to deploy the mobile generators. It is shown that long-term line undergrounding decisions which are cognizant of short-term mobile generator deployments yield superior results relative to undergrounding decisions made without considering mobile generators. Using an extended version of the IEEE 123-bus test system, numerical simulations show that combining the ability to underground distribution lines with the deployment of mobile generators can significantly improve the resilience of the power supply to critical loads. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Harris, A. Banerjee, and

**D.K. Molzahn**, "Synthetic Test Case for Ukraine's Power Grid,"*55th North American Power Symposium (NAPS)*, October 15-17, 2023.

Power systems research requires realistic test cases to demonstrate and benchmark algorithmic innovations. However, to keep power grid information secure, much of the data for actual systems cannot be publicly released. This motivates the creation of synthetic test cases designed to match the characteristics of actual power grids. By only using publicly available data, these synthetic test cases can be freely released for research and educational purposes. Motivated by the ongoing conflict in Ukraine, this paper presents a synthetic test case for the Ukrainian electric transmission system. With the ability to run power flow and other analyses, this test case is relevant to emerging research and educational activities related to power grid security. To develop the test case, we leveraged publicly available data on population centers, generators, and transmission topology and voltage levels along with synthetic test case creation methodologies from recent literature. After describing the process used to develop this test case, this paper presents validation results using several widely accepted criteria for assessing the test case's realism. As an illustrative application, we demonstrate one use case by solving a bilevel linearization of the $N-k$ interdiction problem to identify critical sets of line failures which cause the most load shedding. [ abstract ] [ pdf ] [ Ref: bib txt ] [ GitHub Repository ]B. Taheri and

**D.K. Molzahn**, "Restoring AC Power Flow Feasibility for Solutions to Relaxed and Approximated Optimal Power Flow Problems,"*American Control Conference (ACC)*, May 31 - June 2, 2023.

To address computational challenges associated with power flow nonconvexities, significant research efforts over the last decade have developed convex relaxations and approximations of optimal power flow (OPF) problems. However, benefits associated with the convexity of these relaxations and approximations can have tradeoffs in terms of solution accuracy since they may yield voltage phasors that are inconsistent with the power injections and line flows, limiting their usefulness for some applications. Inspired by state estimation techniques, this paper proposes a new method for obtaining an AC power flow feasible point from the solution to a relaxed or approximated OPF problem. By treating the inconsistent voltage phasors, power injections, and line flows analogously to noisy measurements in a state estimation algorithm, the proposed method yields power injections and voltage phasors that are feasible with respect to the AC power flow equations while incorporating information from many quantities in the solution to a relaxed or approximated OPF problem. We improve this method by adjusting weighting terms with a machine learning based approach. We demonstrate the proposed method using several relaxations and approximations. The result show up to several orders of magnitude improvement in accuracy over traditional methods. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A.D. Owen Aquino, R. Harris, A. Kody, and

**D.K. Molzahn**, "Comparing Machine Learning and Optimization Approaches for the*N-k*Interdiction Problem Considering Load Variability,"*56th Hawaii International Conference on System Sciences (HICSS)*, January 3-6, 2023.

Power grids must be operated, protected, and maintained such that a small number of line failures will not result in significant load shedding. To identify problematic combination of failures, we consider an*N-k*interdiction problem that seeks the set of*k*failed lines (out of*N*total lines) that result in the largest load shed. This is naturally formulated as a bilevel optimization problem with an upper level representing the attacker that selects line failures and a lower level modeling the defender's generator redispatch to minimize the load shedding. Compounding the difficulties inherent to the bilevel nature of interdiction problems, we consider a nonlinear AC power flow model that makes this problem intractable with traditional solution approaches. Furthermore, since the solutions found at a particular load condition may not generalize to other loading conditions, operators may need to quickly recompute these worst-case failures online to protect against them during operations. To address these challenges, we formulate and compare the performance of three simplified methods for solving the*N-k*interdiction problem: a state-of-the-art optimization approach based on a network-flow relaxation of the power flow equations and two newly developed machine learning algorithms that predict load sheds given the state of the network. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Kody, S. Chevalier, S. Chatzivasileiadis, and

**D.K. Molzahn**, "Modeling the AC Power Flow Equations with Optimally Compact Neural Networks: Application to Unit Commitment,"*Electric Power Systems Research*, vol. 212, no. 108282, November 2022. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

Nonlinear power flow constraints render a variety of power system optimization problems computationally intractable. Emerging research shows, however, that the nonlinear AC power flow equations can be successfully modeled using Neural Networks (NNs). These NNs can be exactly transformed into Mixed Integer Linear Programs (MILPs) and embedded inside challenging optimization problems, thus replacing nonlinearities that are intractable for many applications with tractable piecewise linear approximations. Such approaches, though, suffer from an explosion of the number of binary variables needed to represent the NN. Accordingly, this paper develops a technique for training an "optimally compact" NN, i.e., one that can represent the power flow equations with a sufficiently high degree of accuracy while still maintaining a tractable number of binary variables. We show that the resulting NN model is more expressive than both the DC and linearized power flow approximations when embedded inside of a challenging optimization problem (i.e., the AC unit commitment problem). [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Zeng, A. Kody, Y. Kim, K. Kim, and

**D.K. Molzahn**, "A Reinforcement Learning Approach to Parameter Selection for Distributed Optimization in Power Systems,"*Electric Power Systems Research*, vol. 212, no. 108546, November 2022. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

With the increasing penetration of distributed energy resources, distributed optimization algorithms have attracted significant attention for power systems applications due to their potential for superior scalability, privacy, and robustness to a single point-of-failure. The Alternating Direction Method of Multipliers (ADMM) is a popular distributed optimization algorithm; however, its convergence performance is highly dependent on the selection of penalty parameters, which are usually chosen heuristically. In this work, we use reinforcement learning (RL) to develop an adaptive penalty parameter selection policy for the AC optimal power flow (ACOPF) problem solved via ADMM with the goal of minimizing the number of iterations until convergence. We train our RL policy using deep Q-learning, and show that this policy can result in significantly accelerated convergence (up to a 59% reduction in the number of iterations compared to existing, curvature-informed penalty parameter selection methods). Furthermore, we show that our RL policy demonstrates promise for generalizability, performing well under unseen loading schemes as well as under unseen losses of lines and generators (up to a 50% reduction in iterations). This work thus provides a proof-of-concept for using RL for parameter selection in ADMM for power systems applications. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason, S. Misra, and

**D.K. Molzahn**, "A Sample-Based Approach for Computing Conservative Linear Power Flow Approximations,"*Electric Power Systems Research*, vol. 212, no. 108579, November 2022. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

Non-convexities induced by the non-linear power flow equations challenge solution algorithms for many power system optimization and control problems. Linear approximations are often used to address these challenges by trading off modeling accuracy for tractability. The accuracy of a power flow linearization depends on the characteristics of the power system and the operational range where the linearization is applied. However, rather than exploiting knowledge of these characteristics for a particular system, many existing power flow linearizations are based on general assumptions for broad classes of systems, thus limiting their accuracy. Moreover, since existing linearizations do not consistently overestimate or underestimate quantities of interest such as voltage magnitudes and line flows, algorithms based on these linearizations may lead to constraint violations when applied to the system. In contrast, this paper computes*conservative linear approximations*of the power flow equations, i.e., linear approximations that overestimate or underestimate a quantity of interest in order to enable tractable algorithms that avoid constraint violations. Using a sample-based approach, we compute these conservative linearizations by solving a constrained linear regression problem. We analyze and improve the conservative linear approximations via an iterative sampling approach, optimization over functions of the quantities of interest, and a sample-complexity analysis. Considering the relationships between the voltage magnitudes and the active and reactive power injections, we characterize the performance of the conservative linear approximations for a range of test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, C. Menendez, and

**D.K. Molzahn**, "Assessing the Impacts of Nonideal Communications on Distributed Optimal Power Flow Algorithms,"*Electric Power Systems Research*, vol. 212, no. 108297, November 2022. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

Power system operators are increasingly looking toward distributed optimization to address various challenges facing electric power systems. To assess their capabilities in environments with nonideal communications, this paper investigates the impacts of data quality on the performance of distributed optimization algorithms. Specifically, this paper compares the performance of the Alternating Direction Method of Multipliers (ADMM), Analytical Target Cascading (ATC), and Auxiliary Problem Principle (APP) algorithms in the context of DC Optimal Power Flow (DC OPF) problems. Using several test cases, this paper characterizes the performance of these algorithms in terms of their convergence rates and solution quality under three data quality nonidealities: (1) additive Gaussian noise, (2) bad data (large error), and (3) intermittent communication failure. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]N. Patari, V. Venkataramanan, A.K. Srivastava,

**D.K. Molzahn**, N. Li, and A. Annaswamy, "Distributed Optimization in Distribution Systems: Use Cases, Limitations, and Research Needs,"*IEEE Transactions on Power Systems*, vol. 37, no. 5, pp. 3469-3481, September 2022.

Electric distribution grid operations typically rely on both centralized optimization and local non-optimal control techniques. As an alternative, distribution system operational practices can consider distributed optimization techniques that leverage communications among various neighboring agents to achieve optimal operation. With the rapidly increasing integration of distributed energy resources (DERs), distributed optimization algorithms are growing in importance due to their potential advantages in scalability, flexibility, privacy, and robustness relative to centralized optimization. Implementation of distributed optimization offers multiple challenges and also opportunities. This paper provides a comprehensive review of the recent advancements in distributed optimization for electric distribution systems and classifications using key attributes. Problem formulations and distributed optimization algorithms are provided for example use cases, including volt/var control, market clearing process, loss minimization, and conservation voltage reduction. Finally, this paper also presents future research needs for the applicability of distributed optimization algorithms in the distribution system. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]J. Liu, B. Cui,

**D.K. Molzahn**, C. Chen, and X. Lu, "Optimal Power Flow for DC Networks with Robust Feasibility and Stability Guarantees,"*IEEE Transactions on Control of Network Systems*, vol. 9, no. 2, pp. 904-916, June 2022.

With high penetrations of renewable generation and variable loads, there is significant uncertainty associated with power flows in DC networks such that stability and constraint satisfaction are important concerns. Most existing DC network optimal power flow (DN-OPF) formulations assume exact knowledge about loading conditions and do not provide stability guarantees. In contrast, this paper studies a DN-OPF formulation which considers both stability and constraint satisfaction under uncertainty. The need to account for a range of uncertainty realizations in this paper’s robust optimization formulation results in a challenging semi-infinite program (SIP). The proposed solution algorithm reformulates this SIP into a computationally amenable problem whose solution provides a feasible operating point for the SIP. This algorithm constructs a convex stability set using sufficient conditions for the existence of a feasible and stable power flow solution. Solving an optimization problem based on this convex stability set provides generator set points which ensure that the DC network has a stable operating point for any considered uncertainty realization. This optimization problem takes a form similar to a deterministic DN-OPF problem and is therefore tractable. The proposed algorithm is demonstrated using various DC networks adapted from IEEE test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Kody, A. West, and

**D.K. Molzahn**, "Sharing the Load: Considering Fairness in De-energization Scheduling to Mitigate Wildfire Ignition Risk using Rolling Optimization,"*61st IEEE Control and Decision Conference (CDC)*, December 6-9, 2022.

Wildfires are a threat to public safety and have increased in both frequency and severity due to climate change. To mitigate wildfire ignition risks, electric power system operators proactively de-energize high-risk power lines during "public safety power shut-off" (PSPS) events. Line de-energizations can cause communities to lose power, which may result in negative economic, health, and safety impacts. Furthermore, the same communities may repeatedly experience power shutoffs over the course of a wildfire season, which compounds these negative impacts. However, there are often many combinations of power lines whose de-energization will result in about the same reduction of wildfire risk, but the associated power loss affects different communities. Therefore, one may raise concerns regarding the fairness of de-energization decisions. Accordingly, this paper proposes a model-predictive-control-inspired framework to select lines to de-energize in order to balance wildfire risk reduction, total load shedding, and fairness considerations. The goal of the framework is to prevent a small fraction of communities from disproportionally being impacted by PSPS events, and to instead more equally share the burden of power outages. For a geolocated test case in the southwestern United States, we use actual California demand data as well as real wildfire risk forecasts to simulate PSPS events during the 2021 wildfire season and compare the performance of various methods for promoting fairness. Our results demonstrate that the proposed formulation can provide significantly more fair outcomes with limited impacts on system-wide performance. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, R. Harris, S. Litchfield, D. Huggins, and

**D.K. Molzahn**, "Analyzing Malicious Data Injection Attacks on Distributed Optimal Power Flow Algorithms,"*54th North American Power Symposium (NAPS)*, October 9-11, 2022.

The rapid growth of distributed energy resources motivates the application of distributed optimization algorithms for solving optimal power flow (OPF) problems. Using distributed optimization to solve OPF problems requires repeated data sharing between agents responsible for different regions of the power system. The performance of distributed optimization algorithms depends on the integrity of the shared data and the communications between the agents. This paper investigates the vulnerability of a distributed algorithm called Auxiliary Problem Principle (APP) with respect to cyberattacks on the communications between the agents. We propose cyberattack models that manipulate the shared data between neighboring agents to achieve an objective such as driving the algorithm to converge to a suboptimal solution. We investigate the convergence characteristics of the APP algorithm in the context of the DC optimal power flow (DC OPF) problem with and without manipulating the shared data. Last, we demonstrate how trained neural networks can provide highly accurate attack detection. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Kody, R. Piansky, and

**D.K. Molzahn**, "Optimizing Transmission Infrastructure Investments to Support Line De-energization for Mitigating Wildfire Ignition Risk,"*IREP Symposium on Bulk Power System Dynamics and Control-XI. A 100% Renewable Energy Source Bulk Power Grid: Opportunities and Challenges*, July 25-30, 2022.

Wildfires pose a growing risk to public safety in regions like the western United States, and, historically, electric power systems have ignited some of the most destructive wildfires. To reduce wildfire ignition risks, power system operators preemptively de-energize high-risk power lines during extreme wildfire conditions as part of "Public Safety Power Shutoff" (PSPS) events. While capable of substantially reducing acute wildfire risks, PSPS events can also result in significant amounts of load shedding as the partially de-energized system may not be able to supply all customer demands. In this work, we investigate the extent to which infrastructure investments can support system operations during PSPS events by enabling reduced load shedding and wildfire ignition risk. We consider the installation of grid-scale batteries, solar PV, and line hardening or maintenance measures (e.g., undergrounding or increased vegetation management). Optimally selecting the locations, types, and sizes of these infrastructure investments requires considering the line de-energizations associated with PSPS events. Accordingly, this paper proposes a multi-period optimization formulation that locates and sizes infrastructure investments while simultaneously choosing line de-energizations to minimize wildfire ignition risk and load shedding. The proposed formulation is evaluated using two geolocated test cases along with realistic infrastructure investment parameters and actual wildfire risk data from the US Geological Survey. We evaluate the performance of investment choices by simulating de-energization decisions for the entire 2021 wildfire season with optimized infrastructure placements. With investment decisions varying significantly for different test cases, budgets, and operator priorities, the numerical results demonstrate the proposed formulation's value in tailoring investment choices to different settings. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Harris, M. Alkhraijah, D. Huggins, and

**D.K. Molzahn**, "On the Impacts of Different Consistency Constraint Formulations for Distributed Optimal Power Flow,"*Texas Power and Energy Conference (TPEC)*, February 28 - March 1, 2022.

The optimal power flow (OPF) problem finds the least costly operating point which meets the power grid's operational limits and obeys physical power flow laws. Complementing today's centralized optimization paradigm, future power grids may rely on distributed optimization where multiple agents work together to determine an acceptable operating point. In distributed algorithms, local agents solve subproblems to optimize their region of the system and share data to achieve consistency with their neighboring agents' subproblems. This paper investigates how different methods of enforcing power flow consistency constraints between local areas in distributed optimal power flow impact convergence rate and a classifier’s ability to detect malicious cyberattack. The distributed OPF problem is solved with the alternating direction method of multipliers (ADMM) algorithm. First, the ADMM algorithm’s convergence rate is compared for three different consistency constraint formulations. Next, the paper considers a cyberattack in which the integrity of information shared between agents is compromised, causing the algorithm to exhibit unacceptable behavior. A support vector machine (SVM) classifier is trained to detect the presence of manipulated data from such cyberattacks. Results demonstrate that consistency constraint formulation impacts the classifier's detection performance; for certain formulations, detection is highly accurate. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Lee, K. Turitsyn,

**D.K. Molzahn**, and L.A. Roald, "Robust AC Optimal Power Flow with Robust Convex Restriction,"*IEEE Transactions on Power Systems*, vol. 36, no. 6, pp. 4953-4966, November 2021.

Electric power grids regularly experience uncertain fluctuations from load demands and renewables, which poses a risk of violating operational limits designed to safeguard the system. In this paper, we consider the robust AC OPF problem that minimizes the generation cost while requiring a certain level of system security in the presence of uncertainty. The robust AC OPF problem requires that the system satisfy operational limits for all uncertainty realizations within a specified uncertainty set. Guaranteeing robustness is particularly challenging due to the non-convex, nonlinear AC power flow equations, which may not always have a solution. In this work, we extend a previously developed convex restriction to a*robust convex restriction*, which is a convex inner approximation of the non-convex feasible region of the AC OPF problem that accounts for uncertainty in the power injections. We then use the robust convex restriction in an algorithm that obtains robust solutions to AC OPF problems by solving a sequence of convex optimization problems. We demonstrate our algorithm and its ability to control robustness versus operating cost trade-offs using PGLib test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, M. Alowaifeer, M. Alsaleh, A. Alfaris, and

**D.K. Molzahn**, "The Effects of Social Distancing on the Temperature-Demand Relationship,"*Energies*, vol. 14, no. 2, 2021.

To mitigate the spread of the Novel Coronavirus (COVID-19), governments around the world have imposed social distancing policies ranging from minor social activity suspensions to full curfews. These social distancing policies have altered electricity consumption behaviors in numerous countries. This paper studies how strict social distancing policies affect the relationship between electricity demand and ambient temperature. This relationship is especially important since most governments imposed strict social distancing policies during a temperature transition season where the impacts of temperature variations are particularly important for grid operations. In this paper, we first review the expected short- and long-term impacts of social distancing on the electricity demand. We then present a case study on the electricity demand of the Kingdom of Saudi Arabia during strict social distancing policies. The results of this case study suggest that strict social distancing policies result in a stronger dependency between temperature and electricity demand. Additionally, we observe a reduction in the time required for the electricity demand to respond to temperature changes. Power system regulators can use the results in this paper to better design energy policies and cope with rare events. Additionally, power system operators can use the results in this paper to more accurately forecast electricity demands to avoid inefficient and insecure operation of the electric grid. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, and M.L. Crow, "Tightening QC Relaxations of AC Optimal Power Flow Problems via Complex Per Unit Normalization,"*IEEE Transactions on Power Systems*, vol. 36, no. 1, pp. 281-291, January 2021.

Optimal power flow (OPF) is a key problem in power system operations. OPF problems that use the nonlinear AC power flow equations to accurately model the network physics have inherent challenges associated with non-convexity. To address these challenges, recent research has applied various convex relaxation approaches to OPF problems. The QC relaxation is a promising approach that convexifies the trigonometric and product terms in the OPF problem by enclosing these terms in convex envelopes. The accuracy of the QC relaxation strongly depends on the tightness of these envelopes. This paper presents two improvements to these envelopes. The first improvement leverages a polar representation of the branch admittances in addition to the rectangular representation used previously. The second improvement is based on a coordinate transformation via a complex per unit base power normalization that rotates the power flow equations. The trigonometric envelopes resulting from this rotation can be tighter than the corresponding envelopes in previous QC relaxation formulations. Using an empirical analysis with a variety of test cases, this paper suggests an appropriate value for the angle of the complex base power. Comparing the results with a state-of-the-art QC formulation reveals the advantages of the proposed improvements. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Peña Ordieres,

**D.K. Molzahn**, L.A. Roald, and A. Wächter "DC Optimal Power Flow with Joint Chance Constraints,"*IEEE Transactions on Power Systems*, vol. 36, no. 1, pp. 147-158, January 2021.

Managing uncertainty and variability in power injections has become a major concern for power system operators due to increasing levels of fluctuating renewable energy connected to the grid. This work addresses this uncertainty via a joint chance-constrained formulation of the DC optimal power flow (OPF) problem, which satisfies*all*the constraints*jointly*with a pre-determined probability. The few existing approaches for solving joint chance-constrained OPF problems are typically either computationally intractable for large-scale problems or give overly conservative solutions that satisfy the constraints far more often than required, resulting in excessively costly operation. This paper proposes an algorithm for solving joint chance-constrained DC OPF problems by adopting an Sℓ_{1}QP-type trust-region algorithm. This algorithm uses a sample-based approach that avoids making strong assumptions on the distribution of the uncertainties, scales favorably to large problems, and can be tuned to obtain less conservative results. We illustrate the performance of our method using several IEEE test cases. The results demonstrate the proposed algorithm's advantages in computational times and limited conservativeness of the solutions relative to other joint chance-constrained DC OPF algorithms. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and C. Rehtanz, "Forward to the EPSR Special Issue for PSCC 2020,"*Electric Power Systems Research*, vol. 190, no. 106827, January 2021.

[ pdf ] [ link ] [ Ref: bib txt ]A. Venzke,

**D.K. Molzahn**, and S. Chatzivasileiadis, "Efficient Creation of Datasets for Data-Driven Power System Applications,"*Electric Power Systems Research*, vol. 190, January 2021. Presented at*21st Power Systems Computation Conference (PSCC)*, June 29 - July 3, 2020.

Advances in data-driven methods have sparked renewed interest for applications in power systems. Creating datasets for successful application of these methods has proven to be very challenging, especially when considering power system security. This paper proposes a computationally efficient method to create datasets of secure and insecure operating points. We propose an infeasibility certificate based on separating hyperplanes that can a-priori characterize large parts of the input space as insecure, thus significantly reducing both computation time and problem size. Our method can handle an order of magnitude more control variables and creates balanced datasets of secure and insecure operating points, which is essential for data-driven applications. While we focus on N-1 security and uncertainty, our method can extend to dynamic security. For PGLib-OPF networks up to 500 buses and up to 125 control variables, we demonstrate drastic reductions in unclassified input space volumes and computation time, create balanced datasets, and evaluate an illustrative data-driven application. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ presentation ]B. Li, B. Cui, F. Qiu, and

**D.K. Molzahn**, "Balancibility: Existence and Uniqueness of Power Flow Solutions under Voltage Balance Requirements,"*Electric Power Systems Research*, vol. 190, January 2021. Presented at*21st Power Systems Computation Conference (PSCC)*, June 29 - July 3, 2020.

**PSCC 2020 Highlights Award, top 3% of papers at the 21st Power Systems Computation Conference.**

In distribution systems, power injection variability due to growing penetrations of distributed energy resources (DERs) and dispatchable loads can lead to power quality issues such as severe voltage unbalance. To ensure safe operation of phase-balance-sensitive components such as transformers and induction motor loads, the amount of voltage unbalance must be maintained within specified limits for a range of uncertain loading conditions. This paper builds on existing "solvability conditions" that characterize operating regions for which the power flow equations are guaranteed to have a unique high-voltage solution. We extend these existing solvability conditions to be applicable to distribution systems and augment them with a “balancibility” condition which quantifies an operating region within which a unique, adequately balanced power flow solution exists. To build this condition, we consider different unbalance definitions and derive closed-form representations through reformulations or safe approximations. Using case studies, we evaluate these closed-form representations and compare the balancibility conditions associated with different unbalance definitions. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ presentation ]A.D. Owen Aquino, L.A. Roald, and

**D.K. Molzahn**, "Identifying Redundant Constraints for AC OPF: The Challenges of Local Solutions, Relaxation Tightness, and Approximation Inaccuracy,"*53rd North American Power Symposium (NAPS)*, November 14-16, 2021.

To compute reliable and low-cost operating points, electric power system operators solve optimization problems that enforce inequality constraints such as limits on line flows, voltage magnitudes, and generator outputs. A common empirical observation regarding these constraints is that only a small fraction of them are binding (satisfied with equality) during operation. Furthermore, the same constraints tend to be binding across time periods. Recent research efforts have developed constraint screening algorithms that formalize this observation and allow for screening across operational conditions that are representative of longer time periods. These algorithms identify redundant constraints, i.e., constraints that can never be violated if other constraints are satisfied, by solving optimization problems for each constraint separately. This paper investigates how the choice of power flow formulation, represented either by the non-convex AC power flow, convex relaxations, or a linear DC approximation, impacts the results and the computational time of the screening method. This allows us to characterize the conservativeness of convex relaxations in constraint screening and assess the efficacy of the DC approximation in this context. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Xu, R. Ma,

**D.K. Molzahn**, H.L. Hijazi, and C. Josz, "Verifying Global Optimality of Candidate Solutions to Polynomial Optimization Problems using a Determinant Relaxation Hierarchy,"*60th IEEE Conference on Decision and Control (CDC)*, December 13-15, 2021.

We propose an approach for verifying that a given feasible point for a polynomial optimization problem is globally optimal. The approach relies on the Lasserre hierarchy and the result of Lasserre regarding the importance of the convexity of the feasible set as opposed to that of the individual constraints. By focusing solely on certifying global optimality and relaxing the Lasserre hierarchy using necessary conditions for positive semidefiniteness based on matrix determinants, the proposed method is implementable as a computationally tractable linear program. We demonstrate this method via application to the optimal power flow problem used to operate electric power systems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason and

**D.K. Molzahn**, "Analysis of Fast Decoupled Power Flow via Multiple Axis Rotations,"*52nd North American Power Symposium (NAPS)*, April 11-13, 2021.

The Fast Decoupled Power Flow (FDPF) method has been widely adopted due to its computational speed advantages relative to the Newton-Raphson method for many practically relevant power flow problems. The FDPF method relies on the assumption that the lines are highly inductive, i.e., have small R/X ratios. When this assumption does not hold (e.g., for distribution systems and some subtransmission systems), the FDPF method may exhibit slow convergence or fail to converge entirely. To address such cases, previous work has proposed an*axis rotation*method which rescales the complex power injections and the bus admittances by a unit-magnitude complex scalar parameter. Since this complex parameter adjusts the lines' R/X ratios, an appropriate choice for this parameter can improve the FDPF method's performance. In contrast to previous work that only introduces one complex parameter for the entire system (or one complex parameter per large subsystem), we propose and analyze an axis rotation method that introduces different complex parameters at each bus. The additional degrees of freedom provided in this more granular approach are particularly valuable for systems where the lines have wider ranges of R/X ratios. To obtain appropriate values for the complex parameters, we propose to minimize the sum of the squared errors associated with the FDPF approximations. This method can also be adapted for cases with large voltage angle differences. Our simulation results demonstrate the effectiveness of the proposed method. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Ivemeyer, M. Bossart, R.W. Kenyon, A. Sajadi, B.-M. Hodge, and

**D.K. Molzahn**, "Assessing the Accuracy of Balanced Power System Models in the Presence of Voltage Unbalance,"*Power and Energy Conference at Illinois (PECI)*, April 1-2, 2021.**Georgia Institute of Technology 2021 Roger P. Webb ECE Undergraduate Research Award**

Traditional models of electric power systems represent distribution systems with unbalanced three-phase network models and transmission systems with balanced single-phase-equivalent network models. This distinction poses a challenge for coupled models of transmission and distribution systems, which are becoming more prevalent due to the growth of distributed energy resources connected to distribution systems. In order to maintain a balanced network representation, transmission system models typically assume that the voltage phasors at the interface to the distribution system are balanced. Inaccuracies resulting from this assumption during unbalanced operation can lead to erroneous values for line currents in the transmission system model. This paper empirically quantifies the accuracy of this balanced operating assumption during unbalanced operating conditions for both a simple two-bus system along with a more complex transmission and distribution co-simulation. This paper also characterizes the performance of different methods for translating the unbalanced voltage phasors into a balanced representation in order to give recommendations for modeling coupled transmission and distribution systems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, M. Alowaifeer, X. Li, M.J. Till, and

**D.K. Molzahn**, "Reactive Power Planning using Security-Constrained AC Optimal Power Flow and Sensitivity Analyses,"*Power and Energy Conference at Illinois (PECI)*, April 1-2, 2021.

This paper proposes an approach for siting and sizing reactive power sources in order to maintain adequate reactive power reserves in future loading scenarios. We consider both normal and post-contingency operation using a security-constrained optimal power flow (SCOPF) formulation to compute the minimum-required reactive power reserves. To maintain tractability, the SCOPF explicitly considers a small set of important contingencies that are identified using a continuation power flow. The remaining contingencies are evaluated using the SCOPF solution. Any violated contingencies are iteratively added to the considered set in the SCOPF formulation until all contingencies are satisfied. Our approach combines this SCOPF formulation with a bus sensitivity index that identifies potential locations for new reactive power supply. We demonstrate the proposed approach on the IEEE 30- and 118-bus test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Tandon, S. Grijalva, and

**D.K. Molzahn**, "Motivating the Use of Dynamic Line Ratings to Mitigate the Risk of Wildire Ignition,"*Power and Energy Conference at Illinois (PECI)*, April 1-2, 2021.

