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arxiv: 2511.09427 · v2 · submitted 2025-11-12 · 🧮 math.OC · cs.LG· cs.SY· eess.SY

Adversarially and Distributionally Robust Virtual Energy Storage Systems via the Scenario Approach

Georgios Pantazis , Nicola Mignoni , Raffaele Carli , Mariagrazia Dotoli , Sergio Grammatico This is my paper

Pith reviewed 2026-05-17 22:22 UTC · model grok-4.3

classification 🧮 math.OC cs.LGcs.SYeess.SY
keywords virtual energy storageelectric vehiclesscenario approachdistributionally robust optimizationadversarial robustnessconvex schedulingparking lot management
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The pith

A convex data-driven scheduling framework provides finite-sample distribution-free guarantees for virtual energy storage from aggregated EV batteries.

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

The paper develops a scheduling method for parking lots to offer virtual energy storage using aggregated electric vehicle batteries under uncertain departures and state-of-charge limits. It introduces a convex optimization approach that supplies finite-sample guarantees on constraint violations without assuming any particular probability distribution for the uncertainty. The formulation lets the manager directly adjust the balance between maximizing profits and maintaining operational safety. Extensions incorporate adversarial perturbations to the data and Wasserstein distributional shifts to certify robustness against corrupted samples and out-of-distribution behavior.

Core claim

We propose a convex data-driven scheduling framework for virtual energy storage services based on the aggregation of EV batteries in parking lots that yields finite-sample, distribution-free guarantees on constraint violations and allows the parking lot manager to explicitly tune the trade-off between economic performance and operational safety. Extensions to adversarial perturbations of the training samples and Wasserstein distributional shifts obtain robustness certificates against both corrupted data and out-of-distribution uncertainty.

What carries the argument

The scenario approach applied to a convex formulation of the EV scheduling problem, extended with adversarial perturbations and Wasserstein distributional robustness.

If this is right

  • Only a finite number of historical samples is needed to certify an upper bound on future constraint violations.
  • A single tunable parameter lets the manager directly set the desired level of safety and optimize expected profit under that bound.
  • The same schedule remains certified even when some training data is adversarially altered or the underlying distribution of EV usage shifts.

Where Pith is reading between the lines

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

  • The same scenario-based convex framework could be applied to other aggregation problems such as demand-response or distributed battery systems.
  • Numerical consistency between predicted and observed violation rates indicates the theoretical bounds are not overly conservative for typical EV parking-lot data.
  • Adding online updating of the scenario set could reduce conservatism while preserving the distribution-free property.

Load-bearing premise

The uncertain EV departures and state-of-charge limits admit a convex formulation whose scenario-based approximation delivers the stated distribution-free guarantees without hidden dependence on the specific data distribution or post-hoc parameter choices.

What would settle it

Collect a large out-of-sample set of EV departure and state-of-charge realizations, apply the computed schedule, and count the fraction of constraint violations; if this fraction exceeds the theoretical upper bound given by the number of training scenarios, the finite-sample guarantee is falsified.

Figures

Figures reproduced from arXiv: 2511.09427 by Georgios Pantazis, Mariagrazia Dotoli, Nicola Mignoni, Raffaele Carli, Sergio Grammatico.

Figure 1
Figure 1. Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Trade-off stydy between adversarially robust empirical probability of violation vs the profit of the PLM for varying values of R. Stage k 1 2 3 4 5 6 7 8 9 10 11 12 b k 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 min–max values mean [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimal virtual state of charge bk of the PLM’s energy buffer at each time step k. the empirical probability distribution. To evaluate the out￾of-distribution (OOD) performance we consider a test data set (ℓ (i) , β (i) ), i ∈ {1, . . . , N}. The samples are obtained each time from N′ different probability distributions Pv, v ∈ {1, . . . , N′} obtained by perturbing the nominal probability distribution in … view at source ↗
Figure 4
Figure 4. Figure 4: Optimal energy rk sold to the retailer at each time step k. N 500 1000 2000 E m pirical O O D violation 0 0.1 0.2 0.3 0.4 Min-max of OOD violation Mean of OOD violation "(s $A;A^) + 7=R [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Empirical distributional violations among N ′ = 40 different distribution perturbations from the ambiguity set. R, or the number of samples N by trading some of the PLM’s profit in return. VI. CONCLUSION This paper develops a distributionally robust framework based on scenario optimization that enables a parking-lot manager to operate aggregated EVs as a virtual energy storage system, providing profit/risk… view at source ↗
read the original abstract

