## 3 Average Model of MMC-Controlled Wind Turbine Generator
### MMC Circuit Model  
To derive a model representation of the MMC, the equivalent approach model, as shown in Figure 14, is considered. By applying Kirchhoff Voltage Low (KVL) across the phase reactor, differential equations for the three-phase circuit can be expressed as follows [32; 33]:  
\[L_{f}\frac{di_{f}^{(abc)}}{dt}=v_{g}^{(abc)}-v_{c}^{(abc)}-R_{f}i_{f}^{(abc)} \tag{8}\]  
By applying Park Transformation, the three-phase differential equation in the \(dq\)-axis component can be expressed as follows:  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]  
\[L_{f}\frac{di_{f}^{q}}{dt}=v_{g}^{q}-v_{c}^{q}-R_{f}i_{f}^{q}-\omega_{g}L_{f}i_ {f}^{d} \tag{10}\]  
Figure 14: Electrical circuit model of MMC. Figure 13: Average model configuration of a wind turbine generator. The complex grid power (\(S_{g}\)) can be calculated as follows:  
\[S_{g}=\left(v_{g}^{d}+jv_{g}^{d}\right)\left(\dot{i}_{f}^{d}-ji_{f}^{d}\right) \Rightarrow\ S_{g}=\left(v_{g}^{d}\dot{i}_{f}^{d}+v_{g}^{q}\dot{i}_{f}^{d} \right)+j\left(v_{g}^{q}\dot{i}_{f}^{d}-v_{g}^{d}\dot{i}_{f}^{d}\right) \tag{11}\]  
From (11), the active power and reactive power can be written as:  
\[P_{g}=v_{g}^{d}\dot{i}_{f}^{d}+v_{g}^{q}\dot{i}_{f}^{q} \tag{12}\]  
\[Q_{g}=v_{g}^{q}\dot{i}_{f}^{d}-v_{g}^{d}\dot{i}_{f}^{q} \tag{13}\]  
In the same way, active power and reactive power flow to the converter valve can be written as:  
\[P_{c}=v_{c}^{d}\dot{i}_{f}^{d}+v_{c}^{q}\dot{i}_{f}^{q} \tag{14}\]  
\[Q_{g}=v_{c}^{d}\dot{i}_{f}^{d}-v_{c}^{d}\dot{i}_{f}^{q} \tag{15}\]  
As reactive power does not propagate to the DC side of the converter valve, the DC current (\(i_{dc}\)) is obtained with regards to the active power balance between the AC side and the DC side. By assuming that the losses on the converter can be omitted, the following relation can be written:  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]0  
The voltages waveform at the AC side of the MMC depends on the reference voltages fed to each of the six arms. The inserted voltages in the six arms of the MMC are represented by a controlled voltage source. The upper and lower arm currents can be written as follows:  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]1  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]2  
where \(i_{diff}\) is the circulating phase leg current, which can be determined as the average of the upper arm and lower arm currents as:  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]3  
The voltage output at the AC side of the MMC is given by:  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]4  
The DC loop of each MMC arm can be expressed as:  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]5  
The inner difference voltage of each phase is given by:  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]6  
The reference voltage of the upper arm and the lower arm for each phase can be written as follows:  
\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]7\[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]8  
Referring to Figure 14, the proposed simplified MMC circuit model can be represented as depicted by \[L_{f}\frac{di_{f}^{d}}{dt}=v_{g}^{d}-v_{c}^{d}-R_{f}i_{f}^{d}+\omega_{g}L_{f}i_ {f}^{d} \tag{9}\]9 The equivalent circuit for the MMC is represented by a pair of three-phase AC voltage sources and a pair of DC voltage sources for the upper arm and the lower arm, respectively. The AC voltage sources are connected to the MMC AC terminal through the arm inductor (\(L_{arm}\)), arm resistance (\(R_{arm}\)), reactor inductor (\(L_{f}\)), and reactor resistance (\(R_{f}\)).