Conditions for Estimation of Sensitivities of Voltage Magnitudes to Complex Power Injections

TL;DR

This paper investigates the conditions under which sensitivities of voltage magnitudes to complex power injections can be uniquely estimated from sensor data, extending classical power flow models and employing matrix completion techniques.

Contribution

It introduces novel conditions for the uniqueness of sensitivity estimations and develops methods using matrix completion for sensor-limited scenarios.

Findings

Sensitivity of voltage magnitude to active power is unique and differs from reactive power sensitivities.

Derived conditions based on bus power factors and system eigenvalues ensure solution uniqueness.

Simulation results verify the effectiveness of the proposed estimation methods.

Abstract

Voltage phase angle measurements are often unavailable from sensors in distribution networks and transmission network boundaries. Therefore, this paper addresses the conditions for estimating sensitivities of voltage magnitudes with respect to complex (active and reactive) electric power injections based on sensor measurements. These sensitivities represent submatrices of the inverse power flow Jacobian. We extend previous results to show that the sensitivities of a bus voltage magnitude with respect to active power injections are unique and different from those with respect to reactive power. The classical Newton-Raphson power flow model is used to derive a novel representation of bus voltage magnitudes as an underdetermined linear operator of the active and reactive power injections; parameterized by the bus power factors. Two conditions that ensure the existence of unique complex power injections given voltage magnitudes are established for this underdetermined linear system, thereby compressing the solution space. The first is a sufficient condition based on the bus power factors. The second is a necessary and sufficient condition based on the system eigenvalues. We use matrix completion theory to develop estimation methods for recovering sensitivity matrices with varying levels of sensor availability. Simulations verify the results and demonstrate engineering use of the proposed methods.