La'Darius Thomas
Improvements to Peer-to-Peer Energy Trading Markets Under Network Constraints in Electrical Distribution Systems
Assistant Professor in the Department of Electrical and Computer Engineering at Mercer University.
Peer-to-peer (P2P) energy trading has emerged as a decentralized market architecture for integrating distributed energy resources into modern electricity distribution systems. In these markets, prosumers exchange surplus electricity directly through bilateral transactions rather than relying on centralized wholesale market clearing. While decentralized P2P trading offers local energy utilization and consumer participation, it also introduces strategic interactions among self-interested participants operating within the physical constraints of low-voltage, three-phase distribution networks. Existing research has investigated strategic electricity markets, network sensitivity analysis, and three-phase distribution system modeling as largely independent problems. Comparatively, less attention has been devoted to developing a unified decentralized framework in which strategic bilateral market interactions, participants who can be both buyers and sellers, and three-phase network feasibility are jointly represented while preserving decentralized market behavior.
This dissertation develops a decentralized game framework for P2P energy trading that integrates strategic market behavior with distribution network operation. First, a P2P market model is formulated to analyze bilateral trading under fixed buyer and seller roles where strategic pricing and market equilibrium are determined. Then, the framework is extended to allow the market roles emerge endogenously rather than being a fixed assignment. Finally, the decentralized market is coupled with a three-phase AC power flow formulation that explicitly incorporates voltage constraints into the coordination of bilateral energy exchanges.
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Sergio A. Dorado-Rojas
Approximate Internal Models and Transmission Network Reconfiguration for Reliable Power Grid Operations
Northwestern University Sustainability and Energy Postdoctoral Fellow.
Converter-interfaced generation is displacing synchronous machines while data centers add large concentrated loads that can connect to or disconnect from the bulk grid abruptly. This combination exposes the power system to abnormal frequency excursions and to transmission congestion. This dissertation addresses the first issue with a control-theoretic approach and the second with an optimization-based one, together supporting reliable grid operation under the stress imposed by large flexible loads.
At the device level, this work develops a theory of approximate internal models for sinusoids of varying frequency. The construction approximates the instantaneous frequency over a receding horizon and embeds the resulting signal model into the controller, extending the internal model principle beyond fixed, known exogenous signals. Applied to single-phase grid-following inverters, the resulting current controller preserves tracking under frequency deviations well beyond the ranges contemplated by power quality standards.
At the network level, this work formulates optimal transmission switching in the linear-coupled AC power flow model using a node-breaker representation of substations. Under an apparent power congestion metric, the resulting mixed-integer linear program reproduces the congestion outcomes of an AC formulation to within a few percent of branch loading. Coordination constraints among the breakers of each element restrict the switching decisions to practically meaningful combinations.
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Samuel Talkington
Randomness as a Resource for Electric Power Systems
Postdoctoral fellow at Harvard University's John A. Paulson School of Engineering and Applied Sciences; will subsequently begin an Assistant Professor appointment in the Electrical Engineering and Computer Science Department at the University of Michigan.
In the field of electric power engineering, randomness has canonically been viewed as exogenous and uncontrollable uncertainty that purely corrupts or hinders operation, planning, and communication goals. This dissertation, in contrast, presents new methods and principles for power engineering that leverage uncertainty to improve the underlying mathematical and computational tools used to achieve such aims. A variety of fields have shown that randomization provides avenues for computing approximate solutions to difficult large-scale numerical analysis problems more efficiently than deterministic algorithms. Likewise, we demonstrate that such approximation algorithms hold significant promise for improving the efficiency and scalability of the algorithms used to solve a broad swath of power engineering questions.
To begin, we review this intersection of approximation algorithms and computational power systems in Chapter 2, alongside preliminary power engineering principles. Then, as an appetizer, we analyze in Chapter 3 a suite of nonasymptotic concentration inequalities for electric power systems. These inequalities, imported from applied probability theory, provide fundamental worst-case bounds on the fluctuation of electrical quantities of interest to power engineers under generic uncertainty assumptions. This uncertainty can variously be the exogenous uncertainty of interest in classical power engineering tasks, or the endogenous uncertainty that we study in later chapters.
