Optimization of Optimal Power Flow Problems

Abstract

Optimal power flow problems arise in the context of the optimization and secure exploitation of electrical power in alternating current (AC) networks. This optimization problem evaluates the flow on each line and to ensure that they are under line thermal limits. To improve the reliability of the power supply, a secure network is necessary, i.e., it has to be able to cope with some contingencies. Nowadays high performance solution methods, that are based on nonlinear programming algorithms, search for an optimal state while considering certain contingencies. According to the number of contingencies the problem size increases linearly. As the base case can already be large-scaled, the optimization time computation increases quickly. Parallelization seems to be a good way to solve quickly this kind of problem. This paper considers the minimization of an objective function and at least two constraints at each node. This optimization problem is solved by IPOPT, an interior point method, coupled with ADOL-C, an algorithmic differentiation tool, and MA27, a linear solver. Several results on employed parallelizing strategies will be discussed. Problem FormulationThe project parts of which are reported in this paper was motivated by a cooperation of the University of Paderborn with RTE France, one of the most important transmission system operators of Europe. We focus on the analysis of optimal power flow problems in AC networks. Our goal is to obtain an optimal secured network state. This optimization estimates the state variables when injections and the structure of the network are given. Therefore a nonlinear objective function with at least two constraints at each node has to be solved. The general problem can be described as followswhere f : R n → R, h : R n → R m and g : R n → R p . The bounded vector x is divided into control variables (directly controllable) and state variables (not directly controllable). State variables are voltage magnitude and angle. Control variables are fictive active and reactive injection at each node. The objective function consists of a sum of quadratic terms for the least square minimization: the goal is to minimize the fictive active and reactive node injection and the deviation of state voltage to target voltage on nodes where there is a production unit. The equalities are due to Kirchhoff's law and represent the power balance within the network. The inequalites are due to power limits of production units. The weight in the objective function on fictive injections are high so they are only used to solve infeasible problems. The interpretation of these fictive injections is used to point out data errors. Parallel ApproachesInitially, a sequential simulation code, written in C++, was provided by RTE France. The given optimization problem is solved by IPOPT (http://www.coin-or.org/projects/Ipopt.xml) and the required first and second order derivatives are computed by ADOL-C (http://www.coin-or.org/projects/ADOL-C.xml). A linear subproblem formulated by IPOPT is solved by MA27. For the...