Novel grid-forming control of PMSG-based wind turbine for integrating weak AC grid without sacriﬁcing maximum power point tracking

Traditional permanent magnet synchronous generator (PMSG)-based wind turbine (WT) normally utilises phase lock loop (PLL) for obtaining accurate phase angle of AC grid to make sure of maximum power point tracking (MPPT). However, a weak AC system with a low short circuit ratio (SCR) and high control bandwidth of PLL might have impaired effects on the stability of the system. To overcome these, a novel power synchronous control of PMSG-based WT with grid forming ability of PMSG for integrating very weak AC system is ﬁrst proposed in this study. Grid-side converter (GSC) of PMSG emulates the inertia response of synchronous generator (SG) by artiﬁcially coupling the DC-link voltage with the grid frequency based on the certain droop property, while the converter voltage magnitude is regulated according to DC-link voltage deviation by controlling the modulation index of GSC. This novel scheme can realise the grid synchronisation without PLL by mimicking the natural characteristic of SG. Small-signal analysis of the proposed scheme indicates the DC-link capacitor energy can be exerted for system inertia support during system disturbances, while the system damping can be provided by spontaneously alternating converter voltage magnitude. More importantly, MPPT property of PMSG is not compromised while providing inertia and damping for the system, and the related inertia and damping coefﬁcient of WT with the proposed control can be easily tuned for satisfying the system requirement. Non-linear simulations of one PMSG connected with one large SG considering a sudden change of active power reference and variable wind speeds have been studied to verify the effectiveness of the proposed grid-forming control.


INTRODUCTION
The modern power grid has gradually become relatively weak with respect to the large-scale of renewable energy integration [1,2], which imposes high risks and challenges on the stability of the system [3,4]. Traditionally, the grid-connected variable speed wind turbine (VSWT) utilises the rotor-side converter (RSC) to realise the maximum power point tracking (MPPT) algorithm, while the grid-side converter (GSC) aims to maintain DC-link voltage [5]. However, this classical wind turbine (WT) control cannot exert much inertia or frequency support during system disturbances, which leads to the large system frequency deviations in the daily operation [6]. Meanwhile, the short-circuit ratio (SCR) of the system is gradually decreasing since the large synchronous generators (SGs) have been gradually substituted by renewable energy with little ability of providing short-circuit capability [7]. Sub-synchronous oscillation phenomenon [8] induced by the control dynamics of WT appears in the large-scale renewable energy integration into the weak AC system of low equivalent SCR. Therefore, it is very essential to make up novel control schemes of VSWTs for enhancing the stability of high penetration of wind energy integrated into future weak AC system. To improve the overall stability and enhance the strength of voltage source converter (VSC)-embedded power system, virtual synchronous control schemes with different control functions [9-12, 14-16, 19-25] have been proposed nowadays. Generally speaking, there are two main virtual synchronous control approaches of VSCs that emulate the dynamic behaviour of the traditional SG for providing system support during disturbances. The first way that equips VSC with the ability of inertia response [9], frequency regulation [10], and suppressing inter-area oscillation [11] is realised by altering the active or reactive power references in the outer control loop of VSC. These schemes are easy to implement and widely utilised in VSWTs with little control modification. As a result, partially stored energy in DC-link capacitor and rotor kinetic energy of WT can be temporarily exerted for system support, which enhances the stability of the connected AC system [12]. However, these control schemes that improve the transient performance of AC system need to rely on the phase lock loop (PLL) [13] for obtaining the phase angle information of the point of common coupling (PCC) to realise the grid synchronism. On the one hand, utilising VSWT-reserved energy by blade pitching [14], de-loading control [15], and the kinetic energy of WT [16] for system inertia and damping support inevitably leads to the rotor speed of WT deviating from the optimal value, and the significant wind energy is sacrificed during system disturbances. On the other hand, the dynamics of PLL [7,17,18] have a detrimental impact on the stability of the weak system with low SCR. In [18], it is indicated that high control bandwidth of PLL might induce a negative damping coefficient of the close-loop control system by utilising the impedance-based analysis method.
Another virtual synchronous control of VSC, namely, virtual SG (VSG) control, which controls the transmitted active power via utilising the internal synchronisation mechanism in AC system without PLL is proposed in [19]. This control scheme well emulates the behaviour of SG during the transient process, which renders the strong voltage support for the AC system, and the strength of the AC system is largely improved. It has been widely utilised in the situations of twoarea power grid inter-connection via high voltage direct current (HVDC) and power supply for the isolated network [20]. Moreover, [21] improves this strategy by enabling VSC the ability of the system damping and fault ride-through capability. Moreover, in order to improve the response speed and suppress the power fluctuation during system disturbance, [22] introduces a new control scheme called inducverters, which well mimics the dynamic behaviour of the inductive machine. In addition, the inertialess-based power synchronous control (PSC) in [23] and developed in [24] is further proposed and well applicable to the strong and weak power grid with the fast response speed of VSC. However, above strategies that enable the single VSC with the ability of providing system inertia and damping is achieved by active power modulation, which might make WT constantly deviate from MPPT status, and much power loss might be induced in the above control schemes.

