# Circulating Current Reduction in MMC-HVDC System Using Average Model
## 2 MMC-HVDC System Structure
### MMC Control System  
The MMC-HVDC control system comprises the following components:  
1. System level (DC voltage control/active and reactive power);
2. Lower level control (circulating current control);
3. Capacitor voltage balancing control. The system level control can be implemented using the direct or vector control strategy [32], as shown in Figure 2. This paper implements the vector current control strategy, which consists of a cascaded loop structure, i.e., an outer control loop (power/voltage) and an inner control loop (current). The outer loop feeds the current references for the inner current controllers. The current controller introduces decoupled control of reactive and active powers by controlling its \(q\)-axis and \(d\)-axis current components. The circuit model of MMC (shown in Figure 1a) is connected to a symmetrical three-phase balanced grid with a frequency of \(\omega_{0}\). Its voltage and current equations are as follows:  
\[\text{V}_{\text{g(a)}}(\text{t})=\sqrt{2}Vsin(\omega_{o}t) \tag{13}\]  
\[\text{V}_{\text{g(b)}}(\text{t})=\sqrt{2}V\sin(\omega_{o}t-\frac{2\pi}{3}) \tag{14}\]  
\[\text{V}_{\text{g(c)}}(\text{t})=\sqrt{2}\,\text{V}\,\sin(\omega_{0}\text{t}+ \frac{2\pi}{3}) \tag{15}\]  
Figure 2: (a) Inner current control; (b) Outer loop power and voltage. \[\mathrm{I_{g(a)}(t)}=\sqrt{2}\:\mathrm{I}\:\sin(\omega_{o}t-\varphi) \tag{16}\]  
\[\mathrm{I_{g(b)}(t)}=\sqrt{2}\:\mathrm{I}\:\sin\biggl{(}\omega_{o}t-\frac{2\pi}{ 3}-\varphi\biggr{)} \tag{17}\]  
\[\mathrm{I_{g(c)}(t)}=\sqrt{2}\:\mathrm{I}\:\sin\biggl{(}\omega_{o}t+\frac{2\pi} {3}-\varphi\biggr{)} \tag{18}\]  
Each arm voltage can be regulated to get the required current. The output of each arm is half of the phase current. \[L\frac{d}{dt}i_{1u}(t)=-V_{up(a)}(t)-V_{g(a)}(t) \tag{19}\]  
\[L\frac{d}{dt}i_{1l}(t)=-V_{low(a)}(t)+V_{g(a)}(t) \tag{20}\]  
The expression for the DC side current in the upper and lower arms in three leg is:  
\[\text{V}_{\text{g(a)}}(\text{t})=\sqrt{2}Vsin(\omega_{o}t) \tag{13}\]0  
Three-phase AC current signals are transformed into DC signals, which is more appropriate for a control system design. The d-q current components in a rotating reference frame is based on the Park transformation as implemented in Equation (22) [33]. The desired d-q current quantities can be determined by the current magnitude and phase in a three-phase system. \[\left[ \begin{array}{c}\mathrm{i_{du}(t)}\\ \mathrm{i_{qu}(t)}\\ \mathrm{i_{ou}(t)}\end{array} \right]=\frac{2}{3}\left[ \begin{array}{c}\sin( \omega_{o}t)+\sin(\omega_{o}t-\frac{2\pi}{3})+\sin(\omega_{o}t+\frac{2\pi}{3} )\\ \cos(\omega_{o}t)+\cos(\omega_{o}t-\frac{2\pi}{3})+\cos(\omega_{o}t+\frac{2\pi} {3})\\ \frac{1}{2}\frac{1}{2}\frac{1}{2}\end{array} \right].\left[ \begin{array}{c} \mathrm{i_{g(a)}(t)}\\ \mathrm{i_{g(b)}(t)}\\ \mathrm{i_{(c)}(t)}\end{array} \right] \tag{22}\]  
By taking the derivative of Equations (19), (20) and (22), the equations in the d-q reference frame for the inner current dynamics are given as:  
\[L\frac{d}{dt}i_{ud}(t)-\omega_{o}Li_{ud}=-V_{ud}(t)-V_{d}(t) \tag{23}\]  
\[L\frac{d}{dt}i_{uq}(t)+\omega_{o}Li_{uq}=-V_{uq}(t)-V_{q}(t) \tag{24}\]  
According to Equations (23) and (24), direct and quadrature current axes of terms \(\omega_{o}Li_{ud}\) and \(\omega_{o}Li_{uq}\) are coupled. De coupling of these terms can be achieved by using control inputs \(V_{ud}\) and \(V_{uq}\) as shown in Figure 2a. The \(V_{ud}\) and \(V_{uq}\) are the reference voltages used to generate insertion indices for each leg. The outer controllers compare reference values, active power, DC voltage, reactive power \((\mathrm{P^{*}},\mathrm{V^{*}dc},\mathrm{Q^{*}})\), and measured values \((\mathrm{P},\mathrm{V_{dc}},\mathrm{Q})\) using PI controllers, as shown in Figure 2b. The PI compensators are tuned based on the open loop approach [34].