# Electric Vehicle Inverter Electro-Thermal Models Oriented to Simulation Speed and Accuracy Multi-Objective Targets

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## Abstract

**:**

## 1. Introduction

## 2. Semiconductor Modeling

- Conduction voltage drop
- Conduction power losses
- Switching power losses
- Thermal behavior (as described in Section 4)

#### 2.1. Conduction Voltage Drop Variants

_{th}) and an on-resistance (R

_{on}) (Figure 4) [17]. Considering a non-ideal semiconductor-based inverter leg, the total voltage drop between the two terminals of a semiconductor means a slight reduction or deviation from the theoretical obtainable output voltage. This can become an important factor when modeling an inverter in voltage sensitive applications such as the series connection of semiconductor devices, low voltage electric vehicle drives, and high-speed flux weakening control.

_{on}

_{)}and threshold voltage (V

_{th}) dependent voltage drop; junction temperature (T

_{j}) dependent voltage drop; current dependent voltage drop; and current and T

_{j}dependent voltage drop.

#### 2.1.1. Ideal Output Voltage

_{ph-N}) of the inverter will be calculated as follows in Equations (1) and (2):

_{T}refers to the set of gate logic signals that control the state of each leg/half-bridge of the inverter and V

_{DC}is the DC bus voltage.

#### 2.1.2. R_{on} and V_{th} Dependent Voltage Drop

_{th}) and an on-resistance (R

_{on}), according to their forward conduction curves, where both are considered constant. For that purpose, an average value was chosen for each of them as in Equations (3) and (4). In this case, the junction temperature does not affect to the output voltage value.

#### 2.1.3. T_{j} Dependent Voltage Drop

_{on}and the V

_{th}that model the conduction of the semiconductors are junction temperature dependent (Equations (5) and (6)).

_{on}(T

_{j}) refers to the junction temperature-dependent internal resistance of the semiconductor and V

_{th}(T

_{j}) is the junction temperature dependent-voltage threshold.

#### 2.1.4. Current Dependent Voltage Drop

_{d}) is modeled by means of a lookup table dependent on conducted current, according to the forward characteristic of each semiconductor (Equation (7)). Thus, this variant does not consider the dependence on junction temperature.

_{d,1D-LUT}(I) is the conducted current-dependent voltage drop in the semiconductor.

#### 2.1.5. Current and T_{j} Dependent Output Voltage

_{d,2D-LUT}(I,T

_{j}) is the conducted current and junction temperature dependent voltage drop in the semiconductor.

#### 2.2. Conduction Losses

_{T}) switch, which results in a voltage drop V

_{d}through its terminals, as described in Section 2.1. Depending on the required model accuracy, the voltage drop can be modeled taking into account different variables. The antiparallel diode can also be modeled through the same approach. The ideal switching waveforms of a semiconductor and its power losses (conduction and switching) can be seen in Figure 6.

#### 2.2.1. Ideal Conduction

#### 2.2.2. Constant R_{on} and V_{th} Conduction Losses

_{on}and V

_{th}dependent output voltage calculation, each semiconductor is modeled as a set of a voltage drop (V

_{th}) and an on-resistance (R

_{on}), according to their forward conduction curves, where both are considered constant (Equation (11)). An average value is chosen and the junction temperature is not taken into account.

#### 2.2.3. T_{j} Dependent Conduction Losses

_{on}and V

_{th}parameters are not constant, but dependent on the junction temperature. The conduction losses are calculated as follows (Equation (12)):

#### 2.2.4. Current Dependent Conduction Losses

_{d}is modeled by means of a lookup table dependent on the conducted current, according to the forward characteristic of each semiconductor. This variant does not consider the dependence on junction temperature. Therefore, the conduction losses are calculated by multiplying the voltage drop and the current (Equation (13)).

#### 2.2.5. Current and T_{j} Dependent Conduction Losses

#### 2.3. Switching Losses

_{sw,on}, E

_{sw,off}), distributed along a switching step time (Equation (15), Figure 6).

