Terminal energy valuation and end-of-horizon effects in battery arbitrage scheduling

Abstract

Introduction: Battery energy storage systems scheduled over finite horizons may undervalue terminal stored energy, producing the end-of-horizon effect (EOHE). Terminal constraints, salvage values, stochastic optimization, and risk measures address this boundary condition through different economic mechanisms. Materials and methods: Four linked experiments are conducted. A 12-context synthetic benchmark compares five formulations, including stochastic optimization with and without terminal salvage. Hourly dayahead wholesale electricity prices for Italy and Finland in 2024 are used to construct fixed ensembles of 1000 winter-weekday trajectories through a regime-aware, locally smoothed whole-day bootstrap. Terminalvalue ablation evaluates 𝜅 ∈ {0, 0.85, 1}. A fixed-sample day-ahead Conditional Value-at-Risk (CVaR) experiment evaluates 𝛼 ∈ {0.90, 0.95, 0.99} and 𝜆 ∈ {0, 0.25, 0.50, 0.75, 1}. A 30-day controlled rolling-horizon experiment implements only the first decision of each 24 h optimization and carries the realized state of charge forward. Results: Terminal valuation produces market-dependent depletion, restoration, and complete-preservation regimes. In Italy, 𝜅 = 0 and 0.85 select the same depletion policy, while 𝜅 = 1 produces complete preservation. Finland shows a gradual transition from depletion to partial and full restoration. CVaR changes the common day-ahead schedule and generates an expected-return/downside-risk trade-off. Under extreme risk aversion, Finland selects inactivity, whereas Italy may convert operating cash into terminal inventory. In the controlled rolling experiment, the intermediate setting 𝜅 = 0.85 improves Italian realized cash by EUR 58.59 relative to stochastic optimization without salvage, while 𝜅 = 1 suppresses arbitrage and preserves the full inventory. Conclusions: Terminal value and CVaR are not interchangeable corrections. Terminal value determines what economic quantity is preserved, while CVaR determines which part of the payoff distribution is protected. The results support EOHE as a terminal-energy valuation problem, but multi-year data and genuine sequential out-of-sample testing remain necessary before broad market generalization.