# Analytical Selection of Leakage Inductance for Single-Phase Dual Active Bridge Converters
## 2 Steady State Analysis
\begin{array}{c}\frac{v_{pd}^{*}(k)-v_{pd} (k)}{T_{s}}i_{leq}(k)+\frac{i_{leq}^{*}(k)-i_{leq}(k)}{T_{s}}v_{pd}(k)\\ -\frac{v_{pq}^{*}(k)-v_{pq}(k)}{T_{s}}i_{led}(k)-\frac{i_{led}^{*}(k)-i_{led}(k )}{T_{s}}v_{pq}(k)\end{array} \right) \tag{13}\]  
The discrete-time large signal model of the converter in the \(dq\) rotating reference frame can be expressed as  
\[ \begin{bmatrix}\frac{i_{ld}^{*}(k)-i_{led}(k)}{T_{s}}\\ \frac{i_{leq}^{*}(k)-i_{leq}(k)}{T_{s}}\end{bmatrix} = \begin{bmatrix}-\frac{R_{e }}{L_{e}}&-\omega_{s}\\ \omega_{s}&-\frac{R_{e}}{L_{e}}\end{bmatrix}\begin{bmatrix}i_{led}(k)\\ i_{leq}(k)\end{bmatrix} + \begin{bmatrix}\frac{1}{L_{e}}&0\\ 0&\frac{1}{L_{e}}\end{bmatrix}\begin{bmatrix}v_{pd}(k)\\ v_{pq}(k)\end{bmatrix} + \begin{bmatrix}-\frac{1}{L_{e}}&0\\ 0&-\frac{1}{L_{e}}\end{bmatrix}\begin{bmatrix}v_{sd}(k)\\ v_{sd}(k)\end{bmatrix} . \tag{14}\]  
By substituting Equation (9) into Equation (8), we obtain  
\[\Delta P_{1}(k)=0.5\Big{(}M_{1}v_{pd}^{*}(k)+M_{2}v_{pq}^{*}(k)+M_{3}v_{pd}(k)+ M_{4}v_{pq}(k)\Big{)} \tag{15}\]  
\[\Delta Q_{1}(k)=0.5\Big{(}N_{1}v_{pd}^{*}(k)+N_{2}v_{pq}^{*}(k)+N_{3}v_{pd}(k)+ N_{4}v_{pq}(k)\Big{)} \tag{16}\]  
where the mathematical expressions of the coefficients \(M_{1}\), \(M_{2}\), \(M_{3}\), \(M_{4}\), \(N_{1}\), \(N_{2}\), \(N_{3}\), and \(N_{4}\) are provided in Appendix A. A unique voltage command is derived from Equation (10) as a function of \(dq\) voltages and active/reactive powers for the subsequent sampling time, expressed as  
\[ \begin{array}{ll}v_{pd}^{*}(k)&=\Big{(}\frac{M_{1}N_{2}-M_{3}N_{1}}{M_{2}N_{1} -M_{1}N_{2}}\Big{)}v_{pd}(k)\\ &+\Big{(}\frac{M_{4}N_{2}-M_{3}N_{4}}{M_{2}N_{1}-M_{1}N_{2}}\Big{)}v_{pq}(k)- \Big{(}\frac{2N_{2}}{M_{2}N_{1}-M_{1}N_{2}}\Big{)}\Big{(}\frac{P_{1}^{*}(k)-P _{1}(k)}{T_{s}}\Big{)}\\ &+\Big{(}\frac{2M_{3}}{M_{2}N_{1}-M_{1}N_{2}}\Big{)}\Big{(}\frac{Q_{1}^{*}(k)- Q_{1}(k)}{T_{s}}\Big{)}\\ \end{array}  \tag{17}\]  
\[ \begin{array}{ll}v_{pq}^{*}(k)&=\Big{(}\frac{M_{1}N_{2}-M_{3}N_{1}}{M_{1}N_{ 2}-M_{2}N_{1}}\Big{)}v_{pd}(k)\\ &+\Big{(}\frac{M_{4}N_{3}-M_{1}N_{4}}{M_{1}N_{2}-M_{2}N_{1}}\Big{)}v_{pq}(k)- \Big{(}\frac{2N_{1}}{M_{1}N_{2}-M_{2}N_{1}}\Big{)}\Big{(}\frac{P_{1}^{*}(k)- P_{1}(k)}{T_{s}}\Big{)}\\ &+\Big{(}\frac{2M_{1}}{M_{1}N_{2}-M_{2}N_{1}}\Big{)}\Big{(}\frac{Q_{1}^{*}(k)- Q_{1}(k)}{T_{s}}\Big{)}\end{array}  \tag{18}\]  
The unique voltage command \(\boldsymbol{v}^{*}_{pd}\) in Equations (17) and (18) is transformed into a stationary reference frame, with an amplitude of \(V_{m}\) and a phase shift \(\theta\) to achieve the requisite power commands. Then, the ensuing phase shifts \(d_{1}\) and \(d_{3}\) for the subsequent sample are determined via  
\[d_{1}(k+1)=\frac{2}{\pi}\cos^{-1}\bigg{(}\frac{\pi}{4}\cdot\frac{V_{m}}{V_{in }}\bigg{)} \tag{19}\]  
\[d_{3}(k+1)=\frac{\theta}{\pi} \tag{20}\]  
Here, \(d_{1}\) and \(d_{3}\) serve as the control variables enabling independent control over active and reactive powers. This independence is crucial for efficient energy management. The control of active power not only regulates the output voltage but also influences the overall power efficiency of the system. Meanwhile, controlling reactive power minimizes the current flowing through the inductor, which helps in reducing losses and improves the power factor. Together, these controls contribute to optimizing the performance of the converter. The control block diagram of the power level controller is depicted in Figure 4. An active power command is generated by a Proportional-Integral voltage controller to adjust the output voltage, ensuring that it tracks the reference voltage (\(V_{\textit{ref}}\)). Simultaneously, a reactive power generator produces a reactive power command. This command is specifically aimed at reducing the current that flows through the inductor. By minimizing the current, the converter can achieve lower losses and enhanced efficiency.