arXiv subjects

Daniel K. Molzahn

Publications and source records attributed to Daniel K. Molzahn.

At least 37 records · Page 2Linked to original sources

arXiv2025

Error Bounds for Radial Network Topology Learning from Quantized Measurements

We probabilistically bound the error of a solution to a radial network topology learning problem where both connectivity and line parameters are estimated. In our model, data errors are introduced by the precision of the sensors, i.e., quantization. This produces a nonlinear measurement model that embeds the operation of the sensor communication network into the learning problem, expanding beyond the additive noise models typically seen in power system estimation algorithms. We show that the error of a learned radial network topology is proportional to the quantization bin width and grows sublinearly in the number of nodes, provided that the number of samples per node is logarithmic in the number of nodes.

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arXiv2025

VArsity: Can Large Language Models Keep Power Engineering Students in Phase?

This paper provides an educational case study regarding our experience in deploying ChatGPT Large Language Models (LLMs) in the Spring 2025 and Fall 2023 offerings of ECE 4320: Power System Analysis and Control at Georgia Tech. As part of course assessments, students were tasked with identifying, explaining, and correcting errors in the ChatGPT outputs corresponding to power factor correction problems. While most students successfully identified the errors in the outputs from the GPT-4 version of ChatGPT used in Fall 2023, students found the errors from the ChatGPT o1 version much more difficult to identify in Spring 2025. As shown in this case study, the role of LLMs in pedagogy, assessment, and learning in power engineering classrooms is an important topic deserving further investigation.

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arXiv2025

Optimizing Battery and Line Undergrounding Investments for Transmission Systems under Wildfire Risk Scenarios: A Benders Decomposition Approach

With electric power infrastructure posing an increasing risk of igniting wildfires under continuing climate change, utilities are frequently de-energizing power lines to mitigate wildfire ignition risk, which can cause load shedding. Recent research advocates for installing battery energy storage systems as well as undergrounding risky overhead lines to reduce the load shedding during such de-energizations. Since wildfire ignition risk can exhibit substantial geographic and temporal variations, it is important to plan battery installation and line undergrounding investments while considering multiple possible scenarios. This paper presents a scenario-based framework for optimizing battery installation and line undergrounding investments while considering many scenarios, each consisting of a day-long time series of uncertain parameters for the load demand, renewable generation, and wildfire ignition risks. This problem is difficult to solve due to a large number of scenarios and binary variables associated with the battery placements as well as the lines to be undergrounded. To address the computational challenges, we decompose the problem in a two-stage scheme via a Benders decomposition approach. The first stage is a master problem formulated as a mixed integer linear programming (MILP) model that makes decisions on the locations and sizes of batteries as well as the lines to be undergrounded. The second stage consists of a linear programming model that assesses these battery and line undergrounding decisions as modeled by a DC OPF formulation. We demonstrate the effectiveness of the proposed scheme on a large-scale transmission network with real world data on wildfire ignition risks, load, and renewable generation.

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arXiv2025

Evaluating Undergrounding Decisions for Wildfire Ignition Risk Mitigation across Multiple Hazards

With electric power infrastructure increasingly susceptible to impacts from climate-driven natural disasters, there is an increasing need for optimization algorithms that determine where to harden the power grid. Prior work has primarily developed optimal hardening approaches for specific acute disaster scenarios. Given the extensive costs of hardening the grid, it is important to understand how a particular set of resilience investments will perform under multiple types of natural hazards. Using a large-scale test case representing the Texas power system, this paper aims to understand how line undergrounding investment decisions made for wildfire ignition risk mitigation perform during a range of wildfire, hurricane, and wind events. Given the varying geographical spread and damage profile of these events, we show that investment decisions made to address one type of natural disaster do not necessarily improve broader resilience outcomes, supporting the need for co-optimization across a range of hazards.

