The D-PMSG consists of a wind turbine shaft system, a back-to-back converter, and a DC link. A dual-mass module is used in the modeling of the permanent magnet direct-drive wind turbine to describe the mechanical characteristics of the shaft system of the D-PMSG, and the linearized model of the shaft system is as follows: p Δ ω a = T Ja − 1 − D aa + D ab Δ ω a + D ab Δ ω b − K ab Δ δ a + K 12 Δ δ b + Δ T a ] p Δ ω b = T Jb − 1 − D bb + D ab Δ ω b + D ab Δ ω a + K ab Δ δ a − K ab Δ δ b + Δ T b ] p Δ δ a = ω r Δ ω a p Δ δ b = ω r Δ ω b (1) where p is the differential operation factor; subscripts a and b correspond to the wind turbine and generator, respectively; Δω _a, Δω _b, and T _Ja, T _Jb are the rotational speed and inertia time constant, respectively; ω _r is the rotational speed reference value; D _aa, D _bb, and D _ab are the self-damping coefficients as well as mutual damping coefficients; K _ab is the stiffness coefficient of the shaft system; ΔT _a and ΔT _b are the mechanical torque and electromagnetic torque, respectively; Δδ _a and Δδ _b are the electrical angular displacements of the wind turbine rotor and the generator rotor, respectively, relative to the synchronous rotating reference axis of the rated electrical speed.

The remaining parts of the dynamic model can be found in the literature (Chen et al., 2018).

The flexible model is composed of the rectifier inverter and its controller, the inverter and its controller, and the DC transmission link, and the specific structure is shown in Figure 2.

The sender-side converter model and the receiver-side converter model set up the positive sequence d-q coordinate system with the grid voltage u _s1 and u _s2 as the reference, respectively. Then, the dynamic model of the voltage source converter-based high-voltage direct current transmission (VSC-HVDC) system is as follows: ω s 1 b − 1 L 1 p i s 1 d = − R 1 i s 1 d + ω s 1 L 1 i s 1 q + u s 1 d − u c 1 d ω s 1 b − 1 L 1 p i s 1 q = − R 1 i s 1 q − ω s 1 L s 1 d + u s 1 q − u c 1 q C u dc 1 p u dc 1 = u c 1 d i s 1 d + u c 1 q i s 1 q − i dc u dc 1 L dc p i dc = u dc 1 − u dc 2 − R dc i dc C u dc 2 p u dc 2 = − u c 2 d i s 2 d + u c 2 q i s 2 q − i dc u dc 2 ω s 2 b − 1 L 2 p i s 2 d = − R 2 i s 2 d + ω s 2 L 2 i s 2 q + u s 2 d − u c 2 d ω s 2 b − 1 L 2 p i s 2 q = − R 2 i s 2 q − ω s 2 L 2 i s 2 d + u s 2 q − u c 2 q (2) where i _s1d and i _s2d are the sending end current and the receiving end current; i _dc is the direct current; u _dc1 and u _dc2 are the sending end DC voltage and the receiving end DC voltage; ω _s1b and ω _s2b are the grid angular frequencies on the AC side of the sending end and the receiving end; R ₁ and L ₁ are the equivalent resistance and inductance of the outlet transformer of the sending end converter; R ₂ and L ₂ are the equivalent resistance and inductance of the outlet transformer of the receiving end converter, respectively.

Among them, the sender end controller adopts the fixed active and fixed reactive power control strategy, and the receiver end controller adopts the fixed DC voltage and fixed reactive power control strategy.

In building the unified state–space model, the coordinate system of each module must be unified using a coordinate transformation because the reference coordinate system of each module is different. In the grid-side converter model of D-PMSG, the d ₁ -q ₁ rotational coordinate system is set based on the grid voltage u _s, while in the feeder converter model of VSC-HVDC, the d ₂ -q ₂ rotational coordinate system is set based on the voltage u _s1.

The interface dynamic equations, that is, the dynamic equations of the AC transmission line, are obtained by transforming the physical quantities in the VSC-HVDC feeder converter and its controller model based on the d ₁-q ₁ rotational coordinate system to the d ₂ -q ₂ rotational coordinate system: u sd − u s 1 d ′ = R 1 i sd + ω s 1 b − 1 L 1 p i sd − ω s 1 L 1 i sq u sq − u s 1 q ′ = R 1 i sq + ω s 1 b − 1 L 1 p i sq + ω s 1 L 1 i sd (3)

Among them, u s 1 d ′ u s 1 q ′ = T u s 1 d u s 1 q (4)

The non-linear mathematical model of each module is converted into a state–space model by means of linearization as follows: p X = A 1 X + B 1 Y Y = C 1 X (5) where X and Y are the system state variables and algebraic variables, respectively; A ₁, B ₁, and C ₁ are the state matrix, the state and algebraic variable relationship matrix, and the algebraic and state variable relationship matrix, respectively.

