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35 records · Page 2

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DOE OSTI2018

Multi-Timescale Three-Phase Unbalanced Distribution System Operation With Variable Renewable Generations

This paper proposes a multi-timescale operational approach for a three-phase unbalanced distribution system, which contains hourly scheduling at substation level and minutes power flow operation at feeder level. In the substation scheduling model, the objective is to minimize the system cost with variable renewable generations. The given error distribution model of renewable generation is formulated as a chance constraint, and derived into a deterministic form by Gaussian mixture model (GMM) with genetic algorithm-based expectation-maximization (GAEM). In the feeder scheduling model, the system cost is further reduced with the optimal power flow (OPF) at a timescale of minutes. Considering the nonconvexity of the three-phase unbalanced OPF problem in distribution systems, the semidefinite programming (SDP) is used to relax the problem into a convex problem, and a distributed computation approach is built based on alternating direction method of multiplier (ADMM). The IEEE 123-bus distribution system, DU (University of Denver) campus distribution system, and the IEEE 8500-bus distribution system are used as the test bench for the proposed approach. And the numerical results demonstrate the effectiveness and validity of the proposed method.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2019

Optimal Operation of Soft Open Points in Active Distribution Networks Under Three-Phase Unbalanced Conditions

The asymmetric integration of distributed generators (DGs) exacerbates the three-phase unbalanced condition in distribution systems. The serious unbalanced operation causes inefficient utilization of network assets and security risks in the system. Soft open point (SOP) is a flexible power electronic device which can achieve accurate active and reactive power flow control to balance the power flow among phases. This paper proposes an SOPs-based operation strategy for unbalanced active distribution networks (ADNs). By regulating the operation of SOPs, the strategy can reduce power losses and simultaneously mitigate the three-phase unbalance of the upper-level grid. Semidefinite programming (SDP) relaxation is advocated to convert the original non-convex, nonlinear optimization model into an SDP formulation, which can be efficiently solved to meet the requirement of rapid adjustment. Case studies are conducted on the modified IEEE 33-node and IEEE 123-node distribution system to verify the effectiveness and efficiency of the proposed strategy.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2020

Toward a scalable robust security-constrained optimal power flow using a proximal projection bundle method

Robust security-constrained optimal power flow (rSCOPF) aims to find the worst-case contingencies of alternating current optimal power flow (ACOPF) in power systems. With the rise of GPU architectures on the upcoming supercomputer architectures, optimization algorithms that rely on sparse linear algebra and indefinite linear systems are becoming increasingly hard to solve efficiently (e.g. interior-point method). To address this we revisit a maximin optimization formulation of the rSCOPF and the single-level mixed-integer semidefinite programming (MISDP) reformulation, which is obtained by taking the Lagrangian relaxation of the inner minimization ACOPF problem. In this paper, we focus on the development of a proximal projection bundle method (PPBM) for solving continuous relaxation node subproblems of the MISDP problem, based primarily on the well-known alternating direction method of multipliers. Cutting planes reminiscent of bundle method ideas are also applied in coordination with updates of the proximal parameter. The cutting-plane method can generate a large number of linear inequalities, leading to a large scale but decomposable quadratic programming (QP) subproblem that is amenable to GPUs. We present the numerical results on the IEEE 30, 57, 118, and 300-bus systems by using our PBMM method. We discuss the main computational bottleneck of our method, which is the time taken to solve each iteration of a QP subproblem instance of the PPBM, and how GPU architectures can accelerate this solution process.

bundle method↗

DOE OSTI2019

Coordinating Multiple Sources for Service Restoration to Enhance Resilience of Distribution Systems

When a major outage occurs on a distribution system due to extreme events, microgrids, distributed generators, storage devices, and other local resources can be used to restore critical loads and enhance resiliency. This paper proposes a decision-making method to determine the optimal restoration strategy coordinating multiple sources to serve critical loads after a major outage. The critical load restoration problem is formulated as a mixed -integer semidefinite program. The objective is maximizing the number of loads restored, weighted by their priority. The unbalanced three-phase power flow constraint and operational constraints are considered. An iterative algorithm is proposed to deal with integer variables and can attain the global optimum of the critical load restoration problem by solving a few semidefinite programs under mild conditions. The effectiveness of the proposed models and algorithm is validated by numerical simulation with the modified IEEE 13-node test feeder and the modified IEEE123-node test feeder under a large number of scenarios. The results indicate that the optimal restoration strategy can be determined efficiently in most scenarios

Wang, Ying↗

DOE OSTI2021

A Bilevel Approach for Identifying the Worst Contingencies for Nonconvex Alternating Current Power Systems

