# Fast Prediction of Characteristics in Wound Rotor Synchronous Condenser Using Subdomain Modeling
## 2 Subdomain Modeling  
### Governing Partial Differential Equations (PDEs)  
Figure 1 describes the WRSC and its simplified models employed in the subdomain modeling. Initially, the following assumptions were made:  
* The end effects are ignored;
* The problem is two-dimensional (2-D) in polar coordinates;
* Magnetic vector potential \(A\), current density \(J\), and magnetic flux density vector \(B\) have the following non-zero components: \(A=[0,0,A_{z}]\); \(J=[0,0,J_{z}]\); \(B=[B_{r},B_{\theta},0]\);
* The core materials have infinite permeability.  
The WRSC model, as illustrated in Figure 1b, is divided into five subdomains: rotor, rotor opening, air gap, stator opening, and stator slots. Each subdomain is represented by a vector potential of the form \(A_{z}^{p}\), \(A_{z}^{q}\), \(A_{z}^{l}\), \(A_{z}^{i}\), \(A_{z}^{j}\), with respective harmonic components \(h\), \(g\), \(n\), \(k\), \(m\). The model includes \(N_{r}\) rotor and \(N_{s}\) stator slots, where indices \(p\), \(q\), \(i\), and \(j\) denote specific rotor and stator slots with initial positions \(\theta_{p}\), \(\theta_{q}\), \(\theta_{i}\), and \(\theta_{j}\). A 2-D analysis in polar coordinates is used to derive partial differential equations (PDEs) for each region:  
\[\frac{\partial^{2}A_{z}^{p}}{\partial r^{2}}+\frac{\partial A_{z}^{p}}{r \partial r}+\frac{\partial^{2}A_{z}^{p}}{r^{2}\partial\theta^{2}}=-\mu_{0}J_{ z}^{p} \tag{1}\]  
\[\frac{\partial^{2}A_{z}^{q}}{\partial r^{2}}+\frac{\partial A_{z}^{q}}{r \partial r}+\frac{\partial^{2}A_{z}^{q}}{r^{2}\partial\theta^{2}}=0 \tag{2}\]  
\[\frac{\partial^{2}A_{z}^{i}}{\partial r^{2}}+\frac{\partial A_{z}^{i}}{r \partial r}+\frac{\partial^{2}A_{z}^{i}}{r^{2}\partial\theta^{2}}=0 \tag{3}\]  
\[\frac{\partial^{2}A_{z}^{i}}{\partial r^{2}}+\frac{\partial A_{z}^{i}}{r \partial r}+\frac{\partial^{2}A_{z}^{i}}{r^{2}\partial\theta^{2}}=0 \tag{4}\]  
\[\frac{\partial^{2}A_{z}^{i}}{\partial r^{2}}+\frac{\partial A_{z}^{i}}{r \partial r}+\frac{\partial^{2}A_{z}^{i}}{r^{2}\partial\theta^{2}}=-\mu_{0}J_ {z}^{i} \tag{5}\]  
The air gap and slot-opening regions can be modeled using Laplace's equation, while the rotor-stator slot subdomains can be represented by Poisson's equation. To facilitate the Fourier series expansion of the vector potential, the right-hand side of Poisson's equation must be reformulated in Fourier form.  
In the double-layer stator winding, the current density in the \(j\)-th slot is expressed as  
\[\begin{array}{l}\mathbf{j}_{z}^{j}=\left(J_{0}^{j}+\sum\limits_{m=1,2}^{\infty} J_{m}^{j}\text{cos}\big{(}\frac{m\pi}{\delta}\big{(}\theta-\theta_{j}\big{)} \big{)}\right)\mathbf{i}_{z}\\ \text{where}\\ J_{0}^{j}=\frac{J_{0}^{j-layer1}+J_{0}^{j-layer2}}{2}\\ J_{m}^{j}=\left[\begin{array}{l}\frac{2}{m\pi}\Big{(}j_{0}^{j-layer1}-J_{ 0}^{j-layer2}\Big{)}\text{sin}\big{(}\frac{m\pi}{2}\big{)}\leftrightarrow \text{non overlapping}\\ 0\;\leftrightarrow\;\text{overlapping}\end{array}\right.\end{array} \tag{6}\]  
In the single-layer rotor winding, the current sheet is simply described as \(\mathbf{j}_{z}^{p}=J_{0}^{p}\mathbf{i}_{z}\).  
Figure 1: (a) Wound rotor synchronous condenser and (b) a simplified model.