The five-terminal VSC-MTDC system’s structure is presented in Figure 1. VSC1∼VSC5 depict voltage-source type converter stations. The DC side is connected through a DC network in parallel, and the AC side is connected to its corresponding AC grid, and the resistance of the lines between converter stations is 0.25 Ω.

Analogous to the primary frequency regulation characteristics of a conventional generator, droop control avoids the reliance on inter-station communication and leverages the characteristic curves of DC voltage and active power to rapidly distribute unbalanced power and maintain stable control of DC voltage. Figure 2 illustrates the structure of the DC voltage-active power (U _dc-P _s) controller. e t = U dc − U dcref + k P s − P sref . (1) where U _dc and U _dcref represent the measured and reference values of DC-side voltage, respectively; P _s and P _sref represent the measured and reference values of the active power of the converter station, respectively; k is the droop coefficient; and e(t) is the PI controller input (Song et al., 2021; Xiong et al., 2022).

When the system is in a stable state, the input of the PI controller is zero, and Eq. 1 can be simplified to: U dc = U dcref + k P sref − P s . (2)

It is assumed that the DC system has N converter stations, of which 1∼m converter stations use conventional droop control; m+1∼n converter stations use constant active power control, and n+1∼N converter stations use constant AC voltage control.

The sum of the active power reference values of 1∼m droop stations P _r is P r = ∑ i = 1 m P i . (3) where 1 ≤ i ≤ m.

The sum of active power P _t of m+1∼n constant power stations is P t = ∑ b = m + 1 n P b . (4) where m+1 ≤ b ≤ n.

The sum of active power P _w of n+1∼N constant AC voltage converter stations is P w = ∑ j = n + 1 N P j . (5) where n+1 ≤ j ≤ N.

The conventional droop control operating curve is shown in Figure 3.

According to Figure 3, at the initial steady state, the system should be operated in state 1. If a power disturbance, ΔP, occurs in the DC system, the m droop stations will use their own droop characteristics to dissipate the unbalanced power while the system operating point moves. When the steady state is reached again, the system is set to run in state 2 and the DC voltage deviation is ΔU _dc. From Figure 3, it is apparent that the unbalanced power ΔP _i borne by the droop station i has the following relationship with the DC voltage deviation ΔU _dc. Δ P i = Δ U dc k i . (6)

According to the principle of energy conservation, the sum of the active power variation of each droop station should be equal to ΔP Δ P = ∑ i = 1 m Δ P i . (7)

Combining Eq. 6 with Eq. 7, we can obtain Δ U dc = Δ P ∑ i = 1 m 1 k i . (8) Δ P i = Δ P k i ∑ i = 1 m 1 k i . (9)

From Eq. 8, it is apparent that ΔU _dc is proportional to ΔP and inversely proportional to the sum of the reciprocals of the droop coefficients of m droop stations. This shows that when the power of the DC system fluctuates, all droop stations will cooperatively dissipate the unbalanced power according to their own droop curves (Li et al., 2019b; Liu et al., 2023).

From Eq. 9, it is evident that the droop coefficient determines how much unbalanced power is borne by the droop station in the dynamic regulation process. If each converter station adopts the same droop coefficient, all converter stations share the unbalanced power; if each converter station adopts different droop coefficients, converter stations with smaller droop coefficients will bear more unbalanced power, and converter stations with larger droop coefficients will bear less unbalanced power.

Aiming at the DC voltage deviation problem inherent in the conventional droop control for dissipating the unbalanced power at the converter station, this paper proposes an improved coordinated control strategy for the multipoint DC voltage of the VSC-MTDC system, in which the unbalanced power is injected as a feed-forward compensation quantity into the droop control, and quasi non-error DC voltage regulation is realized by shifting the droop curve. The characteristic curve of the improved coordinated control strategy is shown in Figure 4.

Let the system be in state 1 at the initial steady state. When power disturbance ΔP occurs in the DC system, each droop station is given a feed-forward compensation amount of magnitude ΔP _i , at this time, the stable operation point of the system is changed from A _i to D _i to realize the flattening of the droop curve. P _isref and P _i '' are the reference values of active power before and after adjustment of the converter station, respectively, which satisfies P i ″ = P i sref − Δ P i . (10)

And then the converter station dissipates the unbalanced power according to the droop characteristic, and finally the system operates stably near the C _i point, realizing the quasi non-error DC voltage regulation. Compared with the conventional droop control, the active power transmitted by the converter station does not change, while the DC voltage deviation is approximately zero, and the stability of the system is greatly improved (Fu Q. et al., 2021; Liu Q. et al., 2022).

