## 3 DFIG-Based Load Flow Analysis
The power coefficient \(C_{p}\) depends on the values of the pitch angle \(\beta\)\(\left[\mathrm{deg}\right]\) and the tip speed ratio \(\lambda\), defined in Equation (14):  
\[\lambda\ =\ \frac{\mathrm{v}_{t}}{V_{w}} \tag{14}\]  
where \(\mathrm{v}_{t}\) is the blade tip speed \(\left[\mathrm{m}/\mathrm{s}\right]\). The three modes of operating the speed control of a DFIG include:  
* Below the synchronous speed (for low wind speeds). For this case, \(C_{p,opt}\) cannot be reached and the DFIG is operated at a constant speed (\(\omega_{r,\ min}\)) and \(\beta\) = 0;
* At maximum power production;
* Above the synchronous speed (for high wind speed). In this case, the speed control optimizes the power extraction from the wind between the minimum and maximum wind speeds given by Equation (15), using an optimal value for the power coefficient \(C_{p}\ =\ C_{p,opt}\) and \(\beta\) = 0, \[P_{m}\ =\ \frac{1}{2}\rho AV_{w}{}^{3}C_{p,\ opt}\] (15) where \(\gamma\ =\ (1/2)\rho AC_{p,opt}\). In addition, if it is multiplied by the cube of the wind speed, we obtain, \(P_{m}\ =\ \gamma V_{w}{}^{3}\). Now, by substituting the wind speed variable produced in Equation (14), it is obtained that  
\[P_{m}\ =\ \gamma\left(\frac{\mathrm{v}_{t}}{\lambda}\right)^{3}\ =\ \gamma^{\prime}\omega_{r}{}^{3}\ =\ \gamma_{0}(1-s)^{3} \tag{16}\]  
Figure 4: Mechanical power of the DFIG as a rotor speed function. Considering the Equations (1) to (12), a set of equations describing the DFIG in a steady-state is obtained. To solve these, another equation set is derived, taking the real and imaginary parts in Equations (1) and (3), obtaining the following:  
\[ \begin{array}{rcl}\Gamma_{1}&=&V_{s}\cos\delta_{s}-r_{s}I_{s}\cos\varphi_{s}+ \omega_{es}(L_{ls}+L_{M})I_{s}\sin\varphi_{s}+\omega_{es}L_{M}I^{\prime},\sin \varphi_{r}\end{array}  \tag{17}\]  
\[ \begin{array}{rcl}\Gamma_{2}&=&V_{s}\sin\delta_{s}-r_{s}I_{s}\sin\varphi_{s}- \omega_{es}(L_{ls}+L_{M})I_{s}\cos\varphi_{s}-\omega_{es}L_{M}I^{\prime},\cos \varphi_{r}\end{array}  \tag{18}\]  
Similarly, the Equations (19) and (20) are obtained:  
\[ \begin{array}{rcl}\Gamma_{3}&=&V^{\prime},\cos\delta_{r}-r^{\prime},I^{ \prime}r\cos\varphi_{r}+s\omega_{es}(L^{\prime},_{Ir}+L_{M})I^{\prime},\sin \varphi_{r}+s\omega_{es}L_{M}I_{s}\sin\varphi_{s}\end{array}  \tag{19}\]  
\[ \begin{array}{rcl}\Gamma_{4}&=&V^{\prime},_{r}\sin\delta_{r}-r^{\prime},_{r }I^{\prime},_{r}\sin\varphi_{r}-s\omega_{es}(L^{\prime},_{Ir}+L_{M})I^{ \prime},\cos\varphi_{r}-s\omega_{es}L_{M}I_{s}\cos\varphi_{s}\end{array}  \tag{20}\]  
From the Equations (9) and (10), the function \(\Gamma\)5 is  
\[ \begin{array}{rcl}\Gamma_{5}&=&P-V_{s}I_{s}\cos(\delta_{s}-\varphi_{s})-V_{ r}I_{r}\cos(\delta_{r}-\varphi_{r})\end{array}  \tag{21}\]  
Thus, from Equation (11), the function \(\Gamma\)6 is,  
\[ \begin{array}{rcl}\Gamma_{6}&=&Q-P_{s}&=&Q-V_{s}I_{s}\cos(\delta_{s}-\varphi _{s})\end{array}  \tag{22}\]  
Finally, from the Equations (8), (10), and (12) we obtain  
\[ \begin{array}{rcl}\Gamma_{7}&=&P-\gamma_{0}(1-s)^{3}+\left(I_{s}^{2}r_{s}+ I_{r}^{2}r_{r}\right)\end{array}  \tag{23}\]  
In this manner, eleven variables in the previous set of seven nonlinear equations are as following: \(U_{s}\), \(\delta_{s}\), \(Ur\), \(\delta r\), \(I_{s}\), \(\varphi_{s}\), \(Ir\), \(\varphi r\), \(s\), \(P\), and \(Q\). To solve the Equations (17)-(23), four variables with known value are required. By selecting \(V\), \(\delta\), \(P\), and \(Q\) as fixed variables, \(ss\) solutions for the Equations (17)-(23) are obtained using the Decoupled Newton-Raphson method expressed as  
\[ \begin{array}{rcl}\Delta x&=&J^{-1}\,\Delta\Gamma\end{array}  \tag{24}\]  
where \(\Delta x\) and \(\Delta\Gamma\)are the incremental variables defined by the Equations (25) and (26):  
\[ \begin{array}{rcl}\Delta\Gamma&=&\Delta(\Gamma_{1},\,\Gamma_{2},\,\Gamma_{3 }\ldots\Gamma_{7})^{T}\end{array}  \tag{25}\]  
\[\begin{array}{rcl}\Delta x&=&\Delta(V_{r},\,\delta_{r},\,I_{r  
* The system is modeled in steady-state with all derivatives equal to zero.