# Modeling and Control Simulation of Power Converters in Automotive Applications
## 2 Electothermal Modeling and Control Algorithm
### DC/DC Converter Model and Control  
The dual active bridge (DAB) converter, illustrated in Figure 8, is designed with two active bridges, one on the primary side and one on the secondary side of a high-frequency transformer [64; 65; 66; 67; 68; 69; 70; 71; 72; 73]. Its operation can be described in the following steps:  
1. Primary side (DC-AC conversion): The first inverter converts the direct voltage into a high-frequency alternating voltage. The duty cycle \(D_{1}\) regulates the primary voltage \(V_{1}^{sw}\). This voltage is applied to the transformer.
2. Power transfer: The high-frequency alternating voltage flows through the transformer, generating an alternating current on the secondary side and transferring the power through the inductance.
3. Secondary side (AC-DC conversion): The second inverter converts the high-frequency alternating current into direct voltage. The duty cycle \(D_{2}\) regulates the secondary voltage \(V_{2}^{sw}\).
4. Regulation and control: Duty cycles and phase shifts are adjusted to control power flow and output voltage. Control occurs through modulation, which determines the duration and synchronization of the work cycles.  
Phase shift modulation is a key control technique in DAB, regulating the power transmission between the primary and secondary sides [74; 75; 76; 77; 78; 79; 80; 81]. The process can be understood through the following concepts:  
Figure 8: Dual active bridge converter reference circuit.  
Phase and work cycles: The phase \(\phi\) represents the time shift between the voltages \(V_{1}^{sw}\) and \(V_{2}^{sw}\). The phase shift approach often requires that \(D_{1}=D_{2}=D=0.5\), simplifying the control to just one parameter, \(\phi\). Power equation: The power equation (see Equation (3)) relates the phase shift \(\phi\) to the power transfer between the primary and secondary sides. Maximum power transfer occurs when \(\phi=\pi/2\).  
\[\begin{split} P=\frac{nV_{1}V_{2}}{2\pi^{2}f_{sw}L_{tot}}\phi \left(1-\frac{\phi}{\pi}\right)\\ \phi^{*}=\frac{\pi}{2}\left(1-\sqrt{1-\frac{8P^{*}f_{sw}L_{tot}} {nV_{1}V_{2}^{2}}}\right)\end{split} \tag{3}\]  
Advantages and limitations: Phase shift modulation is simple and requires only one control parameter. However, it has limitations such as a narrow working range and higher currents in the transformer.  
The concept of the phase shifting modulation is illustrated in Figure 9. In summary, DAB offers flexible bidirectional power control through the transformer, with phase-shift modulation as the key mechanism for efficient control of power flow.  
\[\begin{split} t_{\phi}^{*}=\left(P_{y}+\frac{l_{y}}{s}\right) \left[\left(P_{i}+\frac{l_{i}}{s}\right)\left(V_{2}^{ref}-V_{2}\right)-I_{2} \right]\frac{T_{sw}}{2\pi}\\ PWM_{2}=PWM_{1}(t-t_{\phi}^{*})\end{split} \tag{4}\]  
As schematized in Figure 10, the objective of the feedback control system is to automatically obtain the temporal shift between the modulation signal of the first stage \(PWM_{1}\) and the one that manages the second conversion stage \(PWM_{2}\). In particular, the control algorithm must manage the voltage request \(V_{2}^{ref}\), and manage the current feedback; this process obtains the phase shift \(\phi^{*}\) between the primary and secondary voltage of the transformer. In Equation (4), the control expressions are explicitly reported.  
Figure 9: DAB switching voltage and inductance current in different operating conditions of the PS modulation.