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A flexible numerical tool for large dynamic DC networks

Erwin Luesink, Juan Giraldo, Bernard Geurts, Johann Hurink, Hans Zwart

TL;DR

This paper addresses the need for fast, scalable time-domain simulations of large DC networks in the context of increasing AC-DC hybrid grids. It introduces a sparse, affine DC-network model built from DGUs and line Pi-models, and evaluates five adaptive time-stepping schemes (RK45, RK23, DOP853, BDF, Radau) on IEEE and PEGASE benchmarks, highlighting the impact of sparsity on computational cost. Key findings show that explicit methods scale linearly with network size (when exploiting sparsity), with RK23 often fastest at modest accuracy and Radau delivering the best overall efficiency for stiff dynamics, while implicit methods are not consistently advantageous for the largest cases. The work provides practical guidance for choosing solvers in grid planning, fault simulation, and communication-optimization tasks, and points to future extensions for integration with AC networks and more extensive fault analysis.

Abstract

DC networks play an important role within the ongoing energy transition. In this context, simulations of designed and existing networks and their corresponding assets are a core tool to get insights and form a support to decision-making. Hereby, these simulations of DC networks are executed in the time domain. Due to the involved high frequencies and the used controllers, the equations that model these DC networks are stiff and highly oscillatory differential equations. By exploiting sparsity, we show that conventional adaptive time stepping schemes can be used efficiently for the time domain simulation of very large DC networks and that this scales linearly in the computational cost as the size of the networks increase.

A flexible numerical tool for large dynamic DC networks

TL;DR

This paper addresses the need for fast, scalable time-domain simulations of large DC networks in the context of increasing AC-DC hybrid grids. It introduces a sparse, affine DC-network model built from DGUs and line Pi-models, and evaluates five adaptive time-stepping schemes (RK45, RK23, DOP853, BDF, Radau) on IEEE and PEGASE benchmarks, highlighting the impact of sparsity on computational cost. Key findings show that explicit methods scale linearly with network size (when exploiting sparsity), with RK23 often fastest at modest accuracy and Radau delivering the best overall efficiency for stiff dynamics, while implicit methods are not consistently advantageous for the largest cases. The work provides practical guidance for choosing solvers in grid planning, fault simulation, and communication-optimization tasks, and points to future extensions for integration with AC networks and more extensive fault analysis.

Abstract

DC networks play an important role within the ongoing energy transition. In this context, simulations of designed and existing networks and their corresponding assets are a core tool to get insights and form a support to decision-making. Hereby, these simulations of DC networks are executed in the time domain. Due to the involved high frequencies and the used controllers, the equations that model these DC networks are stiff and highly oscillatory differential equations. By exploiting sparsity, we show that conventional adaptive time stepping schemes can be used efficiently for the time domain simulation of very large DC networks and that this scales linearly in the computational cost as the size of the networks increase.
Paper Structure (23 sections, 8 equations, 5 figures, 3 tables)

This paper contains 23 sections, 8 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature review
  4. Contributions
  5. Circuits and networks
  6. Circuits.
  7. Networks.
  8. DC networks
  9. Building blocks for a DC network
  10. Control objectives
  11. Proportional current sharing.
  12. Average voltage regulation.
  13. Linear representation
  14. Adaptive time-stepping schemes
  15. RK45.
  16. ...and 8 more sections

Figures (5)

  • Figure 1: The circuit on the left represents distributed generation unit (DGU) $i$ and the circuit on the right represents line $ij$. Line $ij$ connects to DGU $i$ by means of the point of common connection (PCC) indexed by $i$.
  • Figure 2: Circuit diagram describing DGU $i$ and the physical line connecting DGU $i$ to a DGU $j$. Note that the capacitor $C^L_i$ is the lumped capacitor that represents all parallel capacitors at the $i$th point of common connection.
  • Figure 3: The physical graph and communication graph of the IEEE Case 9 network, plotted in the circular layout. Generator nodes are indicated with a star. The color scale is used to visualise the voltage and currents in Figure \ref{['fig:case9response']}.
  • Figure 4: Simulation results of the computational experiment for the IEEE Case 9 network. The colors match with the nodes and lines in Figure \ref{['fig:case9networks']}. The top plot shows the voltage at the point of common connection for each DGU. The middle plot shows the generated current for each DGU. The bottom plot shows the absolute value of the line currents for each line. The initial state does not satisfy the control objectives, but after a short time, the state converges to an equilibrium that does satisfy the objectives of proportional current sharing and average voltage regulation. At time $t=1.5$, the load is increased in one node, and at time $t=2.0$, the load is removed.
  • Figure 5: Simulation results of the DC networks based on power network topologies. For the competitive adaptive schemes, the simulation time is shown versus the state space dimension associated with each network.

Theorems & Definitions (1)

  • Remark 2.1