We then turn our attention to the distribution grid topology learning problem, where the underlying graphical structure and electrical parameters of a radial network are sought to be recovered from sensor data. We show that by using sensor measurements with variable precision, namely dithered quantization—purposefully added random noise—one can predict and improve reconstruction quality in grid topology learning problems. This culminates in Chapter 4, where we show that dithered measurements yield what are, to the knowledge of the authors, the first error bounds of their kind for topology learning problems. Among many other future directions, these results enable applications in strategically allocating limited communication bandwidth for a variety of downstream power system estimation tasks.
Bandwidth allocation then takes center stage as we turn to studying the randomness of the network's own load fluctuations. In Chapter 5, we develop an online sensor sampling algorithm that strategically queries limited subsets of smart meters to expose extreme voltage magnitude fluctuations; the algorithm embeds the spectral structure of the power flow equations in a multi-armed bandit framework whose reward model is supplied by the concentration inequalities of Chapter 3. In Chapter 6, we show that the same load fluctuations can be tracked using incomplete sensitivities that approximately describe the network model. Namely, we establish conditions on the bus power factors under which the complex power injections are uniquely determined by voltage magnitude measurements, without observing voltage phase angles. This also enables estimation of the sensitivities themselves, and we develop matrix completion algorithms that recover these sensitivities under partial observability. In Chapter 7, we extend this estimation theme behind the meter, learning a generative model of customer demand that makes it possible to recover hidden reactive power control laws from net load data. Sensitivities return in Chapter 8, wherein we change the variable of differentiation from the power injections to the network itself. Implicitly differentiating power flow solutions with respect to the admittance parameters yields sensitivities of voltages, currents, and flows to topology changes. These sensitivities supply fast linearized formulations for admittance control and the gradients for the randomized switching algorithms that follow.
Subsequently, we develop an algorithmic framework to efficiently compute nearly optimal solutions to electric power network design problems. The crux of the proposed approach is a family of conditional gradient procedures that unite structural properties of the power flow equations with ideas from differentiable optimization and randomized rounding. Preliminary analyses indicate that this approach admits fast and robust algorithms for a broad class of operational and planning problems. In addition to being simple to implement, the proposed framework comes with accompanying guarantees on accuracy, feasibility, and runtime under clearly stated assumptions. Numerical experiments suggest substantial computational efficiency gains compared to the state of the art. In particular, the proposed implementation is the fastest method at every network size tested. The exact baseline, Gurobi's mixed-integer second order cone programming (MISOCP) solver, exceeds its time limit on the largest cases while the proposed method finishes with a comparable optimality gap, and the median rounded solution stays within 5% of the convex optimum in every case.
We then conclude by introducing a generic bilevel optimization framework that allows engineers to embed goals—such as improving energy affordability, mitigating risk, lowering training error, and producing other positive externalities—directly into power network planning tasks. Common structure shared among these applications reveals what may be a scalable approximation framework for a general class of mixed-integer bilevel cone programs.
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Richard Asiamah
Development and Applications of Synthetic Electric Grid Models for Underrepresented Regions
Lecturer in the School of Electrical and Computer Engineering at the Georgia Institute of Technology.
This thesis aims to develop algorithms and methods for creating synthetic electricity networks across various geographic regions and applications, with a specific focus on regions outside the United States and Europe. It builds on the extensive literature on synthetic network creation and extends to regions with lower access to detailed information and documentation of electricity transmission and distribution infrastructure. We explore the various techniques that may be necessary due to differences in geographical areas and the lack of concrete information on electricity networks. Complementing the existing literature, this dissertation develops models for parts of Africa, South America, and Asia that have previously been neglected in prior synthetic grid development research.
This document first presents a synthetic electric test case for Ghana, a country in West Africa. We provide a detailed description of the steps involved in developing this first-of-its-kind synthetic electric grid model for steady-state analysis. Building on this, we validate the realism of the grid model by demonstrating its accuracy against real-world scenarios in Ghana.
