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Economic Nonlinear Model Predictive Control of Prosumer District Heating Networks: The Extended Version

Max Sibeijn, Saeed Ahmed, Mohammad Khosravi, Tamás Keviczky

Abstract

In this paper, we propose an economic nonlinear model predictive control (MPC) algorithm for district heating networks (DHNs). The proposed method features prosumers, multiple producers, and storage systems, which are essential components of 4th generation DHNs. These networks are characterized by their ability to optimize their operations, aiming to reduce supply temperatures, accommodate distributed heat sources, and leverage the flexibility provided by thermal inertia and storage, all crucial for achieving a fossil-fuel-free energy supply. Developing a smart energy management system to accomplish these goals requires detailed models of highly complex nonlinear systems and computational algorithms able to handle large-scale optimization problems. To address this, we introduce a graph-based optimization-oriented model that efficiently integrates distributed producers, prosumers, storage buffers, and bidirectional pipe flows, such that it can be implemented in a real-time MPC setting. Furthermore, we conduct several numerical experiments to evaluate the performance of the proposed algorithms in closed-loop. Our findings demonstrate that the MPC methods achieved up to 9% cost improvement over traditional rule-based controllers while better maintaining system constraints.

Economic Nonlinear Model Predictive Control of Prosumer District Heating Networks: The Extended Version

Abstract

In this paper, we propose an economic nonlinear model predictive control (MPC) algorithm for district heating networks (DHNs). The proposed method features prosumers, multiple producers, and storage systems, which are essential components of 4th generation DHNs. These networks are characterized by their ability to optimize their operations, aiming to reduce supply temperatures, accommodate distributed heat sources, and leverage the flexibility provided by thermal inertia and storage, all crucial for achieving a fossil-fuel-free energy supply. Developing a smart energy management system to accomplish these goals requires detailed models of highly complex nonlinear systems and computational algorithms able to handle large-scale optimization problems. To address this, we introduce a graph-based optimization-oriented model that efficiently integrates distributed producers, prosumers, storage buffers, and bidirectional pipe flows, such that it can be implemented in a real-time MPC setting. Furthermore, we conduct several numerical experiments to evaluate the performance of the proposed algorithms in closed-loop. Our findings demonstrate that the MPC methods achieved up to 9% cost improvement over traditional rule-based controllers while better maintaining system constraints.

Paper Structure

This paper contains 33 sections, 3 theorems, 52 equations, 15 figures, 3 tables.

Table of Contents

  1. Introduction
  2. State of the Art
  3. Contributions
  4. District Heating Network Model
  5. Graph Model
  6. Edge Dynamics
  7. Nodal Constraints
  8. AROMA: A Benchmark for District Heating Networks
  9. DHN Model: A Tractable Reformulation for the Hydraulics
  10. Fixing the Flow Direction
  11. Independent Flows
  12. Treatment of the Momentum Equation
  13. Convex Reformulation of the Kirchhoff Loop Constraints
  14. DHN Model: A Tractable Reformulation for the Thermal Dynamics
  15. State-space Representation of the DHN Interconnection
  16. ...and 18 more sections

Key Result

Proposition 1

Let Assumption ass:unique_valves hold. Then, for any $q$ that satisfies eq:inequality, there exist ${\nu} \in \mathbb{R}_+^{|\mathcal{V}|}$ and ${r} \in [0, 1]^{|\mathcal{P}|}$ such that eq:equality is satisfied.

Figures (15)

  • Figure 1: A schematic diagram of a district heating network with multiple substations (consumers) and production points.
  • Figure 2: The AROMA district heating network with multiple consumers, a prosumer, a storage buffer, and a loop.
  • Figure 3: A directed graph abstraction of the AROMA district heating network with fixed flow directions. Double-sided arrows indicate two opposite facing edges.
  • Figure 4: The valve checking procedure, described in Remark \ref{['rem:valve_placement']}, illustrated by showing the flow from producers (p) to consumers (c) going through the supply network consisting of merging nodes (m) and splitting nodes (s).
  • Figure 5: Illustration of thermal node model and mesh refinement of the DHN graph. (a) shows the conceptual thermal node model with its dynamics defined by \ref{['eq:dynamics']}, while (b) demonstrates the injection of thermal nodes in a section of the AROMA network.
  • ...and 10 more figures

Theorems & Definitions (19)

  • Remark 1
  • Definition 1: Directed cycle
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 9 more