Research on Adaptive Exponential Droop Control Strategy for VSC-MTDC System

Abstract

To solve the problem of large DC voltage deviation caused by the power fluctuations and poor power distribution characteristics of converters in a voltage source converter multi-terminal DC (VSC-MTDC) system based on traditional droop control, this paper proposes an adaptive exponential droop control strategy. This strategy introduces the relative power deviation factor of the converter, and replaces the traditional linear droop control curve with a nonlinear exponential curve. Under different working conditions, the converter adaptively adjusts the droop control coefficient of the converter according to the relative power deviation factor to realize stability for the DC voltage and a reasonable power distribution for the MTDC system. A simulation model of a three-terminal VSC-MTDC was established in MATLAB/Simulink, and the feasibility and effectiveness of the proposed strategy were verified.

1. Introduction

The voltage source converter multi-terminal DC (VSC-MTDC) system, which is composed of multiple VSC converter stations and DC grids, can easily realize multi-power supply and multi-load power supply, has strong flexibility, and is economical and reliabile. Therefore, this system is a hot research and engineering topic in the areas of large-scale renewable energy grid connection, interconnection between energy systems, and island power supply [1,2,3,4,5]. In recent years, with the rapid development of power electronics technology, research on the VSC-MTDC system has deepened [6,7,8,9], but the current research is mainly aimed at optimizing its stability and power quality. When the distribution network fails to maintain the stability of grid voltage, the system can quickly provide active and reactive power support [10,11,12]. When the active power is unbalanced in the DC network of the system and the fluctuation of the DC side voltage is large, the dynamic response process of the control strategy will activate the protection device to stop the equipment, which will then lead to economic losses. Therefore, it is particularly important to study the control of the DC side voltage of VSC-MTDC, and to improve its power distribution and voltage fluctuation characteristics during faults or disturbances.

At present, the control methods based on VSC-MTDC mainly include master–slave control [13,14], voltage droop control [15,16,17,18,19], voltage margin control [20], and combinations of these [21,22,23]. Different control strategies have different applications and scope. Among them, the voltage droop control strategy is generally applicable to the VSC-MTDC system with multiple converter stations. This strategy both manages the DC voltage control according to the slope relationship between the given DC power and the DC voltage, and realizes a fast, balanced distribution of DC power [24]. However, the traditional voltage droop control strategy has a fixed droop coefficient, which cannot flexibly respond to system failures or power disturbances; it also causes DC voltage fluctuations. Therefore, an improved droop control strategy that considers the power marginwas proposed in [25]. The droop control coefficient is determined by the power margin of the converter station to avoid a full load of the converter and to realize reasonable power distribution, but it is easy for the DC voltage to exceed the limit range. A power influence factor in the droop control was thus introduced in [26], which greatly improved the DC voltage quality and power distribution characteristics. An improved droop control strategy based on DC voltage deviation was proposed in [27], which can cope with flexible operating conditions, but the calculation of the sag factor is too complicated.

To sum up, in order to take into account DC voltage quality and power distribution characteristics, an adaptive exponential droop control strategy is proposed, which introduces the relative power deviation factor of the converter in order to transform the traditional linear droop control into a nonlinear exponential droop control. Under different working conditions, the strategy uses the relative power deviation factor of the converter to adaptively adjust the droop control coefficient of the converter; it also optimizes the DC voltage deviation caused by the power fluctuation of the MTDC system, thus improving its power distribution characteristics.

2. Adaptive Exponential Droop Control

2.1. Traditional Droop Control

For conventional control, the (U–I) curve equation is as follows:

$$U_{dc1}=k{I}_{dc}+U_{dcc\_ref}$$

(1)

where k is the traditional droop control coefficient; I_dc is the DC side current, and $U_{dc\_ref}$ is the rated voltage of the system. With the traditional droop control strategy, both voltage and power are linearly adjusted according to a fixed slope: when the slope is large, the power distribution characteristics of the converter are good, but the voltage deviation is relatively large, and when the slope is small, the voltage deviation is small, but the power distribution is poor.

