# MGRIT-Based Multi-Level Parallel-in-Time Electromagnetic Transient Simulation

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## Abstract

**:**

## 1. Introduction

## 2. The Parallel-in-Time (PinT) Approach

#### Multigrid Reduction in Time

Algorithm 1: 2-level MGRIT algorithm |

1: repeat |

2: Propagate approximate solution ${\mathit{x}}^{\left(l\right)}$, cf. Equation (3) |

3: Compute residual $\mathit{R}\left(l\right)$ on coarse grid, cf. Equation (6) |

4: Solve coarse grid correction problem, cf. Equation (7) |

5: Correct approximate solution $\mathit{X}(l+1)$ at C-points, cf. Equation (8) |

6: until norm of residual $\mathit{R}$ is sufficiently small. |

7: Update solution $\mathit{x}(l+1)$ at F-points, cf. Equation (3) |

## 3. Implementation

#### 3.1. Special Considerations for the Solver

#### 3.2. Modeling of Converters

## 4. Evaluation

#### 4.1. Pi-Model Line

#### 4.2. Converter

#### 4.3. Microgrid

#### 4.4. Multi-Level Scaling Advantage

#### 4.5. Scalability of the Test Case

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Visualization of the time interval $[{t}_{0},{t}_{f}]$ and notation for two different time grids. Step sizes are $\delta $ (fine grid) and $\mathrm{\Delta}$ (coarse grid), the coarsening factor is $\mathrm{cf}=4$. C-points (F-points) are drawn as long (short) vertical lines which define the time grids ${\mathrm{\Theta}}_{\mathrm{\Delta}}$ (${\mathrm{\Theta}}_{\delta}$).

**Figure 2.**Execution flowchart of the implemented PinT version of the resistive-companion type solver in the two-level version.

**Figure 3.**Schematic of the pi-line. For testing the components individually in a simple test case, voltage source and resistive load were added at the respective terminals.

**Figure 4.**Schematic of the converter with output LC-filter. For testing the components individually in a simple test case, voltage source and resistive load were added at the respective terminals.

**Figure 5.**Schematic of the microgrid. Converters are marked by the label “DC/DC”, while the labels “Grid”, “Battery”, and “Household” represent simple models of the given elements. Black boxes represent a pi-model line, as described in Section 4.1.

**Figure 6.**Simulation results of a 16-household microgrid with ramps in each household for (

**a**) PinT and (

**b**) sequential execution. The simulation was performed with a cold-start (all voltages and currents equal to zero). The initial transient of the main bus voltage is shown in the lower left plots, respectively. The development of the current over the whole simulation interval is shown in the respective upper left plots (note the shift of the abscissa by the target voltage, $1000\mathrm{V}$). The main bus voltage stays well within an error interval of $(1000\pm 0.1)\mathrm{V}$. The currents in the households are summarized in the right-hand side plots. Each household ramps up its power consumption from zero to $8.450\mathrm{kW}$ over one second, with randomly chosen switch-on times. (

**a**) Results of PinT simulation. Some minor artifacts resulting from iterating only until the chosen error tolerance was reached are visible. (

**b**) Results of sequential simulation for comparison.

**Figure 7.**Multi-level allows for higher rates of parallelism, resulting in more speed-up in case the available resources are already exhausted by a two-level PinT algorithm. The plots illustrate this comparing speed-ups between the two-level and five-level versions of the MGRIT algorithm for 64, 96, 128, and 256 time-steps on the coarsest level. In all cases, a microgrid with four household-type elements was simulated with a coarsest time-step of $\mathrm{\Delta}t=12.5\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{s}$ and a fine time-step of $\delta t=0.78125\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{s}$. For the five-level version, three intermediate coarsening levels with uniform coarsening factor $\mathrm{cf}=2$ were added, resulting in time-steps of $\mathrm{\Delta}{t}_{\mathrm{lvl}}=\delta t\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{cf}}^{\mathrm{lvl}}$. (

**a**) Absolute runtime per coarse step. For all cases, the multi-level version scales better with increasing number of processors. (

**b**) Illustration of speed-up only resulting from adding processors. Relative runtime compared to that of the first datapoint.

**Table 1.**Runtime of multi-level PinT compared to two-level and sequential time-stepping for the pi-line test case.

Simulated timespan${t}_{f}\phantom{\rule{0.166667em}{0ex}}\left[s\right]$ | 1 | 10 | 100 | |

Sequential runtime$\left[\mathrm{s}\right]$ | 0.01055 | 0.08173 | 0.7922 | |

Parallel runtime and number of iterations | ||||

[% of sequential time/% of 2-lvl time/#iter] | ||||

${N}_{\mathrm{lvl}}=2$ | $\mathrm{cf}=2$ | 73.82/-/3 | 50.24/-/3 | 42.94/-/3 |

$\mathrm{cf}=4$ | 50.38/-/3 | 30.95/-/3 | 22.66/-/3 | |

$\mathrm{cf}=10$ | 50.58/-/4 | 20.43/-/4 | 12.95/-/4 | |

${N}_{\mathrm{lvl}}=3$ | $\mathrm{cf}=2$ | 50.89/68.94/3 | 30.30/60.31/3 | 23.59/54.94/3 |

$\mathrm{cf}=4$ | 47.65/94.58/4 | 16.62/53.70/4 | 11.70/51.63/4 | |

$\mathrm{cf}=10$ | 52.21/103.2/4 | 13.58/66.47/5 | 11.63/89.81/9 | |

${N}_{\mathrm{lvl}}=4$ | $\mathrm{cf}=2$ | 55.76/75.54/5 | 25.60/50.96/4 | 18.14/42.24/4 |

$\mathrm{cf}=4$ | — | 13.25/42.81/5 | 12.16/53.66/6 | |

$\mathrm{cf}=10$ | — | — | 9.567/73.88/8 |

**Table 2.**Runtime of multi-level PinT compared to two-level and sequential time-stepping for the converter test case. Deterioration in convergence occurs for cases in which the coarsest time-step and switching period of the converter are not well-aligned.