Using the input–output relationship between modules, that is, the above interface equations, combined with the method of coordinate transformation, the state equations of different modules are converted to a unified coordinate system to form the state–space model of the whole system. p X = A X + B 1 Y A = A 1 + B 1 C 1 (6)

The dynamic equation of the whole system is as follows: X = X PMSG 1 , X plls 0 , X PMSG 1 − Ctrl , X plls 1 , X HVDC , X plls 2 , ×   X HVDC − Ctrl , X PMSG 2 , X plls 3 , X PMSG 2 − Ctrl X PMSG 1 = Δ ω 3 , Δ ω 4 , Δ δ 3 , Δ δ 4 , Δ i d , Δ i q , Δ u dc , Δ i sld , Δ i slq T X plls 0 = Δ θ plls 0 , Δ Z plls 0 T X PMSG 1 − Ctrl = Δ w 4 , Δ w 5 , Δ w 6 , Δ x 4 , Δ x 5 , Δ x 6 T X plls 1 = Δ θ plls 1 , Δ Z plls 1 T X HVDC = Δ u dc 1 , Δ i dc , Δ u dc 2 , Δ i s 2 d , Δ i s 2 q T X plls 2 = Δ θ plls 2 , Δ Z plls 2 T X HVDC − Ctrl = Δ y 1 , Δ y 2 , Δ y 3 , Δ y 4 , Δ z 1 , Δ z 2 , Δ z 3 , Δ z 4 T X PMSG 2 = Δ ω 1 , Δ ω 2 , Δ δ 1 , Δ δ 2 , Δ i 2 d , Δ i 2 q , Δ u 2 dc , Δ i 2 sd , Δ i 2 sq T X plls 3 = Δ θ plls 3 , Δ Z plls 3 T X PMSG 2 − Ctr l = Δ w 1 , Δ w 2 , Δ w 3 , Δ x 1 , Δ x 2 , Δ x 3 T (7) where X _PMSG1 and X _PMSG2 are the wind turbine generators (WTG) state variables; X _plls is the phase-locked loop state variable; X _PMSG1-Ctrl and X _PMSG2-Ctrl are the WTG control system state variables; and X _HVDC is the VSC-HVDC state variable. Δθ _pllsi and ΔZ _plli are the phase-locked loop i (i = 0, 1, 2, 3), respectively, negative state variables of the output phase and angle instantaneous change values of the controller; Δω ₃ and Δω ₄ are the electrical rotational speeds of the rotor of the PMSG1 wind turbine and generator, respectively; Δω ₁ and Δω ₂ are the electrical rotational speeds of the rotor of the PMSG2 wind turbine and generator, respectively; Δδ ₃ and Δδ ₄ are the electrical angular displacements of the rotor of the PMSG1 wind turbine and generator relative to the reference axis for synchronous rotation at rated electrical speed; Δδ ₁ and Δδ ₂ are the electrical angular displacement of PMSG2 wind turbine and generator rotor relative to the rated electrical speed synchronous rotation reference axis; Δi _d and Δi _q are the d- and q-axis components of the grid-connected current of PMSG1, respectively; Δi _2d and Δi _2q are the d- and q-axis components of the grid-connected current of PMSG2, respectively; Δu _dc and Δu _2dc are the DC voltages of PMSG1 and PMSG2, respectively; Δi _2sd and Δi _2sq are the d-axis and q-axis components of the PMSG2 outlet AC current, respectively; Δw ₄ and Δw ₆ are the d- and q-axis state variables of the PMSG1 machine-side current controller, respectively; Δw ₁ and Δw ₃ are the d- and q-axis state variables of the PMSG2 machine-side current controller, respectively; Δw ₅ is the state variable of the PMSG1 machine-side speed controller; Δw ₂ is the state variable of the PMSG2 machine-side speed controller; Δx ₄ and Δx ₆ are the d-axis state variables of PMSG1 grid-side voltage and the q-axis state variables of current controller, respectively; Δx ₁ and Δx ₃ are the d-axis state variables of PMSG2 grid-side voltage and the q-axis state variables of current controller, respectively; Δx ₅ is the d-axis state variable of the PMSG1 grid-side current controller; Δx ₂ is the d-axis state variable of the PMSG2 grid-side current controller; Δy ₁ and Δy ₃ are the d- and q-axis state variables of the VSC-HVDC external loop active and reactive power controller, respectively; Δy ₂ is the d-axis state variable of the VSC-HVDC inner loop circuit controller; Δy ₄ sends the d-axis state variable of the inner loop current controller at the end; Δz ₁ and Δz ₃ are the external loop voltage and reactive power controller state variables at the VSC-HVDC receiving end, respectively; and Δz ₂ and Δz ₄ are the d- and q-axis state variables of the VSC-HVDC receiver loop current controller, respectively.

