# Shafting Torsional Vibration Analysis of 1000 MW Unit under Electrical Short-Circuit Fault
## 3 Construction of Electrical Short-Circuit Fault Model
### SIMULINK Synchronous Generator Model  
The load part of the rotor system uses the synchronous machine module embedded in the SIMULINK Toolbox, which considers the dynamic characteristics of stator, excitation and winding damping. The equivalent d-axis and q-axis circuits of the model are shown in Figure 5. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **HP** & **IP** & **LP1** & **LP2** \\ \hline Volume time constant & 0.3 & 15 & 0.4 & 0.4 \\ Connection torque factor & 0.3 & 0.4 & 0.15 & 0.15 \\ Connection stiffness coefficient & 48.9 & 39.3 & 36 & 15.8 \\ Moment of inertia \(\times\)\(10^{3}(\text{kg}/\text{m}^{2})\) & 16.5 & 20.8 & 7.52 & 7.52 \\ Torsional rigidity \(\times\)\(10^{8}(\text{Nm}/\text{rad})\) & 2.205 & 1.828 & 1.54 & 1.307 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Input parameters of steam turbine simulation module. Figure 4: Power transmission mode of each rotor of turbo-generator. Figure 3: Turbo-generator simulation module. In Figure 5, R and Rs represent the generator rotor and stator, respectively. \(L\) and \(L_{m}\) represent leakage inductance and excitation inductance, respectively. \(f\) and \(k\) represent excitation and damping windings, respectively [53; 54]. The synchronous generator model is represented by the sixth-order state space model, and the corresponding equation is  
\[V_{d}=R_{s}i_{d}+\frac{d}{dt}\phi_{d}-\omega_{R}\phi_{q};\phi_{d}=L_{d}i_{d}+L_ {md}(i^{\prime}{}_{fd}+i^{\prime}{}_{kl}) \tag{6}\]  
\[V_{q}=R_{s}i_{q}+\frac{d}{dt}\phi_{d}+\omega_{R}\phi_{q};\phi_{q}=L_{d}i_{d}+L _{md}i^{\prime}{}_{kl} \tag{7}\]  
\[V^{\prime}{}_{fd}=R^{\prime}{}_{fd}i^{\prime}{}_{fd}+\frac{d}{dt}\phi^{\prime} {}_{fd}i^{\prime}{}_{fd}=L^{\prime}{}_{fd}i^{\prime}{}_{kl}+L_{md}(i_{d}+i^{ \prime}{}_{kl}) \tag{8}\]  
\[V^{\prime}{}_{kl}=R^{\prime}{}_{kl}i^{\prime}{}_{kl}+\frac{d}{dt}\phi^{\prime }{}_{kl};\phi^{\prime}{}_{kl}=L^{\prime}{}_{fd}i^{\prime}{}_{kl}+L_{md}(i_{d}+i ^{\prime}{}_{fd}) \tag{9}\]  
\[V^{\prime}{}_{kl1}=R^{\prime}{}_{kl1}i^{\prime}{}_{kl1}+\frac{d}{dt}\phi^{ \prime}{}_{kl1};\phi^{\prime}{}_{kl1}=L^{\prime}{}_{kl1}i^{\prime}{}_{kl1}+L_ {md}i_{q} \tag{10}\]  
Figure 5: Synchronous generator model equivalent circuit diagram. \[V^{\prime}{}_{kq2}=R^{\prime}{}_{kq2}i^{\prime}{}_{kq2}+\frac{d}{dt}\phi^{\prime}{ }_{kq2};\phi^{\prime}{}_{kq2}=L^{\prime}{}_{kq2}i^{\prime}{}_{kq2}+L_{mq}i_{q} \tag{11}\]  
The basic equation of the synchronous generator can be expressed as  
\[\left[ \begin{array}{c}V_{a}\\ V_{b}\\ V_{c}\\ -V_{f}\\ 0\end{array} \right]=\left[\begin{array}{c} \begin{array}{c}\dot{\psi}_{a}\\ \dot{\psi}_{b}\\ \dot{\psi}_{f}\\ \dot{\psi}_{d}\\ \dot{\psi}_{q}\end{array} \right]-\left[ \begin{array}{cccccc}R&0&0&0&0&0\\ 0&R&0&0&0&0\\ 0&0&R&0&0&0\\ 0&0&0&R_{f}&0&0\\ 0&0&0&0&R_{d}&0\\ 0&0&0&0&R_{q}\end{array} \right]\cdot\left[ \begin{array}{c}i_{a}\\ i_{b}\\ i_{c}\\ i_{d}\\ i_{q}\end{array} \right] \tag{12}\]  
In Equation (12), \(V_{a}\), \(V_{b}\) and \(V_{c}\) are the voltage of the three-phase winding generator, \(V_{f}\) is the voltage of excitation winding, and \(i\) is the resistance of each phase of the three-phase winding of stator \(a\), \(b\), \(c\). \(\psi\) is the total magnetic chain of each phase winding, \(\dot{\psi}_{x}\) is the derivative of the above magnetic chain to time, and the flux equation corresponding to the synchronous generator can be expressed as  
\[\left[ \begin{array}{c}\psi_{a}\\ \psi_{b}\\ \psi_{c}\\ \psi_{f}\\ \psi_{d}\\ \psi_{q}\end{array} \right]=\left[ \begin{array}{cccccc}L_{aa}&L_{ab}&L_{ac} &L_{af}&L_{ad}&L_{aq}\\ L_{ba}&L_{bb}&L_{bc}&L_{bf}&L_{bd}&L_{bg}\\ L_{ca}&L_{cb}&L_{cc}&L_{cf}&L_{cd}&L_{cq}\\ L_{fa}&L_{fb}&L_{fc}&L_{ff}&L_{fd}&L_{fg}\\ L_{da}&L_{db}&L_{dc}&L_{df}&L_{dd}&L_{dq}\\ L_{qa}&L_{qb}&L_{qc}&L_{df}&L_{qd}&L_{qq}\end{array} \right]\cdot\left[ \begin{array} []{c}i_{a}\\ i_{b}\\ i_{c}\\ i_{f}\\ i_{d}\\ i_{q}\end{array}