# Fast Prediction of Characteristics in Wound Rotor Synchronous Condenser Using Subdomain Modeling
## 2 Subdomain Modeling
### Compounding Curve  
Figure 2 illustrates three compounding curves, where the red, blue, and orange lines represent power factors of \(1\) and \(0.95\) leading (WRSC generates reactive power), and \(0.95\) lagging (WRSC absorbs reactive power), respectively. For instance, at a stator load of \(105\) A, the rotor generates \(86\) A to achieve a power factor of \(1\). If the rotor current  
exceeds 86 A, the machine becomes overexcited, and excess reactive power flows to the grid. Conversely, if the rotor current is below 86 A, the machine becomes under-excited, lowering its operating voltage below the grid voltage, prompting the grid to supply reactive power to compensate for the power deficit in the machine. The compounding curve is essential for assessing WRSC performance, allowing designers to estimate the field current necessary for a given stator load to achieve a desired electrical system power factor. This section presents the procedure for deriving the unity power factor line and applies it to salient and non-salient rotors. The procedure involves:  
* Transforming phase quantities (flux, voltage, and current) obtained from subdomain modeling into the _dq_-frame under current excitation. * Deriving and transforming self- and mutual inductances into the _dq_-frame. * Building and solving a d-axis current function with a reactive power of zero from the voltage and power equations in the _dq_-frame. By aligning the rotor pole with phase A in the initial position, the quantities of voltage, current, flux, and inductance can be transferred to the _dq_-frame using Park and Clarke transformations [21]. By assuming that machine saturation is neglected, Park and Clarke transformations are given as follows:  
\[UI\lambda_{dq_{2}}=\frac{2}{3} \begin{bmatrix}\cos(\theta_{0})&\cos(\theta_{0}- \frac{2\pi}{3})&\cos(\theta_{0}+\frac{2\pi}{3})\\ -\sin(\theta_{0})&-\sin(\theta_{0}-\frac{2\pi}{3})&-\sin(\theta_{0}+\frac{2 \pi}{3})\\ \sfrac{1}{2}&\sfrac{1}{2}&\sfrac{1}{2}\end{bmatrix}\begin{bmatrix}UI\lambda_{a }\\ U\lambda_{b}\\ U\lambda_{c}\end{bmatrix}  \tag{28}\]  
\[ \begin{bmatrix}L_{d}&M_{dq}&M_{dz}\\ M_{qd}&L_{q}&M_{qz}\\ M_{zd}&M_{zq}&L_{z}\end{bmatrix} =\sqrt{\frac{2}{3}} \begin{bmatrix}1&\cos(\frac {2\pi}{3})&\cos(\frac{4\pi}{3})\\ 0&\sin(\frac{2\pi}{3})&\sin(\frac{4\pi}{3})\\ 1/\sqrt{2}&\sfrac{1}{2}/\sqrt{2}\end{bmatrix}\begin{bmatrix}L_{aa}&M_{ab}&M_{ ac}\\ M_{ba}&L_{bb}&M_{bc}\\ M_{ca}&M_{cb}&L_{cc}\end{bmatrix} \sqrt{\frac{2}{3}} \tag{29}\]  
To calculate the winding self-inductance and mutual inductances in (29), the authors apply an excitation current of 1 \(A\) to the first winding, while setting the currents in all other windings to zero, excluding rotor current effects from the analysis. For example, phase A is set to 1 \(A\), while phases B and C and the rotor current are set to zero. The obtained flux linkages of phases A, B, and C (\(\psi_{a},\psi_{b},\psi_{c}\)) are then used to calculate the inductance as follows:  
\[[L_{aa}\ L_{ab}\ L_{ac}]=\frac{[\psi_{a}\ \psi_{b}\ \psi_{c}]}{1A} \tag{30}\]  
Additionally, rotor flux linkage (\(\psi_{f}\)), used to derive rotor inductance, \(M_{df}=\psi_{f}/I_{f}\), is calculated by setting all stator winding currents to zero and applying only a 1 \(A\) current to the rotor. The flux linkage of phase A is considered the rotor flux linkage. Figure 2: Illustration of compounding curves. In high-power machines, reactance is dominant, and stator resistance can be neglected; therefore, the voltage equation is given by:  
\[ \begin{bmatrix}U_{d}\\ U_{q}\end{bmatrix} \approx\omega_{e} \begin{bmatrix}-L_{q}i_{q}\\ L_{d}i_{d}+M_{df}i_{f}\end{bmatrix}  \tag{31}\]  
Consequently, the active and reactive powers are defined as:  
\[ \begin{array}{l}P=1.5(U_{d}i_{d}+U_{d}i_{q})\\ Q=1.5(U_{d}i_{d}-U_{d}i_{q})\end{array}  \tag{32}\]  
Table 1 shows the operating modes of the machine's power. To operate as a capacitor bank, the machine generates reactive power (negative \(Q\)).