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Decision-Focused Learning for Neural Network-Constrained HVAC Scheduling

Pietro Favaro, Jean-François Toubeau, François Vallée, Yury Dvorkin

TL;DR

This work tackles day-ahead HVAC scheduling by embedding a neural network model of building thermal dynamics into a mixed-integer optimization framework. It introduces stochastic-smoothing decision-focused learning (SS-DFL) to train NN parameters directly for improved decision quality, circumventing non-differentiability of the MILP/MIQP through score-function gradients. The approach uses an MIQP formulation with adaptive tightness for ReLU activations and interval analysis, enabling stable and efficient training. A five-zone Denver office case demonstrates that SS-DFL outperforms traditional identify-then-optimize and relaxed DFL methods in ex-post costs and thermal comfort, with notable gains in solving time and grid-services performance.

Abstract

Heating, Ventilation, and Air Conditioning (HVAC) is a major electricity end-use with a substantial potential for providing grid services, such as demand response. Harnessing this flexibility requires accurate modeling of the thermal dynamics of buildings, a difficult task because nonlinear heat transfer and recurring daily cycles make historical data highly correlated and insufficient to generalize to new weather, occupancy, and control scenarios. This paper presents an HVAC management system formulated as a Mixed Integer Quadratic Program (MIQP), where Neural Network (NN) models of thermal dynamics are embedded as exact mixed-integer linear constraints. Unlike traditional training approaches that minimize prediction errors, we employ Decision-Focused Learning (DFL) to learn the NN parameters with the objective of directly improving the HVAC cost performance. However, the discrete nature of MIQP hinders DFL, as it leads to undefined and discontinuous gradients, thus impeding standard gradient-based training. We leverage Stochastic Smoothing (SS) to enable efficient gradient computation without the need to differentiate the MIQP. Experiments on a realistic five-zone building using a high-fidelity simulator demonstrate that the proposed SS-DFL approach outperforms conventional identify-then-optimize (i.e., the thermal dynamics model is identified on historical data then used in optimization) and relaxed DFL methods in both cost savings and grid service performance, highlighting its potential for scalable, grid-aware building control.

Decision-Focused Learning for Neural Network-Constrained HVAC Scheduling

TL;DR

This work tackles day-ahead HVAC scheduling by embedding a neural network model of building thermal dynamics into a mixed-integer optimization framework. It introduces stochastic-smoothing decision-focused learning (SS-DFL) to train NN parameters directly for improved decision quality, circumventing non-differentiability of the MILP/MIQP through score-function gradients. The approach uses an MIQP formulation with adaptive tightness for ReLU activations and interval analysis, enabling stable and efficient training. A five-zone Denver office case demonstrates that SS-DFL outperforms traditional identify-then-optimize and relaxed DFL methods in ex-post costs and thermal comfort, with notable gains in solving time and grid-services performance.

Abstract

Heating, Ventilation, and Air Conditioning (HVAC) is a major electricity end-use with a substantial potential for providing grid services, such as demand response. Harnessing this flexibility requires accurate modeling of the thermal dynamics of buildings, a difficult task because nonlinear heat transfer and recurring daily cycles make historical data highly correlated and insufficient to generalize to new weather, occupancy, and control scenarios. This paper presents an HVAC management system formulated as a Mixed Integer Quadratic Program (MIQP), where Neural Network (NN) models of thermal dynamics are embedded as exact mixed-integer linear constraints. Unlike traditional training approaches that minimize prediction errors, we employ Decision-Focused Learning (DFL) to learn the NN parameters with the objective of directly improving the HVAC cost performance. However, the discrete nature of MIQP hinders DFL, as it leads to undefined and discontinuous gradients, thus impeding standard gradient-based training. We leverage Stochastic Smoothing (SS) to enable efficient gradient computation without the need to differentiate the MIQP. Experiments on a realistic five-zone building using a high-fidelity simulator demonstrate that the proposed SS-DFL approach outperforms conventional identify-then-optimize (i.e., the thermal dynamics model is identified on historical data then used in optimization) and relaxed DFL methods in both cost savings and grid service performance, highlighting its potential for scalable, grid-aware building control.

Paper Structure

This paper contains 21 sections, 1 theorem, 32 equations, 7 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Contributions
  4. Outline
  5. Model and Methods
  6. HVAC Management System
  7. Decision-Focused Learning
  8. Problem Formulation
  9. Gradient Computation
  10. Task Loss
  11. Neural Network MILP Formulation
  12. Improved tightness of the NN reformulation
  13. Case Study
  14. Data
  15. Models and Benchmarks
  16. ...and 6 more sections

Key Result

Proposition 1

Consider a neuron---reformulated as mixed-integer linear constraints using Big-M constants---with preactivation $\hat{y}=w_x x + w_p p + b$, where $x \in \mathbb{R}^X$ are unknown input variables with weights $w^x \in \mathbb{R}^{1 \times X}$, $p \in \mathbb{R}^{P}$ are known parameters with weights where $|\cdot|$ denotes the Lebesgue measure (i.e., length) of the interval.

Figures (7)

  • Figure 1: Comparison of the proposed stochastic smoothing DFL pipeline (red) for HVAC management system with conventional DFL approaches (gray).
  • Figure 2: Layer of neurons with Rectified Linear Unit (ReLU) activation function. The parameters are the weight $w_i$ and the bias $b$.
  • Figure 3: Interval analysis of the bounds for a ReLU neuron with inputs $p$ and $x$, where $p$ is a parameter and $x$ a variable in the overarching optimization.
  • Figure 4: Layout of the building floor.
  • Figure 5: Cluster analysis.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Proposition 1: Preactivation Range Reduction
  • proof