#### 2.3.1. Ideal Switching

#### 2.3.2. Current Dependent Switching Losses

_{sw,on,1D-LUT}(I) and E

_{sw,off,1D-LUT}(I) refer to the current-dependent switching energy losses in a semiconductor.

#### 2.3.3. Voltage and Current Dependent Switching Losses

_{sw,on,2D-LUT}(I,V) and E

_{sw,off,2D-LUT}(I,V) refer to the current and voltage dependent energy losses in a semiconductor.

#### 2.3.4. T_{j} and Current Dependent Switching Losses

_{sw,on,2D-LUT}(I,T

_{j}) and E

_{sw,off,2D-LUT}(I,T

_{j}) refer to the current and junction temperature dependent energy losses.

#### 2.3.5. Voltage, T_{j} and Current Dependent Switching Losses

_{sw,on,3D-LUT}(I,V,T

_{j}) and E

_{sw,off,3D-LUT}(I,V,T

_{j}) refer to the current, voltage, and junction temperature dependent energy losses.

#### 2.3.6. Analytical Switching Losses

_{on}is the turn on time and t

_{off}is the turn off time of the semiconductor.

#### 2.4. Blocking Behavior

_{off}) was introduced to comply with this required relationship to run the simulations. The possible resultant blocking losses were neglected. However, in some applications such as battery powered portable devices, the blocking behavior could be linked to a standby consumption, and could be crucial for the application.

## 3. Inverter Electrical Models

#### 3.1. Hi-Fi Model

#### 3.2. M-Fi Model

#### 3.3. Lo-Fi Model

#### 3.4. Fast Lo-Fi Model

#### 3.5. Model Summary

## 4. Inverter Thermal Variants

_{th,j-cs}(instead of Z

_{th,j-c}and Z

_{th,c-s}) was considered.

#### 4.1. Junction-to-Case Thermal Model

#### 4.1.1. Individual n-Stage Thermal Model (12 × n RC)

_{th,i}) and time constant (τ

_{i}) pairs, each of them referring to a cascaded Foster RC pair.

#### 4.1.2. Simplified Single Stage Thermal Model (12 × 1 RC)

#### 4.1.3. Global Equivalent Inverter Thermal Model (1 × n RC)

#### 4.2. Heatsink-to-Coolant Output Behavior (ΔT_{sa} Calculation)

_{th,sa,n}, τ

_{th,sa,n}) are not constant.

_{th,sa,1}, R

_{th,sa,2}and τ

_{th,sa,1}, τ

_{th,sa,2}are the parameters of a second order foster network.

_{th,sa,1}, R

_{th,sa,2}, and τ

_{th,sa,1}, τ

_{th,sa,2}) with respect to flow rate and inlet coolant temperature was introduced by means of lookup tables, which were derived from the formula given in a SEMIKRON application note [29] via the MATLAB script. A fixed water to glycol ratio for coolant was assumed.

## 5. Analysis of the Inverter Model Accuracy

#### 5.1. Comparison of Electrical Models

#### 5.1.1. Conduction Losses Comparison

#### 5.1.2. Switching Losses Comparison

#### 5.1.3. Output Voltage Comparison

#### 5.2. Thermal Model Comparison

#### 5.2.1. Individual n-Stage Thermal Model (12 × n RC) vs. Simplified Single Stage Thermal Model (12 × 1 RC)

_{R1}active switch and the D

_{R1}diode with the most accurate 12 × 5 RC thermal model (n = 5 RC pairs were considered). The waveforms showed that both semiconductors had similar junction temperatures and evolution in time, but they were not equal, since they dissipated different power losses and their associated thermal circuit parameters were different.