The initial conditions of the system are set in Supplementary Table A1 in Supplementary Appendix A. To distinguish the intrinsic oscillation modes of the two wind turbines to avoid the resonance of the two oscillation modes with similar frequencies, the unit parameters of the two direct-driven wind turbines are slightly different. The specific parameter settings of each part are shown in Supplementary Tables A2–A6 in Supplementary Appendix A. The results of the analysis of the eigenvalues obtained under the initial conditions of the system are shown in Table 1.

The damping ratio of the LFO-1 oscillatory mode is 2.93%, a value below the 3% damping ratio threshold, and the mode is weakly damped with a high risk of instability. In contrast, the damping ratio of the LFO-2 oscillation mode is 4.94%, which exceeds the damping ratio threshold, indicating that this mode is relatively stable. In the sub-synchronous frequency band, except for the SSO-1 mode with a lower damping ratio, the damping ratios of the SSO-2 to SSO-5 modes are all greater than the threshold value of 3%, thus possessing better stability. The damping ratio of the SupSO oscillation mode is 7.49%, which exceeds the threshold value of the damping ratio and, therefore, the mode has a high degree of stability.

In summary, the LFO-1 and SSO-1 modes have a greater risk of instability. Most of the remaining oscillation modes show high damping ratios and relatively stable performances. The participation factors of each oscillation mode are obtained by the eigenvalue analysis method, as shown in Table 2.

In Table 2, Δi _s1dq, Δi _s2dq, and Δi _2sdq represent the communication network between D-PMSG and VSC-HVDC, the VSC-HVDC receiving end communication network, and the variables transmitted by D-PMSG through the communication network, respectively; Δi _s3dq is the sum of the VSC-HVDC output current and the D-PMSG AC line current; Δi _sq is the q-axis state variable of the AC network current component between D-PMSG and VSC-HVDC.

Analyses using Table 2 data show that

(1) The two sets of oscillation modes of LFO are caused by the shaft systems of D-PMSG1 and D-PMSG2, respectively, and the subsystems involved in the oscillations are their respective shaft systems.

The model shown in Figure 1 was built in the PSCAD/EMTDC environment, with a simulation duration of 10.0 s. When the system runs for 2.0 s, a single-phase ground short-circuit fault with a duration of 0.2 s is applied at the point of common coupling (PCC) of the receiving grid. The change in the active power output at the PCC is observed. The waveform of the output active power is shown in Figure 3a The output active power of the established system was analyzed using the Prony analysis method, and the results are shown in Figure 3b.

According to Figure 3, the frequency components of the output active power include sub-synchronous oscillation components at 1.99 Hz and 4.69 Hz. The amplitude of 1.99 Hz is relatively large, indicating strong oscillation. The results in Figure 3 are consistent with those in Table 2, verifying the correctness of the model established in the previous section.

A damping controller uses system electrical quantities (average unit speed, active power, reactive power, etc.) as the input signal. Different frequencies of the oscillating components extracted will be filtered through the bandpass filter. Then, the amplitude and the phase of this group of signals can be moved to form a signal that can offset or compensate for part of the original oscillating components of the damping to achieve the damping control.

To govern the coupled oscillation modes generated by the system with the additional damping controller, the parameter optimization model of the additional damping controller is established by taking the additional damping controller parameters as the optimization variables, setting the coupled oscillation modes as the improvement objective, and taking the damping ratio of the coupled oscillation modes as high as possible as the optimization objective. F = max ∑ i = 1 K η i ξ i (10) where K is the number of target modes; η is the weight coefficient of the target mode; ξ is the damping ratio of the target mode; and F is the objective function.

(1) The damping ratio of the mode to be improved should be greater than a certain threshold. The current study shows that a damping ratio below 3% can be regarded as a low-damping mode, but considering a certain margin, this article takes 5% as the threshold.

Considering that the optimization objective is to maximize the damping ratio of the target modes, additional damping controllers are applied to the grid-side controller of the permanent magnet direct-drive wind turbine and the flexible and straight receiver-end controller. When solving the parameters of the additional damping controller, adjusting the controller parameters leads to changes in the eigenvalues of the system, which may cause significant changes in the frequency of the target oscillation mode. If the frequency of the target mode changes too much, it may lead to the loss of tracking of the optimization target in the subsequent optimization iterations, and thus, the optimization of this target mode cannot be continued. For this reason, a mode tracking technique is introduced in the optimization strategy proposed in this article to ensure that the optimization process always improves on the pre-selected oscillatory modes. The target mode is tracked through the information of the first five participant orderings of the mode, the left and right eigenvectors, and the oscillation frequency to ensure that the optimization is always performed for the given target mode.

