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

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.

💡 Research Summary

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The paper addresses two intertwined challenges in the control of large‑scale building HVAC systems: (1) the computational burden of incorporating the highly nonlinear Predicted Mean Vote (PMV) comfort index into Model Predictive Control (MPC), and (2) the scalability limits of centralized MPC when the number of zones grows to dozens or hundreds. To overcome these issues, the authors propose a novel distributed MPC framework that combines a Piecewise‑Affine (PWA) approximation of the PMV function with an Alternating Direction Method of Multipliers (ADMM) based decomposition.

System Modeling
Each thermal zone is modeled with a nine‑state RC (resistance‑capacitance) representation that captures indoor air temperature and the inner/outer surface temperatures of four walls. The continuous‑time dynamics are discretized into a linear state‑space form: x_i(k+1)=A_i x_i(k)+B_i u_i(k)+d_i(k), where u_i is the HVAC power input. Inter‑zone heat exchange is handled by treating the temperature of adjacent zones as the “outside” temperature for shared walls, thereby preserving the linear structure while accounting for coupling.

PMV Approximation via PWA
The PMV index depends on temperature, humidity, metabolic rate, clothing insulation, air speed, and mean radiant temperature, resulting in a highly nonlinear expression. Rather than using a simple linearization (which can produce large errors far from the operating point), the authors partition the feasible state space into several regions and fit a separate affine function in each region. This Piecewise‑Affine (PWA) model retains high fidelity (average PMV error ≈0.12 °C) while allowing the comfort term to be expressed as a set of linear constraints and a quadratic cost, thus converting the original nonconvex MPC problem into a PWA‑based quadratic program (QP).

Optimization Problem Formulation
The MPC objective for each zone i is a weighted sum of comfort cost C_PMV,i and energy cost C_u,i:
J_i = α·C_PMV,i + C_u,i,
where C_PMV,i = Σ_{l=1}^{N} δ_k(l)·PMV_i(k+l)^2 and C_u,i = Σ_{l=1}^{N} λ_k(l)·(u_i(k+l))^2.
δ_k(l) indicates occupancy (1 if occupied, 0 otherwise) and λ_k(l) reflects time‑varying electricity tariffs. Constraints include input bounds, the linear RC dynamics, and initial conditions. The global problem is to minimize Σ_i J_i subject to coupling constraints (e.g., total power limits, heat exchange between adjacent zones).

Convex ADMM Algorithm
Because the PWA formulation introduces binary variables that select the active affine region, the problem remains nonconvex. The authors propose a “convex ADMM” approach: at each ADMM iteration, the binary region‑selection variables are held fixed, turning the subproblem for each zone into a convex QP. The ADMM steps are:

1. Local Update – Solve a small QP for u_i and x_i given the current estimates of the coupling variables and Lagrange multipliers.
2. Global Consensus – Update auxiliary variables that enforce consistency of shared wall temperatures and total power limits.
3. Multiplier Update – Adjust the Lagrange multipliers using the standard ADMM dual‑ascent rule.
   Because each local QP is low‑dimensional, commercial solvers (e.g., Gurobi, OSQP) solve them in milliseconds. The authors prove that, under standard assumptions (closed‑form proximal operators, bounded penalty parameter), the sequence generated by this algorithm converges to a stationary point of the original nonconvex problem, i.e., a local optimum.

Case Study and Results
A 36‑zone building (four floors, nine zones per floor) is simulated over a 24‑hour horizon using realistic weather data, occupancy schedules, and time‑of‑use electricity tariffs. Three configurations are compared: (a) centralized MPC solving the full nonconvex problem, (b) distributed ADMM with a simple linear PMV approximation, and (c) the proposed PWA‑ADMM distributed scheme. Key findings include:

• Computation Time – Centralized MPC requires ~12 seconds per control step, while the PWA‑ADMM distributed method finishes within ~0.3 seconds per zone (total <30 seconds), representing an 86 % reduction.
• Energy Consumption – By adjusting λ_k(l) according to tariff peaks, the distributed method reduces peak power draw by 12 % and overall electricity usage by ~5 % relative to the centralized baseline.
• Comfort Accuracy – The PWA approximation yields an average PMV error of 0.12 °C versus 0.35 °C for the linear model, keeping all zones within the comfort band (PMV between –0.5 and +0.5).
• Scalability – The number of ADMM iterations needed for convergence stays below 30 across all zones, indicating robustness to problem size.

Discussion of Limitations and Future Work
The approach hinges on a pre‑designed PWA partition; its quality depends on the representativeness of the training data. Increasing the number of affine regions improves accuracy but also enlarges the binary selection space, potentially increasing ADMM overhead. The current implementation uses synchronous ADMM; asynchronous or event‑triggered variants could further reduce communication latency in large‑scale deployments. Future research directions suggested include automated region generation (e.g., via clustering or neural network regression), incorporation of stochastic disturbances (weather forecast uncertainty), and real‑world pilot tests in operational buildings.

Conclusion
The authors present a practical, high‑performance solution for large‑building HVAC control that simultaneously addresses energy efficiency and occupant comfort. By marrying a high‑fidelity Piecewise‑Affine PMV model with a convex ADMM‑based distributed MPC algorithm, they achieve near‑centralized performance while dramatically cutting computational time. The 86 % execution‑time reduction and improved comfort fidelity demonstrated on a 36‑zone case study underscore the method’s suitability for real‑time smart‑building applications.