SG
The schematic diagram of the single synchronous generator (SG) connected to an infinite bus system To improve this, reference [25] proposes a novel synchronous control of VSC via utilising the dynamics of DC-link voltage. However, this control logic is relatively complicated for providing system damping and inertia support. Moreover, the system damping support is still realised by the active power modulation from the stored electrical energy in DC-link capacitor, and it might lead to the large DC-link voltage deviation during system disturbances due to the limited capacitance of the converter. The above control schemes mainly focus on the inertia and damping control of WT via largely sacrificing the MPPT that is the prominent characteristic of WT. Focusing on the control schemes of WTs with self-synchronism ability without the aid of PLL, this study proposes a novel PSC by well emulating the dynamics of traditional SG without sacrificing the MPPT control of WT. In the proposed scheme, GSC of PMSG changes the converter voltage frequency and magnitude based on DClink voltage deviation with the droop property to emulate the inertia response and damping of SG, while MPPT algorithm is still realised by RSC of PMSG. Accordingly, the inertia and system damping provided by WT with the proposed control are quantified based on the small-signal analysis method. The proposed scheme stands out itself by rendering system inertia support via utilising partial DC-link capacitor energy. Meanwhile, system damping is exerted by automatically changing the converter voltage magnitude during system disturbances.

DYNAMIC RESPONSE OF GRID-CONNECTED SINGLE SG DURING DISTURBANCES
Before introducing the proposed PSC of PMSG-based WT for integrating into very weak AC grid, the dynamics of single SG connected to an infinite bus through the external reactance under the small perturbation of variables are described first. Accordingly, some fundamental concepts of the phenomenon of the stability including the damping, oscillation frequency and damping ratio of SG are reviewed in this section.

2.1
Rotor angle dynamics of grid-connected SG Figure 1 describes a simple single SG connected to an infinite bus through the external impendence system. SG is represented by the classical model where the transient inner voltage of SG marked as E G ' is kept constant under the small perturbations of the rotor angle positions. The initial rotor angle position of SG before the system disturbance is marked as δ G0 , and the phase angle of system voltage is kept as zero. X d ' symbol is the transient reactance of SG. V G and V S are the terminal voltage of SG and the infinite source. A transmission line that connects SG and the infinite source is represented by the equivalent external reactance marked as X l .
According to the basic knowledge of electrical circuit, the active power from SG marked as P G can be written as follows: It is assumed that the rotor angle of SG is abruptly changed to δ G1 . As a result, the active power of SG will be suddenly altered to P G ' based on the following equation: In terms of the sudden power generation change of SG, SG will go through a series of rotor dynamics based on the following two differential equations: where ω B and ω G mean the base angular speed of the infinite source system and the rotor speed of SG, respectively. H G and P M are the inertia time constant and the mechanical power from SG, respectively. D G is the damping coefficient of SG. Due to the small disturbance of rotor angle position of SG, Equations (3) and (4) can be linearised around the initial operation points in the increase form as follows: where Δδ G and Δω G are the small perturbation of state variables. Combining Equations (5) and (6), the rotor dynamics of grid-connected SG can be described in Equation (7) as

