## 3 SRPC Mathematical Analysis
### Deadbeat Predictive Current Control  
Unlike the PI and PR controllers that compile integral errors, deadbeat predictive control is a technique that attempts to extract the required control action during each sample interval through calculations based on the circuit model of the system being controlled. The accuracy of the equivalent circuit is a significant factor to maintain the system output values equal to the reference signals [25]. Deadbeat predictive current control methodology can be easily programmed, and it is used when a fast-dynamic response is required. However, this control has some disadvantages in terms of existing errors due to the model assumptions and the unmodeled delay introduced by calculation time and modulation [25]. In this context, according to [28], the current control strategies with the smallest tracking error were the synchronous PI, the feedforward in [29], the sliding mode in [30], and the deadbeat predictive control proposed in [31]. Therefore, reference equations based on the deadbeat predictive control in [31] have been extracted in this paper using a low computational cost after neglecting the semiconductor conduction and switching losses. This current control strategy was selected since the results in [31] have a small steady-state error and a fast transient response. Further steps are applied for the upper arm of section \(x\) converter. The deadbeat predictive current control methodology used in this paper can be defined after knowing the error current of the upper arm of section \(x\) converter as in (15). \[i_{xu\_err}=\ \dot{i}_{rxu}-i_{rxu} \tag{15}\]  
Due to the low-voltage drops across the internal resistors of the MMC arm inductors, the voltage drop across the MMC arm resistors is neglected to simplify the calculations, without introducing significant error. By substituting the first Equation of (14) in (15):  
\[u_{x}=\ -L_{xu}\frac{d(i_{rxu}^{*}-i_{rxu\_err})}{dt}\ -u_{xu} \tag{16}\]  
By separating the differential part of Equation (16), the upper arm voltage of section \(x\) converter can be presented as in Equation (17). \[u_{xu}=\ -L_{xu}\frac{d\ \dot{i}_{rxu}^{*}}{dt}+L_{xu}\frac{d\ \dot{i}_{rxu\_err}}{dt}-u_{x} \tag{17}\]  
It is feasible to assume a linear variation of \(i_{rxu\_err}(k)\) with the time, where \((k)\) denotes the value at the present sample interval and the next sample interval is denoted as \((k+1)\). According to Equation (18), an assumption of \(\dot{i}_{xu}(k)\) derivative in function to the variables value is possible at the instant of \((k)\) and \((k-1)\). Then, it is feasible to get Equation (19), where \(T_{s}\) refers to the sampling period. \[\frac{dx}{dt}\ \cong\ \frac{x\ (k)-x\ (k-1)}{T_{s}} \tag{18}\]  
\[u_{xu}(k)=-\frac{L_{xu}}{T_{s}}[\ \dot{i}_{rxu}(k)-i_{rxu}^{*}(k\ -1)-i_{rxu\_err}(k)]-u_{x}(k) \tag{19}\]  
By substituting Equation (15) in (19):  
\[u_{xu}(k)=\frac{L_{xu}}{T_{s}}[-2\ \dot{i}_{rxu}^{*}(k)+\dot{i}_{rxu}^{*}(k-1)\ +i_{rxu}(k)]-u_{x}(k) \tag{20}\]The SRPC control algorithm should calculate the compensation current references, \(\vec{i_{rx}}\), and \(\vec{i_{ry}}\), as in Equation (10), respectively. Therefore, Equation (20) as it is does not serve that purpose, since it computes the compensation current references of the upper and the lower MMC arms, \(\vec{i_{rxu}}\), and \(\vec{i_{ryu}}\). The main purpose of this control is to have a near zero circulating current, \(i_{circx}\), and \(i_{city}\), in the MMC phases, then from Equation (13), the circulating current is significantly reduced when the upper and the lower arm currents are complementary as in Equation (21). \[i_{circx}=\frac{i_{rxu}+i_{rxl}}{2}\ \cong\ 0\ \Rightarrow\ i_{rxu}\cong-i_{ rxl} \tag{21}\]  
By considering the aforementioned assumptions:  
\[ \begin{array}{l}i_{rxu}=-\frac{i_{rx}}{2}\\ i_{rxl}=\ +\ \frac{i_{rx}}{2}\end{array}