# A PMSG Wind Energy System Featuring Low-Voltage Ride-through via Mode-Shift Control
## 3 System under Consideration
### Inverter DC-LINK  
The inverter DC-link offers a buffering stage between the MSC and the GSC, where the energy balance equation can be represented by (14):  
\[P_{gen}=C_{dc}\frac{dV_{dc}}{dt}+i_{grid}{}^{2}R+P_{grid} \tag{14}\]  
where \(P_{gen}\) and \(P_{grid}\) represent the PMSG generator extracted power and grid injected power, respectively. It should be noted that in case of normal operation, both \(P_{gen}\) and \(P_{grid}\) are equal ignoring converters and grid losses. This should be guaranteed to maintain energy balance at this link and in turn the DC-link voltage is regulated and system robustness is maintained. Thus, this DC-link is considered the system core and the dynamic model at this zone is derived to verify the proposed system stability as follows:  
State-space models give a basic and powerful method for the dynamic modelling of numerous systems (continuous and discrete, linear and non-linear, time-variant and invariant systems). Thus, they are mostly convenient to model the steady-state and the dynamic performances of the power electronic converters. The method of State-Space Averaging was developed to characterize the transfer properties of switching converter power stages. This method has been very well received by the power electronics industry, in contrast to other approaches, which give little or no insight into the design and behavior of the converter [37; 38]. In the considered work, the boost converter output DC-link capacitor is also placed at the input of the three-phase VSI. Hence, it is not only affected by the rippled DC waveforms of the boost converter, but also by the VSI AC waveforms. Therefore, state space is the best modeling technique to model the considered time-invariant non-linear system [39; 40]. It is worth noting that the VSI output signals are AC waveforms while the VSI input signals as well as those of the boost converter are DC waveforms associated with AC component oscillating around the average value as shown in (15):  
\[x=X+\hat{\kappa} \tag{15}\]  
The VSI input and output signals are related as follows:  
\[v_{inv}=\frac{m}{2}\ v_{dc} \tag{16}\]  
where the modulation index \(m=Msin(2\pi 50t+\theta)\), \(v_{inv}\) is the VSI output AC voltage, while \(v_{dc}\) is the VSI input rippled DC voltage. For lossless VSI,  
\[v_{dc}i_{dc}=v_{inv}i_{grid} \tag{17}\]  
\[i_{dc}=\frac{m}{2}\ i_{grid} \tag{18}\]  
where \(i_{grid}\) is the VSI output AC sinusoidal current, while \(i_{dc}\) is the VSI input rippled DC current. The dynamic model of the considered system topology, at the DC-link, is derived using State-Space Averaging. Switching periods of the boost chopper is described by two circuit topologies (continuous conduction mode), i.e., two sub-models each addressing system-specific circuit dynamics as shown in Figure 3. The first sub-model offers the state-space dynamic equation of the system when the boost converter switch (SW1) is at the ON state as shown in Figure 3a. On the other hand, the second sub-model shows the dynamic model when the switch (SW1) is OFF, as demonstrated by Figure 3b. For the sake of simplicity, capacitors and inductors' internal resistances are neglected while the diode is modeled as a short circuit in ON-state and open circuit in OFF-state. * When SW1 is ON, applying [PERSON]'s voltage and current laws, the following results are found: \[L_{dc}\frac{d\;i_{b}}{dt}=v_{r}\] (19) \[C_{dc}\frac{d\;v_{dc}}{dt}=-\frac{m}{2}\;i_{grid}\] (20) \[L\frac{d\;i_{grid}}{dt}=\frac{m}{2}v_{dc}-i_{grid}R\;\;-v_{grid}\] (21) where \(i_{b}\), \(v_{r}\), \(v_{dc}\) are the instantaneous boost converter current, rectifier output voltage and DC-link voltage, respectively. Equations (19)-(21) are rearranged to obtain the state-space sub-model shown in (22): \[\left[ \begin{array}{c}\dot{i_{b}}\\ \dot{v_{dc}}\\ \dot{l_{grid}}\end{array} \right]=A_{1}\left[ \begin{array}{c}\dot{i_{b}}\\ \dot{v_{dc}}\\ \dot{i_{grid}}\end{array} \right]+B_{1}\left[ \begin{array}{c}v_{r}\\ v_{grid}\end{array} \right]\] (22) where \[A_{1}=\left[