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arXiv2025

AC-Informed DC Optimal Transmission Switching Problems via Parameter Optimization

Optimal Transmission Switching (OTS) problems minimize operational costs while treating both the transmission line energization statuses and generator setpoints as decision variables. The combination of nonlinearities from an AC power flow model and discrete variables associated with line statuses makes AC-OTS a computationally challenging Mixed-Integer Nonlinear Program (MINLP). To address these challenges, the DC power flow approximation is often used to obtain a DC-OTS formulation expressed as a Mixed-Integer Linear Program (MILP). However, this approximation often leads to suboptimal or infeasible switching decisions when evaluated with an AC power flow model. This paper proposes an enhanced DC-OTS formulation that leverages techniques for training machine learning models to optimize the DC power flow model's parameters. By optimally selecting parameter values that align flows in the DC power flow model with apparent power flows -- incorporating both real and reactive components -- from AC Optimal Power Flow (OPF) solutions, our method more accurately captures line congestion behavior. Integrating these optimized parameters into the DC-OTS formulation significantly improves the accuracy of switching decisions and reduces discrepancies between DC-OTS and AC-OTS solutions. We compare our optimized DC-OTS model against traditional OTS approaches, including DC-OTS, Linear Programming AC (LPAC)-OTS, and Quadratic Convex (QC)-OTS. Numerical results show that switching decisions from our model yield better performance when evaluated using an AC power flow model, with up to $44\%$ cost reductions in some cases.

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arXiv2025

Discrete Shortest Paths in Optimal Power Flow Feasible Regions

Optimal power flow (OPF) is a critical optimization problem for power systems to operate at points where cost or other operational objectives are optimized. Due to the non-convexity of the set of feasible OPF operating points, it is non-trivial to transition the power system from its current operating point to the optimal one without violating constraints. On top of that, practical considerations dictate that the transition should be achieved using a small number of small-magnitude control actions. To solve this problem, this paper proposes an algorithm for computing a transition path by framing it as a shortest path problem. This problem is formulated in terms of a discretized piece-wise linear path, where the number of pieces is fixed a priori in order to limit the number of control actions. This formulation yields a nonlinear optimization problem (NLP) with a sparse block tridiagonal structure, which we leverage by utilizing a specialized interior point method. An initial feasible path for our method is generated by solving a sequence of relaxations which are then tightened in a homotopy-like procedure. Numerical experiments illustrate the effectiveness of the algorithm.

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arXiv2025

Sample-Based Piecewise Linear Power Flow Approximations Using Second-Order Sensitivities

The inherent nonlinearity of the power flow equations poses significant challenges in accurately modeling power systems, particularly when employing linearized approximations. Although power flow linearizations provide computational efficiency, they can fail to fully capture nonlinear behavior across diverse operating conditions. To improve approximation accuracy, we propose conservative piecewise linear approximations (CPLA) of the power flow equations, which are designed to consistently over- or under-estimate the quantity of interest, ensuring conservative behavior in optimization. The flexibility provided by piecewise linear functions can yield improved accuracy relative to standard linear approximations. However, applying CPLA across all dimensions of the power flow equations could introduce significant computational complexity, especially for large-scale optimization problems. In this paper, we propose a strategy that selectively targets dimensions exhibiting significant nonlinearities. Using a second-order sensitivity analysis, we identify the directions where the power flow equations exhibit the most significant curvature and tailor the CPLAs to improve accuracy in these specific directions. This approach reduces the computational burden while maintaining high accuracy, making it particularly well-suited for mixed-integer programming problems involving the power flow equations.

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arXiv2025

Optimizing Phase Allocation in Unbalanced Power Distribution Networks using a Linearized DistFlow Formulation

Power distribution networks, especially in North America, are often unbalanced but are designed to keep unbalance levels within the limits specified by IEEE, IEC, and NEMA standards. However, rapid integration of unbalanced devices, such as electric vehicle (EV) chargers and single-phase solar plants, can exacerbate these imbalances. This increase can trigger protection devices, increase losses, and potentially damage devices. To address this issue, phase swapping (or phase allocation) has been proposed. Existing approaches predominantly rely on heuristic methods. In this work, we develop a mixed integer linear programming (MILP) approach for phase allocation. Our approach uses linearized DistFlow equations to represent the distribution network and incorporates a phase consistency constraint, enforced with binary variables, to ensure that downstream phase configurations align with upstream configurations. We validate the proposed approach on multiple benchmark test cases and demonstrate that it effectively improves network balance, as quantified by various metrics.