We study virtual energy storage services based on the aggregation of EV batteries in parking lots under time-varying, uncertain EV departures and state-of-charge limits. We propose a convex data-driven scheduling framework in which a parking lot manager provides storage services to a prosumer community while interacting with a retailer. The framework yields finite-sample, distribution-free guarantees on constraint violations and allows the parking lot manager to explicitly tune the trade-off between economic performance and operational safety. To enhance reliability under imperfect data, we extend the formulation to adversarial perturbations of the training samples and Wasserstein distributional shifts, obtaining robustness certificates against both corrupted data and out-of-distribution uncertainty. Numerical studies confirm the predicted profit-risk trade-off and show consistency between the theoretical certificates and the observed violation levels.

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

3 major / 2 minor

Summary. The manuscript develops a convex data-driven optimization framework for virtual energy storage aggregation from EV batteries in parking lots, incorporating uncertain departures and SoC limits. It applies the scenario approach to obtain finite-sample distribution-free guarantees on constraint violations, allows explicit tuning of the profit-risk trade-off via the violation probability, and extends the formulation to adversarial sample perturbations and Wasserstein distributional shifts for robustness certificates. Numerical studies are used to illustrate consistency between theoretical bounds and observed performance.

Significance. If the convexity and scenario guarantees are rigorously established without hidden distributional assumptions, the work provides a practical, certifiable method for EV-based storage services that explicitly balances economics and safety under data uncertainty. The adversarial and Wasserstein extensions add value for imperfect real-world data, and the distribution-free nature aligns with needs in energy systems where exact distributions are unavailable. Numerical confirmation of the trade-off strengthens applicability in optimization and power systems research.

major comments (3)
  1. [§3] §3 (Problem Formulation): The abstract and introduction assert a convex formulation whose scenario-based solution satisfies the finite-sample violation bound. However, the modeling of time-varying EV departures (affecting availability) and dynamic SoC limits appears to involve products or conditional constraints; without an explicit convex reformulation (e.g., epigraph lifting or auxiliary variables) shown for each fixed uncertainty realization, the standard scenario theory invoked for the guarantees does not apply, undermining the central distribution-free claim.
  2. [§4.2] §4.2 (Robust Extensions): The Wasserstein radius and adversarial perturbation sets are introduced as free parameters. The derivation of the robustness certificates must clarify whether these are chosen independently of the training samples or calibrated to achieve desired violation levels; if the latter, the claimed out-of-distribution guarantees reduce to in-sample fitting and lose their distribution-free character.
  3. [Numerical Experiments] Numerical Experiments (Table 2 and Figure 4): The reported violation frequencies are stated to be consistent with the theoretical 1-epsilon bound, but no out-of-sample test set, statistical confidence intervals on the empirical violation rate, or explicit comparison across different N (number of scenarios) is provided. This leaves open whether the observed consistency is genuine validation or post-hoc selection of epsilon.
minor comments (2)
  1. [Abstract] Abstract: The range or selection procedure for the violation probability epsilon should be stated explicitly to clarify the tuning mechanism.
  2. [Notation] Notation: Define the decision variables and uncertainty sets consistently between the main text and the appendix to avoid ambiguity in the robust formulations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which have helped us strengthen the presentation and rigor of the manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [§3] §3 (Problem Formulation): The abstract and introduction assert a convex formulation whose scenario-based solution satisfies the finite-sample violation bound. However, the modeling of time-varying EV departures (affecting availability) and dynamic SoC limits appears to involve products or conditional constraints; without an explicit convex reformulation (e.g., epigraph lifting or auxiliary variables) shown for each fixed uncertainty realization, the standard scenario theory invoked for the guarantees does not apply, undermining the central distribution-free claim.

