arXiv subjects

Russell Bent

Publications and source records attributed to Russell Bent.

At least 37 records · Page 2Linked to original sources

arXiv2025

Microgrids optimal radial reconfiguration via FORWARD algorithm

Microgrids offer a promising paradigm for integrating distributed energy resources, bolstering energy resilience, and reducing the impact of blackouts. However, their inherent decentralization and dynamic operation present substantial energy management complexities. These complexities, including balancing supply and demand, ensuring system stability, and minimizing operational costs, often necessitate solving computationally intractable NP-hard Mixed-Integer Non-Linear Programming (MINLP) problems. Traditional MINLP solvers struggle with the scalability and feasibility guarantees required for these challenges. To address this, this paper tackles the problem of resource allocation and radial configuration design for microgrid power distribution and proposes and abstracted problem which is solved by introducing a permutation-based iterative search method over the recently introduced FORWARD method to efficiently identify feasible, near-optimal radial network structures while inherently respecting physical constraints. Furthermore, this paper investigates the integration of the proposed method as a warm-start strategy for benchmark MINLP solvers offering a scalable solution for comprehensive microgrid design.

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arXiv2025

A General and Streamlined Differentiable Optimization Framework

Differentiating through constrained optimization problems is increasingly central to learning, control, and large-scale decision-making systems, yet practical integration remains challenging due to solver specialization and interface mismatches. This paper presents a general and streamlined framework-an updated DiffOpt.jl-that unifies modeling and differentiation within the Julia optimization stack. The framework computes forward - and reverse-mode solution and objective sensitivities for smooth, potentially nonconvex programs by differentiating the KKT system under standard regularity assumptions. A first-class, JuMP-native parameter-centric API allows users to declare named parameters and obtain derivatives directly with respect to them - even when a parameter appears in multiple constraints and objectives - eliminating brittle bookkeeping from coefficient-level interfaces. We illustrate these capabilities on convex and nonconvex models, including economic dispatch, mean-variance portfolio selection with conic risk constraints, and nonlinear robot inverse kinematics. Two companion studies further demonstrate impact at scale: gradient-based iterative methods for strategic bidding in energy markets and Sobolev-style training of end-to-end optimization proxies using solver-accurate sensitivities. Together, these results demonstrate that differentiable optimization can be deployed as a routine tool for experimentation, learning, calibration, and design-without deviating from standard JuMP modeling practices and while retaining access to a broad ecosystem of solvers.

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arXiv2025

FORWARD: A Feasible Radial Reconfiguration Algorithm for Multi-Source Distribution Networks

This paper considers an optimal radial reconfiguration problem in multi-source distribution networks, where the goal is to find a radial configuration that minimizes quadratic distribution costs while ensuring all sink demands are met. This problem arises in critical infrastructure systems such as power distribution, water networks, and gas distribution, where radial configurations are essential for operational safety and efficiency. Optimal solution for this problem is known to be NP-hard. In this paper, we prove further that constructing a feasible radial distribution configuration is weakly NP-complete, making exact solution methods computationally intractable for large-scale networks. We propose FORWARD (Feasibility Oriented Random-Walk Inspired Algorithm for Radial Reconfiguration in Distribution Networks), a polynomial-time algorithm that leverages graph-theoretic decomposition and random walk principles to construct feasible radial configurations. Our approach introduces novel techniques including strategic graph partitioning at articulation points, dual graph condensation to address greedy shortsightedness, and capacity-aware edge swapping for infeasibility resolution. We provide rigorous theoretical analysis proving feasibility guarantees and establish a compositional framework enabling parallel processing while preserving optimality properties. Comprehensive numerical evaluation on networks ranging from IEEE standard test systems to 400-node small-world networks demonstrates that FORWARD consistently outperforms commercial MINLP solvers, achieving optimal or near-optimal solutions in seconds where traditional methods require hours or fail entirely. The algorithm's polynomial-time complexity and scalability make it particularly suitable for real-time distribution network management and as an effective initialization strategy for iterative optimization solvers.

