Model Predictive Control of Thermo-Hydraulic Systems Using Primal Decomposition

Technical Summary: Model Predictive Control of Thermo-Hydraulic Systems Using Primal Decomposition

Problem Statement
The decarbonization of global energy supply, particularly in heating and cooling systems, necessitates improved operational efficiency. Model Predictive Control (MPC) offers a pathway to optimize these systems but relies heavily on accurate system models. Thermo-hydraulic systems, such as district heating networks (DHNs), are typically modeled using Control Volumes (CVs), resulting in systems described by time-continuous differential-algebraic equations (DAEs). While automated frameworks exist for generating MPCs for such systems, scaling these approaches to large networks remains computationally challenging. Standard optimization methods often struggle with the complexity introduced by time-discretizing nonlinear systems, particularly when higher-order discretization methods alter the system structure or eliminate bilinear properties required for efficient decomposition.

Methodology
The authors propose an automated framework for generating MPCs for thermo-hydraulic systems modeled by CVs. The methodology consists of three primary components:

1. System Modeling: The system is represented by a general dynamic model where temperatures and mass flows are governed by differential equations, while hydraulic constraints and other algebraic relations are handled separately. The model assumes constant fluid properties and negligible enthalpy changes due to pressure variations.
2. Time Discretization: To ensure computational efficiency and preserve the system's structural properties, the authors employ second-order backward differentiation for time discretization. Unlike Taylor or Runge-Kutta methods, which can introduce complex nonlinear terms that obscure the system's structure, backward differentiation maintains a form amenable to decomposition. This results in a time-discrete representation where, assuming constant mass flow, the thermal dynamics become linear.
3. Primal Decomposition (PD): The core innovation is the application of primal decomposition to solve the resulting optimization problem. The approach exploits the specific model structure by separating the problem into a hydraulic sub-problem and a thermal sub-problem.
   • The algorithm iteratively updates the primal variables (mass flows).
   • In each iteration, it solves a hydraulic optimization problem to determine optimal hydraulic states and inputs.
   • Subsequently, it solves a thermal optimization problem using the updated mass flows.
   • The primal variables are updated using subgradients derived from the Lagrangian of the overall problem, employing a diminishing step-size rule to ensure convergence. If infeasibility arises during updates, a backtracking procedure is utilized.

Key Contributions

• Automated Workflow: The paper presents a complete workflow for MPC generation, integrating a general dynamic model for thermo-hydraulic systems, a specific time-discretization procedure based on backward differentiation, and a scalable optimization strategy.
• Scalability via Decomposition: The primary contribution is the demonstration that primal decomposition can be effectively applied to CV-based thermo-hydraulic models. By decoupling hydraulic and thermal constraints, the method aims to reduce computational complexity in large-scale systems.
• Validation: The approach is validated on an underground heating system (representative of underfloor heating or snow-free surfaces) with varying numbers of pipes and control volumes.

Results
The proposed PD-MPC was compared against a standard Interior-Point Method (IPM) MPC using a benchmark Dymola simulation and a custom Matlab implementation.

• Accuracy: The discretized Matlab model showed high accuracy against the Dymola benchmark, with maximum errors of 0.473 K for soil CVs and 0.936 K for pipe CVs (reducible to 0.23 K with smaller time steps).
• Computational Performance:
  • For small systems (1 pipe), the IPM was faster (1.18 s vs. 6.03 s per time step).
  • For medium systems (2 pipes), the PD-MPC showed a significant advantage (7.81 s vs. 12.51 s).
  • For large systems (7 pipes), the PD-MPC demonstrated superior scalability, reducing calculation time from 3706 seconds (IPM) to 124.29 seconds (PD), a reduction to approximately 3.35% of the IPM time.
• Objective Value: While the IPM generally achieved slightly lower objective values (costs), the difference was marginal for smaller systems (nearly equal for 1 and 2 pipes) and approximately 17.2% higher for the 7-pipe system. The authors note that the PD-MPC successfully controlled the system to leverage electricity price fluctuations and minimize operating costs.

Significance and Claims
The paper claims that the proposed approach significantly enhances the scalability of MPC for thermo-hydraulic systems compared to standard monolithic optimization methods. The authors assert that by exploiting the model structure through primal decomposition, it becomes feasible to control large-scale networks that would otherwise be computationally prohibitive.

The authors maintain a modest stance regarding the current scope of their work. They acknowledge that the assumption of constant fluid properties is a simplification and that incorporating temperature-dependent properties would improve accuracy. Furthermore, they state that a full evaluation of the benefits requires more advanced system setups that account for prediction uncertainties. The paper concludes that while the PD approach offers a promising path for scalability, future work is necessary to provide a robust evaluation through comparisons with other automated MPC frameworks and more sophisticated system analyses.