Small-signal stability of grid-connected SG
It can be clearly seen from Equation (7) that the dynamic behaviour of the rotor angle position of SG is very similar to the response of the second-order system. Considering the damping coefficient of SG is zero, the rotor dynamic of SG turns into a pure oscillator with the natural oscillation frequency marked as ω n .
where K 1 is the defined coefficient representing the change in electrical power for the change in rotor angle position of SG. It should be noted that the inertia of the machine determines the oscillation frequency; the larger the inertia of the machine, the slower is the oscillation frequency. Moreover, based on Equations (3) and (4), for the given change of electrical power, the larger the machine inertia, the smaller the variation of the rotor angle and rotor speed of SG, which improves system stability.
Rewrite Equation (7) in a Laplace form, and the resulting characteristic equation of rotor angle position of SG is Accordingly, it gives rise to the damped oscillation with the frequency noted as ω s and the damping ratio symbolised as ζ, which can be expressed as follows: The damping ratio of SG determines how fast the oscillation can be damped, and the larger the damping ratio, the higher the stability of the system. Based on the above analysis of small disturbance stability of SG, the rotor dynamics described in Equation (7) indicate that any small disturbances of rotor angle position can be returned to the original equivalent point only if SG has sufficient damping and inertia. Moreover, the gridconnected SG has natural power-angle property described in Equations (3) and (4) to realise self-synchronism without the need of PLL.

SCR of the system
Normally, SCR defined as Equation (11) is utilised to evaluate the strengths of the connected AC system in many literatures [26,27]:

FIGURE 2
The schematic diagram single converter connected to an infinite bus system where S ac is the short-circuit capacity of AC system, P dN is the rated system power, and U N is the rated AC voltage, and Z denotes the AC system impedance.
As indicated by Equation (11), the value of the AC system impedance determines the value of system SCR under the rated system capacity and voltage. Generally speaking, the larger the system impedance and the lower the value of the SCR, the weaker the connected AC system. Empirically, if SCR of a VSCintegrated AC system is less than 2.5, it can be regarded as the weak AC system.
Oscillation phenomenon [28,29] induced by PLL of WT might appear in the large-scale wind energy integration into such a weak AC system of low SCR. Therefore, it is very essential to design the novel control schemes for WTs to enhance the stability of the connection under the weak AC system, which is the main motivation of the study.

PROPOSED CONTROL OF PMSG-BASED WT WITH GRID FORMING ABILITY
Due to the large-scale variable speed range and no gearbox needed between the wind blade and the generator shaft, compared to the double fed induction generator-based WT, PMSGbased WT has gained more popularity in offshore wind farm under high variable wind speed range. PMSG-based WT has two full-rated back-to-back converters that are connected through DC-link capacitor. Traditional GSC of PMSG-based WT is controlled to maintain DC-link voltage and the reactive power exchanged to AC grid. However, this scheme may require PLL to track the synchronous phase of AC grid, which might have impaired impacts on the stability of the very weak AC system. To well emulate the rotor dynamics of SG, GSC emulates the inertia of SG by artificially coupling DC-link voltage with the AC system frequency with certain droop property. Meanwhile, RSC is controlled to realise the MPPT of WT. The concrete control schemes of GSC and RSC of PMSG-based WT are illustrated below. Figure 2 describes GSC of PMSG-based WT that connects to the infinite source through the converter reactance marked as X C and the transmission line that is simplified by the equivalent line reactance marked as X l . According to the basic operation principle of VSC, any magnitude or the frequency of the converter voltage can be obtained by controlling the switching actions of insulated gate bipolar transistors (IGBTs) in three-phase converter bridge. In order to emulate the converter AC voltage as the transient inner voltage of traditional SG, GSC of PMSG-based WT is controlled as an ideal AC voltage by regulating the modulation index in the grid connection operation mode.

