Algebraic Set Preserving Mappings for Electric Power Grid Models and its Application

This work investigates three different topics in power system engineering with the idea of problem representation which is achieved by a type of mappings called the algebraic set preserving mapping in the context. It alters undesirable features of the original problem to the desirables of the new problem without compromising the original solution.;The first application attempts to identify multiple power flow real-valued solutions efficiently. In this work, an algebraic set preserving mapping is carefully designed to preserve the affine algebraic set of the power flow problem while altering the unbounded manifold of each power flow equation into a high-dimensional ellipse. Then, the branch tracing method is implemented to the power flow problem. It is testified on all the cases of which the solution sets are known, and provides solution sets for larger systems which have never been completely solved before.;The second application attempts to identify multiple local solutions (maybe the global solution) to the optimal power flow (OPF) problem. This work proposes a deterministic way to locate multiple local solutions, in hope of enumerating the global solution. An algebraic variety set preserving mapping is carefully constructed to preserve the affine algebraic set of the Fritz John conditions of the OPF problem. It converts the unbounded manifold of each Fritz John equation into a high-dimensional ellipse. Then, the branch tracing method is applied to locate the algebraic set of the Fritz John conditions. A monotone search strategy is further designated to enforce the objective function value non-increasing at each step. The proposed method succeeded on some special ACOPF cases which fail the SDP relaxation.;The third application designs an encryption strategy to mask the sensitive information of the OPF data for shared computing techniques in the multi-party scenario. To compromise both the security and the computational complexity, this thesis designs a strategy that enables each participant to construct its own encryption mapping to mask the sensitive information, and a computing procedure is developed based on the masked data to obtain the masked solution. Finally, the solution is returned to each party and decrypted by each participant individually.