Reachability Guarantees for Energy Arbitrage

TL;DR

The paper addresses energy arbitrage for battery storage under price uncertainty while enforcing a chance-constrained terminal SoC band. It develops a unified framework combining time-dependent $k$-search thresholds with a forward-propagation probability redistribution to compute the end-of-horizon SoC distribution and a minimum stopping time to satisfy the constraint, alongside a conformalized quantile regression (CQR) validation approach for out-of-sample uncertainty. The key contributions include the introduction of time-varying thresholds, a stopping-time pricing mechanism, a pruning algorithm for feasible SoC trajectories, and a data-driven validation pipeline using PJM data. The results reveal a strong dependence of terminal SoC reachability on initial state and start time, highlight a trade-off between reliability and arbitrage profit, and demonstrate that CQR provides distribution-free coverage guarantees for the terminal SoC under the threshold policy.

Abstract

This paper introduces a unified framework for battery energy arbitrage under uncertain market prices that integrates chance-constrained terminal state-of-charge requirements with online threshold policies. We first cast the multi-interval arbitrage problem as a stochastic dynamic program enhanced by a probabilistic end-of-horizon state-of-charge (SoC) constraint, ensuring with high confidence that the battery terminates within a prescribed energy band. We then apply a $k$-search algorithm to derive explicit charging (buying) and discharging (selling) thresholds with provable worst-case competitive ratio, and compute the corresponding action probabilities over the decision horizon. To compute exact distributions under operational limits, we develop a probability redistribution pruning method and use it to quantify the likelihood of meeting the terminal SoC band. Leveraging the resulting SoC distribution, we estimate the minimum stopping-time required to satisfy the SoC chance constraint. Computational experiments on historical real price data demonstrate that the proposed framework substantially improves the estimation of SoC evolution and supports chance-constraint satisfaction.

Figures (12)

• Figure 1: State-of-charge and action trajectories for the battery arbitrage problem.
• Figure 2: Full-day price uncertainty set based on the PJM real-time prices data from 2011-2016. Dashed orange lines represent the minimum ($\lambda^{\min}$) and maximum ($\lambda^{\max}$) price values across the time horizon, while the dashed blue lines bound the possible realization for the energy price ($\boldsymbol{\lambda}_t$).
• Figure 3: Pruning and probability redistribution algorithm.
• Figure 4: (a) Non-pruning SoC distribution over time. (b) Pruning SoC distribution over time. (c) Non-pruning terminal SoC distribution. (d) Pruning terminal SoC distribution.
• Figure 5: Stopping-time for the battery arbitrage problem.