Reachability Guarantees for Energy Arbitrage
Batteries performing energy arbitrage may overdischarge ahead of a high-demand or critical operating hour when they re-optimize in real time and depart from the day-ahead schedule. This paper introduces a framework for battery energy arbitrage under uncertain market prices that integrates chance-constrained terminal state-of-charge (SoC) 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 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 a 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 improves SoC estimation and supports chance-constraint satisfaction, while also showing that reachability is highly sensitive to a given initial SoC and operating start time.