# Nonlinear Model Predictive Control for Doubly Fed Induction Generator with Uncertainties
## 2 Problem Formulation  
### Dynamic Modeling of DFIG  
In the examination and regulation of the DFIG, the dynamic model of the induction machine is outlined within an arbitrary and rotating reference frame, denoted by the necessary angular velocity \(\Omega\), as illustrated in the equivalent circuit depicted in Figure 1 [15].  
The equations for the stator and rotor voltages \(v_{sdq}\), and \(v_{rdq}\), presented in the arbitrary frame of reference rotating with angular velocity \(\omega\) and referred to the stator side, can be expressed in the following equations.  
\[v_{sdq}=R_{s}i_{sdq}+j\omega\psi_{sdq}+\frac{1}{\omega_{b}}\frac{d\psi_{sdq}}{ dt} \tag{1}\]  
\[v_{rdq}=R_{r}i_{rdq}+j(\omega-\omega_{r})\psi_{rdq}+\frac{1}{\omega_{b}}\frac{d \psi_{rdq}}{dt} \tag{2}\]  
\[\psi_{sdq}=L_{s}i_{sdq}+L_{m}i_{rdq} \tag{3}\]  
\[\psi_{rdq}=L_{r}i_{rdq}+L_{m}i_{sdq} \tag{4}\]  
where \(\psi\), \(v\) and \(i\) represent flux, voltage, and current. The subscripts \(s\) and \(r\) specify the stator and rotor values, respectively. \(L_{s}\) and \(L_{r}\) are the inductances of the stator and rotor, \(L_{m}\) is the mutual inductance, \(\omega_{r}\) is the rotor speed, \(\omega-\omega_{r}\) is the sliding frequency of the rotor, \(\omega_{b}\) is the base angular frequency, \(\omega\) is the speed of the reference frame \(d-q\), and \(R_{s}\) and \(R_{r}\) are the resistances of the stator and rotor, respectively.  
The general dynamic model of the DFIG can be represented in various reference frames, such as the stationary rotor or synchronous reference frames, depending on its application. This choice involves considering different frame angles and speeds. By utilizing the aforementioned equations, the dynamics of the rotor can be depicted, incorporating expressions for rotor current and stator flux, as follows:  
\[v_{rdq}=R_{r}^{\prime}i_{rdq}+\frac{L_{r}^{\prime}}{\omega_{b}}\frac{di_{rdq}}{ dt}+\frac{1}{\omega_{b}}\frac{L_{m}}{L_{s}}\frac{d\varphi_{sdq}}{dt}+j\omega_{2}L_{ r}^{\prime}i_{rdq}+j\omega_{2}\frac{L_{m}}{L_{s}}\varphi_{sdq} \tag{5}\]  
Figure 1: Equivalent circuit of DFIG in an arbitrary reference frame [14].  
\[v_{nld}=R_{r}^{\prime}i_{nld}+\frac{L_{r}^{\prime}}{\omega_{b}}\frac{dit_{nld}}{dt}+ j\omega_{2}L_{r}^{\prime}i_{nld}+E_{ld} \tag{6}\]  
In Equations (5) and (6), \(R_{r}^{\prime}\) and \(L_{r}^{\prime}\) are the transient resistance and inductance of the rotor current dynamics and are as follows:  
\[R_{r}^{\prime}=R_{r}+\left(\frac{L_{m}}{L_{s}}\right)^{2}R_{s} \tag{7}\]  
\[L_{r}^{\prime}=L_{r}-\frac{L_{m}^{2}}{L_{s}} \tag{8}\]  
Also, \(E_{ld}\) in equation 6 is induced back-EMF voltage in the rotor winding and reflects the effects of the stator dynamics on the rotor current dynamics. This expression plays an important role on rotor irnush current, DC link overvoltage and torque increase during voltage drop. \(E_{dq}\) is a function of stator flux and voltage, which can be expressed as follows:  
\[E_{dq}=\frac{L_{m}}{L_{s}}\bigg{(}v_{sdq}-j\omega_{r}\psi_{sdq}-\frac{R_{s}}{ L_{s}}\psi_{sdq}\bigg{)} \tag{9}\]  
The active and reactive powers of the stator are calculated from the following equations:  
\[P_{s}=\text{Re}\Big{\{}v_{sdd}i_{sdq}^{*}\Big{\}}=\{v_{sd}i_{sd}+v_{sq}i_{sq}\} \tag{10}\]  
\[Q_{s}=\text{Re}\Big{\{}v_{sdq}i_{sdq}^{*}\Big{\}}=\big{\{}v_{sq}i_{sd}-v_{sd} i_{sq}\big{\}} \tag{11}\]  
The active and reactive powers of the stator can be obtained in the steady state as follows:  
\[P_{s}=R_{s}|I_{s}|^{2}-\frac{L_{m}}{L_{s}}\big{(}v_{sd}i_{rd}+v_{sq}i_{rq} \big{)} \tag{12}\]  
\[Q_{s}=\frac{L_{m}}{L_{s}}\big{(}v_{sd}i_{rq}-v_{sq}i_{rq}\big{)} \tag{13}\]  
As evident from Equations (12) and (13), the active and reactive power of the stator are influenced by both the d-component and the \(q\)-component of the rotor current. In vector control, it is preferable to employ synchronous reference transformations in a manner that the active and reactive power of the stator depend solely on either the \(d\) or \(q\) component of the rotor current.