# A Review of Modular Multilevel Converters for Stationary Applications
## 3 Mathematical Modeling and Control of Three-Phase MMC
### Output Voltage and Current Control  
The MMC control strategies enable its high performance associated with superior safety, reliability, and efficiency. The control is challenging and sophisticated as involving substantial submodules and multiple control objectives related to the output voltage and current control, submodule voltage or SoC balancing control, and circulating current control. The MMC AC terminal can be connected to a renewable energy system, e.g., a wind turbine farm, or a high-voltage transport grid or medium-voltage distribution grid. Assuming the MMC is connected to the grid, as shown in Figure 7, the control of its output voltage and current vary under different grid conditions such as a balanced grid, unbalanced grid, and distorted grid [4]. In most cases, the AC side is assumed to interface a balanced utility only involving a fundamental component. The MMC output voltage and current control can be achieved by using the outer voltage control and inner current control. The inner current control is required to satisfy the command of active and reactive power based on the instantaneous power theory coordinated in a reference frame [84,85]. Generally, in the synchronous rotating \(dq\)-frame, assuming the \(d\) axis is completely aligned with the rotating vector of source voltage, the active power and reactive power can be expressed by  
\[P=\frac{3}{2}v_{\_}sqi_{\_}d \tag{27}\]  
\[Q=-\frac{3}{2}v_{\_}sqi_{\_}q \tag{28}\]  
where \(P\) and \(Q\) are the active and reactive power, respectively, \(v_{\_}sd\) and \(v_{\_}sq\) represent the \(d\) component and \(q\) component of the source voltage, and \(i_{\_}d\) and \(i_{\_}q\) give the reference currents \(i_{\_}d^{*}\) and \(i_{\_}q^{*}\) in \(dq\) axes. Using Equations (27) and (28), the active and reactive power can be managed by regulating \(i_{\_}d\) and \(i_{\_}q\) independently. The output voltage in \(dq\)-axes can be expressed as  
\[v_{\_}d=v_{\_}sd+L_{\_}s\frac{di_{\_}d}{dt}+r_{\_}si_{\_}d-\omega L_{\_}si_{\_}d \tag{29}\]  
\[v_{\_}q=v_{\_}sq+L_{\_}s\frac{di_{\_}q}{dt}+r_{\_}si_{\_}q+\omega L_{\_}si_{\_}d \tag{30}\]  
Figure 7: Grid-connected MMC. where \(v_{d}\) and \(v_{q}\) represent the \(d\) and \(q\) components of the MMC AC-side voltage, \(L_{s}\) and \(r_{s}\) donate the inductance and resistance at the grid side, and \(\omega\) is the fundamental angular frequency. In addition, the DC-link voltage should maintain at the setpoint, which can be realized by the power balance between the AC terminal and DC terminal. \[P=\frac{3}{2}v_{sd}i_{d}=V_{dc}i_{dc} \tag{31}\]  
From Equation (31), the DC voltage can be used to produce the reference current in the \(d\) component. Therefore, the control block of the output voltage and current is demonstrated in Figure 8.  
When unbalanced loads are connected or main failures occur in the grid, the grid voltage is unbalanced, resulting in an invalid injection of active power and limited restoration of reactive power. The output voltage and current must be controlled with independent control loops regarding positive and negative sequence grid components. A method is referred to as Delayed Signal Cancellation (DSC) [86]; it facilitates the calculation of the positive and negative sequence components. The control scheme of the output voltage and current control under unbalance grid conditions is depicted in Figure 9 [87]. The positive and negative sequence currents in \(d\) components are regulated with the effect of active power, and the command values are set to zero, as the active power ought to be zero. The positive and negative sequence voltages are controlled in the manner of reactive power by using PI controllers. The voltage references for positive and negative sequence are set to the grid voltage nominal value and zero, respectively. Distorted grid is another abnormal condition of the grid system in which the grid voltage involves fundamental component and multiple harmonics.