[Paper Review] Model order reduction of hyperbolic systems at the example of district heating networks

Markus Rein, Jan Möhring+2 more|arXiv (Cornell University)|Mar 8, 2019

Model Reduction and Neural Networks24 references4 citations

TL;DR

This paper presents a stable, computationally efficient reduced-order model (ROM) for hyperbolic differential-algebraic equations (DAEs) in district heating networks by leveraging a parametric linear time-varying (LTV) formulation and frequency-domain greedy Galerkin projection. The method ensures Lyapunov stability via a global energy matrix and achieves up to 10x speedup over full-order models while maintaining accuracy in optimal control-relevant error ranges.

ABSTRACT

In this article a framework for the generation of a computationally fast surrogate model for district heating networks is presented. An appropriate model results in an index-1 hyperbolic, differential algebraic equation quadratic in state, exhibiting several hundred of outputs to be approximated. We show the existence of a global energy matrix which fulfills the Lyapunov inequality ensuring stability of the reduced model. By considering algebraic variables as parameters to the dynamical transport, the reduction of a linear, time varying (LTV) problem results. We present a scheme to efficiently combine linear reductions to a global surrogate model using a greedy strategy in the frequency domain. The numerical effectiveness of the scheme is demonstrated at different, existing, large scale networks.

Motivation & Objective

• To develop a stable, fast surrogate model for large-scale district heating networks governed by hyperbolic DAEs with high-dimensional outputs.
• To address the challenge of slow singular value decay and nonlinearity in hyperbolic systems for model order reduction (MOR).
• To preserve stability and passivity in the reduced model through a globally constructed energy matrix satisfying the Lyapunov inequality.
• To enable efficient online simulation for model predictive control by combining linear reductions across network substructures.
• To demonstrate effectiveness on real-world networks with complex dynamics, including flux reversal and looped topologies.

Proposed method

• Formulate the district heating network as an index-1 hyperbolic DAE with algebraic constraints for energy conservation at junctions.
• Treat time-varying flow velocities as parameters, transforming the system into a parametric linear time-varying (LTV) system for each pipeline.
• Construct a global energy matrix Q that satisfies the Lyapunov inequality, ensuring stability of the reduced model.
• Apply a greedy algorithm in the frequency domain to sample input signals and generate optimal reduced-order bases.
• Use Galerkin projection with frequency-domain snapshots to construct a stable, low-dimensional ROM for each subnetwork.
• Decompose complex networks with flux reversals into subnetworks, reducing only those with constant flow direction, while treating flux-reversing parts in full order.

Experimental results

Research questions

• RQ1 Can a stable, low-dimensional surrogate model be constructed for large-scale hyperbolic DAEs arising in district heating networks?
• RQ2 How can the Lyapunov stability of the reduced model be guaranteed despite nonlinearity and high dimensionality?
• RQ3 Can a global energy matrix be systematically derived to ensure stability across the entire network?
• RQ4 How effective is the frequency-domain greedy sampling strategy in capturing the dynamics with minimal reduced order?
• RQ5 How does the method perform on real-world networks with complex topologies, including loops and flux reversals?

Key findings

• The proposed method achieves up to a 10-fold speedup in simulation runtime compared to full-order models while maintaining accuracy within the error range relevant for optimal control.
• The global energy matrix Q satisfies the Lyapunov inequality, ensuring asymptotic stability of the reduced model across all network configurations.
• For a 32-consumer street network, the reduced model achieves high accuracy across in-sample and out-of-sample inputs, with runtime proportional to the square root of the number of cells.
• In a large-scale 333-consumer city district network with 6 loops and flux reversals, the method successfully captures complex dynamics by isolating the flux-reversing subnetwork and reducing only the remaining parts.
• The speed-up increases with spatial refinement, indicating that the reduced model becomes more efficient at higher resolutions, especially in stiff, synchronizing regimes.
• The greedy frequency-domain sampling strategy enables stable and accurate ROMs even for nonlinear systems with changing flow directions, outperforming standard MOR techniques in this context.