Optimal Adaptive Power Flow Linearizations: Expected Error Minimization using Polynomial Chaos Expansion

The nonlinearity of the power flow equations leads to algorithmic and theoretical challenges for a wide variety of optimization and control problems relevant to electric power systems. Solution algorithms for these problems often use linearization techniques to obtain tractable power flow formulations. Many existing linearizations lack specialization to a given system and operating range of interest, leading to unnecessarily large linearization errors. In contrast, recently proposed “optimal adaptive” linearizations are 1) tailored to specific systems and operating ranges of interest and 2) optimal in the sense that they minimize a selected error metric relative to the nonlinear power flow equations. Existing work proposes optimal adaptive linearizations that minimize the worst-case linearization error. Building on this existing work, this paper uses an uncertainty quantfiation method called Polynomial Chaos Expansion to minimize the expected linearization error for a given probability distribution of the power injections. As validated using Monte Carlo analyses, the capabilities of the expected-error-minimizing linearizations are shown to be superior to first-order Taylor approximations computed at a nominal operating point.