Studies On Stochastic Linear Second-Order Cone Complementarity Problems And Their Applications In Optimal Power Flow

Second-order cone complementarity problems are a class of equilibrium optimization problems,which refer to that,under the second-order cone constraints,two groups of variables meet a "omplementarity" relationship.As the generalization of the complementarity problems and second-order cone programming,the second-order cone complementarity problems have reached a great progress in the theory study with the help of Euclidean Jordan algebra theory,and have wide applications in engineering,economic and so on.However,there are usually uncertain factors in the practical problems,since ignoring these factors would lead to mistakes in decision-making and immeasurable loss,therefore,it has important significance and application value to study the stochastic second-order conecomplementarity problems.On the other hand,the optimal power flow in power system is the application of mathematical optimization theory in power system,it can be described unifily by a mathematical model with the consideration of the power system security and economic issues.With the change of the operation of a power system,especially the direct grid-connection of renewable energy,theinstability of the injection power at the node is more obvious,which would bring great challenge to the power system dispatching and operation,and thus the stochastic optimal power flow appears.How to effectively solve the stochastic optimal power flow is one of the hot topics that the scholars concerned currently.This thesis mainly studies the stochastic linear second-order cone complementarity problem and its solution methods,and further illustrates the effectiveness of the obtained theoretical results and the methods through the application in stochastic optimal power flow.The main contents and innovation points of this paper are as follows:Firstly,a regularized parallel matrix-splitting method is proposed for solving the linear second-order cone complementarity problems.Compared with other similar algorithms,the matrix considered in the problem is symmetric and semidefinite,and the regularization parameter is monotonically decreasing to zero.Under some suitable conditions,the new algorithm has convergence and can be implemented in a parallel way,especially,the subproblems can be solved exactly.Numerical experiments show that the new algorithm is applicable to large-scale problems,especially to the problems with dense,ill-conditioned,symmetric positive definite matrices or semidefinite matrices.Secondly,the stochastic linear second-order cone complementarity problem is considered.Motivated by the expected residual minimization method for the stochastic complementarity problems,the stochastic linear second-order cone complementarity problem is formulated as a unconstrained optimization problem through the second-order cone complementarity function and the expected residual minimization method.Due to the expectation in the objective function,Monte Carlo approximation method is employed to approximate the expected residual minimization problem.Then the existence and convergence of the solutions of the expected residual minimization problem and the approximation problems is discussed,and the sequence of solutions of the approximation problems converges exponentially fast to a solution of the expected residual minimization problem with probability one under suitable conditions.Due to the fact that the approximation problem is a nonconvex optimization problem,the convergence and exponential convergence rate of the sequence of the stationary points are discussed.Furthermore,the robustness of the solution of the expected residual minimization problem to the original stochastic linear second-order cone complementarity problems.Thirdly,the mixed stochastic linear second-order cone complementarity problem is investigated.Because the applications usually contain other constraints,the models obtained are mixed complementarity problems,this paper discusses the mixed stochastic linear second-order cone complementarity problems.Firstly,the coerciveness and the robustness of the expected residual minimization problem and the Monte Carlo approximation problems are discussed.Then the convergence and exponential convergence rate of the sequence of the optimal solutions of the approximation problems are given.Due to that the approximation problems are nonconvex optimization,the convergence and the exponential convergence rate of the sequence of the stationary points of the approximation problems are also given.Finally,the stochastic optimal power flow with radial network structure in power system is considered.It can be reformulated as a stochastic second-order cone programming,which can be deduced by the fact that the convex relaxation of the nonlinear power flow equations has the same form with the rotated second-order cone.Under some conditions,the stochastic second-order cone programming can be solved by its KKT conditions.Since the KKT conditions of the stochastic second-order cone programming optimal power flow problem are a mixed tochastic linear second-order cone complementarity problem,whose solution methods can be employed to solve the stochastic second-order cone programming optimal power flow problem.Numerical experiments show that the proposed method is effective,and depending on their practical situations and actual needs,people may seek for their best strategies at an acceptable level of tolerance by taking different values of the parameter,which is contained in the second-order cone complementarity function considered.