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arxiv: 2604.15227 · v1 · submitted 2026-04-16 · 🧮 math.OC

Pricing Electric Vehicle Charging and Station Access via Copositive Duality

Nanfei Jiang , Yi Zhou , Josh A. Taylor , Mahnoosh Alizadeh This is my paper

Pith reviewed 2026-05-10 10:32 UTC · model grok-4.3

classification 🧮 math.OC
keywords electric vehicle chargingmarginal pricingmixed-integer optimizationrevenue adequacyindividual rationalitystation access constraintsincentive alignmentcopositive duality
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The pith

The first marginal-price mechanism for EV charging with binary station access recovers full revenue for the operator and keeps every user happy with their assigned plan.

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

Electric vehicle charging requires both continuous decisions about energy delivery and binary decisions about which station to use, creating a mixed-integer problem that breaks standard marginal pricing. The paper develops a mechanism by taking the dual of a convex reformulation of that problem, which produces shadow prices for energy and for station access. If the mechanism works as described, operators can set prices that cover their costs exactly while every driver ends up better off accepting the system-optimal assignment than choosing any other feasible option. A reader would care because public charging networks need rules that scale without subsidies or misaligned driver behavior.

Core claim

The authors show that the mixed-integer linear program for optimized EV charging can be expressed in a form whose dual supplies valid marginal prices that capture both continuous capacity limits and discrete station congestion. These prices define a mechanism that is revenue adequate for the operator and individually rational for users, in the strong sense that each user maximizes their welfare by taking the assigned charging plan rather than deviating to any alternative.

What carries the argument

The copositive program that is the dual of the completely positive reformulation of the mixed-integer charging problem; it generates the shadow prices used to price both energy and station access.

If this is right

  • The operator collects enough revenue to cover the costs of every assigned charging plan.
  • No user can improve their own welfare by switching to another station or time profile.
  • Prices internalize the congestion created by binary station access decisions.
  • Inner approximations keep the method tractable for large networks while preserving the revenue and incentive properties.

Where Pith is reading between the lines

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

  • The same dual-based pricing approach could apply to other mixed-integer resource allocation problems in energy systems, such as scheduling discrete loads or storage units.
  • Field tests on real arrival data would show whether the approximated prices still satisfy revenue adequacy and individual rationality when user preferences are noisy.
  • Embedding the mechanism in a dynamic platform could let prices update continuously as vehicles arrive and depart.

Load-bearing premise

The mixed-integer EV charging problem admits an exact convex reformulation whose dual prices remain valid when applied to the original discrete decisions.

What would settle it

Solve a small instance with two stations and a handful of vehicles, compute the prices, then verify whether any vehicle can obtain strictly higher net benefit by choosing a different feasible station or schedule under those prices; if yes, the individual-rationality property fails.

Figures

Figures reproduced from arXiv: 2604.15227 by Josh A. Taylor, Mahnoosh Alizadeh, Nanfei Jiang, Yi Zhou.

Figure 1
Figure 1. Figure 1: Comparison of Payments Under Different Pricing Mechanisms (Users 2 and 5 are not served in the optimal [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
read the original abstract

Optimized charging of electric vehicles (EVs) at public locations consists of two decisions: how much energy to deliver at what times, which is continuous, and where to plug in, which is binary. This makes optimizing EV charging a mixed-integer linear program (MILP). This discreteness undermines traditional marginal pricing methods. In this paper, we develop the first marginal-price-based mechanism for pricing EV charging with binary station access constraints. Using the result of Burer (2009), we express the EV charging as a completely positive program (CPP), whose dual is a copositive program (COP). This convex dual admits valid shadow prices even though the original allocation problem is discrete and nonconvex. By interpreting the COP dual variables as marginal prices, we construct a pricing mechanism that captures EV supply equipment (EVSE) congestion as well as charging-capacity limits. We prove that the resulting mechanism is revenue-adequate for the operator and individually rational for every EV user, in the strong sense that each user maximizes their own welfare by accepting their assigned charging plan rather than deviating to any alternative option. We further develop problem-specific inner-approximation and dimension-reduction techniques that substantially improve the computational tractability of solving the COP in our setting. Numerical experiments on both small and large scale charging instances demonstrate that our pricing mechanism captures discrete congestion effects and aligns user incentives with the system-optimal assignment, outperforming time-of-use (TOU) and convex relaxation benchmarks.

