Probabilistic Optimal PMU Placements under Limited Observability Propagations

The problem of phasor measurement units (PMUs) placement for state estimation in modern power grids has emerged with renewed interest in recent years. Conventional optimal PMU placement (OPP) formulations without limitations on observability propagations pose the power grid at the higher risk of losing its observability under unexpected contingencies or malicious cyberattacks. In order to mitigate these impacts in PMU deployments, by extending our proposed enhanced OPP formulation under the concept of the observability propagation depth (OPD), a probabilistic variation of this enhanced OPP is presented. To further explore the robustness of OPP problems under severe contingencies, a linearized probabilistic observability model is proposed and incorporated with the previous enhanced OPP model into a bilevel optimization framework. Thus, in addition to derive the minimum number of PMUs by solving the enhanced OPP model in the upper level, the optimal PMU configuration with the maximal probabilistic observability can also be depicted in the lower level. By solving the dual problem of the lower-level problem under Karush-Kuhn-Tucker conditions, this bilevel optimization formulation can be reformulated into a single-level mixed-integer linear programming problem, which can be easily solved by commercial optimization tools. Extensive simulations are conducted on several IEEE test-bed systems as well as the Polish-2383 system to demonstrate the feasibility and the effectiveness of the proposed formulation. Simulation results indicate that there is strong evidence that the probability of losing the observability of a bus is positively related to its OPD.