We then conduct a sensitivity analysis of our Ghanaian electric grid model to topology and demand estimation techniques commonly employed in the literature. The purpose of this experiment is to assess the accuracy of applying existing grid estimation techniques in developing countries and regions. We find that the estimation techniques used in Western grids do not apply well, necessitating the development of specialized methods that more accurately represent the characteristics of these non-Western grids.
Ultimately, we devise a detailed methodology for creating global synthetic electric grids that accurately capture the characteristics of the electricity networks regardless of the geographic location. Our method formalizes the process, enabling the creation of models for various regions and different sizes. The methods developed and the open-source tools available allow interested users to create models for various regions for use in research and education. We also provide a repository for daily load and solar profiles to enable multiple-period applications and simulations using these synthetic electric grid models.
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Amanda West
Addressing the Energy Trilemma in Electrical Energy Systems: From Switchgear Dielectrics to Power System Optimization and Community-Level Impacts
Member of the inaugural cohort of the IEEE-USA Congressional Electric Grid Policy Fellowship program.
The world is currently facing an energy trilemma, which is the challenge of simultaneously achieving equitable, secure, and environmentally sustainable energy systems. This dissertation addresses this challenge by providing a set of tools that integrates technical power systems analysis with energy justice principles across most levels of the electricity infrastructure: transmission, distribution, and demand. This dissertation is divided into three volumes to address this trilemma.
Volume I of this dissertation discusses the impact of power system components on the environment. Reducing greenhouse gas emissions in the electrical energy system decreases global warming, which not only helps air quality in local communities, but also helps limit the effects on climate change in global communities. Considering the environmental impact of power system components, this dissertation introduces foundational research on sustainable dielectric materials for medium voltage switchgear, which are an essential safety component of power systems. Volume 1 explores this via the characterization of supercritical fluid viscosity. Such an exploration is necessary to enable the development of next-generation electrical infrastructure for environmentally friendly energy systems.
Volume II of this dissertation presents optimization frameworks for considering fairness and equity in transmission and distribution system operations. A multi-objective optimization framework for wildfire-related public safety power shutoffs (PSPS) balances wildfire risk reduction with fair distribution of service interruptions, preventing disproportionate impacts on vulnerable populations during emergency de-energization events. Complementing this, an energy burden-based rate optimization explicitly incorporates household economic constraints into power system planning, ensuring that affordability considerations drive operational decisions. Furthering the integration of equity into power systems optimization, this volume also introduces Locational Marginal Burden (LMB), a new metric that bridges power systems engineering with energy burden analysis. At the distribution level, a fairness aware load delivery model employs bi-level optimization to simultaneously maximize load delivery fairness and minimize load shed while respecting physical system constraints.
Volume III of this dissertation connects power systems, equity-focused insights, and community development through a regenerative energy community framework. The framework demonstrates how strategic investments in rooftop solar photovoltaics empower communities to revitalize themselves across economic, social, and technical dimensions. It also presents an interdisciplinary approach to community modeling by bridging power systems, control theory, and community development.
In summary, this dissertation advances quantitative assessments of energy justice concerns within the electricity system while maintaining technical and economic feasibility. This research provides perspectives that may enable more equitable energy transitions.
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Daniel Turizo Arteaga
Advances in Interior Point Methods: Power Systems and Privacy Applications
Engineer at SimpleRose, a high-performance computing optimization company.
The objective of the proposed dissertation is to improve the performance of interior point methods in specialized settings related to the optimal power flow (OPF) problem of power engineering and privacy-preserving computation. The OPF is a constrained optimization problem of critical importance for efficient operation of power systems. However, the formulation of the OPF problem involves sensitive data from generators, and also from end users in systems implementing "smart grid" technology. As the OPF problem is traditionally solved by system operators, they have access to this data, which is a privacy concern. Consequently, recent research has focused not only on improving solution methods for the OPF, but also on addressing these privacy concerns. Currently, the most popular techniques for constrained optimization (like OPF and its related problems) are interior point methods. However, to this day there is not an efficient, privacy-preserving implementation of these methods.
