Sequence Impedance Modeling and Optimization of MMC-HVDC Considering DC Voltage Control and Voltage Feedforward Control

Abstract

The dynamic performance of the DC bus significantly influences the impedance characteristics of MMC and the system stability in a high-voltage direct current system. However, most of the existing MMC-HVDC system stability research simplifies the DC side as an ideal voltage source and ignores the impacts of voltage feedforward control, which affects the accuracy and practicability of stability analysis. In this paper, a sequence impedance model considering both DC voltage control and voltage feedforward control is developed, and the necessity of considering DC control and voltage feedforward control for MMC-HVDC stability analysis is illustrated. Then, the impact of control parameters on MMC-HVDC impedance is discussed, and the boundary conditions of control parameters are also derived. Finally, a method of control parameters design and impedance optimization for MMC-HVDC based on the stability boundary is proposed. Compared to the traditional optimization method, the system stability is further improved by the impedance optimization method proposed this paper.

1. Introduction

The modular multilevel converter (MMC) has the advantages of significant redundancy, less harmonic content, and high conversion efficiency [1,2,3], and has been widely used in high-voltage direct current (HVDC) transmission. However, in the actual process, several MMC-HVDC projects have reported broadband resonance accidents with frequencies covering a dozen Hz to several hundred Hz [4,5,6,7,8], which seriously affect the safe and stable operation of the power system and cause huge losses.

Grid-connected system stability analysis based on impedance modeling has now been recognized as an effective method and widely used in the stability analysis of MMC grid-connected systems. Several scholars have established a more accurate MMC impedance modeling using the multi-harmonic linearization method [9,10,11,12] or the multi-harmonic state space method (HSS) [13,14,15]. However, in the above models, the DC side of MMC-HVDC is often simplified as an ideal DC source which means the dynamic of the DC bus voltage control will be ignored. In the analysis of permanent magnet synchronous generators (PMSG) integrated into a power grid system, it has been found that the DC voltage control is an essential factor in determining the impedance characteristics of the machine side PMSG. In practical MMC-HVDC projects, there must be a current-source converter station to support the DC-side voltage [14]; the impact of DC voltage control on the impedance characteristics of the converter station cannot be ignored. Thus, it is worth studying the DC voltage control of the MMC impedance model and its impact on the MMC impedance characteristics as well as grid-connected system stability.

Voltage feedforward control is also commonly used in practical projects to mitigate the imbalance of the grid voltage and improve the performance of the converter [14,15,16]. Most of the existing impedance modeling and stability analysis of MMC-HVDC ignore the influence of voltage feedforward control. However, voltage feedforward control has a non-negligible impact on the impedance characteristics and system stability of MMC-HVDC [17], so it is necessary to analyze the grid stability of MMC-HVDC and its impedance optimization method under the premise of considering voltage feedforward.

The impedance optimization of the converter can effectively improve the impedance characteristics of the system at the resonance frequency and reduce the risk of resonance [15]. Studies have been conducted to optimize the design of control parameters such as current control [18,19] and voltage feedforward control [20,21], which affect the stability of MMC-HVDC integrated systems on high-frequency ranges. Due to the complex structure, the impedance influence factors of MMC-HVDC are quite complex, especially in the sub-/super-synchronous frequency. Numerous control loops, including DC voltage control and voltage feedforward control, are coupled with each other, which makes it challenging to optimize the impedance characteristic of MMC-HVDC and improve grid-connected system stability [22,23].

To address the problem of ignoring DC voltage control and voltage feedforward control in the existing stability analysis of the MMC-HVDC integrated system, an impedance optimization method based on the impedance characteristic considering DC voltage control and voltage feedforward control is proposed in this paper to enhance the system stability. The rest of this paper is organized as follows. An MMC sequence impedance modeling considering DC voltage control and voltage feedforward control is present in Section 2. The influencing factors of control parameters are discussed in Section 3. In Section 4, the effects of DC voltage control and voltage feedforward control on the stability of the system are discussed. In Section 5, an MMC-HVDC impedance optimization method based on stability boundary design is proposed to effectively improve the system stability.

