Traditional droop control strategies do not provide inertia and damping for the DC system. To effectively mitigate fluctuations in DC voltage, this study adopts the strategy of implementing VSG control on the GSVSC side. The governing equation for the VSG control of DC voltage is as follows (Li et al., 2018): C v d v d c v i r d t + D v v d c v i r − v d c n = P m v d c n − P e v d c n (1)

Where C _v and D _v represent the virtual capacitance and damping coefficient, respectively; v _dc ^vri and v _dcn denote the virtual voltage on the DC side and the rated voltage value, respectively. P _m and P _e represent the mechanical input power and electrical output power of the synchronous machine.

According to Eq. 1, it can be seen that the term for simulated inertia power is C _v dv _dc ^vir /dt, while the term for simulated damping power is D _v (v _dc ^vir –v _dcn ). Within the MTDC network, the VSG functions primarily to stabilize the DC voltage. In instances where the DC voltage experiences oscillations, the VSG will produce virtual inertia power to counter these fluctuations. This means that the generation of virtual inertia power is for the purpose of suppressing oscillatory phenomena. The speed of oscillations in DC voltage has a direct relationship with the generation of virtual inertia power within the system. Specifically, faster oscillations lead to an increased production of virtual inertia power, which offers inertia support during transient processes. Additionally, a larger virtual inertia constant equates to a slower rate of change in DC voltage, which is synonymous with a lower frequency of oscillation. The inertia power and damping power of the VSG are adjusted according to the changes in the DC side voltage. When the system tends toward stability, dv _dc ^vir /dt = 0. The damping power of the VSG is proportional to the offset of the DC side voltage, and as long as there is a voltage offset (v _dc ^vir –v _dcn ), there will be a contribution from the damping power.

According to the direction of power flow, the charging and discharging of the DC capacitor control the value of the DC-side voltage, which can be obtained as follows: N C v d c i S V S C i ⋅ d v d c i d t = P d c − P a c = P i n − P o u t = Δ P C (2)

Where N is the number of DC capacitors in the VSC-MTDC; C is the capacitance of a single DC capacitor; v _dci is the actual DC line voltage of the ith GSVSC controlling the DC-side voltage; S _VSCi is the apparent capacity of a single converter station; P _in /P _dc and P _out /P _ac are the input and output power of the VSC-MTDC system respectively; ΔP _C is the per-unit value of the electromagnetic power absorbed or released by the DC capacitor.

Equation 2 reveals that in order to ensure power balance between the GSVSC and the WFVSC, the DC voltage must be controlled to be constant. However, as a capacitor is an energy storage component, it can release or absorb power by adjusting the voltage. Therefore, it is feasible to use this characteristic of the DC capacitors in the VSC-MTDC system to balance power transmission (Liu and Chen, 2015). By regulating the DC link voltage through power compensation, the VSC-MTDC system can inject more or less active power into the onshore grid than the active power collected from the OWF.

Additionally, addressing the frequency regulation problem of OWF, a comprehensive inertia control method enables the wind turbine units to have an inertia response and primary frequency regulation ability (Liu and Chen, 2015; Wang and Tomsovic, 2019). In the active control loop of the wind turbine units, the rate of frequency change and frequency deviation Δf are introduced, thereby realizing the additional active power reference value in the frequency response process, as shown in Eq. 3. Δ P a d d = − K d d f d t − K p Δ f (3)

Where f represents the system frequency; Δf denotes the frequency deviation; K _d is the inertia control coefficient; K _p is the droop control coefficient; ΔP _add is the additional active power.

In cases of increasing frequency, wind turbine units can easily reduce their power output through wind curtailment. However, this study chiefly addresses situations in which units augment their output power by releasing rotor kinetic energy during frequency decreases. Using comprehensive inertia control as an exemplar, this response process primarily comprises two pivotal phases: the frequency support and rotor speed recovery stages, as illustrated in Figure 3. The entire response process can be categorized into two key phases: the frequency support phase and the rotor speed recovery phase. Here, ω ₀ and P ₀ denote the initial rotor speed and unit power, respectively; ω _off signifies the rotor speed at the conclusion of frequency regulation; P _W represents the unit’s power output, while ΔP _W indicates the power variation observed at the conclusion of the frequency support phase.