We address the bilevel optimization problem of identifying the most critical attacks to an alternating current (AC) power flow network. The upper-level binary maximization problem consists of choosing an attack that is treated as a parameter in the lower-level defender minimization problem. Instances of the lower-level global minimization problem by themselves are NP-hard due to the nonconvex AC power flow constraints, and bilevel solution approaches commonly apply a convex relaxation or approximation to allow for tractable bilevel reformulations at the cost of underestimating some power system vulnerabilities. Our main contribution is to provide an alternative branch-and-bound algorithm whose upper bounding mechanism (in a maximization context) is based on a reformulation that avoids relaxation of the AC power flow constraints in the lower-level defender problem. Lower bounding is provided with semidefinite programming (SDP) relaxed solutions to the lower-level problem. We establish finite termination with guarantees of either a globally optimal solution to the original bilevel problem, or a globally optimal solution to the SDP-relaxed bilevel problem which is included in a vetted list of upper-level attack solutions, at least one of which is a globally optimal solution to the bilevel problem. We demonstrate through computational experiments applied to IEEE case instances both the relevance of our contribution, and the effectiveness of our contributed algorithm for identifying power system vulnerabilities without resorting to convex relaxations of the lower-level problem. We conclude with a discussion of future extensions and improvements.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Multistage distributionally robust mixed-integer programming with decision-dependent moment-based ambiguity sets

We study multistage distributionally robust mixed-integer programs under endogenous uncertainty, where the probability distribution of stage-wise uncertainty depends on the decisions made in previous stages. We first consider two ambiguity sets defined by decision-dependent bounds on the first and second moments of uncertain parameters and by mean and covariance matrix that exactly match decision-dependent empirical ones, respectively. For both sets, we show that the subproblem in each stage can be recast as a mixed-integer linear program (MILP). Moreover, we extend the general moment-based ambiguity set in to the multistage decision-dependent setting, and derive mixed-integer semidefinite programming (MISDP) reformulations of stage-wise subproblems. We develop methods for attaining lower and upper bounds of the optimal objective value of the multistage MISDPs, and approximate them using a series of MILPs. We deploy the Stochastic Dual Dynamic integer Programming (SDDiP) method for solving the problem under the three ambiguity sets with risk-neutral or risk-averse objective functions, and conduct numerical studies on multistage facility-location instances having diverse sizes under different parameter and uncertainty settings. Furthermore, our results show that the SDDiP quickly finds optimal solutions for moderate-sized instances under the first two ambiguity sets, and also finds good approximate bounds for the multistage MISDPs derived under the third ambiguity set. We also demonstrate the efficacy of incorporating decision-dependent distributional ambiguity in multistage decision-making processes.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Semidefinite programs at finite fermion density

Not Available

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2026

Correlated purification for restoring 𝑁-representability in quantum simulation

Experimentally measured reduced density matrices (RDMs) often violate constraints that ensure they represent N-electron states—known as N-representability conditions—because of statistical and hardware noise. In this work, we present a correlated purification framework based on semidefinite programming to restore the accuracy of a noisy, unphysical two-electron RDM (2-RDM). The method performs a bi-objective optimization that minimizes both the many-electron energy and the nuclear norm of the correction to the measured 2-RDM. The nuclear norm, often employed in matrix completion, promotes low-rank corrections, while the energy term acts as a regularization term that can improve the purity of the ground state. While the method is particularly effective for ground states, it can also be applied to excited and nonstationary states by decreasing the weight of the energy relative to the error norm. In an application to fermionic shadow tomography of large hydrogen chains, correlated purification yields substantial reductions in both energy and 2-RDM error, achieving chemical accuracy across dissociation curves. This framework provides a robust strategy for tomography in many-body quantum simulations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Certifiably Correct Range-Aided SLAM

We present the first algorithm capable of efficiently computing certifiably optimal solutions to range-aided simultaneous localization and mapping (RA-SLAM) problems. Robotic navigation systems are increasingly incorporating point-to-point ranging sensors, leading state estimation which takes the form of RA-SLAM. However, the RA-SLAM problem is more difficult to solve than traditional pose-graph SLAM; ranging sensor models introduce additional non-convexity, unlike pose-pose or pose-landmark measurements, a single range measurement does not uniquely determine the relative transform between the involved sensors, and RA-SLAM inference is highly sensitive to initial estimates. Our approach relaxes the RA-SLAM problem to a semidefinite program (SDP), which we show how to solve efficiently using the Riemannian staircase methodology. The solution of this SDP provides a high-quality initialization for our original RA-SLAM problem, which is subsequently refined via local optimization, as well as a lower-bound on the RA-SLAM problem's optimal value. Our algorithm, named certifiably correct RA-SLAM (CORA), applies to problems comprised of arbitrary pose-pose, pose-landmark, and ranging measurements. Evaluation on simulated and real-world marine examples shows that our algorithm frequently produces certifiably optimal RA-SLAM solutions; moreover, even suboptimal estimates are typically within 1-2\% of the optimal value.

