arXiv subjects

Xu Andy Sun

Publications and source records attributed to Xu Andy Sun.

At least 19 recordsLinked to original sources

arXiv2026

Modeling Adversarial Wildfires for Power Grid Disruption

Electric power infrastructure faces increasing risk of damage and disruption due to wildfire. Operators of power grids in wildfire-prone regions must consider the potential impacts of unpredictable fires. However, traditional wildfire models do not effectively describe worst-case, or even high-impact, fire behavior. To address this issue, we propose a mixed-integer conic program to characterize an adversarial wildfire that targets infrastructure while respecting realistic fire spread dynamics. We design a wind-assisted fire spread set based on the Rothermel fire spread model and propose principled convex relaxations of this set, including a new relaxation of the inner product over Euclidean balls. We present test cases derived from the recent Park, Eaton, and Palisades fires in California and solve models to identify the minimum time-to-outage of multiple-element contingencies and the maximum load shed associated with a sequence of element outages caused by a realistic wildfire. We use the minimum time-to-outage values to screen contingencies and construct security-constrained optimal power flow models that promote operational resilience against wildfire.

math.OC↗

arXiv2025

On Distributionally Robust Multistage Convex Optimization: Data-driven Models and Performance

This paper presents a novel algorithmic study with extensive numerical experiments of distributionally robust multistage convex optimization (DR-MCO). Following the previous work on dual dynamic programming (DDP) algorithmic framework for DR-MCO, we focus on data-driven DR-MCO models with Wasserstein ambiguity sets that allow probability measures with infinite supports. These data-driven Wasserstein DR-MCO models have out-of-sample performance guarantees and adjustable in-sample conservatism. Then by exploiting additional concavity or convexity in the uncertain cost functions, we design exact single stage subproblem oracle (SSSO) implementations that ensure the convergence of DDP algorithms. We test the data-driven Wasserstein DR-MCO models against multistage robust convex optimization (MRCO), risk-neutral and risk-averse multistage stochastic convex optimization (MSCO) models on multi-commodity inventory problems and hydro-thermal power planning problems. The results show that our DR-MCO models could outperform MRCO and MSCO models when the data size is small.

math.OC↗

arXiv2025

Bus Fleet Electrification Planning Through Logic-Based Benders Decomposition and Restriction Heuristics

To meet sustainability goals and regulatory requirements, transit agencies must plan partial and full transitions to electric bus fleets. This paper presents a comprehensive and computationally efficient multi-period optimization framework integrating the key decisions required to support such electrification initiatives. Our model is formulated as a two-stage integer program with integer subproblems. These two levels optimize, respectively, yearly fleet sizing and charging infrastructure investments, and hourly vehicle scheduling and charging operations. We develop an exact logic-based Benders decomposition algorithm enhanced by several acceleration techniques, including preprocessing, master problem strengthening, and efficient cut separation methods applied to different relaxations of the operational problem. These accelerations achieve speedups of up to three orders of magnitude and provide new theoretical and practical insights into Benders cut selection. We also propose a heuristic tailored for long-term, citywide electrification planning. This approach imposes and progressively relaxes additional scheduling constraints derived from auxiliary problems. It delivers high-quality solutions with optimality gaps below 1% for instances an order of magnitude larger than those considered in prior work. We illustrate our model using real data from the Chicago public bus system, providing managerial insights into optimal investment and operational policies.

math.OC↗

arXiv2025

Alternating Methods for Large-Scale AC Optimal Power Flow with Unit Commitment

Security-constrained unit commitment with alternating current optimal power flow (SCUC-ACOPF) is a central problem in power grid operations that optimizes commitment and dispatch of generators under a physically accurate power transmission model while encouraging robustness against component failures. SCUC-ACOPF requires solving large-scale problems that involve multiple time periods and networks with thousands of buses within strict time limits. In this work, we study a detailed SCUC-ACOPF model with a rich set of features of modern power grids, including price-sensitive load, reserve products, transformer controls, and energy-limited devices. We propose a decomposition scheme and a penalty alternating direction method to find high-quality solutions to this model. Our methodology leverages spatial and temporal decomposition, separating the problem into a set of mixed-integer linear programs for each bus and a set of continuous nonlinear programs for each time period. To improve the performance of the algorithm, we introduce a variety of heuristics, including restrictions of temporal linking constraints, a second-order cone relaxation, and a contingency screening algorithm. We quantify the quality of feasible solutions through a dual bound from a convex second-order cone program. To evaluate our algorithm, we use large-scale test cases from Challenge 3 of the U.S. Department of Energy's Grid Optimization Competition that resemble real power grid data under a variety of operating conditions and decision horizons. The experiments yield feasible solutions with an average optimality gap of 1.33%, demonstrating that this approach generates near-optimal solutions within stringent time limits.

