# A Novel Energy Optimization Control Strategy for Electric Drive System Based on Current Angle
## 2 Electric Drive System Loss Model
### Inverter Loss Model  
The loss of inverter mainly occurs when the voltage and current of IGBT module (IGBT + free-wheeling diode (FWD)) produces overlapping area in the switching process, which is usually defined as conduction loss and switching loss. Figure 3 shows the working process of voltage and current through the IGBT module in a switching cycle.  
\(I_{c}\) is the current between diode collector and emitter. \(V_{\alpha c}\) is collector-emitter voltage. \(t_{r}\) is the time required for the current to reach the target value in the opening process of IGBT. \(t_{f}\) is the time required for the current to reach the target value in the switching-on process of IGBT. \(t_{r}\) is the time required for FWD reverse recovery process. \(I_{rr}\) is the reverse current through FWD in reverse recovery process.  
The relationship of collector current and collector-emitter voltage between IGBT and FWD is shown in Figure 4. The collector-emitter voltage \(V_{\alpha c}\) and the FWD conduction voltage \(V_{F}\) can be fitted with a threshold voltage and a voltage drop with a slope of resistance respectively [32]:  
\[V_{\alpha c}=\frac{V_{\alpha cn}-V_{\alpha c0}}{I_{cn}}I_{c}+V_{\alpha c0} \tag{11}\]  
\[V_{F}=\frac{V_{Fn}-V_{F0}}{I_{cn}}I_{c}+V_{F0} \tag{12}\]  
where \(V_{\alpha cn}\) is the value of \(V_{\alpha c}\) under rated current. \(V_{\alpha c0}\) is the threshold voltage. \(I_{cn}\) is the rated value of \(I_{c}\). \(V_{Fn}\) is the voltage drop of FWD under rated current. \(V_{F0}\) is the threshold voltage of FWD.  
Figure 3: Typical IGBT module working process.  
SVPWM is adopted as the modulation method in this paper, and its AC current is:  
\[I_{c}=I_{cm}\text{sin }(\theta+\psi) \tag{13}\]  
where \(\psi\) is the power factor angle and \(I_{cm}\) is the peak phase current.  
Then, the conduction loss power of IGBT and FWD is [32]:  
\[P_{conduction\_IGBT}=(\frac{1}{8}+\frac{M}{3\pi})\frac{V_{cm}-V_{ccw}}{I_{cn}}{ I_{cm}}^{2}+(\frac{1}{2\pi}+\frac{M}{8}\cos\theta V_{ccw}I_{cm}) \tag{14}\]  
\[P_{conduction\_FWD}=(\frac{1}{8}-\frac{M}{3\pi})\frac{V_{cm}-V_{ccw}}{I_{cn}} {I_{cm}}^{2}+(\frac{1}{2\pi}-\frac{M}{8}\cos\theta V_{ccw}I_{cm}) \tag{15}\]  
They can be simplified as [33]:  
\[P_{conduction\_IGBT}=(\frac{1}{8}+\frac{M}{3\pi}\cos\theta)V_{cc}I_{c} \tag{16}\]  
\[P_{conduction\_FWD}=(\frac{1}{8}-\frac{M}{3\pi}\cos\theta)V_{cc}I_{c} \tag{17}\]  
The switching loss includes the turn on loss \(E_{on}\), turn off loss \(E_{off}\), and reverse recovery loss \(E_{rr}\). \(E_{on}\), \(E_{off}\), \(E_{off}\) are, respectively, as:  
\[E_{on}=\int_{0}^{t_{r}}V_{cc}\frac{l_{c}}{t_{r}}tdt=\frac{1}{2}V_{cc}l_{c}t_{r} \tag{18}\]  
\[E_{off}=\int_{0}^{t_{f}}V_{cc}\frac{l_{c}}{t_{f}}tdt=\frac{1}{2}V_{cc}l_{c}t_{f} \tag{19}\]  
\[E_{rr}=V_{cc}t_{rr}(0.35I_{rm}+0.15\frac{l_{c}}{l_{c}n}I_{rm}+I_{c}) \tag{20}\]  
where \(I_{rm}\) is the value of \(I_{rr}\) under rated current \(I_{c}\).  
\[P_{switch}=(E_{on}+E_{off}+E_{rr})f_{sw}\frac{1}{\pi} \tag{21}\]  
Consequently, the total loss power of the inverter containing six IGBT modules is:  
\[P_{Inverter\_Loss}=6(P_{conduction\_IGBT}+P_{conduction\_FWD}+P_{switch}) \tag{22}\]  
Figure 4: Relationship of collector current and collector-emitter voltage between IGBT and FWD.  
It should be noted that this paper adopted the inverter of model 6 MBI100 XBA120-50 produced by Fuji Electric, and the data are from the website [34].