2. Impedance Modeling of MMC-HVDC Considering DC Voltage Control

A typical power and control circuit of MMC-HVDC is shown in Figure 1. Each phase of MMC-HVDC consists of an upper arm and a lower arm, where both arms contain n submodules (SM₁ ~ SM_n) and an arm inductor L. Each submodule is in a half-bridge topology with a module capacitor C_m. The output voltages and currents of MMC-HVDC are denoted as v_sx and i_sx (x = a, b, c), the voltage of DC side is denoted as v_dc, and the three-phase upper arm currents and lower arm currents are denoted as i_xu and i_xl (x = a, b, c). The typical controls of MMC-HVDC include the DC voltage control, phase current control, PCC voltage feedforward control, circulating current suppression control (CCSC), and phase-locked loop (PLL) to generate θ [12]. H_d(s), H_i(s), H_c(s), and H_θ(s) represent proportional-integral (PI) regulators of DC voltage control, phase current control, CCSC and PLL.

In order to eliminate the adverse effects of high harmonics in the grid, a voltage feedforward controller is usually adopted to filter out the high harmonics of the grid in practical projects; the voltage feedforward controller H_f(s) can be expressed as:

$$H_f\left(s\right)=k_f\frac{{\omega }_f}{s+{\omega }_f}$$

(1)

where k_f is the voltage feedforward coefficient, and ω_f is the cut-off frequency of the low-pass filter.

In addition, existing studies usually take the DC side as an ideal voltage source to simplify modeling, however, this simplification may lead to inaccurate analysis results of MMC-HVDC. This paper also focuses on the effects of DC voltage control on MMC-HVDC impedance characteristics.

Based on the power conservation law of the DC side and AC side, the small-signal instantaneous power of MMC-HVDC can be expressed as:

$${\widehat{p}}_{dc}=2{\widehat{p}}_{Cx}+2{\widehat{p}}_{Lx}+{\widehat{p}}_o$$

(2)

where ${\widehat{p}}_C$, ${\widehat{p}}_L$, and ${\widehat{p}}_o$ represent the instantaneous power of equivalent capacitance, the inductance of the upper arm, and the instantaneous output power of AC side, respectively. ${\widehat{p}}_{dc}$ denotes the side instantaneous power of the DC side which can be expressed as (3)–(6):

$${\widehat{p}}_C=\frac{1}{2}{\displaystyle \sum _{x=a,b,c}\left(Y_c{V}_{Cx}{\widehat{v}}_{Cx}+V_{Cx}{Y}_c{\widehat{v}}_{Cx}\right)}=6T_4{\widehat{v}}_{ca}$$

(3)

$${\widehat{p}}_L=\frac{1}{2}{\displaystyle \sum _{x=a,b,c}\left(Z_l{I}_x{\widehat{i}}_x+I_x{Z}_l{\widehat{i}}_x\right)}=6T_3{\widehat{i}}_a$$

(4)

$${\widehat{p}}_o={\displaystyle \sum _{x=a,b,c}{V}_{sx}{\widehat{i}}_{sx}+I_{sx}{\widehat{v}}_{sx}}=6T_1{\widehat{i}}_a+6T_2{\widehat{v}}_{sa}$$

(5)

$${\widehat{p}}_{dc}=V_{d\mathrm{c}}{\widehat{i}}_{dc}+I_{dc}{\widehat{v}}_{dc}=V_{d\mathrm{c}}{T}_5{\widehat{i}}_a+I_{dc}{\widehat{v}}_{dc}$$

(6)

where Z_l and Y_c respectively denote the arm inductance impedance and equivalent capacitance admittance at small-signal harmonic frequencies. V_Cx, I_x, V_sx, I_sx, V_dc, and I_dc denote the steady-state harmonic matrix obtained by the Toplitz transform of i_xu, v_Cx, i_sx, v_sx, i_dc, and v_dc, respectively. ${\widehat{v}}_{Cx}$, ${\widehat{i}}_x$, ${\widehat{v}}_{sx}$, ${\widehat{i}}_{sx}$, ${\widehat{v}}_{dc}$, and ${\widehat{i}}_{dc}$ denote the small-signal harmonic vectors of MMC equivalent capacitance voltage, arm current, output voltage and current, and voltage and current of the DC side; the nonzero elements in T₁ ~ T₅ are expressed as (A1) (Appendix A).

