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Distributed Model Predictive Control for Energy and Comfort Optimization in Large Buildings Using Piecewise Affine Approximation

Hongyi Li, Jun Xu, Jinfeng Liu

TL;DR

This work tackles the computational difficulty of applying MPC to large buildings with nonlinear comfort models by introducing a distributed MPC framework that uses a piecewise affine (PWA) representation of the PMV comfort index. The PWA model enables the problem to be decomposed via ADMM, and a convex ADMM algorithm solves a sequence of small convex subproblems to converge to a local optimum of the original nonconvex formulation. Case studies on a 36-zone building demonstrate that the distributed PWA approach achieves performance close to centralized methods while delivering an 86% reduction in computation time, validating its scalability and practicality for real-time operation. The results indicate significant potential for real-time, energy-efficient, occupant-friendly control in large building ensembles, with identified avenues for further improving convergence guarantees and adaptive PWA granularity.

Abstract

The control of large buildings encounters challenges in computational efficiency due to their size and nonlinear components. To address these issues, this paper proposes a Piecewise Affine (PWA)-based distributed scheme for Model Predictive Control (MPC) that optimizes energy and comfort through PWA-based quadratic programming. We utilize the Alternating Direction Method of Multipliers (ADMM) for effective decomposition and apply the PWA technique to handle the nonlinear components. To solve the resulting large-scale nonconvex problems, the paper introduces a convex ADMM algorithm that transforms the nonconvex problem into a series of smaller convex problems, significantly enhancing computational efficiency. Furthermore, we demonstrate that the convex ADMM algorithm converges to a local optimum of the original problem. A case study involving 36 zones validates the effectiveness of the proposed method. Our proposed method reduces execution time by 86\% compared to the centralized version.

Distributed Model Predictive Control for Energy and Comfort Optimization in Large Buildings Using Piecewise Affine Approximation

TL;DR

This work tackles the computational difficulty of applying MPC to large buildings with nonlinear comfort models by introducing a distributed MPC framework that uses a piecewise affine (PWA) representation of the PMV comfort index. The PWA model enables the problem to be decomposed via ADMM, and a convex ADMM algorithm solves a sequence of small convex subproblems to converge to a local optimum of the original nonconvex formulation. Case studies on a 36-zone building demonstrate that the distributed PWA approach achieves performance close to centralized methods while delivering an 86% reduction in computation time, validating its scalability and practicality for real-time operation. The results indicate significant potential for real-time, energy-efficient, occupant-friendly control in large building ensembles, with identified avenues for further improving convergence guarantees and adaptive PWA granularity.

Abstract

The control of large buildings encounters challenges in computational efficiency due to their size and nonlinear components. To address these issues, this paper proposes a Piecewise Affine (PWA)-based distributed scheme for Model Predictive Control (MPC) that optimizes energy and comfort through PWA-based quadratic programming. We utilize the Alternating Direction Method of Multipliers (ADMM) for effective decomposition and apply the PWA technique to handle the nonlinear components. To solve the resulting large-scale nonconvex problems, the paper introduces a convex ADMM algorithm that transforms the nonconvex problem into a series of smaller convex problems, significantly enhancing computational efficiency. Furthermore, we demonstrate that the convex ADMM algorithm converges to a local optimum of the original problem. A case study involving 36 zones validates the effectiveness of the proposed method. Our proposed method reduces execution time by 86\% compared to the centralized version.
Paper Structure (26 sections, 2 theorems, 38 equations, 8 figures, 7 tables, 1 algorithm)

This paper contains 26 sections, 2 theorems, 38 equations, 8 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. System descriptions
  3. Overall framework
  4. Predictive model for MPC
  5. Comfort index
  6. Explicit Mapping Between the PMV Index and the System State
  7. PWA-based distributed MPC scheme
  8. Nonlinear optimization problem formulation
  9. PWA-based optimization problem formulation
  10. Effect of PWA Approximation on Comfort Performance
  11. ADMM iteration
  12. Convex ADMM algorithm
  13. Initialization Strategy and Practical Behavior of Algorithm \ref{['explore']}
  14. Case study
  15. Large-scale setup
  16. ...and 11 more sections

Key Result

Lemma 1

The iterates $\{\mathbf{u}^{\tau}_1,\cdots,\mathbf{u}^{\tau}_M,\mathbf{z}^{\tau} ,\bm{\lambda}^{\tau}\}$ from the ADMM algorithm (ADMM) converge linearly to the optimal primal-dual solution of $\mathcal{P}_3$.

Figures (8)

  • Figure 1: Overall framework for distributed optimization of energy and comfort in large buildings.
  • Figure 2: Difference in solution updates.
  • Figure 3: The considered building.
  • Figure 4: Indoor temperature heatmap for 36 zones over 24 hours. Note that the occupied hours are from 10:00 to 20:00.
  • Figure 5: 36 zones PMV.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Remark 4
  • Theorem 1
  • proof