This process can be delineated into four distinct stages:

1. Stage A: This represents the wind turbine unit’s normal operational state, where it operates in Maximum Power Point Tracking (MPPT) mode, producing the optimal power level, denoted as P0.

Figure 3 illustrates that when ΔP _W exceeds a certain threshold, the system inevitably experiences secondary frequency drops.

Accounting for the intrinsic variability of OWF power output and the dynamic switching of grid-side loads, continuous adjustments are needed in the power of the VSC station. Without timely adaptation of the VSC station’s output power command, potential disturbances to the power balance may instigate significant frequency variations, potentially jeopardizing system stability.

Within this context, the OWF is modeled as a SG that provides supplementary power to the AC onshore grid. The balance of power in the SG is governed by the swing equation, P _m -P _e = Jωdω/dt, where J represents the moment of inertia parameter. Upon disturbance in the AC onshore gird, this balance is disrupted, prompting the energy stored in the capacitor to automatically offset the unbalanced power, leading to a reduction or oscillation in the DC voltage. Combining (3), OWF participates in system frequency regulation. From the principle of energy conservation, the following can be deduced: ∫ ω 1 ω 2 J OWF ω d ω d t + ∫ v d c 1 v d c 2 C v d c d v d c d t = ∫ ω 1 ω 2 J AC ω d ω d t (4)

When energy changes are defined as ΔE _c and ΔE _J , the subsequent derivation from Eq. 4 is: Δ E c = 1 2 C v d c 2 2 − 1 2 C v d c 1 2 Δ E J O W F = 1 2 J O W F ω 2 2 − 1 2 J O W F ω 1 2 Δ E J A C = 1 2 J A C ω 2 2 − 1 2 J A C ω 1 2 (5)

Equation 5 encapsulates the actual physical dynamics of the DC-side capacitor. Fundamentally, the energy stored in the rotor must be fully utilized during the control process to emulate the inertia stored in the virtual capacitor. For MTDC systems, this involves harnessing energy both from the OWF grid and the capacitor itself. Furthermore, voltage distribution within the MTDC system demonstrates non-uniform properties across the entire DC system. While some voltages within the MTDC grid may be well-regulated, other ports may display significant deviations from expected values (Kirakosyan et al., 2023).

In this study, to ensure fairness among VSCs participating in control tasks with VSG voltage control, the inertia power ΔP _iner = C _v dv _dc ^vir /dt is introduced as a consensus variable in the voltage control strategy. This variable is incorporated into the VSC station with voltage control as an additional power command, bolstering the DC-side voltage balancing capability, facilitating mutual inertia support, and improving frequency stability.

In the system’s initial operational phase, the power injection of the ith converter, denoted as P _i , aligns with its reference power, P _ref . However, in the event of active power imbalance, the actual power injection will diverge from its reference value. If there is an overabundance of active power in the DC system, the VSC will increase its drawn power from the MTDC system. Conversely, if the DC system’s active power is lacking, the VSC will decrease its drawn power. As such, this paper proposes the introduction of power deviation, as resolved by MPC, as an adjustment variable. This adjustment pre-distributes power injection values among the VSCs, in accordance with the VSC’s output power.

For the voltage control group, let the DC-side voltage it is about to control be v _dci , and this voltage can be measured in real-time. Based on the consensus protocol, the method of updating the supplemental power for the ith VSC in the group is presented as Eq. 6: φ i k + 1 = ∑ j = 1 N i d i j ( φ i k − φ j k ) + ε d i j Δ φ i r e f k (6) Where φ i k is the inertia power variable calculated by VSCi in the voltage control group after the kth iteration, which can be described as φ i = Δ P i n e r i . N _i is the total number of VSCs using DC-side voltage control; ε is the convergence factor (Li et al., 2019). Equation 6 provides interactive power instructions for both the grid side and wind power. d_ij represents the weight, and its value can be calculated using Eq. 7: d i j = a i j / ∑ j = 1 N i a i j (7) If there is a communication link between VSCi and VSCj, then a _ij = 1, otherwise a _ij = 0. Δ φ i r e f is the reference value of the unbalanced power of the ith VSC at the kth iteration. In order to regulate it, the voltage control performance evaluation function is first defined in Eq. 8: f v = 1 2 v d c i − 1 2 (8)

