Arbitrarily Sparse Polynomial Chaos Expansion for High-Dimensional Parametric Problems: Parametric and Probabilistic Power Flow as an Example

Abstract

Parameters, no matter whether they are uncontrollable uncertainty factors or controllable variables, have a great impact on the states and performances of continuous engineering systems. Acquiring an explicit expression of the complicated implicit function between these parameters and system states, called parametric problem in this article, will facilitate immediate analysis of parameters’ impact on the system, optimal parameter design, and uncertainty quantification. Polynomial chaos expansion (PCE) is a globally optimal polynomial approximation method for parametric problems, but it suffers from the curse of dimensionality, i.e., the number of basis functions and consequent computational burden increases rapidly with the number of parameters (i.e., dimensions). This article proposes a PCE method characterized by the arbitrarily sparse basis and novel generalized Smolyak sparse grid quadrature and, hence, effectively relieves the curse of dimensionality for high-dimensional problems. This basis is not restricted by any existing fixed-form truncation and can merely incorporate a few important basis functions. The novel generalized sparse grid can remarkably reduce the number of required collocation points compared with the prevailing classic sparse grid while achieving high accuracy. The proposed method is universal for parametric problems in engineering systems and is exemplified by parametric and probabilistic power flow problems of electrical power systems. Its effectiveness is validated by computational results on the IEEE 30-bus, 118-bus, and 500-parameter 9241pegase systems.