Impact of Power Fluctuations on Frequency Quality

TL;DR

The paper develops a rigorous analytical framework linking stochastic power injections to frequency quality via the Complex Frequency formulation, extending deterministic relations to a stochastic setting and clarifying when the Central Limit Theorem is applicable in power systems. A key result is the frequency relation $\bm{\omega}(t) = \bm{H}(t) \dot{\bm{p}}(t) + \bm{K}(t) \dot{\bm{q}}(t)$, complemented by the CoI frequency expression $\omega_{\rm CoI}(t) = \bm{c}^{\top} \bm{\omega}(t) + \alpha$, and extended to stochastic differential equations for injections. The work demonstrates that frequency deviations are linear in injection noise, with topology and impedance strongly shaping distributions, often invalidating Gaussian assumptions especially at transmission scales. Case studies on the IEEE 14-bus and All-Island Irish systems illustrate improved transient accuracy and reveal non-Gaussian frequency statistics arising from uneven bus weighting. The findings have practical implications for frequency control, wind-generation reconciliation, and risk assessment in modern, low-inertia grids.

Abstract

This paper analyzes how power injections affect frequency quality in power systems. We first derive a general expression linking active and reactive power injections at buses to the system's frequency. This formulation explicitly considers both real and imaginary frequency components, providing a complete description of frequency behavior in power systems during transients. Next, we extend our analysis to incorporate stochastic variations of power injections. Using the frequency divider concept and power-based frequency estimation, we develop analytical relationships linking stochastic load fluctuations to frequency deviations. We discuss under which conditions the Central Limit Theorem cannot be applied to capture the frequency distribution, thereby clarifying how its hypotheses are not satisfied in power system applications. Then, we establish clear criteria for the appropriate use of statistical methods in frequency analysis. Finally, we validate our theoretical results through simulations on modified IEEE 14-bus and all-island Irish transmission test systems, highlighting the accuracy, practical utility, and limitations of our proposed formulation.

Figures (11)

• Figure 1: Stochasticity propagation -- Distribution system.
• Figure 2: Stochasticity propagation -- Transmission system.
• Figure 3: Calculated vs. simulated frequency response using different formulations.
• Figure 4: Weibull-distributed power injection at Bus 10.
• Figure 5: Weibull-distributed power injection at Bus 12.