Abstract
Diode-rectifier-based high-voltage direct current (DR-HVDC) systems are considered an attractive solution for integrating offshore wind farms (OWFs). Grid-forming (GFM) control with a rational reactive power allocation capability is crucial for the safe operation of numerous wind turbines (WTs). Most typical GFM controls aim to share surplus reactive power of the system equally among WTs, easily rendering capacity overloads for WTs that are outputting high levels of active power. In this paper, a novel GFM control for OWFs is proposed, allowing for adaptively allocating the reactive power according to the actual active power output of WTs. Firstly, the reactive power characteristics of the AC collection networks and WTs are analyzed across a wide wind power range. Then, combining the positive correlation of WT active power with the output AC voltage, a Q-θ type GFM control for WTs is presented. The adaptive reactive power allocation mechanism and the parameter design of the Q-θ based reactive power controller are elucidated, ensuring that WTs with lower active power output contribute more reactive power to the system than WTs with higher active power output. The AC impedance models of WTs under various GFM controls are established to evaluate the impact of different reactive power controllers. Finally, the feasibility of the proposed control is validated in PSCAD/EMTDC, accompanied by stability analysis.
OFFSHORE wind power has been widely recognized for its abundant resource potential [
Recently, diode-rectifier-based HVDC (DR-HVDC) systems have garnered considerable attention with the superiority of light-duty and cost-effective converters [
In [
Given the potential destabilizing effects of additional PLLs on OWFs, a Q-f based self-synchronizing type GFM control (i.e., Q-f type GFM control) is explored in [
1) A Q-θ type GFM control of WTs is proposed to optimally and adaptively allocate reactive power according to their actual active power, effectively precluding the overload of the WT GSC without increasing the rated capacity under varying conditions.
2) The impact of different reactive power controllers under different GFM controls on the stability of DR-HVDC systems is revealed, which recommends the conservative reactive power control parameter settings to mitigate negative damping.
3) The potential risk of high-frequency oscillations in the distributed PLL type GFM control, which results from the additional control delay introduced by a PLL-based AC voltage controller, is highlighted.
The remainder of the paper is outlined as follows. Section II illustrates the system configuration and operating characteristics of the DR-HVDC based OWF integration system. In Section III, a Q-θ type GFM control of WTs based on adaptive reactive power allocation is presented, and the mechanism of reactive power allocation among WTs is analyzed. Moreover, Section IV presents the sequence impedance modeling of wind farms under different GFM controls. The feasibility of the proposed control and system stability are validated by simulations in Section V. Finally, Section VI provides the conclusion.

Fig. 1 Topology of DR-HVDC based OWF integration system.
In light of the fact that the reactive power compensation of DRs is implemented by WTs, the rated capacity of WT GSCs and AC transformers must be increased by approximately 6.5% [
Considering the impact of the leakage inductance of the AC transformer TDR on the DR commutation process, the voltage relationship between the AC and DC sides of DRs is expressed in (1) with reference to [
(1) |
where vdcDR and idc are the DC voltage and DC current of DRs, respectively; VPCC is the root mean square (RMS) value of PCC phase voltage vPCC; is the rated angular frequency; kDR is the AC transformer ratio of TDR; and LTDR is the leakage inductance of TDR.
Based on (1), the transmitted active power PDR of DRs can be written in (2), ignoring the loss of the rectifier and the HVDC link [
(2) |
where vdcMMC is the DC voltage of onshore MMC.
According to the characteristics of DRs in [
(3) |
(4) |
where and are the DR commutation angle and power factor angle, respectively.