Combining Equations (2)–(6), the small-signal harmonic vectors of DC voltage can be expressed as:

$${\widehat{v}}_{\mathrm{dc}}=H_1{\widehat{i}}_a+H_2{\widehat{v}}_{sa}+H_3{\widehat{v}}_{ca}$$

(7)

where

$$H_1=I_{dc}^{-1}\left(6T_1+6T_3-V_{dc}{T}_5\right),H_2=6I_{dc}^{-1}{T}_2,H_3=6I_{dc}^{-1}{T}_4$$

Based on the dq control structure of MMC-HVDC, the small-signal vector of the insertion index can be modeled as follows:

$$\{\begin{array}{l}{\widehat{m}}_d=K_{\mathrm{i}}\cdot {\widehat{i}}_{\mathrm{sq}}-H_{\mathrm{i}}\cdot {\widehat{i}}_{\mathrm{sd}}+H_{\mathrm{d}}\cdot {\widehat{v}}_{\mathrm{dc}}+H_{\mathrm{f}}\cdot {\widehat{v}}_{\mathrm{s}}\\ {\widehat{m}}_q=-K_{\mathrm{i}}\cdot {\widehat{i}}_{\mathrm{sd}}-H_{\mathrm{i}}\cdot {\widehat{i}}_{\mathrm{sq}}+H_{\mathrm{v}}\cdot {\widehat{v}}_{\mathrm{s}}\end{array}$$

(8)

where the red part in Equation (8) represents the effects of DC voltage control and feedforward control on the small-signal vector of the insertion index. ${\widehat{i}}_{\mathrm{sd}}$ and ${\widehat{i}}_{\mathrm{sq}}$ denote the small-signal harmonic vectors of the output current in the dq coordinate system. K_i represent dq-frame decoupling coefficient of phase current control. H_i, H_d, and H_f are 7 × 7 matrices as shown:

$$H_{\mathrm{i}}=diag\left(H_i\left(s+jn{\omega }_1\right)|{}_{n=-3,-2,-1,0,1,2,3}\right)$$

(9)

$$H_{\mathrm{d}}=diag\left(H_d\left(s+jn{\omega }_1\right)|{}_{n=-3,-2,-1,0,1,2,3}\right)$$

(10)

$$H_{\mathrm{f}}=diag\left(G_f\left(s+jn{\omega }_1\right)|{}_{n=-3,-2,-1,0,1,2,3}\right)$$

(11)

Combined with Equations (7) and (8), the linearized small-signal harmonic vector of the insertion index in abc coordinates is shown as:

$${\widehat{m}}_u=G_1{\widehat{i}}_u+G_2{\widehat{v}}_s+G_3{\widehat{v}}_s$$

(12)

where

$$\begin{array}{l}{G}_1={\left(U-Q_3{H}_3\right)}^{-1}\left(Q_1+Q_3{H}_1\right)\\ G_2={\left(U-Q_3{H}_3\right)}^{-1}\left(Q_2+Q_3{H}_2\right)\\ G_3={\left(U-Q_3{H}_3\right)}^{-1}{Q}_{v2}\end{array}$$

the nonzero elements in Q₁, Q₂, Q₃, and Q_v2 are expressed as (A2)–(A5) (Appendix A).