math.OC↗

arXiv2025

On the Acceleration of Proximal Bundle Methods

The proximal bundle method (PBM) is a fundamental and computationally effective algorithm for solving nonsmooth optimization problems. In this paper, we present the first variant of the PBM for smooth objectives, achieving an accelerated convergence rate of $O(ε^{-1/2}\log(1/ε))$, where $ε$ is the desired accuracy. Our approach addresses an open question regarding the convergence guarantee of proximal bundle type methods, which was previously posed in two recent papers. We interpret the PBM as a proximal point algorithm and base our proposed algorithm on an accelerated inexact proximal point scheme. Our variant introduces a novel null step test and oracle while maintaining the core structure of the original algorithm. The newly proposed oracle substitutes the traditional cutting planes with a smooth lower approximation of the true function. We show that this smooth interpolating lower model can be computed as a convex quadratic program. We also examine a second setting where Nesterov acceleration can be effectively applied, specifically when the objective is the sum of a smooth function and a piecewise linear one.

math.OC↗

arXiv2024

An Absolute-Error Proximal Bundle Method through the Lens of Frank-Wolf

The proximal bundle method (PBM) is a fundamental and computationally effective algorithm for solving optimization problems with nonsmooth components. In this paper, we conduct a theoretical investigation of a modified proximal bundle method, which we call the Modified Proximal Bundle with Fixed Absolute Accuracy (MPB-FA). MPB-FA modifies the serious-step test of the PBM serious-step test. In MPB-FA, it is based on an absolute accuracy criterion, and the accuracy is fixed over iterations, while the standard PBM uses a relative accuracy in the serious-step test, which changes with iterations. Also, similarly to multiple PBM analyses, the proximal parameter in MPB-FA is also fixed over iterations, while it is permitted to change in the standard PBM. These modifications allow us to build the first link between a proximal bundle method and a Frank-Wolfe algorithm on the Moreau envelope of the dual problem. In light of this correspondence, we first extend the linear convergence of Kelley's method on the sum of a smooth strongly convex function and a convex piecewise linear function from the positive homogeneous to the general case. Building on this result, we propose a novel complexity analysis of MPB-FA when the objective is the sum of a smooth and a piecewise function and derive an $\mathcal{O}(\epsilon^{-8/9})$ iteration complexity, improving upon the best known $\mathcal{O}(\epsilon^{-2})$ guarantee on a related variant of PBM. It is worth-noting that the best known complexity bound for the classical PBM in the general case is $\mathcal{O}(\epsilon^{-3})$ and $\mathcal{O}(\epsilon^{-2})$ when the proximal parameter is fixed. Our approach also reveals new insights on bundle management and empirical behavior of the proximal bundle methods.

math.OC↗

arXiv2024

Nonlinear conjugate gradient methods: worst-case convergence rates via computer-assisted analyses

We propose a computer-assisted approach to the analysis of the worst-case convergence of nonlinear conjugate gradient methods (NCGMs). Those methods are known for their generally good empirical performances for large-scale optimization, while having relatively incomplete analyses. Using our computer-assisted approach, we establish novel complexity bounds for the Polak-Ribière-Polyak (PRP) and the Fletcher-Reeves (FR) NCGMs for smooth strongly convex minimization. In particular, we construct mathematical proofs that establish the first non-asymptotic convergence bound for FR (which is historically the first developed NCGM), and a much improved non-asymptotic convergence bound for PRP. Additionally, we provide simple adversarial examples on which these methods do not perform better than gradient descent with exact line search, leaving very little room for improvements on the same class of problems.

math.OC↗

arXiv2024

BattOpt: Optimal Facility Planning for Electric Vehicle Battery Recycling

The electric vehicle (EV) battery supply chain will face challenges in sourcing scarce and expensive minerals required for manufacturing and in disposing of hazardous retired batteries. Integrating recycling technology into the supply chain has the potential to alleviate these issues; however, players in the battery market must design investment plans for recycling facilities. In this paper, we propose a multistage stochastic optimization model for computing minimum cost recycling capacity decisions, in which retired batteries are recycled and recovered materials are used to manufacture new batteries. We transform EverBatt, a leading evaluation framework for battery recycling cost and environmental impact, into a prescriptive decision-making tool for determining optimal investment strategies. Our model is a separable concave minimization subject to linear constraints, a class for which we describe a finitely convergent global optimization algorithm based on piecewise linear approximation that solves up to 14x faster than comparable algorithms. We propose an equivalent reformulation of the model that eliminates variables by introducing integrality constraints, drastically reducing solve times. To produce a set of operational scenarios, we generate time-series projections for new battery demand, retired battery supply, and material costs, leveraging state-of-the-art econometric models. We apply our model to compute optimal investment plans and quantify the impact of optimal decision-making: optimal investment can reduce battery manufacturing costs by 22% and environmental impacts by 26%. We measure the operational benefits of different recycling technologies and demonstrate how high uncertainty motivates greater investment in recycling capacity. Finally, we analyze the impact of policy instruments to reveal key insights on the efficacy of capacity grants and production credits.

