Chance-Constrained Terminal SoC

• Chance-constrained terminal SoC is a probabilistic requirement ensuring energy storage systems meet prescribed terminal state-of-charge bounds under uncertainty.
• It employs techniques such as analytical tightening, risk measure relaxations, and scenario-based methods to balance economic benefits with reliability and device health.
• Applications span grid scheduling, arbitrage, and electric mobility, integrating stochastic model predictive control to optimize performance in uncertain environments.

A chance-constrained terminal state-of-charge (SoC) constraint is a probabilistic requirement applied at the terminal point of a control or dispatch horizon, enforcing that the SoC of an energy storage system (ESS)—such as a battery, aggregation of distributed resources, or electric bus—lies within prescribed bounds with high probability, despite uncertainties in dynamics, disturbances, or measurement. This concept has become central in stochastic model predictive control (SMPC), multi-stage scheduling, arbitrage, and risk-averse battery management. The goal of such constraints is to explicitly trade off economic benefit, reliability, and device health, while remaining computationally tractable even under high-fidelity uncertainty models.

1. Mathematical Formulation of Terminal SoC Chance Constraints

Chance-constrained terminal SoC enforces that, for a given violation tolerance $\epsilon \in (0,1)$ and target set $\mathcal{E}$ (typically a range $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$), the terminal SoC $x(T)$ (or $e_T$, $s_T$) satisfies

$$\mathbb{P}\bigl(x(T) \in \mathcal{E}\bigr) \geq 1-\epsilon.$$

This formulation is applied in diverse settings, including discrete-time LTI systems under arbitrary stochastic disturbances (Ghosh et al., 2024), scenario-based stochastic programming (Lei et al., 2021), receding horizon battery arbitrage (Tapia et al., 6 Dec 2025, Tapia et al., 17 Jan 2026), bus scheduling (Ricard et al., 25 Mar 2025), and virtual energy storage management (Amini et al., 2018). The chance constraint may be single-sided ($$\mathbb{P}(x(T)\geq\underline{\text{SoC}})\geq 1-\epsilon$$) or double-sided.

Terminal SoC chance constraints serve both to guarantee operational requirements (e.g., grid support, ride completion, scheduled delivery) and to enforce regime constraints supporting battery health (e.g., avoid deep discharge).

2. Reformulation and Tractability: Deterministic Cuts and Risk Approximations

Enforcing such constraints directly is intractable for most non-Gaussian or high-dimensional problems. Convex risk measure approximations or deterministic surrogate constraints are prevalent:

• Analytical tightening: For additive stochastic disturbances or measurement noise $\xi$ with known/estimated moments, Chebyshev- or Gaussian-based quantile bounds yield deterministic surrogates. For example, the tightening (Amini et al., 2018, Qi et al., 2022, Tookanlou et al., 2021) is

$$x(T) \geq \mu_{\underline{\text{SoC}}} + \kappa(\epsilon)\cdot \sigma_{\underline{\text{SoC}}}$$

with $\mathcal{E}$0 (Chebyshev), $\mathcal{E}$1 (Gaussian), or other distribution-specific constants.

• Risk measure relaxation: Using Average Value-at-Risk (AVaR) or other coherent risk measures to convexify the feasible set, as in (Lei et al., 2021). For SoC error $\mathcal{E}$2, AVaR approximations allow tractable conic constraints:

$\mathcal{E}$3

This both bounds violation probability and penalizes its magnitude.

• Scenario approaches: Under empirical distributions or via sample-averaged approximation (SAA), the chance constraint is enforced over a set of $\mathcal{E}$4 scenarios. Typically, variables $\mathcal{E}$5 flag constraint satisfaction per scenario, and

$\mathcal{E}$6

(Tapia et al., 6 Dec 2025). Logical implication is relaxed to linear constraints in mixed-integer form.

• Distributionally-robust formulations: Using ambiguity sets (e.g., all distributions with specified mean/covariance) to ensure robust satisfaction. Second-order cone (SOCP) reformulations provide exact convex constraints, e.g.

$\mathcal{E}$7

(Tookanlou et al., 2021).

3. Adaptive Relaxation and Policy Iteration

Recent advances replace conservative, a priori constraint designs with online adaptive relaxation:

• Adaptive update rule: The adaptive relaxation based method (Ghosh et al., 2024) adjusts the constraint right-hand-side $\mathcal{E}$8 online by tracking the empirical violation rate $\mathcal{E}$9. For each constraint $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$0, at time $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$1,

$[\underline{\text{SoC}},\,\overline{\text{SoC}}]$2

Tightening occurs if empirical violations exceed $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$3, otherwise relaxation increases. Under ideal control, $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$4 almost surely.

• Stopping-time reward and reachability: In arbitrage, a stopping-time reward is designed such that the optimal policy switches from profit-driven to reliability-driven as critical SoC is approached (Tapia et al., 6 Dec 2025, Tapia et al., 17 Jan 2026). This is formalized by a binary indicator and a reward rate, optimizing both arbitrage profit and the terminal SoC chance constraint.
• Policy iteration for DDU bounds: When SoC limits are themselves random and decision-dependent, robust-approximation may be combined with iterative quantile estimation—solving the control problem, simulating realized distributions, updating quantiles, and iterating to convergence (Qi et al., 2022).

4. Applications: Grid, Mobility, and Aggregation

Chance-constrained terminal SoC arises in a spectrum of real-world settings:

• Microgrid and BESS scheduling: SMPC dispatches for grid-connected BESS with PV/load under unknown disturbances, using adaptive constraint relaxation to limit SOC violation probability while saving cost and improving peak shaving (Ghosh et al., 2024, Lei et al., 2021).
• Arbitrage and market participation: Threshold policies and stochastic DP with SoC band targets balance arbitrage profit against terminal SoC reliability, using distribution-aware pruning and stopping-time computation (Tapia et al., 6 Dec 2025, Tapia et al., 17 Jan 2026).
• Mobility and fleet operations: Electric bus scheduling with random energy use enforces that, fleet-wide, the chance no bus under-discharges below a health band is at least $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$5, optimizing both cost and battery wear metrics (Ricard et al., 25 Mar 2025).
• Distributed and virtual storage: Coordinated management of VESS (Amini et al., 2018) and generic storage (Qi et al., 2022) incorporates both exogenous and decision-dependent uncertainty, with loss-of-response and energy-not-served indices to evaluate reliability after implementation.

5. Control, Optimization, and Solution Methodologies

Implementations reflect the underlying system and uncertainty structure:

• Model predictive control: SMPC with analytical or scenario-based terminal SoC chance constraints, often exploiting convex relaxations, scenario decomposition, or conic reformulation (Ghosh et al., 2024, Lei et al., 2021, Amini et al., 2018).
• Dynamic programming and online policies: Stochastic DP recursions modified to embed feasibility via $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$6 terminal cost penalties or auxiliary reachability layers, complemented by provably competitive $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$7-search threshold algorithms (Tapia et al., 17 Jan 2026).
• Convex programming under distributions: Second-order cone and semidefinite relaxations enable exact enforcement under Gaussian mixture or worst-case moment ambiguity models. For instance, all kernels in a mixture must hit the terminal mean, and the mixture covariance must satisfy a Ky Fan-type constraint (Kumagai et al., 2024).
• Mixed-integer and decomposition methods: Large-scale, scenario-based, or multi-agent problems are addressed by branch-and-price (with stochastic pricing/labeling) or iterative decomposition, especially in fleet, e-mobility, or aggregator contexts (Ricard et al., 25 Mar 2025, Tookanlou et al., 2021).

6. Empirical Performance, Trade-Offs, and Tuning

Empirically, chance-constrained terminal SoC enables explicit trade-off between cost, flexibility, and reliability:

• Cost vs. reliability: Allowing small violation probabilities ($[\underline{\text{SoC}},\,\overline{\text{SoC}}]$8 of 0.05–0.1) yields operational cost savings (often several percent) and reduced fleet size, with only marginal impact on device lifetime or risk (Ghosh et al., 2024, Ricard et al., 25 Mar 2025, Lei et al., 2021). There is diminishing return for $[\underline{\text{SoC}},\,\overline{\text{SoC}}]$9.
• Coverage and reachability: Empirical coverage of the SoC band can be directly tuned, and minimum stopping times can be computed to ensure prescribed reachability probabilities (Tapia et al., 6 Dec 2025, Tapia et al., 17 Jan 2026).
• Online adaptation: Adaptive methods quickly drive empirical rates to prescribed thresholds; mid-horizon re-initialization can capture further savings with controlled overshoot that regresses over longer horizons.
• Reliability indices: Post-deployment metrics such as loss of response probability and energy not served provide feedback on real-world performance and model accuracy (Qi et al., 2022).

7. Extensions and Research Directions

Recent work expands chance-constrained terminal SoC to settings with decision-dependent uncertainty, nonlinear device and fleet dynamics, advanced risk metrics (AVaR, distributionally-robust ambiguity sets), and adaptive, data-driven control layers. End-to-end learning integrates uncertainty-aware forecasting with robust optimal dispatch, co-training both components to maximize expected profit while satisfying prescribed reliability targets (Tapia et al., 6 Dec 2025). Further directions include scalable decentralized algorithms, integration with device degradation and market settlement, and higher-order joint and pathwise risk constraints.

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Chance-constrained terminal SoC has thus emerged as a unifying paradigm—mathematically rigorous, operationally meaningful, and computationally tractable—for balancing economic, reliability, and health objectives in stochastic battery management across electric power, transportation, and distributed resource ecosystems (Ghosh et al., 2024, Lei et al., 2021, Tapia et al., 6 Dec 2025, Tapia et al., 17 Jan 2026, Ricard et al., 25 Mar 2025, Amini et al., 2018, Qi et al., 2022, Kumagai et al., 2024, Tookanlou et al., 2021, Bank et al., 2016).