# Fast Prediction of Characteristics in Wound Rotor Synchronous Condenser Using Subdomain Modeling
## 3 FEM Simulation Comparison  
To validate these principles, specifications for a one-pole-pair WRSC, as listed in Table 2, were simulated using a 2-D finite element method (FEM) model with the mesh setting depicted in Figure 2(a). Due to mechanical constraints in MW-class WRSC, the air gap was intentionally designed to be large (up to 100 mm). The FEM simulation took 66 s to compute flux density, flux at the air gap, and induced voltage (see Figures 2(b) and 4). By contrast, the subdomain modeling approach achieved these calculations in 33 s, showcasing the computational efficiency of the proposed method in WRSC pre-design.  
Figure 2(b) illustrates the flux density distribution at current conditions \(i_{f}=1\) kA; \(i_{d}=i_{q}=-0.5\) kA. The salient-pole rotor caused flux linkage to concentrate in the rotor pole and teeth near the pole, inducing mild saturation in the teeth. Under typical conditions, machines avoid saturation, making the assumption of infinite permeability in subdomain modeling suitable. However, when the field current or stator current is increased, saturation intensifies, resulting in accuracy discrepancies in the subdomain method (see Figure 3(e,f)).  
The compounding curves derived from the subdomain method were closely aligned with the FEM results, as seen in Figure 5. To maintain a unity power factor while increasing stator current, the field current must also rise. Additionally, rotor current must increase at higher stator voltages. Calculations for these curves, derived from Equation (33), involve  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mode** & **Power Factor** & \(P\) & \(Q\) \\ \hline Motor & Lagging & \textgreater{}0 & \textgreater{}0 \\ Motor & Leading & \textgreater{}0 & \textless{}0 \\ Generator & Lagging & \textless{}0 & \textgreater{}0 \\ Generator & Leading & \textless{}0 & \textless{}0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Machine mode definition.  
computing the \(dq\)-axis currents for each rotor current, then determining the stator current. The analysis covered phase peak voltages of 10, 15, 18, and 22 kV, revealing that the machine can support stator currents up to 5 kA with a maximum field current of 1.3 kA. The mathematical method obtained these results in under 1 s, which represents a significant time advantage in deriving compounding curves.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Quantity** & **Symbol** & **Unit** & **Value** \\ \hline Inner rotor slot radius & \(R_{1}\) & mm & 269.50 \\ Outer rotor radius & \(R_{2}\) & mm & 425.25 \\ Inner stator slot radius & \(R_{3}\) & mm & 525.25 \\ Outer stator slot radius & \(R_{4}\) & mm & 700.50 \\ Outer stator radius & \(R_{5}\) & mm & 1119.50 \\ Stack length & \(L_{slt}\) & mm & 6402.40 \\ Vacuum permeability & \(\mu_{0}\) & kg\(\cdot\)m\(\cdot\)s\({}^{-2}\)\(\cdot\)A\({}^{-2}\) & \(4\pi\times 10^{-7}\) \\ Rotor slot number & \(N_{r}\) & - & 32 \\ Stator slot number & \(N_{s}\) & - & 42 \\ Rotor slot pitch ratio & \(\alpha\) & \(\pi/180\) rad & 4.64 \\ Rotor opening slot pitch ratio & \(\gamma\) & \(\pi/180\) rad & 4.64 \\ Stator opening slot pitch ratio & \(\beta\) & \(2\pi/N_{s}\) rad & 0.46 \\ Stator slot pitch ratio & \(\delta\) & \(2\pi/N_{s}\) rad & 0.46 \\ Rotor slot winding turn & \(T_{slot}^{\prime}\) & - & 5/7 * \\ Stator slot winding turn & \(T_{slot}^{\prime}\) & - & 2 \\ \hline \hline \end{tabular}  
* The 1 st, 16 th, 17 th, and 32 nd slots each contain five turns. The remaining slots accommodate seven turns.  
\end{table}
Table 2: WRSC parameters.  
Figure 3: (**a**) Mesh operation (29,242 elements) and (**b**) flux density distribution at \(i_{f}=1\) kA\({}_{2}\)\(i_{d}=i_{q}=-0.5\) kA.  
Figure 4: Flux density at the air gap and induced voltage in case of **(a,b)**\(i_{f}=1\) kA;\(i_{d}=i_{q}=0\) kA, **(c,d)**\(i_{f}=1\) kA;\(i_{d}=i_{q}=-0.5\) kA, and **(e,f)**\(i_{f}=1.5\) kA;\(i_{d}=-0.5\);\(i_{q}=-1\) kA, respectively.