Numerical Performance of Different Formulations for Alternating Current Optimal Power Flow

In this paper, we develop an online method that leverages machine learning to obtain feasible solutions to the AC optimal power flow (OPF) problem with negligible optimality gaps on extremely fast timescales (e.g., milliseconds), bypassing solving an AC OPF altogether. This is motivated by the fact that as the power grid experiences increasing amounts of renewable power generation, controllable loads, and other inverter-interfaced devices, faster system dynamics and quicker fluctuations in the power supply are likely to occur. Currently, grid operators typically solve AC OPF every 15 minutes to determine economic generator settings while ensuring grid constraints are satisfied. Due to the computational challenges with solving this nonconvex problem, many efforts have focused on linearizing or approximating the problem in order to solve the AC OPF on faster timescales. However, many of these approximations can be fairly poor representations of the actual system state and still require solving an optimization problem, which can be time consuming for large networks. In this work, we leverage historical data to learn a mapping between the system loading and optimal generation values, enabling us to find near-optimal and feasible AC OPF solutions on extremely fast timescales without actually solving an optimization problem.

A trust-region based sequential linear programming approach for AC optimal power flow problems

Successive linear programming (SLP) is a practical approach for solving large-scale nonlinear optimization problems. Alternating current optimal power flow (ACOPF) is no exception, particularly the large size of real-world networks. However, in order to achieve tractability, it is essential to tune the SLP algorithm presented in the literature. This paper presents a modified SLP algorithm to solve the ACOPF problem, specified by the U.S. Department of Energy’s (DOE) Grid Optimization (GO) Competition Challenge 1, within strict time limits. The algorithm first finds a near-optimal solution for the relaxed problem (i.e., Stage 1). Then, it finds a feasible solution in the proximity of the near-optimal solution (i.e., Stage 2 and Stage 3). The numerical experiments on test cases ranging from 500-bus to 30,000-bus systems show that the algorithm is tractable. The results show that our proposed algorithm is tractable and can solve more than 80% of test cases faster than the well-known Interior Point Method while significantly reduce the number of iterations required to solve ACOPF. The number of iterations is considered an important factor in the examination of tractability which can drastically reduce the computational time required within each iteration.

This paper presents a deterministic approach to compute multiple local extrema for AC Optimal Power Flow (ACOPF) problems. An elliptical representation of the sphere-confined Fritz John conditions is constructed from a quadratic formulation of the ACOPF problem. Application of a branch tracing method to the elliptical formulation enables the calculation of multiple solutions. Further, a monotone search strategy enhances the computational efficiency of finding multiple local solutions with improved objective values. The proposed approach is first illustrated using two small examples with known feasible spaces. Then, four additional local solutions to a 39-bus system are found using the proposed approach.

The AC optimal power flow (OPF) problem is essential for the schedule and operation of power systems. Convex relaxation methods have been studied and used extensively to obtain an optimal solution to the OPF problem. When the exactness of convex relaxations is not guaranteed, it is important to recover a feasible solution for the convex relaxation methods. This paper presents an alternative convex optimisation (ACP) approach that can efficiently recover a feasible solution from the result of second‐order cone programming (SOCP) relaxed OPF in mesh networks. The OPF problem is first formulated as a difference‐of‐convex programming (DCP) problem, then efficiently solved by a penalty convex concave procedure (CCP). By using the solution of a tightened SOCP OPF as an initial point, the proposed algorithm is able to find a global or near‐global optimal solution to the AC OPF problem. Numerical tests show that the proposed method outperforms those semi‐definite programming (SDP) and quadratically constrained quadratic programming (QCQP)–based algorithms.

GPU-resident sparse direct linear solvers for alternating current optimal power flow analysis

Traditionally, electric system operators have dispatched generation to minimise total production costs ignoring the flexibility of the AC transmission system (ACTS). One available option to enhance power system security is to harness the flexibility of the ACTS, where a variety of flexible AC transmission system (FACTS) devices can be incorporated in the ACTS. However, utilisation of FACTS devices is limited today due to the complexities that these devices introduce to the AC optimal power flow (ACOPF) problem. The mathematical representation of the full ACOPF problem, with the added modelling of FACTS devices, is a non‐linear programming (NLP) optimisation problem, which is computationally burdensome for large‐scale systems. This study presents a method to convert this NLP problem into a mixed‐integer linear program (MILP) where a certain level of solution accuracy can be achieved for a time budget. In this regard, this study first proposes a linear AC OPF model, using which the OPF solution with the operation of FACTS devices is obtained. In addition, the loadability of the power systems is utilised to quantify the impacts of FACTS devices on improving the security of system. The OPF problem including FACTS devices based on a linearised model is tested on a 6‐bus and the IEEE 118‐bus test systems.