Fig. 2 Diagram of reactive power distribution within AC collection networks.
In
Denote the reactive power generated by OWFs and AC filters as QWF and Qfilters, respectively. The balance of reactive power on the PCC side is satisfied in (5)-(7). As derived in (6), the reactive power components Qsti in OWFs originate in each series of WT string i.
(5) |
(6) |
(7) |
where QWTij is the reactive power generated by WT j at WT string i; Qcij is the reactive power generated by AC collection cable j at WT string i; n is the number of AFBs in operation; and Qbank is the reactive power capacity per bank.
Take WTij at WT string i as an example. Regardless of the effect of capacitor reactive current on the voltage drop of a series branch, QWTij and Qcij in (6) can be further expressed as:
(8) |
(9) |
where QGij is the reactive power generated in the GSC of WT j at WT string i; Vfij and IGij are the RMS values of phase voltage vfij and current iGij in the GSC of WT j at WT string i, respectively; Lf and Cf are the inductance and capacitance of LC filters in WTs, respectively; and LT is the leakage inductance of AC step-up transformer.
As analyzed in (8) and (9), the reactive power generated in the AC collection networks QAC can be defined as:
(10) |
Based on (3) and (10),

Fig. 3 Reactive power characteristics of DRs and AC collection networks in various wind power scenarios.
It is shown that the reactive power QDR and QAC calculated in (3) and (10) are basically consistent with the simulation results in PSCAD. The AC collection networks exhibit a capacitive characteristic that is enhanced as the wind power PWF decreases [
Quantity | Value | Quantity | Value |
---|---|---|---|
Rated wind power | 1000 MW | Lf | 0.15 p.u. |
Rated frequency f1 | 50 Hz | Cf | 0.08 p.u. |
Rated PCC voltage | 66 kV | Lci1-9, Lci10 | 0.001 H, 0.01 H |
kDR | 3.92 | Cci1-9, Cci10 | 0.33 μF, 3.3 μF |
LTDR | 0.18 p.u. | Rc1-9, Rc10 | 0.038 Ω, 0.38 Ω |
LT | 0.07 p.u. | x, yi | 2 km, 20 km |
Rhp, Lhp, Chp | 5.1 Ω, 0.96 mH, 36.5 μF | L1, C1, R2,L2, C2 | 2 mH, 36.5 μF, 200 Ω, 0.05 mH, 1405 μF |
According to (5)-(10), denote QGtot as the sum of QGij, which can be written in (11). It can be observed that the variation of QGtot contains the variation of QDR, QAC, and Qfilters, which is predominantly attributable to wind power fluctuations and AFB switching.
(11) |
Under the Q-f type GFM control and distributed PLL type GFM control, the total reactive power demand is balanced equally by each WT based on its rated capacity. Since the frequency fG of vPCC is a common feature for all WTs, the equal reactive power allocation of WTs can be achieved by a Q-f droop controller with the same reactive power reference and droop coefficient, shown as:
(12) |
where QGref is the reference of QGij; and kQ is the proportional gain.

Fig. 4 Reactive power characteristics of Q-f droop based GFM-type WTs considering abnormal conditions of AFBs.
It can be observed that WT GSCs are required to absorb the surplus reactive power under normal operation. When the aggregated wind power output of OWFs is less than 0.4 p.u., QGtot equally balanced by each WT falls below -0.15 p.u.. When a conventional Q-f droop controller is used, even WTs operating at the rated wind power are forced to share the reactive power equally. It poses a risk of overloading the WT. In particular, as shown by red dotted lines in
III. Q- Type GFM Control of WTs Based on Adaptive Reactive Power Allocation and Mechanism of Reactive Power Allocation Among WTs
In general, WTs need to achieve the maximum power point tracking (MPPT) of wind power generation and control the DC voltage inside.
Due to the uncontrollable nature of DRs, WTs also need to generate AC voltage and frequency and regulate the balance of reactive power in the AC collection network. Given that the DC voltage of WT is maintained by its machine-side converter (MSC), the voltage of WT DC link can be decoupled from the grid side, minimizing the interaction effects [
As evaluated in