Adaptive droop control based on the power influence factor was proposed in [26], and its control (U–I) curve equation is as follows:

$$U_{dc2}=k\frac{U_{dc\_ref}}{P_m}{I}_{dc}^2+U_{dc\_ref}$$

(2)

where $P_m$ is the rated capacity of the converter.

Adaptive droop control based on the power influence factor realizes the transformation of the traditional linear relationship between DC voltage and power into a nonlinear relationship and optimizes the system voltage quality and power distribution characteristics, which is what provided ideas for this paper regarding adaptive exponential droop control.

2.2. Adaptive Exponential Droop Control

As shown in Figure 1, in order for the MTDC system to have better power distribution characteristics and voltage quality, the power relative deviation factor $\epsilon$ is introduced into the traditional droop control, and a VSC-MTDC system method based on adaptive exponential droop control is proposed, with $\epsilon =\frac{P_i-P_m}{P_m}$ and $P_i=U_{dc\_ref}{I}_{dc}$; the control (U–I) curve equation is expressed as follows:

$$U_{dc3}=k{e}^{2\epsilon }{I}_{dc}+U_{dc\_ref}$$

(3)

2.3. Dc Voltage Deviation Analysis

Usually, the operating reference point of the MTDC system is set as the transmission power equal to zero, that is, $I_{dc\_ref}=0$ [26], while the rated voltage of the system is $I_{d{c}_{r}ef}=0$.

The DC voltage deviation caused by the traditional droop control power change is as follows:

$$\mathsf{\Delta }{U}_{dc1}=k\left(I_{dc\_ref}-I_{dc}\right)$$

(4)

In [26], the DC voltage deviation caused by the adaptive droop control power change is as follows:

$$\mathsf{\Delta }{U}_{dc2}=k\frac{U_{dc\_ref}}{P_m}\left({I_{dc\_ref}}^2-I_{dc}^2\right)$$

(5)

The DC voltage deviation caused by the power change of the adaptive exponential droop control proposed in this paper is expressed by the following equation:

$$\mathsf{\Delta }{U}_{dc3}=k[e^{\frac{2(U_{dc\_ref}{I}_{dc\_ref}-P_m)}{P_m}}{I}_{dc\_ref}-e^{\frac{2(U_{dc\_ref}{I}_{dc}-P_m)}{P_m}}{I}_{dc}]$$

(6)

The range of the DC current $I_{cd}$ in Equations (4)–(6) is $I_{cd}\in (0,P_m/U_{cd\_ref})$, and $P_m/U_{dc\_ref}$ is the maximum allowable current of the converter; the allowable range of the DC side voltage $U_{dc}$ is 5%, that is, $U_{dc}\in (0.95U_{dc\_ref},1.05U_{dc\_ref})$.

• If we compare the voltage deviation between the adaptive exponential droop control and the traditional droop control by mathematical difference, the difference result is as follows:

  $$\mathsf{\Delta }{U}_{dc3}-\mathsf{\Delta }{U}_{dc1}=k{I}_{dc}[1-e^{\frac{2(U_{dc\_ref}{I}_{dc}-P_m)}{P_m}}]$$

  (7)
  Since this is within the range of the converter current variation, that is, $I_{dc}\in (0,P_m/U_{dc\_ref})$, there is always $U_{dc\_ref}{I}_{dc}<P_m$, and so the above formula always has $\mathsf{\Delta }{U}_{dc3}-\mathsf{\Delta }{U}_{dc1}<0$. Thus, the DC voltage deviation is always better for the adaptive exponential droop control than the traditional droop control.
• As shown in Figure 2, adaptive exponential droop control (A), adaptive droop control (B) [26], and the traditional droop control (C) are compared through a curve diagram, and it is easy to see that the deviation of DC voltage is slighty better for the adaptive droop control than the adaptive exponential droop control in a small range; however, the voltage deviation of the exponential droop control is better than that of the adaptive droop control in a large variation range.