Fixed converter voltage control of PMSG-based WT
Based on the basic operation principle of two-level voltage VSC, the relationship between DC-link voltage noted as V DC and RMS line-to-line converter voltage noted as V C can be expressed as follows: where m abc is the modulation index of phase a,b,c voltage of the converter. In this study, the unbalanced AC voltage condition is not considered, and m a = m b = m c = m. It should be noted that all the variables are in per-unit form. In order to maintain the converter voltage as the desired operating value marked as V C * , the modulation index m should be controlled as follows: Certainly, maintaining the converter voltage is not the only control objective for GSC. The converter can be also controlled to make sure that the zero reactive power change between converter and system. However, for WT integration into a very weak AC system, providing strong voltage support has higher priority than sustaining unit power factor for the grid-connected converter, only if there is enough reactive capacity of the converter.

Inertia emulation of traditional SG
In the traditional grid-connected WT control, PLL is essential for obtaining the accurate phase angle information of PCC to make sure of the active power control of WT. Moreover, inertia support of WT can be hardly provided for the weak low-inertia AC system in the classical grid-following control scheme without any ancillary grid-friendly control. However, in the SG-dominated AC system, the voltage and frequency are constructed by SG itself without the aid of PLL, and the normal operation of multi-machines relies on the power synchronism characteristic of each generation described in Equations (5) and (6). Moreover, the AC system frequency fluctuations during system disturbance can be mitigated by temporarily releasing or absorbing the stored rotational mass-energy, which is referred as 'inertia response' of SG. In order to make full use of the merits of traditional SG, the core idea of the proposed GSC control is to emulate the dynamic behaviour of SG described in Equations (5) and (6) as the classical secondorder system. As a result, PMSG-based WT does not need PLL in the normal operation and can exert the partial stored DC-link capacitor energy for system inertia support during disturbances. Similar to the traditional SG, GSC should have the capability of the grid forming without the need of PLL. The converter frequency is controlled as a variable value that can reflect the power imbalance between the generation from RSC and power transmitted to the GSC as the traditional SG does by artificially coupling DC-link voltage changes with the system frequency deviations. Suppose that the converter voltage is operated with certain frequency f C0 , magnitude V C0 and initial phase angle δ C0 before the system disturbance. DC-link voltage dynamics when overlooking the power losses among back-to-back converter can be written in the increase form as follows: where S B is the base value of the system. C DC , C are the total capacitance and the equivalent capacitance in per unit, respectively. ΔP R and ΔP C are the active power change from RSC and GSC of PMSG-based WT. Based on Equation (14), any power imbalance between two-sided converters will reflect the changes of DC-link voltage, which is quite similar to the dynamic of rotor speed of SG described by Equation (4). To well emulate the inertia response of SG, DC-link voltage change is artificially coupled with the system frequency deviations based on the following droop-based control law.
where k DC is the proportional coefficient of the droop control described in Equation (16). Therefore, the control law of the converter frequency can be written as Equation (17), Based on Equations (14) and (16), and assuming that DC-link voltage fluctuates around the nominal value noted as V DC0 , the dynamics of GSC can be written as follows: It can be clearly drawn from Equation (18) that any power imbalance between GSC and RSC will lead to the system frequency changes, which is quite similar to the inertia response of traditional SG. The main difference between SG and converter for smoothing the system frequency deviation is that the traditional SG utilises the rotational mass-energy to decrease the rate of change of frequency (ROCOF), while the converter makes use of static DC-link capacitor energy to mitigate system frequency fluctuations. Therefore, the inertia time constant provided by the static converter noted as H C can be defined in Equation (19) as It can be clearly indicated that the larger the capacitance of DC-link capacitor noted as C and the larger the droop coefficient noted as k DC , the more the inertia of the converter can be provided during system disturbances. The control law of phase angle of converter voltage marked δ C can be written as Therefore, the Equations (13), (17) and (20) construct the control law of GSC in the proposed scheme of WT.

