Data Center Chiller Plant Optimization via Mixed-Integer Nonlinear Differentiable Predictive Control

Abstract: We present a computationally tractable framework for real-time predictive control of multi-chiller plants that involve both discrete and continuous control decisions coupled through nonlinear dynamics, resulting in a mixed-integer optimal control problem. To address this challenge, we extend Differentiable Predictive Control (DPC) -- a self-supervised, model-based learning methodology for approximately solving parametric optimal control problems -- to accommodate mixed-integer control policies. We benchmark the proposed framework against a state-of-the-art Model Predictive Control (MPC) solver and a fast heuristic Rule-Based Controller (RBC). Simulation results demonstrate that our approach achieves significant energy savings over the RBC while maintaining orders-of-magnitude faster computation times than MPC, offering a scalable and practical alternative to conventional combinatorial mixed-integer control formulations.

• The paper extends Differentiable Predictive Control to mixed-integer nonlinear chiller systems, using neural policies, differentiable rounding, and binary-variance regularization to coordinate continuous setpoints with chiller staging.
• MI-DPC reduced simulated chiller-and-pump energy consumption by up to 11.17% versus rule-based control, improving effective COP from 3.88 to 4.38 while increasing relative cooling-tracking error from 5.90% to 9.31%.
• The learned controller inferred actions in approximately 1.9×10⁻⁴ seconds and anticipated abrupt load ramps, but remained simulation-based and lacked formal feasibility guarantees, optimality certificates, and robustness to model uncertainty.

Motivation and problem setting

Data center cooling is a substantial and growing share of electricity demand, with U.S. data centers consuming roughly 150 TWh annually and projected to reach 200–500 TWh by 2030. Chiller plants—the dominant cooling infrastructure—pose a control problem that is inherently mixed-integer: continuous setpoints (chilled water supply temperature, evaporator temperature, mass flow rates) must be coordinated with discrete decisions (chiller on/off staging, pump modes), coupled through nonlinear thermal dynamics. The resulting mixed-integer nonlinear optimal control problem (MI-OCP) is computationally intractable for real-time solution via conventional branch-and-bound solvers, while the industrial default, rule-based control (RBC), relies on fixed part-load-ratio (PLR) staging thresholds and constant setpoints, ignoring future load evolution.

This paper extends Differentiable Predictive Control (DPC)—a self-supervised, model-based scheme that learns explicit neural policies approximating parametric MPC solutions—to the mixed-integer nonlinear case (MI-DPC), and demonstrates it on multi-chiller plant optimization. The contribution builds directly on prior MI-DPC work (Boldocký et al., 24 Jun 2025) and adds: a nonlinear control-oriented chiller model; extension to binary decision variables; a binary-variance regularization technique to suppress high-frequency switching; a systematic benchmark against RBC and an implicit MI-MPC solved with Gurobi; and open-source code.

System model and optimal control problem

The plant consists of $M$ parallel chillers with nonlinear thermal dynamics governing return and supply water temperatures. Delivered cooling per chiller is bilinear in mass flow, the on/off state $\delta^{(i)}\in\{0,1\}$, and temperature difference. The effective thermal load is filtered through a low-pass FIR filter to capture higher-order heat-transfer dynamics. Chiller power is modeled via a quadratic PLR-dependent coefficient of performance (COP), plus a fixed base power when active; pump power scales cubically with flow. Dynamics are discretized with RK4 at $\Delta t = 180$ s.

The OCP minimizes total electrical energy over horizon $N$, penalizing switching ($\|\Delta\delta\|_R^2$) and load-tracking error ($\lambda_W(Q - Q_\text{load})^2$), subject to the dynamics, input/state bounds, capacity limits, and the safety constraint that at least one chiller remains active. The problem is genuinely non-trivial: the binary variable couples bilinearly into both heat-transfer equations, and polynomial power constraints make it a mixed-integer nonlinear program.