The mathematical difference of the adaptive exponential droop control and the voltage deviation of adaptive droop control in [26] is calculated as follows:

$$\mathsf{\Delta }{U}_{dc3}-\mathsf{\Delta }{U}_{dc2}=k\left[\frac{U_{dc\_ref}}{P_m}{I}_{dc}^2-I_{dc}{e}^{\frac{2\left(U_{dc\_ref}{I}_{dc}-P_m\right)}{P_m}}\right]$$

(8)

2.4. Analysis of Power Distribution Characteristics

The power distribution characteristics of MTDC are generally measured by the slope of the curve: the larger the slope, the better the power distribution characteristics, and vice versa. The greater the voltage change under unit power fluctuation, the better the power distribution characteristics.

The $I_{dc}$ partial derivation is performed on the (U–I) equation of the adaptive exponential droop control of Equation (3), and the result is expressed by the equation below:

$$\frac{\partial U_{dc3}}{\partial I_{dc}}=k{e}^{\frac{2\left(U_{dc\_ref}{I}_{dc}-P_m\right)}{P_m}}\left(1+\frac{2U_{dc\_ref}{I}_{dc}}{P_m}\right)$$

(9)

Analyzing the above formula, we obtain the following:

$$\begin{array}{cc}\hfill & k{e}^{\frac{2\left(U_{dc\_ref}{I}_{dc}-P_m\right)}{P_m}}\left(1+\frac{2U_{d{c}_{-}ref}{I}_{dc}}{P_m}\right)>k,I_{dc}\in \left(0,\frac{0.6039U_{d{c}_{-}rf}}{P_m}\right)\hfill \\ \hfill & \hfill k{e}^{\frac{2\left(U_{dc\_ref}{I}_{dc}-P_m\right)}{P_m}}\left(1+\frac{2U_{d{c}_{-}ref}{I}_{dc}}{P_m}\right)<k,I_{dc}\in \left(\frac{0.6039U_{d{c}_{-}ref}}{P_m},\frac{U_{d{c}_{-}ref}}{P_m}\right)\hfill \end{array}$$

(10)

The analysis result shows that when the DC side current is in the interval $(0,\frac{0.6039U_{ref}}{P_m})$, that is, the MTDC system is lightly loaded, the voltage quality of the adaptive exponential droop control within this range is good, and it is more inclined toward the optimization of the voltage quality; when the DC side current is in the interval $(\frac{0.6039U_{ref}}{P_m},\frac{U_{ref}}{P_m})$, that is, the load is heavy, the adaptive exponential droop control does not only have good voltage quality, but it also has better power distribution characteristics.

3. Influence of Adaptive Exponential Sag Coefficient on Stability

Figure 3 shows a control block diagram of the adaptive exponential droop control converter, in which the droop coefficient k₁ = −k is taken, and the droop control outer loop PI controller is $G_0\left(s\right)=\frac{s{k}_{pv}+k_{iv}}{s}$. The current inner loop function is set as $G_1\left(s\right)=G_2\left(s\right)G_3\left(s\right)G_4\left(s\right)$: $G_2\left(s\right)=\frac{s{k}_{pi}+k_{ii}}{s}$ is the current inner loop control PI controller; $G_3\left(s\right)=\frac{k_{PWM}}{1.5T_s+1}$ is the current inner loop sampling transfer function, where $k_{PWM}$ is the PWM equivalent gain and $T_s$ is the sampling period; and $G_4\left(s\right)=\frac{1/R}{1+(L/R)s}$ is the transfer function of the AC filter resistance and inductance link. The equivalent proportional gain is 0.75 and the DC side capacitance transfer function is $\frac{1}{sC}$.

According to the adaptive exponential droop control block diagram in Figure 3, the open-loop transfer function is derived as follows:

$$G\left(s\right)=\frac{0.75\left(k_{ii}+s{k}_{pi}\right)\left(k_{iv}+s{k}_{pv}\right)}{k_{1}Cs{S}^2\left[(k_{ii}+s{k}_{pi})+s\left(1+1.5T_{s}s\right)(1+Ls/R)\right]}$$

(11)

The experimental simulation parameters of the MTDC system in this paper are shown in

Table 1. Parameters of the simulation system.