Fig. 5 Diagram of Q-θ type GFM control based on adaptive reactive power allocation for WT GSCs.
The Q-θ based reactive power controller contains two operating modes, activated by an enabling signal En based on the capacity limit criterion, as shown in (13) and (14).
(13) |
(14) |
where θGij is the voltage angle of WT j at WT string i; and θN is the reference voltage angle of each WT generated by a voltage-controlled oscillator. Since QGij is adjusted via the change of θGij, QGref in Q-θ droop control can be set to be 0.
The activation logic of the enabling controller is concisely outlined as follows. When SGij reaches its upper limit SGup, the enabling controller is activated and the output of the comparator in
In
Take the reactive power allocation among WT strings as an example. For simplicity of analysis, WTi1-WTi10 of WT string i are aggregated as an equivalent WT i. Referring to [
(15) |
(16) |
where Psi and Qsi are the transmitted active and reactive power of equivalent WT i, respectively; θPCC is the voltage angle of vPCC; and Vsi and θsi are the RMS values of the phase voltage and angle at the terminal of equivalent WT i, respectively.
Similar to (13), the reactive power output characteristics of equivalent WT i can be written as:
(17) |
Combining (15)-(17), a key relationship between Vsi and Psi is derived as:
(18) |

Fig. 6 Relationship between Vsi and Psi in (18).
Notably, Psi is positively correlated with Vsi, indicating that equivalent WT i with lower active power output Psi will experience smaller terminal voltage Vsi. Thus, based on (16), these WTs will absorb more reactive power compared to their counterparts with higher Psi. Within a reasonable variation range of LCeqi, the impact on Vsi is minimal, suggesting that the reactive power outputs of WTs with the same active power level are basically equal.
As shown in
Under abnormal conditions where AFBs are not available, WTs with higher active power output are forced to provide the reactive power for DRs, as evident from
Combined with (15), the voltage angle difference between equivalent WTs 1 and 2 is further written in (19), where the length of AC collection cables is nearly the same.
(19) |
Based on (17), the relationship between the voltage angle and reactive power of equivalent WTs 1 and 2 is expressed as:
(20) |
As shown in (19), the voltage angle difference between equivalent WTs 1 and 2 is mainly determined by their active power output, ignoring the slight effect of voltage magnitude. When the active power output of WTs is determined, the voltage angle difference between θs1 and θs2 can be constant. Hence, their reactive power difference in (20) will be inversely proportional to the droop coefficient kQp. Thus, kQp is the key factor in the reactive power allocation among WTs. Substituting (19) into (20), the relationship between active and reactive power of equivalent WTs 1 and 2 can be obtained as:
(21) |
As in (21) is less than , the reactive power difference will be greater than the active power difference, ignoring the slight dissimilarities of Vs1 and Vs2. This will cause the WTs with higher active power output to generate excess reactive power during normal operation, imposing WTs with lower active power output to absorb additional reactive power. Hence, kQp cannot be selected too small. The objective of the lower limit of kQp is to maintain the reactive power output of WTs operating at the rated active power output essentially unchanged under wind power fluctuations of OWFs.
Assume that all equivalent WTs are operating at their rated capacity. At a given time, the active power output of equivalent WT 1 is slightly reduced by . Based on (21), the reactive power fluctuations of equivalent WTs 1 and 2 can be expressed in (22) regardless of the slight variation of voltage variables.
(22) |
where and are the variation of transmitted reactive power and active power of equivalent WT i, respectively.
Based on (3), the variation of QDR is written as:
(23) |
To determine the lower limit kQp,min of kQp, - are ideally set to be 0. Thus, combining (22) with (23), kQp,min can be obtained as:
(24) |
where is the maximum power factor angle of DRs operating at rated active power output (tan ).
Additionally, it is not advisable to set kQp too large. For example, assume that kQp in (21) is 50 times , i.e., . When the difference of active power output between WTs is at its maximum of 1 p.u., the reactive power deviation is only about 0.02 p.u.. This will not only impede WTs with lower active power output from contributing more reactive power, but also result in poor dynamic response. Hence, is set to be 0.75 in this paper.
The sequence impedance models of the DR-HVDC system for OWF integration comprise two parts: DR-HVDC and wind farms. The impedance model of DRs has been established in [
The equivalent circuit of WT string i is established in Fig. SA1 of Supplementary Material A. Referring to [
(25) |
where , , and are the small-signal vector forms of phase voltage vfi1, modulation index dGi1, and current iGi1 in the GSC of WTi1, respectively; VdcG is the rated DC voltage of GSC; and is the harmonic impedance matrix of Lf.
Based on the architecture of the proposed Q-θ type GFM control in
(26) |
where and are the small-signal vector forms of PGi1 and θGi1, respectively; I is the unit matrix; GHv, GHi, and GHP are the transfer function matrices decided by the dual-loop PI controller, the current PI controller, and the active power PI controller, respectively [
and can be expressed as follows. Considering that En is basically not activated under normal operation, it has been omitted from subsequent equations.
(27) |
(28) |
(29) |
(30) |
where is the Toeplitz matrix defined for convolution operations in the frequency domain; is the small-signal vector form of terminal voltage vGi1 in WTi1; is the small-signal vector form of QGi1; and fp is the frequency of the additional positive-sequence small-signal voltage.
Substituting (27)-(30) into (26), the analytical model of the proposed Q- type GFM control can be expressed as:
(31) |
(32) |
(33) |
In conclusion, combining (25) with (31), the relationship between the AC current and voltage of GSC in WTi1 can be derived in (34). Thus, the sequence impedance model of WTi1 containing the impedance of its AC step-up transformer is derived in (35). Moreover, according to the series and parallel structure of WTij and WT string i in
(34) |
(35) |
where Z and Z are the positive- and negative-sequence impedances, respectively, while their frequency-coupling terms are Z and Z; Z and Z are the sequence impedances of Cf and LT, respectively; and // represents the relationship of impedances connected in parallel.
As proposed in [
(36) |
It is observed that the difference between (28) and (36) is mainly attributed to an extra term 1/s in the Q-f type GFM control. Thus, the effects of at medium- and high-frequency bands will be weakened. Conversely, the impedance characteristics at the rated frequency range are amplified.
In comparison to the above two GFM controls, the distributed PLL type GFM control in [

