A Computational Approach to Anticipating Supply Injections and Bus Voltages in Steady-State Power System Analysis

Power flow analysis is the bread-and-butter computational framework where one can assess the steady-state operation of an electric power system by analyzing how the powers supplied in response to those demanded affect the system state through the bus voltages, according to the physical model known as the power flow equations. This thesis presents an analysis tool called anticipatory power flow (APF). It is a two-stage task for finding a set of supply injections and bus voltages that are power-flow feasible under the conditions of an upcoming dispatch (e.g., expected power demands, scheduled supply limits, estimated system parameters), considering information from the preceding system snapshot (e.g., bus voltages, powers demanded and supplied). The first stage—anticipating the supply injections—is formulated as a convex program called extended economic dispatch, modifying the standard economic dispatch model by (i) accounting for reactive supply injections; (ii) adjusting the required power with estimated system loss; and (iii) minimizing a linear combination of the system loss and a tunable regularizer representing the influence of the previous-snapshot injections. The second stage—finding the corresponding bus voltages—amounts to solving the APF equations, which are a modification of the power flow equations to have (i) a user-specified reference bus (with voltage phase angle fixed to an arbitrary value) to avoid rotational degeneracy, and (ii) a distributed slack variable (compensating for inexactness of the anticipated supply injections) to keep the degrees of freedom and the number of equations equal.

APF can be regarded as an extension of the standard power flow problem, deviating from the traditional classification of buses (i.e., into PQ, PV, and slack) which assumes predetermination of a subset of active supply injections and bus voltage magnitudes. APF can also be viewed as a special case of continuation power flow where the demand draws and the entire system parameters are the continuation parameters, and the two APF stages are analogous to a predict-correct cycle that lands on the target values of the continuation parameters.

Experimental results suggest that the APF subproblems are amply handled by existing off-the-shelf solvers. Specifically, simulated scenarios based on a 3374-bus, 4161-branch, 596-generator portion of Poland's transmission network show that the extended economic dispatch subproblem and the APF equations are solved in sub-second run times on a consumer-grade machine.

This work also demonstrates the use of APF notions as auxiliary tools in two optimal power flow (OPF) scenarios: (i) warm-starting OPF solvers with initial iterates set according to the anticipated supply injections and the solution to their corresponding APF equations; and (ii) finding the nearest power-flow feasible values of a set of supply injections and bus voltages that optimize some approximated OPF model.

Lastly, this thesis derives and verifies methods for differentiating through the APF equations, enabling computation of derivatives/sensitivities of the bus voltages, distributed slack, and arbitrary functions thereof, with respect to the corresponding anticipated supply injections. This lays the groundwork for the use of the APF equations to enforce structure in differentiable programming pipelines.