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 influence by 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.

Experimental results show that the APF subproblems are amply handled by existing off-the-shelf solvers. In simulated scenarios based on a portion of Poland's transmission network with 3374 buses, 4161 branches, 596 supply units, and 2344 demand units, the Ye-Todd-Mizuno self-dual embedding (for solving the extended economic dispatch) and the Moré variant of the Levenberg-Marquardt algorithm (for solving the APF equations) together take sub-second run times on a consumer-grade machine.

This work also demonstrates the use of APF notions as auxiliary tools in two scenarios concerning optimal power flow (OPF). First, OPF solvers can be warm-started with initial iterates given by the anticipated supply injections and the solution to their corresponding APF equations. With a disciplined variation of the tunable parameter of the extended economic dispatch, this is a crude method for finding multiple OPF solutions. Second, given a set of supply injections and bus voltages that optimize an approximated OPF model (e.g., DCOPF), one can find their nearest power-flow feasible values simply by solving their corresponding APF equations. This post-processing refinement easily extends to more general scenarios involving time-series approximated-OPF solutions.

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 contexts, the most immediate of which is the notion of the power flow equations as a differentiable computational layer in physics-inspired deep learning.