Increasingly severe wildfires pose significant dangers to ecological, social, and economic systems. To curtail the risk of igniting wildfires, California utilities employ the Public Safety Power Shutoff (PSPS) program, which de-energizes transmission lines that run through wildfire-prone regions. In order to reduce the amount of load shedding required during PSPS events, this paper explores the use of dynamic line ratings (DLR) to manage the current flows on lines that pose a high risk of igniting wildfires. DLR schemes determine line flow limits that change with ambient conditions. This paper uses DLR in combination with more restrictive ground clearance requirements during wildfire-prone conditions to reduce the likelihood of wildfire-igniting faults from conductors contacting vegetation. Using two case studies, this paper demonstrates the potential benefits of DLR in this context by comparing the tradeoffs between completely de-energizing lines versus imposing more stringent ground clearance requirements via reducing current flows with DLR. Results from a large dataset representing WECC with actual data regarding ambient conditions, load demands, and wildfire risks show that the proposed approach can significantly reduce load shedding due to de-energizing lines that pose high risks of igniting wildfires. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, M. Alowaifeer, S. Grijalva, and

**D.K. Molzahn**, "Distributed Multi-Period DCOPF via an Auxiliary Principle Problem Algorithm,"*5th IEEE Texas Power and Energy Conference (TPEC)*, February 2-5, 2021.

Distributed algorithms provide attractive features for solving Optimal Power Flow (OPF) problems in interconnected power systems compared to traditional centralized algorithms. Distributed algorithms help to maintain the control autonomy and data privacy of subsystems, which is particularly relevant in competitive markets and practical control system implementations. This paper analyzes a distributed optimization algorithm known as the "Auxiliary Principle Problem" to solve multiperiod distributed DCOPF problems with energy storage systems. The proposed approach enables multiple interconnected systems with their own sub-objectives to share their resources and to participate in an electricity market without implicitly sharing information about their local generators or internal network parameters. The paper also shows how the proposed approach can enable future microgrids to coordinate their operation, reduce the total operational cost, and avoid internal constraint violations caused by unscheduled flows (USF) while maintaining the subsystems’ autonomy. We used an 11-bus test system consisting of two interconnected subsystems to evaluate the proposed approach and analyze the impact of USF. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Venzke, S. Chatzivasileiadis, and

**D.K. Molzahn**, "Inexact Convex Relaxations of AC Optimal Power Flow Problems: Towards AC Feasibility,"*Electric Power Systems Research*, vol. 187, October 2020.

Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted significant interest as in several instances they provably yield the global optimum to the original non-convex problem. If, however, the relaxation is inexact, the obtained solution is not AC-feasible. The quality of the obtained solution is essential for several practical applications of AC-OPF, but detailed analyses are lacking in existing literature. This paper aims to cover this gap. We provide an in-depth investigation of the solution characteristics when convex relaxations are inexact, we assess the most promising AC feasibility recovery methods for large-scale systems, and we propose two new metrics that lead to a better understanding of the quality of the identified solutions. We perform a comprehensive assessment on 96 different test cases, ranging from 14 to 3120 buses, and we show the following: (i) Despite an optimality gap of less than 1%, several test cases still exhibit substantial distances to both AC feasibility and local optimality and the newly proposed metrics characterize these deviations. (ii) Penalization methods fail to recover an AC-feasible solution in 15 out of 45 test cases. (iii) The computational benefits of warm-starting non-convex solvers have significant variation, but a computational speedup exists in over 75% of the test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Lee, K. Turitsyn,

**D.K. Molzahn**, and L.A. Roald, "Feasible Path Identification in Optimal Power Flow with Sequential Convex Restriction,"*IEEE Transactions on Power Systems*, vol. 35, no. 5, pp. 3648-3659, September 2020.

Nonconvexity induced by the nonlinear AC power flow equations challenges solution algorithms for AC optimal power flow (OPF) problems. While significant research efforts have focused on reliably computing high-quality OPF solutions, it is not always clear that there exists a feasible path to reach the desired operating point. Transitioning between operating points while avoiding constraint violations can be challenging since the feasible space of the OPF problem is nonconvex and potentially disconnected. To address this problem, we propose an algorithm that computes a provably feasible path from an initial operating point to a desired operating point. Given an initial feasible point, the algorithm solves a sequence of convex quadratically constrained optimization problems over conservative convex inner approximations of the OPF feasible space. In each iteration, we obtain a new, improved operating point and a feasible transition from the operating point in the previous iteration. In addition to computing a feasible path to a known desired operating point, this algorithm can also be used to improve the operating point locally. Extensive numerical studies on a variety of test cases demonstrate the algorithm and the ability to arrive at a high-quality solution in few iterations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]Z.J. Zhang, P.T. Mana, D. Yan, Y. Sun, and

**D.K. Molzahn**, "Study of Active Line Flow Constraints in DC Optimal Power Flow Problems,"*IEEE SoutheastCon*, March 12-15, 2020.

Efficiently and reliably operating electric power grids requires the frequent solution of optimization problems such as the DC Optimal Power Flow (DC OPF) problem. Solution of DC OPF problems can be challenging due to the large number of operational constraints imposed in these problems, especially limits on the line flows. Improving solver times can be facilitated by a better understanding of the active line flow constraints in DC OPF problems (i.e., the inequality constraints limiting line flows that are satisfied with equality at the solution). Considering a variety of test cases and a range of loading scenarios, this paper empirically characterizes the sets of active line flow constraints in DC OPF problems. Among other analyses, the paper compares the sets of redundant line flow constraints (i.e., constraints guaranteed to be inactive) that are identified by previously proposed constraint screening methods to the line flow constraints that are actually observed to be active over a range of scenarios. The results indicate that a large fraction of the line flow constraints which are not identified as being redundant by these screening methods are nevertheless inactive in the DC OPF solutions. This observation suggests the potential for improvements to the constraint screening methods. Laying the groundwork for achieving these improvements, this paper also identifies which line characteristics tend to be associated with active flow constraints and studies the relationships among sets of simultaneously active line flow constraints. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and I.A. Hiskens, "A Survey of Relaxations and Approximations of the Power Flow Equations,"*Foundations and Trends in Electric Energy Systems*, vol. 4, no. 1-2, pp. 1-221, February 2019.

The power flow equations relate the power injections and voltages in an electric power system and are therefore key to many power system optimization and control problems. Research efforts have developed a wide variety of relaxations and approximations of the power flow equations with a range of capabilities and characteristics. This monograph surveys relaxations and approximations of the power flow equations, with a particular emphasis on recently proposed formulations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ Tutorial Slides on Power Flow Relaxations ]A. Barzegar,

**D.K. Molzahn**, and R. Su, "A Method for Quickly Bounding the Optimal Objective Value of an OPF Problem Using a Semidefinite Relaxation and a Local Solution,"*Electric Power Systems Research*, vol. 177, December 2019.

Optimal power flow (OPF) is an important problem in the operation of electric power systems. Due to the OPF problem's non-convexity, there may exist multiple local optima. Certifiably obtaining the global solution is important for certain applications of OPF problems. Many global optimization techniques compute an optimality gap that compares the achievable objective value corresponding to the feasible point from a local solution algorithm with the objective value bound from a convex relaxation technique. Rather than the traditional practice of completely separating the local solution and convex relaxation computations, this paper proposes a method that exploits information from a local solution to speed the computation of an objective value bound using a semidefinite programming (SDP) relaxation. The improvement in computational tractability comes with the trade-off of reduced tightness for the resulting objective value bound. Numerical experiments illustrate this trade-off, with the proposed method being faster but weaker than the SDP relaxation and slower but tighter than second-order cone programming (SOCP) and quadratic convex (QC) relaxations for many large test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Yao,

**D.K. Molzahn**, and J.L. Mathieu, "A Multiperiod Optimal Power Flow Approach to Improve Power System Voltage Stability Using Demand Response,"*IEEE Transactions on Control of Network Systems, Special Issue on Analysis, Control, and Optimization of Energy System Networks*, vol. 6, no. 3, pp. 1015-1025, September 2019.

The increasing penetration of renewables has driven power systems to operate closer to their stability boundaries, increasing the risk of instability. We propose a multiperiod optimal power flow approach that uses demand responsive loads to improve steady-state voltage stability, which is measured by the smallest singular value (SSV) of the power flow Jacobian matrix. In contrast to past work that employs load shedding to improve stability, our approach improves the SSV by decreasing and increasing individual loads while keeping the total loading constant to avoid fluctuation of the system frequency. Additionally, an energy payback period maintains the total energy consumption of each load at its nominal value. The objective function balances SSV improvements against generation costs in the energy payback period. We develop an iterative linear programming algorithm using singular value sensitivities to obtain an AC-feasible solution. We demonstrate its performance on two IEEE test systems. The results show that demand response actions can improve static voltage stability, in some cases more cost-effectively than generation actions. We also compare our algorithm's performance to that of an iterative nonlinear programming algorithm from the literature. We find that our approach is approximately 6 times faster when applied to the IEEE 9-bus system, allowing us to demonstrate its performance on the IEEE 118-bus system. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and J. Wang, "Detection and Characterization of Intrusions to Network Parameter Data in Electric Power System Operations,"*IEEE Transactions on Smart Grid,*vol. 10, no. 4, pp. 3919-3928, July 2019.

Combating cyberattacks is an emerging challenge in maintaining the reliable and economic operation of electric power systems. Possible cyberattacks include intrusions to the parameter data at a control center. In this class of attacks, algorithms at the control center are correctly executed, but the attacker's modification of the associated parameter data yields improper results. This paper proposes an algorithm for detecting and characterizing cyberattacks to network parameter data, with specific application to optimal power flow problems. The proposed algorithm evaluates whether historical operating point data are consistent with the network parameters. Inconsistencies indicating potential cyberattacks are characterized using historical operational data (power injections and voltage phasors) along with network parameter data. Simulated test cases illustrate the proposed algorithm's detection and characterization capabilities. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]K. Girigoudar,

**D.K. Molzahn**, and L.A. Roald, "On the Relationships Among Different Voltage Unbalance Definitions,"*51st North American Power Symposium (NAPS)*, October 13-14, 2019.

Growing penetrations of distributed energy resources (DERs) increase the power injection variability in distribution systems, which can result in power quality issues such as voltage unbalance. To measure unbalance, organizations such as IEC, NEMA and IEEE define phase unbalance in their power quality standards. However, the definitions in these different standards are not consistent, and voltages that are considered acceptable by one standard may violate good practices defined by another standard. To address this issue, this paper provides analytical comparisons of the most common voltage unbalance definitions, which are supplemented with numerical simulations. The analytical relationships suggest that it is possible to approximately bound the symmetrical-component-based voltage unbalance factor (which depends on the magnitude and relative phase angle) by limiting the line-to-line voltage unbalance, whereas applying line-to-ground voltage unbalance definitions neglects all information about phase angle offsets. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]L.A. Roald and

**D.K. Molzahn**, "Implied Constraint Satisfaction in Power System Optimization: The Impacts of Load Variations,"*57th Annual Allerton Conference on Communication, Control, and Computing,*September 25-27, 2019.

In many power system optimization problems, we observe that only a small fraction of the line flow constraints ever become active at the optimal solution, despite variations in the load profile and generation costs. This observation has far-reaching implications not only for power system optimization, but also for the practical long-term planning, operation, and control of the system. We formalize this empirical observation for problems involving the DC power flow equations. We use a two-step constraint screening approach to identify constraints whose satisfaction is implied by other constraints in the problem. These constraints are redundant and can therefore can be eliminated from the problem. For the first screening step, we derive analytical bounds that quickly identify redundancies in the flow limits on parallel lines. The second screening step uses optimization-based constraint tightening, where we solve a (relaxed) optimization problem for each constraint to identify redundancies. We specifically focus our approach on large ranges of load variation such that the results are valid for long periods of time, thus justifying the computational overhead required for the screening method. Numerical results for a wide variety of standard test cases show that even with load variations up to ±100% of nominal loading, we are able to eliminate a significant fraction of the flow constraints. This large reduction in constraints may enable a range of possible applications. As one illustrative example, we demonstrate the computational improvements for the unit commitment problem obtained as a result of the reduced number of constraints. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]T. Mühlpfordt,

**D.K. Molzahn**, V. Hagenmeyer, and S. Misra, "Optimal Adaptive Power Flow Linearizations: Expected Error Minimization using Polynomial Chaos Expansion,"*IEEE Milan PowerTech*, June 23-27, 2019.

The nonlinearity of the power flow equations leads to algorithmic and theoretical challenges for a wide variety of optimization and control problems relevant to electric power systems. Solution algorithms for these problems often use linearization techniques to obtain more tractable power flow formulations. Many existing linearizations lack specialization to a given system and operating range of interest, leading to unnecessarily large linearization errors. In contrast, recently proposed "optimal adaptive" linearizations are 1) tailored to specific systems and operating ranges of interest and 2) optimal in the sense that they minimize a selected error metric relative to the nonlinear power flow equations. Previous work proposes optimal adaptive linearizations that minimize the*worst-case*linearization error. Building on previous work, this paper uses an uncertainty quantifiation method called Polynomial Chaos Expansion to minimize the*expected*linearization error for a given probability distribution of the power injections. As validated using Monte Carlo analyses, the capabilities of the expected-error-minimizing linearizations are shown to be superior to first-order Taylor approximations computed at a nominal operating point. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]C. Josz,

**D.K. Molzahn**, M. Tacchi, and S. Sojoudi, "Transient Stability Analysis of Power Systems via Occupation Measures,"*Innovative Smart Grid Technologies (ISGT)*, February 17-20, 2019.

We propose the application of occupation measure theory to the classical problem of transient stability analysis for power systems. This enables the computation of certified inner and outer approximations for the region of attraction of a nominal operating point. In order to determine whether a post-disturbance point requires corrective actions to ensure stability, one would then simply need to check the sign of a polynomial evaluated at that point. Thus, computationally expensive dynamical simulations are only required for post-disturbance points in the region between the inner and outer approximations. We focus on the nonlinear swing equations but voltage dynamics could also be included. The proposed approach is formulated as a hierarchy of semidefinite programs stemming from an infinite-dimensional linear program in a measure space, with a natural dual sum-of-squares perspective. On the theoretical side, this paper lays the groundwork for exploiting the oscillatory structure of power systems by using Hermitian (instead of real) sums-of-squares and connects the proposed approach to recent results from algebraic geometry. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and L.A. Roald, "Grid-Aware versus Grid-Agnostic Distribution System Control: A Method for Certifying Engineering Constraint Satisfaction,"*52nd Hawaii International Conference on System Sciences (HICSS)*, January 8-11, 2019.

Growing penetrations of distributed energy resources (DERs) in distribution systems have motivated the design of controllers that leverage DER capabilities to achieve system-wide objectives. These controllers may be either*grid-agnostic*or*grid-aware*, depending on whether distribution network constraints are considered. Grid-agnostic controllers have the benefit of not requiring network models or system measurements, but may cause dangerous constraint violations. Rather than develop a specific controller, this paper considers the*potential impacts*of DER controllers with respect to network constraint violations. Specifically, this paper develops an optimization-based method to rigorously certify when any grid-agnostic controller can be applied without concern regarding network constraint violations, or, conversely, when grid-aware control may be needed to maintain distribution grid security. The proposed method uses convex optimization techniques to bound the impacts of load variability, given a subset of buses with voltage measurements and control. The method either provides a certificate for secure operation or identifies potentially critical constraints and the need for additional controllability. Numerical tests illustrate the ability to certify secure operation for different ranges of variability. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Babaeinejadsarookolaee, A. Birchfield, R.D. Christie, C. Coffrin, C.L. DeMarco, R. Diao, M. Ferris, S. Fliscounakis, S. Greene, R. Huang, C. Josz, R. Korab, B.C. Lesieutre, J. Maeght,

**D.K. Molzahn**, T.J. Overbye, P. Panciatici, B. Park, J. Snodgrass, and R.D. Zimmerman, "The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms,"*IEEE PES PGLib-OPF Task Force Report*, August 2019.

**2020 Working Group Award for Technical Report from the IEEE Power and Energy Society (PES) Power System Operation, Planning and Economics (PSOPE) Committee.**

In recent years, the power systems research community has seen an explosion of novel methods for formulating the AC power flow equations. Consequently, benchmarking studies using the seminal AC Optimal Power Flow (AC-OPF) problem have emerged as the primary method for evaluating these emerging methods. However, it is often difficult to directly compare these studies due to subtle differences in the AC-OPF problem formulation as well as the network, generation, and loading data that are used for evaluation. To help address these challenges, this IEEE PES Task Force report proposes a standardized AC-OPF mathematical formulation and the PGLib-OPF networks for benchmarking AC-OPF algorithms. A motivating study demonstrates some limitations of the established network datasets in the context of benchmarking AC-OPF algorithms and a validation study demonstrates the efficacy of using the PGLib-OPF networks for this purpose. In the interest of scientific discourse and future additions, the PGLib-OPF benchmark library is open-access and all the of network data is provided under a creative commons license. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ link to PGLib-OPF datasets ]**D.K. Molzahn**, "Identifying and Characterizing Non-Convexities in Feasible Spaces of Optimal Power Flow Problems,"*IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 65, no. 5, pp. 672-676, May 2018. Presented at*IEEE International Symposium on Circuits and Systems (ISCAS), special session on On-line Identification, Control, & Optimization of Electric Power Systems,*May 27-30, 2018.

Optimal power flow (OPF) is an important problem in the operation of electric power systems. The solution to an OPF problem provides a minimum cost operating point that satisfies constraints imposed by both the non-linear power flow equations and engineering limits. These constraints can yield non-convex feasible spaces that result in significant computational challenges. This paper proposes an algorithm that identifies and characterizes non-convexities in OPF feasible spaces. This algorithm searches for a pair of feasible points whose connecting line segment contains an infeasible point. Such points certify the existence of a non-convexity in the OPF feasible space. Moreover, the constraint violations at the infeasible point along the connecting line segment physically characterize a cause of the non-convexity. Numerical demonstrations include a small illustrative example as well as applications to various test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]C. Josz and

**D.K. Molzahn**, "Lasserre Hierarchy for Large Scale Polynomial Optimization in Real and Complex Variables,"*SIAM Journal on Optimization*, vol. 28, no. 2, pp. 1017-1048, 2018.

We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the twenty first century, namely the optimal power flow problem. These notions enable us to find global minimizers on instances with up to 4,500 variables and 14,500 constraints. First, we generalize the Lasserre hierarchy from real to complex to numbers in order to enhance its tractability when dealing with complex polynomial optimization. Complex numbers are typically used to represent oscillatory phenomena, which are omnipresent in physical systems. Using the notion of hyponormality in operator theory, we provide a finite convergence criterion which generalizes the Curto-Fialkow conditions of the real Lasserre hierarchy. Second, we introduce the multi-ordered Lasserre hierarchy in order to exploit sparsity in polynomial optimization problems (in real or complex variables) while preserving global convergence. It is based on two ideas: 1) to use a different relaxation order for each constraint, and 2) to iteratively seek a closest measure to the truncated moment data until a measure matches the truncated data. Third and last, we exhibit a block diagonal structure of the Lasserre hierarchy in the presence of commonly encountered symmetries. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ technical report pdf ] [ technical report link ]**D.K. Molzahn**, "Identifying Redundant Flow Limits on Parallel Lines,"*IEEE Transactions on Power Systems (Letters)*, vol. 33, no. 3, pp. 3210-3212, 2018.

Many power system optimization problems constrain line flows with limits specified in terms of the magnitudes of apparent power or current flows. The set of line-flow constraints may have redundancies (i.e., the feasible space may be unchanged upon removal of a subset of the line-flow constraints), which unnecessarily complicate optimization problems. This letter describes an algorithm for identifying redundant line-flow constraints corresponding to certain parallel lines. After formulating the constraints as ellipsoids in the voltage variables, redundancies are detected from the absence of intersections between pairs of ellipsoids corresponding to parallel lines. This algorithm is demonstrated using several large test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]O. Coss, J.D. Hauenstein, H. Hoon, and

**D.K. Molzahn**, "Locating and Counting Equilibria of the Kuramoto Model with Rank One Coupling,"*SIAM Journal on Applied Algebra and Geometry,*vol. 2, no. 1, pp. 45-71, 2018.

The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phase-coherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, n, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of n. For these, the equilibria can be located by finding the real solutions of a system of polynomial equations utilizing techniques from algebraic geometry. However, typical methods for solving such systems locate all complex solutions even though only the real solutions give equilibria. In this paper, we present an algorithm to locate only the real solutions of the model, thereby shortening computation time by several orders of magnitude in certain situations. This is accomplished by choosing specific equilibria representatives and the consequent algebraic decoupling of the system. The correctness of the algorithm (that it finds only and all the equilibria) is proved rigorously. Additionally, the algorithm can be implemented using interval methods so that the equilibria can be approximated up to any given precision without significantly more computational effort. We also compare this solving approach to other computational algebraic geometric methods. Furthermore, analyzing this approach allows us to prove, asymptotically, that the maximum number of equilibria grows at the same rate as the number of complex solutions of a corresponding polynomial system. Finally, we conjecture an upper bound on the maximum number of equilibria for any number of oscillators which generalizes the known cases and is obtained on a range of explicitly provided natural frequencies. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Wu,

**D.K. Molzahn**, B.C. Lesieutre, and K. Dvijotham, "A Deterministic Method to Identify Multiple Local Extrema for the AC Optimal Power Flow Problem,"*IEEE Transactions on Power Systems,*vol. 33, no. 1, pp. 654-668, January 2018.

This paper presents a deterministic approach to compute multiple local extrema for AC Optimal Power Flow (ACOPF) problems. An elliptical representation of the sphereconfined Fritz John conditions is constructed from a quadratic formulation of the ACOPF problem. Application of a branch tracing method to the elliptical formulation enables the calculation of multiple solutions. Further, a monotone search strategy enhances the computational efficiency of finding multiple local solutions with improved objective values. The proposed approach is first illustrated using two small examples with known feasible spaces. Then, four additional local solutions to a 39-bus system are found using the proposed approach. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, H. Nagarajan, and M.L. Crow, "Comparison of Various Trilinear Monomial Envelopes for Convex Relaxations of Optimal Power Flow Problems,"*IEEE Global Conference on Signal and Information Processing (GlobalSIP),*November 26-29, 2018.

Solutions to optimal power flow (OPF) problems provide operating points for electric power systems that minimize operational costs while satisfying both engineering limits and the power flow equations. OPF problems are non-convex and may have multiple local optima. To search for global optima, recent research has developed a variety of convex relaxations to bound the optimal objective values of OPF problems. Certain relaxations, such as the quadratic convex (QC) relaxation, are derived from OPF representations that contain trilinear monomials. Previous work has considered three techniques for relaxing these trilinear monomials:*recursive McCormick*(RMC) envelopes,*Meyer and Floudas*(MF) envelopes, and*extreme-point*(EP) envelopes. This paper compares the tightness and computational speed of relaxations that employ each of these techniques. Forming the convex hull of a single trilinear monomial, MF and EP envelopes are equivalently tight. Empirical results show that QC formulations using MF and EP envelopes give tighter bounds than those using RMC envelopes. Empirical results also indicate that the EP envelopes have advantages over MF envelopes with respect to computational speed and numerical stability when used with state-of-the-art second-order cone programming solvers. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, D. Wu, and M.L. Crow, "Empirical Investigation of Non-Convexities in Optimal Power Flow Problems,"*American Control Conference (ACC),*June 27-29, 2018.

Optimal power flow (OPF) is a central problem in the operation of electric power systems. An OPF problem optimizes a specified objective function subject to constraints imposed by both the non-linear power flow equations and engineering limits. These constraints can yield non-convex feasible spaces that result in significant computational challenges. Despite these non-convexities, local solution algorithms actually find the global optima of some practical OPF problems. This suggests that OPF problems have a range of difficulty: some problems appear to have convex or "nearly convex" feasible spaces in terms of the voltage magnitudes and power injections, while other problems can exhibit significant non-convexities. Understanding this range of problem difficulty is helpful for creating new test cases for algorithmic benchmarking purposes. Leveraging recently developed computational tools for exploring OPF feasible spaces, this paper first describes an empirical study that aims to characterize non-convexities for small OPF problems. This paper then proposes and analyzes several medium-size test cases that challenge a variety of solution algorithms. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ test cases ]**D.K. Molzahn**and L.A. Roald, "Towards an AC Optimal Power Flow Algorithm with Robust Feasibility Guarantees,"*20th Power Systems Computation Conference (PSCC)*, June 11-15, 2018.

With growing penetrations of stochastic renewable generation and the need to accurately model the network physics, optimization problems that explicitly consider uncertainty and the AC power flow equations are becoming increasingly important to the operation of electric power systems. This paper describes initial steps towards an AC Optimal Power Flow (AC OPF) algorithm which yields an operating point that is guaranteed to be robust to all realizations of stochastic generation within a specified uncertainty set. Ensuring robust feasibility requires overcoming two challenges: 1) ensuring solvability of the power flow equations for all uncertainty realizations and 2) guaranteeing feasibility of the engineering constraints for all uncertainty realizations. This paper primarily focuses on the latter challenge. Specifically, this paper poses the robust AC OPF problem as a bi-level program that maximizes (or minimizes) the constraint values over the uncertainty set, where a convex relaxation of the AC power flow constraints is used to ensure conservativeness. The resulting optimization program is solved using an alternating solution algorithm. The algorithm is illustrated via detailed analyses of two small test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Misra,

**D.K. Molzahn**, and K. Dvijotham, "Optimal Adaptive Linearizations of the AC Power Flow Equations,"*20th Power Systems Computation Conference (PSCC)*, June 11-15, 2018.

The power flow equations are at the heart of many optimization and control problems relevant to power systems. The non-linearity of these equations leads to computational challenges in solving power flow and optimal power flow problems (non-convergence, local optima, etc.). Accordingly, various linearization techniques, such as the DC power flow, are often used to approximate the power flow equations. In contrast to a wide variety of general linearization techniques in the power systems literature, this paper computes a linear approximation that is specific to a given power system and operating range of interest. An "adaptive linearization" developed using this approach minimizes the worst-case error between the output of the approximation and the actual non-linear power flow equations over the operating range of interest. To compute an adaptive linearization, this paper proposes a constraint generation algorithm that iterates between 1) using an optimization algorithm to identify a point that maximizes the error of the linearization at that iteration and 2) updating the linearization to minimize the worst-case error among all points identified thus far. This approach is tested on several IEEE test cases, with the results demonstrating up to a factor of four improvement in approximation error over linearizations based on a first-order Taylor approximation. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, and M.L. Crow, "Improving QC Relaxations of OPF Problems via Voltage Magnitude Difference Constraints and Envelopes for Trilinear Monomials,"*20th Power Systems Computation Conference (PSCC)*, June 11-15, 2018.

AC optimal power flow (AC OPF) is a challenging non-convex optimization problem that plays a crucial role in power system operation and control. Recently developed convex relaxation techniques provide new insights regarding the global optimality of AC OPF solutions. The quadratic convex (QC) relaxation is one promising approach that constructs convex envelopes around the trigonometric and product terms in the polar representation of the power flow equations. This paper proposes two methods for tightening the QC relaxation. The first method introduces new variables that represent the voltage magnitude differences between connected buses. Using "bound tightening" techniques, the bounds on the voltage magnitude difference variables can be significantly smaller than the bounds on the voltage magnitudes themselves, so constraints based on voltage magnitude differences can tighten the relaxation. Second, rather than a potentially weaker "nested McCormick" formulation, this paper applies "Meyer and Floudas" envelopes that yield the convex hull of the trilinear monomials formed by the product of the voltage magnitudes and trignometric terms in the polar form of the power flow equations. Comparison to a state-of-the-art QC implementation demonstrates the advantages of these improvements via smaller optimality gaps. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]J.A. Kersulis, I.A. Hiskens, C. Coffrin, and

**D.K. Molzahn**, "Topological Graph Metrics for Detecting Grid Anomalies and Improving Algorithms,"*20th Power Systems Computation Conference (PSCC)*, June 11-15, 2018.

Power grids are naturally represented as graphs, with buses as nodes and power lines as edges. Graph theory provides many ways to measure power grid graphs, allowing researchers to characterize system structure and optimize algorithms. We apply several topological graph metrics to 33 publiclyavailable power grids. Results show that a straightforward, computationally inexpensive set of checks can quickly identify structural anomalies, especially when a broad set of test networks is available to establish norms. Another application of graph metrics is the characterization of computational behavior. We conclude by illustrating one compelling example: the close connection between clique analysis and semidefinite programming solver performance. These two applications demonstrate the power of purely topological graph metrics when utilized in the right settings. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]

## Journal Papers

## Conference Proceedings

## Journal Papers

## Conference Proceedings

## Journal Papers

## Conference Proceedings

## Journal Papers

## Conference Proceedings

## Journal Papers

## Conference Proceedings

## Monographs

## Journal Papers

## Conference Proceedings

## Journal Papers

## Conference Proceedings

J. Lavaei, S.H. Low, R. Baldick, B. Zhang,

**D.K. Molzahn**, F. Dörfler, and H. Sandberg, "Guest Editorial: Distributed Control and Efficient Optimization Methods for Smart Grid,"*IEEE Transactions on Smart Grid, Special issue on Distributed Control and Efficient Optimization Methods for Smart Grid*, vol. 8, no. 6, pp. 2939-2940, November 2017.

[ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, F. Dörfler, H. Sandberg, S.H. Low, S. Chakrabarti, R. Baldick, and J. Lavaei, "A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems,"*IEEE Transactions on Smart Grid, Special issue on Distributed Control and Efficient Optimization Methods for Smart Grid*, vol. 8, no. 6, pp. 2941-2962, November 2017.

Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, "Computing the Feasible Spaces of Optimal Power Flow Problems,"*IEEE Transactions on Power Systems*, vol. 32, no. 6, pp. 4752-4763, November 2017.

The solution to an optimal power flow (OPF) problem provides a minimum cost operating point for an electric power system. The performance of OPF solution techniques strongly depends on the problem's feasible space. This paper presents an algorithm that is guaranteed to compute the entire feasible spaces of small OPF problems to within a specified discretization tolerance. Specifically, the feasible space is computed by discretizing certain of the OPF problem's inequality constraints to obtain a set of power flow equations. All solutions to the power flow equations at each discretization point are obtained using the Numerical Polynomial Homotopy Continuation (NPHC) algorithm. To improve computational tractability, "bound tightening" and "grid pruning" algorithms use convex relaxations to preclude consideration of many discretization points that are infeasible for the OPF problem. The proposed algorithm is used to generate the feasible spaces of two small test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, "Incorporating Squirrel-Cage Induction Machine Models in Convex Relaxations of OPF Problems,"*IEEE Transactions on Power Systems (Letters)*, vol. 32, no. 6, pp. 4972-4974, November 2017.

The optimal power flow (OPF) problem determines a minimum cost operating point for an electric power system. Recently developed convex relaxations are capable of globally solving certain OPF problems. Using a semidefinite relaxation of the OPF problem as an illustrative example, this letter presents a method for extending convex relaxations of the OPF problem to include steady-state squirrel-cage induction machine models. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]J.F. Marley,

**D.K. Molzahn**, and I.A. Hiskens, "Solving Multiperiod OPF Problems using an AC-QP Algorithm Initialized with an SOCP Relaxation,"*IEEE Transactions on Power Systems*, vol. 32, no. 5, pp. 3538-3548, September 2017.

Renewable generation and energy storage are playing an ever increasing role in power systems. Hence, there is a growing need for integrating these resources into the optimal power flow (OPF) problem. While storage devices are important for mitigating renewable variability, they introduce temporal coupling in the OPF constraints, resulting in a multiperiod OPF formulation. This paper explores a solution method for multiperiod AC OPF that combines a successive quadratic programming approach (AC-QP) with a second-order cone programming (SOCP) relaxation of the OPF problem. The SOCP relaxation’s solution is used to initialize the AC-QP OPF algorithm. Additionally, the lower bound on the objective value obtained from the SOCP relaxation provides a measure of solution quality. This combined method is demonstrated on several test cases with up to 4259 nodes and a time horizon of 8 time steps. A comparison of initialization schemes indicates that the SOCP-based approach offers improved convergence rate, execution time and solution quality. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, C. Josz, I.A. Hiskens, and P. Panciatici, "A Laplacian-Based Approach for Finding Near Globally Optimal Solutions to OPF Problems,"*IEEE Transactions on Power Systems*, vol. 32, no. 1, pp. 305-315, January 2017.

A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Yao, J.L. Mathieu, and

**D.K. Molzahn**, "The Impact of Load Models in an Algorithm for Improving Power System Voltage Stability via Demand Response,"*55th Annual Allerton Conference on Communication, Control, and Computing,*October 4-6, 2017.

Increasing communication and control capabilities will allow future power system operators to exploit large quantities of responsive demand. This paper discusses ongoing work that employs demand response to improve voltage stability via virtual spatial shifting of loads (i.e., altering the locational distribution of power consumption in one time period with an energy payback in a following time period). In this paper, we study the impact of load models on a previously proposed iterative linearization algorithm to determine loading patterns that maximize a voltage stability margin, namely, the smallest singular value (SSV) of the power flow Jacobian matrix. Specifically, we extend the algorithm to enable inclusion of composite load models consisting of both "ZIP" components and a steady-state squirrel-cage induction machine (IM) model. We then investigate the impact of different load models on both the stability margin and the loading pattern. Using the IEEE 14-bus system as an illustrative example, the results show that the type of load model affects the nominal system's SSV, the optimal SSV, and the optimal loading pattern. The maximum-achievable percent change in SSV is larger using IM models than using ZIP models. We also discuss the difficulty in interpreting the stability margin when the system undergoes structural changes resulting from the use of different voltage-dependent load models. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]L.A. Roald,

**D.K. Molzahn**, and A.F. Tobler, "Power System Optimization with Uncertainty and AC Power Flow: Analysis of an Iterative Algorithm,"*IREP Symposium on Bulk Power System Dynamics and Control-X. The Power System of the Future: Global Dynamics arising from Distributed Actions,*August 27 - September 1, 2017.

Many power system optimization problems are inherently affected by uncertainty, as it is generally impossible to exactly forecast load or renewable generation. Finding solutions that lead to low-cost, yet secure operation for a range of conditions is an important concern to system operators. This paper considers an iterative algorithm for solving power system optimization problems that include models of uncertainty and AC power flow physics. This algorithm iterates between solving deterministic nonlinear optimization problems with the AC power flow model and updating constraint tightenings that adjust the constraints to facilitate secure operation. Decoupling the uncertainty and the nonlinear optimization steps provides several advantages, such as tractability for large-scale problems and the ability to separately apply state-of-the-art techniques for handling the uncertainty and the nonlinearity. Using numerical examples and theoretical analyses, this paper discusses the benefits and limitations of the iterative algorithm in solving general power system optimization problems. This paper further characterizes the behavior of the iterative algorithm with respect to convergence, optimality, and infeasibility, and suggests several modifications to improve performance. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Yao, J.L. Mathieu, and

**D.K. Molzahn**, "Using Demand Response to Improve Power System Voltage Stability Margins,"*IEEE PowerTech Manchester*, June 18-22, 2017.**High-quality paper award.**

Electric power systems with high penetrations of fluctuating renewable generation may operate close to their stability limits. Demand response can be used to improve power system stability. For example, it is already used to provide frequency control, improving frequency stability. This paper presents a new demand response strategy that uses fast-acting flexible loads to change the loading pattern while keeping the total load constant, improving steady-state voltage stability without affecting the system frequency. The new loading pattern is maintained only for a short period of time until the generators can be re-dispatched. Our goal is to find a permissible loading pattern that maximizes the smallest singular value of the power flow Jacobian matrix, which serves as a measure of voltage stability. The corresponding non-convex optimization problem is solved with an iterative linear programming approach that uses eigenvalue sensitivities and a linearization of the AC power flow equations. Using two IEEE test cases as illustrative examples, we show that the loading patterns resulting from the proposed approach give smallest singular values that are very close to those obtained from a brute force search. The results are also compared to another voltage stability measure given by the maximum loading margin for uniformly varying power injections. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]

## Journal Papers

## Conference Proceedings

**D.K. Molzahn**and I.A. Hiskens, "Convex Relaxations of Optimal Power Flow Problems: An Illustrative Example,"*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 63, no. 5, pp. 650-660, May 2016.

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite programming (SDP) relaxation globally solves many OPF problems. However, there exist practical problems for which the SDP relaxation fails to yield the global solution. Conditions for the success or failure of the SDP relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the SDP relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The SDP relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the SDP relaxation must involve factors beyond just the network physics. The lack of universal physical conditions for success of the SDP relaxation motivates the development of tighter convex relaxations capable of solving a broader class of problems. Tools from polynomial optimization theory provide a means of developing tighter relaxations. This paper uses the example problem to illustrate relaxations from the Lasserre hierarchy for polynomial optimization and a related "mixed semidefinite/second-order cone programming" hierarchy. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ figures ]K. Dvijotham and

**D.K. Molzahn**, "Error Bounds on the DC Power Flow Approximation: A Convex Relaxation Approach,"*IEEE 55th Annual Conference on Decision and Control (CDC)*, December 12-14, 2016.

Power flow models are fundamental to power systems analyses ranging from short-term market clearing and voltage stability studies to long-term planning. Due to the nonlinear nature of the AC power flow equations and the associated computational challenges, linearized approximations (like the DC power flow) have been widely used to solve these problems in a computationally tractable manner. The linearized approximations have been justified using traditional engineering assumptions that under "normal" operating conditions, voltage magnitudes do not significantly deviate from nominal values and phase differences are "small". However, there is only limited work on rigorously quantifying when it is safe to use these linearized approximations. In this paper, we propose an algorithm capable of computing rigorous bounds on the approximation error in the DC power flow (and, in future extensions, more general linearized approximations) using convex relaxation techniques. Given a set of operational constraints (limits on the voltage magnitudes, phase angle differences, and power injections), the algorithm determines an upper bound on the difference in injections at each bus computed by the AC and DC power flow power flow models within this domain. We test our approach on several IEEE benchmark networks. Our experimental results show that the bounds are reasonably tight (i.e., there are points within the domain of interest that are close to achieving the bound) over a range of operating conditions. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, C. Josz, and I.A. Hiskens, "Moment Relaxations of Optimal Power Flow Problems: Beyond the Convex Hull,"*IEEE Global Conference on Signal and Information Processing (GlobalSIP)*, December 7-9, 2016.

Optimal power flow (OPF) is one of the key electric power system optimization problems. "Moment" relaxations from the Lasserre hierarchy for polynomial optimization globally solve many OPF problems. Previous work illustrates the ability of higher-order moment relaxations to approach the convex hulls of OPF problems' non-convex feasible spaces. Using a small test case, this paper focuses on the ability of the moment relaxations to globally solve problems with objective functions that have unconstrained minima at infeasible points inside the convex hull of the non-convex constraints. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, D. Mehta, and M. Niemerg, "Toward Topologically Based Upper Bounds on the Number of Power Flow Solutions,"*American Control Conference (ACC)*, July 6-8, 2016.**Best presentation in session award.**

The power flow equations, which relate power injections and voltage phasors, are at the heart of many electric power system computations. While Newton-based methods typically find the "high-voltage" solution to the power flow equations, which is of primary interest, there are potentially many "low-voltage" solutions that are useful for certain analyses. This paper addresses the number of solutions to the power flow equations. There exist upper bounds on the number of power flow solutions; however, there is only limited work regarding bounds that are functions of network topology. This paper empirically explores the relationship between the network topology, as characterized by the maximal cliques, and the number of power flow solutions. To facilitate this analysis, we use a numerical polynomial homotopy continuation approach that is guaranteed to find all complex solutions to the power flow equations. The number of solutions obtained from this approach upper bounds the number of real solutions. Testing with many small networks informs the development of upper bounds that are functions of the network topology. Initial results include empirically derived expressions for the maximum number of solutions for certain classes of network topologies. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Mehta,

**D.K. Molzahn**, and K. Turitsyn, "Recent Advances in Computational Methods for the Power Flow Equations,"*American Control Conference (ACC)*, July 6-8, 2016.

The power flow equations are at the core of most of the computations for designing and operating electric grids. This system of multivariate nonlinear equations relate the power injections and voltages in an electric power system. A plethora of methods have been devised to solve these equations, from Newton-based methods to homotopy continuation and other optimization-based methods. Although many of these methods often efficiently find a high-voltage, stable solution, challenges remain for finding low-voltage solutions, which play significant roles in certain stability-related computations. While we do not claim to have exhausted the existing literature on all related methods, this tutorial paper introduces some of the recent advances in power flow solution methods to the wider power systems community as well as bringing attention from the computational mathematics and optimization communities to power systems problems. After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method, Gröbner basis techniques, and moment/sum-of-squares relaxations using semidefinite programming. In passing, we also emphasize the importance of an upper bound on the number of solutions of the power flow equations and review the current status of research in this direction. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, C. Josz, I.A. Hiskens, and P. Panciatici, "Computational Analysis of Sparsity-Exploiting Moment Relaxations of the OPF Problem,"*19th Power Systems Computation Conference (PSCC)*, June 20-24, 2016.

With the potential to find global solutions, significant research interest has focused on convex relaxations of the non-convex OPF problem. Recently, "moment-based" relaxations from the Lasserre hierarchy for polynomial optimization have been shown capable of globally solving a broad class of OPF problems. Global solution of many large-scale test cases is accomplished by exploiting sparsity and selectively applying the computationally intensive higher-order relaxation constraints. Previous work describes an iterative algorithm that indicates the buses for which the higher-order constraints should be enforced. In order to speed computation of the moment relaxations, this paper provides a study of the key parameter in this algorithm as applied to relaxations from both the original Lasserre hierarchy and a recent complex extension of the Lasserre hierarchy. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, I.A. Hiskens, and B.C. Lesieutre, "Calculation of Voltage Stability Margins and Certification of Power Flow Insolvability using Second-Order Cone Programming,"*49th Hawaii International Conference on System Sciences (HICSS)*, January 5-8, 2016.

Reliable power system operation requires maintaining sufficient voltage stability margins. Traditional techniques based on continuation and optimization calculate lower bounds for these margins and generally require appropriate initialization. Building on a previous semidefinite programming (SDP) formulation, this paper proposes a new second-order cone programming (SOCP) formulation which directly yields upper bounds for the voltage stability margin without needing to specify an initialization. Augmentation with integer-constrained variables enables consideration of reactive-power-limited generators. Further, leveraging the ability to globally solve these problems, this paper describes a sufficient condition for insolvability of the power flow equations. Trade-offs between tightness and computational speed of the SDP and SOCP relaxations are studied using large test cases representing portions of European power systems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, M. Niemerg, D. Mehta, and J.D. Hauenstein, "Investigating the Maximum Number of Real Solutions to the Power Flow Equations: Analysis of Lossless Four-Bus Systems," March 2016.

The power flow equations model the steady-state relationship between the power injections and voltage phasors in an electric power system. By separating the real and imaginary components of the voltage phasors, the power flow equations can be formulated as a system of quadratic polynomials. Only the real solutions to these polynomial equations are physically meaningful. This paper focuses on the maximum number of real solutions to the power flow equations. The literature provides upper bounds on the number of real power flow solutions based on the maximum number of complex solutions. There exist two- and three-bus systems which have as many real solutions as the upper bounds given by the maximum number of complex solutions. It is an open question whether this is also the case for larger systems. This paper uses a conjecture and recent algorithmic developments from Galois group theory to suggest a negative answer to this question. Specifically, this paper studies a generic four-bus system composed of PV buses. If a conjecture in Galois group theory holds, the maximum number of real power flow solutions for this system is upper bounded by 16, which is strictly less than 20, the maximum number of complex solutions. This paper also provides parameter values for this system which give rise to 16 real solutions, so this upper bound is achievable. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]

## Journal Papers

## Conference Proceedings

## Technical Reports

**D.K. Molzahn**and I.A. Hiskens, "Sparsity-Exploiting Moment-Based Relaxations of the Optimal Power Flow Problem,"*IEEE Transactions on Power Systems*, vol. 30, no. 6, pp. 3168-3180, November 2015.

Convex relaxations of non-convex optimal power flow (OPF) problems have recently attracted significant interest. While existing relaxations globally solve many OPF problems, there are practical problems for which existing relaxations fail to yield physically meaningful solutions. This paper applies moment relaxations to solve many of these OPF problems. The moment relaxations are developed from the Lasserre hierarchy for solving generalized moment problems. Increasing the relaxation order in this hierarchy results in "tighter" relaxations at the computational cost of larger semidefinite programs. Low-order moment relaxations are capable of globally solving many small OPF problems for which existing relaxations fail. By exploiting sparsity and only applying the higher-order relaxation to specific buses, global solutions to larger problems are computationally tractable through the use of an iterative algorithm informed by a heuristic for choosing where to apply the higher-order constraints. With standard semidefinite programming solvers, the algorithm globally solves many test systems with up to 300 buses for which the existing semidefinite relaxation fails to yield globally optimal solutions. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, C. Josz, I.A. Hiskens, and P. Panciatici, "Solution of Optimal Power Flow Problems using Moment Relaxations Augmented with Objective Function Penalization,"*IEEE 54th Annual Conference on Decision and Control (CDC)*, December 15-18, 2015.

The optimal power flow (OPF) problem minimizes the operating cost of an electric power system. Applications of convex relaxation techniques to the non-convex OPF problem have been of recent interest, including work using the Lasserre hierarchy of "moment" relaxations to globally solve many OPF problems. By preprocessing the network model to eliminate low-impedance lines, this paper demonstrates the capability of the moment relaxations to globally solve large OPF problems that minimize active power losses for portions of several European power systems. Large problems with more general objective functions have thus far been computationally intractable for current formulations of the moment relaxations. To overcome this limitation, this paper proposes the combination of an objective function penalization with the moment relaxations. This combination yields feasible points with objective function values that are close to the global optimum of several large OPF problems. Compared to an existing penalization method, the combination of penalization and the moment relaxations eliminates the need to specify one of the penalty parameters and solves a broader class of problems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, Z.B. Friedman, B.C. Lesieutre, C.L. DeMarco, and M.C. Ferris, "Estimation of Constraint Parameters in Optimal Power Flow Data Sets,"*47th North American Power Symposium (NAPS)*, October 4-6, 2015.

Large-scale electric power system analysis depends upon representation of vast numbers of components whose individual models must be populated with parameters. The challenge of populating such component models is particularly apparent in optimal power flow applications, in which incorrect parameters and/or constraint limits can yield overall system representations with either unrealistically large feasible regions or an empty feasible set. Unfortunately, many data sets, particularly those of publicly available test cases, were originally developed to illustrate simpler "power flow only" applications, and may contain unrealistic values or wholly omit important constraint limits. This paper describes engineering-based approaches to obtain credible estimates for parameters and limits associated with line-flow constraints and generator capability curves, as may be employed in a number of steady state analyses such as the optimal power flow. These can substitute for missing or unrealistic data in test systems for which more fully detailed, "real-world" component specifications and limits are not available, and thereby make such test systems more valuable as research tools. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and I.A. Hiskens, "Mixed SDP/SOCP Moment Relaxations of the Optimal Power Flow Problem,"*IEEE PowerTech Eindhoven*, June 29 - July 2, 2015.

Recently, convex "moment" relaxations developed from the Lasserre hierarchy for polynomial optimization problems have been shown capable of globally solving many optimal power flow (OPF) problems. The moment relaxations, which take the form of semidefinite programs (SDP), generalize a previous SDP relaxation of the OPF problem. This paper presents an approach for improving the computational performance of the moment relaxations for many problems. This approach enforces second-order cone programming (SOCP) constraints that establish necessary (but not sufficient) conditions for satisfaction of the SDP constraints arising from the higher-order moment relaxations. The resulting "mixed SDP/SOCP" formulation implements the first-order relaxation using SDP constraints and the higher-order relaxations using SOCP constraints. Numerical results demonstrate that this mixed SDP/SOCP relaxation is capable of solving many problems for which the first-order moment relaxation fails to yield a global solution. For several examples, the mixed SDP/SOCP relaxation improves computational speed by factors from 1.13 to 18.7. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, S.S. Baghsorkhi, and I.A. Hiskens, "Semidefinite Relaxations of Equivalent Optimal Power Flow Problems: An Illustrative Example,"*IEEE International Symposium on Circuits and Systems (ISCAS)*, May 24-27, 2015.

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite relaxation globally solves many OPF problems. However, there exist practical problems for which the semidefinite relaxation fails to yield the global solution. Conditions for the success or failure of the semidefinite relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the semidefinite relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The semidefinite relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the semidefinite relaxation must involve factors beyond just the network physics. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]

## Journal Papers

## Conference Proceedings

**D.K. Molzahn**, B.C. Lesieutre, and C.L. DeMarco, "Approximate Representation of ZIP Loads in a Semidefinite Relaxation of the OPF Problem,"*IEEE Transactions on Power Systems (Letters)*, vol. 29, no. 4, pp. 1864-1865, July 2014.

Recent research has applied semidefinite programming (SDP) to the optimal power flow (OPF) problem. Extending SDP formulations to include the ZIP load model, which consists of constant impedance, constant current, and constant power components, is necessary for many practical problems. Existing SDP formulations consider constant power components and can be easily extended to incorporate constant impedance components. With linear dependence on voltage magnitude, constant current components are not trivially formulated in a manner amenable to SDP. This letter details an approximate representation of ZIP loads in a SDP relaxation of the OPF problem. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and C.L. DeMarco, "A Sufficient Condition for Global Optimality of Solutions to the Optimal Power Flow Problem,"*IEEE Transactions on Power Systems (Letters)*, vol. 29, no. 2, pp. 978-979, March 2014.

Recent applications of a semidefinite programming (SDP) relaxation to the optimal power flow (OPF) problem offers a polynomial time method to compute a global optimum for a large subclass of OPF problems. In contrast, prior OPF solution methods in the literature guarantee only local optimality for the solution produced. However, solvers employing SDP relaxation remain significantly slower than mature OPF solution codes. This letter seeks to combine the advantages of the two methods. In particular, we develop a SDP-inspired sufficient condition test for global optimality of a candidate OPF solution. This test may then be easily applied to a candidate solution generated by a traditional, only-guaranteed-locally-optimal OPF solver. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Panciatici, M.C. Campi, S. Garatti, S.H. Low,

**D.K. Molzahn**, A.X. Sun, and L. Wehenkel, "Advanced Optimization Methods for Power Systems,"*18th Power Systems Computation Conference (PSCC),*August 18-22, 2014.

Power system planning and operation offers multitudinous opportunities for optimization methods. In practice, these problems are generally large-scale, non-linear, subject to uncertainties, and combine both continuous and discrete variables. In the recent years, a number of complementary theoretical advances in addressing such problems have been obtained in the field of applied mathematics. The paper introduces a selection of these advances in the fields of non-convex optimization, in mixed-integer programming, and in optimization under uncertainty. The practical relevance of these developments for power systems planning and operation are discussed, and the opportunities for combining them, together with high-performance computing and big data infrastructures, as well as novel machine learning and randomized algorithms, are highlighted. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and I.A. Hiskens, "Moment-Based Relaxation of the Optimal Power Flow Problem,"*18th Power Systems Computation Conference (PSCC),*August 18-22, 2014.

The optimal power flow (OPF) problem minimizes power system operating cost subject to both engineering and network constraints. With the potential to find global solutions, significant research interest has focused on convex relaxations of the non-convex AC OPF problem. This paper investigates "moment-based" relaxations of the OPF problem developed from polynomial optimization theory. At the cost of increased computational requirements, moment relaxations are generally tighter than relaxations employed in previous research, thus resulting in global solutions for a broader class of OPF problems. Exploration of the feasible spaces of test systems illustrates the effectiveness of the moment relaxations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and C.L. DeMarco, "Investigation of Non-Zero Duality Gap Solutions to a Semidefinite Relaxation of the Optimal Power Flow Problem,"*47th Hawaii International Conference on System Sciences (HICSS)*, January 6-9, 2014.

Recently, a semidefinite programming relaxation of the power flow equations has been applied to the optimal power flow problem. When this relaxation is "tight" (i.e., the solution has zero duality gap), a globally optimal solution is obtained. Existing literature investigates sufficient conditions whose satisfaction guarantees zero duality gap solutions. However, there is limited study of non-zero duality gap solutions. By illustrating the feasible spaces for optimal power flow problems and their semidefinite relaxations, this paper investigates examples of nonzero duality gap solutions. Results for large system models suggest that non-convexities associated with small subsections of the network are responsible for non-zero duality gap solutions. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]B.C. Lesieutre and

**D.K. Molzahn**, "Optimization and Control of Electric Power Systems,"*Final Technical Report, DOE Grant #DE-SC0002319*, October 17, 2014.

The Electric Power Grid is a critical infrastructure upon which the nation relies for delivery of much of its energy use. It supports the needs of industry, commerce, and consumer end use. Reliability and economic efficiency are essentials. Disruptions are inconvenient and can be very costly. The research presented here is directed towards optimal use of grid resources to ensure reliability and minimize cost. The analysis and optimization needs for planning and operation of the electric power system are challenging to meet due to the scale and form of model representations. The connected network spans the continent and the mathematical models are inherently nonlinear. Traditionally, computational limits have necessitated the use of very simplified models for grid analysis, and this has resulted in either less secure operation, or less efficient operation, or both. The research conducted in this project advances techniques for power system optimization problems that will enhance reliable and efficient operation. The results of this work appear in numerous publications and address different application problems include Optimal Power Flow (OPF) [8, 9, 15, 52, 53], unit commitment [41], demand response [46, 49], reliability margins [32, 33, 38], planning [42, 47, 48], transmission expansion [50, 51], and general tools and algorithms [43, 44, 45]. Importantly, the research has been successfully transferred into the practical application domain by its inclusion in the most recent release of the widely-used MATPOWER suite of power system analysis tools. [ abstract ] [ link ] [ Ref: bib txt ]

## Journal Papers

## Conference Proceedings

## Technical Reports

**D.K. Molzahn**, J.T. Holzer, B.C. Lesieutre, and C.L. DeMarco, "Implementation of a Large-Scale Optimal Power Flow Solver Based on Semidefinite Programming,"*IEEE Transactions on Power Systems*, vol. 28, no. 4, pp. 3987-3998, November 2013.

The application of semidefinite programming to the optimal power flow (OPF) problem has recently attracted significant research interest. This paper provides advances in modeling and computation required for solving the OPF problem for large-scale, general power system models. Specifically, a semidefinite programming relaxation of the OPF problem is presented that incorporates multiple generators at the same bus and parallel lines. Recent research in matrix completion techniques that decompose a single large matrix constrained to be positive semidefinite into many smaller matrices has made solution of OPF problems using semidefinite programming computationally tractable for large system models. We provide three advances to existing decomposition techniques: a matrix combination algorithm that further decreases solver time, a modification to an existing decomposition technique that extends its applicability to general power system networks, and a method for obtaining the optimal voltage profile from the solution to a decomposed semidefinite program. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and C.L. DeMarco, "A Sufficient Condition for Power Flow Insolvability with Applications to Voltage Stability Margins,"*IEEE Transactions on Power Systems,*vol. 28, no. 3, pp. 2592-2601, August 2013.

For the nonlinear power flow problem specified with standard PQ, PV, and slack bus equality constraints, we present a sufficient condition under which the specified set of nonlinear algebraic equations has no solution. This sufficient condition is constructed in a framework of an associated feasible, convex optimization problem. The objective employed in this optimization problem yields a measure of distance (in a parameter set) to the power flow solution boundary. In practical terms, this distance is closely related to quantities that previous authors have proposed as voltage stability margins. A typical margin is expressed in terms of the parameters of system loading (injected powers); here we additionally introduce a new margin in terms of the parameters of regulated bus voltages. [ abstract ] [ pdf ] [ link ] [ extended version ] [ Ref: bib txt ]**D.K. Molzahn**and B.C. Lesieutre, "Initializing Dynamic Power System Simulations Using Eigenvalue Formulations of the Induction Machine and Power Flow Models,"*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 60, no. 3, pp. 690-702, March 2013.

Initializing internal variables in dynamic power system simulations is a two-stage process. First, a power flow model is used to find the steady state bus variables. Second, values for the internal variables associated with bus connected components are determined such that the components' terminal values match the bus variables calculated from the power flow model. Initializing most components' variables is a straightforward, direct process. However, initializing induction machine variables traditionally uses an indirect, iterative process. In this paper, eigenvalue formulations are detailed for both the induction machine initialization and power flow models, which provide a direct method for determining all possible sets of induction machine initializations and offer a novel model for the power flow equations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and H. Chen, "Counterexample to a Continuation-Based Algorithm for Finding All Power Flow Solutions,"*IEEE Transactions on Power Systems (Letters)*, vol. 28, no. 1, pp. 564-565, February 2013.

Existing literature claims that an algorithm based on continuation is capable of finding all solutions to the power flow equations for all power systems. This claim is demonstrated to be incorrect through the use of a five-bus system counterexample. Existing literature also claims that a similar algorithm is capable of finding all Type-1 solutions (i.e., solutions where the power flow Jacobian has a single eigenvalue with positive real part) to the power flow equations for all systems. This claim is also shown to be incorrect using the five-bus system counterexample. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A.R. Borden,

**D.K. Molzahn**, B.C. Lesieutre, and P. Ramanathan, "Power System Structure and Confidentiality Preserving Transformation of Optimal Power Flow Problem,"*51st Annual Allerton Conference on Communication, Control, and Computing*, October 2-4, 2013.

In the field of power system engineering, the optimal power flow problem is essential in planning and operations. With increasing system size and complexity, the computational requirements needed to solve practical optimal power flow problems continues to grow. Increasing computational requirements make the possibility of performing these computations remotely with cloud computing appealing. However, power system structure and component values are often confidential; therefore, the problem cannot be shared. To address this issue of confidential information in cloud computing, some techniques for masking optimization problems have been developed. The work of this paper builds upon these techniques for optimization problems but is specifically developed for addressing the DC and AC optimal power flow problems. We study the application of masking a sample OPF using the IEEE 14-bus network. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, V. Dawar, B.C. Lesieutre, and C.L. DeMarco, "Sufficient Conditions for Power Flow Insolvability Considering Reactive Power Limited Generators with Applications to Voltage Stability Margins,"*IREP Symposium on Bulk Power System Dynamics and Control - IX. Optimization, Security and Control of the Emerging Power Grid*, 25-30 August 2013.