After considering the DC voltage control, the MMC-HVDC small-signal of the power stage is changed to:

$$\{\begin{array}{l}{\widehat{v}}_u=Z_c\left(M_u{\widehat{i}}_u+I_u{\widehat{m}}_u\right)\\ \widehat{i}=Y_l\left(\frac{1}{2}{\widehat{v}}_{dc}-{\widehat{v}}_s-M_u{\widehat{v}}_u-V_u{\widehat{m}}_u\right)\end{array}$$

(13)

where M_u, I_u, and V_u denote the matrices of steady-state signals of the insertion index, arm current, and equivalent capacitive voltage obtained by the Toeplitz transform, and ${\widehat{v}}_u$ denotes the small-signal harmonic vector of the equivalent capacitive voltage.

Combining Equations (12) and (13), MMC-HVDC admittance matrix Y considering DC voltage control and voltage feedforward control can be obtained.

$$\begin{array}{r}Y={\left(U-Y_l\left(0.5H_1-M{Z}_{c}M+\left(0.5H_3-M{Z}_{c}I-V\right)G_1\right)\right)}^{-1}\cdot \\ Y_l\left(0.5H_2-U+\left(0.5H_3-M{Z}_{c}I-V\right)G_2\right)\end{array}$$

(14)

where the MMC-HVDC conductance matrix Y is a 7 × 7 matrix and the (4, 4) elements of Y represent the transfer function between the MMC-HVDC output side voltage at f_p frequency and the bridge arm current at the same frequency, so the positive and negative sequence impedance of the MMC-HVDC considering DC voltage control can be defined as

$$Z_p\left(s\right)=-\frac{1}{2Y\left(4,4\right)}$$

(15)

$$Z_n\left(s\right)=Z_p^{\ast }\left(-s\right)$$

(16)

In order to verify the accuracy of the established MMC-HVDC impedance modeling, the MMC-HVDC model including DC voltage control, current control, voltage feedforward control, loop control, and phase lock control was built based on Matlab/Simulink, and the specific simulation parameters are shown in

Table 1. Parameters of MMC-HVDC in simulation.

Parameters	Value	Parameters	Value
Rated power	255 MW	Hd(s)	0.021 + 1.743/s
Output L-L voltage	375 kV	Hi(s)	7.169 × 10−5 + 0.024/s
DC-side voltage	±350 kV	Hc(s)	6.089 × 10−5 + 0.024/s
Number of submodules per arm	500	Hθ(s)	4.961 × 10−5 + 1.062 × 10−3/s
Arm inductance L	100 mH	Hf(s)	0.7 × 200π/(200π + s)
Module capacitance Cm	13.6 mF	Simulation step	4 µs

. Figure 2 shows the comparison between the analytical and measurement results of both the positive and negative MMC-HVDC impedance. The simulated results are consistent with the analytical results, which confirms the accuracy of the developed MMC-HVDC impedance model.

It can be observed that there are apparent differences between the positive and negative sequence impedance of MMC-HVDC; the positive sequence impedance increases with the frequency and the impedance amplitude and phase angle curve change significantly. In the frequency range of 20–80 Hz, the MMC-HVDC impedance is mainly affected by DC voltage control and PLL, and the impedance shows two symmetrical resonance peaks centered on the fundamental frequency. In the range of 80–400 Hz, the MMC-HVDC impedance is dominated by the current loop integrator. In the frequency range above 400 Hz, the MMC-HVDC impedance is mainly influenced by the filter and shows the damping characteristic. The positive and negative sequence impedance of the MMC-HVDC differs significantly in the low-frequency band, but with the increase in frequency both impedance curves gradually coincide.

3. MMC-HVDC Impedance Characteristic Analysis

3.1. Sensitivity Analysis of MMC-HVDC Impedance

Multiple controllers of MMC-HVDC are coupled with each other to affect the impedance characteristics of MMC-HVDC. Different controls dominate impedance characteristics of MMC-HVDC in their respective frequency ranges. In this section, impedance sensitivity is introduced to quantify the impacts of controllers on MMC-HVDC impedance.

