Risk-Averse Storage Arbitrage Model

• Risk-averse storage arbitrage is a stochastic self-scheduling framework that incorporates explicit risk controls like chance constraints and CVaR to manage downside profit exposure.
• The model couples energy storage state-of-charge dynamics with convex control and dynamic programming techniques to optimize arbitrage while ensuring reliability.
• Adaptive mechanisms such as stopping rewards, SoC range penalties, and conformal learning enable robust decision-making under various price uncertainty conditions.

Searching arXiv for the central paper and closely related risk-averse storage arbitrage work.

First, I’ll retrieve the main paper by arXiv id (Tapia et al., 6 Dec 2025), then a few adjacent papers on CVaR, conformal risk control, and robust/chance-constrained storage arbitrage to ground the article.

Searching for the main paper and related storage arbitrage research on arXiv.

Main query: (Tapia et al., 6 Dec 2025)

Related queries: "risk-averse storage arbitrage CVaR", "conformal storage arbitrage", "chance-constrained storage arbitrage"

Risk-averse analytical storage arbitrage models are formulations of energy storage self-scheduling in which arbitrage decisions over time are coupled to explicit controls on downside profit risk, reserve adequacy, or opportunity-cost uncertainty. In recent arXiv literature, the topic includes stochastic, online inventory optimization with chance-constrained terminal state of charge (SoC), CVaR-based self-scheduling, robust max–min dispatch under uncertainty sets, dynamic quantile-based risk measures in Markov decision processes, and conformal controllers that adjust conservativeness online without distributional assumptions (Tapia et al., 6 Dec 2025, Yurdakul et al., 2022, Wu et al., 14 Jan 2025, Jiang et al., 2015, Wu et al., 2 Nov 2025).

1. Formal problem class and core storage dynamics

A central formulation treats energy storage arbitrage as a stochastic, online inventory optimization problem over a horizon $t=1,\dots,T$. The storage chooses charge $c_t$, discharge $d_t$, and SoC $e_t$ under uncertain electricity prices $\boldsymbol{\lambda}_t$, with intertemporal dynamics, power limits, and an explicit reliability constraint on terminal SoC (Tapia et al., 6 Dec 2025). A representative model is

$$\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$$

Here $e_t$ is SoC, $P$ is the maximum charge/discharge rate, $\eta_c,\eta_d$ are charge/discharge efficiencies, $\hat\eta$ is the self-discharge factor, and $c_t$0 is a target SoC band for reliability (Tapia et al., 6 Dec 2025).

The same intertemporal structure also appears in analytical convex-control models in which the net addition to storage is written as $c_t$1, with capacity constraints $c_t$2, rate bounds $c_t$3, and convex cost functions $c_t$4 that can encode bid–ask spreads, inefficiencies, and market impact (Cruise et al., 2014). In that setting, the arbitrage problem is posed as

$c_t$5

which is equivalent to profit maximization up to sign convention (Cruise et al., 2014). A related stochastic dynamic programming formulation writes a Bellman recursion

$c_t$6

with $c_t$7, thereby making the marginal value of energy explicit in the control law (Zheng et al., 2021).

Within this problem class, “risk-averse” does not remove arbitrage; it reweights arbitrage against future reliability value, adverse price realizations, or tail-loss exposure. The resulting model remains an intertemporal control problem, but the terminal or continuation value of stored energy is no longer determined by expected price spread alone.

2. Principal risk-aversion mechanisms

A first mechanism is the chance-constrained terminal SoC. In the reachability formulation, the requirement

$c_t$8

makes the reliability level explicit: smaller $c_t$9 implies higher reliability and more conservative arbitrage (Tapia et al., 6 Dec 2025). The same paper uses a sample-average approximation with binary scenario indicators $d_t$0, so that at least a fraction $d_t$1 of scenarios must remain in the target band.

A second mechanism is the stopping-time reward. To discourage myopic discharge before critical hours, a binary stopping indicator $d_t$2 is introduced, with monotonicity $d_t$3 and charging/discharging disabled after stopping. The objective includes

$d_t$4

or, to pay only at the stopping time,

$d_t$5

so that earlier reserve-preserving behavior sacrifices arbitrage profit but receives higher reward (Tapia et al., 6 Dec 2025). This makes reliability an economically interpretable incentive rather than only an administrative hard constraint.

A third mechanism is the SoC range target penalty. In reinforcement learning form, the terminal reward is penalized when $d_t$6 by a piecewise quadratic term with factor $d_t$7; in the differentiable dispatch layer, the penalty becomes $d_t$8, which is convex quadratic and suitable for backpropagation (Tapia et al., 6 Dec 2025). These are soft reliability constraints: violating the target SoC costs profit.

A fourth mechanism is CVaR on stochastic profit. In risk-averse self-scheduling, the decision vector includes charge, discharge, SoC, a VaR threshold $d_t$9, and auxiliary variables $e_t$0, with

$e_t$1

subject to

$e_t$2

This is the Rockafellar–Uryasev linear programming representation of CVaR of profits and directly penalizes the worst $e_t$3 tail of outcomes (Yurdakul et al., 2022).

A fifth mechanism is robust feasibility. In joint arbitrage and ancillary service bidding, the operator chooses a schedule and capacity bids that must remain feasible for all regulation signals $e_t$4 in an uncertainty set $e_t$5. The risk-averse requirement is operational feasibility for all $e_t$6, which protects against SoC violations or non-delivery under worst-case activation patterns allowed by market rules (Lauinger et al., 12 Oct 2025).

Taken together, these formulations show that risk aversion in storage arbitrage is not limited to a single metric. In the cited models it appears as chance constraints on SoC, stopping incentives, terminal penalties, CVaR of profits, and robust worst-case feasibility.

3. Analytical control structure and dynamic programming

The analytical core of storage arbitrage is the marginal value of stored energy. In convex-control formulations, there exists a shadow-price sequence $e_t$7 such that $e_t$8 minimizes

$e_t$9

over feasible $\boldsymbol{\lambda}_t$0, while complementary slackness links $\boldsymbol{\lambda}_t$1 to the capacity boundaries (Cruise et al., 2014). When $\boldsymbol{\lambda}_t$2, the shadow price evolves deterministically; when the store is at 0 or $\boldsymbol{\lambda}_t$3, the shadow price can kink. The same line of work proves that the optimal management decision depends only a finite, and typically short, time horizon (Cruise et al., 2013, Cruise et al., 2014). This finite-horizon locality is analytically important because it limits how far ahead uncertainty must be represented with high fidelity.

In stochastic dynamic programming, prices can be modeled as a Markov process with discretized price nodes and transition matrices. Under variable charge and discharge efficiencies, the value-to-go function is approximated as piecewise linear in SoC, and the derivative of the Q-function admits an explicit threshold structure: very low prices induce full charging, intermediate prices create partial charging or idling, and sufficiently high prices induce partial or full discharge (Zheng et al., 2021). This preserves computational tractability while keeping the physical SoC dynamics and efficiency dependence explicit.

Risk-averse dynamic programming generalizes the Bellman operator by replacing expectation with a one-step risk mapping. In a finite-horizon MDP, the objective can be written as a dynamic quantile-based risk measure,

$\boldsymbol{\lambda}_t$4

where the one-step mappings include VaR and CVaR as quantile-based risk measures (Jiang et al., 2015). The corresponding Bellman equations preserve a state–action value function, but the continuation value becomes a tail-sensitive functional rather than a linear expectation.

A complementary steady-state MDP formulation places CVaR on the distribution of stage costs induced by the stationary occupancy measure $\boldsymbol{\lambda}_t$5. The resulting mean–risk model is bilinear in the occupancy variables and the VaR threshold $\boldsymbol{\lambda}_t$6, but admits both a DC reformulation and an exact global method based on the structural result that an optimal $\boldsymbol{\lambda}_t$7 equals one of finitely many stage-cost values (Khojaste et al., 5 Jan 2026). This is analytically notable because it converts a seemingly nonconvex risk-averse control problem into a sequence of linear programs.

The common analytical pattern is therefore clear: SoC remains the endogenous inventory state, prices remain exogenous uncertainty, and risk aversion enters through the continuation operator or terminal value. What changes is not the intertemporal nature of arbitrage, but the valuation rule applied to future scenarios.

4. Learning-based and distribution-free risk-aware formulations

Recent work embeds the analytical dispatch problem inside end-to-end learning. In the reachability framework, a predictor outputs both a center $\boldsymbol{\lambda}_t$8 and an uncertainty set $\boldsymbol{\lambda}_t$9, while the dispatch layer solves

$$\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$$0

subject to relaxed stopping logic $$\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$$1, SoC bounds, and linear dynamics (Tapia et al., 6 Dec 2025). Conformal calibration of residuals provides finite-sample coverage of price realizations, and gradients are propagated through the dispatch layer by implicit differentiation. In this setting, risk aversion is tuned by the stopping reward sequence $$\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$$2, the SoC penalty weight $$\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$$3, and the uncertainty set design.

A distinct distribution-free approach constructs conformal prediction intervals for real-time prices and then samples multiple price paths within those intervals. For each sampled path, a deterministic arbitrage problem is solved; the controller charges only if all scenarios agree on charging, discharges only if all scenarios agree on discharging, and otherwise idles (Alghumayjan et al., 2024). The method is explicitly conservative: it only acts when directional agreement holds across plausible price scenarios. Under a good forecaster, the aggressive mode earns $\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$44,438.98, compared with $\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$512,936.43 for the point-forecast policy; this is described as about 34% of the point-forecast’s purchases (Alghumayjan et al., 2024).

An online conformal controller extends this logic to value-function uncertainty. The controller builds a symmetric prediction set around the forecast marginal value $\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$6, modifies the analytical charge/discharge thresholds accordingly, and updates a conformal control variable $\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$7 by

$\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$8

where the loss $\begin{aligned} \max_{e,c,d}\quad & \mathbb{E}_{\boldsymbol{\lambda}}\Bigg[\sum_{t=1}^T \boldsymbol{\lambda}_t (d_t - c_t)\Bigg] \ \text{s.t.}\quad & e_{0} = e_0 \ & e^{\min} \le e_t \le e^{\max}, \quad \forall t \ & e_t = \hat{\eta} e_{t-1} - \frac{d_t}{\eta_d} + c_t \eta_c, \quad \forall t \ & 0 \le d_t \le P,\quad 0 \le c_t \le P,\quad \forall t \ & \mathbb{P}_{\boldsymbol{\lambda}}\{ e_{\tau} \in \mathcal{E}^{\ell} \} \ge 1 - \epsilon,\quad \forall \tau \in \mathcal{T}^{\rm target}. \end{aligned}$9 is based on either prediction error or value error (Wu et al., 2 Nov 2025). Because online profit loss feedback is unobservable, the paper establishes that a temporal difference error serves as a measurable proxy. The resulting controller proves bounded long-term risk with convergence guarantees in temporal difference error, which further effectively manages risk exposure in potential profit losses (Wu et al., 2 Nov 2025).

These learning-based models do not replace analytical arbitrage structure; they parameterize it. The decision rule still depends on SoC dynamics, continuation values, and threshold comparisons, but the uncertainty representation is learned, calibrated, or updated online rather than fixed ex ante.

5. Reported trade-offs between profit, variance, and reliability

The empirical literature consistently reports a profit–risk trade-off rather than a single dominant operating point. In the reachability study, the same 24-hour case is solved by scenario-based sample-average approximation (SAA), DQN, and end-to-end learning, with target band $e_t$0 MWh and target SoC $e_t$1 MWh (Tapia et al., 6 Dec 2025).

Method	Illustrative operating point	Reported risk effect
SAA	$e_t$2	deviation $e_t$3
SAA	$e_t$4	std dev reduces to $e_t$5
DQN	$e_t$6	std dev $e_t$7
DQN	$e_t$8	$e_t$9 hr; std deviation $P$0
E2E	$P$1	strong robustness
E2E	$P$2: profit $P$3	std dev $P$4
E2E	$P$5	std dev $P$6; coverage above 94%

These results show reduced profit variance under E2E, consistent stopping times, and high reliability of SoC (Tapia et al., 6 Dec 2025). They also show that a stopping reward can change the feasible operating region itself, not merely re-rank otherwise identical schedules.