Simulated timespan${t}_{f}\phantom{\rule{0.166667em}{0ex}}\left[s\right]$ | 0.1 | 1 | 10 | |

Sequential runtime$\left[\mathrm{s}\right]$ | 3.909 | 39.27 | 394.3 | |

Parallel runtime and number of iterations | ||||

[% of sequential time/% of 2-lvl time/#iter] | ||||

${N}_{\mathrm{lvl}}=2$ | $\mathrm{cf}=2$ | 136.0/-/3 | 133.8/-/3 | 133.8/-/3 |

$\mathrm{cf}=5$ | 76.26/-/3 | 74.00/-/3 | 73.89/-/3 | |

$\mathrm{cf}=10$ | 41.83/-/4 | 38.82/-/4 | 38.70/-/4 | |

${N}_{\mathrm{lvl}}=3$ | $\mathrm{cf}=2$ | NC | NC | NC |

$\mathrm{cf}=5$ | 24.21/31.75/6 | 19.86/26.84/6 | 19.67/26.62/6 | |

$\mathrm{cf}=10$ | NC | NC | NC | |

${N}_{\mathrm{lvl}}=4$ | $\mathrm{cf}=2$ | NC | NC | NC |

$\mathrm{cf}=5$ | NC | NC | NC | |

$\mathrm{cf}=10$ | NC | NC | NC |

**Table 3.**Runtime of multi-level PinT compared to two-level and sequential time-stepping for the microgrid test case with four household-type elements. The results summarized here display the expected behavior of no convergence whenever the coarsest time-step does not align with the switching period. On the other hand, when alignment is given, the algorithm converges and, for higher coarsening factors and amounts of levels, speeds up the simulation somewhat.

Simulated timespan${t}_{f}\phantom{\rule{0.166667em}{0ex}}\left[s\right]$ | 0.1 | 1 | 10 | |

Sequential runtime$\left[s\right]$ | 10.10 | 100.4 | 1000 | |

Parallel runtime and number of iterations | ||||

[% of sequential time/% of 2-lvl time/#iter] | ||||

${N}_{\mathrm{lvl}}=2$ | $\mathrm{cf}=2$ | 184.2/-/4 | 184.7/-/4 | 186.0/-/4 |

$\mathrm{cf}=5$ | 95.93/-/6 | 95.99/-/6 | 96.74/-/6 | |

$\mathrm{cf}=10$ | 61.59/-/20 | 60.83/-/20 | 61.32/-/20 | |

${N}_{\mathrm{lvl}}=3$ | $\mathrm{cf}=2$ | NC | NC | NC |

$\mathrm{cf}=5$ | 32.84/34.23/8 | 36.69/38.22/8 | 36.97/38.22/8 | |

$\mathrm{cf}=10$ | NC | NC | NC | |

${N}_{\mathrm{lvl}}=4$ | $\mathrm{cf}=2$ | NC | NC | NC |

$\mathrm{cf}=5$ | NC | NC | NC | |

$\mathrm{cf}=10$ | NC | NC | NC |

**Table 4.**Runtimes for sufficiently small time-step to allow convergence in the microgrid test case. Time-steps of this size would usually not be used in applications, but the results prove that better speed-ups are possible if the alignment of coarse step size and switching period is respected.

Timestep $\mathit{\delta}\mathit{t}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]$ | Sim. Time ${\mathit{t}}_{\mathit{f}}\phantom{\rule{0.166667em}{0ex}}\left[\mathbf{s}\right]$ | Seq. Runtime $\mathbf{\left[}\mathbf{s}\mathbf{\right]}$ | Par. Runtime [% Seq. Time/#iter] | ||
---|---|---|---|---|---|

${\mathit{N}}_{\mathbf{lvl}}=4$ | |||||

$\mathbf{cf}=2$ | $\mathbf{cf}=5$ | $\mathbf{cf}=10$ | |||

2.0$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 0.1 | 51.63 | NC | 15.04/33 | NC |

2.5$\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ | 0.1 | 4109 | 25.45/3 | 13.49/4 | 13.05/7 |

**Table 5.**Microgrid: Runtime Comparison Between PinT and Sequential Timestepping for Different Grid Sizes.

Households | Runtime | |
---|---|---|

Seq. [s] | PinT [% Seq.] | |

4 | 29.71 | 33.24 |

8 | 57.23 | 32.86 |

16 | 148.8 | 33.72 |

32 | 545.0 | 34.63 |

64 | 2798 | 35.31 |

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**MDPI and ACS Style**

Strake, J.; Döhring, D.; Benigni, A.
MGRIT-Based Multi-Level Parallel-in-Time Electromagnetic Transient Simulation. *Energies* **2022**, *15*, 7874.
https://doi.org/10.3390/en15217874

**AMA Style**

Strake J, Döhring D, Benigni A.
MGRIT-Based Multi-Level Parallel-in-Time Electromagnetic Transient Simulation. *Energies*. 2022; 15(21):7874.
https://doi.org/10.3390/en15217874

**Chicago/Turabian Style**

Strake, Julius, Daniel Döhring, and Andrea Benigni.
2022. "MGRIT-Based Multi-Level Parallel-in-Time Electromagnetic Transient Simulation" *Energies* 15, no. 21: 7874.
https://doi.org/10.3390/en15217874