For the non-linear power flow problem with PQ and reactive power limited slack and PV buses, we present two sufficient conditions under which the specified set of nonlinear algebraic equations has no solution. The first condition uses a semidefinite programming relaxation of the power flow equations along with binary variables to model the generators' reactive power capabilities. As a byproduct, this condition yields a voltage stability margin to the power flow solvability boundary. The second condition formulates the power flow equations, including generator reactive power limits, as a system of polynomials and uses real algebraic geometry and sum of squares programming to create infeasibility certificates which prove power flow insolvability. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, "Application of Semidefinite Optimization Techniques to Problems in Electric Power Systems,"*Ph.D. Dissertation, University of Wisconsin-Madison Department of Electrical and Computer Engineering*, August 2013.

**2014 Harold A. Peterson Best Dissertation Award, Second Place in the UW-Madison ECE Department.**

Motivated by the potential for improvements in electric power system economics and reliability, this dissertation investigates applications of a semidefinite programming relaxation of the power flow equations, which model the steady-state relationship between power injections and voltages in a power system. In contrast to other optimal power flow solution techniques, semidefinite program solvers can reliably find a global optimum in polynomial time when the semidefinite relaxation is "tight" (i.e., the solution has zero relaxation gap). Semidefinite relaxations have been applied to a variety of computationally difficult problems in many fields. The power systems literature details limited success in applying semidefinite programming to the optimal power flow (OPF) problem, which minimizes operating cost under engineering and network constraints. Semidefinite programming holds significant promise for application to other power systems problems.

This dissertation first investigates power system economics using a semidefinite relaxation of the OPF problem. In contrast to claims in the literature, this dissertation shows that the semidefinite relaxation may fail to give a physically meaningful solution (i.e., the solution has non-zero relaxation gap) for some practical problems. This dissertation also provides computational and modeling advances required for solving large-scale, realistic OPF problems. Modeling advances include allowing multiple generators at the same bus; parallel transmission lines and transformers; and an approximate method for modeling constant impedance, constant current, constant power (ZIP) loads. Existing methods exploit system sparsity to speed computation using matrix completion techniques that decompose the large positive semidefinite matrix constraint into constraints on many smaller matrices. Computational advances include modifying the matrix decomposition techniques for further reductions in solver times, extending the applicability of an existing decomposition, and a method for obtaining an optimal voltage profile from a solution to a decomposed semidefinite program. This dissertation also provides a sufficient condition test for global optimality of a candidate OPF solution obtained by any method.

This dissertation next provides techniques for improving power system reliability. The first reliability-related research develops a sufficient condition under which the power flow equations have no solution. This sufficient condition is evaluated using a feasible semidefinite optimization problem. The objective employed in this optimization problem yields a measure of distance (in a parameter set) to the power flow solvability boundary. This distance is closely related to quantities that previous authors have proposed as voltage stability margins. A typical margin is expressed in terms of the parameters of system loading (injected powers); this dissertation additionally introduces a new margin in terms of the parameters of regulated bus voltages. This dissertation considers generators modeled as ideal voltage sources and generators with reactive power limits, which require either mixed-integer semidefinite programming or sum of squares programming formulations.

Also related to power system reliability is the problem of finding multiple solutions to the power flow equations. Power systems typically operate at a high-voltage, stable power flow solution; however, other solutions are used for stability calculations. Existing literature claims that a continuation-based algorithm can reliably find all power flow solutions. This claim is demonstrated to be incorrect with a small counterexample system. Thus, no computationally tractable algorithms exist for reliably finding all power flow solutions. Methods for calculating multiple power flow solutions therefore deserve further research. This dissertation describes a method for finding multiple power flow solutions using semidefinite programming. Results suggest the method's promise for identifying large numbers of power flow solutions.

Although the semidefinite relaxation of the power flow equations is often "tight," non-zero relaxation gap solutions can occur. This dissertation investigates examples of non-zero relaxation gap solutions to semidefinite formulations for the optimal power flow problem, for the power flow insolvability condition, and for determining multiple solutions to the power flow equations. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]

## Journal Papers

## Conference Proceedings

## Ph.D. Dissertation

A.R. Borden,

**D.K. Molzahn**, P. Ramanathan, and B.C. Lesieutre, "Confidentiality-Preserving Optimal Power Flow for Cloud Computing,"*50th Annual Allerton Conference on Communication, Control, and Computing,*pp. 1300-1307, October 1-5, 2012.

In the field of power system engineering, the optimal power flow problem is essential in planning and operations. With increasing system size and complexity, the computational requirements needed to solve practical optimal power flow problems continues to grow. Increasing computational requirements make the possibility of performing these computations remotely with cloud computing appealing. However, power system structure and component values are often confidential; therefore, the problem cannot be shared. To address this issue of confidential information in cloud computing, some techniques for masking optimization problems have been developed. The work of this paper builds upon these techniques for optimization problems but is specifically developed for addressing the DC and AC optimal power flow problems. We study the application of masking a sample OPF using the IEEE 14-bus network. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B.C. Lesieutre, C.L. DeMarco, and

**D.K. Molzahn**, "Power System Deliverability Ranking of Critical Elements to Support Select Facilities,"*Report to the Federal Energy Regulatory Commission*, January 2012.

[ not publicly available ]B.C. Lesieutre, C.L. DeMarco, and

**D.K. Molzahn**, "Power System Ranking of Nodal Locations,"*Report to the Federal Energy Regulatory Commission*, January 2012.

[ not publicly available ]

## Conference Proceedings

## Technical Reports

**D.K. Molzahn**and C. Singletary, "An Empirical Investigation of Speculation in the MISO Financial Transmission Rights Auction Market,"*The Electricity Journal*, vol. 24, issue 5, pp. 57-68, June 2011.

Although financial transmissions rights (FTRs) are intended to allow physical participants in electricity markets to hedge the locational price risk associated with their bilateral contracts, both physical participants and speculators take part in FTR auctions. While speculators likely increase the liquidity of FTR auction markets, physical participants and ultimately ratepayers bear the cost of the profits made by speculators. Given the regulated nature of physical participants in contrast to the largely unregulated speculators, it is hypothesized that speculators in FTR auction markets are obtaining substantial profits at the expense of physical participants. This study details a statistical model for characterizing market participants in the Midwest Independent System Operator (MISO) FTR auctions as either speculators or physical participants. Estimates of the total profit made by speculators and measures of the liquidity provided by speculators in FTR auctions are then determined. A sensitivity analysis is used to provide reasonable bounds for these estimates. The best estimates determined in this analysis show speculator profits of 152% of total market profits, equal to $283 million over three years, with speculator transactions accounting for 74% of total market transactions on a megawatt basis. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B.C. Lesieutre,

**D.K. Molzahn**, A.R. Borden, and C.L. DeMarco, "Examining the Limits of the Application of Semidefinite Programming to Power Flow Problems,"*49th Annual Allerton Conference on Communication, Control, and Computing*, pp. 1492-1499, September 28-30, 2011.

The application of semidefinite programming (SDP) to power system problems has recently attracted substantial research interest. Specifically, a recent SDP formulation offers a convex relaxation to the well-known, typically nonconvex "optimal power flow" (OPF) problem. This new formulation was demonstrated to yield zero duality gap for several standard power systems test cases, thereby ensuring a globally optimal OPF solution in each. The first goal of the work here is to investigate this SDP algorithm for the OPF, and show by example that it can fail to give a physically meaningful solution (i.e., it has a nonzero duality gap) in some scenarios of practical interest. The remainder of this paper investigates a SDP approach utilizing modified objective and constraints to compute all solutions of the nonlinear power flow equations. Several variants are described. Results suggest SDP's promise as an efficient algorithm for identifying large numbers of solutions to the power flow equations. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]D.R. Schwarting,

**D.K. Molzahn**, C.L. DeMarco, and B.C. Lesieutre. "Topological and Impedance Element Ranking (TIER) of the Bulk-Power System,"*44th Hawaii International Conference on System Sciences (HICSS)*, pp. 1-10, January 4-7, 2011.

The Electricity Modernization Act of the Energy Policy Act of 2005 requires the creation of enforceable reliability standards for users, owners, and operators of the “Bulk-Power System” (BPS). This paper introduces an algorithm based on electrical properties of an electric power network that yields a numeric ranking of its branch elements, as a step towards identifying which elements should be considered part of the BPS. This ranking depends solely on network topology and element electrical characteristics, and is wholly independent of generator cost functions and system operating point. Two perspectives for deriving and justifying this ranking algorithm are described: one based on the sensitivity of bus-to-branch Lagrange multipliers and another based on generation shift factors. Summary results obtained from applying the algorithm to a detailed model of the US Eastern Interconnection are reported (with suitable anonymity for any facility specific data, in keeping with Critical Energy Infrastructure Information (CEII) protections). [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]

## Journal Papers

## Conference Proceedings

**D.K. Molzahn**and B.C. Lesieutre, "An Eigenvalue Formulation for Determining Initial Conditions of Induction Machines in Dynamic Power System Simulations,"*IEEE International Symposium on Circuits and Systems (ISCAS)*, pp. 2311-2313, May 30 - June 2, 2010.

The problem of determining the initial conditions for induction machine variables in a non-linear dynamic power system analysis is posed as an eigenvalue problem. The eigenvalue formulation can then be solved using standard linear algebra techniques. This method has the advantage of determining all possible solutions for the internal variables rather than the single solution obtained from traditional iterative methods. Additionally, the method can easily determine when the problem has no solutions through the absence of non-zero real eigenvalues. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, "Power System Models Formulated as Eigenvalue Problems and Properties of Their Solutions,"*Master's Thesis, University of Wisconsin-Madison ECE Department*, December 2010.

Eigenvalue problems incorporate multiplicative nonlinearities in otherwise linear equations. Two power systems problems with multiplicative nonlinearities are reformulated as eigenvalue problems. Reformulation allows for the application of eigenvalue theory and solution techniques to these problems.

The first problem involves determination of the initial conditions for induction machine internal variables in a non-linear dynamic power system analysis. The internal variables of stator and rotor currents, rotor speed, and mechanical torque must be determined from the given values of input real power, stator voltage magnitude, and stator voltage angle. This problem is posed in an eigenvalue formulation that can be solved using standard linear algebra techniques. The eigenvalue formulation allows for determination of all possible solutions for the internal variables rather than the single solution obtained from traditional iterative methods. Additionally, the absence of non-zero real eigenvalues indicates that the problem has no solution. Both single-cage and double-cage induction machine models are considered, and numeric examples are presented.

The second problem involves reformulation of the power flow equations as a multiparameter eigenvalue problem. In this formulation, both the eigenvalues and the eigenvectors are composed of the d and q orthogonal components of the bus voltages. The two parameter formulation of the power flow equations for two bus systems can be solved directly by decomposing the problem into two generalized eigenvalue problems that must be simultaneously satisfied. Since n bus systems require 2(n-1) parameter eigenvalue problems, which do not yet have a general solution method for n > 2, systems with more than two buses are not yet directly solvable from the multiparameter eigenvalue formulation.

In addition to direct solution, two other research avenues for the multiparameter eigenvalue formulation of the power flow equations are pursued. First, motivated by eigenvalue problem structure, a reformulation of the multiparameter eigenvalue form of the power flow equations is presented. Although it is without known practical applications, this reformulation and the intermediate results in its derivation are interesting from a theoretical standpoint. Second, an eigenvalue sensitivity analysis is performed on the multiparameter eigenvalue formulation of the power flow equations. The linearization obtained from the eigenvalue sensitivity analysis is equivalent to the linearization obtained from the power flow Jacobian.

Future developments in multiparameter eigenvalue theory may provide additional insights into solutions of the power flow equations. General solution techniques for multiparameter eigenvalue problems with more than two parameters may enable direct solution of the power flow equations. A method for determining the number of solutions to multiparameter eigenvalue problems would be useful as a stopping condition for continuation power flows. Conditions for the existence of any solutions to multiparameter eigenvalue problems would be useful for finding the point of voltage collapse and for analyzing power systems in heavily loaded conditions. [ abstract ][ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, S.P. Williams, and R. Srinivasan, "Plug-In Vehicles in Madison, WI: Consumer Preferences, Charging Stations, and Distribution System Impacts,"*Energy Analysis and Policy Capstone Project Report to Madison Gas and Electric*, May 2010.

[ not publicly available ]

## Conference Proceedings

## Master's Thesis

## Technical Reports

**D.K. Molzahn**and I.A. Hiskens, "A Survey of Relaxations and Approximations of the Power Flow Equations,"*Foundations and Trends in Electric Energy Systems*, vol. 4, no. 1-2, pp. 1-221, February 2019.

The power flow equations relate the power injections and voltages in an electric power system and are therefore key to many power system optimization and control problems. Research efforts have developed a wide variety of relaxations and approximations of the power flow equations with a range of capabilities and characteristics. This monograph surveys relaxations and approximations of the power flow equations, with a particular emphasis on recently proposed formulations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ Tutorial Slides on Power Flow Relaxations ]

## 2019

M. Pollack, R. Piansky, S. Gupta, A. Kody, and

**D.K. Molzahn**, "Equitably Allocating Wildfire Resilience Investments for Power Grids – The Curse of Aggregation and Vulnerability Indices," submitted.

Wildfires ignited by power systems infrastructure are among the most destructive wildfires; hence some utility companies in wildfire-prone regions have pursued a proactive policy of emergency power shutoffs. These shutoffs, while mitigating the risk of disastrous ignition events, result in power outages that could negatively impacts vulnerable communities. In this paper, we consider how to equitably allocate funds to underground and effectively de-risk power lines in transmission networks. We explore the impact of the 2021 White House resource allocation policy called the Justice40 initiative, which states that 40% of the benefits of federally-funded climate-related investments should go to socially vulnerable communities. The definition of what constitutes a vulnerable community varies by organization, and we consider two major recently proposed vulnerability indices: the Justice40 index created under the 2021 White House and the Social Vulnerability Index (SVI) developed by the Center for Disease Control and Prevention (CDC). We show that allocating budget according to these two indices fails to reduce power outages for indigenous communities and those subject to high wildfire ignition risk using a high-fidelity synthetic power grid dataset that matches the key features of the Texas transmission system. We discuss how aggregation of communities and "one size fits all" vulnerability indices might be the reasons for the misalignment between the goals of vulnerability indices and their realized impact in this particular case study. We provide a method of achieving an equitable investment plan by adding group-level protections on percentage of load that is shed across each population group of interest. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri, R. Gupta and

**D.K. Molzahn**, "Optimized LinDistFlow for High-Fidelity Power Flow Modeling of Distribution Networks," submitted.

The DistFlow model accurately represents power flows in distribution systems, but the model's nonlinearities result in computational challenges for many optimization applications. Accordingly, a linear approximation known as LinDistFlow is commonly employed. This paper introduces an algorithm for enhancing the accuracy of the LinDistFlow approximation, with the goal of aligning the outputs more closely with those from the nonlinear DistFlow model. Using sensitivity information, our algorithm optimizes the LinDistFlow approximation's coefficient and bias parameters to minimize discrepancies in predictions of voltage magnitudes relative to the nonlinear DistFlow model. The algorithm employs the Truncated Newton Conjugate-Gradient (TNC) optimization method to fine-tune coefficients and bias parameters during an offline training phase in order to improve the LinDistFlow approximation's accuracy in optimization applications. Numerical results underscore the algorithm's efficacy, showcasing accuracy improvements in L1-norm and L∞-norm losses of up to 92% and 88%, respectively, relative to the traditional LinDistFlow model. We assess how the optimized parameters perform under changes in the network topology and also validate the optimized LinDistFlow approximation's efficacy in a hosting capacity optimization problem. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Gupta and

**D.K. Molzahn**, "Improving Fairness in Photovoltaic Curtailments via Daily Topology Reconfiguration for Voltage Control in Power Distribution Networks," submitted.

In PV-rich power distribution systems, over-voltage issues are often addressed by curtailing excess generation from PV plants (in addition to reactive power control), raising fairness concerns. Existing fairness-aware control schemes tackle this problem by incorporating fairness objectives into the cost function. However, such schemes result in increased overall curtailments. This paper proposes a solution through daily topology reconfiguration, ensuring that different PV plants face varying grid conditions each day, leading to different curtailment levels and enhancing fairness. We illustrate that implementing this approach enhances overall fairness without significantly increasing overall curtailments. The optimization problem involves two stages. The day-ahead stage optimizes the network topology using day-ahead forecasts of PV generation and demand, minimizing net curtailment and accounting for fairness based on curtailments from prior days. The real-time stage implements the optimized topology and computes active and reactive power setpoints for the PV plants. Day-ahead grid constraints are modeled using LinDistFlow, and real-time control employs a linearized model with a first-order Taylor approximation. The proposed scheme is numerically validated on several benchmark test cases. Results are compared using the Jain Fairness Index, considering fairness and reconfiguration scenarios. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah and

**D.K. Molzahn**, "A Fault-Tolerant Distributed Termination Method for Distributed Optimization Algorithms," submitted.

This paper proposes a fully distributed termination method for distributed optimization algorithms solved by multiple agents. The proposed method guarantees terminating a distributed optimization algorithm after satisfying the global termination criterion using information from local computations and neighboring agents. The proposed method requires additional iterations after satisfying the global terminating criterion to communicate the termination status. The number of additional iterations is bounded by the diameter of the communication network. This paper also proposes a fault-tolerant extension of this termination method that prevents early termination due to faulty agents or communication errors. We provide a proof of the method's correctness and demonstrate the proposed method by solving the optimal power flow problem for electric power grids using the alternating direction method of multipliers. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, K.R. Davis, and M.L. Crow, "Tightening QC Relaxations of AC Optimal Power Flow through Improved Linear Convex Envelopes," submitted.

AC optimal power flow (AC OPF) is a fundamental problem in power system operations. Accurately modeling the network physics via the AC power flow equations makes AC OPF a challenging nonconvex problem. To search for global optima, recent research has developed various convex relaxations that bound the optimal objective values of AC OPF problems. The QC relaxation convexifies the AC OPF problem by enclosing the non-convex terms within convex envelopes. The QC relaxation's accuracy strongly depends on the tightness of these envelopes. This paper proposes two improvements for tightening QC relaxations of OPF problems. We first consider a particular nonlinear function whose projections are the nonlinear expressions appearing in the polar representation of the power flow equations. We construct a polytope-shaped convex envelope around this nonlinear function and derive convex expressions for the nonlinear terms using its projections. Second, we use sine and cosine expression properties, along with changes in their curvature, to tightness this convex envelope. We also propose a coordinate transformation to tighten the envelope by rotating power flow equations based on individual bus-specific angles. We compare these enhancements to a state-of-the-art QC relaxation method using PGLib-OPF test cases, revealing improved optimality gaps in 68% of the cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, R. Harris, S. Litchfield, D. Huggins, and

**D.K. Molzahn**, "Detecting Shared Data Manipulation in Distributed Optimization Algorithms," submitted.

This paper investigates the vulnerability of the Alternating Direction Method of Multipliers (ADMM) algorithm to shared data manipulation, with a focus on solving optimal power flow (OPF) problems. Deliberate data manipulation may cause the ADMM algorithm to converge to suboptimal solutions. We derive two sufficient conditions for detecting data manipulation based on the theoretical convergence trajectory of the ADMM algorithm. We evaluate the detection conditions' performance on three data manipulation strategies we previously proposed: simple, feedback, and bilevel optimization attacks. We then extend these three data manipulation strategies to avoid detection by considering both the detection conditions and a neural network (NN) detection model in the attacker's decision process. We also propose an adversarial NN training framework to detect shared data manipulation. We illustrate the performance of our data manipulation strategy and detection framework on OPF problems. The results show that the proposed detection conditions successfully detect most of the data manipulation attacks. However, a bilevel optimization attack strategy that incorporates the detection methods may avoid being detected. Countering this, our proposed adversarial training framework detects all the instances of the bilevel optimization attack. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Harris, M. Alkhraijah, and

**D.K. Molzahn**, "Optimally Managing the Impacts of Convergence Tolerance for Distributed Optimal Power Flow," submitted.

The future power grid may rely on distributed optimization to determine the set-points for huge numbers of distributed energy resources. There has been significant work on applying distributed algorithms to optimal power flow (OPF) problems, which require separate computing agents to agree on shared boundary variable values. Looser tolerances for the mismatches in these shared variables generally yield faster convergence at the expense of exacerbating constraint violations, but there is little quantitative understanding of how the convergence tolerance affects solution quality. To address this gap, we first quantify how convergence tolerance impacts constraint violations when the distributed OPF generator dispatch is applied to the power system. Using insights from this analysis, we then develop a bound tightening algorithm which guarantees that operating points from distributed OPF algorithms will not result in violations despite the possibility of shared variable mismatches within the convergence tolerance. We also explore how bounding the cumulative shared variable mismatches can prevent unnecessary conservativeness in the bound tightening. The proposed approach enables control of the trade-off between computational speed, which improves as the convergence tolerance increases, and distributed OPF solution cost, which increases with convergence tolerance due to tightened constraints, while ensuring feasibility. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason, S. Misra, J.P. Watson, and

**D.K. Molzahn**, "Adaptive Power Flow Approximations with Second-Order Sensitivity Insights," submitted.

The power flow equations are fundamental to power system planning, analysis, and control. However, their inherent non-linearity and non-convexity can pose significant challenges when solving problems that involve these equations. To address these challenges, linear approximations are frequently used to simplify computations at the potential cost of compromising accuracy and feasibility. Typical power flow approximations rely on general assumptions that are not always valid. Additionally, many existing linear approximations, such as the first-order Taylor series, overlook important information contained within higher-order terms. To gain a deeper understanding of the relationships among variables within a specific operational range and to achieve more accurate approximations, this paper considers a second-order sensitivity analysis of the power flow equations. By modeling the curvature of the power flow manifold, this paper uses the second-order sensitivities in an importance sampling method to improve the accuracy of adaptive power flow linearizations. Furthermore, this paper explores approximations formulated as rational functions with linear numerators and denominators that can be rewritten as linear constraints in power system optimization problems. These approximations, inspired by the Pad\'e approximant, utilize second-order power flow sensitivities. This approach is extended to enhance accuracy beyond what linear functions can achieve across various operational scenarios, and can also be constructed to be conservative (over- or under-estimate a quantity of interest). [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]C. Hettle, S. Gupta, and

**D.K. Molzahn**, "Fair and Reliable Reconnections for Temporary Disruptions in Electric Distribution Networks using Submodularity," submitted.

The economic cost of power outages has been estimated to be between $35 billion and $50 billion annually in the United States. We analyze and advance methodologies for electrical distribution network reconfiguration in order to quickly restore power, an important lever for improving reliability. For short-term network disruptions, we give a novel model for reconfiguration by considering switches that can be sequentially closed upon detection of a network anomaly upstream. The order in which switches close has an impact on reliability metrics such as SAIDI, which are the basis for regulator-imposed financial performance incentives. We introduce the "Minimum Reconnection Time (MRT)" problem, which aims to find an optimal switch ordering that minimizes the cost of outages and show that it generalizes metrics like SAIDI. We first show that MRT is a special case of the well-known minimum linear ordering problem from submodular optimization literature and that MRT is also NP-hard. We show how to approximate MRT using recent kernel-based randomized rounding approaches, and in turn improve the state-of-the-art for a broad class of MLOP instances. Finally, we note that the choice of reliability metric can result in significant differences for low and medium voltage buses. We therefore consider optimizing multiple metrics simultaneously using local search methods that reconfigure the system's base tree to optimize service disruptions, reconnection times, and energy losses. We computationally validate our reconfiguration methods on the NREL SMART-DS Greensboro synthetic urban-suburban network and show significant improvement in reliability metrics. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Jalilian, B. Taheri, and

**D.K. Molzahn**, "Co-Optimization of Damage Assessment and Restoration: A Resilience-Driven Dynamic Crew Allocation for Power Distribution Systems," to appear in*IEEE Transactions on Power Systems*.

This study introduces a mixed-integer linear programming (MILP) model, effectively co-optimizing patrolling, damage assessment, fault isolation, repair, and load re-energization processes. The model is designed to solve a vital operational conundrum: deciding between further network exploration to obtain more comprehensive data or addressing the repair of already identified faults. As information on the fault location and repair timelines becomes available, the model allows for dynamic adaptation of crew dispatch decisions. In addition, this study proposes a conservative power flow constraint set that considers two network loading scenarios within the final network configuration. This approach results in the determination of an upper and a lower bound for node voltage levels and an upper bound for power line flows. To underscore the practicality and scalability of the proposed model, we have demonstrated its application using IEEE 123-node and 8500-node test systems, where it delivered promising results. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason, S. Misra, and

**D.K. Molzahn**, "A Data-Driven Sensor Placement Approach for Detecting Voltage Violations in Distribution Systems,"*Electric Power Systems Research*, vol. 232, no. 110387, July 2024.

Stochastic fluctuations in power injections from distributed energy resources (DERs) combined with load variability can cause constraint violations (e.g., exceeded voltage limits) in electric distribution systems. To monitor grid operations, sensors are placed to measure important quantities such as the voltage magnitudes. In this paper, we consider a sensor placement problem which seeks to identify locations for installing sensors that can capture all possible violations of voltage magnitude limits. We formulate a bilevel optimization problem that minimizes the number of sensors and avoids false sensor alarms in the upper level while ensuring detection of any voltage violations in the lower level. This problem is challenging due to the nonlinearity of the power flow equations and the presence of binary variables. Accordingly, we employ recently developed conservative linear approximations of the power flow equations that overestimate or underestimate the voltage magnitudes. By replacing the nonlinear power flow equations with conservative linear approximations, we can ensure that the resulting sensor locations and thresholds are sufficient to identify any constraint violations. Additionally, we apply various problem reformulations to significantly improve computational tractability while simultaneously ensuring an appropriate placement of sensors. Lastly, we improve the quality of the results via an approximate gradient descent method that adjusts the sensor thresholds. We demonstrate the effectiveness of our proposed method for several test cases, including a system with multiple switching configurations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Piansky, G. Stinchfield, A. Kody,

**D.K. Molzahn**, and J.P. Watson, "Long Duration Battery Sizing, Siting, and Operation Under Wildfire Risk Using Progressive Hedging," to appear in*Electric Power Systems Research*, to be presented at*23rd Power Systems Computation Conference (PSCC)*, June 4-7, 2024.

Battery sizing and siting problems are computationally challenging due to the need to make long-term planning decisions that are cognizant of short-term operational decisions. This paper considers sizing, siting, and operating batteries in a power grid to maximize their benefits, including price arbitrage and load shed mitigation, during both normal operations and periods with high wildfire ignition risk. We formulate a multi-scenario optimization problem for long duration battery storage while considering the possibility of load shedding during Public Safety Power Shutoff (PSPS) events that de-energize lines to mitigate severe wildfire ignition risk. To enable a computationally scalable solution of this problem with many scenarios of wildfire risk and power injection variability, we develop a customized temporal decomposition method based on a progressive hedging framework. Extending traditional progressive hedging techniques, we consider coupling in both placement variables across all scenarios and state-of-charge variables at temporal boundaries. This enforces consistency across scenarios while enabling parallel computations despite both spatial and temporal coupling. The proposed decomposition facilitates efficient and scalable modeling of a full year of hourly operational decisions to inform the sizing and siting of batteries. With this decomposition, we model a year of hourly operational decisions to inform optimal battery placement for a 240-bus WECC model in under 70 minutes of wall-clock time. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri and

**D.K. Molzahn**, "Optimizing Parameters of the DC Power Flow," to appear in*Electric Power Systems Research*, to be presented at*23rd Power Systems Computation Conference (PSCC)*, June 4-7, 2024.

Many power system operation and planning problems use the DC power flow approximation to address computational challenges from the nonlinearity of the AC power flow equations. The DC power flow simplifies the AC power flow equations to a linear form that relates active power flows to phase angle differences across branches, parameterized by coefficients based on the branches' susceptances. Inspired by techniques for training machine learning models, this paper proposes an algorithm that seeks optimal coefficient and bias parameters to improve the DC power flow approximation's accuracy. Specifically, the proposed algorithm selects the coefficient and bias parameter values that minimize the discrepancy, across a specified set of operational scenarios, between the power flows given by the DC approximation and the power flows from the AC equations. Gradient-based optimization methods like Broyden-Fletcher-Goldfarb-Shanno (BFGS), Limited-Memory BFGS (L-BFGS), and Truncated Newton Conjugate-Gradient (TNC) enable solution of the proposed algorithm for large systems. After an off-line training phase, the optimized parameters are used to improve the accuracy of the DC power flow during on-line computations. Numerical results show several orders of magnitude improvements in accuracy relative to a hot-start DC power flow approximation across a range of test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri and

**D.K. Molzahn**, "AC Power Flow Feasibility Restoration via a State Estimation-Based Post-Processing Algorithm," to appear in*Electric Power Systems Research*, to be presented at*23rd Power Systems Computation Conference (PSCC)*, June 4-7, 2024.