_{j,TR1}) of both modeling approaches are compared, it can be concluded that both waveforms are similar. From the zoom shown in Figure 26, it can be deduced that the 12 × 1 thermal model approach filters the high frequency (f

_{sw}= 5 kHz) ripple that appears in the 12 × 5 thermal model. Moreover, the fundamental frequency ripple (f

_{o}= 50 Hz) was also attenuated, and a slight delay phenomenon was observed, as seen with the different electrical approaches in this paper. However, the accuracy error of the average value was less than 1% along all transient states and in the steady state. Hence, it can be concluded that a low accuracy loss is guaranteed when simplifying from a 12 × n RC model to a 12 × 1 RC model.

#### 5.2.2. Individual n-Stage Thermal Model (12 × n RC) vs. Global Equivalent Inverter Thermal Model (1 × n RC)

_{sw}= 5 kHz) and fundamental (f

_{o}= 50 Hz) frequencies. However, an accurate averaging effect among the different semiconductors temperatures can be clearly seen.

#### 5.3. Accuracy Analysis in Different Operating Conditions

## 6. Analysis of the Inverter Model Simulation Speed

#### 6.1. Speed Analysis of the Electrical Models

#### 6.2. Speed Analysis of the Thermal Models

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Electric car power train scheme [6].

**Figure 2.**Two level three-phase inverter comprising the DC link (with V

_{DC}, I

_{DC}voltage, and current highlighted), six bidirectional switches (six active switches and six antiparallel diodes), and the output phase terminals (with phase current and voltages I

_{RST}, V

_{RST-N}, V

_{RST-n}, highlighted).

**Figure 3.**Electro-thermal model coupling represented by the electrical and thermal model. V

_{DC}and I

_{DC}are the DC bus voltage and current, V

_{Rn}, V

_{Sn}, and V

_{Tn}and I

_{R}, I

_{S}and I

_{T}are the AC side voltages and currents, Ploss

_{n}and Tj

_{n}are the semiconductors (diodes and switches) power losses and junction temperature, respectively.

**Figure 6.**Electric behavior modeling, ideal conduction and switching waveforms and conduction and switching losses estimation [19].

**Figure 9.**Model simplification resume scheme, presented in order of decreasing accuracy and increasing speed. Four different models have been developed.

**Figure 10.**Hi-Fi model block. (

**Left**) DC voltage nodes and the gate input Pulse Width Modulation (PWM) signal, g

_{Tx}. (

**Right**) The AC R,S,T terminals and the AC voltage waveform. On the lower side, the thermal and power variables are shown.

**Figure 11.**M-Fi model block. (

**Left**) DC voltage and current, and the gate input PWM signal. (

**Right**) AC voltage and currents. On the lower side, the thermal and power variables are shown.

**Figure 12.**Lo-Fi model block. (

**Left**) DC voltage and current, and the gate input low frequency PWM signal. (

**Right**) AC voltage and currents. On the lower side, the thermal and power variables are shown.

**Figure 13.**Lo-Fi model block. (

**Left**) DC voltage and current, and the gate input low frequency PWM signal. (

**Right**) AC voltage and currents. On the lower side, the thermal and power variables are shown.

**Figure 14.**n layer Foster thermal circuit of a semiconductor. From junction temperature to ambient temperature, where the semiconductor is marked by a grey area.

**Figure 15.**Equivalent thermal circuit of the two level three-phase inverter, where each semiconductor Foster circuit is marked with a grey area.

**Figure 16.**n layer semiconductor’s Foster circuit, where each pair of RCs can be transformed into a transfer function.

**Figure 17.**Semiconductor temperature calculation via transfer functions. The software under use is Simulink.

**Figure 20.**12 × n RC to 1 × n RC simplification. Each semiconductor’s thermal model is shaded in dark grey.

**Figure 25.**Comparative between individual n = 5-Stage (12 × 5 RC) (

**above**) and simplified single stage (12 × 1) (

**below**) thermal models.

**Figure 26.**Zoom of the evolution of T

_{R1}semiconductor junction temperature with the individual 5-stage 12 × 5 thermal model and simplified single stage 12 × 1 thermal model.