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 the first marginal-price-based mechanism for EV charging that incorporates binary station-access decisions. It reformulates the underlying MILP as a completely positive program via Burer (2009), takes the copositive dual, interprets the dual variables as congestion and capacity prices, proves revenue adequacy for the operator and strong individual rationality for users (each prefers the assigned plan over any deviation), introduces problem-specific inner approximations and dimension reductions for tractability, and reports numerical results on small and large instances showing superiority to TOU and convex-relaxation benchmarks.

Significance. If the claimed mechanism properties survive the inner approximations, the work supplies a theoretically grounded pricing method for a practically important non-convex resource-allocation problem. The combination of an exact reformulation result, explicit proofs of revenue adequacy and strong IR, and scalable numerical validation on realistic charging instances would constitute a useful contribution to mechanism design in energy systems.

major comments (3)
  1. [§4] §4 (inner-approximation and dimension-reduction techniques): The revenue-adequacy and strong-IR proofs (stated in §3 for the exact COP) rely on dual variables satisfying exact complementarity and supporting-hyperplane conditions with respect to the original discrete feasible set. The paper does not demonstrate that these conditions continue to hold once the COP is replaced by its inner approximation; if the approximation is strict, the computed prices used in the numerical experiments may fail to support the claimed incentive properties.
  2. [Numerical experiments] Numerical experiments section: The reported outperformance over TOU and convex-relaxation benchmarks is presented without quantitative measures of the approximation error (e.g., distance to the copositive cone or duality gap relative to the exact CPP). Without such controls it is impossible to verify that the prices remain sufficiently accurate to preserve the theoretical guarantees on the tested instances.
  3. [§3.2] §3.2 (strong individual rationality): The proof assumes that each user can evaluate any alternative charging plan under the posted prices. With approximated dual variables this comparison may no longer be exact, so the “strong sense” claim (user strictly prefers the assigned plan) does not automatically transfer to the prices actually computed and deployed.
minor comments (2)
  1. [Preliminaries] The definition of the copositive cone and its relation to the completely positive cone could be stated explicitly in the preliminaries rather than only referenced via Burer (2009).
  2. [Figures] Figure captions for the large-scale instances should include the specific values of the approximation parameters used in each run.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the referee's thorough and constructive review. We address each major comment point by point below, acknowledging where the manuscript requires clarification or additional reporting. We outline the revisions we will make to improve precision without overstating the results.

read point-by-point responses

  1. Referee: [§4] §4 (inner-approximation and dimension-reduction techniques): The revenue-adequacy and strong-IR proofs (stated in §3 for the exact COP) rely on dual variables satisfying exact complementarity and supporting-hyperplane conditions with respect to the original discrete feasible set. The paper does not demonstrate that these conditions continue to hold once the COP is replaced by its inner approximation; if the approximation is strict, the computed prices used in the numerical experiments may fail to support the claimed incentive properties.

    Authors: We agree that the proofs of revenue adequacy and strong individual rationality in §3 are established only for the exact copositive program, relying on exact complementarity and supporting hyperplanes with respect to the discrete feasible set. The inner approximations and dimension reductions in §4 produce feasible dual variables that may be conservative. We do not claim the exact conditions transfer automatically. In the revised manuscript we will add an explicit remark after the approximation techniques stating that the theoretical guarantees apply to the exact COP, while the inner approximations are used for tractability and their incentive properties are assessed empirically in the experiments. We will also note conditions under which the approximation is expected to remain tight. revision: partial

  2. Referee: [Numerical experiments] Numerical experiments section: The reported outperformance over TOU and convex-relaxation benchmarks is presented without quantitative measures of the approximation error (e.g., distance to the copositive cone or duality gap relative to the exact CPP). Without such controls it is impossible to verify that the prices remain sufficiently accurate to preserve the theoretical guarantees on the tested instances.

