Assessment of Methods for the Real-Time Simulation of Electronic and Thermal Circuits
Abstract
:1. Introduction
2. Numerical Methods
2.1. Formulation of the Problem as a System of ODEs
2.2. Solvers for Systems of DAE
2.3. Implementation
3. Benchmark Problems
3.1. RLC Circuit
3.2. Scalable RLC Circuit
3.3. Full Wave Rectifier
3.4. Synchronous Motor Thermal Equivalent Circuit
3.5. Reference Solutions
3.5.1. RLC Circuit
3.5.2. Scalable RLC Circuit
3.5.3. Full Wave Rectifier
3.5.4. Synchronous Motor
3.6. Error Measurement and Performance Evaluation
4. Numerical Experiments
4.1. Simulation Environments
- a conventional desktop PC,
- a Beaglebone Black, and
- a Raspberry Pi 4.
4.2. Numerical Results
4.2.1. RLC Circuit
4.2.2. Scalable RLC Circuit
4.2.3. Full Wave Rectifier
- the update of the dynamic terms , , , and their derivatives;
- the evaluation of the residual and the tangent matrix in Equation (7);
- the factorization and backsubstitution of the linear solver in Equation (7);
- the projections in Equation (16); and
- other operations, such as the update of variables and regulation of code execution.
4.2.4. Synchronous Motor
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Abbreviations
BDF | Backward Differentiation Formula |
CCS | Column-Compressed Storage |
DAE | Differential-Algebraic Equation |
HiL | Hardware-in-the-Loop |
ODE | Ordinary Differential Equation |
PMSM | Permanent-Magnet Synchronous Motor |
RT | Real Time |
SITL | System-in-the-Loop |
TR | Trapezoidal Rule |
Appendix A. Airgap Resistor
References
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1 | 1 | - | - | |
2 | 2 | - | ||
3 | 3 |
Resistor | Value | Capacitor | Value | Loss | Value |
---|---|---|---|---|---|
[K/W] | [J/K] | [W] | |||
1 | 2000 | 150 | |||
1 | 5000 | 2000 | |||
0.025 | 2000 | 200 | |||
0.015 | 1500 | 50 | |||
0.025 | 3000 | 50 | |||
0.01 | 300 | ||||
0.01 | 300 | ||||
0.001 | 500 | ||||
0.05 | 500 | ||||
0.0025 | |||||
0.25 | |||||
0.25 | |||||
0.001 | |||||
0.5 | |||||
0.5 | |||||
3 | |||||
3 |
Problem | Variables | Algebraic ctr. | Differential ctr. |
---|---|---|---|
RLC | 8 | 5 | 3 |
Sc. RLC | |||
Rectifier | 20 | 19 | 1 |
Thermal | 49 | 40 | 9 |
Circuit | Simulation Time [s] | Sample Points |
---|---|---|
RLC | 10 | 1000 |
Sc. RLC | 1 | 100 |
Rectifier | 1 | 1000 |
Thermal | 5000 | 5000 |
Circuit | Precision | Error V or T | Error I or Q |
---|---|---|---|
RLC / Sc. RLC | Low | 1 mV | 1 mA |
High | 10 μV | 10 μA | |
Rectifier | Low | 1 mV | 1 mA |
High | 10 μV | 10 μA | |
Thermal | Low | 0.1 K | 1 W |
High | 0.01 K | 0.1 W |
Platform | CPU | Clock Freq. | Cores/ | RAM | L1 Cache | L2 Cache | L3 Cache | OS |
---|---|---|---|---|---|---|---|---|
[GHz] | Threads | [GB] | [kB] | [MB] | [MB] | |||
PC | Intel i7-8700K | 3.7 | 6/12 | 8 | 192/192 | 1.5 | 12 | Win10 |
RPi4 | ARM A-72 | 1.5 | 4/4 | 4 | 48/32 | 1 | 0 | Raspbian 4.19.57 |
BBB | ARM A-8 | 1.0 | 1/1 | 0.5 | 32/32 | 0.25 | 0 | Debian 9.9 IoT |
Method | h | Error V | Error I | |
---|---|---|---|---|
[ms] | [mV] | [mA] | ||