**Figure 27.**Comparison between the junction temperatures of semiconductors T

_{R1}and D

_{R1}obtained with the Individual 5-stage 12 × 5 thermal model and the equivalent T

_{j}temperature obtained with the global equivalent 1 × 5 RC thermal model.

**Figure 28.**Zoom of the evolution of T

_{R1}and D

_{R1}semiconductors junction temperature obtained with individual 5-stage 12 × 5 thermal model and equivalent T

_{j}temperature obtained with the global equivalent 1 × 5 RC thermal model.

Main Characteristics | Hi-Fi | M-Fi | Lo-Fi | Fast Lo-Fi |
---|---|---|---|---|

Model type | High detail electric simulator. | Equation based simulator. | Equation based simulator. | Equation based simulator. |

Input signal | Pulsed signal. | Pulsed signal. | Low frequency signal. | Low frequency signal. |

Output signal | Switched (PWM) phase voltage. | Switched (PWM) phase voltage. | Low frequency phase voltage. | Averaged low frequency phase voltage. |

Time-step order | Faster than switching frequency. | Faster than switching frequency. | Same as the switching frequency. | Slower than switching frequency. |

Other Features | Accurate physics of semiconductors. Short-circuit simulation. | Accurate physics of semiconductors. Faster than Hi-Fi. No short-circuit. | No Semiconductor physics. Faster than M-Fi No short-circuit. | No Semiconductor physics. Even faster than Lo-Fi No short-circuit. |

Parameter | Value |
---|---|

Selected semiconductor: | Infineon FF600R07ME4_B11 |

Selected variant: | Full detailed model |

Switching frequency: | 5000 Hz |

Fundamental frequency: | 50 Hz |

Bus voltage: | 600 V |

Load: | I_{ph,rms}_{,1} = 144 A, cos ϕ = 0.85 (lagging) |

Simulation step: | M-Fi: 1 × 10^{–5} sLo-Fi: 2 × 10 ^{–4} sFast Lo-Fi: 1 × 10 ^{–3} s |

Model | Step Time [s] | Accuracy Error (with Respect to Hi-Fi Model) [%] | ||||
---|---|---|---|---|---|---|

Total Power Losses | Semiconductor Power Losses | Semiconductor Junction Temperature | Voltage Fundamental Harmonic | DC-link Current | ||

Hi-Fi | 1 × 10^{−5} | - | - | - | - | - |

M-Fi | 1 × 10^{−5} | 2.03% | 0.81% | 0.75% | 0.02% | 0.55% |

Lo-Fi | 2 × 10^{−4} | 5.29% | 2.63% | 0.82% | 4.58% | 3.49% |

Fast Lo-Fi | 1× 10^{−3} | 6.49% | 1.50% | 0.45% | 4.76% | 33.24% |

Model | Average Conduction Losses [W] | Error with Respect to Hi-Fi Model [%] |
---|---|---|

Hi-Fi | 47.26 | - |

Lo-Fi | 49.49 | 4.7 |

Fast Lo-Fi | 43.98 | 6.9 |

Model | Average Switching Losses [W] | Error with Respect to Hi-Fi Model [%] |
---|---|---|

Hi-Fi | 78.20 | - |

Lo-Fi | 78.25 | 0.06 |

Fast Lo-Fi | 78.62 | 0.53 |

**Table 6.**Accuracy comparison between different inverter physical models for a braking scenario (V

_{ll,rms,}

_{1}= 294 V, I

_{ph,rms,}

_{1}= 600 A, cos ϕ = −0.97 (regenerating), f

_{o}= 50 Hz.

Model | Step Time [s] | Accuracy Error (with Respect to Hi-Fi Model) [%] | ||||
---|---|---|---|---|---|---|

Total Power Losses | Semiconductor Power Losses | Semiconductor Junction Temperature | Voltage Fundamental Harmonic | DC-Link Current | ||

Hi-Fi | 1 × 10^{−5} | - | - | - | - | - |

M-Fi | 1 × 10^{−5} | 6.94% | 2.72% | 2.44% | 0.07% | 0.08% |

Lo-Fi | 2 × 10^{−4} | 6.89% | 1.43% | 2.35% | 4.55% | 2.89% |

Fast Lo-Fi | 1× 10^{−3} | 6.27% | 6.18% | 2.11% | 4.47% | 8.95% |

**Table 7.**Speed comparison of the electrical models where the relation between the real time and the simulated time is shown.

