Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Scenario-Based Approach for Stochastic Economic Model Predictive Control with an Expected Shortfall Constraint

Alireza Arastou, Algo Carè, Ye Wang, Marco Campi, Erik Weyer

TL;DR

This work tackles risk management in stochastic economic model predictive control by enforcing an empirical expected shortfall constraint to bound high-cost events. It employs a scenario-based SEMPC framework, optimizing the expected economic cost while constraining the risk via EES and deriving probabilistic guarantees that depend on the number of support elements in the input domain. A heuristic algorithm estimates the requisite number of support elements, and a convex reformulation reduces computational burden without sacrificing the risk bounds. The approach is validated on the Richmond water network, showing effective risk control at the cost of higher average economic cost, and demonstrating practical applicability to risk-averse operation under price uncertainty.

Abstract

This paper presents a novel approach to stochastic economic model predictive control (SEMPC) that minimizes average economic cost while satisfying an empirical expected shortfall (EES) constraint to manage risk. A new scenario-based problem formulation ensuring controlled risk with high confidence while minimizing the average cost is introduced. The probabilistic guarantees is dependent on the number of support elements over the entire input domain, which is difficult to find for high-dimensional systems. A heuristic algorithm is proposed to find the number of support elements. Finally, an efficient method is presented to reduce the computational complexity of the SEMPC problem with an EES constraint. The approach is validated on a water distribution network, showing its effectiveness in balancing performance and risk.

A Scenario-Based Approach for Stochastic Economic Model Predictive Control with an Expected Shortfall Constraint

TL;DR

This work tackles risk management in stochastic economic model predictive control by enforcing an empirical expected shortfall constraint to bound high-cost events. It employs a scenario-based SEMPC framework, optimizing the expected economic cost while constraining the risk via EES and deriving probabilistic guarantees that depend on the number of support elements in the input domain. A heuristic algorithm estimates the requisite number of support elements, and a convex reformulation reduces computational burden without sacrificing the risk bounds. The approach is validated on the Richmond water network, showing effective risk control at the cost of higher average economic cost, and demonstrating practical applicability to risk-averse operation under price uncertainty.

Abstract

This paper presents a novel approach to stochastic economic model predictive control (SEMPC) that minimizes average economic cost while satisfying an empirical expected shortfall (EES) constraint to manage risk. A new scenario-based problem formulation ensuring controlled risk with high confidence while minimizing the average cost is introduced. The probabilistic guarantees is dependent on the number of support elements over the entire input domain, which is difficult to find for high-dimensional systems. A heuristic algorithm is proposed to find the number of support elements. Finally, an efficient method is presented to reduce the computational complexity of the SEMPC problem with an EES constraint. The approach is validated on a water distribution network, showing its effectiveness in balancing performance and risk.

Paper Structure

This paper contains 12 sections, 5 theorems, 22 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. PROBLEM STATEMENT
  3. Motivating Example
  4. Risk Measures
  5. Scenario Theory for General Decision Making garatti2022risk
  6. Probabilistic Guarantees
  7. Finding the Number of Support Elements
  8. Computationally Tractable SEMPC Problem
  9. Case Study: Richmond Water Network
  10. Effect of Design Parameters on the Obtained Guarantees
  11. SEMPC with EES constraint
  12. Conclusion

Key Result

Theorem 1

Assume that Assumptions As: Map properties and As: Non-degeneracy assumption hold. Let $\beta \in (0,1)$ be a confidence parameter. For each $k = 0, 1, \dots, m-1$, consider the polynomial equation which has exactly two solutions in $[0, +\infty)$, denoted by $\underline{{t}}(k)$ and $\overline{t}(k)$, with $\underline{{t}}(k) \leq \overline{t}(k)$. For $k = m$, consider the polynomial equation

Figures (4)

  • Figure 1: Region $D$ in \ref{['eq: feasibility problem']} with $N_s=4$, $k=2$. The blue line is the second-largest cost, while gray lines are $L_i(u)$, $i=1,2,3,4$.
  • Figure 2: Electricity price and the demand multiplier
  • Figure 3: The simulation results for $u_A$ and $x_A$ (dashed red lines indicate constraints)
  • Figure 4: Average cost and EES with and without the EES constraint

Theorems & Definitions (10)

  • Theorem 1: garatti2022risk
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof