Mixed-integer nonlinear differentiable predictive control for chiller plant optimization

This repository contains the implementation and experimental results for the paper: Data center chiller plant optimization via mixed-integer nonlinear differentiable predictive control.

A condensed version of the paper is available here: https://xboldocky.github.io/chiller-plant-MIDPC/

Overview

This repository implements Mixed-Integer Differentiable Predictive Control (MI-DPC) for optimal control of multi-chiller plants used for data center cooling. MI-DPC embeds mixed-integer model predictive control principles within a self-supervised deep learning framework. Thus, we are able to achieve energy-efficient chiller operation, satisfy cooling demand and process constraints, while perserving computational scalability. The approach is compared against the nominal Mixed-Integer Model Predictive Control (MI-MPC) in terms of computational time and a Rule-Based Control (RBC) in terms of control performance.

The code models a chiller plant with multiple chillers ($M$) that must meet time-varying cooling loads while minimizing energy consumption. The control problem involves discrete decisions (chiller on/off status) ($\delta$) and continuous decisions (mass flow rates $\dot{m}$, evaporation temperatures ($T_\mathrm{e}$), making it a mixed-integer optimization problem. Moreover, the optimal control problem at hand also features bilinear dynamics and nonlinear performance curves, making it formally a mixed-integer nonlinear optimal control problem, which is notoriously expensive to solve in real time.

Instructive live Colab example: This repository provides a step-by-step implementation of MI-DPC for a chiller plant, using a parameterization similar to that presented in the paper.

Graphical abstract

Mixed-Integer Differentiable Predictive Control

The methodology that enables the DPC to handle mixed-integer decision variables is described in "Learning to Solve Parametric Mixed-Integer Optimal Control Problems via Differentiable Predictive Control" paper, available at (https://arxiv.org/abs/2506.19646)

Repository Structure

MI-DPC/
├── MIDPC.py                          # MIDPC policy training and inference module (neuromancer)
├── MIMPC.py                          # Mixed-Integer MPC implementation (pyomo)
├── RBC.py                            # Rule-Based Control implementation (torch)
├── chiller_system.py                 # multi-chiller plant dynamics model
├── simulate_chiller.py               # Simulation framework for evaluating all controllers (MIDPC, MIMPC, RBC)
├── analysis.py                       # Performance analysis and plotting
├── utils.py                          # Utilities (plotting, thermal load data generation, custom MLP class)
├── init.py                           # Chiller system parameters and configuration
├── MIDPC_BVR_experiment.py           # Binary-variance regularization experiments
├── run_training.sh                   # Batch training script for training MIDPC policies (runs MIDPC.py)
├── run_simulation.sh                 # Batch simulation script for all controllers (runs simulate_chiller.py)
├── chiller_staging_with_ramp_limits/ # Experiments with ramp limits on delivered cooling
├── results/                          # Trained models and simulation results
│   ├── MIDPC/                        # MIDPC trained policies and simulation data
│   ├── MIMPC/                        # MIMPC simulation results
│   └── RBC/                          # RBC simulation results
├── plots/                            # Generated figures and visualizations
└── logs/                             # Training and simulation logs

Note: The chiller_staging_with_ramp_limits/ directory contains the same experimental setup as the main repository, with the addition of ramp rate constraints on the delivered cooling. This variant explores the impact of rate-of-change limitations on cooling delivery. This part of the implementation is still in development.

Major Files

Core Control Implementations

• MIDPC.py: MIDPC policy implementation with training loop, and inference interface.
• MIMPC.py: Mixed-Integer MPC baseline using Pyomo with gurobi solver for mixed-integer optimization.
• RBC.py: Rule-Based Control baseline with threshold-based chiller staging.

System Modeling

• chiller_system.py: Physics-based chiller dynamics including thermal models, power consumption, and cooling delivery calculations.
• init.py: System parameters and operating bounds.

Simulation and Analysis

• simulate_chiller.py: Closed-loop simulation framework for evaluating control policies.
• analysis.py: Performance evaluation, metrics computation, and visualization generation.

Utilities

• utils.py: Custom MLP implementation, load signal generation, and plotting utilities.
• MIDPC_BVR_experiment.py: Binary variable regularization experiments.

Scripts

• run_training.sh: Batch training script for multiple configurations.
• run_simulation.sh: Batch simulation script for all control policies.

Key Results

The following figures illustrate the main results:

Control Trajectory

Closed-loop simulation results of a two-chiller system obtained with MI-DPC for a prediction horizon of $N = 15$.

Inference Time Analysis of MIDPC

Mean inference time (MIT) and training time (TT) as a function of prediction horizon length ($N$) for different numbers of chillers ($M$) with Mixed-Integer Differentiable Predictive Control. The plot indicates an approximate linear trend of problem size and computational complexity.

The impact of Binary-Variance Regularization on relaxed integer variable

This plot shows how scaling up the magnitude of binary-variance regularization ($\Lambda$) affects the relaxed integer variable ($\tilde\delta$). With higher magnitudes the regularization penalizes low confidence of the binary value, i.e., values near rounding threshold $0.5$; $\tilde\delta = 0.5$.

Usage

Training MIDPC Policies

python MIDPC.py -nsteps 15 -Ts 180 -M 2

Parameters:

• -nsteps: Prediction horizon length (N)
• -Ts: Sampling time in seconds (default: 180)
• -M: Number of chillers (default: 2)

Running Simulations

python simulate_chiller.py -policy MIDPC -nsteps 15 -Ts 180 -M 2 -n_days 7

Parameters:

• -policy: Control strategy (MIDPC, MIMPC, or RBC)
• -nsteps: Prediction horizon (for MIDPC/MIMPC)
• -Ts: Sampling time in seconds
• -M: Number of chillers
• -n_days: Simulation duration in days

Key Features of MIDPC for chiller plant optimization

• Mixed-Integer Control: Handles discrete (chiller on/off) and continuous (flow rates, temperatures) decisions
• Computational Scalability: Linear scaling of training time and inference time with respect to the number of chillers and prediction horizon length
• Fast Inference: Sub-millisecond inference times, suitable for real-time control
• Energy Efficiency: Optimizes chiller and pump power consumption while satisfying cooling demand
• Physics-Informed Policy Learning: Incorporates system dynamics directly into the optimal control problem
• Constraints Handling: Enforces input constraints via element-wise clipping, while state constraints are treated as penalties within the control objective loss function.

Dependencies

• PyTorch
• Neuromancer
• Pyomo (for MIMPC optimization)
• Gurobi or SCIP (for MIMPC solver)
• NumPy, Matplotlib
• Pandas, Tabulate (for analysis)

Citations

If you use this code in your research, please cite:

@article{boldocky2026data,
  title={Data center chiller plant optimization via mixed-integer nonlinear differentiable predictive control},
  author={Boldock{\'y}, J{\'a}n and Faulkner, Cary and Michael, Elad and Gulan, Martin and Tuor, Aaron and Drgo{\v{n}}a, J{\'a}n},
  journal={Control Engineering Practice},
  volume={174},
  pages={107063},
  year={2026},
  publisher={Elsevier},
 doi = {https://doi.org/10.1016/j.conengprac.2026.107063},
}

If you use Mixed-Integer Differentiable Predictive Control methodology, please cite:

[Paper citation will be added upon publication]
https://arxiv.org/abs/2506.19646

Acknowledgment

This research was primarily supported by the U.S. Department of Energy Building Technologies Office (BTO), USA within the Office of Energy Efficiency and Renewable Energy (EERE), USA under Contract No. DE-AC05-76RL01830.

This work was also supported by the Ralph O’Connor Sustainable Energy Institute (ROSEI) at Johns Hopkins University.