A Sample-Based Approach for Computing Conservative Linear Power Flow Approximations

Non-convexities induced by the non-linear power flow equations challenge solution algorithms for many power system optimization and control problems. Linear approximations are often used to address these challenges by trading off modeling accuracy for tractability. The accuracy of a power flow linearization depends on the characteristics of the power system and the operational range where the linearization is applied. However, rather than exploiting knowledge of these characteristics for a particular system, many existing power flow linearizations are based on general assumptions for broad classes of systems, thus limiting their accuracy. Moreover, since existing linearizations do not consistently overestimate or underestimate quantities of interest such as voltage magnitudes and line flows, algorithms based on these linearizations may lead to constraint violations when applied to the system. In contrast, this paper computes conservative linear approximations of the power flow equations, i.e., linear approximations that overestimate or underestimate a quantity of interest in order to enable tractable algorithms that avoid constraint violations. Using a samplebased approach, we compute these conservative linearizations by solving a constrained linear regression problem. We analyze and improve the conservative linear approximations via an iterative sampling approach, optimization over functions of the quantities of interest, and a sample-complexity analysis. Considering the relationships between the voltage magnitudes and the active and reactive power injections, we characterize the performance of the conservative linear approximations for a range of test cases.