Parallel Distributed-Memory Implementation of the Corrective Switching Problem

LMC-IIVLAG EDF-DER Abstract. For the past 20 years, an increasing interest has been devoted to the sequential Conjugate Gradient Method for solving large linear systems arising from the modeling of physical problems (especially for very large systems with sparse matrices). This paper deals with the implementation on parallel supercomputers of a preconditioned conjugate gradient method for solving the corrective switching problem obtained while modeling the behavior of power systems in electrical networks. This problem consists in finding the successive solutions of many close linear systems (not too large) with very ill-conditioned matrices (sometimes even singular). We present a new method based on the Preconditioned Conjugate Gradient algorithm with an original preconditioning and study its parallelization on both shared and distributed memory computers. 1. Setting of the problem During the control of electrical networks, the operator must ensure the system to bc in a safc state (i.e. to be able to protect the system against incidents liable to occur in real time). The demand and the possibility of the plants are such that nuclear energy between two plants flows from various nodes of the network. The loss of one element could jeopardize the security of the whole system by a chain tripping: in such case, an overload line occurs and without any operation the protective devices will act and the line will trip out. In actual operations conditions, the switching actions that the operator applies to the electrical network ensure that overloads will disappear before the delayed protective devices go into action. Such actions are shown on the picture at the end of the paper. The computation of switching actions is a combinatorial problem, very hard to solve. The connections of the switching elements are described as discrete variables. The corrective switching problem corresponds to determine the various possible solutions of the load flow calculation. Each such situation requires to solve a linear system where the matrices have only a few elements which differ from each other. Let us consider the N consecutive linear systems below: (Si) Ajx; = b;, lG