Studies On Deterministic Models For Stochastic Second-Order Cone Complementarity Problems And Their Applications

The complementarity problems are an important topic in optimization theory,which have been widely applied in supply chain management,engineering mechanics,game theory,and so on.Second-order cone complementarity problems are an important generalization of the complementarity problems.Based on the Jordan algebra theory,The second-order cone complementarity problems have achieved fruitful theoretical results and have been widely used in mechanics,economy,transportation,communication,and so on.However,there are various uncertain factors in many practical problems and ignoring these random factors would lead to mistakes in decision-making.Therefore,due to the needs of theory and practical applications,random variables are introduced into the second-order cone complementarity systems,which leads to the proposal of the stochastic second-cone complementarity problems.At present,the research on the stochastic second-order cone complementarity problems is still in its infancy,and there are many problems that need to be studied systematically.On the other hand,the optimal power flow is the application of optimization theory in the power systems.Mathematical models can be used to describe the safety and economy of the power systems,so as to provide effective solutions for the power systems scheduling operation.With the market-oriented reform of power system and the continuous access of renewable energy to power generation,the instability of the injection power at the node becomes more and more obvious.Therefore,the stochastic optimal power flow has attracted more and more attention.Under certain constraint qualifications,the stochastic optimal power flow problems can be transformed into the stochastic second-order cone complementarity problems,and then effective algorithms can be developed by using the stochastic second-order cone complementarity theory.This is a new exploration for solving the stochastic optimal power flow problems in power systems.This thesis mainly studies the stochastic second-order cone complementarity problems,and proposes a new cone complementarity function and a new merit function,and establishes the expected residual minimization model and the expected value model and their solution methods for the stochastic second-order cone complementarity problems.The application of these models to the stochastic optimal power flow under wind power access in power system is considered to provide theoretical support for the safe operation and economic scheduling of the power systems.The main contents of this paper are as follows:Firstly,based on the characteristics of cone"complementary"relationship,a termwise residual complementarity function and its corresponding new merit function are proposed.By the Jordan algebra techniques,the new complementarity function and the new merit function are shown to be continuously differentiable and semismooth everywhere.Some results related to stationary,coerciveness,and error bounds are established.The numerical results show that the new merit function is comparable with other traditional merit functions.In particular,the values of the new merit function drops more quickly at the beginning of the iteration for the tested problems.Secondly,the expected residual minimization model for the stochastic second-order cone complementarity problems is considered by using the termwise residual complementarity function,and it is proved that the expected residual minimization model has bounded level sets under the stochastic weak₀R-property.Some error bound results are derived under the strong monotonicity or the NNAMCQ constraint qualifications.Furthermore,the approximation problems of the expected residual minimization model are given by Monte Carlo approximation technique.The convergence results of global optimal solutions sequence and stationary points sequence for the approximation problems are discussed.The optimal solution of the expected residual minimization model can be regarded as robust solutions for the the stochastic second-order cone complementarity problems,and the exponential convergence of the proposed approximation method are obtained.Thirdly,with the help of the natural residual complementary function and the Fischer-Burmeister complementariy function,the expected value model for the stochastic second-order cone complementarity problems is established,and its error bound analysis is given.Due to the expectation and the non-smooth in the objective function,Monte Carlo approximation method and the smoothing technology are employed to obtain the approximation problems.The convergence results and the exponential convergence rate of global optimal solutions sequence and stationary points sequence for the approximation problems are given.Finally,the application of expected residual minimization model and the expected value model to the stochastic optimal power flow in power systems is studied.Considering the impact of wind power uncertainty on the power systems,two kinds of stochastic optimal flow models under wind power access for the radial distribution network system and the high voltage transmission network system are established,which can be effectively solved by the expected residual minimization model and the expected value model.At the same time,the SCE47-bus and IEEE 30-bus systems are tested.The numerical results of stable convergence are obtained,and the expexted residual values of the stochastic optimal flow can be maintained within a small acceptable range.It shows that the dispatchi-ng results can withstand the disturbance of wind power output uncertainty,which can provide a strong theoretical support for the safe and economic dispatching of the power systems under wind power access.