# Enhanced Efficient EMT-Type Model of the MMCs Based on Arm Equivalence
## 3 Equivalent Circuit of the MMC
### Equivalent Voltage of the Arm  
Similarly to the average capacitor current, the output voltage of the \(x\)th SM on the \(r\) arm in phase \(j\)\(v_{\text{sm},\,rj\_x}\) is expressed as:  
\[v_{\text{sm},rj\_x}=S_{rj\_x}v_{\text{C},rj\_x} \tag{16}\]  
Sum up the voltage of all \(N\) SMs, then:  
\[\sum_{x=1}^{N}v_{\text{sm},rj\_x}=\sum_{x=1}^{N}S_{rj\_x}v_{\text{C},rj\_x} \tag{17}\]  
Suppose that all \(N\) SMs on an arm are identical, and all SM capacitor voltages are perfectly balanced [27; 28], then the average capacitor voltage of each SM on the \(r\) arm in phase \(j\)\(v_{\text{C},\,rj}\) is expressed as:  
\[v_{\text{C},rj}=v_{\text{C},rj\_x} \tag{18}\]  
Substitute Equations (12), (17) into Equation (16), and Equation (16) is rewritten as:  
\[\sum_{x=1}^{N}v_{\text{sm},rj\_x}=S_{rj}\Big{(}Nv_{\text{C},rj}\Big{)} \tag{19}\]  
From Equation (18), the equivalent output voltage on the \(r\) arm in phase \(j\)\(v_{\text{EQ},\,rj}\) is defined as:  
\[v_{\text{EQ},rj}=S_{rj}\Big{(}Nv_{\text{C},rj}\Big{)} \tag{20}\]  
### Equivalent Voltage of the Arm  
Based on the relationship between the voltage and current of the capacitor, the \(x\)th SM capacitor current \(i_{\text{C},\,rj\_x}\) can also be written as:  
\[i_{\text{C},rj\_x}=S_{rj\_x}C_{0}\frac{dv_{\text{C},rj\_x}}{dt} \tag{21}\]Sum up the capacitor current of all \(N\) SMs on the specific arm, then:  
\[\sum_{x=1}^{N}i_{C,rj,x}=C_{0}\sum_{x=1}^{N}S_{rj,x}\frac{dv_{C,rj,x}}{dt} \tag{22}\]  
Combine with Equation (18), then  
\[\frac{1}{N}\sum_{x=1}^{N}i_{C,rj,x}=C_{0}\frac{dv_{C,rj}}{dt}\Bigg{(}\frac{1}{ N}\sum_{x=1}^{N}S_{rj,x}\Bigg{)} \tag{23}\]  
Substitute Equations (13), (14), (20) into Equation (23), and Equation (23) is rewritten as:  
\[i_{C,rj}=\frac{C_{0}}{N}\frac{dv_{\text{EQ},rj}}{dt} \tag{24}\]  
According to Equation (23), the arm equivalent circuit based on the average capacitor current are shown in Figure 5b, which models the charging and discharging of the arm equivalent capacitor \(C_{\text{EQI},\,rj}\), and can accurately represent the voltage ripples of the equivalent capacitor. The resistances \(R_{1}\) and \(R_{2}\) in Figure 5b,c are employed to take the losses of the power devices into account, whose values will be discussed later. In Figure 5b,c, the arm current mainly flows through the path marked in blue or purple color, and extremely weak current flows through \(R_{2}\) due to the very large \(R_{2}\).  
\(C_{\text{EQI},\,rj}\) is expressed as:  
\[C_{\text{EQI},rj}=\frac{C_{0}}{N} \tag{25}\]  
From (19), Equation (21) is re-derived as:  
\[\frac{C_{0}}{NS_{rj}}\frac{dv_{\text{EQ},rj}}{dt}=i_{\text{Arm},rj} \tag{26}\]  
According to Equation (26), the arm equivalent circuit based on arm current is replaced as in Figure 5c, which reflects the relationship between the arm equivalent output voltage and arm current, with an ASF-related equivalent capacitor \(C_{\text{EQ2},\,rj}\), and can exhibit the dynamics of the arm currents.  
Figure 5: The arm equivalence: (**a**) The general structure of the arm; (**b**) The arm equivalent circuit based on average capacitor current; (**c**) The arm equivalent circuit based on arm current.  
In Figure 5c, the circuit \(i^{\prime}_{\text{Arm},rj}\) is very approximate to the arm current \(i_{\text{Arm},rj}\) due to the \(R_{2}\) shunt effect. \(C_{\text{EQ2},\,rj}\) is defined as:  
\[C_{\text{EQ2},rj}=\frac{C_{0}}{NS_{rj}} \tag{27}\]  
As shown in Figure 5b,c, compared with DSMs, the arm equivalence could drastically reduce the size of the nodes in the entire circuit and the computational intensity. The application for two arm equivalent circuits will be discussed in next Section.