To this end, new specialized interior point methods aimed at tackling some of the aforementioned issues are presented in this dissertation. First, we consider the problem of driving the state of a power system to a desired state (usually an OPF solution). This is equivalent to a discretized shortest path problem constrained by the feasible region of a non-linear OPF. We propose an interior point method that exhibits block tri-diagonal structure, and we show how this structure can be used to solve this problem efficiently. Next, we study the theoretical properties of an interior point method designed to have reduced computation time in privacy-preserving settings. Lastly, we develop an interior point method for linear programs that uses a novel type of constraint barrier, where the cost of matrix factorizations is reduced in proportion to the amount of constraints away from the current iterate.
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Ryan Piansky
Optimization Methods for Wildfire-Resilient Transmission Grid Operations and Infrastructure Planning
Member of the research staff at Lawrence Livermore National Laboratory.
Climate change-driven natural disasters pose an increasing threat to the power grid. Wildfires pose a unique challenge as power systems can ignite these destructive events, exposing utilities to liability. To mitigate the risk of ignitions, system operators proactively de-energize high-risk transmission lines in Public Safety Power Shutoff (PSPS) events. While effective for ignition risk mitigation, PSPS events can cause significant load shed. Utilities can also pursue infrastructure investments, such as line hardening or batteries, to mitigate the risk of ignition or maintain service but these decisions add complexity to already challenging optimization problems.
The contributions of this thesis include advances in modeling for optimally mitigating wildfire ignition risks, improved computational methods that leverage decomposition and machine-learning techniques for scalable algorithms on large and realistic test networks, and applications in infrastructure investments, climate resilience, and equity relevant to policymakers and utilities. This thesis provides detailed optimal power shutoff formulations in work evaluating sensitivity of decisions to ignition risk aggregation metrics and power flow formulations. Optimal power shutoff results achieve an order of magnitude reduction in load shed relative to methods comparable to industry standards. Extensions to infrastructure investment planning to support PSPS events are presented through tractable optimization algorithms, including flexibility to consider policy, equity, and alternative extreme weather events. Computational improvements are introduced through machine learning techniques and a novel temporal decomposition method that enables long time horizons to be modeled, resulting in fast, high-quality solutions that outperforms what was previously possible. These methods provide results on the order of an hour of computing time compared to days required under previous methods. In summary, this dissertation presents a detailed understanding of optimal transmission switching for PSPS events, offering further insights for engineers, utilities, and policymakers through flexible tools with realistic simulations.
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Rachel Harris
Computational Methods for Fast and Secure Distributed Optimal Power Flow
Transmission planning at Dominion Energy.
Distributed optimization algorithms may be used to coordinate large numbers of distributed energy resources (DERs) across transmission and distribution networks in the future power grid. These distributed algorithms have the potential to preserve privacy and autonomy while coordinating interconnected systems, and to improve scalability for large problems. However, there are challenges that must be addressed before implementing distributed algorithms for real-world system operation. Distributed optimal power flow (OPF) requires repeated communication between controllers, which creates vulnerability to cyberattacks. In addition, distributed OPF often takes many iterations to converge for large-scale power systems. Another challenge for optimally coordinating transmission and distribution networks is that traditional distributed optimization methods lack convergence guarantees for mixed-integer problems. This makes solving large-scale integrated transmission-distribution problems with discrete decisions, such as line switching, difficult.
This dissertation explores methods to improve data security and reduce computation time for distributed optimal power flow. First, we present a method to detect and mitigate data integrity attacks on information shared between controllers. We train neural network models of controllers' OPF subproblems, which can then be used to detect anomalies in shared data, and to generate predictions to replace corrupted data in the event of an attack. Second, we explore how the choice of distributed OPF convergence tolerance impacts constraint violations at the operating point selected by the distributed OPF solution. We introduce a bound tightening algorithm to ensure that distributed OPF algorithms, when converged to looser tolerances, do not cause violations when their solutions are applied to the power system. Using looser convergence tolerances reduces the number of iterations required to converge, and thus reduces the total computation time.