From the above impedance modeling analysis, it can be seen that the impedance characteristics of the MMC-HVDC are affected by multiple controllers. The influence of controllers on impedance characteristics can be quantified by the impedance sensitivity. Impedance sensitivity can be refined as impedance magnitude sensitivity H_Mωα and impedance phase sensitivity H_Pωα to represent the influence of different controller parameters on magnitude and phase of MMC-HVDC impedance, respectively. The impedance magnitude and phase sensitivity of controller α are shown in Equations (17) and (18).

$$H_{M{\omega }_{\alpha }}\left(s\right)=\underset{\Delta {\omega }_{\alpha }\to 0}{\lim }\frac{\left|\left|Z\left({\omega }_{\alpha }+\Delta {\omega }_{\alpha },s\right)\right|-\left|Z\left({\omega }_{\alpha },s\right)\right|\right|}{\Delta {\omega }_{\alpha }}$$

(17)

$$H_{P{\omega }_{\alpha }}\left(s\right)=\underset{\Delta {\omega }_{\alpha }\to 0}{\lim }\frac{\left|\angle Z\left({\omega }_{\alpha }+\Delta {\omega }_{\alpha },s\right)-\angle Z\left({\omega }_{\alpha },s\right)\right|}{\Delta {\omega }_{\alpha }}$$

(18)

where Z(ω_α,s) represents the MMC-HVDC impedance with the parameter of controller α is ω_α, and Z(ω_α + Δω_α,s) represents that the parameter of controller α changes to ω_α + Δω_α.

Figure 3 depicts the impedance magnitude and phase sensitivity of the voltage feedforward coefficient, bandwidth of the current controller, DC voltage controller, and PLL control of the MMC-HVDC. The larger value of the sensitivity curve indicates that the corresponding controller has a greater effect on the impedance characteristics of MMC-HVDC.

In Figure 3, the impedance sensitivity of voltage feedforward control is significantly greater than other controllers in the frequency bands of 1 Hz ~ 46 Hz and 57 Hz ~ 1000 Hz, which means that the voltage feedforward coefficient dominates the impedance characteristics of MMC-HVDC in this frequency range. Voltage feedforward control will more greatly affect the magnitude of MMC-HVDC impedance at the frequency closer to the fundamental frequency. Similarly, the above rule can also be found on the effect of voltage feedforward control on the phase of MMC-HVDC impedance in the high-frequency range.

The impedance sensitivity of PLL and DC voltage control are more significant in the frequency band near the fundamental frequency (46 Hz ~ 57 Hz), indicating that the impedance characteristics of the MMC-HVDC are mainly influenced by PLL and DC voltage control, which is consistent with the existing findings.

The impedance sensitivity of the PLL and DC voltage control is approximately zero on the frequency range (1 Hz ~ 46 Hz, 57 Hz ~ 1000 Hz), where the voltage feedforward control plays a dominant role in the MMC-HVDC impedance characteristics mentioned above. This indicates that different controllers dominate impedance characteristics of MMC-HVDC in the different frequency ranges.

The output current control bandwidth is also located in the frequency range dominated by the voltage feedforward. At the cut-off frequency of the control bandwidth, the impedance sensitivity of the output current control is relatively improved, but still less than the voltage feedforward control, indicating that the MMC-HVDC impedance characteristics may be affected by more than one controller in some frequency ranges, but the influence degree of controllers is different.

In summary, the impedance characteristics of MMC-HVDC is affected by multiple control loops; the effect of different controls on MMC-HVDC impedance characteristics are different. Voltage feedforward control plays a dominant role in most frequency ranges.

3.2. Impacts of DC Voltage Control on Impedance Characteristics

Figure 4a compares MMC-HVDC impedance response with and without DC voltage control under the power and control parameters shown in Table 1. The DC voltage control affects the impedance characteristic mostly around the fundamental frequency (20 ~ 65 Hz): the impedance considering DC voltage control introduces a primary resonance peak at the fundamental frequency and a pair of secondary resonance peaks at both sides of the fundamental frequency (46.8 Hz and 56.5 Hz). Furthermore, the distance of frequency of the two secondary resonance peaks (9.7 Hz) is almost the same as the bandwidth of the DC voltage control (10 Hz). The impedance magnitude of MMC-HVDC increases significantly, while the impedance angle jumps from 180° to −180° at the resonance frequency.