Using deep neural networks to predict the solutions of AC optimal power flow (ACOPF) problems has been an active direction of research. However, because the ACOPF is nonconvex, it is difficult to construct a good data set that contains mostly globally optimal solutions. To overcome the challenge that the training data may contain suboptimal solutions, we propose a Lagrangian-based approach. First, we use a neural network to learn the dual variables of the ACOPF problem. Then we use a second neural network to predict solutions of the partial Lagrangian from the predicted dual variables. Since the partial Lagrangian has a much better optimization landscape, we use the predicted solutions from the neural network as a warm start for the ACOPF problem. Using standard and modified IEEE 22-bus, 39-bus, and 118-bus networks, we show that our approach is able to obtain the globally optimal cost even when the training data is mostly comprised of suboptimal solutions.

This paper proposes a data-driven Alternating Current Optimal Power Flow (AC-OPF) method assisted by Lagrange multipliers. Stacked Extreme Learning Machine (SELM) is introduced for AC-OPF learning to avoid the time-consuming training and hyperparameter adjustment process of deep neural networks. Instead of incorporating the prior physical information into the neural networks algorithm, we developed a new neural network structure for the SELM learning based on the Lagrange multipliers of the AC-OPF problem. Case studies of several IEEE benchmark systems demonstrate that the AC-OPF learning performance is improved by introducing additional Lagrange multipliers information, while the proposed method outperforms other alternatives with almost 99% learning accuracy and fast computation.

A Real-Time Optimization with Warm-Start of Multiperiod AC Optimal Power Flows

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

We develop DeepOPF-Stoc as a neural network approach to efficiently solve the two-stage stochastic AC optimal power flow (AC-OPF) problem, which emphasizes reliable and cost-effective grid operation while accounting for uncertainties in renewable energy generation and load demand. We first identify a crucial partial permutation-invariance property of the mapping from load to first-stage solution. We then leverage this property in DeepOPF-Stoc to design (i) PPNN as a principled architectural framework that orchestrates standard neural network designs to learn such mappings, and (ii) a PPNN-aware sampling algorithm to prepare training data efficiently. DeepOPF-Stoc effectively addresses the dimensionality explosion issue inherent in vanilla designs, significantly reducing both the required number of neurons and the volume of training data. Our theoretical analysis confirms the universal approximation capability of PPNNs for our application. We further prove that the PPNN-aware sampling algorithm improves sampling efficiency by a factor of K! compared to uniform sampling, where K represents the number of second-stage scenarios. Simulation results on IEEE 118-bus and synthetic 793-bus test systems demonstrate the superiority of DeepOPF-Stoc over state-of-the-art alternatives. It achieves two orders of magnitude speedup compared to iterative solvers while generating feasible first-stage solutions with comparable second-stage out-of-sample feasibility and 0.95% cost difference.

In the context of active distribution networks, innovative solution techniques based on reformulating the original non-convex AC Optimal Power Flow (AC-OPF) problem have received increasing attention. However, their accuracy is often sensitive to the objective function and grid size. This paper first investigates the performance of a selection of state-of-the-art convex approximations and relaxations of the AC-OPF problem for a range of engineering and economic objective functions in distribution grids. Then, it proposes a novel convex approximation based on the Taylor expansion, which is applicable to a wide range of objective functions as it yields high-quality solutions with small AC-violations. The performance is evaluated via optimality gaps and power mismatches on several active grids varying in size and the penetration of active energy storage systems.

Strengthened SOCP Relaxations for ACOPF with McCormick Envelopes and Bounds Tightening

During the energy transition, the significance of collaborative management among institutions is rising, confronting challenges posed by data privacy concerns. Prevailing research on distributed approaches, as an alternative to centralized management, often lacks numerical convergence guarantees or is limited to single-machine numerical simulation. To address this, we present a distributed approach for solving AC Optimal Power Flow (OPF) problems within a geographically distributed environment. This involves integrating the energy system Co-Simulation (eCoSim) module in the eASiMOV framework with the convergence-guaranteed distributed optimization algorithm, i.e., the Augmented Lagrangian based Alternating Direction Inexact Newton method (aladin). Comprehensive evaluations across multiple system scenarios reveal a marginal performance slowdown compared to the centralized approach and the distributed approach executed on single machines—a justified trade-off for enhanced data privacy. This investigation serves as empirical validation of the successful execution of distributed AC OPF within a geographically distributed environment, highlighting potential directions for future research.