Multi-Timescale Control of Variable-Speed Wind Turbine for Inertia Provision

: The increasing deployment of power converters has led to a signiﬁcant reduction in the power system inertia and consequently resulted in frequency stability issues. To improve the robustness of the grid against frequency disturbances, it is becoming more expected in many countries that renewable energy generation, such as wind turbine power systems, should provide equivalent inertia support to the power system. This can be achieved through advanced control of power converters, in addition to adding extra energy storage devices, e.g., batteries. In wind turbine systems, although the ancillary service of inertia provision can be realized by coupling the rotor speed with the grid frequency, the rotor speed recovery process affects the inertia response if the controller is not properly designed or well-tuned. To address this issue, in this paper, we propose a multi-timescale control strategy for a doubly fed induction generator (DFIG) wind turbine system. Synthetic inertia control and speed recovery control are simultaneously incorporated into the controller of the rotor-side converter, whereas their dynamics are decoupled under different timescales to avoid control conﬂict. Extensive simulation results are provided, which validate the efﬁcacy of the proposed inertia emulation scheme.


Introduction
The increasing penetration of renewable energy resources (RESs), such as the photovoltaic (PV) panels and wind turbines, has fundamentally changed power system characteristics. These RESs are usually integrated into the power grid through power-electronic converters, which, naturally, do not possess the attribute of mechanical inertia. Conventional synchronous generators (SG) are gradually being replaced by inertia-less power converters (low-inertia systems), which will result in a reduction in power system inertia to a large extent [1][2][3][4][5][6][7][8][9][10]. The decrease in the power system inertia, on one hand, will result in a higher rate of change of frequency (RoCoF). On the other hand, the reduction in the power system inertia can cause a low-frequency nadir [2]. As a consequence, frequency stability issues are more likely to happen [3,4]. One recently reported incident is the 2016 Australia blackout, which affected millions of people and caused considerable economic loss. It is believed that a lack of power system inertia is one of the reasons behind the blackout [5]. Therefore, it is of importance in many countries to increase power system inertia and enhance the frequency stability of future power-electronics-dominated power system, where more and more renewable energy resources are adopted.
To achieve this objective, a number of research activities have explored the possibility of using power-electronics converters to provide grid inertia support [6][7][8], in addition to the use of extra energy resources, e.g., batteries. Among them, the virtual synchronous generator (VSG) technique stands out as a promising solution. The VSG control strategy enables the power converter to emulate the rotor swing equation of a conventional SG.
As a consequence, the dynamic of the converter is similar to that of the SG. Nevertheless, the successful implementation of the VSG control strategy usually requires a bulky and costly energy storage system to provide active power support. Alternatively, the electrical energy stored in DC-link capacitors can also be utilized to provide inertia support [10]. To achieve this objective, large DC capacitors are normally required, which may not conform with the power-electronics converter design principle. In [11,12], the voltage across a supercapacitor was proportionally linked with the AC power grid. Through this effort, the supercapacitor can be controlled to behave similarly to a virtual synchronous generator to support the power system frequency response. Alternatively, the inertia emulation effect can be equivalently achieved by demand-side management techniques. In [13], the power consumptions of certain noncritical loads, such as electric heating, lighting, and cooling, were adaptively regulated to realize synthetic inertia and suppress the power grid RoCoF and frequency deviation.
The inertia emulation target can also be achieved by PV generation systems. Conventionally, the PV inverter operates under maximum power point tracking (MPPT) mode, which harvests the maximum solar energy and does not participate in grid frequency regulation [14]. Recently, power reserve control (PRC) and active power curtailment control were developed to intentionally reduce the PV output power and spare a certain amount of active power for inertia provision [15][16][17][18]. Through this effort, the requirement of energy storage units, capacitors, and supercapacitors becomes unnecessary and largely alleviated. Accordingly, an event-triggering control strategy was developed to switch the PV system between different operation modes to optimize the frequency support from the reserved power [19]. A coordination scheme was further proposed to achieve the virtual inertia and frequency-damping effect in [20]. These advanced control algorithms have made PV generation systems more grid-friendly and actively contribute to power system frequency regulation.
In comparison with PV generation systems, wind turbines have even greater potential to provide inertia support. This is because the significant mechanical energy stored in the rotor can be temporarily released to respond to a power grid frequency event. The earlier application can be traced back to the wind turbine system in the Hydro-Quebec system, as well as that in Ontario [21]. For the normal stage, the wind turbine operates at the maximum power point (MPP). When a grid-frequency drop is detected, the reference power of the wind turbine is increased by 5-10% to provide active power support. After a certain amount of time, the wind turbine decreases its output power reference by 30% until the rotor speed is gradually recovered. Similar techniques have also been reported in recent publications. In [22], a supplementary electromagnetic torque component proportionally linked with the grid RoCoF was included to achieve the desired inertia emulation. In [23,24], a high-pass filter was implemented to avoid the high-frequency noise amplification concern. In [25], the dynamics between a type 3 wind turbine and the SG were compared. On this basis, the phase motion of the SG was reconstructed in a wind turbine system to enable the inertia function. In [26], frequency droop control was implemented as an alternative for a synthetic inertia strategy through control parameter optimization. Moreover, in [27], a cascaded control structure was proposed to regulate DC-link energy and subsequently control the rotor kinetic energy to support the grid frequency, which minimizes the impacts on wind energy harvesting. In addition to this, the inertia provision function of a doubly fed induction generator (DFIG) unit was enabled by rotor current control and optimized through status assessment in [28]. In [29], a power management strategy was developed that blends the energy stored in the high-voltage direct current (HVDC) link and the energy of a wind turbine to provide quick frequency responses. Consequently, the power capability requirement of wind turbines can be relieved.