Across uncertainty-set models, efficient frontiers highlight the tradeoff between risk and profit. Robust and chance-constrained optimization approaches are both used, but robust strategies perform better in risk management across varying levels of conservativeness, especially under highly volatile market conditions (Wu et al., 14 Jan 2025). In that study, risk is operationalized as the number of days with negative realized profit per year, and the uncertainty budget $P$7 or confidence level $P$8 acts as the control knob moving the strategy along the frontier (Wu et al., 14 Jan 2025).

A recurring policy interpretation is that non-response to strong price signals need not indicate malfunction or weak incentives. In stochastic self-scheduling on Australian market data, risk-averse storage resources tend to have a myopic operational perspective, that is, they typically engage in near-term price arbitrage and chase only few extreme price spikes and troughs, thus remaining idle in several time periods with markedly high and low prices (Yurdakul et al., 2022). This is consistent with the reachability motivation that, without accounting for the future reliability value of stored energy, batteries may discharge too early or fail to preserve reserves during critical hours (Tapia et al., 6 Dec 2025).

6. Assumptions, controversies, and extension paths

The current analytical literature makes several simplifying assumptions. In the reachability model, the battery is a single storage asset with simple linear SoC dynamics, no degradation, no cycle limits, and no network constraints or multi-node interactions (Tapia et al., 6 Dec 2025). Critical hours $P$9 are assumed known ex ante, often reduced to $\eta_c,\eta_d$0, and the chance constraint is imposed only on final SoC rather than on a multi-stage reliability event (Tapia et al., 6 Dec 2025). In end-to-end learning, the stopping variable $\eta_c,\eta_d$1 is relaxed to $\eta_c,\eta_d$2, which approximates but does not exactly reproduce binary stopping (Tapia et al., 6 Dec 2025).

Another limitation is the treatment of uncertainty. Some formulations use historical scenarios, some assume normal or lognormal approximations, some use i.i.d. training samples, and some use conformal sets without a structural price model (Wu et al., 14 Jan 2025, Alghumayjan et al., 2024). This has led to a mild methodological controversy over whether risk should be represented probabilistically, robustly, or distribution-free. The literature does not resolve this by declaring one universal metric; instead, it shows that the choice depends on the operational question. Reliability of terminal SoC, worst-case market feasibility, tail profit loss, and long-run calibration error are distinct objects.

Extensions now move in three main directions. One direction is richer risk modeling. A tri-objective newsvendor framework for capacity reservation under non-normal uncertainty jointly maximizes expected profit, minimizes CVaR tail loss, and minimizes maximum regret over candidate non-normal distributions; in stylized log-NMVM experiments with matched mean and variance, a moment-matched normal policy over-reserves capacity by up to 1.337 units (Abudurexiti, 6 Jul 2026). This suggests that reduced-form pre-commitment layers can be attached to multi-period arbitrage when the key decision is how much capacity to withhold for an uncertain future window.

A second direction is degradation-aware and multi-service coordination. A degradation-infused energy portfolio allocation framework embeds a closed-form marginal degradation profile into a multi-battery, multi-market dispatch problem and equips the portfolio formulation with a CVaR-based mechanism to bring risk-averseness against uncertainty in market prices (Pareek et al., 2024). In that setting, arbitrage is no longer evaluated only against price spreads or SoC reachability, but against long-run asset wear and fair allocation of coalition profit.

A third direction is explicit operator-risk quantification and predictive control. The reachability paper suggests future work toward predictive control with stopping rewards, extending stopping-reward design to quantify operator’s risk more explicitly, and applying the framework to other time-series dynamics and resilience attributes (Tapia et al., 6 Dec 2025). A plausible implication is that the analytical storage arbitrage model is increasingly becoming a modular object: a physical inventory model, a risk operator, and a learned or calibrated uncertainty representation, all coupled through the marginal value of energy.