Impact of Power Fluctuations on Frequency Quality

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.