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arXiv2025

LEVIS: Large Exact Verifiable Input Spaces for Neural Networks

The robustness of neural networks is crucial in safety-critical applications, where identifying a reliable input space is essential for effective model selection, robustness evaluation, and the development of reliable control strategies. Most existing robustness verification methods assess the worst-case output under the assumption that the input space is known. However, precisely identifying a verifiable input space $\mathcal{C}$, where no adversarial examples exist, is challenging due to the possible high dimensionality, discontinuity, and non-convex nature of the input space. To address this challenge, we propose a novel framework, **LEVIS**, consisting of **LEVIS-α** and **LEVIS-\b{eta}**. **LEVIS-α** identifies a single, large verifiable ball that intersects at least two boundaries of a bounded region $\mathcal{C}$, while **LEVIS-\b{eta}** systematically captures the entirety of the verifiable space by integrating multiple verifiable balls. Our contributions include: (1) introducing a verification framework that uses mixed-integer programming (MIP) to compute nearest and directional adversarial points, (2) integrating complementarity-constrained (CC) optimization with a reduced MIP formulation for scalability, achieving up to a 6 times runtime reduction, (3) theoretically characterizing the properties of the verifiable balls obtained by **LEVIS-α**, and (4) validating the approach across applications including electrical power flow regression and image classification, with demonstrated performance gains and geometric insights into the verifiable region.

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arXiv2025

Sobolev Training of End-to-End Optimization Proxies

Optimization proxies - machine learning models trained to approximate the solution mapping of parametric optimization problems in a single forward pass - offer dramatic reductions in inference time compared to traditional iterative solvers. This work investigates the integration of solver sensitivities into such end to end proxies via a Sobolev training paradigm and does so in two distinct settings: (i) fully supervised proxies, where exact solver outputs and sensitivities are available, and (ii) self supervised proxies that rely only on the objective and constraint structure of the underlying optimization problem. By augmenting the standard training loss with directional derivative information extracted from the solver, the proxy aligns both its predicted solutions and local derivatives with those of the optimizer. Under Lipschitz continuity assumptions on the true solution mapping, matching first order sensitivities is shown to yield uniform approximation error proportional to the training set covering radius. Empirically, different impacts are observed in each studied setting. On three large Alternating Current Optimal Power Flow benchmarks, supervised Sobolev training cuts mean squared error by up to 56 percent and the median worst case constraint violation by up to 400 percent while keeping the optimality gap below 0.22 percent. For a mean variance portfolio task trained without labeled solutions, self supervised Sobolev training halves the average optimality gap in the medium risk region (standard deviation above 10 percent of budget) and matches the baseline elsewhere. Together, these results highlight Sobolev training whether supervised or self supervised as a path to fast reliable surrogates for safety critical large scale optimization workloads.

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arXiv2025

Transient Stability-Constrained OPF: Neural Network Surrogate Models and Pricing Stability

A Transient Stability-Constrained Optimal Power Flow (TSC-OPF) problem is proposed that enforces frequency stability constraints using Neural Network (NN) surrogate models. NNs are trained using a novel model-driven active sampling algorithm that iteratively generates NN training data located near the stability boundary and contained within the feasible set of the Alternating Current Optimal Power Flow (AC-OPF) problem. In the context of wholesale electricity markets, pricing structures are analyzed along with their dependencies on the selected input features to the NN surrogate model. An important insight identifies a trade-off between the accuracy of the NN surrogate model and sensible locational pricing structures. NN surrogate models for frequency stability are validated by ensuring the resulting TSC-OPF solution is stable over randomly generated load samples using a small Hawaii test case. The proposed TSC-OPF problem is shown to significantly enhance frequency stability at low computational cost and low financial cost to the system. For certain selections of NN inputs, the TSC-OPF problem is able to stabilize all load scenarios for which the solution to the AC-OPF problem resulted in instability.

