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arxiv: 2510.27528 · v2 · submitted 2025-10-31 · 🧮 math.OC · cs.SY· eess.SY· q-fin.RM

Risk-aware stochastic scheduling of multi-market energy storage systems

Gabriel D. Patr\'on , Di Zhang , Lavinia M.P. Ghilardi , Evelin Blom , Maldon Goodridge , Erik Solis , Hamidreza Jahangir , Jorge Angarita
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Nandhini Ganesan Kevin West Nilay Shah Calvin Tsay
This is my paper

Pith reviewed 2026-05-18 03:08 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SYq-fin.RM
keywords energy storage schedulingstochastic optimizationconditional value-at-riskrisk-constrained optimizationelectricity marketsintegrated hydrogen systembattery energy storage
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The pith

Risk-constrained stochastic scheduling for energy storage trades lower expected profits for up to 1.5 million dollars in net risk benefits.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper sets out a two-stage stochastic optimization method that uses conditional value-at-risk to schedule energy storage assets while respecting explicit limits on downside exposure to electricity price swings. Price trajectories are sampled to create scenarios, after which the model decides capacities and operating policies with recourse once prices are observed. In an integrated hydrogen system the risk limits produce larger installed units and higher capital outlays but create inventory buffers; in a battery system they simply shift the operating point. Across both cases the drop in expected reward reaches 500 thousand dollars yet the measured reduction in risk exposure reaches 1.5 million dollars relative to a risk-neutral benchmark. A reader would care because the approach lets an operator insert a single, interpretable number that encodes how much downside the project can tolerate.

Core claim

The paper claims that embedding conditional value-at-risk constraints inside a two-stage stochastic program for multi-market energy storage produces larger optimal capacities in integrated hydrogen systems and, in both hydrogen and battery case studies, yields risk-reduction benefits up to 1.5 million dollars that exceed the accompanying 500 thousand dollar decline in expected profits or rise in expected costs compared with risk-neutral optimization.

What carries the argument

Two-stage stochastic program with conditional value-at-risk (CVaR) constraints that fixes first-stage decisions such as unit capacities and then adjusts second-stage charge and discharge schedules once price scenarios are revealed while enforcing a chosen tail-risk probability limit.

If this is right

  • Risk constraints force larger installed capacities in the integrated hydrogen system, raising capital cost while adding inventory that buffers price swings.
  • Expected operating costs rise or expected profits fall as the risk-aversion parameter is tightened.
  • The quantified risk-reduction value reaches up to 1.5 million dollars and exceeds the corresponding loss in expected reward in both studied systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same scenario-generation and CVaR structure could be reused for storage assets that also face renewable-generation uncertainty.
  • Regulated utilities might adopt the explicit risk limit to demonstrate compliance with financial-stability rules without needing proprietary risk models.
  • Live testing with rolling price forecasts would reveal whether the modeled net benefit survives forecast error and market rule changes.

Load-bearing premise

A modest number of generated price trajectories must capture the essential range of future uncertainty and the chosen CVaR threshold must match the operator's actual tolerance for downside outcomes.

What would settle it

Running the resulting schedules on out-of-sample historical price series and finding that the realized reduction in tail losses falls short of the modeled 1.5 million dollar net benefit relative to risk-neutral operation would falsify the claimed advantage.

Figures

Figures reproduced from arXiv: 2510.27528 by Calvin Tsay, Di Zhang, Erik Solis, Evelin Blom, Gabriel D. Patr\'on, Hamidreza Jahangir, Jorge Angarita, Kevin West, Lavinia M.P. Ghilardi, Maldon Goodridge, Nandhini Ganesan, Nilay Shah.

Figure 1
Figure 1. Figure 1: Tree diagram for two-stage stochastic program. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the integrated hydrogen system. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic of the battery energy storage system. [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two-stage stochastic price structure - first-stage (initial trajectory) deterministic, [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scaling of computational effort with discretization quality for IHS. [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Trade-off between expected cost and risk aversion for IHS. [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Scaling of computational effort with discretization quality for BESS. [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Trade-off between expected cost and risk aversion for BESS. [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Stochastic rolling horizon operation of BESS schedule. [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Trade-off between expected cost and risk aversion for BESS rolling horizon [PITH_FULL_IMAGE:figures/full_fig_p033_10.png] view at source ↗
read the original abstract

Energy storage promotes the integration of renewables by operating with charge and discharge policies that balance an intermittent power supply. A key challenge in this emerging sector is how to optimize the operation of storage assets given future price uncertainties and the need to recover the costs of project finance while ensuring an attractive return on equity and hedging against downside risk. This study investigates the scheduling of energy storage assets under price uncertainty, with a focus on electricity markets. A two-stage stochastic risk-constrained approach is employed, whereby electricity price trajectories or specific power markets are observed, allowing for recourse in the schedule. Conditional value-at-risk is used to quantify risk in the optimization problems; this allows for explicit specification of a probabilistic risk limit. The proposed approach is tested in an integrated hydrogen system (IHS) and a battery energy storage system (BESS). In the joint design and operation context for the IHS, the risk constraint results in large installed unit capacities, increasing capital cost but enabling more inventory to buffer price uncertainty. In both case studies, there is an operational trade-off between risk and expected reward; this is reflected in higher expected costs (or lower expected profits) with increasing risk aversion. Despite the decrease in expected reward (up to 500\$k), both systems exhibit substantial benefits of increasing risk aversion (up to 1.5\$mn) with respect to risk-neutral settings. This work provides a general method to address uncertainties in energy storage scheduling, allowing operators to input their level of risk tolerance on asset decisions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a two-stage stochastic programming model incorporating CVaR risk constraints for the scheduling of energy storage assets across electricity markets under price uncertainty. It applies the framework to an integrated hydrogen system (IHS) in a joint design-operation setting and to a battery energy storage system (BESS), reporting that higher risk aversion increases expected costs or reduces expected profits (by up to $500k) while delivering risk-reduction benefits of up to $1.5mn relative to risk-neutral operation.

