In a HVDC transmission system, the MMC can realize the AC-DC power conversation. Figure 1 shows the control scheme of the MMC with the outer DC voltage closed loop, inner current closed loop, and circulating current control loop.

As shown in Figure 1, u _a , u _b , and u _c are the AC system voltage; i _a , i _b , and i _c are the AC system current; θ is the phase angle of three-phase grid voltage; u _dc is the voltage of the DC bus; u _dc-ref is the voltage reference of the DC bus; u _d and u _q are the d-axis voltage and q-axis voltage, respectively; i _d and i _q are the d-axis current and q-axis current, respectively; u _{a_ref} , u _{b_ref} , and u _{c_ref} are the output midpoint voltage reference of the bridge arm; i _au , i _bu , and i _cu are the current flowing through the upper bridge arm; i _al , i _bl , and i _cl are the current flowing through the lower bridge arm; u _{adiff_ref} , u _{bdiff_ref} , and u _{cdiff_ref} are the voltage drops of the bridge arm reactance, which are caused by a circulating current; u _{au_ref} , u _{bu_ref} , and u _{cu_ref} are the voltage reference of the upper bridge arm; and u _{al_ref} , u _{bl_ref}, and u _{cl_ref} are the voltage reference of the lower bridge arm.

L in Figure 1 is expressed as: L = L s + L a r m 2 (1) where L _s is the interface inductance between the MMC and grid, and L _arm is the arm inductance.

The voltage references of the upper bridge arm and lower bridge arm are expressed as: u a u _ r e f = u d c 2 − u a _ r e f − u a d i f f _ r e f u b u _ r e f = u d c 2 − u b _ r e f − u b d i f f _ r e f u c u _ r e f = u d c 2 − u c _ r e f − u c d i f f _ r e f u a l _ r e f = u d c 2 + u a _ r e f − u a d i f f _ r e f u b l _ r e f = u d c 2 + u b _ r e f − u b d i f f _ r e f u c l _ r e f = u d c 2 + u c _ r e f − u c d i f f _ r e f (2)

As shown in Figure 1, θ has an important influence on the control system of the MMC. The three-phase grid voltage is assumed as: u a t = ∑ h = 1,5,7 … … U h + ⁡ cos h ω 1 t + θ h + + U h − ⁡ cos − h ω 1 t + θ h − u b t = ∑ h = 1,5,7 … … U h + ⁡ cos h ω 1 t + θ h + − 2 3 π + U h − ⁡ cos − h ω 1 t + θ h − + 2 3 π u c t = ∑ h = 1,5,7 … … U h + ⁡ cos h ω 1 t + θ h + + 2 3 π + U h − ⁡ cos − h ω 1 t + θ h − − 2 3 π (3) where U _h ⁺ and U _h ⁻ are the amplitude of the positive-sequence component and negative-sequence component, respectively, θ _h ⁺ and θ _h ⁻ are the phase angle of the positive-sequence component and negative-sequence component, respectively, h is the harmonic order, and ω ₁ is the angular frequency of the fundamental harmonic.

With the regard to the DC component, the PLL applies the moving average filter to eliminate its influence, and with regard to the even-order harmonic component, the amplitudes are much lower. Thus, (3) neglects the DC component and even-order harmonic component.

By applying the Clarke transformation, (3) can be expressed as: u α t = ∑ h = 1,5,7 … … U h + ⁡ cos h ω 1 t + θ h + + U h − ⁡ cos − h ω 1 t + θ h − u β t = ∑ h = 1,5,7 … … U h + ⁡ sin h ω 1 t + θ h + − U h − ⁡ sin − h ω 1 t + θ h − (4)

By applying the Park transformation, (4) can be expressed as: u d t = ∑ h = 1,5,7 … … U h + ⁡ cos h ω 1 t + θ h + − θ ^ 1 + + U h − ⁡ cos − h ω 1 t + θ h − + θ ^ 1 + u q t = ∑ h = 1,5,7 … … U h + ⁡ cos h ω 1 t + θ h + − θ ^ 1 + − U h − ⁡ cos − h ω 1 t + θ h − + θ ^ 1 + (5)

Once the output of PLL is θ 1 + , u _q (t) is expressed as: u q t = U 1 + θ 1 + − θ ⌢ 1 + + f ′ (6) where f ^’ is the high-frequency disturbance.

Thus, the transfer function of the SRF-PLL with the first-order low-pass filter is expressed as: G s = U 1 + K p s + K i s 2 T l f s + 1 (7) where K _p and K _i are the proportional coefficient and integral coefficient, respectively, and T _lf is the time constant of the first-order low-pass filter.

According to (7), the bandwidth of the SRF-PLL is highly related with U ₁ ⁺. Thus, voltage sags will affect the stability of the SRF-PLL.