In addition, we reduce computation time by developing a secure warm start for distributed OPF. Using privacy-preserving neural networks, we compute the distributed OPF initialization without revealing sensitive local power demand data. Combining the bound tightening method for looser convergence tolerance with the secure warm start enables distributed OPF to converge very quickly. By drastically reducing the number of iterations required to converge, we significantly reduce computation time and also improve data security by reducing communication between controllers.
In addition, the dissertation explores making large-scale integrated transmission-distribution problems more tractable. In particular, we explore solving an ITD optimal switching problem for wildfire risk mitigation, over distribution networks with many battery energy storage devices. We propose developing reduced network models of distribution networks with an adaptive LinDistFlow power flow approximation. These reduced networks preserve important topology information, and the adaptive LinDistFlow model is optimized to match power flow across the full network as closely as possible.
Optimizing over reduced networks significantly reduces computational burden, compared to optimizing over full distribution networks. The reduced networks can be used as part of a hierarchical optimization scheme for large-scale ITD optimization. First, an upper-level ITD problem using the reduced networks selects setpoints for the aggregate device models. Then lower-level problems over individual distribution networks, or over device aggregations, find setpoints for individual devices to match the aggregate flexibility requested by the upper-level ITD problem. When solving the proposed ITD optimal switching problem, the reduced ITD formulations remain tractable for some problems for which the full ITD problem is not tractable, while producing high-quality and near-feasible solutions.
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Babak Taheri
Improving Power System Approximations Through Machine Learning-Inspired Optimization Methods
Research scientist at Hitachi Energy.
This dissertation aims to improve electric power system optimization algorithms using optimization and techniques inspired by machine learning. Power system optimization problems are inherently nonlinear and involve large-scale computations, making them challenging for real-time applications and for scenarios requiring complex models, such as bilevel formulations, mixed-integer nonlinear programs, and stochastic programs. To address these challenges, researchers and practitioners frequently simplify these problems using methods like relaxations, approximations, machine learning models, and reduced networks. However, such simplifications often introduce approximation errors, potentially leading to suboptimal or infeasible operational decisions. Drawing inspiration from computational methods used in machine learning, this dissertation develops algorithms to optimize parameter selection in power flow approximations, construct reduced network models, and restore feasibility in alternating current (AC) power flow solutions derived from simplified models. First, an improved version of the direct current (DC) power flow model is proposed. This model adaptively selects coefficients and bias parameters through machine learning-inspired techniques. The result is a significant improvement in the accuracy of the DC power flow approximation while preserving the model's simple structure, enabling seamless integration into existing computational workflows. Next, the dissertation introduces a novel algorithm for network reduction. This method optimizes the process of creating reduced network models, ensuring that the DC power flow solutions for the reduced networks align closely with the AC power flow results of the original, larger networks across a variety of operational scenarios. This advancement enhances the accuracy of inter-zonal flow predictions, providing a more dependable tool for power system analysis. The dissertation also tackles challenges associated with the nonlinearities of the DistFlow model, commonly used for distribution systems. A parameter optimization algorithm is developed to enhance the accuracy of the linearized DistFlow approximation for both single-phase equivalent and three-phase distribution network models. By optimizing the coefficients and bias parameters in the linearized model using sensitivity information, the algorithm reduces errors in voltage magnitude predictions compared to the nonlinear DistFlow model. Furthermore, this work proposes an algorithm to improve the accuracy of DC optimal power flow (DC-OPF) solutions relative to nonlinear AC optimal power flow (AC-OPF) solutions under various operating conditions. Using machine learning-inspired methods, this algorithm adjusts coefficients and bias parameters in the DC-OPF model, yielding more accurate generator set points and better alignment with the AC-OPF results. Additionally, the dissertation enhances the DC optimal transmission switching (DC-OTS) model. Traditional DC-OTS formulations, which simplify the AC optimal transmission switching (AC-OTS) problem into a mixed-integer linear program, often result in suboptimal or infeasible outcomes due to errors in the DC power flow approximation. The proposed DC-OTS algorithm addresses this issue by optimizing the parameters in the DC-OPF model to better represent AC-OPF results. Specifically, it captures both real and reactive power flows, improving congestion modeling and enhancing the accuracy of transmission switching decisions. This improvement reduces approximation errors, ultimately enhancing system reliability and operational efficiency. Finally, the dissertation introduces an AC power flow feasibility restoration algorithm. This algorithm employs a state estimation-based post-processing approach to adjust solutions from simplified optimization problems, ensuring they satisfy the AC power flow equations. By leveraging techniques inspired by machine learning, the algorithm learns the reliability of outputs from simplified optimization models, optimizing weight and bias parameters to improve the accuracy of these adjustments.
