# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 5 Case Study
### Three-Phase Electrical System  
A three-phase electrical system formed by two sources and a transmission line for each phase is shown in Figure 11.  
Park's transformation offers a great reduction in the mathematical modeling of three-phase systems. This Parktransformation changes the parameters and variables from phases \((a,b,c)\) to new variables of the reference frame [42; 43]. The new variables are \((d,q,0)\) with \(d\) and \(q\) axis components, and a stationary current that is the zero-sequence current is the third variable. Thus, by definition:  
\[\left[ \begin{array}{c}i_{d}(t)\\ i_{q}(t)\\ i_{0}(t)\end{array} \right]=\sqrt{\frac{2}{3}}\left[ \begin{array}{cc}\cos\theta &\cos(\theta-\frac{2\pi}{3})&\cos(\theta+\frac{2\pi}{3})\\ \sin\pi&\sin(\theta-\frac{2\pi}{3})&\sin(\theta+\frac{2\pi}{3})\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{array} \right] \left[ \begin{array}{c}i_{a}(t)\\ i_{b}(t)\\ i_{c}(t)\end{array} \right] \tag{47}\]  
The angle \(\theta\) is given by:  
\[\theta=wt+\frac{\pi}{2} \tag{48}\]  
where \(w\) denotes the rated angular frequency in rad/s. In a similar way, to change the voltages and flux linkages:  
\[v^{dq0}(t) = \mathbf{p}v^{abc}(t) \tag{49}\] \[\lambda^{dq0}(t) = \mathbf{p}\lambda^{abc}(t) \tag{50}\]  
By knowing the importance of [PERSON]'s transformation for the analysis of three-phase electrical systems, the multibond graph model of this case study is shown in Figure 12. The dynamic model of the system in the (\(d\),\(q\),\(0\)) axis is modeled by a multibond graph. However, the supply voltages to the system are (\(a\),\(b\),\(c\)) and the multiport transformers to convert (\(a,b,c\)) to (\(d,q,0\)) are applied. Furthermore, the three-phase output current by using a multiport transformer with the inverse Parktransformation is obtained. Figure 11: Three-phase electrical system. The multiport storage elements in an integral causality assignment are expressed by the following key vectors:  
\[\underline{x}(t)=\left[ \begin{array}{c}\underline{p_{3}(t)}\\ \underline{p_{4}(t)}\end{array} \right]\text{; }\frac{\bullet}{\underline{x}}(t)=\left[ \begin{array}{c}\underline{e_{3}(t)}\\ \underline{e_{4}(t)}\end{array} \right]\text{; }\underline{z}(t)=\left[ \begin{array}{c}\underline{f_{3}(t)}\\ \underline{f_{4}(t)}\end{array} \right]\]  
with the constitutive relationship:  
\[\mathbf{F}^{-1}=diag\left\{\mathbf{L}_{1}^{dq0},\mathbf{L}_{2}^{dq0}\right\} \tag{51}\]  
In a derivative causality assignment, they are described by:  
\[\underline{x_{d}}(t)=\underline{p_{10}}(t)\text{; }\frac{\bullet}{\underline{x }_{d}}(t)=\underline{e_{10}}(t)\text{; }\underline{z_{d}}(t)=\underline{f_{10}}(t)\]  
with the constitutive relationship:  
\[\mathbf{F}_{d}^{-1}=\mathbf{L}_{3} \tag{52}\]  
The multiport gyrators are defined by:  
\[\underline{I_{GY}}(t)=\left[ \begin{array}{c}\underline{f_{8}}(t)\\ \underline{f_{9}}(t)\\ \underline{f_{11}}(t)\end{array} \right]\text{; }\underline{O_{GY}}(t)=\left[ \begin{array}{c}\underline{e_{8}}(t)\\ \underline{e_{9}}(t)\\ \underline{e_{11}}(t)\end{array} \right]\]  
with:  
\[\mathbf{X}_{GY}=diag\left\{\mathbf{X}_{1}(wL_{1}),\ \mathbf{X}_{2}(wL_{2}),\ \mathbf{X}_{3}(wL_{3})\right\} \tag{53}\]  
For the multiport dissipation elements:  
\[\underline{D_{in}}(t)=\underline{f_{12}}(t)\text{; }\underline{D_{out}}(t)= \underline{e_{12}}(t)\]  
Figure 12: MBGI of the electrical system. with:  
\[\textbf{L}=\textbf{R}_{1}^{dq0} \tag{54}\]  
and the inputs and outputs are:  
\[\underline{u}(t)=\left[ \begin{array}{c}\underline{e_{1}}(t)\\ \underline{e_{2}}(t)\end{array} \right]\text{; }\underline{y}(t)=\left[ \begin{array}{c} \underline{f_{21}}(t)\\ \underline{f_{25}}(t)\\ \underline{f_{23}}(t)\end{array} \right]\]  
where the matrices for resistors, inductors, and gyrators are:  
\[\textbf{L}_{i}^{dq0} = diag\Big{\{}L_{i}^{d},\ L_{i}^{q},\ L_{i}^{0}\Big{\}};i=1,2,3 \tag{55}\] \[\textbf{R}_{1}^{dq0} = diag\Big{\{}R_{1}^{d},\ R_{1}^{q},\ R_{1}^{0}\Big{\}}\] (56) \[\textbf{X}_{i}(wL_{i}) = \left[