The common principle behind these proposals is to link the power grid frequency (or the RoCoF) with the rotor speed (or the electromagnetic torque) of the wind turbine. It should be mentioned that the wind turbine needs to recover after the settling of a frequency event (i.e., post-event operation). However, the speed recovery process can significantly affect the inertia-provision effect if it is not properly designed [30]. It is likely that a subsequent frequency drop, as the second frequency event, will occur due to the interaction between speed recovery control and inertia control [2]. For the worst scenarios, the coupling between inertia provision and the dynamic characteristic of a wind turbine system may even affect the small-signal stability of the entire power system and lead to power oscillations [31,32].
To address this problem, a multi-timescale control scheme for inertia provision of a DFIG wind conversion system is proposed in this paper. A schematic diagram of the DFIG power conversion system is shown in Figure 1. By properly selecting the rotor-side converter (RSC) control parameters, the dynamics of inertia control and those of speed recovery control can be decoupled in the frequency domain to avoid control conflict. As a benefit, inertia provision, as well as speed recovery objectives, can be simultaneously achieved. should be mentioned that the wind turbine needs to recover after the settling of a frequency event (i.e., post-event operation). However, the speed recovery process can significantly affect the inertia-provision effect if it is not properly designed [30]. It is likely that a subsequent frequency drop, as the second frequency event, will occur due to the interaction between speed recovery control and inertia control [2]. For the worst scenarios, the coupling between inertia provision and the dynamic characteristic of a wind turbine system may even affect the small-signal stability of the entire power system and lead to power oscillations [31,32].
To address this problem, a multi-timescale control scheme for inertia provision of a DFIG wind conversion system is proposed in this paper. A schematic diagram of the DFIG power conversion system is shown in Figure 1. By properly selecting the rotor-side converter (RSC) control parameters, the dynamics of inertia control and those of speed recovery control can be decoupled in the frequency domain to avoid control conflict. As a benefit, inertia provision, as well as speed recovery objectives, can be simultaneously achieved. Compared to prior-art approaches, the contributions of this paper are summarized as: (1). Revealing the coupling effect between virtual inertia control and speed recovery control of the wind energy conversion system with an investigation of the corresponding impact on the performance of RoCoF suppression; (2). Decoupling of rotor speed recovery control and inertia control into different timescales; as a result, the inertia provision and speedy post-event recovery objectives can be simultaneously achieved; (3). Designing a noise-free approach to acquire the real-time grid RoCoF, the response time of which is less than 0.1 s; (4). Conducting simulations to verify the theoretical analysis and control performance.
With the wind turbine virtual inertia being 5 s, the maximum RoCoF is successfully reduced by around 23%.
The rest of this paper is organized as follows. Section 2 introduces the basic inertia emulation principle. Section 3 discusses the DFIG control strategy, including RSC control, grid-side converter (GSC) control, and acquisition of the power grid RoCoF. Extensive simulation results are provided in Section 4 to verify the theoretical findings and the proposed control scheme. Finally, Section 5 concludes the paper. Figure 2 shows the power system primary frequency regulation framework. For the convenience of a statement, all the variables are displayed in per-unit values. R is the frequency droop coefficient; TG refers to the speed governor time constant; FHP, TRH, and TCH are the reheat turbine time constants; ΔPref_pu is the input-power reference variation; ΔPm_pu is the mechanical power variation; and ΔPl_pu is the load power variation. According to the well-known swing equation [8], the power grid frequency, Δfg_pu, is given by: Compared to prior-art approaches, the contributions of this paper are summarized as:

Power System Frequency Response
(1). Revealing the coupling effect between virtual inertia control and speed recovery control of the wind energy conversion system with an investigation of the corresponding impact on the performance of RoCoF suppression; (2). Decoupling of rotor speed recovery control and inertia control into different timescales; as a result, the inertia provision and speedy post-event recovery objectives can be simultaneously achieved; (3). Designing a noise-free approach to acquire the real-time grid RoCoF, the response time of which is less than 0.1 s; (4). Conducting simulations to verify the theoretical analysis and control performance.
With the wind turbine virtual inertia being 5 s, the maximum RoCoF is successfully reduced by around 23%.
The rest of this paper is organized as follows. Section 2 introduces the basic inertia emulation principle. Section 3 discusses the DFIG control strategy, including RSC control, grid-side converter (GSC) control, and acquisition of the power grid RoCoF. Extensive simulation results are provided in Section 4 to verify the theoretical findings and the proposed control scheme. Finally, