Matrix minor reformulation and SOCP-based spatial branch-and-cut method for the AC optimal power flow problem

International audience

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

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.

Optimal power dispatch in hybrid power system for medium- and large-scale practical power systems using self-adaptive bonobo optimizer algorithm

This paper proposes a novel computing paradigm aimed at solving constrained optimization problems in electric power systems (a.k.a. optimal power flow analysis). The insight is to formulate the OPF problem by a proper set of ordinary differential equations, whose equilibrium points correspond to the optimization problem solutions. Starting from the Lyapunov theory, it will be shown that this artificial dynamic system could be designed to be stable with an exponential asymptotic convergence to equilibrium points. This important feature allows the analyst to overcome some of the inherent limitations of the traditional iterative minimization algorithms that can fail to converge due to the highly nonlinearities of the first-order conditions. Extensive simulation studies aimed at assessing the effectiveness of the proposed computing paradigm are presented and discussed.

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.

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.

This paper proposes the comparison results of various optimal power flow algorithms (OPF) to calculate the locational marginal prices (LMP) of the unreduced full scale Korean transmission system. Five different types of optimal power flow models are employed: Full AC OPF, Cubic AC OPF, Quadratic AC OPF, Linear AC OPF and DC OPF. As the results, full AC OPF and cubic AC OPF model provides LMP calculation results very similar to each other while the calculation time of cubic AC OPF model is faster than that of the Full AC OPF. Other simplified OPF models, quadratic AC OPF, linear AC OPF and DC OPF offer erroneous results even though the calculation times are much faster than the Full AC OPF and the Cubic AC OPF. Given the condition that the OPF models sometimes fail to find the optimal solution due to the severe complexity of the Korean transmission power system, the Full AC OPF should be used as the primary OPF model while the Cubic AC OPF can be a promising backup OPF model for the LMP calculations and/or real-time operation.

With the increasing penetration of intermittent renewable energy sources into the electric power grid, there is an emerging need to develop stochastic optimization methods to enable the reliable and efficient operation of power systems having a large fraction of their power supplied form uncertain resources. In this paper, we formulate the stochastic AC optimal power flow (OPF) problem as a two-stage stochastic program with robust constraints. This problem amounts to an infinite-dimensional nonconvex optimization problem. We develop a finite-dimensional inner approximation as a semidefinite program. Its solution yields an affine recourse policy that is guaranteed to be feasible for the stochastic AC OPF problem.

Recent years have witnessed the success of employing convex relaxations of the AC optimal power flow (OPF) problem to find global or near-global optimal solutions. The majority of the effort has focused on solving problem formulations where variables live in continuous spaces. Our focus here is in the extension of these results to the co-optimization of network topology and the OPF problem. We employ binary variables to model topology reconfiguration in the standard semidefinite programming (SDP) formulation of the OPF problem. This makes the problem non-convex, not only because the variables are binary, but also because of the presence of bilinear products between the binary and other continuous variables. Our proposed convex relaxation to this problem incorporates the bilinear terms in a novel way that improves over the commonly used McCormick approximation. We also address the exponential complexity associated with the discrete variables by partitioning the network graph in a way that minimizes the impact on the optimal value of the relaxation. As a result, the problem is broken down into several parallel mixed-integer problems, reducing the overall computational complexity. Simulations in the IEEE 118-bus test case demonstrate that our approach converges to solutions which are very close to the lower bound of the mixed-integer OPF problem.

This paper presents an optimization problem of siting capacitors in a power network to achieve economic benefits. The formulated optimization problem is a nonlinear mixed integer programming problem. In order to find an approximate optimal solution, Benders decomposition is adopted. The optimization problem will be decomposed into a main problem and a subproblem. The subproblem determines dispatch patterns for generators to minimize an objective function if the sites of the capacitors are given. The subproblem is a conventional AC optimal power flow problem. Existing toolboxes such as MATPOWER is used to solve such problems. The subproblem helps to create Benders cuts for the main problem which decides where to put capacitors with a set of cuts introduced accumulatively from the subproblems at each iteration. The main problem is solved by MATLAB CVX toolbox. Numerical results for IEEE 118-bus system are given.

A mathematical model is presented for optimizing power flows in an electric network with a flexible controlled alternating current power transmission device - FACTS. The range of questions on the problem of optimizing the power flow in electric networks with one of the FACTS devices, the SVC static power factor compensator, has been expanded and investigated. The Lagrange function for the SVC device is proposed, which serves as the basis for obtaining a linearized equation and determining the optimum of the objective function.The method and algorithm of the optimization problem in the electric power system containing devices of the FACTS technology have been synthesized. The mathematical model allows for flexible and reliable optimization of the electrical power system. Flexibility is explained by the universality of the model, and reliability is explained by the high convergence rate of the Newton method used.

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.

To cope with increasing uncertainty from renewable generation and flexible load, grid operators need to solve alternative current optimal power flow (AC-OPF) problems more frequently for efficient and reliable operation. In this article, we develop a deep neural network (DNN) approach, called DeepOPF, for solving AC-OPF problems in a fraction of the time used by conventional iterative solvers. A key difficulty for applying machine learning techniques for solving AC-OPF problems lies in ensuring that the obtained solutions respect the equality and inequality physical and operational constraints. Generalized a prediction-and-reconstruction procedure in our previous studies, DeepOPF first trains a DNN model to predict a set of independent operating variables and then directly compute the remaining ones by solving the power flow equations. Such an approach not only preserves the power-flow balance equality constraints but also reduces the number of variables to be predicted by the DNN, cutting down the number of neurons and training data needed. DeepOPF then employs a penalty approach with a zero-order gradient estimation technique in the training process toward guaranteeing the inequality constraints. We also drive a condition for tuning the DNN size according to the desired approximation accuracy, which measures its generalization capability. It provides theoretical justification for using DNN to solve AC-OPF problems. Simulation results for IEEE 30/118/300-bus and a synthetic 2000-bus test cases demonstrate the effectiveness of the penalty approach. They also show that DeepOPF speeds up the computing time by up to two orders of magnitude as compared to a state-of-the-art iterative solver, at the expense of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$&lt; $</tex-math></inline-formula> 0.2% cost difference.