Storage Participation in Electricity Markets: Time Discretization through Robust Optimization

Referee: [§3] §3 (equivalence derivation): the reduction from infinite nonconvex functional constraints to a finite deterministic set must explicitly enumerate or bound the worst-case trajectories arising from arbitrary sequences of charge/discharge mode switches. Because efficiencies differ by mode and binary variables select the sign and value of the efficiency factor in the state equation, each switching sequence produces a distinct candidate trajectory; if the derivation considers only a fixed mode or a restricted subset of sequences, the finite constraint set is incomplete and the claimed exact equivalence does not hold.

Authors: We appreciate the referee drawing attention to the handling of mode switches in the equivalence proof. The derivation in §3 obtains the finite deterministic constraints by propagating the state equation over the discretized time horizon while taking the worst-case fluctuation signal from the uncertainty set at each step. Because the binary mode variables are decision variables in the overall model, the robust reformulation implicitly accounts for every admissible switching sequence: any feasible solution to the mixed-integer bilinear program satisfies the original infinite constraints for all signals and all possible mode paths, as the state bounds are derived from the extremal cumulative efficiency effects under the binary selection. Explicit enumeration of the 2^T sequences is avoided by using the mixed-integer structure to bound the trajectory without loss of equivalence. We will add a clarifying paragraph in the revised §3 that explicitly states how arbitrary switching sequences are bounded rather than restricted to a fixed mode. revision: yes

Referee: [§4.1–4.2] §4.1–4.2 (uncertainty-set construction and bilinear program): the paper must verify that the chosen uncertainty set, while inspired by market rules, contains all relevant real-world fluctuation signals that storage must honor, and that the subsequent bilinear relaxations remain tight for the reported negligible optimality gap. Any post-hoc tightening or parameter choices that affect tightness should be stated explicitly.

Authors: We thank the referee for this request for additional verification. The uncertainty set is constructed directly from the FCR market rules (maximum deviation magnitude, response time, and activation thresholds), which by definition define the signals that storage must honor. In the revision we will insert a short verification paragraph in §4.1 showing that all fluctuation signals recorded in the July 2020–June 2024 European market data used for the backtest lie inside the set. For the bilinear program, the relaxations are obtained solely via standard McCormick envelopes on the bilinear terms arising from the product of binary mode variables and continuous power variables; no additional tightening parameters or post-hoc adjustments are applied. The reported optimality gaps below 1 % across all instances already demonstrate practical tightness. We will state this explicitly in the revised §4.2. revision: yes