# Voltage Ride through Control Strategy of Modular Multilevel Converter under Unbalanced Voltage Sag
## 2 Modeling of the Grid-Connected Modular Multilevel Converter  
The classic formation of an MMC that is integrated to photovoltaic (PV) units as a renewable energy source is presented in Figure 1. To explore the dynamic behavior features of a GC-MMC, a precise model is needed. In fact, the developed model in this paper is based on the average modeling technique which has been popularly used in many research papers, i.e., the accelerated switching function model of MMCs used in Reference [18]. This model improves the simulation speed significantly and the Central Processing Unit (CPU) running time is reduced by a third for the tested two-terminal HVDC-link. This can provide significant benefits when studying large HVDC networks that involve many converter stations, as well as various operating conditions, including normal operation, AC fault, DC fault, etc. The mathematic equations of an N-cell MMC are described as follows [14]:  
\[\begin{array}{l}\frac{d_{itk}}{dt}=\frac{1}{L}\left[\frac{V_{d_{c}}}{2}-\sum \limits_{  
characteristics of sliding mode control (SMC) and the adaptive nature of fuzzy systems, it was possible to combine fuzzy control and SMC. This combination was employed to reduce the chattering problem of SMC and the results were found to be effective. A complementary control strategy for a voltage source converter based on positive and negative symmetrical components was introduced in Reference [19] to investigate the voltage sag ride-through. In this paper, this idea was extended for the MMC and is explained as follows:  
The first part in the planning of the fuzzy sliding control strategy was describing the current tracking errors, which were realized for positive and negative sequence components as follows:  
\[\begin{array}{l}\begin{array}{l}\begin{array}{l}e_{d_{p}}^{u}=(i_{d_{p-ref}}^ {u}-i_{d_{p}}^{u})\\ e_{dn}^{u}=(i_{d_{n-ref}}^{u}-i_{dn}^{u})\\ e_{qr}^{u}=(i_{qp-ref}^{u}-i_{qp}^{u})\\ e_{qn}^{u}=(i_{qn-ref}^{u}-i_{qp}^{u})\\ e_{d_{p}}^{i}=(i_{d_{p-ref}}^{i}-i_{dp}^{i})\\ e_{dn}^{l}=(i_{d_{n-ref}}^{l}-i_{dn}^{l})\\ e_{d_{p}}^{i}=(i_{d_{p-ref}}^{j}-i_{dp}^{j})\\ e_{qn}^{l}=(i_{qn-ref}^{l}-i_{qn}^{l})\\ \end{array}\end{array} \tag{2}\]  
Figure 2: The proposed control strategy for the grid-connected MMC (GC-MMC).  
where \(i_{dp}^{u}\), \(i_{dn}^{u}\), \(i_{qp}^{u}\), \(i_{qp}^{u}\), \(i_{dp}^{l}\), \(i_{dn}^{l}\), \(i_{qp}^{l}\) and \(i_{qp}^{l}\) are currents of symmetrical \(dq\) components which are defined for reference, upper and lower currents, respectively.  
Commonly, after modulation, the next step is the activation and deactivation of a number of SMs. Therefore, an algorithm was developed for sorting the capacitor voltages of SMs. On that account, the voltage error between the SM capacitor voltages one step ahead and their references should be considered as follows:  
\[e_{j}=\left|V_{c,j}(t+T_{s})-V_{c,ref}\right| \tag{3}\]  
where \(V_{c,ref}\) is the reference SM capacitor voltage.  
Then for each SM, the estimated capacitor voltage \(V_{c,j}\) was obtained by the formula below:  
\[V_{c,j}(t+T_{s})=V_{c,j}(t)+\frac{i_{kj}}{C}T_{s} \tag{4}\]  
where \(i_{kj}\) is the arm current and \(T_{s}\) is the control time period.  
Therefore, the concluding vector for forming the error matrix can be described as follows:  
\[e=\left[\begin{array}{c}e_{dp}^{u}\\ e_{qp}^{l}\\ e_{dn}^{u}\\ e_{pa}^{l}\\ e_{qp}^{l}\\ e_{dn}^{l}\\ e_{pa}^{l}\\ e_{p}^{l}\\ e_{pa}^{l}\\ e_{j}^{l}\end{array}\right] \tag{5}\]  
Then the sliding surface for the nonlinear system was written in Reference [18] as:  
\[s(X(t),t)=\lambda e \tag{6}\]  
where \(\lambda\) is a strict positive parameter which is regulated by the systems' bandwidth.  
From this stage, the proposed control strategy was implemented as shown in Figure 2. The details of the fuzzy sliding mode control strategy are explained in Reference [20].