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Alejandro Owen Aquino
Offline Simplification and Reduction Strategies for Online Solution of Power System Optimization Problems
Member of the research staff at Sandia National Laboratories.
The objective of the research presented in this dissertation is to push forward the boundaries of the types of power systems optimization problems that can be solved, especially when fast solutions are required for online decision making.
There is no shortage of complex optimization problems in the area of Power Systems. The inherent physics of power flow through an electric power system, as well as the need to model networks that encompass anything from a local three-phase distribution feeder to a country's entire transmission system make some of these optimization problems very computationally challenging. Furthermore, the answers to some of these problems may also need to be computed quickly during real-time operation or under other time constraints, thereby adding additional difficulties to already complicated problems. These are the challenges that motivate the work presented in this document. This dissertation proposes, investigates, and validates offline computing strategies to reduce the size and complexity of power system optimization problems used online. The objective of the proposed strategies is to leverage the increased computational power usually available when solution time is not critical, to come up with tailor-made reduced, simplified, or surrogate models that can produce fast, yet accurate results. These techniques are proposed and then applied to different challenging problems that serve as case studies and validation.
To that end, this dissertation presents a non-convex constraint screening methodology, a single-level reformulation strategy for bilevel optimization problems using surrogate neural networks, and an application of a linearizing approach to represent large three-phase unbalanced distribution networks. Furthermore, the techniques presented here are used to solve modern day complex problems, showcasing possible applications including the ACOPF problem, the N-k Interdiction problem, and an Emergency Electric Vehicle Charging problem.
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Mohannad Alkhraijah
Cybersecurity-Aware Distributed Optimization for Optimal Power Flow
Postdoctoral researcher at the National Renewable Energy Laboratory.
Distributed optimization algorithms have many attractive features for coordinating systems with multiple agents. Using distributed optimization allows agents to collaborate in solving large-scale optimization problems while maintaining their autonomy. However, distributed algorithms may be vulnerable to cyberattacks due to their dependency on communication. This dissertation proposes a general cybersecurity-aware distributed optimization implementation framework for solving power systems optimization problems.
To streamline the process of testing and experimenting with distributed optimization algorithms, this dissertation presents an open-source framework implementation via PowerModelsADA, a software package for solving Optimal Power Flow (OPF) problems using distributed optimization algorithms. The framework expedites and automates the process of experimenting with distributed optimization algorithms via standardizing data, communication, and computational requirements of existing and new distributed optimization algorithms. All the results in this dissertation uses PowerModelsADA.
The dissertation then presents the results of an empirical analysis of three distributed optimization algorithms with ideal and nonideal communication models. The nonideal communication models are additive Gaussian noise, bad data (large errors), and intermittent communication loss. The numerical results indicate that distributed algorithms with additive Gaussian noise converge to the optimal solution but with noise in the final solutions that is proportional to the standard deviation of the communication noise. The results also show that bad data can prevent distributed optimization algorithms from converging even with low probabilities of occurrence, while intermittent communication loss may cause distributed optimization algorithms to converge to suboptimal solutions.
