# A Review of Modular Multilevel Converters for Stationary Applications
## 3 Mathematical Modeling and Control of Three-Phase MMC
### Nonlinear and Predictive Control  
The MMC as a switching converter is a nonlinear system associated with multiple coupled variables. The conventional control methods are easy to implement in digital controllers by using linear techniques. However, these methods have a limitation on the dynamic response in terms of multiple control objectives such as AC current and voltage, leg/arm voltage, and circulating currents. To achieve simultaneous control and dynamic response, nonlinear and predictive control strategies are promising alternatives for MMC applications.  
A nonlinear decoupling control based on feedback linearization [109] is proposed for MMC converters with the remaining linearized variables and linear controller to achieve desired transient responses and stability. Another nonlinear control method based on sliding mode control [110] is investigated with the faster dynamic response by splitting the state space into several subspaces, which are further controlled in an individual control structure. Passivity-based control [111; 112],  
Figure 14: Circulating current control: (**a**) Decoupling control in the \(dq\)-frame; (**b**) Proportional resonant control; (**c**) Quasi proportional resonant control.  
belonging to the nonlinear control strategies is applied to the MMC system based on an energy function to acquire a better dynamic and steady-state performance. The passivity-based control and sliding mode control are further combined in the MMC control system to improve the transient performance and robustness to system changes [113]. A nonlinear back-stepping control scheme [114], based on energy equations, is explored for single- and three-phase MMC by using Lyapunov theory with a simplified control structure in the _abc_ frame. The back-stepping method is armed with the inherent feature of circulating current suppression and validated in simulation and experiments showing good dynamic response and robustness.  
Predictive control, often referred to as model predictive control (MPC), is a preferable choice for power converters as it is a nonlinear control method. The MPC method can deal with multiple constraints in a single cost function and provide fast dynamic response and robustness against parameter variation and external disturbance. The finite control set MPC (FCS-MPC), also known as direct MPC, is a popular predictive approach used in MMC control. The implementation of FCS-MPC requires a discrete-time model of the MMC system which can be obtained from the discretization of the continuous-time model. The conversion from continuous-time to discrete-time can be achieved by using the forward Euler approximation or backward Euler approximation. A flow chart of the FCS-MPC [115] is depicted in Figure 15. One disadvantage of the FSC-MPC is the computational burden to obtain the optimal switching states, which worsens with increasing voltage levels. To deal with this issue, a fast FCS control scheme is investigated. Fast FCS-MPC [116], also known as indirect MPC, remains the submodule balancing strategy along with the predictive control. The predictive control is employed as the primary controller to control the output and circulating currents. Meanwhile, the submodule balancing strategy is applied as the secondary controller to equalize the submodule voltages or SoCs within the arm. The computational effort of the indirect FCS-MPC is minimized to a large extent, as only N+1 switching states are required to be evaluated for the optimal cost function.  
Figure 15: Flowchart of finite control set model predictive control.