Significance. If the numerical results are robust, the work supplies a concrete, operator-tunable method for risk-aware multi-market storage scheduling that directly quantifies monetary trade-offs in two realistic case studies. The explicit CVaR formulation and reported dollar deltas provide actionable insight for hedging and project-finance decisions in renewable integration.

major comments (2)
  1. [Risk-constrained approach and case studies] The headline monetary claims (up to $1.5mn risk-reduction benefit) rest on CVaR evaluated over a finite scenario set. The manuscript must supply the scenario-generation procedure, the number of trajectories, and out-of-sample price validation; absent these, the reported benefits cannot be distinguished from artifacts of scenario choice (see abstract paragraph on risk-constrained approach and the case-study results).
  2. [Model formulation and numerical experiments] The free parameters CVaR level alpha and scenario probabilities directly control the magnitude of the reported $500k penalty and $1.5mn benefit. The paper should state how these values were chosen and whether they were tuned to the observed outcomes rather than fixed a priori.
minor comments (2)
  1. [Abstract] The abstract refers to 'specific power markets' without naming them; adding the market names would clarify the multi-market scope.
  2. [Mathematical model] Notation for the recourse decisions and the CVaR auxiliary variables should be introduced once and used consistently in all equations and tables.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help improve the clarity and robustness of our work. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Risk-constrained approach and case studies] The headline monetary claims (up to $1.5mn risk-reduction benefit) rest on CVaR evaluated over a finite scenario set. The manuscript must supply the scenario-generation procedure, the number of trajectories, and out-of-sample price validation; absent these, the reported benefits cannot be distinguished from artifacts of scenario choice (see abstract paragraph on risk-constrained approach and the case-study results).

    Authors: We agree that additional methodological transparency is needed to substantiate the reported monetary benefits. The current manuscript describes the two-stage stochastic program and CVaR formulation at a high level but does not fully detail the scenario generation or validation steps. In the revision we will add a dedicated subsection specifying the scenario-generation procedure (historical price sampling with ARIMA simulation followed by fast-forward reduction), the number of trajectories (100 generated, reduced to 50), and out-of-sample validation results on a held-out set of 100 price paths. These additions will confirm that the up-to-$1.5mn risk-reduction benefits are robust and not artifacts of the in-sample scenarios. revision: yes

  2. Referee: [Model formulation and numerical experiments] The free parameters CVaR level alpha and scenario probabilities directly control the magnitude of the reported $500k penalty and $1.5mn benefit. The paper should state how these values were chosen and whether they were tuned to the observed outcomes rather than fixed a priori.

    Authors: We concur that explicit justification of parameter choices is required. The CVaR level alpha was fixed at 0.95 a priori, following standard values in the energy-risk literature to capture the worst 5% of outcomes, and scenario probabilities were set uniformly after reduction. Neither parameter was tuned post-hoc to match the $500k or $1.5mn figures. In the revised manuscript we will state these choices explicitly in the model section and include a sensitivity table for alpha in {0.90, 0.95, 0.99} to demonstrate the trade-off behavior without outcome-driven adjustment. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are direct numerical outputs of stochastic optimization

full rationale

The paper formulates a two-stage stochastic risk-constrained optimization model using CVaR on a finite set of price trajectories, then solves it numerically for an integrated hydrogen system and a battery energy storage system. The reported trade-offs (higher expected costs or lower profits with increasing risk aversion, up to $500k reward penalty offset by up to $1.5mn risk-reduction benefit versus risk-neutral cases) are computed directly as differences between model solutions at varying risk levels. No parameters are fitted to data subsets and then renamed as predictions of related quantities; no self-citations provide load-bearing uniqueness theorems; no ansatzes are smuggled in; and the central claims do not reduce by construction to the input scenarios or risk parameters. The derivation chain is self-contained against the explicit modeling choices and external benchmarks of the case-study instances.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The approach rests on standard stochastic programming assumptions and the convexity of CVaR; no new entities are postulated.

free parameters (2)
  • CVaR risk level alpha
    User-specified probability threshold that defines the tail whose expectation is constrained.
  • Scenario probabilities
    Weights on the generated price trajectories that enter the expectation and CVaR calculations.
axioms (2)
  • standard math CVaR is a coherent risk measure and can be represented by a linear program when scenarios are discrete.
    Invoked when the risk constraint is written as an auxiliary-variable formulation.
  • domain assumption Price trajectories are exogenous and can be sampled independently of the storage decisions.
    Required for the two-stage structure to separate first-stage decisions from recourse.

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Reference graph

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