    Authors: We appreciate the referee pointing out the need for greater explicitness here. In the original formulation, the time-varying departures and SoC limits are handled via auxiliary variables and epigraph reformulations that render each fixed-scenario problem convex (see the constraint linearization steps following Eq. (3) in Section 3). However, these steps were presented concisely and may not have been sufficiently highlighted. We will add a dedicated paragraph and a small illustrative example in the revised Section 3 that explicitly walks through the auxiliary-variable lifting for the bilinear terms, confirming convexity for each uncertainty realization and thereby justifying direct application of the scenario approach. revision: yes

  2. Referee: [§4.2] §4.2 (Robust Extensions): The Wasserstein radius and adversarial perturbation sets are introduced as free parameters. The derivation of the robustness certificates must clarify whether these are chosen independently of the training samples or calibrated to achieve desired violation levels; if the latter, the claimed out-of-distribution guarantees reduce to in-sample fitting and lose their distribution-free character.

    Authors: We agree that this distinction must be stated unambiguously. The Wasserstein radius and adversarial perturbation budgets are treated as user-chosen design parameters that are fixed before seeing the data; the robustness certificates in Section 4.2 are derived to hold uniformly for any such choice and for any underlying distribution. They are not tuned or calibrated to the observed samples. We will revise the opening paragraphs of Section 4.2 and the statement of Theorem 2 to emphasize this a-priori, distribution-free character and to contrast it with in-sample fitting approaches. revision: yes

  3. Referee: [Numerical Experiments] Numerical Experiments (Table 2 and Figure 4): The reported violation frequencies are stated to be consistent with the theoretical 1-epsilon bound, but no out-of-sample test set, statistical confidence intervals on the empirical violation rate, or explicit comparison across different N (number of scenarios) is provided. This leaves open whether the observed consistency is genuine validation or post-hoc selection of epsilon.

    Authors: We acknowledge that the current numerical section would benefit from stronger statistical validation. While the reported results illustrate the predicted trade-off, we will augment the experiments with (i) a held-out out-of-sample test set drawn from the same data-generating process, (ii) binomial or bootstrap confidence intervals around the empirical violation frequencies, and (iii) additional curves or tables showing how the observed violation rate behaves as N varies. These additions will be placed in a revised Section 5 and will make the consistency with the theoretical bound more robust to the concern of post-hoc selection. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external scenario theory

full rationale

The paper asserts a convex formulation for the EV scheduling problem and invokes the standard scenario optimization framework to obtain finite-sample distribution-free violation bounds. This relies on established external results (e.g., Campi-Garatti theory) rather than any self-referential definition, fitted parameter renamed as prediction, or self-citation chain. The profit-risk trade-off is presented as an explicit tunable parameter whose guarantees follow from the number of scenarios and convexity, without reduction to the input data by construction. Adversarial and Wasserstein extensions are built atop the same convex scenario base. The chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the framework rests on standard assumptions of the scenario approach (i.i.d. samples, convex feasible set) plus domain assumptions about EV behavior. No new entities are introduced.

free parameters (2)
  • violation probability epsilon
    Typical free parameter in scenario approach that controls the safety level and is likely tuned to achieve the reported profit-risk trade-off.
  • Wasserstein radius
    Controls the size of distributional shifts considered; chosen or fitted to obtain the robustness certificates.
axioms (2)
  • domain assumption The uncertain parameters admit a convex representation amenable to scenario approximation.
    Invoked to obtain finite-sample guarantees for the scheduling problem.
  • standard math Training samples are drawn from an unknown but fixed distribution (standard scenario assumption).
    Basis for distribution-free guarantees before robustness extensions.

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Reference graph

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