Figure 4b compares the impedance response of the MMC-HVDC under different DC voltage control bandwidths. All the impedances of MMC-HVDC with different DC voltage bandwidths have a primary resonance peak at the fundamental frequency and a pair of secondary resonance peaks. The frequency offset between the two secondary resonant peaks increases with the increase in the DC voltage control bandwidth.

3.3. Impacts of Voltage Feedforward Control Coefficient on Admittance Characteristics

Figure 5a compares the characteristics of MMC-HVDC admittance (the reciprocal of impedance) considering and ignoring voltage feedforward control with the control parameters shown in Table 1. The amplitude of MMC-HVDC admittance with voltage feedforward control (denoted as Y_MMCo in Figure 5a) is significantly reduced compared with that without feedforward control admittance (denoted as Y_O). Near the fundamental frequency, both admittance curves almost overlap. With frequency increases, the effect of voltage feedforward control gradually decreases, and the magnitude of Y_MMCo approaches that of Y_O. It should be noted that the phase of equivalent admittance of voltage feedforward control (denoted as Y_lf) is greater than 90° in the low-frequency range (1 Hz ~ 34 Hz) and less than −90° in the middle-frequency range (72 Hz ~ 160 Hz), which will lead Y_MMCo to exhibit extra negative-damping characteristics (1 Hz ~ 8 Hz and 74 Hz ~ 133 Hz). Similarly, in the frequency band where Y_lf provides damping effects (34 Hz ~ 44 Hz), Y_MMCo also changes from negative-damping to positive-damping characteristics when voltage feedforward is considered.

Figure 5b compares the impedance responses of the MMC-HVDC with different voltage feedforward coefficient k_f. As k_f decreases, the magnitude of Y_lf gradually decreases, which leads the magnitude of Y_MMCo closer to that of Y_o. Similarly, with a smaller k_f, the impacts of voltage feedforward control on the phase of impedance also decrease, and the frequency ranges where Y_lf shows the negative-damping characteristic become narrower.

4. Stability Analysis of MMC-HVDC Integrated System

Based on the developed impedance model in Section 2, the impedance-based analysis method can be used to evaluate the stability of the MMC-HVDC integrated system. The configuration of the MMC-HVDC integrated system used to analyze the stability is presented in Figure 6, where the impedance of MMC-HVDC is denoted as Z_MMC(s) and the grid impedance is denoted as Z_g(s), and the parameters of MMC-HVDC are presented in Table 1.

Figure 7a depicts the eigenloci curve of the return ratio matrix Z_MMC(s)/Z_g(s) of three cases. If the DC voltage control or voltage feedforward control is ignored in the system stability analysis, both trajectories will not enclose the (−1, j 0) point, indicating that the system is stable. However, when both DC voltage control and voltage feedforward control are considered in the stability analysis as an actual control structure, the trajectories will enclose the (−1, j 0) point clockwise, implying instability. The opposite analysis result indicated that both DC voltage control and voltage feedforward control are not negligible for stability analysis.

Figure 7b illustrates the frequency responses of impedance of MMC-HVDC considering DC voltage control and voltage feedforward control Z_MMC(s) and Z_g(s). The comparison of the impedance that ignores DC voltage control or voltage feedforward control is also shown in Figure 7b. The interaction characteristic of the MMC-HVDC ignoring DC voltage control or voltage feedforward control with grid impedance implies that the system will be stable. However, the MMC-HVDC impedance shows a negative-damping characteristic at the intersection frequency of 90 Hz, which makes the phase margin of the MMC-HVDC integrated system less than 0°. Those impedance responses may result in a resonance of the MMC-HVDC at the intersection frequency when DC voltage control and voltage feedforward control are considered.

A simulation developed in MATLAB/Simulink was used to verify the above analysis results, and the time-domain simulated results are shown in Figure 8. It can be found that there are obvious resonance exits in the injected currents of the MMC-HVDC. The corresponding FFT result of the injected current in Figure 8b confirms the resonance frequency is 90 Hz, which is in agreement with the impedance-based stability analysis.