This paper presents an algorithm for restoring AC power flow feasibility from solutions to simplified optimal power flow (OPF) problems, including convex relaxations, power flow approximations, and machine learning (ML) models. The proposed algorithm employs a state estimation-based post-processing technique in which voltage phasors, power injections, and line flows from solutions to relaxed, approximated, or ML-based OPF problems are treated similarly to noisy measurements in a state estimation algorithm. The algorithm leverages information from various quantities to obtain feasible voltage phasors and power injections that satisfy the AC power flow equations. Weight and bias parameters are computed offline using an adaptive stochastic gradient descent method. By automatically learning the trustworthiness of various outputs from simplified OPF problems, these parameters inform the online computations of the state estimation-based algorithm to both recover feasible solutions and characterize the performance of power flow approximations, relaxations, and ML models. Furthermore, the proposed algorithm can simultaneously utilize combined solutions from different relaxations, approximations, and ML models to enhance performance. Case studies demonstrate the effectiveness and scalability of the proposed algorithm, with solutions that are both AC power flow feasible and much closer to the true AC OPF solutions than alternative methods, often by several orders of magnitude in the squared two-norm loss function. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]G. Nilsson, A.D. Owen Aquino, S. Coogan, and

**D.K. Molzahn**, "GreenEVT: Greensboro Electric Vehicle Testbed,"*IEEE Systems Journal*, vol. 18, no. 1, pp. 600-611, March 2024.

The ongoing electrification of the transportation fleet will increase the load on the electric power grid. Since both the transportation network and the power grid already experience periods of significant stress, joint analyses of both infrastructures will most likely be necessary to ensure acceptable operation in the future. To enable such analyses, this paper presents an open-source testbed that jointly simulates high-fidelity models of both the electric distribution system and the transportation network. The testbed utilizes two open-source simulators, OpenDSS to simulate the electric distribution system and the microscopic traffic simulator SUMO to simulate the traffic dynamics. Electric vehicle charging links the electric distribution system and the transportation network models at vehicle locations determined using publicly available parcel data. Leveraging high-fidelity synthetic electric distribution system data from the SMART-DS project and transportation system data from OpenStreetMap, this testbed models the city of Greensboro, NC down to the household level. Moreover, the methodology and the supporting scripts released with the testbed allow adaption to other areas where high-fidelity geolocated OpenDSS datasets are available. After describing the components and usage of the testbed, we exemplify applications enabled by the testbed via two scenarios modeling the extreme stresses encountered during evacuations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ GitHub Repository ]M. Alkhraijah, R. Harris, C. Coffrin, and

**D.K. Molzahn**, "PowerModelsADA: A Framework for Solving Optimal Power Flow using Distributed Algorithms,"*IEEE Transactions on Power Systems (Letters)*, vol. 39, no. 1, pp. 2357-2360, January 2024.

This letter presents PowerModelsADA, an open-source framework for solving Optimal Power Flow (OPF) problems using Alternating Distributed Algorithms (ADA). PowerModelsADA provides a framework to test, verify, and benchmark both existing and new ADAs. This paper demonstrates use cases for PowerModelsADA and validates its implementation with multiple OPF formulations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ GitHub Repository ]S. Talkington, D. Turizo, S. Grijalva, J. Fernandez, and

**D.K. Molzahn**, "Conditions for Estimation of Sensitivities of Voltage Magnitudes to Complex Power Injections,"*IEEE Transactions on Power Systems*, vol. 39, no. 1, pp. 478-491, January 2024.

Voltage phase angle measurements are often unavailable from sensors in distribution networks and transmission network boundaries. Therefore, this paper addresses the conditions for estimating sensitivities of voltage magnitudes with respect to complex (active and reactive) electric power injections based on sensor measurements. These sensitivities represent submatrices of the inverse power flow Jacobian. We extend previous results to show that the sensitivities of a bus voltage magnitude with respect to active power injections are unique and different from those with respect to reactive power. The classical Newton-Raphson power flow model is used to derive a novel representation of bus voltage magnitudes as an underdetermined linear operator of the active and reactive power injections—parameterized by the bus power factors. Two conditions that ensure the existence of unique complex power injections given voltage magnitudes are established for this underdetermined linear system, thereby compressing the solution space. The first is a sufficient condition based on the bus power factors. The second is a necessary and sufficient condition based on the system eigenvalues. We use matrix completion theory to develop estimation methods for recovering sensitivity matrices with varying levels of sensor availability. Simulations verify the results and demonstrate engineering use of the proposed methods. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]O. Kuryatnikova, B. Ghaddar, and

**D.K. Molzahn**, "Two-Stage Robust Quadratic Optimization with Equalities and Its Application to Optimal Power Flow,"*SIAM Journal on Optimization*, vol. 33, no. 4, pp. 2830-2857, December 2023.

In this work, we consider two-stage quadratic optimization problems under ellipsoidal uncertainty. In the first stage, one needs to decide upon the values of a subset of optimization variables (control variables). In the second stage, the uncertainty is revealed, and the rest of the optimization variables (state variables) are set up as a solution to a known system of possibly nonlinear equations. This type of problem occurs, for instance, in optimization for dynamical systems, such as electric power systems as well as gas and water networks. We propose a convergent iterative algorithm to build a sequence of approximately robustly feasible solutions with an improving objective value. At each iteration, the algorithm optimizes over a subset of the feasible set and uses affine approximations of the second-stage equations while preserving the nonlinearity of other constraints. We implement our approach and demonstrate its performance on Matpower instances of AC optimal power flow. Although this paper focuses on quadratic problems, the approach is suitable for more general setups. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]I. Aravena,

**D.K. Molzahn**, S. Zhang, F.E. Curtis, S. Tu, A. Waechter, E. Wei, E. Wong, A. Gholami, K. Sun, X.A. Sun, S.T. Elbert, J.T. Holzer, and A. Veeramany, "Recent Developments in Security-Constrained AC Optimal Power Flow: Overview of Challenge 1 in the ARPA-E Grid Optimization Competition,"*Operations Research*, special issue on*Computational Advances in Short-Term Power System Operations*, vol. 71, no. 6, pp. 1997-2014, November-December 2023.

The optimal power flow problem is central to many tasks in the design and operation of electric power grids. This problem seeks the minimum cost operating point for an electric power grid while satisfying both engineering requirements and physical laws describing how power flows through the electric network. By additionally considering the possibility of component failures and using an accurate AC power flow model of the electric network, the security-constrained AC optimal power flow (SC-AC-OPF) problem is of paramount practical relevance. To assess recent progress in solution algorithms for SC-AC-OPF problems and spur new innovations, the U.S. Department of Energy's Advanced Research Projects Agency-Energy (ARPA-E) organized Challenge 1 of the Grid Optimization (GO) competition. This paper describes the SC-AC-OPF problem formulation used in the competition, overviews historical developments and the state of the art in SC-AC-OPF algorithms, discusses the competition, and summarizes the algorithms used by the top three teams in Challenge 1 of the GO Competition (Teams gollnlp, GO-SNIP, and GMI-GO). [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]F.E. Curtis,

**D.K. Molzahn**, S. Tu, A. Wächter, E. Wei, and E. Wong, "A Decomposition Algorithm for Large-Scale Security-Constrained AC Optimal Power Flow,"*Operations Research*, special issue on*Computational Advances in Short-Term Power System Operations*, vol. 71, no. 6, pp. 2031-2044, November-December 2023.

**Description of Team GO-SNIP's algorithm which finished second overall in the Department of Energy's Grid Optimization Competition Challenge 1.**

A decomposition algorithm for solving large-scale security-constrained AC optimal power flow problems is presented. The formulation considered is the one used in the ARPA-E Grid Optimization (GO) Competition, Challenge 1, held from November 2018 through October 2019. The techniques found to be most effective in terms of performance in the challenge are presented, including strategies for contingency selection, fast contingency evaluation, handling complementarity constraints, avoiding issues related to degeneracy, and exploiting parallelism. The results of numerical experiments are provided to demonstrate the effectiveness of the proposed techniques as compared to alternative strategies. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Turizo and

**D.K. Molzahn**, "Invertibility Conditions for the Admittance Matrices of Balanced Power Systems,"*IEEE Transactions on Power Systems*, vol. 38, no. 4, pp. 3841-3853, July 2023.

The admittance matrix encodes the network topology and electrical parameters of a power system in order to relate the current injection and voltage phasors. Since admittance matrices are central to many power engineering analyses, their characteristics are important subjects of theoretical studies. This paper focuses on the key characteristic of*invertibility*. Previous literature has presented an invertibility condition for admittance matrices. This paper first identifies and fixes a technical issue in the proof of this previously presented invertibility condition. This paper then extends this previous work by deriving new conditions that are applicable to a broader class of systems with lossless branches and transformers with off-nominal tap ratios. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]F. Milano and

**D.K. Molzahn**, "Forward to the EPSR Special Issue for PSCC 2022,"*Electric Power Systems Research*, vol. 217, no. 109148, April 2023.

[ pdf ] [ link ] [ Ref: bib txt ]L.A. Roald, D. Pozo, A. Papavasiliou,

**D.K. Molzahn**, J. Kazempour, and A. Conejo, "Power Systems Optimization under Uncertainty: A Review of Methods and Applications,"*Electric Power Systems Research*, vol. 214, Part A, no. 108725, January 2023. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

Electric power systems and the companies and customers that interact with them are experiencing increasing levels of uncertainty due to factors such as renewable energy generation, market liberalization, and climate change. This raises the important question of how to make optimal decisions under uncertainty. This paper aims to provide an overview of existing methods for modeling and optimization of problems affected by uncertainty, targeted at researchers with a familiarity with power systems and optimization. We also review some important applications of optimization under uncertainty in power systems and provide an outlook to future directions of research. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri,

**D.K. Molzahn**, and S. Grijalva, "Improving Distribution System Resilience by Undergrounding Lines and Deploying Mobile Generators,"*Electric Power Systems Research*, vol. 214, Part A, no. 108804, January 2023.

To improve the resilience of electric distribution systems, this paper proposes a stochastic multi-period mixed-integer linear programming model that determines where to underground new distribution lines and how to coordinate mobile generators in order to serve critical loads during extreme events. The proposed model represents the service restoration process using the linearized DistFlow approximation of the AC power flow equations as well as binary variables for the undergrounding decisions of the lines, the configurations of switches, and the locations of mobile generators during each time period. The model also enforces a radial configuration of the distribution network and considers the transportation times needed to deploy the mobile generators. It is shown that long-term line undergrounding decisions which are cognizant of short-term mobile generator deployments yield superior results relative to undergrounding decisions made without considering mobile generators. Using an extended version of the IEEE 123-bus test system, numerical simulations show that combining the ability to underground distribution lines with the deployment of mobile generators can significantly improve the resilience of the power supply to critical loads. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Kody, S. Chevalier, S. Chatzivasileiadis, and

**D.K. Molzahn**, "Modeling the AC Power Flow Equations with Optimally Compact Neural Networks: Application to Unit Commitment,"*Electric Power Systems Research*, vol. 212, no. 108282, November 2022. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

Nonlinear power flow constraints render a variety of power system optimization problems computationally intractable. Emerging research shows, however, that the nonlinear AC power flow equations can be successfully modeled using Neural Networks (NNs). These NNs can be exactly transformed into Mixed Integer Linear Programs (MILPs) and embedded inside challenging optimization problems, thus replacing nonlinearities that are intractable for many applications with tractable piecewise linear approximations. Such approaches, though, suffer from an explosion of the number of binary variables needed to represent the NN. Accordingly, this paper develops a technique for training an "optimally compact" NN, i.e., one that can represent the power flow equations with a sufficiently high degree of accuracy while still maintaining a tractable number of binary variables. We show that the resulting NN model is more expressive than both the DC and linearized power flow approximations when embedded inside of a challenging optimization problem (i.e., the AC unit commitment problem). [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Zeng, A. Kody, Y. Kim, K. Kim, and

**D.K. Molzahn**, "A Reinforcement Learning Approach to Parameter Selection for Distributed Optimization in Power Systems,"*Electric Power Systems Research*, vol. 212, no. 108546, November 2022. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

With the increasing penetration of distributed energy resources, distributed optimization algorithms have attracted significant attention for power systems applications due to their potential for superior scalability, privacy, and robustness to a single point-of-failure. The Alternating Direction Method of Multipliers (ADMM) is a popular distributed optimization algorithm; however, its convergence performance is highly dependent on the selection of penalty parameters, which are usually chosen heuristically. In this work, we use reinforcement learning (RL) to develop an adaptive penalty parameter selection policy for the AC optimal power flow (ACOPF) problem solved via ADMM with the goal of minimizing the number of iterations until convergence. We train our RL policy using deep Q-learning, and show that this policy can result in significantly accelerated convergence (up to a 59% reduction in the number of iterations compared to existing, curvature-informed penalty parameter selection methods). Furthermore, we show that our RL policy demonstrates promise for generalizability, performing well under unseen loading schemes as well as under unseen losses of lines and generators (up to a 50% reduction in iterations). This work thus provides a proof-of-concept for using RL for parameter selection in ADMM for power systems applications. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason, S. Misra, and

**D.K. Molzahn**, "A Sample-Based Approach for Computing Conservative Linear Power Flow Approximations,"*Electric Power Systems Research*, vol. 212, no. 108579, November 2022. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

Non-convexities induced by the non-linear power flow equations challenge solution algorithms for many power system optimization and control problems. Linear approximations are often used to address these challenges by trading off modeling accuracy for tractability. The accuracy of a power flow linearization depends on the characteristics of the power system and the operational range where the linearization is applied. However, rather than exploiting knowledge of these characteristics for a particular system, many existing power flow linearizations are based on general assumptions for broad classes of systems, thus limiting their accuracy. Moreover, since existing linearizations do not consistently overestimate or underestimate quantities of interest such as voltage magnitudes and line flows, algorithms based on these linearizations may lead to constraint violations when applied to the system. In contrast, this paper computes*conservative linear approximations*of the power flow equations, i.e., linear approximations that overestimate or underestimate a quantity of interest in order to enable tractable algorithms that avoid constraint violations. Using a sample-based approach, we compute these conservative linearizations by solving a constrained linear regression problem. We analyze and improve the conservative linear approximations via an iterative sampling approach, optimization over functions of the quantities of interest, and a sample-complexity analysis. Considering the relationships between the voltage magnitudes and the active and reactive power injections, we characterize the performance of the conservative linear approximations for a range of test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, C. Menendez, and

**D.K. Molzahn**, "Assessing the Impacts of Nonideal Communications on Distributed Optimal Power Flow Algorithms,"*Electric Power Systems Research*, vol. 212, no. 108297, November 2022. Presented at*22nd Power Systems Computation Conference (PSCC)*, June 27 - July 1, 2022.

Power system operators are increasingly looking toward distributed optimization to address various challenges facing electric power systems. To assess their capabilities in environments with nonideal communications, this paper investigates the impacts of data quality on the performance of distributed optimization algorithms. Specifically, this paper compares the performance of the Alternating Direction Method of Multipliers (ADMM), Analytical Target Cascading (ATC), and Auxiliary Problem Principle (APP) algorithms in the context of DC Optimal Power Flow (DC OPF) problems. Using several test cases, this paper characterizes the performance of these algorithms in terms of their convergence rates and solution quality under three data quality nonidealities: (1) additive Gaussian noise, (2) bad data (large error), and (3) intermittent communication failure. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]N. Patari, V. Venkataramanan, A.K. Srivastava,

**D.K. Molzahn**, N. Li, and A. Annaswamy, "Distributed Optimization in Distribution Systems: Use Cases, Limitations, and Research Needs,"*IEEE Transactions on Power Systems*, vol. 37, no. 5, pp. 3469-3481, September 2022.

Electric distribution grid operations typically rely on both centralized optimization and local non-optimal control techniques. As an alternative, distribution system operational practices can consider distributed optimization techniques that leverage communications among various neighboring agents to achieve optimal operation. With the rapidly increasing integration of distributed energy resources (DERs), distributed optimization algorithms are growing in importance due to their potential advantages in scalability, flexibility, privacy, and robustness relative to centralized optimization. Implementation of distributed optimization offers multiple challenges and also opportunities. This paper provides a comprehensive review of the recent advancements in distributed optimization for electric distribution systems and classifications using key attributes. Problem formulations and distributed optimization algorithms are provided for example use cases, including volt/var control, market clearing process, loss minimization, and conservation voltage reduction. Finally, this paper also presents future research needs for the applicability of distributed optimization algorithms in the distribution system. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]J. Liu, B. Cui,

**D.K. Molzahn**, C. Chen, and X. Lu, "Optimal Power Flow for DC Networks with Robust Feasibility and Stability Guarantees,"*IEEE Transactions on Control of Network Systems*, vol. 9, no. 2, pp. 904-916, June 2022.

With high penetrations of renewable generation and variable loads, there is significant uncertainty associated with power flows in DC networks such that stability and constraint satisfaction are important concerns. Most existing DC network optimal power flow (DN-OPF) formulations assume exact knowledge about loading conditions and do not provide stability guarantees. In contrast, this paper studies a DN-OPF formulation which considers both stability and constraint satisfaction under uncertainty. The need to account for a range of uncertainty realizations in this paper’s robust optimization formulation results in a challenging semi-infinite program (SIP). The proposed solution algorithm reformulates this SIP into a computationally amenable problem whose solution provides a feasible operating point for the SIP. This algorithm constructs a convex stability set using sufficient conditions for the existence of a feasible and stable power flow solution. Solving an optimization problem based on this convex stability set provides generator set points which ensure that the DC network has a stable operating point for any considered uncertainty realization. This optimization problem takes a form similar to a deterministic DN-OPF problem and is therefore tractable. The proposed algorithm is demonstrated using various DC networks adapted from IEEE test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Lee, K. Turitsyn,

**D.K. Molzahn**, and L.A. Roald, "Robust AC Optimal Power Flow with Robust Convex Restriction,"*IEEE Transactions on Power Systems*, vol. 36, no. 6, pp. 4953-4966, November 2021.

Electric power grids regularly experience uncertain fluctuations from load demands and renewables, which poses a risk of violating operational limits designed to safeguard the system. In this paper, we consider the robust AC OPF problem that minimizes the generation cost while requiring a certain level of system security in the presence of uncertainty. The robust AC OPF problem requires that the system satisfy operational limits for all uncertainty realizations within a specified uncertainty set. Guaranteeing robustness is particularly challenging due to the non-convex, nonlinear AC power flow equations, which may not always have a solution. In this work, we extend a previously developed convex restriction to a*robust convex restriction*, which is a convex inner approximation of the non-convex feasible region of the AC OPF problem that accounts for uncertainty in the power injections. We then use the robust convex restriction in an algorithm that obtains robust solutions to AC OPF problems by solving a sequence of convex optimization problems. We demonstrate our algorithm and its ability to control robustness versus operating cost trade-offs using PGLib test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, M. Alowaifeer, M. Alsaleh, A. Alfaris, and

**D.K. Molzahn**, "The Effects of Social Distancing on the Temperature-Demand Relationship,"*Energies*, vol. 14, no. 2, 2021.

To mitigate the spread of the Novel Coronavirus (COVID-19), governments around the world have imposed social distancing policies ranging from minor social activity suspensions to full curfews. These social distancing policies have altered electricity consumption behaviors in numerous countries. This paper studies how strict social distancing policies affect the relationship between electricity demand and ambient temperature. This relationship is especially important since most governments imposed strict social distancing policies during a temperature transition season where the impacts of temperature variations are particularly important for grid operations. In this paper, we first review the expected short- and long-term impacts of social distancing on the electricity demand. We then present a case study on the electricity demand of the Kingdom of Saudi Arabia during strict social distancing policies. The results of this case study suggest that strict social distancing policies result in a stronger dependency between temperature and electricity demand. Additionally, we observe a reduction in the time required for the electricity demand to respond to temperature changes. Power system regulators can use the results in this paper to better design energy policies and cope with rare events. Additionally, power system operators can use the results in this paper to more accurately forecast electricity demands to avoid inefficient and insecure operation of the electric grid. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, and M.L. Crow, "Tightening QC Relaxations of AC Optimal Power Flow Problems via Complex Per Unit Normalization,"*IEEE Transactions on Power Systems*, vol. 36, no. 1, pp. 281-291, January 2021.

Optimal power flow (OPF) is a key problem in power system operations. OPF problems that use the nonlinear AC power flow equations to accurately model the network physics have inherent challenges associated with non-convexity. To address these challenges, recent research has applied various convex relaxation approaches to OPF problems. The QC relaxation is a promising approach that convexifies the trigonometric and product terms in the OPF problem by enclosing these terms in convex envelopes. The accuracy of the QC relaxation strongly depends on the tightness of these envelopes. This paper presents two improvements to these envelopes. The first improvement leverages a polar representation of the branch admittances in addition to the rectangular representation used previously. The second improvement is based on a coordinate transformation via a complex per unit base power normalization that rotates the power flow equations. The trigonometric envelopes resulting from this rotation can be tighter than the corresponding envelopes in previous QC relaxation formulations. Using an empirical analysis with a variety of test cases, this paper suggests an appropriate value for the angle of the complex base power. Comparing the results with a state-of-the-art QC formulation reveals the advantages of the proposed improvements. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Peña Ordieres,

**D.K. Molzahn**, L.A. Roald, and A. Wächter "DC Optimal Power Flow with Joint Chance Constraints,"*IEEE Transactions on Power Systems*, vol. 36, no. 1, pp. 147-158, January 2021.

Managing uncertainty and variability in power injections has become a major concern for power system operators due to increasing levels of fluctuating renewable energy connected to the grid. This work addresses this uncertainty via a joint chance-constrained formulation of the DC optimal power flow (OPF) problem, which satisfies*all*the constraints*jointly*with a pre-determined probability. The few existing approaches for solving joint chance-constrained OPF problems are typically either computationally intractable for large-scale problems or give overly conservative solutions that satisfy the constraints far more often than required, resulting in excessively costly operation. This paper proposes an algorithm for solving joint chance-constrained DC OPF problems by adopting an Sℓ_{1}QP-type trust-region algorithm. This algorithm uses a sample-based approach that avoids making strong assumptions on the distribution of the uncertainties, scales favorably to large problems, and can be tuned to obtain less conservative results. We illustrate the performance of our method using several IEEE test cases. The results demonstrate the proposed algorithm's advantages in computational times and limited conservativeness of the solutions relative to other joint chance-constrained DC OPF algorithms. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and C. Rehtanz, "Forward to the EPSR Special Issue for PSCC 2020,"*Electric Power Systems Research*, vol. 190, no. 106827, January 2021.

[ pdf ] [ link ] [ Ref: bib txt ]A. Venzke,

**D.K. Molzahn**, and S. Chatzivasileiadis, "Efficient Creation of Datasets for Data-Driven Power System Applications,"*Electric Power Systems Research*, vol. 190, January 2021. Presented at*21st Power Systems Computation Conference (PSCC)*, June 29 - July 3, 2020.

Advances in data-driven methods have sparked renewed interest for applications in power systems. Creating datasets for successful application of these methods has proven to be very challenging, especially when considering power system security. This paper proposes a computationally efficient method to create datasets of secure and insecure operating points. We propose an infeasibility certificate based on separating hyperplanes that can a-priori characterize large parts of the input space as insecure, thus significantly reducing both computation time and problem size. Our method can handle an order of magnitude more control variables and creates balanced datasets of secure and insecure operating points, which is essential for data-driven applications. While we focus on N-1 security and uncertainty, our method can extend to dynamic security. For PGLib-OPF networks up to 500 buses and up to 125 control variables, we demonstrate drastic reductions in unclassified input space volumes and computation time, create balanced datasets, and evaluate an illustrative data-driven application. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ presentation ]B. Li, B. Cui, F. Qiu, and

**D.K. Molzahn**, "Balancibility: Existence and Uniqueness of Power Flow Solutions under Voltage Balance Requirements,"*Electric Power Systems Research*, vol. 190, January 2021. Presented at*21st Power Systems Computation Conference (PSCC)*, June 29 - July 3, 2020.

**PSCC 2020 Highlights Award, top 3% of papers at the 21st Power Systems Computation Conference.**

In distribution systems, power injection variability due to growing penetrations of distributed energy resources (DERs) and dispatchable loads can lead to power quality issues such as severe voltage unbalance. To ensure safe operation of phase-balance-sensitive components such as transformers and induction motor loads, the amount of voltage unbalance must be maintained within specified limits for a range of uncertain loading conditions. This paper builds on existing "solvability conditions" that characterize operating regions for which the power flow equations are guaranteed to have a unique high-voltage solution. We extend these existing solvability conditions to be applicable to distribution systems and augment them with a “balancibility” condition which quantifies an operating region within which a unique, adequately balanced power flow solution exists. To build this condition, we consider different unbalance definitions and derive closed-form representations through reformulations or safe approximations. Using case studies, we evaluate these closed-form representations and compare the balancibility conditions associated with different unbalance definitions. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ presentation ]A. Venzke, S. Chatzivasileiadis, and

**D.K. Molzahn**, "Inexact Convex Relaxations of AC Optimal Power Flow Problems: Towards AC Feasibility,"*Electric Power Systems Research*, vol. 187, October 2020.

Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted significant interest as in several instances they provably yield the global optimum to the original non-convex problem. If, however, the relaxation is inexact, the obtained solution is not AC-feasible. The quality of the obtained solution is essential for several practical applications of AC-OPF, but detailed analyses are lacking in existing literature. This paper aims to cover this gap. We provide an in-depth investigation of the solution characteristics when convex relaxations are inexact, we assess the most promising AC feasibility recovery methods for large-scale systems, and we propose two new metrics that lead to a better understanding of the quality of the identified solutions. We perform a comprehensive assessment on 96 different test cases, ranging from 14 to 3120 buses, and we show the following: (i) Despite an optimality gap of less than 1%, several test cases still exhibit substantial distances to both AC feasibility and local optimality and the newly proposed metrics characterize these deviations. (ii) Penalization methods fail to recover an AC-feasible solution in 15 out of 45 test cases. (iii) The computational benefits of warm-starting non-convex solvers have significant variation, but a computational speedup exists in over 75% of the test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Lee, K. Turitsyn,

**D.K. Molzahn**, and L.A. Roald, "Feasible Path Identification in Optimal Power Flow with Sequential Convex Restriction,"*IEEE Transactions on Power Systems*, vol. 35, no. 5, pp. 3648-3659, September 2020.

Nonconvexity induced by the nonlinear AC power flow equations challenges solution algorithms for AC optimal power flow (OPF) problems. While significant research efforts have focused on reliably computing high-quality OPF solutions, it is not always clear that there exists a feasible path to reach the desired operating point. Transitioning between operating points while avoiding constraint violations can be challenging since the feasible space of the OPF problem is nonconvex and potentially disconnected. To address this problem, we propose an algorithm that computes a provably feasible path from an initial operating point to a desired operating point. Given an initial feasible point, the algorithm solves a sequence of convex quadratically constrained optimization problems over conservative convex inner approximations of the OPF feasible space. In each iteration, we obtain a new, improved operating point and a feasible transition from the operating point in the previous iteration. In addition to computing a feasible path to a known desired operating point, this algorithm can also be used to improve the operating point locally. Extensive numerical studies on a variety of test cases demonstrate the algorithm and the ability to arrive at a high-quality solution in few iterations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Barzegar,

**D.K. Molzahn**, and R. Su, "A Method for Quickly Bounding the Optimal Objective Value of an OPF Problem Using a Semidefinite Relaxation and a Local Solution,"*Electric Power Systems Research*, vol. 177, December 2019.

Optimal power flow (OPF) is an important problem in the operation of electric power systems. Due to the OPF problem's non-convexity, there may exist multiple local optima. Certifiably obtaining the global solution is important for certain applications of OPF problems. Many global optimization techniques compute an optimality gap that compares the achievable objective value corresponding to the feasible point from a local solution algorithm with the objective value bound from a convex relaxation technique. Rather than the traditional practice of completely separating the local solution and convex relaxation computations, this paper proposes a method that exploits information from a local solution to speed the computation of an objective value bound using a semidefinite programming (SDP) relaxation. The improvement in computational tractability comes with the trade-off of reduced tightness for the resulting objective value bound. Numerical experiments illustrate this trade-off, with the proposed method being faster but weaker than the SDP relaxation and slower but tighter than second-order cone programming (SOCP) and quadratic convex (QC) relaxations for many large test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Yao,

**D.K. Molzahn**, and J.L. Mathieu, "A Multiperiod Optimal Power Flow Approach to Improve Power System Voltage Stability Using Demand Response,"*IEEE Transactions on Control of Network Systems, Special Issue on Analysis, Control, and Optimization of Energy System Networks*, vol. 6, no. 3, pp. 1015-1025, September 2019.

The increasing penetration of renewables has driven power systems to operate closer to their stability boundaries, increasing the risk of instability. We propose a multiperiod optimal power flow approach that uses demand responsive loads to improve steady-state voltage stability, which is measured by the smallest singular value (SSV) of the power flow Jacobian matrix. In contrast to past work that employs load shedding to improve stability, our approach improves the SSV by decreasing and increasing individual loads while keeping the total loading constant to avoid fluctuation of the system frequency. Additionally, an energy payback period maintains the total energy consumption of each load at its nominal value. The objective function balances SSV improvements against generation costs in the energy payback period. We develop an iterative linear programming algorithm using singular value sensitivities to obtain an AC-feasible solution. We demonstrate its performance on two IEEE test systems. The results show that demand response actions can improve static voltage stability, in some cases more cost-effectively than generation actions. We also compare our algorithm's performance to that of an iterative nonlinear programming algorithm from the literature. We find that our approach is approximately 6 times faster when applied to the IEEE 9-bus system, allowing us to demonstrate its performance on the IEEE 118-bus system. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and J. Wang, "Detection and Characterization of Intrusions to Network Parameter Data in Electric Power System Operations,"*IEEE Transactions on Smart Grid,*vol. 10, no. 4, pp. 3919-3928, July 2019.