The dissertation then analyzes cyberattacks on distributed optimization algorithms and proposes a cyberattack detection method. The results demonstrate that cyberattacks can successfully drive the solutions of distributed optimization algorithms to malicious targets by manipulating the shared data between agents. Since the agents repeatedly solve the same subproblems, we exploit the subproblem structure to detect shared data manipulations by deriving two sufficient detection conditions. The dissertation also shows that a sophisticated attacker with knowledge of the detection methods may avoid detection by embedding the detection conditions in their attacks. To counter that, this dissertation proposes an adversarial neural network training framework to enhance the detectability of data manipulation attacks even when the attacker knows the detection methods.
The dissertation then proposes a mitigation strategy to reduce the impact of cyberattacks on distributed optimization. The mitigation strategy uses a robust optimization approach via solving an admissible bounds problem. The admissible bounds problem finds bounds on the power flows from the branches connected to attacked systems while ensuring the feasibility of the other areas' constraints. This proposed mitigation strategy allows agents to coordinate their responses and share their local resources to reduce the impact of cyberattacks.
All potential benefits of distributed optimization require a fully distributed termination method that eliminates the need for a central coordinator to check the termination statuses of all agents, which is lacking in the literature. This dissertation proposes a method for terminating distributed optimization algorithms solved by multiple computing agents. The proposed method uses three simple rules for each agent that rely solely on local information and information from neighboring agents. Furthermore, the dissertation proposes a fault-tolerant extension to the termination method via three additional rules. The fault-tolerant termination method prevents distributed optimization from early termination due to faulty agents or communication errors.
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Paprapee Buason
Sample-Based Power Flow Approximations: Computational Methods, Analysis, and Applications
Faculty member in the Department of Mechatronics at Rajamangala University of Technology Phra Nakhon (RMUTP) in Bangkok, Thailand (previously a postdoc with the Advanced Network Science Initiative at Los Alamos National Laboratory).
The non-convex nature of the power flow equations poses a challenge for solving various power system optimization and control problems. To address these challenges, linear approximations are often employed. However, the accuracy of these linearizations can vary depending on the specific characteristics of the power system and the operational range in which they are applied. Existing linearizations typically rely on general assumptions that apply to broad classes of systems, which can limit their accuracy and result in constraint violations when applied to specific systems.
In contrast to these existing approaches, we introduce conservative linear approximations of the power flow equations. These conservative linearizations intentionally overestimate or underestimate quantities of interest, aiming to make algorithms more tractable while avoiding constraint violations. We compute these conservative linear approximations through a sample-based method, involving the solution of a constrained linear regression problem.
Additionally, we introduce a class of approximations based on rational functions with linear numerators and denominators. This choice is motivated by the resulting linear inequality constraints, making these approximations well-suited for optimization formulations, while still providing enhanced accuracy compared to linear functions.
We enhance the conservativeness and accuracy of our approximations through an iterative sampling method, optimizing these functions with respect to the relevant quantities. We also conduct a sample-complexity analysis. To further develop our approach, we establish an importance sampling method for constructing linear and conservative linear approximations. This method's objective is to efficiently improve approximation quality by selecting the most informative samples. It does so by drawing samples from a relatively low-dimensional subspace exhibiting high curvature. This approach allows us to obtain highly accurate linear approximations with significantly fewer samples than random selection. By examining the relationships between the voltage magnitudes and the active and reactive power injections, we characterize the performance of our proposed power flow approximations for a range of test cases.
Furthermore, we examine applications of conservative linear approximations to prove their effectiveness in an optimal sensor placement problem that we formulate as a bilevel program. In the optimal sensor placement problem, our goal is to place a minimal number of sensors and avoid false sensor alarms in the upper level while the lower level ensures that these sensors will detect any voltage violations. We replace the nonlinear power flow equations with conservative linear approximations to make the bilevel problem tractable. 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.
Numerical tests demonstrate that our power flow approximations enhance accuracy compared to other linear approximations and prove effective in optimization problems. In future research, we plan to leverage machine learning techniques for our power flow approximations, extend these methods to other parameters (e.g., current flows), and apply them to additional applications like capacity expansion planning problems.
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