Discussion of DC-link capacitor
The power system may have strict standards for the power quality concerning the voltage and the frequency, and so forth. This only means that at rare circumstances, either voltage or frequency disturbance may occur; special control strategies such as our proposed scheme should be activated to provide needed support. In this study, it is assumed that DC-link voltage will not exceed its limitations under the severest frequency disturbance. Accordingly, the following inequality constraint should be satisfied: where ΔV DCmax and Δf max are the tolerated maximum DC-link voltage deviation and system frequency deviation, respectively. Normally, the maximum DC-link voltage deviation can reach 0.15 p.u. The exact value depends on the insulation requirement and pulse width modulation (PWM) pattern. Based on Equation (21), the proportional coefficient of the droop control noted as k DC should be satisfied as follows: To enhance the frequency stability of PMSG-based WTembedded power system, the system operator normally requires WT to provide a certain amount of synthetic inertia during frequency disturbance. Accordingly, the system required synthetic inertia marked as H Cr with the proposed control scheme can be written as follows: Combining Equations (22) and (23), the inequality constraint for the DC-link capacitance can be deduced as follows: Even though the stored energy in the DC-link capacitor of single WT is very limited, the accumulated energy of one wind farm for system inertia support is significant. Moreover, the supercapacitor, energy storage system [30] can be also installed between back-to-back converters of PMSG-based WT to improve the synthetic inertia provided by the WT.

Small-signal analysis of the proposed control
As shown in Figure 2, the active power from GSC noted as P C can be written as follows: The converter voltage magnitude is regulated as constant value based on Equation (13), which can be expressed as follows: Moreover, the DC-link voltage dynamics and DC-link voltage droop-based control can be rewritten as follows: Accordingly, the converter phase dynamics can be written as Therefore, Equations (25) to (29) constitute the mathematical model of WT with the proposed control scheme. Considering the small disturbances around the equivalent operation points of variables, the active power change from GSC of PMSGbased WT marked as ΔP C can be linearised around the operation points as follows: Combining Equations (18), (19), and (30), the dynamics of the converter voltage frequency can be written as follows: Combining Equations (29) and (31), the small-signal model of above GSC control can be expressed as follows:

OSCILLATION DAMPING CONTROL OF PMSG-BASED WT
It can be clearly seen from Equation (32) that the above GSC control scheme makes the system a pure oscillator, which has no additional damping provided. Actually, there are two control schemes to provide system damping, namely, active power modulation and converter voltage regulation.

Damping provision by active power modulation
The traditional aim of RSC of the grid-connected PMSGbased WT is to realise MPPT and regulate the reactive power exchanged between PMSG and RSC. The MPPT algorithm is implemented to set the optimal active power reference based on the current rotor speed. The rotor dynamics of WT can be expressed by Equation (33): where ω r and H WT are the rotor speed and inertia time constant of WT, respectively. P W and P R are the captured wind power and active power from RSC. It is noted that due to the fast active power control of VSC, the actual power output of RSC is equal to the power reference value, that is, P R = P R * . The active power reference P R * is the order from the MPPT algorithm. Normally, it is the cube of the current rotor speed, which can be expressed as where k max are the concerned coefficient by MPPT algorithm. Naturally, an easy and feasible way for providing damping of grid-connected PMSG-based WT is by adding the auxiliary damping power modulation loop in the active power control of RSC, which can be indicated as where D C is the damping coefficient provided by RSC power modulation. Combining Equations (18), (19) and (35), the dynamics of GSC incorporating the damping control can be expressed as where ω r0 is the initial rotor speed value before system disturbances. It can be clearly seen from Equation (36), the dynamics of PMSG-based WT of proposed control are very similar to the dynamic behaviour of traditional SG described in Equation (4). Unfortunately, the damping loop provided by the active power modulation from RSC of WT has two main drawbacks: (1) The rotor speed dynamics of WT have strong impacts on the damping control of WT as shown in Equation (36). More specifically, if overlooking the captured wind power change during rotor speed variations, that is, ΔP W = 0, during system dynamics, then rotor dynamics of WT described in Equation (33) can be rewritten as the following equation by combining the additional damping control in Equation (35): It can be seen from Equation (37) that any system frequency change will lead to the rotor speed variations of WT, which has the impaired influences on the damping control effects of WT in Equation (36). (2) The additional damping control via active power modulation of RSC will definitely enable WT to deviate from its maximum power capture status once there is frequency disturbances during system dynamics. As a result, it may not be an economical way in daily operation.