Combating cyberattacks is an emerging challenge in maintaining the reliable and economic operation of electric power systems. Possible cyberattacks include intrusions to the parameter data at a control center. In this class of attacks, algorithms at the control center are correctly executed, but the attacker's modification of the associated parameter data yields improper results. This paper proposes an algorithm for detecting and characterizing cyberattacks to network parameter data, with specific application to optimal power flow problems. The proposed algorithm evaluates whether historical operating point data are consistent with the network parameters. Inconsistencies indicating potential cyberattacks are characterized using historical operational data (power injections and voltage phasors) along with network parameter data. Simulated test cases illustrate the proposed algorithm's detection and characterization capabilities. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, "Identifying and Characterizing Non-Convexities in Feasible Spaces of Optimal Power Flow Problems,"*IEEE Transactions on Circuits and Systems II: Express Briefs*, vol. 65, no. 5, pp. 672-676, May 2018. Presented at*IEEE International Symposium on Circuits and Systems (ISCAS), special session on On-line Identification, Control, & Optimization of Electric Power Systems,*May 27-30, 2018.

Optimal power flow (OPF) is an important problem in the operation of electric power systems. The solution to an OPF problem provides a minimum cost operating point that satisfies constraints imposed by both the non-linear power flow equations and engineering limits. These constraints can yield non-convex feasible spaces that result in significant computational challenges. This paper proposes an algorithm that identifies and characterizes non-convexities in OPF feasible spaces. This algorithm searches for a pair of feasible points whose connecting line segment contains an infeasible point. Such points certify the existence of a non-convexity in the OPF feasible space. Moreover, the constraint violations at the infeasible point along the connecting line segment physically characterize a cause of the non-convexity. Numerical demonstrations include a small illustrative example as well as applications to various test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]C. Josz and

**D.K. Molzahn**, "Lasserre Hierarchy for Large Scale Polynomial Optimization in Real and Complex Variables,"*SIAM Journal on Optimization*, vol. 28, no. 2, pp. 1017-1048, 2018.

We propose general notions to deal with large scale polynomial optimization problems and demonstrate their efficiency on a key industrial problem of the twenty first century, namely the optimal power flow problem. These notions enable us to find global minimizers on instances with up to 4,500 variables and 14,500 constraints. First, we generalize the Lasserre hierarchy from real to complex to numbers in order to enhance its tractability when dealing with complex polynomial optimization. Complex numbers are typically used to represent oscillatory phenomena, which are omnipresent in physical systems. Using the notion of hyponormality in operator theory, we provide a finite convergence criterion which generalizes the Curto-Fialkow conditions of the real Lasserre hierarchy. Second, we introduce the multi-ordered Lasserre hierarchy in order to exploit sparsity in polynomial optimization problems (in real or complex variables) while preserving global convergence. It is based on two ideas: 1) to use a different relaxation order for each constraint, and 2) to iteratively seek a closest measure to the truncated moment data until a measure matches the truncated data. Third and last, we exhibit a block diagonal structure of the Lasserre hierarchy in the presence of commonly encountered symmetries. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ technical report pdf ] [ technical report link ]**D.K. Molzahn**, "Identifying Redundant Flow Limits on Parallel Lines,"*IEEE Transactions on Power Systems (Letters)*, vol. 33, no. 3, pp. 3210-3212, 2018.

Many power system optimization problems constrain line flows with limits specified in terms of the magnitudes of apparent power or current flows. The set of line-flow constraints may have redundancies (i.e., the feasible space may be unchanged upon removal of a subset of the line-flow constraints), which unnecessarily complicate optimization problems. This letter describes an algorithm for identifying redundant line-flow constraints corresponding to certain parallel lines. After formulating the constraints as ellipsoids in the voltage variables, redundancies are detected from the absence of intersections between pairs of ellipsoids corresponding to parallel lines. This algorithm is demonstrated using several large test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]O. Coss, J.D. Hauenstein, H. Hoon, and

**D.K. Molzahn**, "Locating and Counting Equilibria of the Kuramoto Model with Rank One Coupling,"*SIAM Journal on Applied Algebra and Geometry,*vol. 2, no. 1, pp. 45-71, 2018.

The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phase-coherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, n, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of n. For these, the equilibria can be located by finding the real solutions of a system of polynomial equations utilizing techniques from algebraic geometry. However, typical methods for solving such systems locate all complex solutions even though only the real solutions give equilibria. In this paper, we present an algorithm to locate only the real solutions of the model, thereby shortening computation time by several orders of magnitude in certain situations. This is accomplished by choosing specific equilibria representatives and the consequent algebraic decoupling of the system. The correctness of the algorithm (that it finds only and all the equilibria) is proved rigorously. Additionally, the algorithm can be implemented using interval methods so that the equilibria can be approximated up to any given precision without significantly more computational effort. We also compare this solving approach to other computational algebraic geometric methods. Furthermore, analyzing this approach allows us to prove, asymptotically, that the maximum number of equilibria grows at the same rate as the number of complex solutions of a corresponding polynomial system. Finally, we conjecture an upper bound on the maximum number of equilibria for any number of oscillators which generalizes the known cases and is obtained on a range of explicitly provided natural frequencies. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Wu,

**D.K. Molzahn**, B.C. Lesieutre, and K. Dvijotham, "A Deterministic Method to Identify Multiple Local Extrema for the AC Optimal Power Flow Problem,"*IEEE Transactions on Power Systems,*vol. 33, no. 1, pp. 654-668, January 2018.

This paper presents a deterministic approach to compute multiple local extrema for AC Optimal Power Flow (ACOPF) problems. An elliptical representation of the sphereconfined Fritz John conditions is constructed from a quadratic formulation of the ACOPF problem. Application of a branch tracing method to the elliptical formulation enables the calculation of multiple solutions. Further, a monotone search strategy enhances the computational efficiency of finding multiple local solutions with improved objective values. The proposed approach is first illustrated using two small examples with known feasible spaces. Then, four additional local solutions to a 39-bus system are found using the proposed approach. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]J. Lavaei, S.H. Low, R. Baldick, B. Zhang,

**D.K. Molzahn**, F. Dörfler, and H. Sandberg, "Guest Editorial: Distributed Control and Efficient Optimization Methods for Smart Grid,"*IEEE Transactions on Smart Grid, Special issue on Distributed Control and Efficient Optimization Methods for Smart Grid*, vol. 8, no. 6, pp. 2939-2940, November 2017.

[ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, F. Dörfler, H. Sandberg, S.H. Low, S. Chakrabarti, R. Baldick, and J. Lavaei, "A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems,"

Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems. [ abstract ] [ link ] [ pdf ] [ Ref: bib txt ]**D.K. Molzahn**, "Computing the Feasible Spaces of Optimal Power Flow Problems,"*IEEE Transactions on Power Systems*, vol. 32, no. 6, pp. 4752-4763, November 2017.

The solution to an optimal power flow (OPF) problem provides a minimum cost operating point for an electric power system. The performance of OPF solution techniques strongly depends on the problem's feasible space. This paper presents an algorithm that is guaranteed to compute the entire feasible spaces of small OPF problems to within a specified discretization tolerance. Specifically, the feasible space is computed by discretizing certain of the OPF problem's inequality constraints to obtain a set of power flow equations. All solutions to the power flow equations at each discretization point are obtained using the Numerical Polynomial Homotopy Continuation (NPHC) algorithm. To improve computational tractability, "bound tightening" and "grid pruning" algorithms use convex relaxations to preclude consideration of many discretization points that are infeasible for the OPF problem. The proposed algorithm is used to generate the feasible spaces of two small test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, "Incorporating Squirrel-Cage Induction Machine Models in Convex Relaxations of OPF Problems,"*IEEE Transactions on Power Systems (Letters)*, vol. 32, no. 6, pp. 4972-4974, November 2017.

The optimal power flow (OPF) problem determines a minimum cost operating point for an electric power system. Recently developed convex relaxations are capable of globally solving certain OPF problems. Using a semidefinite relaxation of the OPF problem as an illustrative example, this letter presents a method for extending convex relaxations of the OPF problem to include steady-state squirrel-cage induction machine models. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]J.F. Marley,

**D.K. Molzahn**, and I.A. Hiskens, "Solving Multiperiod OPF Problems using an AC-QP Algorithm Initialized with an SOCP Relaxation,"*IEEE Transactions on Power Systems*, vol. 32, no. 5, pp. 3538-3548, September 2017.

Renewable generation and energy storage are playing an ever increasing role in power systems. Hence, there is a growing need for integrating these resources into the optimal power flow (OPF) problem. While storage devices are important for mitigating renewable variability, they introduce temporal coupling in the OPF constraints, resulting in a multiperiod OPF formulation. This paper explores a solution method for multiperiod AC OPF that combines a successive quadratic programming approach (AC-QP) with a second-order cone programming (SOCP) relaxation of the OPF problem. The SOCP relaxation’s solution is used to initialize the AC-QP OPF algorithm. Additionally, the lower bound on the objective value obtained from the SOCP relaxation provides a measure of solution quality. This combined method is demonstrated on several test cases with up to 4259 nodes and a time horizon of 8 time steps. A comparison of initialization schemes indicates that the SOCP-based approach offers improved convergence rate, execution time and solution quality. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, C. Josz, I.A. Hiskens, and P. Panciatici, "A Laplacian-Based Approach for Finding Near Globally Optimal Solutions to OPF Problems,"*IEEE Transactions on Power Systems*, vol. 32, no. 1, pp. 305-315, January 2017.

A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and I.A. Hiskens, "Convex Relaxations of Optimal Power Flow Problems: An Illustrative Example,"*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 63, no. 5, pp. 650-660, May 2016.

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite programming (SDP) relaxation globally solves many OPF problems. However, there exist practical problems for which the SDP relaxation fails to yield the global solution. Conditions for the success or failure of the SDP relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the SDP relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The SDP relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the SDP relaxation must involve factors beyond just the network physics. The lack of universal physical conditions for success of the SDP relaxation motivates the development of tighter convex relaxations capable of solving a broader class of problems. Tools from polynomial optimization theory provide a means of developing tighter relaxations. This paper uses the example problem to illustrate relaxations from the Lasserre hierarchy for polynomial optimization and a related "mixed semidefinite/second-order cone programming" hierarchy. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ figures ]**D.K. Molzahn**and I.A. Hiskens, "Sparsity-Exploiting Moment-Based Relaxations of the Optimal Power Flow Problem,"*IEEE Transactions on Power Systems*, vol. 30, no. 6, pp. 3168-3180, November 2015.

Convex relaxations of non-convex optimal power flow (OPF) problems have recently attracted significant interest. While existing relaxations globally solve many OPF problems, there are practical problems for which existing relaxations fail to yield physically meaningful solutions. This paper applies moment relaxations to solve many of these OPF problems. The moment relaxations are developed from the Lasserre hierarchy for solving generalized moment problems. Increasing the relaxation order in this hierarchy results in "tighter" relaxations at the computational cost of larger semidefinite programs. Low-order moment relaxations are capable of globally solving many small OPF problems for which existing relaxations fail. By exploiting sparsity and only applying the higher-order relaxation to specific buses, global solutions to larger problems are computationally tractable through the use of an iterative algorithm informed by a heuristic for choosing where to apply the higher-order constraints. With standard semidefinite programming solvers, the algorithm globally solves many test systems with up to 300 buses for which the existing semidefinite relaxation fails to yield globally optimal solutions. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and C.L. DeMarco, "Approximate Representation of ZIP Loads in a Semidefinite Relaxation of the OPF Problem,"*IEEE Transactions on Power Systems (Letters)*, vol. 29, no. 4, pp. 1864-1865, July 2014.

Recent research has applied semidefinite programming (SDP) to the optimal power flow (OPF) problem. Extending SDP formulations to include the ZIP load model, which consists of constant impedance, constant current, and constant power components, is necessary for many practical problems. Existing SDP formulations consider constant power components and can be easily extended to incorporate constant impedance components. With linear dependence on voltage magnitude, constant current components are not trivially formulated in a manner amenable to SDP. This letter details an approximate representation of ZIP loads in a SDP relaxation of the OPF problem. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and C.L. DeMarco, "A Sufficient Condition for Global Optimality of Solutions to the Optimal Power Flow Problem,"*IEEE Transactions on Power Systems (Letters)*, vol. 29, no. 2, pp. 978-979, March 2014.

Recent applications of a semidefinite programming (SDP) relaxation to the optimal power flow (OPF) problem offers a polynomial time method to compute a global optimum for a large subclass of OPF problems. In contrast, prior OPF solution methods in the literature guarantee only local optimality for the solution produced. However, solvers employing SDP relaxation remain significantly slower than mature OPF solution codes. This letter seeks to combine the advantages of the two methods. In particular, we develop a SDP-inspired sufficient condition test for global optimality of a candidate OPF solution. This test may then be easily applied to a candidate solution generated by a traditional, only-guaranteed-locally-optimal OPF solver. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, J.T. Holzer, B.C. Lesieutre, and C.L. DeMarco, "Implementation of a Large-Scale Optimal Power Flow Solver Based on Semidefinite Programming,"*IEEE Transactions on Power Systems*, vol. 28, no. 4, pp. 3987-3998, November 2013.

The application of semidefinite programming to the optimal power flow (OPF) problem has recently attracted significant research interest. This paper provides advances in modeling and computation required for solving the OPF problem for large-scale, general power system models. Specifically, a semidefinite programming relaxation of the OPF problem is presented that incorporates multiple generators at the same bus and parallel lines. Recent research in matrix completion techniques that decompose a single large matrix constrained to be positive semidefinite into many smaller matrices has made solution of OPF problems using semidefinite programming computationally tractable for large system models. We provide three advances to existing decomposition techniques: a matrix combination algorithm that further decreases solver time, a modification to an existing decomposition technique that extends its applicability to general power system networks, and a method for obtaining the optimal voltage profile from the solution to a decomposed semidefinite program. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and C.L. DeMarco, "A Sufficient Condition for Power Flow Insolvability with Applications to Voltage Stability Margins,"*IEEE Transactions on Power Systems,*vol. 28, no. 3, pp. 2592-2601, August 2013.

For the nonlinear power flow problem specified with standard PQ, PV, and slack bus equality constraints, we present a sufficient condition under which the specified set of nonlinear algebraic equations has no solution. This sufficient condition is constructed in a framework of an associated feasible, convex optimization problem. The objective employed in this optimization problem yields a measure of distance (in a parameter set) to the power flow solution boundary. In practical terms, this distance is closely related to quantities that previous authors have proposed as voltage stability margins. A typical margin is expressed in terms of the parameters of system loading (injected powers); here we additionally introduce a new margin in terms of the parameters of regulated bus voltages. [ abstract ] [ pdf ] [ link ] [ extended version ] [ Ref: bib txt ]**D.K. Molzahn**and B.C. Lesieutre, "Initializing Dynamic Power System Simulations Using Eigenvalue Formulations of the Induction Machine and Power Flow Models,"*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 60, no. 3, pp. 690-702, March 2013.

Initializing internal variables in dynamic power system simulations is a two-stage process. First, a power flow model is used to find the steady state bus variables. Second, values for the internal variables associated with bus connected components are determined such that the components' terminal values match the bus variables calculated from the power flow model. Initializing most components' variables is a straightforward, direct process. However, initializing induction machine variables traditionally uses an indirect, iterative process. In this paper, eigenvalue formulations are detailed for both the induction machine initialization and power flow models, which provide a direct method for determining all possible sets of induction machine initializations and offer a novel model for the power flow equations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and H. Chen, "Counterexample to a Continuation-Based Algorithm for Finding All Power Flow Solutions,"*IEEE Transactions on Power Systems (Letters)*, vol. 28, no. 1, pp. 564-565, February 2013.

Existing literature claims that an algorithm based on continuation is capable of finding all solutions to the power flow equations for all power systems. This claim is demonstrated to be incorrect through the use of a five-bus system counterexample. Existing literature also claims that a similar algorithm is capable of finding all Type-1 solutions (i.e., solutions where the power flow Jacobian has a single eigenvalue with positive real part) to the power flow equations for all systems. This claim is also shown to be incorrect using the five-bus system counterexample. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and C. Singletary, "An Empirical Investigation of Speculation in the MISO Financial Transmission Rights Auction Market,"*The Electricity Journal*, vol. 24, issue 5, pp. 57-68, June 2011.

Although financial transmissions rights (FTRs) are intended to allow physical participants in electricity markets to hedge the locational price risk associated with their bilateral contracts, both physical participants and speculators take part in FTR auctions. While speculators likely increase the liquidity of FTR auction markets, physical participants and ultimately ratepayers bear the cost of the profits made by speculators. Given the regulated nature of physical participants in contrast to the largely unregulated speculators, it is hypothesized that speculators in FTR auction markets are obtaining substantial profits at the expense of physical participants. This study details a statistical model for characterizing market participants in the Midwest Independent System Operator (MISO) FTR auctions as either speculators or physical participants. Estimates of the total profit made by speculators and measures of the liquidity provided by speculators in FTR auctions are then determined. A sensitivity analysis is used to provide reasonable bounds for these estimates. The best estimates determined in this analysis show speculator profits of 152% of total market profits, equal to $283 million over three years, with speculator transactions accounting for 74% of total market transactions on a megawatt basis. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]

## Preprints

## 2024

## 2023

## 2022

## 2021

## 2020

## 2019

## 2018

## 2017

## 2016

## 2015

## 2014

## 2013

## 2011

R. Gupta and

**D.K. Molzahn**, "Analysis of Fairness-promoting Optimization Schemes of Photovoltaic Curtailments for Voltage Regulation in Power Distribution Networks," submitted.

Active power curtailment of photovoltaic (PV) generation is commonly exercised to mitigate over-voltage issues in power distribution networks. However, fairness concerns arise as certain PV plants may experience more significant curtailments than others depending on their locations within the network. Existing literature tackles this issue through fairness-promoting/aware optimization schemes. These schemes can be broadly categorized into two types. The first type maximizes an additional fairness objective along with the main objective of curtailment minimization. The second type is formulated as a feedback controller, where fairness is accounted for by assigning different weights (as feedback) in the curtailment minimization objective for each PV plant based on previous curtailment actions. In this work, we combine these two schemes and provide extensive analyses and comparisons of these two fairness schemes. We compare the performance in terms of fairness and net curtailments for several benchmark test networks. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Talkington, R. Gupta, R. Asiamah, P. Buason, and

**D.K. Molzahn**, "Strategic Electric Distribution Network Sensing via Spectral Bandits," submitted.

Despite their wide-scale deployment and ability to make accurate, high-frequency voltage measurements, communication network limitations have largely precluded the use of smart meters for real-time monitoring purposes in electric distribution systems. While smart meter communication networks have very limited bandwidth available per meter, they also have the ability to dedicate higher bandwidth to varying subsets of meters. Leveraging this capability in order to enable real-time monitoring from smart meters, this paper proposes an online bandwidth-constrained sensor sampling algorithm that exploits the graphical structure inherent in the power flow equations. The key idea is to use a spectral bandit framework where the estimated parameters are the graph Fourier transform coefficients of the nodal voltages. The structure provided by this framework promotes a sampling policy that strategically accounts for electrical distance. Maxima of sub-Gaussian random variables model the policy rewards, which relaxes distributional assumptions common in prior work. The scheme is implemented on realistic electrical networks to dynamically identify meters exposing violations of voltage magnitude limits and illustrating the effectiveness of the proposed method. [ abstract ] [ pdf ] [ Ref: bib txt ]P. Buason, S. Misra, and

**D.K. Molzahn**, "Sample-Based Conservative Bias Linear Power Flow Approximations," to appear in*IEEE IAS Industrial and Commercial Power System Asia (IEEE I&CPS Asia)*, July 9-12, 2024.

The power flow equations are central to many problems in power system planning, analysis, and control. However, their inherent non-linearity and non-convexity present substantial challenges during problem-solving processes, especially for optimization problems. Accordingly, linear approximations are commonly employed to streamline computations, although this can often entail compromises in accuracy and feasibility. This paper proposes an approach termed*Conservative Bias Linear Approximations (CBLA)*for addressing these limitations. By minimizing approximation errors across a specified operating range while incorporating conservativeness (over- or under-estimating quantities of interest), CBLA strikes a balance between accuracy and tractability by maintaining linear constraints. By allowing users to design loss functions tailored to the specific approximated function, the bias approximation approach significantly enhances approximation accuracy. We illustrate the effectiveness of our proposed approach through several test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Gupta, P. Buason, and

**D.K. Molzahn**, "Fairness-aware Photovoltaic Generation Limits for Voltage Regulation in Power Distribution Networks using Conservative Linear Approximations,"*8th Texas Power and Energy Conference (TPEC)*, February 12-13, 2024.

This paper proposes a framework for fairly curtailing photovoltaic (PV) plants in response to the over-voltage problem in PV-rich distribution networks. The framework imposes PV generation limits to avoid overvoltages. These limits are computed a day ahead of real-time operations by solving an offline stochastic optimization problem using forecasted scenarios for PV generation and load demand. The framework minimizes the overall curtailment while considering fairness by reducing disparities in curtailments among different PV owners. We model the distribution grid constraints using a conservative linear approximation (CLA) of the AC power flow equations which is computed using a set of sampled power injections from the day-ahead predicted scenarios. The proposed framework is numerically validated on a CIGRE benchmark network interfaced with a large number of PV plants. We compare the performance of the proposed framework versus an alternative formulation that does not incorporate fairness considerations. To this end, we assess tradeoffs between fairness, as quantified with the Jain Fairness Index (JFI), and the total curtailed energy. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A.D. Owen Aquino, S. Talkington, and

**D.K. Molzahn**, "Managing Vehicle Charging during Emergencies via Conservative Distribution System Modeling,"*8th Texas Power and Energy Conference (TPEC)*, February 12-13, 2024.

Combinatorial distribution system optimization problems, such as scheduling electric vehicle (EV) charging during evacuations, present significant computational challenges. These challenges stem from the large numbers of constraints, continuous variables, and discrete variables, coupled with the unbalanced nature of distribution systems. In response to the escalating frequency of extreme events impacting electric power systems, this paper introduces a method that integrates sample-based conservative linear power flow approximations (CLAs) into an optimization framework. In particular, this integration aims to ameliorate the aforementioned challenges of distribution system optimization in the context of efficiently minimizing the charging time required for EVs in urban evacuation scenarios. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B. Taheri and

**D.K. Molzahn**, "AC Power Flow Informed Parameter Learning for DC Power Flow Network Equivalents,"*8th Texas Power and Energy Conference (TPEC)*, February 12-13, 2024.

This paper presents an algorithm to optimize the parameters of power systems equivalents to enhance the accuracy of the DC power flow approximation in reduced networks. Based on a zonal division of the network, the algorithm produces a reduced power system equivalent that captures inter-zonal flows with aggregated buses and equivalent transmission lines. The algorithm refines coefficient and bias parameters for the DC power flow model of the reduced network, aiming to minimize discrepancies between inter-zonal flows in DC and AC power flow results. Using optimization methods like BFGS, L-BFGS, and TNC in an offline training phase, these parameters boost the accuracy of online DC power flow computations. In contrast to existing network equivalencing methods, the proposed algorithm optimizes accuracy over a specified range of operation as opposed to only considering a single nominal point. Numerical tests demonstrate substantial accuracy improvements over traditional equivalencing and approximation methods. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Harris and

**D.K. Molzahn**, "Detecting and Mitigating Data Integrity Attacks on Distributed Algorithms for Optimal Power Flow using Machine Learning,"*57th Hawaii International Conference on System Sciences (HICSS)*, January 3-6, 2024.

Using distributed algorithms, multiple computing agents can coordinate their operations by jointly solving optimal power flow problems. However, cyberattacks on the data communicated among agents may maliciously alter the behavior of a distributed algorithm. To improve cybersecurity, this paper proposes a machine learning method for detecting and mitigating data integrity attacks on distributed algorithms for solving optimal power flow problems. In an offline stage with trustworthy data, agents train and share machine learning models of their local subproblems. During online execution, each agent uses the trained models from neighboring agents to detect cyberattacks using a reputation system and then mitigate their impacts. Numerical results show that this method reliably, accurately, and quickly detects data integrity attacks and effectively mitigates their impacts to achieve near-feasible and near-optimal operating points. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Harris, A. Banerjee, and

**D.K. Molzahn**, "Synthetic Test Case for Ukraine's Power Grid,"*55th North American Power Symposium (NAPS)*, October 15-17, 2023.

Power systems research requires realistic test cases to demonstrate and benchmark algorithmic innovations. However, to keep power grid information secure, much of the data for actual systems cannot be publicly released. This motivates the creation of synthetic test cases designed to match the characteristics of actual power grids. By only using publicly available data, these synthetic test cases can be freely released for research and educational purposes. Motivated by the ongoing conflict in Ukraine, this paper presents a synthetic test case for the Ukrainian electric transmission system. With the ability to run power flow and other analyses, this test case is relevant to emerging research and educational activities related to power grid security. To develop the test case, we leveraged publicly available data on population centers, generators, and transmission topology and voltage levels along with synthetic test case creation methodologies from recent literature. After describing the process used to develop this test case, this paper presents validation results using several widely accepted criteria for assessing the test case's realism. As an illustrative application, we demonstrate one use case by solving a bilevel linearization of the $N-k$ interdiction problem to identify critical sets of line failures which cause the most load shedding. [ abstract ] [ pdf ] [ Ref: bib txt ] [ GitHub Repository ]B. Taheri and

**D.K. Molzahn**, "Restoring AC Power Flow Feasibility for Solutions to Relaxed and Approximated Optimal Power Flow Problems,"*American Control Conference (ACC)*, May 31 - June 2, 2023.

To address computational challenges associated with power flow nonconvexities, significant research efforts over the last decade have developed convex relaxations and approximations of optimal power flow (OPF) problems. However, benefits associated with the convexity of these relaxations and approximations can have tradeoffs in terms of solution accuracy since they may yield voltage phasors that are inconsistent with the power injections and line flows, limiting their usefulness for some applications. Inspired by state estimation techniques, this paper proposes a new method for obtaining an AC power flow feasible point from the solution to a relaxed or approximated OPF problem. By treating the inconsistent voltage phasors, power injections, and line flows analogously to noisy measurements in a state estimation algorithm, the proposed method yields power injections and voltage phasors that are feasible with respect to the AC power flow equations while incorporating information from many quantities in the solution to a relaxed or approximated OPF problem. We improve this method by adjusting weighting terms with a machine learning based approach. We demonstrate the proposed method using several relaxations and approximations. The result show up to several orders of magnitude improvement in accuracy over traditional methods. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A.D. Owen Aquino, R. Harris, A. Kody, and

**D.K. Molzahn**, "Comparing Machine Learning and Optimization Approaches for the*N-k*Interdiction Problem Considering Load Variability,"*56th Hawaii International Conference on System Sciences (HICSS)*, January 3-6, 2023.

Power grids must be operated, protected, and maintained such that a small number of line failures will not result in significant load shedding. To identify problematic combination of failures, we consider an*N-k*interdiction problem that seeks the set of*k*failed lines (out of*N*total lines) that result in the largest load shed. This is naturally formulated as a bilevel optimization problem with an upper level representing the attacker that selects line failures and a lower level modeling the defender's generator redispatch to minimize the load shedding. Compounding the difficulties inherent to the bilevel nature of interdiction problems, we consider a nonlinear AC power flow model that makes this problem intractable with traditional solution approaches. Furthermore, since the solutions found at a particular load condition may not generalize to other loading conditions, operators may need to quickly recompute these worst-case failures online to protect against them during operations. To address these challenges, we formulate and compare the performance of three simplified methods for solving the*N-k*interdiction problem: a state-of-the-art optimization approach based on a network-flow relaxation of the power flow equations and two newly developed machine learning algorithms that predict load sheds given the state of the network. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Kody, A. West, and

**D.K. Molzahn**, "Sharing the Load: Considering Fairness in De-energization Scheduling to Mitigate Wildfire Ignition Risk using Rolling Optimization,"*61st IEEE Control and Decision Conference (CDC)*, December 6-9, 2022.

Wildfires are a threat to public safety and have increased in both frequency and severity due to climate change. To mitigate wildfire ignition risks, electric power system operators proactively de-energize high-risk power lines during "public safety power shut-off" (PSPS) events. Line de-energizations can cause communities to lose power, which may result in negative economic, health, and safety impacts. Furthermore, the same communities may repeatedly experience power shutoffs over the course of a wildfire season, which compounds these negative impacts. However, there are often many combinations of power lines whose de-energization will result in about the same reduction of wildfire risk, but the associated power loss affects different communities. Therefore, one may raise concerns regarding the fairness of de-energization decisions. Accordingly, this paper proposes a model-predictive-control-inspired framework to select lines to de-energize in order to balance wildfire risk reduction, total load shedding, and fairness considerations. The goal of the framework is to prevent a small fraction of communities from disproportionally being impacted by PSPS events, and to instead more equally share the burden of power outages. For a geolocated test case in the southwestern United States, we use actual California demand data as well as real wildfire risk forecasts to simulate PSPS events during the 2021 wildfire season and compare the performance of various methods for promoting fairness. Our results demonstrate that the proposed formulation can provide significantly more fair outcomes with limited impacts on system-wide performance. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, R. Harris, S. Litchfield, D. Huggins, and

**D.K. Molzahn**, "Analyzing Malicious Data Injection Attacks on Distributed Optimal Power Flow Algorithms,"*54th North American Power Symposium (NAPS)*, October 9-11, 2022.

