# A Review of Modular Multilevel Converters for Stationary Applications
## 3 Mathematical Modeling and Control of Three-Phase MMC  
### Mathematical Modeling  
Assuming that the arm and leg structures of three-phase MMC are perfectly symmetrical, the mathematical model [3; 75; 76; 77; 78] can be established based on a widely accepted equivalent circuit depicted in Figure 6.  
The equivalent AC electromotive force (EMF) is symmetrical in the upper and lower arm, resulting in equally distributed output AC current in the arms. The upper and lower currents can be expressed by  
\[i_{xu}=\frac{i_{x}}{2}+i_{cirx} \tag{1}\]  
\[i_{xl}=-\frac{i_{x}}{2}+i_{cirx} \tag{2}\]  
where \(i_{xu}\) and \(i_{xl}\) represent the upper and lower currents of phase \(x\)\((x=a,b,c)\), \(i_{cirx}\) is the circulating current in each phase, and \(i_{x}\) represents the phase-x output AC current. From Equations (1) and (2), the output and circulating currents can be deduced as:  
\[i_{x}=i_{xu}-i_{xl} \tag{3}\]  
\[i_{cirx}=\frac{i_{xu}+i_{xl}}{2} \tag{4}\]  
Figure 6: Equivalent circuit of three-phase MMC. The circulating current contains DC and harmonic components. Due to the three-phase symmetrical structure, the equivalent DC EMF is equal for each leg, leading to an evenly distributed direct current in each phase. Thus, the DC component of the circulating current can be expressed by  
\[i_{cirxu\_dc}=i_{cirx\_dc}=i_{cirx\_dc}=\frac{i_{dc}}{3} \tag{5}\]  
where \(i_{cirxu\_dc}\), \(i_{cirx\_dc}\), and \(i_{cirxu\_dc}\) are the DC components of cirrulating currents in the upper/lower arm and phase-x. The DC component of circulating currents is naturally eliminated when the DC link voltage is not practically employed such as operating in the motor drive mode in the battery-based MMC. In this case, the circulating currents consist of only harmonic components. According to [PERSON]'s voltage law, the upper and lower voltages can be given by  
\[v_{xu}=\frac{V_{dc}}{2}-v_{x}-L_{arm}\frac{di_{xu}}{dt}-r_{arm}i_{xu} \tag{6}\]  
\[v_{xl}=\frac{V_{dc}}{2}+v_{x}-L_{arm}\frac{di_{xl}}{dt}-r_{arm}i_{xl} \tag{7}\]  
where \(v_{xu}\) and \(v_{xl}\) are the upper and lower voltages of phase \(x\), \(v_{x}\) is the output voltage, \(L_{arm}\) and \(r_{arm}\) give the inductance and resistance of the arm inductor, and \(V_{DC}\) is the DC-link voltage. By separately adding and subtracting Equations (6) and (7), and using the relationship in Equations (1) and (2), the outer and inner dynamic equations of MMC can be expressed as  
\[\frac{L_{arm}}{2}\frac{di_{x}}{dt}=-\frac{r_{arm}}{2}i_{x}+\frac{v_{xl}-v_{xu}}{2} \tag{8}\]  
\[i_{xu}=\frac{i_{x}}{2}+i_{cirx} \tag{1}\]0  
According to Equations (8) and (9), the output current \(i_{x}\) can be regulated by adjusting the voltage difference between the upper arm and the lower arm, and the circulating current \(i_{cirx}\) can be controlled by changing the sum of the upper arm and the lower arm voltages. These two driving voltages of the output current and circulating current can be defined as  
\[i_{xu}=\frac{i_{x}}{2}+i_{cirx} \tag{1}\]1  
\[i_{xu}=\frac{i_{x}}{2}+i_{cirx} \tag{1}\]2  
In addition, the dynamic equations of the submodule capacitor voltages are given by  
\[i_{xu}=\frac{i_{x}}{2}+i_{cirx} \tag{1}\]3  
\[i_{xu}=\frac{i_{x}}{2}+i_{cirx} \tag{1}\]4  
where \(C_{SM}\) is the capacitor capacitance, \(v_{SMuk}\) and \(v_{SMlk}\) are the submodule voltages in the upper and lower arms, and \(S_{SMuk}\) and \(S_{SMlk}\) represent the submodule gating signals, respectively. When \(S_{SMk}\)=1, the submodule is inserted, and the charging or discharging is realized according to the direction of current. When \(S_{SMk}=0\), the submodule is bypassed, and the capacitor voltage remains the same. Equations (8), (9), (12), and (13) provide a generalized dynamic model of the MMC. The specific dynamic modeling principle can be based on the perspective of either voltage-based control or energy-based control [79]. When the submodule capacitors are substituted by batteries as the energy source, the state of charge (SoC) management turns to the key factor correspondingly [80; 81].