Model | Simulated Time | Variant | $\frac{\mathit{R}\mathit{e}\mathit{a}\mathit{l}\mathit{t}\mathit{i}\mathit{m}\mathit{e}}{\mathit{S}\mathit{i}\mathit{m}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}\mathit{t}\mathit{i}\mathit{m}\mathit{e}}$ | Faster than real time? |
---|---|---|---|---|

Hi-Fi | 2 s | Full detail | 449 | No |

2 s | Null detail | 154 | No | |

M-Fi | 50 s | Full detail | 3.44 | No |

50 s | Null detail | 2.26 | No | |

Lo-Fi | 500 s | Full detail | 0.16 | Yes |

500 s | Null detail | 0.10 | Yes | |

Fast Lo-Fi | 500 s | Full detail | 0.08 | Yes |

500 s | Null detail | 0.05 | Yes |

**Table 8.**A speed comparison of the thermal models, where the relation between the real time and simulated time is shown.

Model | Simulated Time | Variant | $\frac{\mathit{R}\mathit{e}\mathit{a}\mathit{l}\mathit{t}\mathit{i}\mathit{m}\mathit{e}}{\mathit{S}\mathit{i}\mathit{m}\mathit{u}\mathit{l}\mathit{a}\mathit{t}\mathit{e}\mathit{d}\mathit{t}\mathit{i}\mathit{m}\mathit{e}}$ | Faster than Real Time? |
---|---|---|---|---|

Hi-Fi (Full detail) | 2 s | 12 × 5 RC | 449 | No |

2 s | 12 × 1 RC | 341 | No | |

2 s | 1 × 5 RC | 328 | No | |

M-Fi (Full detail) | 50 s | 12 × 5 RC | 3.44 | No |

50 s | 12 × 1 RC | 2.32 | No | |

50 s | 1 × 5 RC | 2.23 | No | |

Lo-Fi (Full detail) | 500 s | 12 × 5 RC | 0.16 | Yes |

500 s | 12 × 1 RC | 0.10 | Yes | |

500 s | 1 × 5 RC | 0.09 | Yes | |

Fast Lo-Fi (Full detail) | 500 s | 12 × 5 RC | 0.08 | Yes |

500 s | 12 × 1 RC | 0.03 | Yes | |

500 s | 1 × 5 RC | 0.03 | Yes |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Urkizu, J.; Mazuela, M.; Alacano, A.; Aizpuru, I.; Chakraborty, S.; Hegazy, O.; Vetten, M.; Klink, R. Electric Vehicle Inverter Electro-Thermal Models Oriented to Simulation Speed and Accuracy Multi-Objective Targets. *Energies* **2019**, *12*, 3608.
https://doi.org/10.3390/en12193608

**AMA Style**

Urkizu J, Mazuela M, Alacano A, Aizpuru I, Chakraborty S, Hegazy O, Vetten M, Klink R. Electric Vehicle Inverter Electro-Thermal Models Oriented to Simulation Speed and Accuracy Multi-Objective Targets. *Energies*. 2019; 12(19):3608.
https://doi.org/10.3390/en12193608

**Chicago/Turabian Style**

Urkizu, June, Mikel Mazuela, Argiñe Alacano, Iosu Aizpuru, Sajib Chakraborty, Omar Hegazy, Marco Vetten, and Roberto Klink. 2019. "Electric Vehicle Inverter Electro-Thermal Models Oriented to Simulation Speed and Accuracy Multi-Objective Targets" *Energies* 12, no. 19: 3608.
https://doi.org/10.3390/en12193608