Solution of optimal power flow problems using moment relaxations augmented with objective function penalization

This study proposes an improved particle swarm optimization method to find optimal power flow by using the power transmission loss as an objective function. In the literature, PSO is a well — known intelligent search method to handle the solution of optimal power flow problem. A novel scheme which is called Improved PSO (IPSO) is defined by modifying the initialization step of PSO algorithm and choosing the load bus voltages, generator active and reactive powers, line flow capacities as penalty functions in the objective function. PSO and IPSO-based optimal power flow solutions are compared with each other on IEEE 118 and 300 bus systems. According to the test results, the power loss obtained by IPSO-based solution has less power loss than PSO-based solution. This proposed method can be used to obtain faster desirable solutions and better power loss results for the optimal power flow problem in case of power loss minimization.

Summary This paper presents a creature-inspired stud krill herd (SKH) algorithm consisting of stud selection and crossover operators, which are augmented in the krill herd algorithm to solve the optimal power flow (OPF) problems in a power system. Due to the nonlinear and nonconvex nature of the OPF problem, it is quite difficult to find optimal solution using conventional optimization algorithms. Here in SKH, a stud-selection operator is introduced in the krill herd algorithm to further enhance the quality of the solution, and crossover operator is embedded to avoid premature convergence. To investigate the performance, the proposed SKH algorithm has been demonstrated on the OPF problems of IEEE 30-bus, Algerian 59-bus, and IEEE 118-bus systems considering various objective functions such as total production cost, L-index, power loss, and emission pollution to be minimized. The OPF results have been compared with those obtained with other methods, and it has been found that SKH algorithm is successful in providing better quality solutions in reasonable execution times.

For operating electrical power networks, the Optimal Power Flow (OPF) problem plays a central role. The problem is nonconvex and NP hard. Therefore, designing efficient solution algorithms is crucial, though their global optimality is not guaranteed. Existing semi-definite programming relaxation based approaches are restricted to OPF problems for which zero duality holds, whereas for non-convex problems there is a lack of solution methods of provable performance. In this paper, an efficient novel method to address the general nonconvex OPF problem is investigated. The proposed method is based on alternating direction method of multipliers combined with sequential convex approximations. The global OPF problem is decomposed into smaller problems associated to each bus of the network, the solutions of which are coordinated via a light communication protocol. Therefore, the proposed method is highly scalable. The convergence properties of the proposed algorithm are mathematically and numerically substantiated.

In this paper, we consider the Optimal Power Flow (OPF) problem which consists in determining the power production at each bus of an electric network by minimizing the production cost. Our contribution is an exact solution algorithm for the OPF problem. It consists in a spatial branch-and-bound algorithm based on a compact quadratically-constrained convex relaxation. It is computed by solving the semidefinite rank relaxation of OPF once at the root node of the algorithm. An important result is that the optimal value of our compact relaxation is equal to the rank relaxation value. Then, at every sub-nodes of our branch-and-bound, the lower bound is obtained by solving a quadratic convex problem instead of an SDP. Another contribution is that we add only O(n+m) variables that model the squares of the initial variables, where n is the number of buses in the power system and m the number of transmission lines, to construct our relaxation. Then, since the relations between the initial and auxiliary variables are non-convex, we relax them to get a quadratic convex relaxation. Finally, in our branch-and-bound algorithm, we only have to force a reduced number of equalities to prove global optimality. This quadratic convex relaxation approach is here tailored to the OPF problem, but it can address any application whose formulation is a quadratic optimization problem subject to quadratic equalities and ring constraints. Our first experiments on instances of the OPF problem show that our new algorithm Compact OPF (COPF) is more efficient than the standard solvers and other quadratic convex relaxation based methods we compare it with.

The optimal power flow (OPF) problem is a well-known non-convex optimization problem that aims to minimize the cost of electric power generation subject to consumer demand, the physics of power flow, and technological constraints. To find an optimal solution to this problem, local search techniques such as interior point methods are typically used. However, due to the non-convex nature of the problem, these methods are likely to result in a sub-optimal solution. The goal of this paper is to characterize the worst-case performance of local search on the OPF problem. To accomplish this, we formulate the OPF problem as a canonical quadratically-constrained quadratic program (QCQP). Then, we study the problem of finding the worst-case local minimum of this QCQP, which is non-convex and hard to solve in general. We find a relaxation of this problem into a semidefinite program (SDP) and show that it is exact for certain cases. Using some test cases which are known to have multiple local minima, we demonstrate the effectiveness of the proposed relaxation to bound the worst-case local minimum. We compare the obtained upper bound on local minima to the lower bound provided by the standard SDP relaxation of the OPF problem to understand how much SDP outperforms local search for a given 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.

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.

In this paper, a new method to incorporating FACTS devices in optimal power flow (OPF) problem is presented. This method can provide an enhanced economic solution with the use of controllable FACTS devices. Two types of FACTS devices, thyristor controlled series compensators (TCSC) and thyristor controlled phase shifters (TCPS), are considered in this method. The specified power flow control constraints due to the use of FACTS devices are included in the OPF problem. The proposed method decomposes the solution of such modified OPF problem into two subproblems' iteration. The first subproblem is a power flow control problem and the second subproblem is a conventional OPF problem. The two subproblems are solved iteratively until convergence. Case studies are presented to show the effectiveness of the proposed method.

