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Model Predictive Control of District Heating Grids Using Stabilizing Terminal Ingredients

Max Rose, Hannes Gernandt, Juan E. Machado, Johannes Schiffer

TL;DR

This paper tackles stabilizing model predictive control for district heating grids (DHGs) incorporating thermal energy storage to support decarbonization. It develops an ordinary differential equation–based thermo-hydraulic DHG model on a graph, including mass and energy balances and Kirchhoff's laws, and discretizes it for MPC. A key theoretical contribution is a sufficient stabilizability condition based on grid topology, enabling MPC with terminal ingredients (terminal region and terminal cost) to achieve asymptotic stabilization for DHGs. A numerical case study demonstrates practical MPC with terminal ingredients for a two-TES DHG, showing successful convergence to two steady states and highlighting improved performance when using a predictive, piecewise-constant reference strategy. The work lays a foundation for scalable, forecast-informed control of RES-based DHGs with provable stability guarantees and constrained operation.

Abstract

The transformation of fossil fuel-based district heating grids (DHGs) to CO$_2$-neutral DHGs requires the development of novel operating strategies. Model predictive control (MPC) is a promising approach, as knowledge about future heat demand and heat supply can be incorporated into the control, operating constraints can be ensured and the stability of the closed-loop system can be guaranteed. In this paper, we employ MPC for DHGs to control the system mass flows and injected heat flows. Following common practice, we derive terminal ingredients to stabilize given steady state temperatures and storage masses in the DHG. To apply MPC with terminal ingredients, it is crucial that the system under control is stabilizable. By exploiting the particular system structure, we give a sufficient condition for the stabilizability in terms of the grid topology and hence, for the applicability of the MPC scheme to DHGs. Furthermore, we demonstrate the practicability of the application of MPC to an exemplary DHG in a numerical case study.

Model Predictive Control of District Heating Grids Using Stabilizing Terminal Ingredients

TL;DR

This paper tackles stabilizing model predictive control for district heating grids (DHGs) incorporating thermal energy storage to support decarbonization. It develops an ordinary differential equation–based thermo-hydraulic DHG model on a graph, including mass and energy balances and Kirchhoff's laws, and discretizes it for MPC. A key theoretical contribution is a sufficient stabilizability condition based on grid topology, enabling MPC with terminal ingredients (terminal region and terminal cost) to achieve asymptotic stabilization for DHGs. A numerical case study demonstrates practical MPC with terminal ingredients for a two-TES DHG, showing successful convergence to two steady states and highlighting improved performance when using a predictive, piecewise-constant reference strategy. The work lays a foundation for scalable, forecast-informed control of RES-based DHGs with provable stability guarantees and constrained operation.

Abstract

The transformation of fossil fuel-based district heating grids (DHGs) to CO-neutral DHGs requires the development of novel operating strategies. Model predictive control (MPC) is a promising approach, as knowledge about future heat demand and heat supply can be incorporated into the control, operating constraints can be ensured and the stability of the closed-loop system can be guaranteed. In this paper, we employ MPC for DHGs to control the system mass flows and injected heat flows. Following common practice, we derive terminal ingredients to stabilize given steady state temperatures and storage masses in the DHG. To apply MPC with terminal ingredients, it is crucial that the system under control is stabilizable. By exploiting the particular system structure, we give a sufficient condition for the stabilizability in terms of the grid topology and hence, for the applicability of the MPC scheme to DHGs. Furthermore, we demonstrate the practicability of the application of MPC to an exemplary DHG in a numerical case study.
Paper Structure (5 sections, 2 theorems, 26 equations, 3 figures, 1 table)

This paper contains 5 sections, 2 theorems, 26 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Thermo-hydraulic model
  3. Model predictive controller
  4. Case Study
  5. Conclusion

Key Result

Lemma III.5

For the steady state mass flows $\overline{q}_{\mathrm{c}}$, the matrix $\tilde{A}$ given by eq:def_tilde_A fulfills $\tilde{A}(\overline{q}_{\mathrm{c}})+\tilde{A}(\overline{q}_{\mathrm{c}})^\top\leq 0$.

Figures (3)

  • Figure 1: Schematic of an exemplary DHG.
  • Figure 2: Graph representation of the exemplary DHG from Figure \ref{['fig:exmplDHG_skizze']} with definitions of associated sets $\mathcal{E}_{\mathrm{d}}$, $\mathcal{E}_{\mathrm{pr}}$, $\mathcal{V}_{\mathrm{h}}$ and $\mathcal{V}_{\mathrm{c}}$. Dotted rectangles frame vertices of a corresponding TES.
  • Figure 3: Masses and temperatures at TESs with corresponding mass flow rates as well as heat flow injections for MPC 1 and MPC 2. Set points are indicated by the dotted lines.

Theorems & Definitions (4)

  • Definition III.2: Stabilizability Heij2021
  • Lemma III.5
  • Proposition III.6
  • proof