Damping provision by converter voltage regulation
From Equation (32) it can be clearly seen that controlling converter voltage as a constant value during system dynamics cannot provide any additional system damping. Therefore, in order to provide the system damping without using the active power modulation by RSC and get rid of sacrificing MPPT of WT for making sure of the economic operation, system damping can be provided by spontaneously altering the converter voltage magnitude during the disturbances, which is very similar to the PSS mechanism of the traditional SG. Accordingly, the GSC voltage can be regulated based on the DC-link voltage deviation, which can be expressed as follows: More specifically, based on Equation (23), the converter voltage is controlled according to the system frequency deviation as follows: Therefore, the modulation index of the GSC is controlled as follows: As indicated by Equation (39), the converter voltage will be altered when there is any system frequency deviation so that the transmitted active power is changed and the system damping is provided accordingly. The damping coefficient can be obtained through the small-signal analysis method in the following section. Therefore, in the proposed control scheme, the RSC of WT still works with MPPT algorithm, while GSC of WT controls converter voltage based on Equation (39). The overall control diagram of the proposed grid-forming control is shown in Figure 3. It can be seen from Figure 3 that once the three variables are controlled based on Equations (17), (20) and (40), the reference voltage of phase a can be generated, which will be compared with the triangular carrier signals with the frequency of several thousand Hz. As a result, the specific IGBT is controlled to be turned on or off based on the generated gating signals.

Damping coefficient with the proposed control
Combining Equations (28) and (39), the small alternation of converter voltage magnitude noted as ΔV C can be written below: Accordingly, the active power change from GSC can be rewritten based on Equations 30) and (41) as To simplify the following deductions, the concerned parameters are defined as follows: Therefore, combining Equations (18), (19), (42) and (43), the dynamics of the converter voltage frequency can be written as follows: Combining Equations (29) and (44), the small-signal model of GSC with the proposed PSC can be expressed as It can be clearly seen from Equation (45)  to the dynamic behaviour of the traditional SG. Therefore, in the proposed control, GSC of WT well mimics the property of SG during system disturbances, which can realise the synchronisation without PLL while providing certain inertia and damping support for the weak AC system during system disturbance.
Transferring Equation (45) into a Laplace form, the transfer function between the change in phase angle of converter voltage and the change in the active power of RSC is written as follows in Equation (46): Accordingly, the damped oscillation induced by the proposed control with the damping ratio noted as ζ C can be indicated as follows: From Equation (47), the damping from GSC of WT with the proposed control scheme is achieved by automatically alternating the converter voltage magnitude during system dynamics.
It can be concluded from the above analysis that in our proposed grid-forming control, MPPT algorithm can be realised through RSC. Meanwhile, GSC can provide the system inertia and damping support without the need of PLL. Moreover, the natural inertia response and system damping of WT can be provided during system frequency disturbance, which is well suitable for future large-scale wind farm integration with low-inertia AC system with low SCR. It should be noted that when there is limited capacity for DC-link or the converter, the active power modulation from RSC control can be activated for providing FIGURE 4 Simple diagram of the studied system the required inertia or damping support, which is not further discussed in this study.