For an oscillatory pattern m, the oscillatory pattern similarity is defined by drawing on the paradigm similarity search as S m R a , R b = ∑ i = 1 K w i 1 − a i − b i F i 2 (12) where R _a and R _b are the sets of corresponding feature information (including the eigenvalues of the mode, the ordering of the participation factors, the frequency of oscillation, and the left and right eigenvectors and the angle between them) of the oscillatory mode m in the optimization process and the initial conditions, respectively; a _i and b _i are the ith feature information of R _a and R _b, respectively; w _i and F _i are the weights and the ranges of values of the ith feature information of the oscillatory mode m, respectively; S _m is the similarity of oscillation modes.

PGSA is an algorithm to simulate the phototropism (optimal growth) of plants, but the conventional PGSA can only be used for problems in integer programming. Therefore, the cloud model theory is incorporated into the conventional method. The cloud model, with its ability to find certainty in uncertainty and to achieve dynamic changes in stability, simulates the fundamental mechanism of plant growth in nature well and can change the accuracy of the results of each calculation, targeting the problem that can only be dealt with in integer planning. At the same time, it adopts the method of varying the step size to improve the convergence speed and to shorten the computation time. The flow of optimizing the damping controller parameters using the I-PGSA algorithm is shown in Figure 7.

For SSO-1, SSO-2 (coupled oscillation mode between flexible direct and permanent magnet direct-drive wind turbines), its related controllers (grid-side converter (GSC), sending-end controller (SEC)), and additional damping controllers are governed, and its calculation results are compared with those of the undamped controller to determine the feasibility of optimization. The initial parameters of the additional damping controller are shown in Table 3. In Table 3, K _s1 and K _s2 are the gain coefficients of the GSC additional damping controller; T _s11, T _s12, T _s13, T _s14, T _s21, T _s22, T _s23, and T _s24 are the time constants of the GSC additional damping controller.

I-PGSA is used to solve the above controller parameters, with the maximum number of iterations being 20; the other parameters of the system remain unchanged. The final iteration results are shown in Table 4, while the eigenvalue calculations are performed under this parameter, and the results are shown in Table 5 and Figure 8.

From Table 5 and Figure 8, it can be seen that, in the absence of the additional damping controller, SSO-1, SSO-2 oscillation mode damping is relatively low. With the additional damping controller parameters gained through the optimization, the SSO-1, SSO-2 oscillation mode damping ratio has a considerable increase. The impact of the remaining oscillation mode is smaller, and it is possible to accurately sort out some of the oscillation modes. Therefore, it has better feasibility.

The additional damping controller initial parameters for the SSO-5, SupSO (coupled oscillation mode between AC and DC subsystems) and its related control system’a (flexible direct receiver control system) additional damping controller for governance are shown in Table 6. The final iteration results are shown in Table 7. The eigenvalue calculations are carried out under this parameter, and the results are shown in Table 8 and Figure 9.

As can be seen from Table 8 and Figure 9, when there is no additional damping controller, the damping ratio of the SSO-5 and SupSO oscillation modes is relatively low. After the optimization of the parameters of the additional damping controller, the damping ratio of the SSO-5 and SupSO oscillation modes has risen considerably. The impact on the rest of the oscillation modes is small, so that it is feasible to accurately sort out certain oscillation modes.

The model shown in Figure 1 was built in PSCAD/EMTDC software, and a damping controller was attached to the grid-side controller of the permanent magnet direct-drive wind turbine. The PCC of the grid was set at the receiving end to cause a single-phase ground short-circuit fault at 2–2.2 s. The active power output of the system was observed. The comparative analysis chart of the obtained active power is shown in Figure 10.

From Figure 10, it can be seen that the system is oscillating, and it is not easy to converge when a small disturbance is added before the damping controller is put into use. However, after the damping controller is put into use and the parameters are optimized, the oscillation of the system is quickly calmed after a small disturbance is added, which verifies the feasibility of the damping controller design method in this article.

The damping controller is attached to the flexible direct receiver controller, and the same simulation conditions as above are set. The resulting active power comparison analysis is shown in Figure 10.

From Figure 11, it can be seen that after the damping controller is put into operation and the parameters are optimized, adding small disturbances does not flatten the oscillation of the system. Considering that the damping ratios of the coupled oscillation modes related to the receiving end of the VSC-HVDC are affected by various factors, adjusting only the parameters of the VSC-HVDC controller may improve a single oscillation mode, but it may also deteriorate other oscillation modes, thus obtaining the simulation results. However, the previous verification showed that adjusting the parameters of the additional damping controller can effectively improve both sets of coupled oscillation modes simultaneously. It also proved that the damping controller design method proposed in this article has excellent governance capabilities for coupled oscillation modes.