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arXiv2024

Formulations and scalability of neural network surrogates in nonlinear optimization problems

We compare full-space, reduced-space, and gray-box formulations for representing trained neural networks in nonlinear constrained optimization problems. We test these formulations on a transient stability-constrained, security-constrained alternating current optimal power flow (SCOPF) problem where the transient stability criteria are represented by a trained neural network surrogate. Optimization problems are implemented in JuMP and trained neural networks are embedded using a new Julia package: MathOptAI.jl. To study the bottlenecks of the three formulations, we use neural networks with up to 590 million trained parameters. The full-space formulation is bottlenecked by the linear solver used by the optimization algorithm, while the reduced-space formulation is bottlenecked by the algebraic modeling environment and derivative computations. The gray-box formulation is the most scalable and is capable of solving with the largest neural networks tested. It is bottlenecked by evaluation of the neural network's outputs and their derivatives, which may be accelerated with a graphics processing unit (GPU). Leveraging the gray-box formulation and GPU acceleration, we solve our test problem with our largest neural network surrogate in 2.5$\times$ the time required for a simpler SCOPF problem without the stability constraint.

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arXiv2024

A Parametric, Second-Order Cone Representable Model of Fairness for Decision-Making Problems

The article develops a parametric model of fairness called "$\varepsilon$-fairness" that can be represented using a single second-order cone constraint and incorporated into existing decision-making problem formulations without impacting the complexity of solution techniques. We develop the model from the fundamental result of finite-dimensional norm equivalence in linear algebra and show that this model has a closed-form relationship to an existing metric for measuring fairness widely used in the literature. Finally, a simple case study on the optimal operation of a damaged power transmission network illustrates its effectiveness.

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arXiv2024

FORWARD: Feasibility Oriented Random-Walk Inspired Algorithm for Radial Reconfiguration in Distribution Networks

We consider an optimal flow distribution problem in which the goal is to find a radial configuration that minimizes resistance-induced quadratic distribution costs while ensuring delivery of inputs from multiple sources to all sinks to meet their demands. This problem has critical applications in various distribution systems, such as electricity, where efficient energy flow is crucial for both economic and environmental reasons. Due to its complexity, finding an optimal solution is computationally challenging and NP-hard. In this paper, we propose a novel algorithm called FORWARD, which leverages graph theory to efficiently identify feasible configurations in polynomial time. By drawing parallels with random walk processes on electricity networks, our method simplifies the search space, significantly reducing computational effort while maintaining performance. The FORWARD algorithm employs a combination of network preprocessing, intelligent partitioning, and strategic sampling to construct radial configurations that meet flow requirements, finding a feasible solution in polynomial time. Numerical experiments demonstrate the effectiveness of our approach, highlighting its potential for real-world applications in optimizing distribution networks.

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arXiv2024

A Review of the GIC Blocker Placement Problem

Space weather poses a tremendous threat to power systems: geomagnetic disturbances could result in widespread disruptions and long-duration blackouts, including severe damage to system components. To mitigate their impacts, a handful of strategies exist, with the most promising being the deployment of transformer neutral blocking devices. The high cost of these devices, however, precludes their installation at all substations; this motivates the development of effective solutions for the cost-effective placement of such devices. While the current state-of-the-art in blocker placement methods is insufficient to be applied to real-sized power grids, ongoing research continues to increase the size of networks for which the placement problem remains tractable. Along these lines, the contributions of this paper are two fold: first, a comprehensive overview of the current state-of-the-art in blocker placement methods is provided; and second, a complete optimization formulation - implemented and benchmarked in an open-source software - for the blocker placement problem is presented.