Let Δ φ i r e f k be the negative value of the partial derivative of the performance evaluation function f _v with respect to φ i . Then, we have: Δ φ i r e f = − ∂ f v ∂ φ i | φ i k = − ∂ f v ∂ v d c i ⋅ ∂ v d c i ∂ φ i = − v d c i − 1 ∂ v d c i ∂ Δ P i n e r (9)

Where ∂ v d c i / ∂ φ i represents the sensitivity of voltage to changes in injected active power, reflecting the sensitivity of VSCi’s voltage to variations in injected active power. The reflection value of active power on voltage change can be obtained from Eq. 9. As Eq. 1 indicates, the direct voltage of the VSC station plays a similar role in reflecting power balance within the DC grid. Considering that the direct voltage determines the flow of DC power, the correlation between voltage and injected power can be determined by constructing objective functions for voltage and power. Therefore, a method combining MPC for solving VSG has been proposed.

According to Eq. 1, the rotor motion equation of the VSG can be described as a state equation, as shown in Eq. 10. v ˙ t = P m t C v v d c n − P e t C v v d c n − D v C v v t y t = v t (10)

Where v(t) is defined as the difference between v _dc ^vir and v _dcn , which represents the voltage-related term in the state equation. P _m represents the control input variable, and P _e can be considered as the disturbance variable.

The discrete state equation is presented in Eq. 11: Δ v k + 1 = A z Δ v k + B u Δ P m k + B d Δ P e k y c k + 1 = C c Δ v k + 1 + y c k (11)

The coefficients in Eq. 11 are detailed in Eq. 12. A z = e − D v C v T s , B u = 1 C v v d c n ∫ 0 T s e − D v C v τ d τ , B d = 1 C v v d c n ∫ 0 T s e − D v C v τ d τ , C c = 1 (12)

Where T _s represents the sampling period. In this study, a three-step model prediction method due to its capacity for an extended prediction horizon is utilized. This allows for a more accurate representation of the system’s dynamic behavior and a stronger response to disturbances, as it can foresee changes further ahead and adjust the control strategy accordingly. The three-step prediction output vector is presented in Eq. 13: Δ Y p k + 1 k ≝ y c k + 1 k y c k + 2 k y c k + 3 k (13)

The voltage prediction equation is presented in Eq. 14: Δ Y p k + 1 k = S z Δ v k + I p y c k + S u Δ P m k + S d Δ P e k (14)

The coefficients in Eq. 14 are detailed in Eq. 15: S z = C c A z ∑ i = 1 2 C c A z i ∑ i = 1 3 C c A z i I p = 1 1 1 T S d = C c B d ∑ i = 1 2 C c A z i − 1 B d ∑ i = 1 3 C c A z i − 1 B d T S u = C c B u 0 0 ∑ i = 1 2 C c A z i − 1 B u C c B u 0 ∑ i = 1 3 C c A z i − 1 B u ∑ i = 1 2 C c A z i − 1 B u C c B u (15)

The objective function aims to minimize the weighted sum of the squared deviations of the voltage, Δv, and the change in the rated power, ΔP _m , as shown in Eq. 16: V p = ∑ i = 1 3 τ Δ v k + i | k 2 + υ Δ P m k + i | k 2 (16)

Where τ and υ are the weighting coefficients for voltage and power variations, respectively. Additionally, Δv (k+i|k) and ΔP _m (k+i|k) represent the errors between the predicted voltage and power at instant k. Based on the predicted inputs, Eq. 16 can be described as follows: V p Δ v k , Δ P m k = ⁡ min T y Y p k + 1 k − R k + 1 2 + T P Δ P m k 2 (17)

Where T _y = diag (τ, τ, τ) and T _P = diag (υ, υ, υ). T _y and T _P are the error weighting matrices for voltage and active power, respectively. R (k+1) represents the reference values for the control outputs at time k+1. The desired steady-state Δv is zero, hence R (k+1) = [0,0,0].