Papalia, Alan↗

DOE OSTI2019

Distribution System State Estimation: A Semidefinite Programming Approach

Distribution system state estimation (DSSE) is one of the vital components in the next-generation distribution management system which allows operators to monitor the entire system's operating conditions. Due to the lack of real-time measurements, DSSE has to process measurements whose quality varies significantly across different sources, which causes a convergence issue to the Gauss-Newton solver. In this paper, a semidefinite programming (SDP) framework is proposed to reformulate the DSSE problem into a rank- constrained SDP problem. One challenge of this technique is the nonconvex rank-one constraint, which is generally relaxed. However, the relaxed SDP-DSSE problem cannot guarantee a rank-one solution and hence loses optimality. Therefore, we propose two solution approaches to obtain rank-one solutions for the SDPDSSE problem: the rank reduction approach and the convex iteration approach. The model and the effectiveness of the proposed solution approaches are numerically demonstrated on the IEEE 13-bus, 34-bus, and 123-bus distribution systems.

convex iteration↗

DOE OSTI2019

A moment approach for entropy solutions to nonlinear hyperbolic PDEs

We propose to solve hyperbolic partial differential equations(PDEs) with polynomial flux using a convex optimization strategy. This approach is based on a very weak notion of solution of the nonlinear equation, namely the measure-valued (mv) solution, satisfying a linear equation in the space of Borel measures. The aim of this paper is, first, to provide the conditions that ensure the equivalence between the two formulations and, second, to introduce a method which approximates the infinite-dimensional linear problem by a hierarchy of convex, finite-dimensional, semidefinite programming problems. This result is then illustrated on the celebrated Burgers equation. Here, we also compare our results with an existing numerical scheme, namely the Godunov scheme.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Convex Optimization for Nonequilibrium Steady States on a Hybrid Quantum Processor

Finding the transient and steady state properties of open quantum systems is a central problem in various fields of quantum technologies. Here, in this work, we present a quantum-assisted algorithm to determine the steady states of open system dynamics. By reformulating the problem of finding the fixed point of Lindblad dynamics as a feasibility semidefinite program, we bypass several well-known issues with variational quantum approaches to solving for steady states. We demonstrate that our hybrid approach allows us to estimate the steady states of higher dimensional open quantum systems and discuss how our method can find multiple steady states for systems with symmetries.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Matrix Completion Using Alternating Minimization for Distribution System State Estimation

This paper examines the problem of state estimation in power distribution systems under low-observability conditions. The recently proposed constrained matrix completion method which combines the standard matrix completion method and power flow constraints has been shown to be effective in estimating voltage phasors under low-observability conditions using single-snapshot information. However, the method requires solving a semidefinite programming (SDP) problem, which becomes computationally infeasible for large systems and if multiple-snapshot (time-series) information is used. This paper proposes an efficient algorithm to solve the constrained matrix completion problem with time-series data. This algorithm is based on reformulating the matrix completion problem as a bilinear (non-convex) optimization problem, and applying the alternating minimization algorithm to solve this problem. This paper proves the summable convergence of the proposed algorithm, and demonstrates its efficacy and scalability via IEEE 123-bus system and a real utility feeder system. This paper also explores the value of adding more data from the history in terms of computation time and estimation accuracy.

41 EE - Solar Energy Technologies Office (EE-4S)↗

DOE OSTI2020

Matrix Completion Using Alternating Minimization for Distribution System State Estimation: Preprint

41 EE - Solar Energy Technologies Office (EE-4S)↗

DOE OSTI2023

Semidefinite programming algorithm for the quantum mechanical bootstrap

Here, we present a semidefinite programming algorithm to find eigenvalues of Schrödinger operators within the bootstrap approach to quantum mechanics. The bootstrap approach involves two ingredients: a nonlinear set of constraints on the variables (expectation values of operators in an energy eigenstate), plus positivity constraints (unitarity) that need to be satisfied. By fixing the energy we linearize all the constraints and show that the feasibility problem can be presented as an optimization problem for the variables that are not fixed by the constraints and one additional slack variable that measures the failure of positivity. To illustrate the method we are able to obtain high-precision, sharp bounds on eigenenergies for arbitrary confining polynomial potentials in one dimension.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Thermal bootstrap of matrix quantum mechanics

We implement a bootstrap method that combines stationary state conditions, thermal inequalities, and semidefinite relaxations of matrix logarithm in the ungauged one-matrix quantum mechanics, at finite rank N as well as in the large N limit, and determine finite temperature observables that interpolate between available analytic results in the low and high temperature limits respectively. We also obtain bootstrap bounds on thermal phase transition as well as preliminary results in the ungauged two-matrix quantum mechanics.

1/N Expansion↗

DOE OSTI2021

Computing partition functions in the one-clean-qubit model

We present a method to approximate partition functions of quantum systems using mixed-state quantum computation. For positive-semidefinite Hamiltonians, our method has an expected running-time that is almost linear in [M/( ε rel Z )] 2 , where M is the dimension of the quantum system, Z is the partition function, and ε rel is the relative precision. It is based on approximations of the exponential operator as linear combinations of certain operators related to block-encoding of Hamiltonians or Hamiltonian evolutions. The trace of each operator is estimated using a standard algorithm in the one-clean-qubit model. For large values of Z , our method may run faster than exact classical methods, whose complexities are polynomial in M . We also prove that a version of the partition function estimation problem within additive error is complete for the so-called DQC1 complexity class, suggesting that our method provides a superpolynomial speedup for certain parameter values. Overall, to attain a desired relative precision, we develop a classical procedure based on a sequence of approximations within predetermined additive errors that may be of independent interest.

97 MATHEMATICS AND COMPUTING↗