No.	Parameter Name	Parameter Value
1	Outer ring proportional $k_{pv}$	0.55
2	Outer loop integral $k_{iv}$	25
3	Inner ring proportional $k_{pi}$	2.5
4	Inner loop integral $k_{ii}$	20
5	Sampling period $T_s$	5 × 10−6 s
6	DC side capacitance C	2000 uF
7	AC filter inductor L	3 mH
8	AC filter resistor R	0.03 Ω

.

The values of the experimental simulation parameters of the MTDC system were used in Formula (11), and a root locus diagram of the open-loop transfer function was obtained, as shown in Figure 4.

As shown in Figure 4, the left half plane is the system stability region in the root locus diagram of the open-loop transfer function for the adaptive exponential droop control. The critical point open-loop gain is 0.948, and the k₁ value is equal to 1.2. As the stable point of the system moves to the left half plane, k₁ gets bigger and goes to positive infinity. Therefore, the range of the droop coefficient k can be obtained as $k_1\in (1.2,+\infty )$. However, to ensure that the MTDC system maintains steady-state operation, the DC voltage quality must have $\mathsf{\Delta }{U}_{dc}>0.95U_{dc}^{ref}$, and the value range of k₁ can be obtained as $k_1\in (-\infty ,5)$. Based on the above analysis, the range of the equivalent sag coefficient k₁ is: $k_1\in (1.2,5)$.

4. Simulation

As shown in Figure 5, a flexible three-terminal radial DC transmission system was simulated and built, in which two ends were the power-sending ends and one end was the power-receiving end. VSC1 and VSC2 were both rectifier stations, and adaptive exponential droop control was adopted. The rated capacity was 400 MVA, the inverter station was VSC3, the equivalent receiving end load adopted constant active power control, and the converter power was 600 MVA; the DC side voltage was ±100 kV, the line equivalent resistance value was 0.5 Ω, and the line equivalent inductance value was 2 mH.

In this paper, the simulation of different droop coefficients and the power change of the receiving end, that is, the fluctuation of the equivalent load, are analyzed through the use of an example, and the DC voltage quality and power distribution characteristics of traditional droop control, adaptive droop control, and adaptive exponential droop control are compared.

4.1. Simulation Results of Different Droop Coefficients

To verify the superiority of the proposed adaptive exponential droop strategy in terms of voltage characteristics under different initial droop coefficients, this paper simulates the effects of different initial droop coefficients on the steady-state voltage of the DC side when the receiving end power of the MTDC system fluctuates. When it is 0.9 s, the power at the receiving end changes from 240 MW to 480 MW; the simulation waveform is shown in Figure 6, Figure 7 and Figure 8.

In the simulation waveforms in Figure 6, Figure 7 and Figure 8, it can be clearly seen that compared with the traditional droop control strategy and the adaptive droop control strategy, the adaptive exponential droop control proposed in this paper is less affected by the initial droop coefficient, and that the voltage deviation in the whole period is smaller.

As can be seen in

Table 2. Comparison table of simulation results with different droop coefficients.

Different Droop Coefficients	DC Side Steady-State Voltage (kV)
	Traditional Droop Control	Adaptive Droop Control [26]	Adaptive Exponential Droop Control
$k_1=2$	196.9	197.9	198.3
$k_1=3$	195.6	197.1	197.7
$k_1=4$	194.4	196.4	197.2
$k_1=5$	193.1	195.7	196.5

, according to the simulation waveforms in Figure 6, Figure 7 and Figure 8, the specific DC steady-state voltage values of adaptive exponential droop control, adaptive droop control, and traditional droop control are obtained through simulation.

Table 3. Comparison table of calculation results with different droop coefficients.

Droop Coefficient	$\Delta {\mathit{U}}_{\mathit{dc}1}(/\mathbf{kV})$	$\Delta {\mathit{U}}_{\mathit{dc}2}(/\mathbf{kV})$	$\Delta {\mathit{U}}_{\mathit{dc}3}(/\mathbf{kV})$	${\mathit{\eta }}_1(/\%)$	${\mathit{\eta }}_2/(\%)$
$k_1=2$	3.1	2.1	1.7	32.26	45.16
$k_1=3$	4.4	2.9	2.3	34.09	47.73
$k_1=4$	5.6	3.6	2.8	35.71	50
$k_1=5$	6.9	4.3	3.5	37.68	49.28