Due to the presence of constraints, it is not possible to obtain an analytical solution for the optimization problem in Eq. 17. Therefore, a numerical solution method is employed, transforming the constrained MPC optimization problem into a quadratic programming (QP) formulation. To facilitate the solution of the optimization problem in Eq. 17, auxiliary variables are defined: E p k + 1 k = R k + 1 − S z Δ v k − I p y c k − S d Δ P e k (18)

By substituting Eq. 14 into Eq. 17, and in conjunction with Eq. 18, it can be obtained that: V p Δ v k , Δ P m k = T y S u Δ P m k − E p k + 1 k 2 + T P Δ P m k 2 = Δ P m k T S u T T y T T y S u Δ P m k + Δ P m k T T P T T P Δ P m − 2 E p k + 1 k T T y T T y S u Δ P m k + E p k + 1 k T T y T T y E p k + 1 k (19)

Given that E p k + 1 k T T y T T y E p k + 1 k is independent of the variable ΔP _m (k), for the optimization problem, Eq. 19 is equivalent to Eq. 20: V p = Δ P m k T H P m k − G k + 1 k T Δ P m k (20)

The coefficients in Eq. 20 are detailed in Eq. 21

Where H = S u T T y T T y S u + T P T T P G k + 1 k = 2 S u T T y T T y E p k + 1 k (21)

By transforming the voltage constraint in Eq. 14 into the form C z ≥ b, Eq. 13 can be reformulated as Eq. 22: Δ Y p m i n k + 1 < Δ Y p k + 1 k < Δ Y p m a x k + 1 (22) − S u S u Δ P m k ≥ b k b k = S z Δ v k + I p y c k + S d Δ P e k − Δ Y p m a x k + 1 − S z Δ v k + I p y c k + S d Δ P e k + Δ Y p m i n k + 1 (23)

From Eq. 23, it can be seen that the MPC optimization problem is transformed into the following QP problem description: min Δ P m k Δ P m k T H Δ P m k − G m k + 1 k Δ P m k s a t i s f y : C u Δ P m k ≥ b k (24)

The coefficients in Eq. 24 are detailed in Eq. 25. C u = − S u S u (25)

By combining Eq. 21 and Eq. 24, the solution for MPC, denoted as ΔP _m ^*(k), can be obtained. According to the working principle of MPC, an initial control sequence will be applied to the system. In the next sampling period, the constraint optimization problem will be updated, and the solution to ΔP _m ^*(k) will be recalculated. According to Eq. 24, the power increment in a single VSG can be quickly solved, which is the correlation value of voltage and injected power. The variation of power increment for VSG can be written as: Δ P k = Δ P m * k (26)

Through Eq. 26, the association value between voltage and active power can be obtained, where Δ φ i r e f = v d c i − 1 Δ P k . The inertia of the system serves as a buffer against abrupt shifts in DC voltage, aiding in damping oscillations and allowing adequate time for the MTDC network to adjust active power. Based on the aforementioned analysis, it can be deduced that surplus power from consistency calculations always exhibits an inverse relationship with the rate of DC voltage change. Therefore, it is concluded that this supplemental power functions as a counterforce to DC voltage fluctuations, supplying inertia power to the system. This mechanism assists in mitigating voltage and power instabilities. Consequently, the proposed cooperative method can timely adjust the output power of GSVSC on the VSC-MTDC side when it is disturbed, increasing the system’s inertia, and reducing the rate of frequency change and voltage deviation. At the same time, the supplemental power provides second power error tracking for control, enhancing the convergence of power output. The proposed DC Voltage cooperative control based on VSG is illustrated as in Figure 4.

Figure 4 illustrates the control schematic of the proposed method. In addition, in accordance with the description in Eq. 5, to fully utilize the energy from the OWF, the frequency regulation equation for the OWF is further depicted as: Δ P a d d = − K d d f d t − K p Δ f − K φ ∑ j = 1 N i φ i (27) P r e f = − K d d f d t − K p Δ f − K φ ∑ j = 1 N i φ i + P m p p t (28)

Where K φ is the adjustment coefficient. P _ref denotes the reference value of wind farm units. Equation 27 represents the formula for the OWF to participate in frequency modulation, while Eq. 28 defines the operating reference value for the OWF. The introduction of the optimal power value P _mppt enables these units to effectively respond to variations in wind speed during the frequency support phase, particularly when coordinated with the interactive power parameter K φ ∑ j = 1 N i φ i . In comparison to Figure 3, the inclusion of cooperative power in Figure 5 leads to a further decrease in the parameter ΔP _W , consequently resulting in a diminished amplitude of SFD. As the system frequency gradually returns to normalcy and the rotor speed decreases, reaching a pivotal point C, P _m aligns with P _e . Subsequently, as P _e falls below P _m , the wind turbine’s rotational speed ascends gradually, ultimately returning to a stable state.

