# Circulating Current Reduction in MMC-HVDC System Using Average Model
## 5 Direct and Proposed Direct Modulation  
Figure 3 shows as angle phase equivalent circuit model based on an average model developed in Equation (43). The purpose of the modulation scheme is to provide insertion indices for all the arms of an MMC to achieve control system objectives. The insertion indices \(n_{u}\) and \(n_{l}\) determines the number of SMs inserted in each arm out of n number of SMs, having a continuous interval between 0-1. The inserted voltages can be varied between 0 and \(V_{u}^{\Sigma}\). The direct modulation is based on the sinusoidal PWM generation; here, it is implemented with the vector current control method, which regulates inner-generated voltage \(e_{a}\) as given in Equation (5),  
Figure 3: Single phase equivalent circuit. since it assumes an average capacitor voltage for all the SMs, but actually due to current flow capacitor voltage and its energy changes. The arm voltages are controlled by two complementary voltage references for upper and lower arms:  
\[\mathrm{V_{upa\_ref}}=\frac{\mathrm{V_{dc}}}{2}[1-\mathrm{mcos}(\omega_{0}t)]-e _{a} \tag{44}\]  
\[\mathrm{V_{lowa\_ref}}=\frac{\mathrm{V_{dc}}}{2}[1+\mathrm{mcos}(\omega_{0}t)] -\ e_{a} \tag{45}\]  
The term \(m[m=\frac{\mathrm{I_{1}}/2}{\mathrm{I_{dc}}/2}]\) represents the modulation index (amplitude), and \(\omega_{0}\) is angular frequency of the output voltage. If \(n\) is the number of utilized SMs in each arm, then voltage output has n + 1 level [39]. Since the duty ratio of the SMs in each arm is equal to its insertion index, voltage and capacitor currents are directly proportional to each arm insertion index. Since \(V_{C}^{\sum}(t)=\mathrm{V_{dc}}\), the circulating current is generated as per Equations (32) to (34). The circulating current is not controlled in the direct modulation approach. Traditionally, compensated modulation is used to control arm voltage oscillations generated in the direct modulation method. The insertion-index selection accounts for the capacitor voltage ripples, but its drawback is the delay in the measurement of the output current and DC link voltage [40; 41]. The traditional approach of controlling the MMC-HVDC system consists of an AC side current control system and a separate circulating current suppression controller, which regulates unbalanced voltages inside each leg of a three-phase MMC converter. Both controllers operate in a cascade manner, which increases system complexity [42]. Unlike Reference [42], which set the inner differential current equal to zero in order to reduce capacitor voltage oscillations, a differential current controller is proposed in References [43; 44], which regulates upper and lower arm capacitor energies in one phase of MMC. In this paper, AC side current is controlled according to the active and reactive power references based on Equations (28) and (29). However, unlike Reference [45], which uses two separate PR controllers for controlling the AC side current and arm current of a converter during unbalanced voltage conditions, this paper proposes a new method of reducing the circulating current by the addition of 2 nd and 4 th harmonics in each arm current of an MMC, which reduces arm voltage oscillations due to reduction in excessive negative currents in a period. The waveform is shifted 120\({}^{\circ}\) and 240\({}^{\circ}\) for the other two legs. As upper and lower arm voltages have a constant DC voltage component in each leg during part of a period, the capacitor energy variations also reduce, thus the flow of circulating current decreases in each leg of a three-phase MMC. The voltage references in the upper and lower arms are generated by a vector current control using Equations (23) and (24).