# Verification Methodology for Simulation Models of the Synchronous Generator on Transients Analysis
## 2 Modeling Methods for Transients Calculation
### Sudden Short-Circuit by a Dynamic Model
However, the zero component of the current cannot flow in the armature winding, thus the zero variables are not present in the next system of equations:  
\[\frac{\text{d}\psi_{\text{d}}}{\text{dt}}=\text{u}_{\text{d}}-\frac{1}{\text {T}^{\prime\prime}_{\text{ds}}}\bigg{(}\psi_{\text{d}}-\frac{\text{k}_{\text {D}}\text{L}_{\text{\sigma D}}}{\text{L}^{\prime}_{\text{f}}}\psi_{\text{f}} -\frac{\text{k}_{\text{f}}\text{L}_{\text{\sigma f}}}{\text{L}^{\prime}_{ \text{Df}}}\psi_{\text{D}}\bigg{)}+\omega_{\text{el}}\psi_{\text{q}} \tag{5}\]  
\[\frac{\text{d}\psi_{\text{f}}}{\text{dt}}=\text{u}_{\text{f}}-\frac{1}{\text {T}^{\prime}_{\text{f}}}\bigg{(}\psi_{\text{f}}-\frac{\text{k}_{\text{D}} \text{L}_{\text{\sigma D}}}{\text{L}^{\prime}_{\text{dD}}}\psi_{\text{d}}- \frac{\text{k}_{\text{d}}\text{L}_{\text{\sigma s}}}{\text{L}^{\prime}_{ \text{Df}}}\psi_{\text{D}}\bigg{)} \tag{6}\]\[\frac{\mathrm{d}\psi_{\mathrm{D}}}{\mathrm{d}t}=-\frac{1}{T_{\mathrm{d}}^{\prime }}\bigg{(}\psi_{\mathrm{D}}-\frac{k_{\mathrm{f}}L_{\mathrm{of}}}{L_{\mathrm{df} }^{\prime}}\psi_{\mathrm{d}}-\frac{k_{\mathrm{d}}L_{\mathrm{os}}}{L_{\mathrm{ fd}}^{\prime}}\psi_{\mathrm{f}}\bigg{)} \tag{7}\]  
\[\frac{\mathrm{d}\psi_{\mathrm{q}}}{\mathrm{d}t}=\mathrm{u}_{\mathrm{q}}-\frac{ 1}{T_{\mathrm{qs}}^{\prime\prime}}\bigg{(}\psi_{\mathrm{q}}-k_{\mathrm{Q}} \psi_{\mathrm{Q}}\bigg{)}-\omega_{\mathrm{el}}\psi_{\mathrm{d}} \tag{8}\]  
\[\frac{\mathrm{d}\psi_{\mathrm{Q}}}{\mathrm{d}t}=-\frac{1}{T_{\mathrm{q}}^{ \prime\prime}}\bigg{(}\psi_{\mathrm{Q}}-k_{\mathrm{q}}\psi_{\mathrm{q}}\bigg{)} \tag{9}\]  
\[\frac{\mathrm{d}\omega_{\mathrm{el}}}{\mathrm{d}t}=\frac{1}{2\mathrm{H}}\bigg{[} \frac{k_{\mathrm{d}}L_{\mathrm{o}D}}{L_{\mathrm{q}D}^{\prime}L_{\mathrm{d}}^{ \prime\prime}}\psi_{\mathrm{f}}\psi_{\mathrm{q}}+\left(\frac{1}{L_{\mathrm{q} }^{\prime\prime}}-\frac{1}{L_{\mathrm{df}}^{\prime\prime}}\right)\psi_{ \mathrm{d}}\psi_{\mathrm{q}}\bigg{]}+\frac{1}{2\mathrm{H}}\bigg{[}\frac{k_{ \mathrm{f}}L_{\mathrm{of}}}{L_{\mathrm{d}}^{\prime}L_{\mathrm{d}}^{\prime \prime}}\psi_{\mathrm{D}}\psi_{\mathrm{q}}-\frac{k_{\mathrm{Q}}}{L_{\mathrm{q} }^{\prime\prime}L_{\mathrm{d}}^{\prime\prime}}\psi_{\mathrm{Q}}\psi_{\mathrm{ d}}\bigg{]} \tag{10}\]  
where  
\(\psi_{\mathrm{d}}\) is the magnetic flux in armature winding in d-axis,  
\(\psi_{\mathrm{f}}\) is the magnetic flux in excitation winding,  
\(\psi_{\mathrm{D}}\) is the magnetic flux in damping winding in d-axis,  
\(\psi_{\mathrm{q}}\) is the magnetic flux in armature winding in q-axis,  
\(\psi_{\mathrm{Q}}\) is the magnetic flux in damping winding in q-axis,  
\(\psi_{\mathrm{d}}\) is the transformed armature winding in d-axis,  
\(\psi_{\mathrm{d}}\) is the relaxation voltage,  
\(\psi_{\mathrm{d}}\) is the linkage factor of armature winding in d-axis,  
\(k_{\mathrm{f}}\) is the linkage factor of excitation winding,  
\(k_{\mathrm{D}}\) is the linkage factor of damping winding in d-axis,  
\(k_{\mathrm{q}}\) is the linkage factor of armature winding in q-axis,  
\(k_{\mathrm{Q}}\) is the linkage factor of damping winding in q-axis,  
\(L_{\mathrm{df}}^{\prime\prime}\) is the transient inductance of damping and excitation windings,  
\(L_{\mathrm{qD}}^{\prime\prime}\) is the transient inductance of armature and damping windings,  
\(L_{\mathrm{df}}^{\prime\prime}\) is the transient inductance of excitation and damping windings,  
\(L_{\mathrm{df}}^{\prime\prime}\) is the transient inductance of armature and excitation windings,  
\(L_{\mathrm{fd}}^{\prime\prime}\) is the transient inductance of excitation and armature windings,  
\(L_{\mathrm{fd}}^{\prime\prime}\) is the transient inductance of damping and armature windings,  
\(L_{\mathrm{fd}}^{\prime\prime}\) is the subtransient inductance of armature winding in q-axis,  
\(L_{\mathrm{df}}^{\prime\prime}\) is the subtransient time constant of the armature winding at short-circuit in d-axis,  
\(T_{\mathrm{q}}^{\prime\prime}\) is the subtransient time constant of the damping winding at short-circuit in q-axis,  
\(T_{\mathrm{q}}^{\prime\prime}\) is the subtransient time constant of the damping winding at short-circuit in d-axis,  
\(T_{\mathrm{f}}^{\prime\prime}\) is the subtransient time constant of the excitation winding at short-circuit, and  
\(H\) is the inertia constant.