SIMULATION STUDIES
To effectively validate the proposed PSC scheme for the gridconnected PMSG-based WT during system disturbances, a simple network consisting of one 2 MW PMSG-based WT connected to the power grid represented by one SG of 20 MVAR with the classical seventh-order model is described in Figure 4. To well emulate the infinite power source, the system frequency is sustained by the governor of SG with large frequency regulation droop and small-time constant to eliminate the dynamic process of SG governor during disturbances. Two local loads (L 1 and L 2 ) are located near the SG.   Figure 5 shows the dynamic response of the system when SCR is suddenly changed from 4.8 to 1.5 at t = 1 s. Three different control schemes, namely, the proposed control scheme, classical PLL-based control with high bandwidth (set as 27 Hz) and with low bandwidth (set as 12 Hz) are compared in detail, respectively. The DC-link voltage droop control parameters k DC for the proposed scheme is set as eight and the damping control coefficient k D is set as 0.5. As shown in Figure 5(a), PCC voltage can be well sustained with the proposed control scheme after system disturbances, while with the classical PLL-based control, an apparent oscillation of PCC voltage begins to appear. This can be well explained that with low system SCR, the large proportion integration (PI) parameter of PLL (high bandwidth of control) will magnify the error of q-axis PCC voltage, which leads to a faster apparent voltage oscillation, compared to the small PI parameter adopted. It is noted from Figure 5(b) that in the proposed scheme, the synchronism can be automatically satisfied by spontaneously increasing the phase angle difference between the PCC voltage and the SG terminal voltage with the sudden decrease of system SCR, which is very similar to the dynamics of SG during system disturbances. However, with the PLL-based control scheme, the accurate phase measurement of PCC voltage cannot be fast obtained during system dynamics with low system SCR. As a result, the active power from the converter will also experience a large range of oscillation after system SCR changes with the classical PLL-based scheme. In contrast, with the proposed control, active power generated from WT will not be interrupted after the short period transient process as clearly indicated in Figure 5(b).

Comparison with PLL-based control
In the proposed control scheme, the DC-link voltage deviation as shown in Figure 5(c) is artificially coupled with the system frequency deviation using certain droop property, which results in the profiles of DC-link voltage and the similar system frequency during system dynamic process. It can be clearly seen from Figure 5(d) that the initial change of system frequency is effectively suppressed with the proposed control scheme since the partial DC-link capacitor energy can be released out for system support during system disturbances. From Figure 5, it can be concluded that the proposed control scheme can achieve self-synchronism without the aid of PLL and render strong support for weak AC system, which has the great potential for the future WT applications.

5.2
Sudden change of active power reference of RSC k DC and the adopted damping control parameter k D . Due to the sudden power reference reduction of RSC of PMSG-based WT, the active power from converter decreases as shown in Figure 6(b). It should be noted that the active power from the converter decreases more slowly with large DC-link voltage droop parameter since more electrical energy stored in the DC-link capacitor can be temporally exerted with the proposed scheme. As a result, DC-link voltage deviation as shown in Figure 6(c) during system disturbance is relatively large when utilising the large droop coefficient. Because of more system inertia provided with large DC droop parameter of the proposed control, ROCOF and system frequency nadir are relatively small compared with small DC droop parameter adopted as clearly indicated in Figure 6(d). With the proposed control, the effective system damping can be provided by GSC as shown in Figure 6 via spontaneously alternating the converter voltage  Figure 6, and the larger the adopted damping control parameter k D , the more the system damping can be exerted for system support. The proposed control distinguishes itself for rendering the system inertia and damping support without sacrificing the MPPT of the WT. Figure 7 shows the dynamic response of the PMSG-based WT with the proposed control scheme under the variable wind speeds, of which the mean value is 11 m/s and the standard deviation is 1.5 m/s. The dynamic performances of WT under different control parameters are compared. It can be clearly seen that the proposed control enables the grid-connected PMSGbased WT to realise the MPPT algorithm without the need of PLL under the variable wind speeds situations. It can be clearly observed from Figures 7(a) and (b) that with the increase of the damping control parameter noted as k D , the converter voltage and the active power from the converter are effectively mitigated since more system damping can be provided by automatically regulating the converter voltage. Meanwhile, with the increase of the DC-droop parameter noted as k DC , the fluctuation of DC-link voltage as shown in Figure 7(c) and the converter voltage as shown in Figure 7(a) are apparently increased because more energy from DC-link capacitor is utilised for the system inertia support. As a result, the grid frequency fluctuations with larger control parameters as shown in Figure 7(d) are apparently mitigated, compared to the smaller adopted control parameters. It should be noted that with the proposed control scheme, the inertia response of WT can be provided by automatically exerting the DC-link capacitor energy, and the system damping can be also provided by spontaneously changing the converter voltage magnitude. More importantly, this control scheme stands out itself by providing system inertia and damping support without sacrificing MPPT. It is verified again that the proposed control scheme can be well suitable for the future large-scale wind farm integration into weak AC system.

