Exact value functions and sensitivities for energy storage by fast Legendre transforms
Abstract
Long chains of decisions coupled only through a bounded cumulative state, such as the operation of an energy-storage device, form convex programs that generic solvers treat as arbitrary sparse models. We present an exact solver for this class that propagates the complete convex cost-to-reach function through restrictions and infimal convolutions, each evaluated by fast Legendre-Fenchel conjugation as an ordered merge of slopes: no state grid, no value approximation, and a function rather than a single primal-dual pair as output. Price-taking storage instances of 5,000 to 15,000 hourly steps solve 35-93 times faster than Gurobi. The value function is what pays: exact capacity marginals in one differentiated pass, recourse assembled rather than sampled on scenario trees, and a certified sizing loop whose cut count does not grow with the number of weather years. We then widen the carried state. Two storages at one node can be eliminated rather than coordinated by carrying the value function of the pair, and we prove that this object is combinatorially rigid: every breakline the recursion realizes is vertical, horizontal, or borne by a line through the origin, with no genericity assumption. For d co-located stores sharing one charge-discharge coefficient, every realized wall is a position hyperplane or an indifference hyperplane {s_i = s_j}. A catalogue law maps where speed holds: one-dimensional chains beat the frontal solve by 15-30x and growing, while the planar block loses by 13.5-107x on rich network curves. Those curves are themselves exact, traced by a parametric dual simplex. The open-source package FLToptim, every benchmark driver and all data accompany the paper.
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BibTeX
@misc{girard2026exact,
title = {{Exact value functions and sensitivities for energy storage by fast Legendre transforms}},
author = {Robin Girard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31213},
eprint = {2610.31213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31213},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact value functions and sensitivities for energy storage by fast Legendre transforms
AU - Robin Girard
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31213
UR - https://arxiv.org/abs/2610.31213
AB - Long chains of decisions coupled only through a bounded cumulative state, such as the operation of an energy-storage device, form convex programs that generic solvers treat as arbitrary sparse models. We present an exact solver for this class that propagates the complete convex cost-to-reach function through restrictions and infimal convolutions, each evaluated by fast Legendre-Fenchel conjugation as an ordered merge of slopes: no state grid, no value approximation, and a function rather than a single primal-dual pair as output. Price-taking storage instances of 5,000 to 15,000 hourly steps solve 35-93 times faster than Gurobi. The value function is what pays: exact capacity marginals in one differentiated pass, recourse assembled rather than sampled on scenario trees, and a certified sizing loop whose cut count does not grow with the number of weather years. We then widen the carried state. Two storages at one node can be eliminated rather than coordinated by carrying the value function of the pair, and we prove that this object is combinatorially rigid: every breakline the recursion realizes is vertical, horizontal, or borne by a line through the origin, with no genericity assumption. For d co-located stores sharing one charge-discharge coefficient, every realized wall is a position hyperplane or an indifference hyperplane {s_i = s_j}. A catalogue law maps where speed holds: one-dimensional chains beat the frontal solve by 15-30x and growing, while the planar block loses by 13.5-107x on rich network curves. Those curves are themselves exact, traced by a parametric dual simplex. The open-source package FLToptim, every benchmark driver and all data accompany the paper.
ER -