# A Review of Modular Multilevel Converters for Stationary Applications
## 3 Mathematical Modeling and Control of Three-Phase MMC
### Submodule Balancing Control  
The submodule balancing control in the MMC mainly involves leg/arm voltage balancing control, submodule capacitor voltage balancing control, or SoC balancing control if based on a battery submodule. Various submodule balancing schemes have been discussed and presented in the literature. These approaches are mainly based on individual voltage/SoC control or a sorting algorithm to  
Figure 10: Output voltage and current control under distorted grid condition: (**a**) in \(\alpha\beta\)-\(dq\) frame; (**b**) in \(\alpha\beta\) frame. equilibrate between submodules. Generally, when a small number of submodules are explored in the converter, the individual voltage/SoC control can be applied to realize the individual voltage/SoC regulation; otherwise, the sorting algorithm is preferred in the application with increased submodules by selecting the inserted/bypassed submodules in order of priority [4]. The individual balancing control and sorting control are also well-known as distributed balancing control and centralized balancing control [5; 93]. Due to the inconsistency of actual submodule parameters and the charging and discharging influence, the submodule capacitor voltage is likely to deviate from the setpoint. In addition, the frequent switching states also contribute to the capacitor voltage ripple. The inconsistent submodule capacitor voltage results in unequal arm and leg voltage, which will further lead to a circulating current increasing the power losses and worsening the submodule voltage balancing [3; 75]. The individual submodule balancing can be achieved by using a closed-loop controller for each cell [94; 95]. The leg voltage balancing can be achieved by involving PI controllers as depicted in Figure 10(a). The average submodule voltage is compared with the setpoint and manipulated by a PI controller outputting the DC component of circulating currents reference. The actual DC component of circulating currents can be obtained through a low-pass filter (LPF) or by averaging the upper arm and lower arm current. The individual voltage balancing algorithm regulates each submodule voltage with a PI controller separately (Figure 10(b)), and thus, the value of the individual control signal is relatively small. The signed block donates the direction of the current. The submodule balancing control can be realized by using either the above-mentioned methods referred to as voltage-based control or the approaches involving energy-based control [79]. When the submodule capacitors are substituted by batteries as the energy source, the control objective changes to SoC balancing control, correspondingly. SoC balancing control involves SoC balancing between legs, arms, and individuals [80; 81]. The control algorithm of SoC balancing resembles that of capacitor voltage control except for the SoC calculation. The arm SoC balancing control is realized by controlling the fundamental component of circulating currents shown in Figure 11(b). The average SoCs of the upper and lower arms are compared and regulated by a PI controller yielding the amplitude of the fundamental component. The fundamental component is in-step phase with the reference and controlled by a proportional resonant (PR) controller to generate the arm SoC balancing control signal. On the basis of leg SoC and arm SoC balancing, a further individual SoC balancing control is conducted on each submodule battery, as demonstrated in Figure 11(c). Apart from employing PI controllers to realize the voltage balancing control or SoC balancing control, another widely used method refers to sorting balancing control [75; 77], as illustrated in Figure 13. In the first step, the instantaneous value of the submodule capacitor voltage/battery SoC is measured, and the current direction is defined. According to the actual value, the submodule capacitor voltage/battery SoC is compared in a logic comparator to acquire an index number and further arranged in ascending or descending order depending on the current direction.