High-dimensional Sparse Global Polynomial Approximation Method And Its Applications To Power System Parametric Small Signal Stability

Small signal stability is the prerequisite of normal operation of power systems.With largescale access of volatile renewable resources and construction of long-distance UHV transmission systems,uncertainty factors are increasing and random characteristics are also becoming conspicuous.Accompanying power electronic devices also pose new challenges to the stability.Under this circumstance,the equilibrium calculation and stability analysis at the normal operating point cannot reflect stability at various operating conditions,so it is necessary to study small signal stability under random scenarios and further utilize controllable measures to maintain stability.At present,the impact of random and controllable variables(called varying parameters collectively)on system states or performances is usually analyzed by sensitivity method,whose accuracy will deteriorate when parameters vary in a large range,because of the strong nonlinear characteristics of power systems.Therefore,this thesis proposes a calculation method of the small signal stability region boundary by using the global sparse polynomial approximation method and bifurcation theory,and thus accurately and explicitly describes the relationship between small signal stability and multiple parameters;based on this,a small signal stability constrained stochastic optimal power flow calculation method is also proposed to ensure stability and economics simultaneously.The main work of this thesis is described as follows.(1)With the aid of the concept of parametric problems,classify a series of power system parametric(or stochastic)steady-state and dynamic problems into three general problems:parametric nonlinear algebraic equations,parametric nonlinear programming,and parametric differential-algebraic equations,which provides a unified perspective for solving various power system problems.Elaborate global polynomial approximatiaon methods such as Galerkin method,numerical integration method,interpolation and fitting methods for parametric problems under the framework of polynomial chaos expansion theory.The summarization and comparison of these methods are made,and the challenges and some promising work are pointed out.(2)To relieve the curse of dimensionality(COD)for high-dimensional parametric or stochastic problems,a global polynomial approximation method based on arbitrarily sparse basis and generalized Smolyak sparse grid quadrature rule(AS-PCE)is proposed.The arbitrarily sparse basis is extremely flexible,so important basis functions can be found gradually whereas unimportant ones are omitted.As result,the number of terms is significantly reduced,and COD is effectively alleviated.In the sparse grid,the number of collocation points is proportional to the number of basis functions,which ensures the accuracy and avoids excessive computational burden.In addition,a sparse Galerkin method is proposed,and a principal component analysis-based parameter dimension reduction method is introduced.Computational results of parametric and stochastic power flow validate the effectiveness of the above general mathematical methods.(3)Aimed at power system small signal stability in the multi-dimensional parameter space,polynomial approximations of local-bifurcation hypersurfaces are constructed by combining the global polynomial approximation method and implicit function theorem,which are much more accurate than those obtained by Taylor expansion.The obtained hypersurfaces representing saddle-node,Hopf,singularity-induced,and limit-induced bifurcations together constitute the small signal stability region(SSSR)boundary in the parameter space,by which the small signal stability can be immediately judged when power outputs of renewable resources and synchronous generators are taken as parameters and vary.In addition,the validity scope of the method is discussed,and bifurcation hypersurface calculation considering multiple AVR limits is presented.(4)Aimed at potential instability caused by random power variation of renewable resource and loads,a small signal stability constrained stochastic optimal power flow method is proposed.It optimizes power dispatch schemes of traditional synchronous generators at both nominal and stochastic cases,and enhances economy while maintaining stability.In the method,the stability constraint is constructed via calculating the SSSR boundary of control and random variables by AS-PCE.Chance-constrained models of the stability margin and power flow quantities are established to express stochastic constraints,and are converted into deterministic constraints by approximating the relationship between their quantiles and control variables with polynomial functions.The primal-dual interior point method is used to solve the resulting optimization model.