5. Impedance Optimization Method for MMC-HVDC

From the above analysis, it can be seen that both DC voltage control and voltage feedforward control greatly influence the grid-connected system’s stability. Since the parameters of DC voltage control and voltage feedforward control impact each other, an impedance optimization method based on the stability boundary design is proposed in this section to improve the system stability.

5.1. Stability Boundary for the Control Parameters

According to the above analysis, both DC voltage control and voltage feedforward control have impacts on the MMC-HVDC impedance characteristics and grid-connected system stability in sub-/super-synchronous frequency ranges. The boundary conditions of voltage feedforward control parameters that ensure the grid-connected system stability will be different when the bandwidth of the DC voltage control changes, and vice versa. Therefore, the DC voltage control and voltage feedforward control need to be designed cooperatively. Figure 9a depicts the mapping of the stability margin of MMC-HVDC grid-connected system with different pairs of parameters of DC voltage control and voltage feedforward control. When the parameter combination (K_f, ξ) is located in the stable area (the lower area below the solid blue line), the corresponding stability margin is larger than 0. Otherwise, when the parameter combination (K_f, ξ) is located in the unstable area (the upper area higher than the solid blue line), the corresponding stability margin is less than 0. The solid blue line with zero stability margin represents the stability boundary of two control parameters to ensure the grid-connected system stability; the blue line is defined as ξ_max(k_f), k_f is between K_fmin ~ K_fmax that can ensure the self-stability of MMC-HVDC.

Taking the three cases shown in Figure 9a as an example, Case A is on the blue line (the same case used in Ch. 4), whose DC voltage control bandwidth is 29 Hz and the voltage feedforward coefficient is 0.7, indicating the system is critical stable. If the system changes from Case A to Case B, i.e., the voltage feedforward coefficient decreases from 0.7 to 0.45, then the operating point changes from a stable area to an unstable area. If the system changes from Case B to Case C, i.e., the bandwidth of DC voltage control decreases from 29 Hz to 13 Hz, then the operating point changes from an unstable area to a stable area again.

The injected current waveforms of MMC-HVDC are shown in Figure 9b. From 2 s to 3 s, the bandwidth of DC voltage control is 29 Hz, and the voltage feedforward coefficient is 0.7 (Case A). The voltage feedforward coefficient changes from 0.7 to 0.45 at 3 s (Case B), and the bandwidth of DC voltage control decreases from 29 Hz to 13 Hz at 3.5 s (Case C). As is shown, injected currents of MMC-HVDC are critically stable/stable in Case A and Case C, and resonance occurs in case B. The simulation results are consistent with the impedance-based stability analysis.

5.2. Impedance Optimization Method Based on Parameters Stability Boundary

All the parameter combinations (K_f, ξ) inside the stability boundary can maintain the MMC-HVDC system stable, but the corresponding phase margin will be different. The impedance optimization method of MMC-HVDC is to maximize the phase margin of the system. The relationship between the parameter combination (K_f, ξ) and system stability margin is illustrated in Figure 10. The (K_f, ξ) that lead to a zero-stability margin will be on the pink plane as Case A. The (K_f, ξ) that lead to a positive stability margin will be above the pink plane as Case C, otherwise, it will be under the pink plane as Case B.

It is difficult to quantify the effect of the parameter combination (K_f, ξ) on the phase margin of the integrated system. Thus, the traversal algorithm is used in this section to calculate every phase margin θ_s of the MMC-HVDC system under all the (K_f, ξ) within the boundary obtained above. Maximizing the phase margin is taken as the aim of impedance optimization. The process of the impedance optimization is shown in Figure 11. The impedance optimization can be divided into the control parameter stability boundary calculation as shown in Figure 12a, and traversal searching of optimized control parameters as shown in Figure 11b.

