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Robust Stochastic Optimal Control via variance penalization: Application to Energy Management Systems

Paul Malisani, Adrien Spagnol, Vivien Smis-Michel

TL;DR

The paper tackles robust stochastic optimal control under convexity, addressing the optimizer's curse by introducing a variance-penalized objective and a Douglas–Rachford–based solver, the Variance-Penalized Progressive Hedging Algorithm ($\mathrm{VPPHA}$). It develops a data-driven framework consisting of scenario generation and reduction to enable tractable, scenario-based control, and proves convergence of the VPPHA to the optimal solution under convexity. The approach is instantiated in a rolling-horizon energy management system for a stationary battery, using real consumption and production data to compare against MPC and standard PHA; results show that VPPHA yields superior out-of-sample performance and greater bill reductions, especially during volatile pricing periods. The work demonstrates that variance penalization can enhance robustness without increasing computational burden, offering a practical path to robust EMS deployments through scalable, scenario-based optimization. Together, these contributions advance robust stochastic control with scalable algorithms and data-driven uncertainty representation for energy systems.

Abstract

This paper addresses a class of robust stochastic optimal control problems. Its main contribution lies in the introduction of a general optimization model with variance penalization and an associated solution algorithm that improves out-of-sample robustness while preserving numerical complexity. The proposed variance-penalized model is inspired by a well-established machine learning practice that aims to limit overfitting and extends this idea to stochastic optimal control. Using the Douglas--Rachford splitting method, the authors develop a Variance-Penalized Progressive Hedging Algorithm (VPPHA) that retains the computational complexity of the standard PHA while achieving superior out-of-sample performance. In addition, the authors propose a three-step control framework comprising (i) a random scenario generation method, (ii) a scenario reduction algorithm, and (iii) a scenario-based optimal control computation using the VPPHA. Finally, the proposed method is validated through simulations of a stationary battery Energy Management System (EMS) using ground-truth electricity consumption and production measurements from a predominantly commercial building in Solaize, France. The results demonstrate that the proposed approach outperforms a classical Model Predictive Control (MPC) strategy, which itself performs better than the standard PHA.

Robust Stochastic Optimal Control via variance penalization: Application to Energy Management Systems

TL;DR

The paper tackles robust stochastic optimal control under convexity, addressing the optimizer's curse by introducing a variance-penalized objective and a Douglas–Rachford–based solver, the Variance-Penalized Progressive Hedging Algorithm (). It develops a data-driven framework consisting of scenario generation and reduction to enable tractable, scenario-based control, and proves convergence of the VPPHA to the optimal solution under convexity. The approach is instantiated in a rolling-horizon energy management system for a stationary battery, using real consumption and production data to compare against MPC and standard PHA; results show that VPPHA yields superior out-of-sample performance and greater bill reductions, especially during volatile pricing periods. The work demonstrates that variance penalization can enhance robustness without increasing computational burden, offering a practical path to robust EMS deployments through scalable, scenario-based optimization. Together, these contributions advance robust stochastic control with scalable algorithms and data-driven uncertainty representation for energy systems.

Abstract

This paper addresses a class of robust stochastic optimal control problems. Its main contribution lies in the introduction of a general optimization model with variance penalization and an associated solution algorithm that improves out-of-sample robustness while preserving numerical complexity. The proposed variance-penalized model is inspired by a well-established machine learning practice that aims to limit overfitting and extends this idea to stochastic optimal control. Using the Douglas--Rachford splitting method, the authors develop a Variance-Penalized Progressive Hedging Algorithm (VPPHA) that retains the computational complexity of the standard PHA while achieving superior out-of-sample performance. In addition, the authors propose a three-step control framework comprising (i) a random scenario generation method, (ii) a scenario reduction algorithm, and (iii) a scenario-based optimal control computation using the VPPHA. Finally, the proposed method is validated through simulations of a stationary battery Energy Management System (EMS) using ground-truth electricity consumption and production measurements from a predominantly commercial building in Solaize, France. The results demonstrate that the proposed approach outperforms a classical Model Predictive Control (MPC) strategy, which itself performs better than the standard PHA.

Paper Structure

This paper contains 21 sections, 3 theorems, 43 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations
  3. On sets and spaces
  4. On functions
  5. On random variables
  6. Robust stochastic optimization via variance penalization
  7. Problem presentation
  8. Variance penalized model and solving algorithm
  9. Robust Stochastic Optimal Control
  10. Problem presentation
  11. VPPHA implementation for \ref{['prblm:general_socp']}
  12. Computing ${\rm SOCP}({\boldsymbol{\xi}}, {\boldsymbol{\zeta}}, {\boldsymbol{\lambda}})$
  13. Reduced scenario tree generation
  14. Scenario generation
  15. Scenario reduction
  16. ...and 6 more sections

Key Result

Theorem 1

Let ${\boldsymbol{\lambda}}^0 \in {\mathcal{N}_\delta^\bot}$, let $r>0$, and let $\alpha \geq 0$. If $f$ is convex, proper, and lower semi-continuous, the following sequence weakly converges to a fixed-point $(\bar{{\boldsymbol{x}}}, \bar{{\boldsymbol{\lambda}}}, \bar{{\boldsymbol{z}}})$ such that $\bar{{\boldsymbol{x}}}$ is an optimal solution of prlbm:general_stoc_multistage_problem.

Figures (6)

  • Figure 1: Quantiles curves obtained for $\alpha \in \{0.01, 0.5, 0.99\}$ for electrical production.
  • Figure 2: Quantiles curves obtained for $\alpha \in \{0.01, 0.5, 0.99\}$ for electrical consumption. The pics in consumption are due to the occasional operations of the building's glass factory.
  • Figure 3: schematic diagram of a domestic system with a stationary battery controlled by an EMS
  • Figure 4: Influence of the weighting parameter $\alpha$ on the performance ratio $\eta(\alpha)$ with an actualization period $H = 24$ hours and a scenario tree of 225-scenarios.
  • Figure 5: time-evolution of the performance ratio $\eta(\alpha)$ from the 2022-01-22 to the 2024-01-22.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Definition 1: $\delta$-adaptation
  • Theorem 1: Variance-Penalized PHA
  • proof
  • Remark 1
  • Definition 2
  • Theorem 2
  • proof
  • Lemma 1
  • proof