TR | 2 | |||
BDF1 | 1 | |||
BDF2 | 2 | |||
BDF3 | 2 |
Method | h | Error V | Error I | |
---|---|---|---|---|
[ms] | [μV] | [μA] | ||
TR | 2 | |||
BDF1 | 1 | |||
BDF2 | 1 | |||
BDF3 | 1 |
Method | Prec. | Elapsed Time | Elapsed Time | Elapsed Time |
---|---|---|---|---|
PC [ms] | BBB [ms] | RPi [ms] | ||
TR | Low | |||
TRnP | ||||
BDF1 | ||||
BDF2 | ||||
BDF3 | ||||
TR | High | |||
TRnP | ||||
BDF1 | 16,764.9 | 496,852.0 | 77,287.4 | |
BDF2 | ||||
BDF3 |
Method | h | Error V | Error I | |
---|---|---|---|---|
[ms] | [mV] | [mA] | ||
TR | 2 | |||
BDF1 | 1 | |||
BDF2 | 1 | 2 | ||
BDF3 | 2 |
Method | h | Error V | Error I | |
---|---|---|---|---|
[ms] | [μV] | [μA] | ||
TR | 2 | |||
BDF1 | 1 | |||
BDF2 | 2 | |||
BDF3 | 2 |
Method | Prec. | Elapsed Time | Elapsed Time | Elapsed Time |
---|---|---|---|---|
PC [ms] | BBB [ms] | RPi [ms] | ||
TR | Low | |||
TRnP | ||||
BDF1 | ||||
BDF2 | ||||
BDF3 | ||||
TR | High | |||
TRnP | ||||
BDF1 | 65,542.3 | |||
BDF2 | ||||
BDF3 |
Method | Precision | Loops | System Size |
---|---|---|---|
[] | [] | ||
TR | Low | 2053 | 10,268 |
BDF1 | 48 | 243 | |
BDF2 | 1790 | 8953 | |
BDF3 | 3661 | 18,308 | |
TR | High | 248 | 1243 |
BDF1 | - | - | |
BDF2 | 186 | 933 | |
BDF3 | 447 | 2238 |
Method | h | Error V | Error I | |
---|---|---|---|---|
[ms] | [mV] | [mA] | ||
TR | 16 | |||
BDF1 | 3 | |||
BDF2 | 7 | |||
BDF3 | 9 |
Method | h | Error V | Error I | |
---|---|---|---|---|
[ms] | [μV] | [μA] | ||
TR | 6 | |||
BDF1 | 1 | |||
BDF2 | 9 | |||
BDF3 | 9 |
Method | Prec. | Elapsed Time | Elapsed Time | Elapsed Time |
---|---|---|---|---|
PC [ms] | BBB [ms] | RPi [ms] | ||
TR | Low | |||
TRnP | ||||
BDF1 | ||||
BDF2 | ||||
BDF3 | ||||
TR | High | |||
TRnP | ||||
BDF1 | 14,009.5 | 141,564.0 | 61,429.0 | |
BDF2 | ||||
BDF3 |
Method | h | Error T | Error Q | |
---|---|---|---|---|
[ms] | [K] | [W] | ||
TR | 500 | 2 | ||
BDF1 | 25 | 2 | ||
BDF2 | 250 | 2 | ||
BDF3 | 500 | 2 |
Method | h | Error T | Error Q | |
---|---|---|---|---|
[ms] | [K] | [W] | ||
TR | 100 | 2 | ||
BDF1 | 2 | |||
BDF2 | 100 | 2 | ||
BDF3 | 250 | 2 |
Method | Prec. | Elapsed Time | Elapsed Time | Elapsed Time |
---|---|---|---|---|
PC [s] | BBB [s] | RPi [s] | ||
TR | Low | |||
TRnP | ||||
BDF1 | ||||
BDF2 | ||||
BDF3 | ||||
TR | High | |||
TRnP | ||||
BDF1 | ||||
BDF2 | ||||
BDF3 |
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Rodríguez, B.; González, F.; Naya, M.Á.; Cuadrado, J. Assessment of Methods for the Real-Time Simulation of Electronic and Thermal Circuits. Energies 2020, 13, 1354. https://doi.org/10.3390/en13061354
Rodríguez B, González F, Naya MÁ, Cuadrado J. Assessment of Methods for the Real-Time Simulation of Electronic and Thermal Circuits. Energies. 2020; 13(6):1354. https://doi.org/10.3390/en13061354
Chicago/Turabian StyleRodríguez, Borja, Francisco González, Miguel Ángel Naya, and Javier Cuadrado. 2020. "Assessment of Methods for the Real-Time Simulation of Electronic and Thermal Circuits" Energies 13, no. 6: 1354. https://doi.org/10.3390/en13061354