Methodology

The DPC reformulation replaces online optimization with offline training of three separate fully connected neural policy modules mapping the parameter vector $\xi$—initial states plus an $N$-step preview of server load—to mass flow, evaporator temperature, and a relaxed binary $\tilde\delta$. Key algorithmic elements:

• Constraint handling: input/state constraints enter as differentiable penalties; the "at least one chiller on" constraint is enforced by fixing $\delta^{(2)}=1$, reducing the relaxed binary dimension to $\delta^{(i)}\in\{0,1\}$0.
• Differentiable rounding: integrality is imposed by thresholding $\delta^{(i)}\in\{0,1\}$1 at 0.5, with gradients approximated via a straight-through estimator using a scaled sigmoid surrogate centered at 0.5 (Bengio et al., 2013).
• Binary-variance regularization (BVR): the penalty $\delta^{(i)}\in\{0,1\}$2 pushes relaxed values toward polarity and away from the rounding threshold. The authors report this term is crucial: without it, $\delta^{(i)}\in\{0,1\}$3 clusters near 0.5, causing high-frequency switching; yet tuning is delicate—low $\delta^{(i)}\in\{0,1\}$4 yields chattering while excessively large values destabilize training.
• Policy optimization: single-shooting rollouts through a differentiable ODE solver, with gradients propagated via backpropagation through time or adjoint sensitivity, optimized by SGD. Global optimality is explicitly not guaranteed given the non-convex parameter space.

Numerical results

Experiments cover systems of $\delta^{(i)}\in\{0,1\}$5 and $\delta^{(i)}\in\{0,1\}$6 identical 500 kW chillers over seven-day simulations with synthetically generated diurnal load profiles. Three findings stand out:

Energy savings. MI-DPC achieves up to 11.17% energy savings over RBC (e.g., 15.52 MWh vs. 17.53 MWh chiller-plus-pump consumption for $\delta^{(i)}\in\{0,1\}$7, $\delta^{(i)}\in\{0,1\}$8), accompanied by improved effective COP (4.38 vs. 3.88). Savings stem from adaptive, more frequent chiller staging and sharply reduced pump power (0.05 MWh vs. 0.39 MWh), but come at the cost of a higher relative control error (9.31% vs. 5.90%)—an explicit trade-off the authors acknowledge rather than hide.

Computational scalability. Mean inference time is ~1.9×10⁻⁴ s regardless of $\delta^{(i)}\in\{0,1\}$9 or $\Delta t = 180$0, since network architecture is fixed across scenarios; training time grows approximately linearly with horizon (~110–120 s total). By contrast, the implicit MI-MPC formulation—reformulated via McCormick relaxations into a mixed-integer quadratic program solvable by Gurobi—is reported as computationally intractable in closed loop, with solver times capped at 180 s (the sampling period itself). This is several orders of magnitude slower than MI-DPC inference, supporting deployment on edge hardware.

Predictive constraint satisfaction. Under cooling ramp-rate constraints observed in real plants, RBC can fail to stage chillers in time during steep load ramps, violating return-temperature safety bounds. MI-DPC anticipates load changes via its preview vector and stages proactively, maintaining feasibility even for abrupt load changes absent from training data—a robustness result beyond mere energy efficiency.

Two additional observations refine the picture. First, horizons beyond $\Delta t = 180$1 yield no further savings for this system, attributed to fast thermal dynamics; longer horizons would likely benefit only systems with larger thermal masses such as storage tanks. Second, closed-loop analysis shows the policy deliberately keeps $\Delta t = 180$2 below the rounding threshold when activating another unit would worsen load-tracking error more than it improves COP—an interpretable behavior consistent with the objective's structure.

Limitations and open questions

The paper is candid about scope. Policy synthesis assumes a fully white-box, deterministic model with known future load; practical deployment requires grey-box identification and must contend with model–plant mismatch, unmodeled dynamics, sensor/actuator noise, and delays, which can degrade performance and compromise constraint satisfaction unless addressed via disturbance modeling, estimation, and constraint tightening. State-constraint feasibility is enforced only softly via penalties—strict feasibility is not guaranteed—and solutions are approximate without optimality certificates. Hyperparameter sensitivity is nontrivial: BVR weight $\Delta t = 180$3, learning rate, gradient clipping, and sigmoid slope all required manual tuning, with failure modes (switching, instability, conservative actuation) documented. Finally, validation is entirely simulation-based; no experimental demonstration on physical hardware is provided.

Conclusion

This work demonstrates that mixed-integer nonlinear predictive control of chiller plants can be cast as self-supervised policy learning, yielding explicit neural controllers with near-constant inference time, up to 11% energy savings over industrial RBC, and proactive constraint satisfaction under ramp-rate limits—at the cost of higher tracking error, manual hyperparameter tuning, and the absence of formal feasibility or optimality guarantees. The open questions left by the study are concrete: whether the framework retains performance under identified models with realistic uncertainty, whether hard state-feasibility guarantees can be incorporated, and whether the approach transfers to plants with cooling towers, economizers, and thermal storage where additional discrete decisions arise.