The process of parameter boundary calculation should be:

Step 1: Input the working conditions of the MMC-HVDC integrated system, including the parameters of the power grid and control parameters and the boundary of parameter combination (K_f, ξ) that can ensure the self-stability of MMC-HVDC denoted as K_fmin ~ K_fmax and ξ_pcmin ~ ξ_pcmax.

Step 2: Keep k_f as a constant K_fc and traverse all the possible values of bandwidth of DC voltage control ξ within ξ_pcmin ~ ξ_pcmax by the step of Δξ_c and calculate every phase margin under each (K_fc, ξ).

Step 3: Denoted the maximum of ξ that leads to a positive stability margin as ξ_max(K_fc) under k_f as a constant K_fc.

Step 4: Traversal all the possible value of k_f within the K_fmin ~ K_fmax by the step of ΔK_fc and repeat steps 2–4.

Step 5: Output stability boundary ξ_max(k_f), k_f ∈ [K_fmin, K_fmax].

The process of parameter optimization should be:

Step 1: Input the working conditions of the MMC-HVDC integrated system as well as the parameter boundary ξ_max(k_f).

Step 2: Traversal all the possible parameter combination (K_f, ξ) within the ξ_max(k_f) by the step of ΔK_fs and Δξ_s.

Step 3: Output the maximum of the phase margin and corresponding parameters as the optimal outputs (K_fo, ξ_o).

In the application shown in Figure 11, the optimal control parameters of (K_fo, ξ_o) are 5Hz and 0.4, which is shown as Case D, and the maximum phase margin is 26.7°. The system impedance characteristic in Case D (ξ_o = 5 Hz, K_fo = 0.4), Case A (ξ = 25 Hz, K_f = 0.7), and Case C (ξ = 13 Hz, K_f = 0.45) are compared in Figure 12. It can be observed that both the amplitude intersection frequency of MMC-HVDC and grid impedance in Case C and Case D change from 90 Hz (Case A) to 78 Hz, where the MMC-HVDC impedance characteristics change from negative-damping to positive-damping. The system stability margin increased more in Case D than in Case C. This verifies that the parameters within the boundary can ensure the grid-connected system stability, and the parameters searched by the proposed optimization method can maximize the phase margin of the system.

The injected current waveforms of MMC-HVDC system are shown in Figure 13. At the beginning of the simulation, the MMC-HVDC is operated as in Case A. At 2.5 s, the system changes from Case A to Case D (ξ = 5 Hz, K_f = 0.4, obtained by the proposed optimization method above). As is shown in Figure 13a, the injected current of MMC-HVDC contains significant harmonics within 2.5 s and becomes stable after impedance optimization. The THD of the stable current waveforms after optimization is less than 1%. Compared with the optimized control parameter (Case D), Case C of other nonoptimized stability parameters, shown in Figure 13b, has significantly more current harmonics during the switching processing, indicating that Case C is more likely to introduce resonance due to disturbance. Thus, the proposed impedance optimization method can effectively improve the system stability of MMC-HVDC.

6. Conclusions

In this paper, the MMC-HVDC impedance modeling and influence of the control parameter on the impedance characteristics and the stability of the grid integrated system considering both DC voltage control and voltage feedforward control are studied, and the following can be concluded:

(1)

In the multi-terminal flexible direct transmission system, there must be a DC voltage control in the MMC-HVDC converter station to maintain the DC voltage stability. Moreover, the voltage feedforward control cannot be ignored either. An MMC impedance modeling considering DC voltage control and voltage feedforward control is derived by using the multi-harmonic linearization method.

(2)

The DC voltage control and voltage feedforward control have great impacts on the impedance characteristics as well as the system stability analysis. The different control loops of MMC-HVDC interact on the premise of guaranteeing the grid-connected system stability. The boundary condition of MMC-HVDC control parameters is given to guide the design of DC voltage control and voltage feedforward control to ensure the system stability.

(3)

An MMC-HVDC impedance optimization method based on stability boundary is proposed. The optimal stability margin is achieved by searching the control parameters within the stability boundary. The impedance optimization method of MMC-HVDC can ensure the stability of the integrated system and maximize the phase margin of the system.