is the comparison of the DC steady-state voltage deviation of the three droop control strategies mentioned above, in which ${\eta }_1$ and ${\eta }_2$ are expressed as the reduction rates of the adaptive droop and the adaptive exponential droop control voltage deviation relative to the traditional droop control in the literature [26]; the higher the relative reduction rate, the better the DC voltage steady-state performance is. In Table 3, it can be seen that the adaptive droop control can effectively improve the DC voltage quality of the VSC-MTDC system, and the adaptive exponential droop control proposed in this paper can further improve effectively. Therefore, the method proposed in this paper is better than the other two methods in terms of DC voltage quality.

4.2. Power Fluctuation Simulation Results

With the droop control coefficient $k_1=3.5$, the load power fluctuated in the range of 240–400 MW, 240–480 MW, and 240–600 MW. The results are shown in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14. Lines A, B, and C in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14 represent the adaptive exponential droop control strategy, the adaptive droop control strategy, and the traditional droop control strategy, respectively, which are consistent with other simulation figures.

Similarly, the specific DC steady-state voltage values of the three control strategies shown in

Table 4. Comparison of simulation results with different load fluctuations.

VSC3 Active Power Variation	DC Side Steady-State Voltage (kV)
	Traditional Droop Control	Adaptive Droop Control [26]	Adaptive Exponential Droop Control
240–400 MW	195.8	197.7	198.2
240–480 MW	195.0	196.7	197.4
240–600 MW	193.8	195.1	196.2

were obtained through simulation, according to the simulation waveforms in Figure 9, Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14; as shown in

Table 5. Comparison of calculation results with different load fluctuations.

VSC3 Active Power Variation	$\Delta {\mathit{U}}_{\mathit{dc}1}(/\mathbf{kV})$	$\Delta {\mathit{U}}_{\mathit{dc}2}(/\mathbf{kV})$	$\Delta {\mathit{U}}_{\mathit{dc}3}(/\mathbf{kV})$	${\mathit{\eta }}_1(/\%)$	${\mathit{\eta }}_2(/\%)$
240–400 MW	4.2	2.3	1.8	45.24	57.14
240–480 MW	5	3.3	2.6	34	48
240–600 MW	6.2	4.9	3.8	20.97	38.71

, the DC steady-state voltage deviation was calculated, and the proposed adaptive exponential droop control could further reduce the DC steady-state voltage deviation caused by the load power fluctuation, thereby improving the DC voltage quality.

4.3. Three-Terminal System N-1 Verification

The N-1 operation of the VSC-MTDC system means that after any converter station is removed due to a fault, the rest of the system maintains stable operation. In order to verify the application reliability of the proposed adaptive exponential droop control strategy in the VSC-MTDC system as well as its superiority compared with other droop control strategies, the simulation was set to cut off the VSC1 converter at 0.9 s in order to check the line voltage deviation.

Figure 15 is the adaptive exponential droop control of the DC current of each side converter when VSC1 de-runs. When VSC1 de-runs, the power of VSC1 at the sending end is 0 at this time, and the transmission power of VSC2 gradually reaches the maximum until it is twice the original value. Furthermore, in the process of VSC2 power rising, VSC3 power first decreases and then gradually increases to the original due to the influence of nonlinear control. Figure 16 shows the line voltage during N-1 operation under the three control strategies. According to the characteristics of the droop control curve, when the power of the converter on the droop control side increases, the corresponding line voltage will drop, but the deviation will stay within the allowable range so that it conforms to the VSC -MTDC system N-1 runs. Consequently, the reliability of the strategy in system application is verified.

5. Conclusions

Aiming at resolving the problems of large DC voltage deviation and the poor power distribution characteristics of converters caused by power fluctuations in MTDC systems, based on traditional droop control, this paper proposed an automatic control coefficient of droop control that uses power deviation through mathematical theory derivation. Finally, based on the three-terminal flexible HVDC transmission system built for this study, the advantages and disadvantages of the adaptive exponential droop control were verified according to the DC voltage and converter power simulation under different droop coefficients and different load power fluctuations.