# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 5 Case Study
### Synchronous Generator
\right]=\left[ \begin{array}{cccccc}0&0&- \mathbf{I}&-\mathbf{I}&0&\mathbf{I}&0\\ 0&0&0&0&-1&0&1\\ \mathbf{0}&\mathbf{I}&0&0&0&0&0\\ \mathbf{0}&\mathbf{I}&0&0&0&0&0\end{array} \right]\left[ \begin{array}{c} \underline{f_{4}(t)}\\ \underline{e_{3}(t)}\\ \underline{e_{2}(t)}\\ \underline{e_{6}(t)}\\ \underline{e_{1}(t)}\\ \underline{e_{7}(t)}\end{array} \right] \tag{80}\]  
Figure 18: Multibond graph of a synchronous generator. From (80):  
\[\mathbf{S}_{22}=0;\ \mathbf{S}_{11}^{22}=0;\ \mathbf{S}_{12}^{21}=0;\ \mathbf{S}_{21}^{12}=0 \tag{81}\]  
and substituting (81) into (15), (14), (12), and (13), the algebraic loop matrices are:  
\[\mathbf{M}_{L} = \mathbf{Q}_{L}=\mathbf{L} \tag{82}\] \[\mathbf{M}_{X} = \mathbf{Q}_{X}=\mathbf{X}_{CY} \tag{83}\]  
From (81)-(83), the state matrix (7) is reduced to:  
\[\mathbf{A}=\Big{(}\mathbf{S}_{11}^{11}+\mathbf{S}_{11}^{12}\mathbf{Q}_{X} \mathbf{S}_{11}^{21}+\mathbf{S}_{12}^{11}\mathbf{Q}_{L}\mathbf{S}_{21}^{11} \Big{)}\mathbf{F} \tag{84}\]  
and the input matrix (8) is written as:  
\[\mathbf{B}=\mathbf{S}_{13}^{11}+\mathbf{S}_{11}^{12}\mathbf{Q}_{X}\mathbf{S}_ {13}^{21}+\mathbf{S}_{12}^{11}\mathbf{Q}_{L}\mathbf{S}_{23} \tag{85}\]  
From (77)-(80) and (82)-(85), the state space representation of the synchronous generator is given by:  
\[\left[ \begin{array}{c}\underline{\epsilon_{4}}(t)\\ \overline{\epsilon_{5}}(t)\end{array} \right]=\left[ \begin{array}{cc}-\mathbf{ R}_{tdp}&0\\ 0&-R_{f}\end{array} \right]\left[ \begin{array}{c}\underline{f_{4}}(t)\\ \overline{f_{5}}(t)\end{array} \right]+\left[ \begin{array}{cc}\mathbf{X}(-d_ {0},-q_{0},0)&0\\ 0&0\end{array} \right]\left[ \begin{array}{c}\underline{f_{4}}(t)\\ \overline{f_{5}}(t)\end{array} \right]+\left[ \begin{array}{c}\underline{ \epsilon_{1}}(t)\\ \overline{\epsilon_{7}}(t)\end{array} \right] \tag{86}\]  
To determine the steady state response, the MBGD of the synchronous generator is shown in Figure 19.  
By assigning a derivative causality to the multiport storage elements defined by the bonds 4 and 7, then the bonds 2 and 6 have to change their causality in order to obtain the correct multibond graph. Hence, the new vectors for the multiport dissipation elements are given by:  
\[\underline{D_{\mathit{int}}^{d}}(t)=\left[ \begin{array}{c}\underline{ \epsilon_{2}}(t)\\ \overline{\epsilon_{6}}(t)\end{array} \right];\ \underline{D_{\mathit{out}}^{d}}(t)=\left[ \begin{array}{c}\underline{f_{2}}(t)\\ \overline{f_{6}}(t)\end{array} \right]\]  
and its constitutive relation is:  
\[\mathbf{L}^{d}=diag\Big{\{}\mathbf{R}_{d\mathbf{D}^{\prime}}^{-1},R_{f}^{-1} \Big{\}} \tag{87}\]  
and for the multiport gyrator is:  
\[\mathbf{X}_{CY}^{d}=\mathbf{X}_{CY} \tag{88}\]  
Figure 19: MBGD of a synchronous generator. The multiport junction structure for the MBGD is defined by:  
\[\left[ \begin{array}{c}f_{4}(t)\\ \overline{f_{5}}(t)\\ f_{2}(t)\\ \overline{f_{6}}(t)\\ \overline{f_{3}}(t)\end{array} \right]=\left[ \begin{array}{cccccccc}0&0&0&{ \bf I}&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&{\bf I}&0&0&0\\ -{\bf I}&0&-{\bf I}&0&0&{\bf I}&0\\ 0&-1&0&0&0&0&1\end{array} \right]\left[ \begin{array}{c}f_{4}(t)\\ \overline{f_{5}}(t)\\ \underline{e_{2}}(t)\\ \underline{e_{6}}(t)\\ \underline{e_{3}}(t)\\ \overline{e_{1}}(t)\\ \overline{e_{7}}(t)\end{array} \right] \tag{89}\]  
From (30) and (89), \({\bf B}^{*}\) is reduced to:  
\[{\bf B}^{*}={\bf F}^{-1}\left({\bf J}_{12}^{11}{\bf P}_{L}{\bf J}_{LB}\right) \tag{90}\]  
From (2), (45), and (90), the steady state response is defined by:  
\[\underline{z_{\rm ss}}={\bf J}_{12}^{11}{\bf P}_{L}{\bf J}_{LB}\underline{u_ {\rm ss}} \tag{91}\]  
also, \({\bf J}_{13}^{21}=0\) then \({\bf J}_{LB}\) is  
\[{\bf J}_{LB}={\bf J}_{23} \tag{92}\]  
From (43), (87)-(89):  
\[{\bf P}_{L}=\left[ \begin{array}{cc}{\bf R}_{d1D}^{-1}\left({\bf I}+{\bf X}_ {GY}{\bf R}_{d0D}^{-1}\right)^{-1}&0\\ 0&R_{f}^{-1}\end{array} \right] \tag{93}\]  
and substituting (89), (92) and (93) into (91), the steady state in a symbolic form is defined by:  
\[\underline{z_{\rm ss}}=\left[