# Robust Model-Free Control for MMC Inverters in Cold Ironing Systems
## 2 Description and Modeling of the MMC Converter
### Modeling of the MMC Converter  
Modeling the modular multilevel converter (MMC) is a crucial step in analyzing and optimizing power electronics systems used in high-voltage transmission applications. Throughout the modeling process, we assume that the output current is equally divided between the upper and lower arms and that switching losses for the power devices are  
Figure 3: Three-phase MMC inverter topology.  
not considered. The total output voltages of the SMs in the lower (or upper) arm of each phase can be represented as controlled voltage sources _vpij_ (or _vnj_), as shown in Figure 4. Applying [PERSON]'s law to phase \(j\) (where \(j\) = \(a\), \(b\), _c_), we derive circulating current \(i^{j}_{cir}\) and voltage \(v^{j}_{cir}\) in Equations (1) and (2):  
\[i^{j}_{cir}(t)=\frac{\left(i^{j}_{p}(t)+i^{j}_{n}(t)\right)}{2} \tag{1}\]  
\[v^{j}_{cir}(t)=v^{j}_{p}(t)+v^{j}_{n}(t) \tag{2}\]  
Lower and upper lower loops are described by Equations (3) and (4), respectively:  
\[-\frac{v_{dc}(t)}{2}+v^{j}_{n}(t)+L_{s}\frac{d^{j}_{n}(t)}{dt}+R_{s}i^{j}_{n}( t)-l_{f}\frac{di_{j}(t)}{dt}-r_{f}i_{j}(t)-v_{cj}(t)=0 \tag{3}\]  
\[\frac{v_{dc}(t)}{2}-v^{j}_{p}(t)-L_{s}\frac{d^{j}_{p}(t)}{dt}-R_{s}i^{j}_{p}( t)-l_{f}\frac{di_{j}(t)}{dt}-r_{f}i_{j}(t)-v_{cj}(t)=0 \tag{4}\]  
where  
\(v^{j}_{n}\) : lower part voltage;  
\(i^{j}_{n}\) : lower part current;  
\(i^{j}_{p}\) : upper part current;  
\(v^{j}_{p}\) : upper part current;  
\(v^{j}_{p}\) : upper part voltage;  
\(l_{f}\), \(r_{f}\) and \(C_{f}\) are, respectively, the inductance, the resistance and the capacitor of the filter;  
\(i_{c}\): the inductor current in abc frame \(i_{a},i_{b},i_{c}\);  
\(v_{cj}\): the voltage of the capacitor of phase \(j\);  
\(i_{jj}\): the load output current of phase \(j\);  
Figure 4: Circulating current of MMC.  
Summing Equations (3) and (4) gives Equation (5) which describes the AC-side dynamics of the Arm Average Model:  
\[v_{j}(t)-v_{cj}(t)=L_{eq}\frac{di_{j}(t)}{dt}+R_{eq}i_{j}(t) \tag{5}\]  
where: \(i_{j}(t)=i_{p}^{j}(t)-i_{n}^{j}(t)\), \(v_{j}(t)=\frac{v_{n}^{j}(t)-v_{p}^{j}(t)}{2}\), \(R_{eq}=r_{f}+\frac{R_{s}}{2}\) and \(L_{eq}=L_{f}+\frac{L_{s}}{2}\)  
Subtracting Equation (3) from (4) gives Equation (6), which describes the DC-side dynamics of the Arm Average Model:  
\[v_{dc}(t)-v_{cir}^{j}(t)=2L_{s}\frac{d\,i_{cir}^{j}(t)}{dt}+2R_{s}\,i_{cir}^{j} (t) \tag{6}\]  
The MMC equations for the AC and DC part transferred into the d,q frame, using Park transformation matrix, are given by the following equations [40]:  
The AC-side dynamics (5) can be decomposed into  
\[v_{d}(t)-v_{cd}(t)=L_{eq}\frac{di_{d}(t)}{dt}+R_{eq}i_{d}(t)-L_{eq}\omega i_{q }(t) \tag{7}\]  
\[v_{q}(t)-v_{eq}(t)=L_{eq}\frac{di_{q}(t)}{dt}+R_{eq}i_{q}(t)+L_{eq}\omega i_{d }(t) \tag{8}\]  
\[C_{f}\frac{dv_{cd}(t)}{dt}=i_{d}(t)-i_{ld}(t)+C_{f}\omega v_{cq}(t) \tag{9}\]  
\[C_{f}\frac{dv_{cq}(t)}{dt}=i_{q}(t)-i_{dq}(t)-C_{f}\omega v_{cd}(t) \tag{10}\]  
With  
\(v_{cd},v_{cq}\): the dq components of the voltage at point PCC;  
\(v_{d},v_{q}\): are the dq components of the voltage inputs of the inverter;  
\(i_{d},i_{q}\): the dq components of the line currents;  
\(i_{ld},i_{q}\): the components dq of the load currents;  
\(\omega\): is the angular frequency.  
The DC-side Equation (6) is simplified as follows:  
\[v_{dc}(t)-v_{cdir}(t)=2L_{s}\frac{di_{cir}\,(t)}{dt}+2R_{s}i_{d}(t) \tag{11}\]  
\[v_{qc}(t)-v_{qcir}(t)=2L_{s}\frac{di_{qcir}(t)}{dt}+2R_{s}i_{q}(t) \tag{12}\]