The SRF-PLL usually applies the PI controller instead of the filter. Thus, the dynamic speed of the SRF-PLL is relatively low. In (Golestan et al., 2014), the trap filter is applied to eliminate the specific-order harmonic, while its performance is insufficient to meet the time-varying three-phase grid. In (Golestan et al., 2015b; Hamed Hany et al., 2016), the second-order generalized integrator is applied in the PLL, which lacks the suppression of the DC component. Thus, the PLL will result in fundamental frequency oscillations and phase angle oscillations.

The transfer function of the moving average filter is expressed as: G m a f s = 1 − e − T ω s T ω s ≈ 1 T ω s 2 + 1 (8) (8) shows that the moving average filter is equivalent to an infinite number of cascaded trap filters. It can eliminate any integer-order harmonic component by setting the T _ω . Furthermore, compared with the PID controller, trap filter, and second-order generalized integrator, the moving average filter is much easier to implement in software.

The single degree of the freedom PID controller cannot meet the needs of both the tracking performance and anti-disturbance performance. Figure 2 shows a PLL with the 2DoFs-PID controller.

As shown in Figure 2, H _C1(s) is the transfer function of the PID controller in the feedback loop, and H _C2(s) is the transfer function of the PID controller in the feed-forward loop. H _C1(s) and H _C2(s) are expressed as: H C 1 s = α K P + K I s + β K D s H C 2 s = 1 − α K P + 1 − β K D s (9) where K _P , K _I , and K _D are the proportional coefficient, integral coefficient, and differential coefficient, respectively, and α and β are the coefficients of the two degrees of freedom.

As shown in Figure 2, the transfer function between the input signal and output signal is expressed as (10), and the transfer function between the disturbance signal and output signal is expressed as (11): H YR s = H C 1 s H p s 1 + H C 1 s + H C 2 s H p s (10) H YR s = H p s 1 + H C 1 s + H C 2 s H p s (11)

Figure 3 shows the structural schematic diagram of the 2DoFs-PLL with the moving average filter. In order to improve the performance of the tracking acceleration signal, an extra integrator is applied in the forward path. Thus, there are a total of three integrators in the improved 2DoFs-PLL.

Based on the internal model principle, the 2DoFs-PLL can track an acceleration signal without a structurally steady error.

The extra integrator results in a smaller phase margin, and the moving average filter introduces an additional pole, which will impact the stability of the 2DoFs-PLL. Thus, to eliminate the influence of the extra integrator and moving average filter, a compensator with a pole and a zero is introduced. Its transfer function is expressed as: H e x s = 1 + T ω 1 2 s s (12)

The structural diagram of the improved 2DoFs-PLL is shown in Figure 4.

Thus, the closed-loop transfer function of the improved 2DoFs-PLL is expressed as: H YR s = β K D s 2 + α K P s + K I s 3 + K D s 2 + K P s + K I (13) H YR s = s s 3 + K D s 2 + K P s + K I (14)

Based on (14), if K _P >0, K _I >0 and K _D >0 is satisfied, the improved 2DoFs-PLL is stable.

Moreover, according to (10) and (11), H _C1(s) + H _C2(s) is equivalent to the traditional PID controller. In terms of integrated time and absolute error, (13) should be expressed as: H Y R s = 2.97 β ω 0 s 2 + 4.94 α ω 0 2 s + ω 0 3 s 3 + 2.97 ω 0 s 2 + 4.94 ω 0 2 s + ω 0 3 (15) where ω0 is the bandwidth of the PLL.

Thus, according to (13) and (15), KP, KI, and KD are achieved.

Then, based on the integrated time and absolute error and HYR(s), the parameters α and β are achieved.

To verify the performance of the improved 2DoFs-PLL, a simulated model is built, and the parameters are shown in Table 1.

According to (13) and (14) and Table 1, the settling time is 0.03s, the gain margin is 26.5dB, and the phase angle margin is 54°.

Figure 5 shows the response diagrams of the disturbance signals and reference signals.

As shown in Figure 5A, the improved 2DoFs-PLL can eliminate the influences of the disturbance signals on the output. The responses of the step disturbance signal, constant speed disturbance signal, and constant speed disturbance signal are relatively small. The unit step response is approximately equal to 5E-06. The responses of the constant speed disturbance signal and constant speed disturbance signal can even be ignored.

Furthermore, the improved 2DoFs-PLL shows a rapid responsibility of the step disturbance signal, constant speed disturbance signal, and constant speed disturbance signal. The decay time is about 0.06 s, which is equal to three power-frequency periods.

As shown in Figure 5B, the improved 2DoFs-PID can track the step reference, constant speed reference, and constant acceleration reference without the steady-state error, and for the step reference, the response is without overshoot.