It is worth stating that when the DC network has a power surplus, the system DC voltage will increase, and under the improved droop control strategy in this paper, the droop curve will be shifted to the upper right, and the DC voltage will first rise and then fall to about the initial value. When the system has a power deficit, the system DC voltage decreases, and under the improved droop control strategy in this paper, the droop curve will level off to the lower left, and the DC voltage first decreases and then increases, and finally stabilizes at the initial value.

To reasonably allocate the unbalanced power in the DC system to each droop station, this paper introduces the available power margin of the converter station to the power allocation coefficients and quickly adjusts the active power reference value. The power balance allocation scheme is shown in Eq. 11. P i ″ = P i sref − P i ⁡ max − P i sref × P t t + Δ t + P w t + Δ t − P t t + P w t ∑ i = 1 m P i ⁡ max − P i sref . (11) where Δt system sampling time. Under this power balance allocation scheme, the control system only needs to collect the active power value of the non-droop station, and update the active power reference value to the droop station when, and only when, the DC system current changes. In the rest of the cases, the droop station only needs to operate stably according to the latest updated active power reference value. At the same time, this power balance allocation scheme requires very low inter-station communication, and even when the communication of each converter station is interrupted, it can still operate normally according to the conventional droop control method.

According to the above analysis, the unbalanced power of the converter station is injected into the droop control as a feed-forward compensation quantity, and the quasi-differential-free regulation of the DC voltage can be realized by shifting the droop curve. However, when the system returns to the steady state again, the DC voltage deviation still exists, which is not conducive to the stable operation of the system. To address this problem, Res (Liu H. Y. et al., 2022) designs an additional DC voltage stabilizer with the control structure shown in Figure 5. Δ P refu = K u d Δ U dc d t + K v Δ U dc . (12) where ΔP _refu represents the value of additional power generated by the controller; ΔK _u refers to the equivalent inertia coefficient; and ΔK _v is the equivalent damping coefficient.

Analogous to the synchronous generator inertia and damping, the controller takes ΔU _dc as the input, and reduces the DC voltage deviation by adjusting the active power reference value during the power perturbation process. However, the DC voltage deviation is generally large in the early stage of power disturbance occurrence, and if ΔK _u and ΔK _v take larger values, it is easy to trigger overshooting of the converter station and generate power oscillations; if ΔK _u and ΔK _v take smaller values, the improvement of DC voltage is generally effective.

Aiming at the above problems, this section proposes an improved control method for the secondary regulation of DC voltage to restore the DC voltage to the initial value in case of slight changes in the active power of the converter station to realize the non-error DC voltage regulation. The structure of the improved DC voltage secondary regulation controller designed in this paper are shown in Figure 6 and Eq. (13): Δ P refu = K p 2 + K i 2 / s U dc − U dcref . (13)

Similarly, the DC voltage secondary regulation controller takes ΔU _dc as input and utilizes the steady state static-free characteristic of the PI controller to realize the static-free regulation of the DC voltage. Unlike the Res. (Liu H. Y. et al., 2022), the DC voltage secondary regulation controller does not have ΔK _u and ΔK _v parameters, so there is no need to worry about the influence of ΔK _u and ΔK _v values on the effect of improving the DC voltage. Meanwhile, to avoid the effect of ΔP _refu on the droop control at the early stage of the disturbance occurrence, the DC voltage secondary regulation controller can be operated again to restore the DC voltage to the initial value after some time of the improved droop control action in Section 3.1. Therefore, in this paper, the PI parameters for the droop control and the improved PI parameters for the DC voltage secondary regulation are set reasonably, K _p1 and K _i1 are set to 0.2 and 0.5 respectively, and K _p2 and K _i2 are set to 0.05 and 0.1 respectively. The step response times of the two PI controllers are differentiated to achieve the purpose of the successive response of the two controllers.

To verify the effectiveness of the improved droop control strategy proposed in Section 3.1 of this paper, the comparative simulations set up in this section are as follows

Control Method 1 (CM1): conventional droop control; Control Method 2 (CM2): the control method proposed in Res. (Li and Gao, 2020); Control Method 3 (CM3): the improved droop control proposed in this paper.

