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Temporal and Spatial Decomposition for Prospective Studies in Energy Systems under Uncertainty

Camila Martinez Parra, Michel de Lara, Jean-Philippe Chancelier, Pierre Carpentier, Jean-Marc Janin

TL;DR

The paper tackles the challenge of evaluating storage usage values in a large-scale, uncertain European electricity network by formulating a two-timescale, multinode stochastic optimization problem. It introduces a spatio-temporal decomposition via Dual Approximate Dynamic Programming (DADP) to decouple nodal subproblems from arc transport, enabling tractable cost-to-go approximations that yield nodal usage values. The authors benchmark DADP against Stochastic Dual Dynamic Programming (SDDP) on a 30-node system, showing that 8-hour and weekly price aggregations provide competitive lower bounds and comparable policy performance with significantly reduced computation times; hourly aggregation, while feasible in small systems, struggles with scalability. The work demonstrates the potential of DADP-based approaches for prospective studies, highlighting scalability, parallel nodal resolution, and robust policy evaluation under RTE-like uncertainty chronicles. Overall, the results support using spatio-temporal decomposition to generate practical, scalable usage-value estimates for energy storage in large-scale, uncertain power systems.

Abstract

The increasing penetration of renewable energy requires greater use of storage resources to manage system intermittency. As a result, there is growing interest in evaluating the opportunity cost of stored energy, or usage values, which can be derived by solving a multistage stochastic optimization problem. Stochasticity arises from net demand (the aggregation of demand and non-dispatchable generation), the availability of dispatchable generation, and inflows when the storage facilities considered are hydroelectric dams. We aim to compute these usage values for each market zone of the interconnected European electricity system, in the context of prospective studies currently conducted by RTE, the French TSO. The energy system is mathematically modelled as a directed graph, where nodes represent market zones and arcs represent interconnection links. In large energy systems, spatial complexity (thirty nodes in the system, each with at most one aggregated storage unit) compounds temporal complexity (a one-year horizon modelled with two timescales: weekly subproblems with hourly time steps). This work addresses three main sources of complexity: temporal, spatial, and stochastic. We tackle the multinode multistage stochastic optimisation problem by incorporating a spatio-temporal decomposition scheme. To efficiently compute usage values, we apply Dual Approximate Dynamic Programming (DADP), which enables tractable decomposition across both time and space. This approach yields nodal usage values that depend solely on the local state of each node, independently of the others. We conduct numerical studies on a realistic system composed of thirty nodes (modelling part of Europe) and show that DADP obtains competitive results when comparing with traditional methods like Stochastic Dual Dynamic Programming (SDDP).

Temporal and Spatial Decomposition for Prospective Studies in Energy Systems under Uncertainty

TL;DR

The paper tackles the challenge of evaluating storage usage values in a large-scale, uncertain European electricity network by formulating a two-timescale, multinode stochastic optimization problem. It introduces a spatio-temporal decomposition via Dual Approximate Dynamic Programming (DADP) to decouple nodal subproblems from arc transport, enabling tractable cost-to-go approximations that yield nodal usage values. The authors benchmark DADP against Stochastic Dual Dynamic Programming (SDDP) on a 30-node system, showing that 8-hour and weekly price aggregations provide competitive lower bounds and comparable policy performance with significantly reduced computation times; hourly aggregation, while feasible in small systems, struggles with scalability. The work demonstrates the potential of DADP-based approaches for prospective studies, highlighting scalability, parallel nodal resolution, and robust policy evaluation under RTE-like uncertainty chronicles. Overall, the results support using spatio-temporal decomposition to generate practical, scalable usage-value estimates for energy storage in large-scale, uncertain power systems.

Abstract

The increasing penetration of renewable energy requires greater use of storage resources to manage system intermittency. As a result, there is growing interest in evaluating the opportunity cost of stored energy, or usage values, which can be derived by solving a multistage stochastic optimization problem. Stochasticity arises from net demand (the aggregation of demand and non-dispatchable generation), the availability of dispatchable generation, and inflows when the storage facilities considered are hydroelectric dams. We aim to compute these usage values for each market zone of the interconnected European electricity system, in the context of prospective studies currently conducted by RTE, the French TSO. The energy system is mathematically modelled as a directed graph, where nodes represent market zones and arcs represent interconnection links. In large energy systems, spatial complexity (thirty nodes in the system, each with at most one aggregated storage unit) compounds temporal complexity (a one-year horizon modelled with two timescales: weekly subproblems with hourly time steps). This work addresses three main sources of complexity: temporal, spatial, and stochastic. We tackle the multinode multistage stochastic optimisation problem by incorporating a spatio-temporal decomposition scheme. To efficiently compute usage values, we apply Dual Approximate Dynamic Programming (DADP), which enables tractable decomposition across both time and space. This approach yields nodal usage values that depend solely on the local state of each node, independently of the others. We conduct numerical studies on a realistic system composed of thirty nodes (modelling part of Europe) and show that DADP obtains competitive results when comparing with traditional methods like Stochastic Dual Dynamic Programming (SDDP).

Paper Structure

This paper contains 67 sections, 2 theorems, 55 equations, 6 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Context and motivation
  3. Mathematical formulations and numerical challenges
  4. Contributions and outline of the paper
  5. Multistage multinode stochastic optimisation problem
  6. Definitions and notations
  7. Timescale, graph and variables definition
  8. Timescale definition.
  9. Directed graph modelling the multinode system.
  10. Nodal uncertain variables.
  11. Nodal recourse control variables.
  12. Nodal storage level variables.
  13. Nodal flows: import and export variables.
  14. Physical and economic constraints in a multinode system
  15. Nodal storage dynamics.
  16. ...and 52 more sections

Key Result

Proposition 3

For any deterministic price decomposition process $\price$, if the sequence $({\mathbf{W}_{\rrbracket \underline{s}\rrbracket}, \dots, \mathbf{W}_{\rrbracket s\rrbracket}, \dots, \mathbf{W}_{\rrbracket \overline{s}\rrbracket}})$ of uncertainties is weekly independent as in Assumption assumption:week and hence a lower bound for the initial global cost-to-go function (see Equation eq:GlobalProblemOp

Figures (6)

  • Figure 1: Price updates scheme
  • Figure 2: Market zones and their interconnections. In blue, the nodes with long-term storage.
  • Figure 3: Operational cost distributions for the 30-node system under the weekly hazard-decision information structure, including the mean cost value for each method. The distributions are reported with and without outliers (\ref{['fig:costDistributionOutliers30HD']} and \ref{['fig:costDistributionNoOutliers30HD']} respectively).
  • Figure 4: ENS distributions for the 30-node system, including the mean ENS value for each method. The distributions are reported with and without outliers (\ref{['fig:ensDistributionOutliers30HD']} and \ref{['fig:ensDistributionNoOutliers30HD']} respectively).
  • Figure 5: ENS distributions without outliers for the 30-node system under the weekly hazard-decision information structure, including the mean ENS value for SDDP, DADP with 8-hour prices, and DADP with weekly prices.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Remark 1
  • Remark 2
  • Proposition 3
  • Proposition 4
  • Remark 5
  • Remark 6
  • Remark 7
  • Remark 8
  • Remark 9