The rapid growth of distributed energy resources motivates the application of distributed optimization algorithms for solving optimal power flow (OPF) problems. Using distributed optimization to solve OPF problems requires repeated data sharing between agents responsible for different regions of the power system. The performance of distributed optimization algorithms depends on the integrity of the shared data and the communications between the agents. This paper investigates the vulnerability of a distributed algorithm called Auxiliary Problem Principle (APP) with respect to cyberattacks on the communications between the agents. We propose cyberattack models that manipulate the shared data between neighboring agents to achieve an objective such as driving the algorithm to converge to a suboptimal solution. We investigate the convergence characteristics of the APP algorithm in the context of the DC optimal power flow (DC OPF) problem with and without manipulating the shared data. Last, we demonstrate how trained neural networks can provide highly accurate attack detection. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Kody, R. Piansky, and

**D.K. Molzahn**, "Optimizing Transmission Infrastructure Investments to Support Line De-energization for Mitigating Wildfire Ignition Risk,"*IREP Symposium on Bulk Power System Dynamics and Control-XI. A 100% Renewable Energy Source Bulk Power Grid: Opportunities and Challenges*, July 25-30, 2022.

Wildfires pose a growing risk to public safety in regions like the western United States, and, historically, electric power systems have ignited some of the most destructive wildfires. To reduce wildfire ignition risks, power system operators preemptively de-energize high-risk power lines during extreme wildfire conditions as part of "Public Safety Power Shutoff" (PSPS) events. While capable of substantially reducing acute wildfire risks, PSPS events can also result in significant amounts of load shedding as the partially de-energized system may not be able to supply all customer demands. In this work, we investigate the extent to which infrastructure investments can support system operations during PSPS events by enabling reduced load shedding and wildfire ignition risk. We consider the installation of grid-scale batteries, solar PV, and line hardening or maintenance measures (e.g., undergrounding or increased vegetation management). Optimally selecting the locations, types, and sizes of these infrastructure investments requires considering the line de-energizations associated with PSPS events. Accordingly, this paper proposes a multi-period optimization formulation that locates and sizes infrastructure investments while simultaneously choosing line de-energizations to minimize wildfire ignition risk and load shedding. The proposed formulation is evaluated using two geolocated test cases along with realistic infrastructure investment parameters and actual wildfire risk data from the US Geological Survey. We evaluate the performance of investment choices by simulating de-energization decisions for the entire 2021 wildfire season with optimized infrastructure placements. With investment decisions varying significantly for different test cases, budgets, and operator priorities, the numerical results demonstrate the proposed formulation's value in tailoring investment choices to different settings. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]R. Harris, M. Alkhraijah, D. Huggins, and

**D.K. Molzahn**, "On the Impacts of Different Consistency Constraint Formulations for Distributed Optimal Power Flow,"*Texas Power and Energy Conference (TPEC)*, February 28 - March 1, 2022.

The optimal power flow (OPF) problem finds the least costly operating point which meets the power grid's operational limits and obeys physical power flow laws. Complementing today's centralized optimization paradigm, future power grids may rely on distributed optimization where multiple agents work together to determine an acceptable operating point. In distributed algorithms, local agents solve subproblems to optimize their region of the system and share data to achieve consistency with their neighboring agents' subproblems. This paper investigates how different methods of enforcing power flow consistency constraints between local areas in distributed optimal power flow impact convergence rate and a classifier’s ability to detect malicious cyberattack. The distributed OPF problem is solved with the alternating direction method of multipliers (ADMM) algorithm. First, the ADMM algorithm’s convergence rate is compared for three different consistency constraint formulations. Next, the paper considers a cyberattack in which the integrity of information shared between agents is compromised, causing the algorithm to exhibit unacceptable behavior. A support vector machine (SVM) classifier is trained to detect the presence of manipulated data from such cyberattacks. Results demonstrate that consistency constraint formulation impacts the classifier's detection performance; for certain formulations, detection is highly accurate. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A.D. Owen Aquino, L.A. Roald, and

**D.K. Molzahn**, "Identifying Redundant Constraints for AC OPF: The Challenges of Local Solutions, Relaxation Tightness, and Approximation Inaccuracy,"*53rd North American Power Symposium (NAPS)*, November 14-16, 2021.

To compute reliable and low-cost operating points, electric power system operators solve optimization problems that enforce inequality constraints such as limits on line flows, voltage magnitudes, and generator outputs. A common empirical observation regarding these constraints is that only a small fraction of them are binding (satisfied with equality) during operation. Furthermore, the same constraints tend to be binding across time periods. Recent research efforts have developed constraint screening algorithms that formalize this observation and allow for screening across operational conditions that are representative of longer time periods. These algorithms identify redundant constraints, i.e., constraints that can never be violated if other constraints are satisfied, by solving optimization problems for each constraint separately. This paper investigates how the choice of power flow formulation, represented either by the non-convex AC power flow, convex relaxations, or a linear DC approximation, impacts the results and the computational time of the screening method. This allows us to characterize the conservativeness of convex relaxations in constraint screening and assess the efficacy of the DC approximation in this context. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Xu, R. Ma,

**D.K. Molzahn**, H.L. Hijazi, and C. Josz, "Verifying Global Optimality of Candidate Solutions to Polynomial Optimization Problems using a Determinant Relaxation Hierarchy,"*60th IEEE Conference on Decision and Control (CDC)*, December 13-15, 2021.

We propose an approach for verifying that a given feasible point for a polynomial optimization problem is globally optimal. The approach relies on the Lasserre hierarchy and the result of Lasserre regarding the importance of the convexity of the feasible set as opposed to that of the individual constraints. By focusing solely on certifying global optimality and relaxing the Lasserre hierarchy using necessary conditions for positive semidefiniteness based on matrix determinants, the proposed method is implementable as a computationally tractable linear program. We demonstrate this method via application to the optimal power flow problem used to operate electric power systems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]P. Buason and

**D.K. Molzahn**, "Analysis of Fast Decoupled Power Flow via Multiple Axis Rotations,"*52nd North American Power Symposium (NAPS)*, April 11-13, 2021.

The Fast Decoupled Power Flow (FDPF) method has been widely adopted due to its computational speed advantages relative to the Newton-Raphson method for many practically relevant power flow problems. The FDPF method relies on the assumption that the lines are highly inductive, i.e., have small R/X ratios. When this assumption does not hold (e.g., for distribution systems and some subtransmission systems), the FDPF method may exhibit slow convergence or fail to converge entirely. To address such cases, previous work has proposed an*axis rotation*method which rescales the complex power injections and the bus admittances by a unit-magnitude complex scalar parameter. Since this complex parameter adjusts the lines' R/X ratios, an appropriate choice for this parameter can improve the FDPF method's performance. In contrast to previous work that only introduces one complex parameter for the entire system (or one complex parameter per large subsystem), we propose and analyze an axis rotation method that introduces different complex parameters at each bus. The additional degrees of freedom provided in this more granular approach are particularly valuable for systems where the lines have wider ranges of R/X ratios. To obtain appropriate values for the complex parameters, we propose to minimize the sum of the squared errors associated with the FDPF approximations. This method can also be adapted for cases with large voltage angle differences. Our simulation results demonstrate the effectiveness of the proposed method. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A. Ivemeyer, M. Bossart, R.W. Kenyon, A. Sajadi, B.-M. Hodge, and

**D.K. Molzahn**, "Assessing the Accuracy of Balanced Power System Models in the Presence of Voltage Unbalance,"*Power and Energy Conference at Illinois (PECI)*, April 1-2, 2021.**Georgia Institute of Technology 2021 Roger P. Webb ECE Undergraduate Research Award**

Traditional models of electric power systems represent distribution systems with unbalanced three-phase network models and transmission systems with balanced single-phase-equivalent network models. This distinction poses a challenge for coupled models of transmission and distribution systems, which are becoming more prevalent due to the growth of distributed energy resources connected to distribution systems. In order to maintain a balanced network representation, transmission system models typically assume that the voltage phasors at the interface to the distribution system are balanced. Inaccuracies resulting from this assumption during unbalanced operation can lead to erroneous values for line currents in the transmission system model. This paper empirically quantifies the accuracy of this balanced operating assumption during unbalanced operating conditions for both a simple two-bus system along with a more complex transmission and distribution co-simulation. This paper also characterizes the performance of different methods for translating the unbalanced voltage phasors into a balanced representation in order to give recommendations for modeling coupled transmission and distribution systems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, M. Alowaifeer, X. Li, M.J. Till, and

**D.K. Molzahn**, "Reactive Power Planning using Security-Constrained AC Optimal Power Flow and Sensitivity Analyses,"*Power and Energy Conference at Illinois (PECI)*, April 1-2, 2021.

This paper proposes an approach for siting and sizing reactive power sources in order to maintain adequate reactive power reserves in future loading scenarios. We consider both normal and post-contingency operation using a security-constrained optimal power flow (SCOPF) formulation to compute the minimum-required reactive power reserves. To maintain tractability, the SCOPF explicitly considers a small set of important contingencies that are identified using a continuation power flow. The remaining contingencies are evaluated using the SCOPF solution. Any violated contingencies are iteratively added to the considered set in the SCOPF formulation until all contingencies are satisfied. Our approach combines this SCOPF formulation with a bus sensitivity index that identifies potential locations for new reactive power supply. We demonstrate the proposed approach on the IEEE 30- and 118-bus test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Tandon, S. Grijalva, and

**D.K. Molzahn**, "Motivating the Use of Dynamic Line Ratings to Mitigate the Risk of Wildfire Ignition,"*Power and Energy Conference at Illinois (PECI)*, April 1-2, 2021.

Increasingly severe wildfires pose significant dangers to ecological, social, and economic systems. To curtail the risk of igniting wildfires, California utilities employ the Public Safety Power Shutoff (PSPS) program, which de-energizes transmission lines that run through wildfire-prone regions. In order to reduce the amount of load shedding required during PSPS events, this paper explores the use of dynamic line ratings (DLR) to manage the current flows on lines that pose a high risk of igniting wildfires. DLR schemes determine line flow limits that change with ambient conditions. This paper uses DLR in combination with more restrictive ground clearance requirements during wildfire-prone conditions to reduce the likelihood of wildfire-igniting faults from conductors contacting vegetation. Using two case studies, this paper demonstrates the potential benefits of DLR in this context by comparing the tradeoffs between completely de-energizing lines versus imposing more stringent ground clearance requirements via reducing current flows with DLR. Results from a large dataset representing WECC with actual data regarding ambient conditions, load demands, and wildfire risks show that the proposed approach can significantly reduce load shedding due to de-energizing lines that pose high risks of igniting wildfires. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Alkhraijah, M. Alowaifeer, S. Grijalva, and

**D.K. Molzahn**, "Distributed Multi-Period DCOPF via an Auxiliary Principle Problem Algorithm,"*5th IEEE Texas Power and Energy Conference*, February 2-5, 2021.

Distributed algorithms provide attractive features for solving Optimal Power Flow (OPF) problems in interconnected power systems compared to traditional centralized algorithms. Distributed algorithms help to maintain the control autonomy and data privacy of subsystems, which is particularly relevant in competitive markets and practical control system implementations. This paper analyzes a distributed optimization algorithm known as the "Auxiliary Principle Problem" to solve multiperiod distributed DCOPF problems with energy storage systems. The proposed approach enables multiple interconnected systems with their own sub-objectives to share their resources and to participate in an electricity market without implicitly sharing information about their local generators or internal network parameters. The paper also shows how the proposed approach can enable future microgrids to coordinate their operation, reduce the total operational cost, and avoid internal constraint violations caused by unscheduled flows (USF) while maintaining the subsystems’ autonomy. We used an 11-bus test system consisting of two interconnected subsystems to evaluate the proposed approach and analyze the impact of USF. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]Z.J. Zhang, P.T. Mana, D. Yan, Y. Sun, and

**D.K. Molzahn**, "Study of Active Line Flow Constraints in DC Optimal Power Flow Problems,"*IEEE SoutheastCon*, March 12-15, 2020.

Efficiently and reliably operating electric power grids requires the frequent solution of optimization problems such as the DC Optimal Power Flow (DC OPF) problem. Solution of DC OPF problems can be challenging due to the large number of operational constraints imposed in these problems, especially limits on the line flows. Improving solver times can be facilitated by a better understanding of the active line flow constraints in DC OPF problems (i.e., the inequality constraints limiting line flows that are satisfied with equality at the solution). Considering a variety of test cases and a range of loading scenarios, this paper empirically characterizes the sets of active line flow constraints in DC OPF problems. Among other analyses, the paper compares the sets of redundant line flow constraints (i.e., constraints guaranteed to be inactive) that are identified by previously proposed constraint screening methods to the line flow constraints that are actually observed to be active over a range of scenarios. The results indicate that a large fraction of the line flow constraints which are not identified as being redundant by these screening methods are nevertheless inactive in the DC OPF solutions. This observation suggests the potential for improvements to the constraint screening methods. Laying the groundwork for achieving these improvements, this paper also identifies which line characteristics tend to be associated with active flow constraints and studies the relationships among sets of simultaneously active line flow constraints. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]K. Girigoudar,

**D.K. Molzahn**, and L.A. Roald, "On the Relationships Among Different Voltage Unbalance Definitions,"*51st North American Power Symposium (NAPS)*, October 13-14, 2019.

Growing penetrations of distributed energy resources (DERs) increase the power injection variability in distribution systems, which can result in power quality issues such as voltage unbalance. To measure unbalance, organizations such as IEC, NEMA and IEEE define phase unbalance in their power quality standards. However, the definitions in these different standards are not consistent, and voltages that are considered acceptable by one standard may violate good practices defined by another standard. To address this issue, this paper provides analytical comparisons of the most common voltage unbalance definitions, which are supplemented with numerical simulations. The analytical relationships suggest that it is possible to approximately bound the symmetrical-component-based voltage unbalance factor (which depends on the magnitude and relative phase angle) by limiting the line-to-line voltage unbalance, whereas applying line-to-ground voltage unbalance definitions neglects all information about phase angle offsets. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]L.A. Roald and

**D.K. Molzahn**, "Implied Constraint Satisfaction in Power System Optimization: The Impacts of Load Variations,"*57th Annual Allerton Conference on Communication, Control, and Computing,*September 25-27, 2019.

In many power system optimization problems, we observe that only a small fraction of the line flow constraints ever become active at the optimal solution, despite variations in the load profile and generation costs. This observation has far-reaching implications not only for power system optimization, but also for the practical long-term planning, operation, and control of the system. We formalize this empirical observation for problems involving the DC power flow equations. We use a two-step constraint screening approach to identify constraints whose satisfaction is implied by other constraints in the problem. These constraints are redundant and can therefore can be eliminated from the problem. For the first screening step, we derive analytical bounds that quickly identify redundancies in the flow limits on parallel lines. The second screening step uses optimization-based constraint tightening, where we solve a (relaxed) optimization problem for each constraint to identify redundancies. We specifically focus our approach on large ranges of load variation such that the results are valid for long periods of time, thus justifying the computational overhead required for the screening method. Numerical results for a wide variety of standard test cases show that even with load variations up to ±100% of nominal loading, we are able to eliminate a significant fraction of the flow constraints. This large reduction in constraints may enable a range of possible applications. As one illustrative example, we demonstrate the computational improvements for the unit commitment problem obtained as a result of the reduced number of constraints. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]T. Mühlpfordt,

**D.K. Molzahn**, V. Hagenmeyer, and S. Misra, "Optimal Adaptive Power Flow Linearizations: Expected Error Minimization using Polynomial Chaos Expansion,"*IEEE Milan PowerTech*, June 23-27, 2019.

The nonlinearity of the power flow equations leads to algorithmic and theoretical challenges for a wide variety of optimization and control problems relevant to electric power systems. Solution algorithms for these problems often use linearization techniques to obtain more tractable power flow formulations. Many existing linearizations lack specialization to a given system and operating range of interest, leading to unnecessarily large linearization errors. In contrast, recently proposed "optimal adaptive" linearizations are 1) tailored to specific systems and operating ranges of interest and 2) optimal in the sense that they minimize a selected error metric relative to the nonlinear power flow equations. Previous work proposes optimal adaptive linearizations that minimize the*worst-case*linearization error. Building on previous work, this paper uses an uncertainty quantifiation method called Polynomial Chaos Expansion to minimize the*expected*linearization error for a given probability distribution of the power injections. As validated using Monte Carlo analyses, the capabilities of the expected-error-minimizing linearizations are shown to be superior to first-order Taylor approximations computed at a nominal operating point. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]C. Josz,

**D.K. Molzahn**, M. Tacchi, and S. Sojoudi, "Transient Stability Analysis of Power Systems via Occupation Measures,"*Innovative Smart Grid Technologies (ISGT)*, February 17-20, 2019.

We propose the application of occupation measure theory to the classical problem of transient stability analysis for power systems. This enables the computation of certified inner and outer approximations for the region of attraction of a nominal operating point. In order to determine whether a post-disturbance point requires corrective actions to ensure stability, one would then simply need to check the sign of a polynomial evaluated at that point. Thus, computationally expensive dynamical simulations are only required for post-disturbance points in the region between the inner and outer approximations. We focus on the nonlinear swing equations but voltage dynamics could also be included. The proposed approach is formulated as a hierarchy of semidefinite programs stemming from an infinite-dimensional linear program in a measure space, with a natural dual sum-of-squares perspective. On the theoretical side, this paper lays the groundwork for exploiting the oscillatory structure of power systems by using Hermitian (instead of real) sums-of-squares and connects the proposed approach to recent results from algebraic geometry. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and L.A. Roald, "Grid-Aware versus Grid-Agnostic Distribution System Control: A Method for Certifying Engineering Constraint Satisfaction,"*52nd Hawaii International Conference on System Sciences (HICSS)*, January 8-11, 2019.

Growing penetrations of distributed energy resources (DERs) in distribution systems have motivated the design of controllers that leverage DER capabilities to achieve system-wide objectives. These controllers may be either*grid-agnostic*or*grid-aware*, depending on whether distribution network constraints are considered. Grid-agnostic controllers have the benefit of not requiring network models or system measurements, but may cause dangerous constraint violations. Rather than develop a specific controller, this paper considers the*potential impacts*of DER controllers with respect to network constraint violations. Specifically, this paper develops an optimization-based method to rigorously certify when any grid-agnostic controller can be applied without concern regarding network constraint violations, or, conversely, when grid-aware control may be needed to maintain distribution grid security. The proposed method uses convex optimization techniques to bound the impacts of load variability, given a subset of buses with voltage measurements and control. The method either provides a certificate for secure operation or identifies potentially critical constraints and the need for additional controllability. Numerical tests illustrate the ability to certify secure operation for different ranges of variability. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, H. Nagarajan, and M.L. Crow, "Comparison of Various Trilinear Monomial Envelopes for Convex Relaxations of Optimal Power Flow Problems,"*IEEE Global Conference on Signal and Information Processing (GlobalSIP),*November 26-29, 2018.

Solutions to optimal power flow (OPF) problems provide operating points for electric power systems that minimize operational costs while satisfying both engineering limits and the power flow equations. OPF problems are non-convex and may have multiple local optima. To search for global optima, recent research has developed a variety of convex relaxations to bound the optimal objective values of OPF problems. Certain relaxations, such as the quadratic convex (QC) relaxation, are derived from OPF representations that contain trilinear monomials. Previous work has considered three techniques for relaxing these trilinear monomials:*recursive McCormick*(RMC) envelopes,*Meyer and Floudas*(MF) envelopes, and*extreme-point*(EP) envelopes. This paper compares the tightness and computational speed of relaxations that employ each of these techniques. Forming the convex hull of a single trilinear monomial, MF and EP envelopes are equivalently tight. Empirical results show that QC formulations using MF and EP envelopes give tighter bounds than those using RMC envelopes. Empirical results also indicate that the EP envelopes have advantages over MF envelopes with respect to computational speed and numerical stability when used with state-of-the-art second-order cone programming solvers. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, D. Wu, and M.L. Crow, "Empirical Investigation of Non-Convexities in Optimal Power Flow Problems,"*American Control Conference (ACC),*June 27-29, 2018.

Optimal power flow (OPF) is a central problem in the operation of electric power systems. An OPF problem optimizes a specified objective function subject to constraints imposed by both the non-linear power flow equations and engineering limits. These constraints can yield non-convex feasible spaces that result in significant computational challenges. Despite these non-convexities, local solution algorithms actually find the global optima of some practical OPF problems. This suggests that OPF problems have a range of difficulty: some problems appear to have convex or "nearly convex" feasible spaces in terms of the voltage magnitudes and power injections, while other problems can exhibit significant non-convexities. Understanding this range of problem difficulty is helpful for creating new test cases for algorithmic benchmarking purposes. Leveraging recently developed computational tools for exploring OPF feasible spaces, this paper first describes an empirical study that aims to characterize non-convexities for small OPF problems. This paper then proposes and analyzes several medium-size test cases that challenge a variety of solution algorithms. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ test cases ]**D.K. Molzahn**and L.A. Roald, "Towards an AC Optimal Power Flow Algorithm with Robust Feasibility Guarantees,"*20th Power Systems Computation Conference (PSCC)*, June 11-15, 2018.

With growing penetrations of stochastic renewable generation and the need to accurately model the network physics, optimization problems that explicitly consider uncertainty and the AC power flow equations are becoming increasingly important to the operation of electric power systems. This paper describes initial steps towards an AC Optimal Power Flow (AC OPF) algorithm which yields an operating point that is guaranteed to be robust to all realizations of stochastic generation within a specified uncertainty set. Ensuring robust feasibility requires overcoming two challenges: 1) ensuring solvability of the power flow equations for all uncertainty realizations and 2) guaranteeing feasibility of the engineering constraints for all uncertainty realizations. This paper primarily focuses on the latter challenge. Specifically, this paper poses the robust AC OPF problem as a bi-level program that maximizes (or minimizes) the constraint values over the uncertainty set, where a convex relaxation of the AC power flow constraints is used to ensure conservativeness. The resulting optimization program is solved using an alternating solution algorithm. The algorithm is illustrated via detailed analyses of two small test cases. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]S. Misra,

**D.K. Molzahn**, and K. Dvijotham, "Optimal Adaptive Linearizations of the AC Power Flow Equations,"*20th Power Systems Computation Conference (PSCC)*, June 11-15, 2018.

The power flow equations are at the heart of many optimization and control problems relevant to power systems. The non-linearity of these equations leads to computational challenges in solving power flow and optimal power flow problems (non-convergence, local optima, etc.). Accordingly, various linearization techniques, such as the DC power flow, are often used to approximate the power flow equations. In contrast to a wide variety of general linearization techniques in the power systems literature, this paper computes a linear approximation that is specific to a given power system and operating range of interest. An "adaptive linearization" developed using this approach minimizes the worst-case error between the output of the approximation and the actual non-linear power flow equations over the operating range of interest. To compute an adaptive linearization, this paper proposes a constraint generation algorithm that iterates between 1) using an optimization algorithm to identify a point that maximizes the error of the linearization at that iteration and 2) updating the linearization to minimize the worst-case error among all points identified thus far. This approach is tested on several IEEE test cases, with the results demonstrating up to a factor of four improvement in approximation error over linearizations based on a first-order Taylor approximation. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M.R. Narimani,

**D.K. Molzahn**, and M.L. Crow, "Improving QC Relaxations of OPF Problems via Voltage Magnitude Difference Constraints and Envelopes for Trilinear Monomials,"*20th Power Systems Computation Conference (PSCC)*, June 11-15, 2018.

AC optimal power flow (AC OPF) is a challenging non-convex optimization problem that plays a crucial role in power system operation and control. Recently developed convex relaxation techniques provide new insights regarding the global optimality of AC OPF solutions. The quadratic convex (QC) relaxation is one promising approach that constructs convex envelopes around the trigonometric and product terms in the polar representation of the power flow equations. This paper proposes two methods for tightening the QC relaxation. The first method introduces new variables that represent the voltage magnitude differences between connected buses. Using "bound tightening" techniques, the bounds on the voltage magnitude difference variables can be significantly smaller than the bounds on the voltage magnitudes themselves, so constraints based on voltage magnitude differences can tighten the relaxation. Second, rather than a potentially weaker "nested McCormick" formulation, this paper applies "Meyer and Floudas" envelopes that yield the convex hull of the trilinear monomials formed by the product of the voltage magnitudes and trignometric terms in the polar form of the power flow equations. Comparison to a state-of-the-art QC implementation demonstrates the advantages of these improvements via smaller optimality gaps. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]J.A. Kersulis, I.A. Hiskens, C. Coffrin, and

**D.K. Molzahn**, "Topological Graph Metrics for Detecting Grid Anomalies and Improving Algorithms,"*20th Power Systems Computation Conference (PSCC)*, June 11-15, 2018.

Power grids are naturally represented as graphs, with buses as nodes and power lines as edges. Graph theory provides many ways to measure power grid graphs, allowing researchers to characterize system structure and optimize algorithms. We apply several topological graph metrics to 33 publiclyavailable power grids. Results show that a straightforward, computationally inexpensive set of checks can quickly identify structural anomalies, especially when a broad set of test networks is available to establish norms. Another application of graph metrics is the characterization of computational behavior. We conclude by illustrating one compelling example: the close connection between clique analysis and semidefinite programming solver performance. These two applications demonstrate the power of purely topological graph metrics when utilized in the right settings. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Yao,

**D.K. Molzahn**, and J.L. Mathieu, "The Impact of Load Models in an Algorithm for Improving Power System Voltage Stability via Demand Response,"*55th Annual Allerton Conference on Communication, Control, and Computing,*October 4-6, 2017.

Increasing communication and control capabilities will allow future power system operators to exploit large quantities of responsive demand. This paper discusses ongoing work that employs demand response to improve voltage stability via virtual spatial shifting of loads (i.e., altering the locational distribution of power consumption in one time period with an energy payback in a following time period). In this paper, we study the impact of load models on a previously proposed iterative linearization algorithm to determine loading patterns that maximize a voltage stability margin, namely, the smallest singular value (SSV) of the power flow Jacobian matrix. Specifically, we extend the algorithm to enable inclusion of composite load models consisting of both "ZIP" components and a steady-state squirrel-cage induction machine (IM) model. We then investigate the impact of different load models on both the stability margin and the loading pattern. Using the IEEE 14-bus system as an illustrative example, the results show that the type of load model affects the nominal system's SSV, the optimal SSV, and the optimal loading pattern. The maximum-achievable percent change in SSV is larger using IM models than using ZIP models. We also discuss the difficulty in interpreting the stability margin when the system undergoes structural changes resulting from the use of different voltage-dependent load models. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]L.A. Roald,

**D.K. Molzahn**, and A.F. Tobler, "Power System Optimization with Uncertainty and AC Power Flow: Analysis of an Iterative Algorithm,"*IREP Symposium on Bulk Power System Dynamics and Control-X. The Power System of the Future: Global Dynamics arising from Distributed Actions,*August 27 - September 1, 2017.

Many power system optimization problems are inherently affected by uncertainty, as it is generally impossible to exactly forecast load or renewable generation. Finding solutions that lead to low-cost, yet secure operation for a range of conditions is an important concern to system operators. This paper considers an iterative algorithm for solving power system optimization problems that include models of uncertainty and AC power flow physics. This algorithm iterates between solving deterministic nonlinear optimization problems with the AC power flow model and updating constraint tightenings that adjust the constraints to facilitate secure operation. Decoupling the uncertainty and the nonlinear optimization steps provides several advantages, such as tractability for large-scale problems and the ability to separately apply state-of-the-art techniques for handling the uncertainty and the nonlinearity. Using numerical examples and theoretical analyses, this paper discusses the benefits and limitations of the iterative algorithm in solving general power system optimization problems. This paper further characterizes the behavior of the iterative algorithm with respect to convergence, optimality, and infeasibility, and suggests several modifications to improve performance. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]M. Yao, J.L. Mathieu, and

**D.K. Molzahn**, "Using Demand Response to Improve Power System Voltage Stability Margins,"*IEEE PowerTech Manchester*, June 18-22, 2017.**High-quality paper award.**

Electric power systems with high penetrations of fluctuating renewable generation may operate close to their stability limits. Demand response can be used to improve power system stability. For example, it is already used to provide frequency control, improving frequency stability. This paper presents a new demand response strategy that uses fast-acting flexible loads to change the loading pattern while keeping the total load constant, improving steady-state voltage stability without affecting the system frequency. The new loading pattern is maintained only for a short period of time until the generators can be re-dispatched. Our goal is to find a permissible loading pattern that maximizes the smallest singular value of the power flow Jacobian matrix, which serves as a measure of voltage stability. The corresponding non-convex optimization problem is solved with an iterative linear programming approach that uses eigenvalue sensitivities and a linearization of the AC power flow equations. Using two IEEE test cases as illustrative examples, we show that the loading patterns resulting from the proposed approach give smallest singular values that are very close to those obtained from a brute force search. The results are also compared to another voltage stability measure given by the maximum loading margin for uniformly varying power injections. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]K. Dvijotham and

**D.K. Molzahn**, "Error Bounds on the DC Power Flow Approximation: A Convex Relaxation Approach,"*IEEE 55th Annual Conference on Decision and Control (CDC)*, December 12-14, 2016.