    Authors: We accept that quantitative controls on approximation quality are needed. In the revised numerical section we will report duality gaps between the inner-approximated COP and the original MILP (or tight bounds on the copositive cone) for all small instances where exact or near-exact solutions are computable. For larger instances we will additionally report the gap to the convex relaxation and the resulting price deviations. These additions will allow readers to evaluate how closely the deployed prices approximate the exact theoretical prices. revision: yes

  3. Referee: [§3.2] §3.2 (strong individual rationality): The proof assumes that each user can evaluate any alternative charging plan under the posted prices. With approximated dual variables this comparison may no longer be exact, so the “strong sense” claim (user strictly prefers the assigned plan) does not automatically transfer to the prices actually computed and deployed.

    Authors: The strong individual rationality result in §3.2 is proven under exact dual variables from the COP, enabling precise welfare comparison for any alternative plan. With inner-approximated prices the strict preference holds only approximately. In the revision we will qualify the claims in the numerical experiments section to state that the mechanism satisfies strong IR for the exact COP and exhibits approximate strong IR under the computed prices, as supported by the observed welfare gains relative to the benchmarks. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation uses external Burer theorem and standard duality

full rationale

The paper starts from a MILP for EV charging with binary access, invokes the external Burer (2009) result to obtain an exact CPP reformulation, takes its copositive dual, and interprets the dual variables as marginal prices. Revenue adequacy and strong individual rationality are then proved directly from complementarity and supporting-hyperplane properties of the exact dual. Inner approximations and dimension reduction are introduced only for computational tractability and are not used in the theoretical claims. No derived quantity is defined in terms of itself, no fitted parameter is relabeled as a prediction, and the load-bearing citation is external rather than a self-citation chain. The mechanism properties therefore stand as independent consequences of the convex reformulation rather than tautological restatements of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on one standard mathematical result for reformulation and standard duality theory; no free parameters or new postulated entities are introduced in the abstract.

axioms (1)
  • standard math Burer (2009) result allowing certain mixed-integer quadratic programs to be equivalently expressed as completely positive programs
    Invoked to convert the EV charging MILP into a CPP whose dual is the COP used for pricing.

pith-pipeline@v0.9.0 · 5568 in / 1311 out tokens · 74141 ms · 2026-05-10T10:32:03.964141+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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    Maxfield, J.E., Minc, H., 1962

    doi:10.1109/TSG.2022.3219109. Maxfield, J.E., Minc, H., 1962. On the matrix equationx′x=a. Proceedings of the Edinburgh Mathematical Society 13, 125–129. doi:10.1017/S0013091500014681. Moghaddam, Z., Ahmad, I., Habibi, D., Masoum, M.A., 2019. A coordinated dynamic pricing model for electric vehicle charging stations. IEEE Transactions on Transportation El...

    work page doi:10.1109/tsg.2022.3219109 2022

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    IEEE Transactions on Intelligent Transportation Systems 20, 3386–3396

    doi:10.1109/TITS.2018.2876287. 18 Appendix A. Helping Lemmas Proof of Lemma 1 Proof.We prove Lemma 1 by considering the dual feasible setFCOP defined in (B.2). Observe that (B.2) forces the dual variableΩto be a copositive matrix. Thus for all non-negative vectorsα, we haveα ⊤Ωα≥0. This also holds for any principal submatrix ofΩ. We first prove thatλ2 jt,...

    work page doi:10.1109/tits.2018.2876287 2018