math.OC↗

arXiv2023

An ADMM-based Distributed Optimization Method for Solving Security-Constrained AC Optimal Power Flow

In this paper, we study efficient and robust computational methods for solving the security-constrained alternating current optimal power flow (SC-ACOPF) problem, a two-stage nonlinear optimization problem with disjunctive constraints, that is central to the operation of electric power grids. The first-stage problem in SC-ACOPF determines the operation of the power grid in normal condition, while the second-stage problem responds to various contingencies of losing generators, transmission lines, and transformers. The two stages are coupled through disjunctive constraints, which model generators' active and reactive power output changes responding to system-wide active power imbalance and voltage deviations after contingencies. Real-world SC-ACOPF problems may involve power grids with more than 30k buses and 22k contingencies and need to be solved within 10-45 minutes to get a base case solution with high feasibility and reasonably good generation cost. We develop a comprehensive algorithmic framework to solve SC-ACOPF that meets the challenge of speed, solution quality, and computation robustness. In particular, we develop a smoothing technique to approximate disjunctive constraints into a smooth structure which can be handled by interior-point solvers; we design a distributed optimization algorithm to efficiently generate first-stage solutions; we propose a screening procedure to prioritize contingencies; and finally, we develop a reliable and parallel architecture that integrates all algorithmic components. Extensive tests on industry-scale systems demonstrate the superior performance of the proposed algorithms.

math.OC↗

arXiv2022

Algorithms for Difference-of-Convex (DC) Programs Based on Difference-of-Moreau-Envelopes Smoothing

In this paper we consider minimization of a difference-of-convex (DC) function with and without linear constraints. We first study a smooth approximation of a generic DC function, termed difference-of-Moreau-envelopes (DME) smoothing, where both components of the DC function are replaced by their respective Moreau envelopes. The resulting smooth approximation is shown to be Lipschitz differentiable, capture stationary points, local, and global minima of the original DC function, and enjoy some growth conditions, such as level-boundedness and coercivity, for broad classes of DC functions. We then develop four algorithms for solving DC programs with and without linear constraints based on the DME smoothing. In particular, for a smoothed DC program without linear constraints, we show that the classic gradient descent method as well as an inexact variant can obtain a stationary solution in the limit with a convergence rate of $\mathcal{O}(K^{-1/2})$, where $K$ is the number of proximal evaluations of both components. Furthermore, when the DC program is explicitly constrained in an affine subspace, we combine the smoothing technique with the augmented Lagrangian function and derive two variants of the augmented Lagrangian method (ALM), named LCDC-ALM and composite LCDC-ALM, focusing on different structures of the DC objective function. We show that both algorithms find an $ε$-approximate stationary solution of the original DC program in $\mathcal{O}(ε^{-2})$ iterations. Comparing to existing methods designed for linearly constrained weakly convex minimization, the proposed ALM-based algorithms can be applied to a broader class of problems, where the objective contains a nonsmooth concave component. Finally, numerical experiments are presented to demonstrate the performance of the proposed algorithms.

math.OC↗

arXiv2022

Recent Developments in Security-Constrained AC Optimal Power Flow: Overview of Challenge 1 in the ARPA-E Grid Optimization Competition

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

math.OC↗

arXiv2022

Stochastic Dual Dynamic Programming for Multistage Stochastic Mixed-Integer Nonlinear Optimization

In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a $(T+1)$-stage stochastic MINLP with $d$-dimensional state spaces, to obtain an $ε$-optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by $\mathcal{O}((\frac{2T}ε)^d)$, and is lower bounded by $\mathcal{O}((\frac{T}{4ε})^d)$ for the general case or by $\mathcal{O}((\frac{T}{8ε})^{d/2-1})$ for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends polynomially on the number of stages. We further show that the iteration complexity depends linearly on $T$, if all the state spaces are finite sets, or if we seek a $(Tε)$-optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with $T$. To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs.

math.OC↗

arXiv2021

A Two-level ADMM Algorithm for AC OPF with Global Convergence Guarantees

This paper proposes a two-level distributed algorithmic framework for solving the AC optimal power flow (OPF) problem with convergence guarantees. The presence of highly nonconvex constraints in OPF poses significant challenges to distributed algorithms based on the alternating direction method of multipliers (ADMM). In particular, convergence is not provably guaranteed for nonconvex network optimization problems like AC OPF. In order to overcome this difficulty, we propose a new distributed reformulation for AC OPF and a two-level ADMM algorithm that goes beyond the standard framework of ADMM. We establish the global convergence and iteration complexity of the proposed algorithm under mild assumptions. Extensive numerical experiments over some largest test cases from NESTA and PGLib-OPF (up to 30,000-bus systems) demonstrate advantages of the proposed algorithm over existing ADMM variants in terms of convergence, scalability, and robustness. Moreover, under appropriate parallel implementation, the proposed algorithm exhibits fast convergence comparable to or even better than the state-of-the-art centralized solver.