Power flow models are fundamental to power systems analyses ranging from short-term market clearing and voltage stability studies to long-term planning. Due to the nonlinear nature of the AC power flow equations and the associated computational challenges, linearized approximations (like the DC power flow) have been widely used to solve these problems in a computationally tractable manner. The linearized approximations have been justified using traditional engineering assumptions that under "normal" operating conditions, voltage magnitudes do not significantly deviate from nominal values and phase differences are "small". However, there is only limited work on rigorously quantifying when it is safe to use these linearized approximations. In this paper, we propose an algorithm capable of computing rigorous bounds on the approximation error in the DC power flow (and, in future extensions, more general linearized approximations) using convex relaxation techniques. Given a set of operational constraints (limits on the voltage magnitudes, phase angle differences, and power injections), the algorithm determines an upper bound on the difference in injections at each bus computed by the AC and DC power flow power flow models within this domain. We test our approach on several IEEE benchmark networks. Our experimental results show that the bounds are reasonably tight (i.e., there are points within the domain of interest that are close to achieving the bound) over a range of operating conditions. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, C. Josz, and I.A. Hiskens, "Moment Relaxations of Optimal Power Flow Problems: Beyond the Convex Hull,"*IEEE Global Conference on Signal and Information Processing (GlobalSIP)*, December 7-9, 2016.

Optimal power flow (OPF) is one of the key electric power system optimization problems. "Moment" relaxations from the Lasserre hierarchy for polynomial optimization globally solve many OPF problems. Previous work illustrates the ability of higher-order moment relaxations to approach the convex hulls of OPF problems' non-convex feasible spaces. Using a small test case, this paper focuses on the ability of the moment relaxations to globally solve problems with objective functions that have unconstrained minima at infeasible points inside the convex hull of the non-convex constraints. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, D. Mehta, and M. Niemerg, "Toward Topologically Based Upper Bounds on the Number of Power Flow Solutions,"*American Control Conference (ACC)*, July 6-8, 2016.**Best presentation in session award.**

The power flow equations, which relate power injections and voltage phasors, are at the heart of many electric power system computations. While Newton-based methods typically find the "high-voltage" solution to the power flow equations, which is of primary interest, there are potentially many "low-voltage" solutions that are useful for certain analyses. This paper addresses the number of solutions to the power flow equations. There exist upper bounds on the number of power flow solutions; however, there is only limited work regarding bounds that are functions of network topology. This paper empirically explores the relationship between the network topology, as characterized by the maximal cliques, and the number of power flow solutions. To facilitate this analysis, we use a numerical polynomial homotopy continuation approach that is guaranteed to find all complex solutions to the power flow equations. The number of solutions obtained from this approach upper bounds the number of real solutions. Testing with many small networks informs the development of upper bounds that are functions of the network topology. Initial results include empirically derived expressions for the maximum number of solutions for certain classes of network topologies. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]D. Mehta,

**D.K. Molzahn**, and K. Turitsyn, "Recent Advances in Computational Methods for the Power Flow Equations,"*American Control Conference (ACC)*, July 6-8, 2016.

The power flow equations are at the core of most of the computations for designing and operating electric grids. This system of multivariate nonlinear equations relate the power injections and voltages in an electric power system. A plethora of methods have been devised to solve these equations, from Newton-based methods to homotopy continuation and other optimization-based methods. Although many of these methods often efficiently find a high-voltage, stable solution, challenges remain for finding low-voltage solutions, which play significant roles in certain stability-related computations. While we do not claim to have exhausted the existing literature on all related methods, this tutorial paper introduces some of the recent advances in power flow solution methods to the wider power systems community as well as bringing attention from the computational mathematics and optimization communities to power systems problems. After briefly reviewing some of the traditional computational methods used to solve the power flow equations, we focus on three emerging methods: the numerical polynomial homotopy continuation method, Gröbner basis techniques, and moment/sum-of-squares relaxations using semidefinite programming. In passing, we also emphasize the importance of an upper bound on the number of solutions of the power flow equations and review the current status of research in this direction. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, C. Josz, I.A. Hiskens, and P. Panciatici, "Computational Analysis of Sparsity-Exploiting Moment Relaxations of the OPF Problem,"*19th Power Systems Computation Conference (PSCC)*, June 20-24, 2016.

With the potential to find global solutions, significant research interest has focused on convex relaxations of the non-convex OPF problem. Recently, "moment-based" relaxations from the Lasserre hierarchy for polynomial optimization have been shown capable of globally solving a broad class of OPF problems. Global solution of many large-scale test cases is accomplished by exploiting sparsity and selectively applying the computationally intensive higher-order relaxation constraints. Previous work describes an iterative algorithm that indicates the buses for which the higher-order constraints should be enforced. In order to speed computation of the moment relaxations, this paper provides a study of the key parameter in this algorithm as applied to relaxations from both the original Lasserre hierarchy and a recent complex extension of the Lasserre hierarchy. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, I.A. Hiskens, and B.C. Lesieutre, "Calculation of Voltage Stability Margins and Certification of Power Flow Insolvability using Second-Order Cone Programming,"*49th Hawaii International Conference on System Sciences (HICSS)*, January 5-8, 2016.

Reliable power system operation requires maintaining sufficient voltage stability margins. Traditional techniques based on continuation and optimization calculate lower bounds for these margins and generally require appropriate initialization. Building on a previous semidefinite programming (SDP) formulation, this paper proposes a new second-order cone programming (SOCP) formulation which directly yields upper bounds for the voltage stability margin without needing to specify an initialization. Augmentation with integer-constrained variables enables consideration of reactive-power-limited generators. Further, leveraging the ability to globally solve these problems, this paper describes a sufficient condition for insolvability of the power flow equations. Trade-offs between tightness and computational speed of the SDP and SOCP relaxations are studied using large test cases representing portions of European power systems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, C. Josz, I.A. Hiskens, and P. Panciatici, "Solution of Optimal Power Flow Problems using Moment Relaxations Augmented with Objective Function Penalization,"*IEEE 54th Annual Conference on Decision and Control (CDC)*, December 15-18, 2015.

The optimal power flow (OPF) problem minimizes the operating cost of an electric power system. Applications of convex relaxation techniques to the non-convex OPF problem have been of recent interest, including work using the Lasserre hierarchy of "moment" relaxations to globally solve many OPF problems. By preprocessing the network model to eliminate low-impedance lines, this paper demonstrates the capability of the moment relaxations to globally solve large OPF problems that minimize active power losses for portions of several European power systems. Large problems with more general objective functions have thus far been computationally intractable for current formulations of the moment relaxations. To overcome this limitation, this paper proposes the combination of an objective function penalization with the moment relaxations. This combination yields feasible points with objective function values that are close to the global optimum of several large OPF problems. Compared to an existing penalization method, the combination of penalization and the moment relaxations eliminates the need to specify one of the penalty parameters and solves a broader class of problems. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, Z.B. Friedman, B.C. Lesieutre, C.L. DeMarco, and M.C. Ferris, "Estimation of Constraint Parameters in Optimal Power Flow Data Sets,"*47th North American Power Symposium (NAPS)*, October 4-6, 2015.

Large-scale electric power system analysis depends upon representation of vast numbers of components whose individual models must be populated with parameters. The challenge of populating such component models is particularly apparent in optimal power flow applications, in which incorrect parameters and/or constraint limits can yield overall system representations with either unrealistically large feasible regions or an empty feasible set. Unfortunately, many data sets, particularly those of publicly available test cases, were originally developed to illustrate simpler "power flow only" applications, and may contain unrealistic values or wholly omit important constraint limits. This paper describes engineering-based approaches to obtain credible estimates for parameters and limits associated with line-flow constraints and generator capability curves, as may be employed in a number of steady state analyses such as the optimal power flow. These can substitute for missing or unrealistic data in test systems for which more fully detailed, "real-world" component specifications and limits are not available, and thereby make such test systems more valuable as research tools. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and I.A. Hiskens, "Mixed SDP/SOCP Moment Relaxations of the Optimal Power Flow Problem,"*IEEE PowerTech Eindhoven*, June 29 - July 2, 2015.

Recently, convex "moment" relaxations developed from the Lasserre hierarchy for polynomial optimization problems have been shown capable of globally solving many optimal power flow (OPF) problems. The moment relaxations, which take the form of semidefinite programs (SDP), generalize a previous SDP relaxation of the OPF problem. This paper presents an approach for improving the computational performance of the moment relaxations for many problems. This approach enforces second-order cone programming (SOCP) constraints that establish necessary (but not sufficient) conditions for satisfaction of the SDP constraints arising from the higher-order moment relaxations. The resulting "mixed SDP/SOCP" formulation implements the first-order relaxation using SDP constraints and the higher-order relaxations using SOCP constraints. Numerical results demonstrate that this mixed SDP/SOCP relaxation is capable of solving many problems for which the first-order moment relaxation fails to yield a global solution. For several examples, the mixed SDP/SOCP relaxation improves computational speed by factors from 1.13 to 18.7. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, S.S. Baghsorkhi, and I.A. Hiskens, "Semidefinite Relaxations of Equivalent Optimal Power Flow Problems: An Illustrative Example,"*IEEE International Symposium on Circuits and Systems (ISCAS)*, May 24-27, 2015.

Recently, there has been significant interest in convex relaxations of the optimal power flow (OPF) problem. A semidefinite relaxation globally solves many OPF problems. However, there exist practical problems for which the semidefinite relaxation fails to yield the global solution. Conditions for the success or failure of the semidefinite relaxation are valuable for determining whether the relaxation is appropriate for a given OPF problem. To move beyond existing conditions, which only apply to a limited class of problems, a typical conjecture is that failure of the semidefinite relaxation can be related to physical characteristics of the system. By presenting an example OPF problem with two equivalent formulations, this paper demonstrates that physically based conditions cannot universally explain algorithm behavior. The semidefinite relaxation fails for one formulation but succeeds in finding the global solution to the other formulation. Since these formulations represent the same system, success (or otherwise) of the semidefinite relaxation must involve factors beyond just the network physics. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]P. Panciatici, M.C. Campi, S. Garatti, S.H. Low,

**D.K. Molzahn**, A.X. Sun, and L. Wehenkel, "Advanced Optimization Methods for Power Systems,"*18th Power Systems Computation Conference (PSCC),*August 18-22, 2014.

Power system planning and operation offers multitudinous opportunities for optimization methods. In practice, these problems are generally large-scale, non-linear, subject to uncertainties, and combine both continuous and discrete variables. In the recent years, a number of complementary theoretical advances in addressing such problems have been obtained in the field of applied mathematics. The paper introduces a selection of these advances in the fields of non-convex optimization, in mixed-integer programming, and in optimization under uncertainty. The practical relevance of these developments for power systems planning and operation are discussed, and the opportunities for combining them, together with high-performance computing and big data infrastructures, as well as novel machine learning and randomized algorithms, are highlighted. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and I.A. Hiskens, "Moment-Based Relaxation of the Optimal Power Flow Problem,"*18th Power Systems Computation Conference (PSCC),*August 18-22, 2014.

The optimal power flow (OPF) problem minimizes power system operating cost subject to both engineering and network constraints. With the potential to find global solutions, significant research interest has focused on convex relaxations of the non-convex AC OPF problem. This paper investigates "moment-based" relaxations of the OPF problem developed from polynomial optimization theory. At the cost of increased computational requirements, moment relaxations are generally tighter than relaxations employed in previous research, thus resulting in global solutions for a broader class of OPF problems. Exploration of the feasible spaces of test systems illustrates the effectiveness of the moment relaxations. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, B.C. Lesieutre, and C.L. DeMarco, "Investigation of Non-Zero Duality Gap Solutions to a Semidefinite Relaxation of the Optimal Power Flow Problem,"*47th Hawaii International Conference on System Sciences (HICSS)*, January 6-9, 2014.

Recently, a semidefinite programming relaxation of the power flow equations has been applied to the optimal power flow problem. When this relaxation is "tight" (i.e., the solution has zero duality gap), a globally optimal solution is obtained. Existing literature investigates sufficient conditions whose satisfaction guarantees zero duality gap solutions. However, there is limited study of non-zero duality gap solutions. By illustrating the feasible spaces for optimal power flow problems and their semidefinite relaxations, this paper investigates examples of nonzero duality gap solutions. Results for large system models suggest that non-convexities associated with small subsections of the network are responsible for non-zero duality gap solutions. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]A.R. Borden,

**D.K. Molzahn**, B.C. Lesieutre, and P. Ramanathan, "Power System Structure and Confidentiality Preserving Transformation of Optimal Power Flow Problem,"*51st Annual Allerton Conference on Communication, Control, and Computing*, October 2-4, 2013.

In the field of power system engineering, the optimal power flow problem is essential in planning and operations. With increasing system size and complexity, the computational requirements needed to solve practical optimal power flow problems continues to grow. Increasing computational requirements make the possibility of performing these computations remotely with cloud computing appealing. However, power system structure and component values are often confidential; therefore, the problem cannot be shared. To address this issue of confidential information in cloud computing, some techniques for masking optimization problems have been developed. The work of this paper builds upon these techniques for optimization problems but is specifically developed for addressing the DC and AC optimal power flow problems. We study the application of masking a sample OPF using the IEEE 14-bus network. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, V. Dawar, B.C. Lesieutre, and C.L. DeMarco, "Sufficient Conditions for Power Flow Insolvability Considering Reactive Power Limited Generators with Applications to Voltage Stability Margins,"*IREP Symposium on Bulk Power System Dynamics and Control - IX. Optimization, Security and Control of the Emerging Power Grid*, 25-30 August 2013.

For the non-linear power flow problem with PQ and reactive power limited slack and PV buses, we present two sufficient conditions under which the specified set of nonlinear algebraic equations has no solution. The first condition uses a semidefinite programming relaxation of the power flow equations along with binary variables to model the generators' reactive power capabilities. As a byproduct, this condition yields a voltage stability margin to the power flow solvability boundary. The second condition formulates the power flow equations, including generator reactive power limits, as a system of polynomials and uses real algebraic geometry and sum of squares programming to create infeasibility certificates which prove power flow insolvability. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]A.R. Borden,

**D.K. Molzahn**, P. Ramanathan, and B.C. Lesieutre, "Confidentiality-Preserving Optimal Power Flow for Cloud Computing,"*50th Annual Allerton Conference on Communication, Control, and Computing,*pp. 1300-1307, October 1-5, 2012.

In the field of power system engineering, the optimal power flow problem is essential in planning and operations. With increasing system size and complexity, the computational requirements needed to solve practical optimal power flow problems continues to grow. Increasing computational requirements make the possibility of performing these computations remotely with cloud computing appealing. However, power system structure and component values are often confidential; therefore, the problem cannot be shared. To address this issue of confidential information in cloud computing, some techniques for masking optimization problems have been developed. The work of this paper builds upon these techniques for optimization problems but is specifically developed for addressing the DC and AC optimal power flow problems. We study the application of masking a sample OPF using the IEEE 14-bus network. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B.C. Lesieutre,

**D.K. Molzahn**, A.R. Borden, and C.L. DeMarco, "Examining the Limits of the Application of Semidefinite Programming to Power Flow Problems,"*49th Annual Allerton Conference on Communication, Control, and Computing*, pp. 1492-1499, September 28-30, 2011.

The application of semidefinite programming (SDP) to power system problems has recently attracted substantial research interest. Specifically, a recent SDP formulation offers a convex relaxation to the well-known, typically nonconvex "optimal power flow" (OPF) problem. This new formulation was demonstrated to yield zero duality gap for several standard power systems test cases, thereby ensuring a globally optimal OPF solution in each. The first goal of the work here is to investigate this SDP algorithm for the OPF, and show by example that it can fail to give a physically meaningful solution (i.e., it has a nonzero duality gap) in some scenarios of practical interest. The remainder of this paper investigates a SDP approach utilizing modified objective and constraints to compute all solutions of the nonlinear power flow equations. Several variants are described. Results suggest SDP's promise as an efficient algorithm for identifying large numbers of solutions to the power flow equations. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]D.R. Schwarting,

**D.K. Molzahn**, C.L. DeMarco, and B.C. Lesieutre. "Topological and Impedance Element Ranking (TIER) of the Bulk-Power System,"*44th Hawaii International Conference on System Sciences (HICSS)*, pp. 1-10, January 4-7, 2011.

The Electricity Modernization Act of the Energy Policy Act of 2005 requires the creation of enforceable reliability standards for users, owners, and operators of the “Bulk-Power System” (BPS). This paper introduces an algorithm based on electrical properties of an electric power network that yields a numeric ranking of its branch elements, as a step towards identifying which elements should be considered part of the BPS. This ranking depends solely on network topology and element electrical characteristics, and is wholly independent of generator cost functions and system operating point. Two perspectives for deriving and justifying this ranking algorithm are described: one based on the sensitivity of bus-to-branch Lagrange multipliers and another based on generation shift factors. Summary results obtained from applying the algorithm to a detailed model of the US Eastern Interconnection are reported (with suitable anonymity for any facility specific data, in keeping with Critical Energy Infrastructure Information (CEII) protections). [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**and B.C. Lesieutre, "An Eigenvalue Formulation for Determining Initial Conditions of Induction Machines in Dynamic Power System Simulations,"*IEEE International Symposium on Circuits and Systems (ISCAS)*, pp. 2311-2313, May 30 - June 2, 2010.

The problem of determining the initial conditions for induction machine variables in a non-linear dynamic power system analysis is posed as an eigenvalue problem. The eigenvalue formulation can then be solved using standard linear algebra techniques. This method has the advantage of determining all possible solutions for the internal variables rather than the single solution obtained from traditional iterative methods. Additionally, the method can easily determine when the problem has no solutions through the absence of non-zero real eigenvalues. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]**D.K. Molzahn**, "Application of Semidefinite Optimization Techniques to Problems in Electric Power Systems,"*Ph.D. Dissertation, University of Wisconsin-Madison Department of Electrical and Computer Engineering*, August 2013.

**2014 Harold A. Peterson Best Dissertation Award, Second Place in the UW-Madison ECE Department.**

Motivated by the potential for improvements in electric power system economics and reliability, this dissertation investigates applications of a semidefinite programming relaxation of the power flow equations, which model the steady-state relationship between power injections and voltages in a power system. In contrast to other optimal power flow solution techniques, semidefinite program solvers can reliably find a global optimum in polynomial time when the semidefinite relaxation is "tight" (i.e., the solution has zero relaxation gap). Semidefinite relaxations have been applied to a variety of computationally difficult problems in many fields. The power systems literature details limited success in applying semidefinite programming to the optimal power flow (OPF) problem, which minimizes operating cost under engineering and network constraints. Semidefinite programming holds significant promise for application to other power systems problems.

This dissertation first investigates power system economics using a semidefinite relaxation of the OPF problem. In contrast to claims in the literature, this dissertation shows that the semidefinite relaxation may fail to give a physically meaningful solution (i.e., the solution has non-zero relaxation gap) for some practical problems. This dissertation also provides computational and modeling advances required for solving large-scale, realistic OPF problems. Modeling advances include allowing multiple generators at the same bus; parallel transmission lines and transformers; and an approximate method for modeling constant impedance, constant current, constant power (ZIP) loads. Existing methods exploit system sparsity to speed computation using matrix completion techniques that decompose the large positive semidefinite matrix constraint into constraints on many smaller matrices. Computational advances include modifying the matrix decomposition techniques for further reductions in solver times, extending the applicability of an existing decomposition, and a method for obtaining an optimal voltage profile from a solution to a decomposed semidefinite program. This dissertation also provides a sufficient condition test for global optimality of a candidate OPF solution obtained by any method.

This dissertation next provides techniques for improving power system reliability. The first reliability-related research develops a sufficient condition under which the power flow equations have no solution. This sufficient condition is evaluated using a feasible semidefinite optimization problem. The objective employed in this optimization problem yields a measure of distance (in a parameter set) to the power flow solvability boundary. This distance is closely related to quantities that previous authors have proposed as voltage stability margins. A typical margin is expressed in terms of the parameters of system loading (injected powers); this dissertation additionally introduces a new margin in terms of the parameters of regulated bus voltages. This dissertation considers generators modeled as ideal voltage sources and generators with reactive power limits, which require either mixed-integer semidefinite programming or sum of squares programming formulations.

Also related to power system reliability is the problem of finding multiple solutions to the power flow equations. Power systems typically operate at a high-voltage, stable power flow solution; however, other solutions are used for stability calculations. Existing literature claims that a continuation-based algorithm can reliably find all power flow solutions. This claim is demonstrated to be incorrect with a small counterexample system. Thus, no computationally tractable algorithms exist for reliably finding all power flow solutions. Methods for calculating multiple power flow solutions therefore deserve further research. This dissertation describes a method for finding multiple power flow solutions using semidefinite programming. Results suggest the method's promise for identifying large numbers of power flow solutions.

Although the semidefinite relaxation of the power flow equations is often "tight," non-zero relaxation gap solutions can occur. This dissertation investigates examples of non-zero relaxation gap solutions to semidefinite formulations for the optimal power flow problem, for the power flow insolvability condition, and for determining multiple solutions to the power flow equations. [ abstract ] [ pdf ] [ link ] [ errata ] [ Ref: bib txt ]**D.K. Molzahn**, "Power System Models Formulated as Eigenvalue Problems and Properties of Their Solutions,"*Master's Thesis, University of Wisconsin-Madison ECE Department*, December 2010.

Eigenvalue problems incorporate multiplicative nonlinearities in otherwise linear equations. Two power systems problems with multiplicative nonlinearities are reformulated as eigenvalue problems. Reformulation allows for the application of eigenvalue theory and solution techniques to these problems.

The first problem involves determination of the initial conditions for induction machine internal variables in a non-linear dynamic power system analysis. The internal variables of stator and rotor currents, rotor speed, and mechanical torque must be determined from the given values of input real power, stator voltage magnitude, and stator voltage angle. This problem is posed in an eigenvalue formulation that can be solved using standard linear algebra techniques. The eigenvalue formulation allows for determination of all possible solutions for the internal variables rather than the single solution obtained from traditional iterative methods. Additionally, the absence of non-zero real eigenvalues indicates that the problem has no solution. Both single-cage and double-cage induction machine models are considered, and numeric examples are presented.

The second problem involves reformulation of the power flow equations as a multiparameter eigenvalue problem. In this formulation, both the eigenvalues and the eigenvectors are composed of the d and q orthogonal components of the bus voltages. The two parameter formulation of the power flow equations for two bus systems can be solved directly by decomposing the problem into two generalized eigenvalue problems that must be simultaneously satisfied. Since n bus systems require 2(n-1) parameter eigenvalue problems, which do not yet have a general solution method for n > 2, systems with more than two buses are not yet directly solvable from the multiparameter eigenvalue formulation.

In addition to direct solution, two other research avenues for the multiparameter eigenvalue formulation of the power flow equations are pursued. First, motivated by eigenvalue problem structure, a reformulation of the multiparameter eigenvalue form of the power flow equations is presented. Although it is without known practical applications, this reformulation and the intermediate results in its derivation are interesting from a theoretical standpoint. Second, an eigenvalue sensitivity analysis is performed on the multiparameter eigenvalue formulation of the power flow equations. The linearization obtained from the eigenvalue sensitivity analysis is equivalent to the linearization obtained from the power flow Jacobian.

Future developments in multiparameter eigenvalue theory may provide additional insights into solutions of the power flow equations. General solution techniques for multiparameter eigenvalue problems with more than two parameters may enable direct solution of the power flow equations. A method for determining the number of solutions to multiparameter eigenvalue problems would be useful as a stopping condition for continuation power flows. Conditions for the existence of any solutions to multiparameter eigenvalue problems would be useful for finding the point of voltage collapse and for analyzing power systems in heavily loaded conditions. [ abstract ][ pdf ] [ link ] [ Ref: bib txt ]S. Babaeinejadsarookolaee, A. Birchfield, R.D. Christie, C. Coffrin, C.L. DeMarco, R. Diao, M. Ferris, S. Fliscounakis, S. Greene, R. Huang, C. Josz, R. Korab, B.C. Lesieutre, J. Maeght,

**D.K. Molzahn**, T.J. Overbye, P. Panciatici, B. Park, J. Snodgrass, and R.D. Zimmerman, "The Power Grid Library for Benchmarking AC Optimal Power Flow Algorithms,"*IEEE PES PGLib-OPF Task Force Report*, August 2019.

**2020 Working Group Award for Technical Report from the IEEE Power and Energy Society (PES) Power System Operation, Planning and Economics (PSOPE) Committee.**

In recent years, the power systems research community has seen an explosion of novel methods for formulating the AC power flow equations. Consequently, benchmarking studies using the seminal AC Optimal Power Flow (AC-OPF) problem have emerged as the primary method for evaluating these emerging methods. However, it is often difficult to directly compare these studies due to subtle differences in the AC-OPF problem formulation as well as the network, generation, and loading data that are used for evaluation. To help address these challenges, this IEEE PES Task Force report proposes a standardized AC-OPF mathematical formulation and the PGLib-OPF networks for benchmarking AC-OPF algorithms. A motivating study demonstrates some limitations of the established network datasets in the context of benchmarking AC-OPF algorithms and a validation study demonstrates the efficacy of using the PGLib-OPF networks for this purpose. In the interest of scientific discourse and future additions, the PGLib-OPF benchmark library is open-access and all the of network data is provided under a creative commons license. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ] [ link to PGLib-OPF datasets ]**D.K. Molzahn**, M. Niemerg, D. Mehta, and J.D. Hauenstein, "Investigating the Maximum Number of Real Solutions to the Power Flow Equations: Analysis of Lossless Four-Bus Systems," March 2016.

The power flow equations model the steady-state relationship between the power injections and voltage phasors in an electric power system. By separating the real and imaginary components of the voltage phasors, the power flow equations can be formulated as a system of quadratic polynomials. Only the real solutions to these polynomial equations are physically meaningful. This paper focuses on the maximum number of real solutions to the power flow equations. The literature provides upper bounds on the number of real power flow solutions based on the maximum number of complex solutions. There exist two- and three-bus systems which have as many real solutions as the upper bounds given by the maximum number of complex solutions. It is an open question whether this is also the case for larger systems. This paper uses a conjecture and recent algorithmic developments from Galois group theory to suggest a negative answer to this question. Specifically, this paper studies a generic four-bus system composed of PV buses. If a conjecture in Galois group theory holds, the maximum number of real power flow solutions for this system is upper bounded by 16, which is strictly less than 20, the maximum number of complex solutions. This paper also provides parameter values for this system which give rise to 16 real solutions, so this upper bound is achievable. [ abstract ] [ pdf ] [ link ] [ Ref: bib txt ]B.C. Lesieutre and

**D.K. Molzahn**, "Optimization and Control of Electric Power Systems,"*Final Technical Report, DOE Grant #DE-SC0002319*, October 17, 2014.

The Electric Power Grid is a critical infrastructure upon which the nation relies for delivery of much of its energy use. It supports the needs of industry, commerce, and consumer end use. Reliability and economic efficiency are essentials. Disruptions are inconvenient and can be very costly. The research presented here is directed towards optimal use of grid resources to ensure reliability and minimize cost. The analysis and optimization needs for planning and operation of the electric power system are challenging to meet due to the scale and form of model representations. The connected network spans the continent and the mathematical models are inherently nonlinear. Traditionally, computational limits have necessitated the use of very simplified models for grid analysis, and this has resulted in either less secure operation, or less efficient operation, or both. The research conducted in this project advances techniques for power system optimization problems that will enhance reliable and efficient operation. The results of this work appear in numerous publications and address different application problems include Optimal Power Flow (OPF) [8, 9, 15, 52, 53], unit commitment [41], demand response [46, 49], reliability margins [32, 33, 38], planning [42, 47, 48], transmission expansion [50, 51], and general tools and algorithms [43, 44, 45]. Importantly, the research has been successfully transferred into the practical application domain by its inclusion in the most recent release of the widely-used MATPOWER suite of power system analysis tools. [ abstract ] [ link ] [ Ref: bib txt ]B.C. Lesieutre, C.L. DeMarco, and

**D.K. Molzahn**, "Power System Deliverability Ranking of Critical Elements to Support Select Facilities,"*Report to the Federal Energy Regulatory Commission*, January 2012.

[ not publicly available ]B.C. Lesieutre, C.L. DeMarco, and

**D.K. Molzahn**, "Power System Ranking of Nodal Locations,"*Report to the Federal Energy Regulatory Commission*, January 2012.

[ not publicly available ]**D.K. Molzahn**, S.P. Williams, and R. Srinivasan, "Plug-In Vehicles in Madison, WI: Consumer Preferences, Charging Stations, and Distribution System Impacts,"*Energy Analysis and Policy Capstone Project Report to Madison Gas and Electric*, May 2010.

[ not publicly available ]