Fig. 7 PLL-based frequency controller of distributed PLL type GFM control.
The added small-signal model can be written as:
(37) |
where and are the small-signal vector forms of vfqref and , respectively; and GHPLL is the transfer function matrix decided by the disturbance of PLL in dq transformation.
The small-signal models for and can be further expressed in (38) and (39). It is noted that the disturbance of PLL in (38) tends to diminish the damping characteristics of the GSC near f1 [
(38) |
(39) |
where is the transfer function of PLL; is the small-signal vector form of vfq; Tv4 is the conversion gain matrix from to , which can be referred to [
To verify the feasibility of the proposed control, a ±320 kV/1000 MW DR-HVDC system shown in
Component | Parameter | Value |
---|---|---|
GFM control | Proportional and integral parameters of AC current controller | , |
Proportional and integral parameters of AC voltage controller | , | |
Proportional and integral parameters of active power controller | , | |
Proposed Q-θ type GFM control | Proportional and integral parameters of reactive power controller | , |
Q-f type GFM control | Proportional parameter of reactive power controller | |
Distributed PLL type GFM control | Proportional parameters of reactive power controller and frequency controller | , |
Proportional and integral parameters of PI controller of PLL | , |
To test the adaptive capability of the proposed control across a wide power range, the wind power is adjusted as follows. OWFs operate in grid-connected mode at 2 s, where the active power of WT11-WT110 at WT string 1 is 1 p.u.. WT_2 and WT_3 are in the initial stages of power generation. During 2-16 s, PG11-PG15 remain unchanged, and PG16-PG110 decrease by 0.05 p.u. at 8 s. WT11-WT110 are used to simulate some WTs with higher active power output in OWFs. The wind power of WT_2 and WT_3 fluctuates disorderly, representing the majority of WTs.

Fig. 8 Simulation results of wind power fluctuations in OWFs under proposed control. (a) Active power of WTs. (b) Active and reactive power of DRs. (c) Reactive power response characteristics of DRs, AFBs, AC collection networks, and WTs.
It can be observed in

Fig. 9 Comparisons of simulation results of wind power fluctuations under different GFM controls. (a) Reactive and apparent power of WTs under proposed control. (b) Reactive and apparent power of WTs under Q-f type GFM control.
It is indicated in
However, in
The simulation setup for wind power fluctuations under AFB maloperation conditions is the same as that in

Fig. 10 Reactive power response characteristics of DRs, AFBs, AC collection networks, and WTs under AFB maloperation conditions.

