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Adaptive Relaxation based Non-Conservative Chance Constrained Stochastic MPC

Avik Ghosh, Cristian Cortes-Aguirre, Yi-An Chen, Adil Khurram, Jan Kleissl

TL;DR

This paper addresses non-conservative chance-constrained SMPC for discrete-LTI systems with unknown uncertainty distributions by introducing an online adaptive relaxation, OA-SMPC, that relaxes state constraints based on the time-average of past violations. It proves convergence properties of the time-average violation to the allowable level under an ideal control policy and shows martingale-like behavior for practical implementations, all without requiring a-priori uncertainty statistics or historical samples. The approach is demonstrated on a grid-connected microgrid with PV, load, and a BESS, showing notable cost savings over traditional EMPC and a state-of-the-art chance-constrained method while maintaining SOC within probabilistic bounds via post-processing. The work provides a scalable, distribution-free, and less-conservative framework for economic microgrid operation, with potential applicability to other energy-routing and process-control problems under uncertainty.

Abstract

Chance constrained stochastic model predictive controllers (CC-SMPC) trade off full constraint satisfaction for economical plant performance under uncertainty. Previous CC-SMPC works are over-conservative in constraint violations leading to worse economic performance. Other past works require a-priori information about the uncertainty set, limiting their application. This paper considers a discrete LTI system with hard constraints on inputs and chance constraints on states, with unknown uncertainty distribution, statistics, or samples. This work proposes a novel adaptive online update rule to relax the state constraints based on the time-average of past constraint violations, to achieve reduced conservativeness in closed-loop. Under an ideal control policy assumption, it is proven that the time-average of constraint violations asymptotically converges to the maximum allowed violation probability. The method is applied for optimal battery energy storage system (BESS) dispatch in a grid connected microgrid with PV generation and load demand, with chance constraints on BESS state-of-charge (SOC). Realistic simulations show the superior electricity cost saving potential of the proposed method as compared to the traditional economic MPC without chance constraints, and a state-of-the-art approach with chance constraints. We satisfy the chance constraints non-conservatively in closed-loop, effectively trading off increased cost savings with minimal adverse effects on BESS lifetime.

Adaptive Relaxation based Non-Conservative Chance Constrained Stochastic MPC

TL;DR

This paper addresses non-conservative chance-constrained SMPC for discrete-LTI systems with unknown uncertainty distributions by introducing an online adaptive relaxation, OA-SMPC, that relaxes state constraints based on the time-average of past violations. It proves convergence properties of the time-average violation to the allowable level under an ideal control policy and shows martingale-like behavior for practical implementations, all without requiring a-priori uncertainty statistics or historical samples. The approach is demonstrated on a grid-connected microgrid with PV, load, and a BESS, showing notable cost savings over traditional EMPC and a state-of-the-art chance-constrained method while maintaining SOC within probabilistic bounds via post-processing. The work provides a scalable, distribution-free, and less-conservative framework for economic microgrid operation, with potential applicability to other energy-routing and process-control problems under uncertainty.

Abstract

Chance constrained stochastic model predictive controllers (CC-SMPC) trade off full constraint satisfaction for economical plant performance under uncertainty. Previous CC-SMPC works are over-conservative in constraint violations leading to worse economic performance. Other past works require a-priori information about the uncertainty set, limiting their application. This paper considers a discrete LTI system with hard constraints on inputs and chance constraints on states, with unknown uncertainty distribution, statistics, or samples. This work proposes a novel adaptive online update rule to relax the state constraints based on the time-average of past constraint violations, to achieve reduced conservativeness in closed-loop. Under an ideal control policy assumption, it is proven that the time-average of constraint violations asymptotically converges to the maximum allowed violation probability. The method is applied for optimal battery energy storage system (BESS) dispatch in a grid connected microgrid with PV generation and load demand, with chance constraints on BESS state-of-charge (SOC). Realistic simulations show the superior electricity cost saving potential of the proposed method as compared to the traditional economic MPC without chance constraints, and a state-of-the-art approach with chance constraints. We satisfy the chance constraints non-conservatively in closed-loop, effectively trading off increased cost savings with minimal adverse effects on BESS lifetime.
Paper Structure (23 sections, 5 theorems, 48 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 23 sections, 5 theorems, 48 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature Review
  4. Contributions
  5. Mathematical preliminaries
  6. Notations
  7. Standard Definitions
  8. Problem formulation
  9. System Description
  10. Online Adaptive SMPC (OA-SMPC)
  11. Observed Violations
  12. Convergence Properties of Yt
  13. h update rule
  14. Deviation of the practical control policy from the ideal control policy
  15. Asymptotic Behavior of the Practical System
  16. ...and 8 more sections

Key Result

Theorem 1

Let Assumptions assumption_system, assumption_uncertainty, assumption_control_inputs, and assumption_ideal_control_policy hold. Given $\alpha_i\!>\!\frac{1}{2(t_0+1)}$, $Z_i^*(t)$, which is the ideal surrogate of the desired supermartingale $Z_i(t)$, is monotonically decreasing a.s. $\forall t \geq

Figures (3)

  • Figure 1: OA-SMPC operational framework with post-processing.
  • Figure 2: Yearly time-series for the: (a) time-average of state constraint violations ($Y$) for the OA-SMPC, SMPC Lit IEEE_1, and Traditional EMPC 2 case studies, and the maximum allowable state constraint violation probability ($\alpha=0.1$), (b) adaptive state constraint relaxing parameters ($h_{1}$ and $\tilde{h}_1$, with $h_1\!=\!h_2$, and $\tilde{h}_1=\tilde{h}_2$ in these case studies) for the OA-SMPC and SMPC Lit, (c) closed-loop behavior of the control inputs for the OA-SMPC, (d) closed-loop behavior of the state for the OA-SMPC.
  • Figure 3: Yearly time-series for the time-average of state constraint violations ($Y_{\alpha}$) for the OA-SMPC for the maximum allowable state constraint violation probability $\alpha\in\{0.05,0.15,0.2\}.$

Theorems & Definitions (24)

  • Definition 1: Filtered probability space williams1991probability
  • Definition 2: Almost surely williams1991probability
  • Definition 3: Adapted stochastic process williams1991probability
  • Definition 4: Supermartingale williams1991probability
  • Definition 5: Monotone convergence theorem for decreasing sequencebartle1964elements
  • Remark 1: Over-conservativeness of previous approaches
  • Remark 2: Structure of the input matrix
  • Remark 3: Behavior of the $h$ update rule
  • Definition 6: Ideal surrogate of a variable
  • Theorem 1
  • ...and 14 more