# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 5 Case Study
### Synchronous Generator
\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]\underline{u_{\rm ss}} \tag{94}\]  
By using physical variables:  
\[\underline{z_{\rm ss}}=\left[ \begin{array}{c}\left(\underline{f_{4}}\right) _{ss}\\ \left(\overline{f_{5}}\right)_{ss}\end{array} \right]=\left[ \begin{array}{c} \left(\underline{f_{4}}\right)_{ss}\\ \left(\overline{f_{4}^{4}}\right)_{ss}\\ \left(\overline{f_{4}^{4}}\right)_{ss}\\ \left(\overline{f_{5}}\right)_{ss}\end{array} \right]=\left[ \begin{array}{c} \left(i_{d}\right)_{ss}\\ \left(i_{d}\right)_{ss}\\ \left(w\right)_{ss}\\ \left(i_{f}\right)_{ss}\end{array} \right]\]  
and system inputs:  
\[\underline{u_{\rm ss}}=\left[ \begin{array}{c}\left(\underline{e_{1}}\right) _{ss}\\ \left(\overline{e_{7}}\right)_{ss}\end{array} \right]=\left[ \begin{array}{c} \left(e_{1}^{d}\right)_{ss}\\ \left(e_{1}^{d}\right)_{ss}\\ \left(e_{1}^{T_{L}}\right)_{ss}\\ \left(e_{7}\right)_{ss}\end{array} \right]=\left[ \begin{array}{c}\left(V_{d }\right)_{ss}\\ \left(V_{d}\right)_{ss}\\ \left(T_{L}\right)_{ss}\\ \left(V_{f}\right)_{ss}\end{array} \right]\]  
and (79), (87) and (94), the steady state for each state variable is described by:  
\[\left[ \begin{array}{c}\left(f_{4}^{d}\right)_{ss}\\ \left(f_{4}^{d}\right)_{ss}\\ \left(f_{4}^{d}\right)_{ss}\\ \left(f_{4}^{d}\right)_{ss}\\ \left(f_{5}^{d}\right)_{ss}\end{array} \right]=\left[ \begin{array}{cccc}P_{d0} ^{2}+DR_{q}&P_{d0}P_{q0}&-P_{q0}R_{q}&0\\ P_{d0}P_{q0}&P_{q0}^{2}+DR_{d}&P_{d0}R_{d}&0\\ P_{q0}R_{q}&-P_{d0}R_{d}&R_{d}R_{q}&0\\ 0&0&\frac{\Delta}{R_{f}}\end{array} \right]\frac{1}{\Delta}\left[ \begin{array}[ ]{c}\left(e_{1}^{d}\right)_{ss}\\ \left(e_{1}^{d}\right)_{ss}\\ \left(e_{1}^{T_{L}}\right)_{ss}\\ \left(e_{7}\right)_{ss}\end{array} \right] \tag{95}\]where:  
\[\Delta=DR_{d}R_{q}+P_{d0}^{2}R_{d}+P_{q0}^{2}R_{q}\]  
Considering balanced conditions applied to the synchronous generator:  
\[v^{abc}(t)=\left[ \begin{array}{c}100\cos(377t)\\ 100\cos(377t-120)\\ 100\cos(377t+120)\end{array} \right]\,\mathrm{V}\]  
and using [PERSON]'s transformation:  
\[\left[ \begin{array}{c}\left(e_{1}^{d}\right)_{ss}\\ \left(e_{1}^{d}\right)_{ss}\\ \left(e_{1}^{T_{1}}\right)_{ss}\\ \left(e_{7}\right)_{ss}\end{array} \right]=\left[ \begin{array}{c}122.4744\, \mathrm{V}\\ 3.1903\times 10^{-10}\,\mathrm{V}\\ 100\,\mathrm{N}\cdot\mathrm{m}\\ 30\,\mathrm{V}\end{array} \right]\]  
the numerical parameters of the elements for the \(d\)-axis are \(R_{d}=0.1\)\(\Omega\) and \(L_{d}=1.7\)\(\mathrm{H}\), for the \(q\)-axis are \(R_{q}=1\)\(\Omega\) and \(L_{q}=1.64\)\(\mathrm{H}\), the mutual inductance between \(d\) and \(q\) axes is \(M=1.55\)\(\mathrm{H}\), for the field circuit are \(R_{f}=1\)\(\Omega\), \(L_{f}=1.65\)\(\mathrm{H}\), and the mechanical part is given by \(J=2.37N-m-s^{2}\) and \(D=1\). The initial flux linkages are: \(P_{d0}=0.002\)\(\mathrm{A}\cdot\mathrm{H}\) and \(P_{q0}=0.1279\)\(\mathrm{A}\cdot\mathrm{H}\). By substituting all the numerical values of the parameters into (95), the steady state of the co-energy vector of the synchronous generator is given by:  
\[\left[ \begin{array}{c}\left(f_{4}^{d}\right)_{ss}\\ \left(f_{4}^{d}\right)_{ss}\\ \left(f_{4}^{l}\right)_{ss}\\ \left(f_{4}^{l}\right)_{ss}\\ \left(f_{5}\right)_{ss}\end{array} \right]=\left[ \begin{array}{c}-46.642\, \mathrm{A}\\ 1.988\,\mathrm{A}\\ 994.03\,\mathrm{rad}/\mathrm{s}\\ 30.00\,\mathrm{A}\end{array} \right] \tag{96}\]  
In order to prove the obtained results, the simulation of the synchronous generator using the 20-SIM software is shown in Figure 20.  
The dynamic behavior based on the MBGI of Figure 18 for the synchronous generator is obtained and shown in Figure 20. The steady state for the variables \(i_{d}\), \(i_{q}\), \(w\), and \(i_{f}\) is illustrated, and the results given in (96) are verified. Figure 20: Physical variables’ performance for the synchronous generator: \(i_{d}\), \(i_{q}\), \(w\), and \(i_{f}\).