The Welfare Gap of Strategic Storage: Universal Bounds and Price Non-Linearity

TL;DR

The paper addresses welfare efficiency in electricity markets with battery storage under stochastic demand, characterizing the efficiency loss from centralized versus profit-maximizing operation. It adopts a general continuous-time framework and derives a universal $PoA=4/3$ bound for linear price functions, while showing that convex price functions can yield unbounded inefficiency; it then analyzes monomial price functions, revealing that price curvature controlled by the degree $d$ yields an upper bound $PoA\leq 2$, with tight results for $d=2$ ($PoA=27/19$) and a degree-dependent lower bound $1/(1-(\frac{d}{d+1})^{d+1})$, illustrating the price of non-linearity. The results establish a clear boundary between the well-behaved linear regime and the problematic non-linear regime, offering policy insights on price design and regulatory interventions to safeguard welfare. They also provide a roadmap for extending the analysis to multi-agent storage, transmission losses, and broader non-linear price structures. Overall, the work delivers analytically robust guarantees for system operators and policymakers navigating inter-temporal storage in uncertain demand environments.

Abstract

This paper studies the efficiency of battery storage operations in electricity markets by comparing the social welfare gain achieved by a central planner to that of a decentralized profit-maximizing operator. The problem is formulated in a generalized continuous-time stochastic setting, where the battery follows an adaptive, non-anticipating policy subject to periodicity and other constraints. We quantify the efficiency loss by bounding the ratio of the optimal welfare gain to the gain under profit maximization. First, for linear price functions, we prove that this ratio is tightly bounded by $4/3$. We show that this bound is a structural invariant: it is robust to arbitrary stochastic demand processes and accommodates general convex operational constraints. Second, we demonstrate that the efficiency loss can be unbounded for general convex price functions, implying that convexity alone is insufficient to guarantee market efficiency. Finally, to bridge these regimes, we analyze monomial price functions, where the degree controls the curvature. For specific discrete demand scenarios, we demonstrate that the ratio is bounded by $2$, independent of the degree.

Figures (2)

• Figure 1: The figure shows the stochastic process defined in Example \ref{['example:demand']}. The gray dashed line depicts the deterministic mean trend. The solid blue curves show two realizations of the stochastic process, while the gray shaded region defines the fluctuation envelope.
• Figure 2: Illustration of Example \ref{['exp:battery']}. The left panel compares the original demand with the net demand adjusted by Centralized (CB) and Decentralized (DCB) batteries. The middle panel depicts the corresponding battery operation policies, while the right panel shows the instantaneous social generation costs before and after adjustment. Note that the revenue of the social welfare maximizing solution is zero, since the demand is completely smoothed.

Theorems & Definitions (21)

• Example 1
• Definition 1
• Remark 1
• Example 2
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Definition 2