At t = 3 s, the active power command value of VSC3 increased from 115 MW to 175 MW. Figure 7 presents a comparison of the simulation outcomes under the three control methods.

(a) Active power of VSC1

As can be seen in Figures 7A–D, in the initial state, the actual power values of VSC1 and VSC2 under the three control methods are close to the power command value because of the existence of losses in the system, but there is a slight difference. After the change of the active power command value of VSC3 at the moment of 3 s, it is equivalent to the power surplus of the DC system, and the DC voltage of the system is gradually reduced. Under the control of CM1, the droop stations VSC1 and VSC2 adjust their active power command values according to the amount of change in DC voltage by a fixed droop coefficient, and the transmitted power changes from −219.5 MW and −100 MW to −239.5 MW and −140 MW, respectively, with a sharing amount of 20 MW and 40 MW, and a DC voltage deviation of 6.08 kV.

Under CM2 control, the droop coefficient is obtained by calculating according to the fixed margin of the converter station, which optimizes the power distribution of the DC system compared to CM1 control, thus reducing the deviation of the DC voltage. The sharing of VSC1 and VSC2 is 15.2 MW and 44.8 MW, respectively, and the deviation of the DC voltage is 4.57 kV.

Under the control of CM3, it can be seen from Figure 7D that the DC voltage first rises and then decreases, which is in line with the analysis in Section 3.1 of this paper. By feedforward compensation of unbalanced power to the active power command value of VSC1 and VSC2, the sharing of VSC1 and VSC2 are 20 MW and 40 MW, respectively, and the DC voltage deviation is 0.12 kV. Compared with CM1, the active power sharing of VSC1 and VSC2 does not change, while the DC voltage deviation is approximately zero, and quasi non-error DC voltage regulation is realized. The simulation results are shown in Table 2.

To verify the effectiveness of the power balance allocation scheme proposed in Section 3.2 of this paper, the comparison simulation set up in this section is as follows

Control Method 1 (CM1): Improved droop control; Control Method 2 (CM2): Improved droop control + power balance allocation scheme.

At t = 3 s, the VSC3 active power instruction value has increased from 115 MW to 175 MW. Figure 8 illustrates the comparison between simulation outcomes obtained with the two control methods.

As can be seen from Figures 8A–D, under CM1 control, the system’s unbalanced power is allocated to VSC1 and VSC2 through a fixed ratio without considering the actual power margin of the converter stations, which may easily cause some converter stations to be fully loaded and thus lose the ability to control the DC voltage when the unbalanced power is too large. The unbalanced power allocated to VSC1 and VSC2 is 20 MW and 40 MW, respectively, and the DC voltage deviation is 0.12 kV.

Under CM2 control, the system’s unbalanced power is reasonably allocated to VSC1 and VSC2 according to the power margin of the converter station, which avoids the problem that VSC1, which has a small power margin, can easily reach full load. The unbalanced power allocated to VSC1 and VSC2 are 14 MW and 46 MW, respectively, and the deviation of the DC voltage is 0.12 kV the simulation results are shown in Table 3.

To verify the effectiveness of the DC voltage secondary regulation control strategy proposed in Section 3.3 of this paper, the comparative simulations set up in this section are as follows:

At t = 3 s, the active power command value for VSC3 is reduced from 115 MW to 45 MW. Figure 9 presents a comparison of the simulation outcomes across the three control methods.

From Figures 9A–D, it can be seen that when the system has a power deficit, the DC voltage first decreases and then increases, which is consistent with the analysis in Section 3.1 of this paper. Under CM1 control, since the system does not have a DC voltage recovery controller, the DC voltage stabilization value still deviates from the initial value, with a deviation value of 0.11 kV. Under CM2 control, it can be seen from Figure 9D that the additional DC voltage stabilizer can further reduce the DC voltage deviation, with a deviation value of 0.08 kV. Since the ΔK _u and ΔK _v parameters are fixed values, which limits the DC voltage recovery effect, the DC voltage is gradually recovered under CM3 control by adjusting the active power reference values of VSC1 and VSC2 in small increments using the static-free characteristic of the PI controller. When the system reaches the steady state again, the DC voltage deviation is 0.02 kV. Simulation results are shown in Table 4.