## 2 Small-Signal Modeling of the Distributed WEDES System
### Derivation of Transfer Functions
#### 2.3.1 Transfer Function of DC-DC Boost Stage  
The circuit diagram of the DC-DC boost stage is shown in Figure 4. When the lower side switch \(\mathsf{S_{L-r}}\) is on and the upper side switch \(\mathsf{S_{U-r}}\) is off, the differential equation of the boost converter can be derived as follows:  
\[\left\{ \begin{array}{c}L_{MXr}\frac{d_{i_{intr}}}{dt}=v_{intr}\\ C_{MXr}\frac{d\
u_{MXr}}{dt}=-i_{MXr}\end{array} \right., \tag{5}\]  
where \(i_{intr}\) and \(v_{intr}\) are the input current and input voltage, respectively, and \(i_{MXr}\) and \(v_{MXr}\) are the output current and output voltage of the boost converter, respectively. \(L_{MXr}\) is the inductor value, and \(C_{MXr}\) is the output capacitor. The state-space form of Equation (5) can be rewritten as Equation (6). \[K\left[ \begin{array}{c}\dot{x_{1}}\\ \dot{x_{2}}\end{array} \right]=A_{1}\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]+B_{1}\left[ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right]=\left[ \begin{array}{cc}0&0\\ 0&0\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]+\left[ \begin{array}{cc}1&0\\ 0&-1\end{array} \right]\left[ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right], \tag{6}\]  
where the variables are \(x_{1}=i_{intr}\), \(x_{2}=v_{MXr}\), \(u_{1}=v_{intr}\), \(u_{2}=i_{MXr}\), respectively. \(K=\left[ \begin{array}{cc}L&0\\ 0&C\end{array} \right]\). Similarly, when the lower side switch \(\mathsf{S_{L-r}}\) is off and the upper side switch \(\mathsf{S_{U-r}}\) is on, the differential equation of the boost converter can be derived as follows:  
\[\left\{ \begin{array}{c}L_{MXr}\frac{d_{i_{intr}}}{dt}=V_{intr}-V_{MXr}\\ C_{MXr}\frac{d\
u_{MXr}}{dt}=i_{intr}-I_{MXr}\end{array} \right.. \tag{7}\]  
The state-space form of Equation (7) can be rewritten as Equation (8). \[K\left[ \begin{array}{c}\dot{x_{1}}\\ \dot{x_{2}}\end{array} \right]=A_{2}\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]+B_{2}\left[ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right]=\left[ \begin{array}{cc}0&-1\\ -1&0\end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\end{array} \right]+\left[ \begin{array}{cc}1&0\\ 0&-1\end{array} \right]\left[ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right] \tag{8}\]  
By using the state-space averaging method [20], the average matrix A and B are calculated as given by Equations (9) and (10). \[A=A_{1}D_{r}+A_{2}(1-D_{r}), \tag{9}\]  
Figure 4: Circuit diagram of DC-DC boost stage in the WEDES system. \[B=B_{1}D_{r}+B_{2}(1-D_{r}), \tag{10}\]  
where \(D_{r}\) is the duty cycle of the boost converter. The steady-state X is calculated as follows:  
\[X=A^{-1}BU=\Big{[} \begin{array}{cc}I_{MXr}\\ 1-D_{r}\end{array}\begin{array}{c}V_{intr}\\ \end{array} \Big{]}^{-1}. \tag{11}\]  
The small-signal equation becomes as given by Equation (12):  
\[\hat{X}=A\hat{x}+B\hat{u}+[(A_{1}-A_{2})X+(B_{1}-B_{2})U]\hat{d_{r}}, \tag{12}\]  
where \(A_{1}-A_{2}=\left[ \begin{array}{cc}0&1\\ -1&0\end{array} \right]\), \(B_{1}-B_{2}=0\), and \(\hat{d_{r}}\) is the small signal variation of duty cycle around its steady state operation point. Equation (12) can be rewritten as Equation (13):  
\[\left\{ \begin{array}{c}L_{MXr}\frac{d_{intr}^{2}}{dt}=-(1-D_{r})v_{MXr}+v_{ intr}^{*}+\frac{V_{intr}}{1-D_{r}}\hat{d_{r}}\\ C_{MXr}\frac{dV_{MXr}}{dt}=(1-D_{r})\hat{i_{intr}}-\hat{i_{MXr}}-\frac{I_{MXr} }{1-D_{r}}\hat{d_{r}}\end{array} \right.. \tag{13}\]  
To simply the analysis, the AC small-signal variation of \(v_{intr}^{*}\) and \(i_{MXN}\) is assumed to be 0 (negligible) because the dynamic variation of battery voltage and battery module output current is very slow compared to the dynamic variation of the control signal \(\hat{d_{r}}\) (duty cycle) of the power converter. Therefore, by performing the Laplace transformation, Equation (14) can be derived as:  
\[\left\{ \begin{array}{c}sL_{MXr}i_{intr}(s)=-(1-D_{r})v_{MXr}(s)+\frac{V_{intr }}{1-D_{r}}d_{r}(s)\\ sC_{MXr}v_{MXr}(s)=(1-D_{r})i_{intr}(s)-\frac{I_{MXN}}{1-D_{r}}d_{r}(s)\end{array}