Comparison with the existing VSG control
To verify the advantage of the proposed control scheme, the simulation study in terms of comparing to the existing VSG control [21] is performed as follows. The dump load noted as P L2 + Q L2 (0.1 MW + 0.03 MVAR) that is 10% of the fixed load marked as P L1 + Q L1 (1 MW + 0.1 MVAR), as shown in Figure 4, is suddenly switched on at t = 1 s. Three different cases, namely, MPPT control, traditional VSG with frequency and damping control, and the proposed control are fully compared in Figure 8. In order to fairly compare the two control schemes, the traditional VSG control and the proposed control, the frequency nadirs under both control schemes are nearly the same of 49.55 Hz as shown in Figure 8(d) under the system frequency disturbance via properly turning the control parameters. In this way, the inertia and damping support capability from WT under both control schemes are nearly the same. As clearly indicated in Figure 8(a), rotor speed of PMSG with the traditional VSG control largely deviates from the optimal rotor speed value as 1.1 p.u. This can be well explained that in the traditional VSG control, the partially stored rotor kinetic energy is released for providing system inertia and damping support during the system frequency disturbance. As a result, the captured wind power is largely compromised with the traditional VSG control, which is clearly marked as the shadow area in Figure 8(b), noted as S1 at 0.1842 MJ. On the contrary, with the proposed control scheme, the system inertia and damping support are provided by the DC-link capacitor energy, which leads to the DC-link voltage   apparently decreasing as seen in Figure 8(c). Therefore, it can be well verified that MPPT property of PMSG is not compromised while providing the inertia and damping for the system with the proposed control scheme.

CONCLUSION
In this study, a novel PSC of PMSG-based WT for integrating very weak AC system is proposed. Different from the traditional PLL-based WT control, the proposed novel scheme enables WT with the grid forming ability without PLL, which largely improves the overall stability of the AC system. In the proposed scheme, GSC of PMSG artificially couples the grid frequency with the certain droop property, and the converter voltage magnitude is regulated based on the DC-link voltage deviation.
Small-signal analysis of the proposed scheme concludes that the partially stored electrical energy in DC-link capacitor can be released out for system inertia support, and the system damping can be supplied by regulating the converter voltage magnitude during system disturbances without sacrificing MPPT. The proposed PSC of PMSG-based WT stands out itself without the need of PLL for grid-forming and spontaneously provides the inertia and damping support with little control impacts on MPPT, which is beneficial for future large-scale wind power integration into weak AC system.

APPENDIX Control equations for the rotor-side converter (RSC) and grid-side converter (GSC)
The typical vector current control is utilised for RSC to achieve maximum power point tracking (MPPT). The control equations for RSC is q are d-q axis current reference value of RSC, which are determined by the outer power control loop. Q * R and Q R denote the reactive power reference value and actual value, and normally Q * R is set as zero for the maximum active power transmission from WT.K P1 , K P2 , K I1 , K I2 are the correlated parameters of the PI controller of the outer control loop. K P3 , K P4 , K I3 , K I4 are the correlated parameters of the PI controller of the inner control loop.
The proposed control as illustrated in Sections 3 and 4 is adopted for GSC. The control equations for GSC is listed as

Relevant parameters of the studied permanent magnet synchronous generator (PMSG)-based wind turbine (WT)
See Table A1-A3.