# Optimal Design and Control of MMC STATCOM for Improving Power Quality Indicators
## 3 Model of the Modular Multilevel Converter (MMC)  
The modular multilevel converter (MMC) has been introduced firstly by Marquardt in 2001, and is presented as the developed configuration including a number of subsystems for achieving the desired voltage level [15, 18, 24]. This type of converter is designed for use in medium and high transmission voltages. The overall configuration of the modular multilevel converter is illustrated in Figure 5, where  
Figure 4: A representation of the harmonic component in the output voltage of the AC side cascaded configuration using 4H-bridges [26].  
Figure 3: Voltage for a single AC side cascaded configuration including 4H-bridges [26].  
each leg of this converter consists of a lower and an upper arm connected to each other by a DC voltage link.  
Each arm includes a number of SMs, and in turn, every sub-system includes two switches and two reverse diodes connected to each other via a DC capacitor. The switches S1 and S2 are operated in reverse. Each SM could be configured as a half-bridge, full-bridge, H-bridge or clamp-double circuit [18]. Due to its low cost and high efficiency, the MMC based on the half-bridge SMs shown in Figure 6 has been widely used in industrial applications and high voltage DC projects. It is possible to perform a selectable individual control for each SM in the converter leg. Principally, the converter legs provided a controllable Voltage Source Converter (VSC) for each phase. The desired waveform voltage can easily be achieved at the terminal by adjusting the terminal voltage of the SM's leg.  
The equivalent circuit presentation of the typical MMC using half-bridge SMs is shown in Figure 7, where \(u_{\gamma j}\) is the phase converter output voltage and \(i_{\gamma j}\) denotes the line current; \(i_{pj}\) and \(i_{nj}\) are the upper and lower leg currents, respectively, that are presented as [8; 10; 27]:  
\[\begin{array}{l}i_{pj}=\frac{i_{\gamma j}}{2}+i_{diffj}\\ i_{nj}=\frac{i_{\gamma j}}{2}-i_{diffj}\end{array} \tag{1}\]  
Figure 5: The circuit configuration of the MMC.  
where \(i_{diffj}\) denotes the phase \(j\) inter-circulating current that circulates between the upper and lower legs and is expressed as below:  
\[i_{diffj}=\frac{i_{pj}-i_{nj}}{2} \tag{2}\]  
As discussed in [7], the MMC operates according to the following equations:  
\[u_{\gamma j}=e_{j}-\frac{R_{arm}}{2}i_{\gamma j}-\frac{L_{arm}}{2}\cdot\frac{ di_{ij}}{dt}\ (j=a,b,c) \tag{3}\]  
\[R_{arm}\cdot i_{diffj}+L_{arm}\cdot\frac{di_{diffj}}{dt}=\frac{V_{dc}}{2}-\frac{ u_{nj}+u_{pj}}{2}\ (j=a,b,c) \tag{4}\]  
Figure 6: A schematic of the configuration of the MMC based on half-bridge and full-bridge sub-modules (SMs).  
Figure 7: The one phase equivalent circuit.  
where \(e_{j}\) denotes the inner _enf_ produced in phase \(a\) and is given as follows:  
\[e_{j}=\frac{u_{nj}-u_{pj}}{2} \tag{5}\]  
The internal dynamic performance of the MMC is explained in Equation (4) and can be characterized as follows:  
\[u_{diffj}=R_{arm}\dot{u}_{diffj}+L_{arm}\cdot\frac{di_{diffj}}{dt}=\frac{V_{dc}}{ 2}-\frac{u_{nj}+u_{pj}}{2}\ (j=a,b,c) \tag{6}\]  
where \(u_{diffj}\) denotes the phase j inner unbalance voltage.  
The MMC is characterized by its small size and simplicity in installation, and as the number of SMs increases, there is a decrease in harmonics and a matching of the required power quality specifications. However, the multilevel converter suffers from converter inter-circulating current, which increases the switches' conduction loss and increases the thermal stress on the capacitors and switches' IGBTs. Therefore, maintaining the balance of the capacitor voltages is one of the main rules of the design of the MMC.