math.OC↗

arXiv2021

Stability of Multi-Microgrids: New Certificates, Distributed Control, and Braess's Paradox

This paper investigates the theory of resilience and stability in multi-microgrid networks. We derive new sufficient conditions to guarantee small-signal stability of multi-microgrids in both lossless and lossy networks. The new stability certificate for lossy networks only requires local information, thus leads to a fully distributed control scheme. Moreover, we study the impact of network topology, interface parameters (virtual inertia and damping), and local measurements (voltage magnitude and reactive power) on the stability of the system. The proposed stability certificate suggests the existence of Braess's Paradox in the stability of multi-microgrids, i.e. adding more connections between microgrids could worsen the multi-microgrid system stability as a whole. We also extend the presented analysis to structure-preserving network models, and provide a stability certificate as a function of original network parameters, instead of the Kron reduced network parameters. We provide a detailed numerical study of the proposed certificate, the distributed control scheme, and a coordinated control approach with line switching. The simulation shows the effectiveness of the proposed stability conditions and control schemes in a four-microgrid network, IEEE 33-bus system, and several large-scale synthetic grids.

eess.SY↗

arXiv2021

A Distributed Scheme for Stability Assessment in Large-Scale Structure-Preserving Models via Singular Perturbation

Assessing small-signal stability of power systems composed of thousands of interacting generators is a computationally challenging task. To reduce the computational burden, this paper introduces a novel condition to assess and certify small-signal stability. Using this certificate, we can see the impact of network topology and system parameters (generators' damping and inertia) on the eigenvalues of the system. The proposed certificate is derived from rigorous analysis of the classical structure-preserving swing equation model and has a physically insightful interpretation related to the generators' parameters and reactive power. To develop the certificate, we use singular perturbation techniques, and in the process, we establish the relationship between the structure-preserving model and its singular perturbation counterpart. As the proposed method is fully distributed and uses only local measurements, its computational cost does not increase with the size of the system. The effectiveness of the scheme is numerically illustrated on the WSCC system.

eess.SY↗

arXiv2020

On Distributionally Robust Multistage Convex Optimization: New Algorithms and Complexity Analysis

This paper presents an algorithmic study and complexity analysis for solving distributionally robust multistage convex optimization (DR-MCO) problems. Our main contribution is a novel nonconsecutive dual dynamic programming (NDDP) algorithm which explores different stages in an adaptive fashion. In contrast with the usual consecutive dual dynamic programming (CDDP) algorithm, we show that NDDP reduces the subproblem complexity from quadratic to linear dependency on the number of stages. Two different DR-MCO examples are also presented to show the efficiency and effectiveness of the proposed NDDP algorithm.

math.OC↗

arXiv2020

A Fast Certificate for Power System Small-Signal Stability

Swing equations are an integral part of a large class of power system dynamical models used in rotor angle stability assessment. Despite intensive studies, some fundamental properties of lossy swing equations are still not fully understood. In this paper, we develop a sufficient condition for certifying the stability of equilibrium points (EPs) of these equations, and illustrate the effects of damping, inertia, and network topology on the stability properties of such EPs. The proposed certificate is suitable for real-time monitoring and fast stability assessment, as it is purely algebraic and can be evaluated in a parallel manner. Moreover, we provide a novel approach to quantitatively measure the degree of stability in power grids using the proposed certificate. Extensive computational experiments are conducted, demonstrating the practicality and effectiveness of the proposal.

math.OC↗

arXiv2019

Solvability of Power Flow Equations Through Existence and Uniqueness of Complex Fixed Point

Variations of loading level and changes in system topological property may cause the operating point of an electric power systems to move gradually towards the verge of its transmission capability, which can lead to catastrophic outcomes such as voltage collapse blackout. From a modeling perspective, voltage collapse is closely related to the solvability of power flow equations. Determining conditions for existence and uniqueness of solution to power flow equations is one of the fundamental problems in power systems that has great theoretical and practical significance. In this paper, we provide strong sufficient condition certifying the existence and uniqueness of power flow solutions in a subset of state (voltage) space. The novel analytical approach heavily exploits the contractive properties of the fixed-point form in complex domain, which leads to much sharper analytical conditions than previous ones based primarily on analysis in the real domain. Extensive computational experiments are performed which validate the correctness and demonstrate the effectiveness of the proposed condition.

eess.SY↗