## 2 The Equivalent Model of MMC Considering AC Side Power and MMC Control
### The Equivalent MMC Model Considering AC Active Power and MMC Controllers  
In the generic RMS model, the equivalent MMC model is shown in Figure 2[24]. The behavior of the MMC on the DC side is equal to a controlled current source \(i_{\rm s}\). The effect of \(p_{\rm ac}\) is reflected by the controlled current source. The part of \(p_{\rm CONV}\) estimated by the terminal DC voltage is reflected in the derivative of energy of the equivalent capacitance. Similarly, the power from the derivative of energy of the equivalent inductance and equivalent resistance is \(p_{\rm LC}\). The expression of the controlled current source, the equivalent capacitance, inductance and resistor is given in Equation (2).  
\[i_{s}=\frac{p_{\rm ac}}{u_{c}},L_{\rm eq}=\frac{2}{3}L_{\rm arm},R_{\rm eq}= \frac{2}{3}R_{\rm arm},\ C_{\rm eq}=\frac{6C_{0}}{N} \tag{2}\]  
where \(p_{\rm ac}\) is power transmitted from the AC side. \(u_{\rm c}\) is the voltage across the equivalent capacitor.  
However, the controlled DC voltage is absent in this model, that is, \(p_{\rm CONV}\) related to the DC control mode is not involved. Since MMC controllers take effect during the fault, the control mode would affect the output voltage on the AC side of MMC and then the transient characteristics of the DC side. However, the control strategy of the DC side  
Figure 1: Basic structure of MMC.  
could be implemented via the controlled DC voltage as a constituent component of the controlled arm voltages. That is, taking into account the control mode in computing the output voltage \(u_{\mathrm{v}}\), the unaccounted power could be compensated by this improvement.  
According to the nearest level modulation (NLM) strategy [25; 26], MMC AC output voltage could be calculated as follows.  
\[\left\{\begin{array}{c}u_{\mathrm{v}j}(t)=\frac{m}{2}v\\ con\\ \\ *(t_{0})sin\big{(}\omega t+\varphi_{j}(t_{0})\big{)}_{dc}\underset{v_{\mathrm{ cam}}=\sqrt{\frac{\varphi_{\mathrm{ref}}^{2}(t_{0})}{2\omega t^{\prime}(t_{0})}}}{v_{ \mathrm{cam}}=\sqrt{\frac{\varphi_{\mathrm{ref}}^{2}(t_{0})}{2\omega t^{\prime }(t_{0})}+\varphi_{\mathrm{ref}}^{2}(t_{0})}}\{ \tag{3}\]  
where, \(m\) is the modulation ratio of the converter. \(\omega\) is the angular frequency of the AC side. \(v_{\mathrm{ref}}^{*}\) and \(v_{\mathrm{ref}}^{*}\) are the outputs of the inner loop current controller.\(\varphi_{j}\) is the phase difference between the MMC AC output voltage and the AC voltage in phase \(j\). \(t_{0}\) represents the last moment.  
Then the power injected into the converter from the AC side could be computed as follows:  
\[p_{ac}=\frac{3u_{sj}u_{vj}}{X_{s}+0.5X_{arm}sin\varphi_{j}} \tag{4}\]  
From Equations (3) and (4) and Figure 2, it could be seen that the DC control mode affects the MMC AC output voltage and then the AC active power. In fact, the AC active power and MMC control influence the discharge of submodule capacitors and thus the DC transient voltages and currents, which could roughly be estimated as follows.  
It is assumed there is an error \(\Delta p_{\mathrm{ac}}\) in estimating the AC active power. Then the error in estimating the controlled current source would be \(\Delta i_{\mathrm{s}}=\Delta p_{\mathrm{ac}}/u_{\mathrm{c}}\). Here, it is assumed that the voltage of the capacitors is equal to the DC voltage terminal of the MMC. Then the error in node voltages would be  
\[\begin{array}{c}\underset{\frac{u_{\mathrm{c}}}{C_{\mathrm{eq}}-\frac{ \Delta u_{\mathrm{dc}}}{C_{\mathrm{eq}}-\frac{\Delta p_{\mathrm{ac}}}{C_{ \mathrm{eq}}-\frac{\Delta p_{\mathrm{ac}}}{C_{\mathrm{eq}}}}}}{u_{dc}}}{u_{dc}} \\ \Delta\quad\quad\quad\quad\quad\quad\quad\quad\quad  
one kA/ms. Then in several ms after the fault, the maximum error in \(i_{L}\) would be about one kA.  
In summary, by combining the control strategy with the controlled current source, the power balance is well expressed. And not only the branch currents but also the node voltages could be estimated accurately by the improved equivalent MMC model shown in Figure 2.