In response to frequency fluctuations in the AC grid caused by disturbances, the VSC station will adjust its output power. This adjustment is made by leveraging the energy from the OWF and the voltage of the DC-side capacitor to compensate for power imbalance, thereby enabling quick convergence of frequency. Simultaneously, leveraging the V−P characteristic, the total power generated by the OWF will be automatically distributed to the VSC station based on the state of DC voltage and AC power.

Different DC-side capacitors can contribute differently to the system’s power, thus the relationships between various virtual capacitors C _v , damping coefficients D _v , and DC-side capacitor C are analyzed. The given power of VSG1 is taken as the input variable, while the active power of VSG1 is set as the output variable. The feasible parameter domain for variables is solved based on Eqs 32 and 33, and the system’s resonance peak Mr under various parameters is obtained using the direct transmission matrix of the linearized model, as shown in Figure 6. The color gradient serves as a visual representation denoting the magnitude of the resonance peak Mr, which is applied here as an indicator for assessing the system’s comparative stability. Typically, a heightened resonance peak tends to correlate with diminished relative stability.

Figure 6 reveals the significant impact of capacitor capacity on system stability. As the capacitor size increases, the system’s stability correspondingly decreases. This phenomenon occurs because, despite a larger DC link capacitor’s ability to greatly reduce DC voltage fluctuations, under these conditions, the GSVSC unable absorb more power from the DC network to provide the necessary power support. Furthermore, grid frequency deviation increases with the growth of the DC link capacitor, which further exacerbates the decline in stability.

Figure 7 shows the schematic diagram of the effects of changes in C _v1 and D _v1 parameters on system stability. Although the system can maintain asymptotic stability within a large range, larger inertia parameters require matching with larger capacitors. Overall, the size of the capacitor, virtual capacitor, and damping coefficient directly affect the system’s stability. In order to achieve optimal performance, these factors need to be considered comprehensively and adjusted appropriately. In this paper, C _v1 = 1.3,D _v1 = 7.5; C _v2 = 1.1, D _v2 = 2.5; C _v3 = 1.2, D _v3 = 5 are selected.

To assess the effectiveness of the control strategy, a load switching experiment was conducted at 8 s. The assigned values for the parameters in this context are as follows: K _p = 1.5,K _d = 0.5, K φ = 1. Figure 8 depicts a comparison of power output curves for VSC stations in response to load switching between the conventional method and the proposed approach. Figure 8A presents a global comparison of the active power output curves. Amplified views of the output power for corresponding VSCs are displayed in Figures 8B, C. Notably, the red dashed line is indicative of the output corresponding to the proposed method. It is important to note that in subsequent test results, the red dashed line consistently represents the output curve associated with the proposed method. At t = 8 s, the load was introduced into AC onshore grid 1. As delineated in Figure 8, subsequent to the application of the proposed voltage cooperative control, the GSVSC side gains additional active power from the OWF for equilibrium adjustment, as depicted in Figures 8B, C.

Figure 9 presents the frequency variations during load switching in the system. Load switching in AC onshore grid 1 introduces noticeable frequency fluctuations in GSVSC1. Traditional control strategies maintain a minimum frequency point at 49.8 Hz, while the proposed cooperative control approach achieves 49.83 Hz, as shown in Figure 9B. When OWF performs secondary frequency adjustment for GSVSC, relying solely on traditional comprehensive inertia control may lead to SFD. However, the introduction of the proposed voltage cooperative control effectively mitigates SFD, as shown in Figures 9B–D. These figures display before-and-after comparisons of the frequencies for GSVSC1-3 when employing the proposed method. After the adoption of the proposed method, the overall frequency variations in the system exhibit improved convergence, accompanied by a reduction in the impact of SFD.

Regarding fluctuations in DC voltage, the effectiveness of the proposed coordinated control method in leveraging the stored energy within DC-side capacitors to mitigate power variations is illustrated in Figure 10. Through coordinated control, there is efficient utilization of the energy reserves stored in the DC-side capacitors for power regulation. This results in reduced frequency variations, thereby achieving a smoother output.

Figure 11 illustrates the corresponding supplemental power output. It is evident from Figure 11 that when the system attains stability, the supplemental power reaches zero, having no impact on the system’s output. Furthermore, by utilizing the correlation values between voltage and injected power, calculated through MPC, the system effectively tracks changes in active power. This verification underscores the efficacy of the proposed method under load switching conditions and its robustness in response to variations in load.

During the regulation process, only OWF1 participates in frequency regulation. In previous research, the frequency dead zone is generally selected to be (±0.02∼0.05) Hz, while in this paper, ±0.02 Hz is chosen. As shown in Figure 8, at t = 11.8 s, OWF1 exits frequency regulation work. Meanwhile, to ensure that the unit has certain frequency regulation capabilities and to reduce instances of unit stall and shutdown, this study stipulates that the output power of the unit involved in frequency response from OWF1 should not be less than 20% of the rated capacity.

Figures 12–15 showcase the assessment of the proposed method’s performance in the case of agent loss. Within this set, Figure 12A presents a global comparison chart of active power output, while Figures 12B, C offers a detailed enlargement of the power. At t = 10 s, GSVSC2 loses communication with others, signifying an agent loss scenario. It is noteworthy that GSVSG2 can continue to function normally based on the predefined default reference (0 kW). Concurrently, the remaining VSGs can continue to function normally. Under cooperative control, the OWF also contributes additional active power for system balance regulation, as depicted in Figure 12C.

Figure 13 illustrates the frequency variation output curve. Given GSVSC2’s limited frequency fluctuations during power output changes, traditional control strategies yield reduced SFD. Nonetheless, the adoption of the proposed coordinated control method improves frequency convergence for GSVSC1 and GSVSC3 while mitigating the frequency drop in GSVSC2. This results in an overall enhancement of frequency convergence and system stability.

Figure 14 displays a comparative plot of voltage output curves. With the adoption of the proposed coordinated control method, DC-side capacitors release additional energy to fulfill the system’s power demands and expedite convergence to a stable state when the system reaches equilibrium. This highlights the efficient utilization of energy stored in the DC-side capacitors. Figure 15 illustrates the corresponding supplemental power output.

The simulation results reveal the effectiveness and robustness of the proposed method in cases of agent loss. Although the communication capabilities of the agent may be interrupted due to malfunctions, the proposed method ensures the stable operation of the power system and successfully suppresses excessive voltage and frequency fluctuations. This also highlights the superiority and potential of the method when faced with system imbalances and uncertainties.

In a communication system, latency encompasses various components, including system fault detection and trip time, transmission delay, and control total delay. These components can be quantified as follows: fault detection time is 3 ms, and the average trip time is 6 ms. Transmission delay is directly proportional to the length of the communication line, represented as 2 times the line length divided by the speed of light. The control total delay, which encompasses network power flow calculations, is fixed at 10 ms. For instance, assuming a VSC interstation line length of 150 km, the resulting communication delay τ _ij is calculated as 20 ms. To assess the algorithm’s effectiveness in the presence of communication delays, a total delay of τ _ij = 1,000 ms is designated within the MTDC for validation purposes. In a suboptimal communication setting, the reception of information from node j to node i can be described as follows: x i j k = x j k − ξ i j k + μ i j k (34) Where ξ _ij (k) is the communication delay; μ _ij (k) is the channel noise using Gaussian noise simulation. Integrate (34) into the system's input signal to simulate the scenario of operating in a non-ideal communication environment.

Figure 16 provides a robustness assessment in a non-ideal communication network environment, specifically with ξ _ij = 1000 ms. Figure 16A visualizes μ _ij (k). As illuminated by Figures 16B–D, the proposed method sustains its performance effectively amidst substantial communication delays. These figures underscore that the system retains its stability notwithstanding the pronounced communication delays. By demanding less data transmission, the proposed method enhances the communication delay margin, demonstrating the robustness and resilience of this control strategy under non-ideal network conditions.