This paper presents the semismooth Newton-type algorithms for solving optimal power flow (OPF) problems. Considering that there exist plentiful bounded constraints in OPF, the paper treats general inequality constraints and bounded constraints separately. By introducing a diagonal matrix and the nonlinear complementarity function, the Karush-Kuhn-Tucker (KKT) system of the OPF is transformed equivalently to a system of nonsmooth bounded constrained equations. Comparing with the classical OPF methods and the nonlinear complementarity problem (NCP) method, this treatment has two remarkable advantages. First, it has a strong ability to handle the inequality constraints in OPF problems. Second, it reduces the number of dual variables in the KKT system. Based on the reformulated equations, the paper designs a projected semismooth Newton algorithm which has nice global and local convergence property. Furthermore, according to the weak-coupling property in power systems, the paper presents a decoupled semismooth Newton-type algorithm. The decoupled method solves the system of equations via solving two lower dimension problems. Therefore the method saves the computing cost in theory. Some standard IEEE systems are used to test the two algorithms, and numerical results show that the proposed algorithms are promising for solving OPF problems

This paper investigates potential of modified differential evolution (MDE) algorithm, to find solution for optimal power flow (OPF) problem of power system. The performance of the proposed MDE algorithm is experienced on the modified IEEE-30 bus test systems integrating flexible ac transmission systems (FACTS) devices, that is, thyristor controlled phase shifter (TCPS) and thyristor controlled series capacitor (TCSC) at preset locations. The OPF problem of the present work is formulated with five different objective functions viz. (a) fuel cost (with valve point effect) minimization, (b) power transmission losses minimization, (c) emission reduction (d) fuel cost (without valve point effect) minimization and (e) minimization of combined economic and environmental cost. The proposed implementation exploits MDE, resulting in a much shorter execution time while retaining the best possible solution of OPF problem.

This paper presents a study based on versatile bio-inspired metaheuristic stud krill herd (SKH) algorithm to tackle the optimal power flow (OPF) problems in a power system network. SKH consists of stud selection and crossover operator that is incorporated into the original krill herd algorithm to improve the quality of the solution and especially to avoid being trapped in local optima. In order to investigate the performance, the proposed algorithm is demonstrated on the optimal power flow problems of IEEE 14-bus, IEEE 30-bus and IEEE 57-bus systems. The different objective functions considered are minimization of total production cost with and without valve point loading effect, minimization of active power loss, minimization of L-index and minimization of emission pollution. The OPF results obtained with the proposed approach are compared with the other evolutionary algorithms recently reported in the literature.

Optimal power flow using gravitational search algorithm

This paper illustrates an application of adaptive tabu search (ATS) to optimal power flow (OPF) problems in comparison with some effective mathematical and evolutionaryoptimization methods. Although, the ATS was originally developed for solving a combinatorial optimization problem whose parameters are discrete, it has the ability to handle continuous variables by treating them as discrete ones with a very small variable step-size to gain accuracy. The proposed algorithm was tested with 9-bus and 300-bustest systems to represent a small-scale and a comparatively large-scale power system, respectively. Each test power system was challenged by performing three test cases. The first test case was given by applying a quadratic function to generators’ fuel-cost curve, whereas a non-smooth fuel-cost function was assigned to the second. In addition, the system voltage profile was considered and set as the objective function to be minimized in the last test case. The comparisons among solutions obtained by sequential quadratic programming (SQP), evolutionary programming (EP) and the ATS were carried out, from which satisfactory results and the selection of solution methods to OPF problems were summarized. Key words: Optimal power flow problem, sequential quadratic programming, evolutionary programming, adaptive tabu search, quadratic fuel cost, non-smooth fuel cost.

A method to solve a multi‐objective optimal power flow (MOOPF) problem with multiple and competing objective functions (OF) is presented. The modified flower pollination algorithm and the normal boundary intersection method are used in a complementary way to determine the Pareto front solution of the MOOPF problem. To help in the decision making process, an intuitive criterion based on the centroid concept is proposed to select the best compromise solution from the Pareto frontier. To demonstrate the capabilities of the proposed method, different OFs are combined to calculate the Pareto front solution on the IEEE 30 bus test system. Finally, a comparison of the proposed centroid based method against the well‐known fuzzy membership and entropy criterions is provided in the results section.

In this study, symbiotic organisms search (SOS) stochastic method is proposed to solve the optimal power flow (OPF) problem with valve-point effect and prohibited zones, which is one of the most important problems of the modern power system. The SOS approach is defined as the symbiotic relationships observed between two organisms in the ecosystem, which do not need the control parameters unlike other meta-heuristic algorithms in the literature. The effectiveness of the proposed SOS method is tested on modified IEEE 30-bus test system. The OPF problem is considered with four different test cases, such as (1) without valve-point effect and prohibited zones, (2) with valve-point effect, (3) with prohibited zones and (4) with valve-point effect and prohibited zones. The obtained results from the SOS algorithm are compared with the other optimization techniques in the literature. The obtained comparison results indicate that proposed approach is effective to reach optimal solution for the OPF problem.