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arXiv2024

Physics-Informed Heterogeneous Graph Neural Networks for DC Blocker Placement

The threat of geomagnetic disturbances (GMDs) to the reliable operation of the bulk energy system has spurred the development of effective strategies for mitigating their impacts. One such approach involves placing transformer neutral blocking devices, which interrupt the path of geomagnetically induced currents (GICs) to limit their impact. The high cost of these devices and the sparsity of transformers that experience high GICs during GMD events, however, calls for a sparse placement strategy that involves high computational cost. To address this challenge, we developed a physics-informed heterogeneous graph neural network (PIHGNN) for solving the graph-based dc-blocker placement problem. Our approach combines a heterogeneous graph neural network (HGNN) with a physics-informed neural network (PINN) to capture the diverse types of nodes and edges in ac/dc networks and incorporates the physical laws of the power grid. We train the PIHGNN model using a surrogate power flow model and validate it using case studies. Results demonstrate that PIHGNN can effectively and efficiently support the deployment of GIC dc-current blockers, ensuring the continued supply of electricity to meet societal demands. Our approach has the potential to contribute to the development of more reliable and resilient power grids capable of withstanding the growing threat that GMDs pose.

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arXiv2024

Exact and Heuristic Approaches for the Stochastic N-k Interdiction in Power Grids

The article introduces the stochastic N-k interdiction problem for power grid operations and planning that aims to identify a subset of k components (out of N components) that maximizes the expected damage, measured in terms of load shed. Uncertainty is modeled through a fixed set of outage scenarios, where each scenario represents a subset of components removed from the grid. We formulate the stochastic N-k interdiction problem as a bi-level optimization problem and propose two algorithmic solutions. The first approach reformulates the bi-level stochastic optimization problem to a single level, mixed-integer linear program (MILP) by dualizing the inner problem and solving the resulting problem directly using a MILP solver to global optimality. The second is a heuristic cutting-plane approach, which is exact under certain assumptions. We compare these approaches in terms of computation time and solution quality using the IEEE-Reliability Test System and present avenues for future research.

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arXiv2023

Joint Expansion Planning of Power and Water Distribution Networks

This research explores the joint expansion planning of power and water distribution networks, which exhibit interdependence at various levels. We specifically focus on the dependency arising from the power consumption of pumps and develop models to seamlessly integrate new components into existing networks. Subsequently, we formulate the joint expansion planning as a Mixed Integer Nonlinear Program (MINLP). Through the application of this MINLP to a small-scale test network, we demonstrate the advantages of combining expansion planning, including cost savings and reduced redundancy, in comparison to independently expanding power and water distribution networks

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arXiv2023

Heuristic Algorithms for Placing Geomagnetically Induced Current Blocking Devices

We propose a new heuristic approach for solving the challenge of determining optimal placements for geomagnetically induced current blocking devices on electrical grids. Traditionally, these determinations are approached by formulating the problem as mixed-integer nonlinear programming models and solving them using optimization solvers based on the spatial branch-and-bound algorithm.However, computing an optimal solution using the solvers often demands substantial computational time due to their inability to leverage the inherent problem structure. Therefore, in this work we propose a new heuristic approach based on a three-block alternating direction method of multipliers algorithm, and we compare it with an existing stochastic learning algorithm. Both heuristics exploit the structure of the problem of interest. We test these heuristic approaches through extensive numerical experiments conducted on the EPRI-21 and UIUC-150 test systems. The outcomes showcase the superior performance of our methodologies in terms of both solution quality and computational speed when compared with conventional solvers.

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arXiv2023

Fairly Extreme: Minimizing Outages Equitably

This paper focuses on the problem of minimizing the outages due to extreme events on the power grid equitably among all customers of the grid. The paper presents two ways of incorporating fairness into the existing formulations that seek to minimize the total outage in the power grid. The first method is motivated by existing literature on incorporating fairness in optimization problems and this is done by modifying the problem's objective function. The second method introduces a novel notion of fairness, termed $\varepsilon$-fairness, that can be incorporated into existing problem formulations through a single second-order cone constraint. Both these methods are very general and can be used to incorporate fairness in existing planning and operational optimization problems in the power grid and beyond. Extensive computational case studies that examine the effectiveness of both these methods to characterize fairness is presented followed by conclusions and ways forward.

