Optimal PMU Placement for State Estimation with Grid Parameter Uncertainty

We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by power flow models. The inverse problem consists of inferring the values of the unknown (or uncertain) susceptance parameters and voltage angles parameter that characterize the power flow from available observations (i.e., power flow and potential Phasor Measurement Units (PMU) measurements). To quantify the uncertainties in the reconstructions, we invoke a Bayesian framework. Under the assumption of Gaussian noise and prior probability densities (for both susceptance and voltage angles) and after linearizing the parameter-to-observable map (describing the power flow), the posterior density becomes Gaussian and can therefore be characterized by its mean and covariance. The mean is given by the solution of a nonlinear least squares optimization problem, which is solved via an inexact Newton method. The posterior covariance matrix is given (in this case) by the inverse of the Hessian of the least squares cost objective function. Following an A-optimal experimental design strategy, we then seek to minimize the posterior variance of the parameter estimates, which is given by the trace of the posterior covariance (i.e., the inverse Hessian operator). To solve the A-optimal design problem, we adopt a greedy approach to cope with the binary structure of the weights. We demonstrate the effectiveness of the A-optimal design approach by comparing the OED results with random designs for 14- and 118-bus power flow problems. The results reveal the OED's potential to significantly reduce the uncertainty in the estimation by optimally placing PMUs.