Quadratic Performance Analysis of Secondary Frequency Controllers

TL;DR

This paper analyzes the input-output performance of secondary frequency controllers using $H_2$ norms, comparing aggregated averaging, distributed averaging, and primal-dual controllers in terms of scalability and efficiency.

Contribution

It provides exact analytical formulas for controller performance and compares different controller types, highlighting their scalability and optimality conditions.

Findings

Aggregated averaging controllers have size-independent performance driven by control gain.

Plain primal-dual controllers scale poorly and do not outperform feedforward methods.

Distributed averaging controllers scale sub-linearly and are size-independent at high gain.

Abstract

This paper investigates the input-output performance of secondary frequency controllers through the control-theoretic notion of $H_2$ norms. We consider a quadratic objective accounting for the cost of reserve procurement and provide exact analytical formulae for the performance of continuous-time aggregated averaging controllers. Then, we contrast it with distributed averaging controllers -- seeking optimality conditions such as identical marginal costs -- and primal-dual controllers which have gained attention as systematic techniques to design distributed algorithms solving convex optimization problems. Our conclusion is that while the performance of aggregated averaging controllers, such as gather & broadcast, is independent of the system size and driven predominantly by the control gain, the plain vanilla closed-loop primal-dual controllers scale poorly with size and do not offer any improvement over feedforward primal-dual controllers. Finally, distributed averaging-based controllers scale sub-linearly with size and are independent of system size in the high-gain limit.