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arXiv2023

Relaxations of the Steady Optimal Gas Flow Problem for a Non-Ideal Gas

Natural gas ranks second in consumption among primary energy sources in the United States. The majority of production sites are in remote locations, hence natural gas needs to be transported through a pipeline network equipped with a variety of physical components such as compressors, valves, etc. Thus, from the point of view of both economics and reliability, it is desirable to achieve optimal transportation of natural gas using these pipeline networks. The physics that governs the flow of natural gas through various components in a pipeline network is governed by nonlinear and non-convex equality and inequality constraints and the most general steady-flow operations problem takes the form of a Mixed Integer Nonlinear Program. In this paper, we consider one example of steady-flow operations -- the Optimal Gas Flow (OGF) problem for a natural gas pipeline network that minimizes the production cost subject to the physics of steady-flow of natural gas. The ability to quickly determine global optimal solution and a lower bound to the objective value of the OGF for different demand profiles plays a key role in efficient day-to-day operations. One strategy to accomplish this relies on tight relaxations to the nonlinear constraints of the OGF. Currently, many nonlinear constraints that arise due to modeling the non-ideal equation of state either do not have relaxations or have relaxations that scale poorly for realistic network sizes. In this work, we combine recent advancements in the development of polyhedral relaxations for univariate functions to obtain tight relaxations that can be solved within a few seconds on a standard laptop. We demonstrate the quality of these relaxations through extensive numerical experiments on very large scale test networks available in the literature and find that the proposed relaxation is able to prove optimality in 92% of the instances.

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arXiv2022

Modeling and Rapid Prototyping of Integrated Transmission-Distribution OPF Formulations with PowerModelsITD.jl

Conventional electric power systems are composed of different unidirectional power flow stages of generation, transmission, and distribution, managed independently by transmission system and distribution system operators. However, as distribution systems increase in complexity due to the integration of distributed energy resources, coordination between transmission and distribution networks will be imperative for the optimal operation of the power grid. However, coupling models and formulations between transmission and distribution is non-trivial, in particular due to the common practice of modeling transmission systems as single-phase, and distribution systems as multi-conductor phase-unbalanced. To enable the rapid prototyping of power flow formulations, in particular in the modeling of the boundary conditions between these two seemingly incompatible data models, we introduce PowerModelsITD.jl, a free, open-source toolkit written in Julia for integrated transmission-distribution (ITD) optimization that leverages mature optimization libraries from the InfrastructureModels.jl-ecosystem. The primary objective of the proposed framework is to provide baseline implementations of steady-state ITD optimization problems, while providing a common platform for the evaluation of emerging formulations and optimization problems. In this work, we introduce the nonlinear formulations currently supported in PowerModelsITD.jl, which include AC-polar, AC-rectangular, current-voltage, and a linear network transportation model. Results are validated using combinations of IEEE transmission and distribution networks.

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arXiv2022

Polyhedral Relaxations for Optimal Pump Scheduling of Potable Water Distribution Networks

The classic pump scheduling or Optimal Water Flow (OWF) problem for water distribution networks (WDNs) minimizes the cost of power consumption for a given WDN over a fixed time horizon. In its exact form, the OWF is a computationally challenging mixed-integer nonlinear program (MINLP). It is complicated by nonlinear equality constraints that model network physics, discrete variables that model operational controls, and intertemporal constraints that model changes to storage devices. To address the computational challenges of the OWF, this paper develops tight polyhedral relaxations of the original MINLP, derives novel valid inequalities (or cuts) using duality theory, and implements novel optimization-based bound tightening and cut generation procedures. The efficacy of each new method is rigorously evaluated by measuring empirical improvements in OWF primal and dual bounds over forty-five literature instances. The evaluation suggests that our relaxation improvements, model strengthening techniques, and a thoughtfully selected polyhedral relaxation partitioning scheme can substantially improve OWF primal and dual bounds, especially when compared with similar relaxation-based techniques that do not leverage these new methods.

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