Fig. 11 Comparisons of simulation results under AFB maloperation conditions. (a) Reactive and apparent power of WTs under proposed control. (b) Reactive and apparent power of WTs under Q-f type GFM control. (c) AC currents of WTs under proposed control. (d) AC currents of WTs under Q-f type GFM control.
Thus, the proposed Q- type GFM control based on adaptive reactive power allocation is an effective scheme for ensuring the safe operation of OWFs.
Since the fault ride-through capability of DR-HVDC systems is fully evaluated under an offshore fault at the PCC bus and an onshore fault [
When an AC fault occurs on the WT110 side shown in

Fig. 12 Simulation results of a three-phase solid short-circuit fault occurring at head end of WT110. (a) DC voltages and AC currents of WTs under proposed control. (b) AC voltage at PCC under different GFM controls. (c) Active power of WTs under different GFM controls. (d) Reactive power of AC filters under different GFM controls.
After 0.1 s, the AC breaker is triggered to disconnect WT string 1. The wind power fluctuations are the same as those in
The AC currents of WT11-WT110 will rise temporarily, which can be suppressed quickly by the fault ride-through control of the WTs. While WT_2 and WT_3 can maintain the DC voltages and AC currents within the normal range after a transient shock. As
The validation of the sequence impedance model under the proposed control is shown in Fig. SA4 of Supplementary Material A. It can be observed that the simulation results of in the red circles match well with their analytical results in the blue line, validating the correctness of established models. It is noted that the frequency-coupling terms and have negligible impedance amplitudes in the high-frequency bands above 1.5 kHz, so there is an allowable measurement error.
Considering that dual-loop and active power PI controllers are common components in the proposed control, Q-f type GFM control and distributed PLL type GFM control, this paper mainly analyzes the impacts of various reactive power controllers on system stability. Referring to [

Fig. 13 Equivalent sequence impedances of DR-HVDC and OWFs under proposed control with different kQp.

Fig. 14 Equivalent sequence impedances of DR-HVDC and OWFs under Q-f type GFM control with different kQ.

Fig. 15 Equivalent sequence impedances of DR-HVDC and OWFs under distributed PLL type GFM control with different .

Fig. 16 Time-domain simulation results when kω is increased from 0.2 to 0.46 at 7 s. (a) Waveforms of vPCC. (b) FFT of vPCC when kω=0.46 (fundamental is 10 Hz).
This paper proposes a Q- type GFM control based on adaptive reactive power allocation for OWFs connected to a DR-HVDC system. The active power controller of WTs coordinates the AC voltage amplitude at the PCC bus, while the reactive power controller automatically allocates reactive power according to the voltage angle difference among WTs. The capacity of WTs with lower active power output can be optimally utilized, thereby allowing WTs with higher active power output to absorb less reactive power.
Together with the enabling controller for capacity limitation, the proposed control prevents the overload risk of WT GSCs without increasing their rated capacity, and even reserves capacity for other critical needs such as frequency support for the onshore grid.
The effectiveness of the proposed control is demonstrated under normal conditions and AFB maloperation conditions by comparing it with Q-f type GFM controls. In addition, the DR-HVDC system can continue to transmit most of the active power under an AC fault on the WT side by adopting the AC fault control scheme for AFBs. Based on the established impedance models under three GFM controls, it is indicated that the corresponding three types of reactive power controllers are not conducive to improving system stability, due to the potential introduction of negative damping around 50 Hz. Therefore, it is recommended to select a relatively small reactive power droop coefficient. Nevertheless, the DR-HVDC systems still show favorable stability characteristics. Furthermore, the stability analysis suggests that the distributed PLL-type GFM control may introduce high-frequency oscillations due to the additional control delay in the PLL-based frequency controller.
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