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arXiv2025

Quantifying Metrics for Wildfire Ignition Risk from Geographic Data in Power Shutoff Decision-Making

Faults on power lines and other electric equipment are known to cause wildfire ignitions. To mitigate the threat of wildfire ignitions from electric power infrastructure, many utilities preemptively de-energize power lines, which may result in power shutoffs. Data regarding wildfire ignition risks are key inputs for effective planning of power line de-energizations. However, there are multiple ways to formulate risk metrics that spatially aggregate wildfire risk map data, and there are different ways of leveraging this data to make decisions. The key contribution of this paper is to define and compare the results of employing six metrics for quantifying the wildfire ignition risks of power lines from risk maps, considering both threshold- and optimization-based methods for planning power line de-energizations. The numeric results use the California Test System (CATS), a large-scale synthetic grid model with power line corridors accurately representing California infrastructure, in combination with real Wildland Fire Potential Index data for a full year. This is the first application of optimal power shutoff planning on such a large and realistic test case. Our results show that the choice of risk metric significantly impacts the lines that are de-energized and the resulting load shed. We find that the optimization-based method results in significantly less load shed than the threshold-based method while achieving the same risk reduction.

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arXiv2024

Adaptive Power Flow Approximations with Second-Order Sensitivity Insights

The power flow equations are fundamental to power system planning, analysis, and control. However, the inherent non-linearity and non-convexity of these equations present formidable obstacles in problem-solving processes. To mitigate these challenges, recent research has proposed adaptive power flow linearizations that aim to achieve accuracy over wide operating ranges. The accuracy of these approximations inherently depends on the curvature of the power flow equations within these ranges, which necessitates considering second-order sensitivities. In this paper, we leverage second-order sensitivities to both analyze and improve power flow approximations. We evaluate the curvature across broad operational ranges and subsequently utilize this information to inform the computation of various sample-based power flow approximation techniques. Additionally, we leverage second-order sensitivities to guide the development of rational approximations that yield linear constraints in optimization problems. This approach is extended to enhance accuracy beyond the limitations of linear functions across varied operational scenarios.

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arXiv2024

Strategic Electric Distribution Network Sensing via Spectral Bandits

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. Although smart meter communication networks have limited bandwidth available per meter, they also have the ability to dedicate higher bandwidth to varying subsets of meters. Using this capability to enable real-time monitoring from smart meters, this paper proposes an online bandwidth-constrained sensor sampling algorithm that takes advantage of 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 a synthetic electrical network to dynamically identify meters exposing violations of voltage magnitude limits, illustrating the effectiveness of the proposed method.

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arXiv2024

Long Duration Battery Sizing, Siting, and Operation Under Wildfire Risk Using Progressive Hedging

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.

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arXiv2024

Tightening QC Relaxations of AC Optimal Power Flow through Improved Linear Convex Envelopes

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 a variety of convex relaxations that bound the optimal objective values of AC OPF problems. The well-known QC relaxation convexifies the AC OPF problem by enclosing the non-convex terms (trigonometric functions and products) within convex envelopes. The accuracy of this method 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 convex envelope around this nonlinear function that takes the form of a polytope and then use projections of this envelope to obtain convex expressions for the nonlinear terms. Second, we use certain characteristics of the sine and cosine expressions along with the changes in their curvature to tighten this convex envelope. We also propose a coordinate transformation that rotates the power flow equations by an angle specific to each bus in order to obtain a tighter envelope. We demonstrate these improvements relative to a state-of-the-art QC relaxation implementation using the PGLib-OPF test cases. The results show improved optimality gaps in 68% of these cases.

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arXiv2024

A Data-Driven Sensor Placement Approach for Detecting Voltage Violations in Distribution Systems

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.

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arXiv2024

Analysis of Fairness-promoting Optimization Schemes of Photovoltaic Curtailments for Voltage Regulation in Power Distribution Networks

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.

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arXiv2024

AC Power Flow Feasibility Restoration via a State Estimation-Based Post-Processing Algorithm

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.

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arXiv2024

Optimizing Parameters of the DC Power Flow

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.

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arXiv2024

Improving Fairness in Photovoltaic Curtailments via Daily Topology Reconfiguration for Voltage Control in Power Distribution Networks

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.

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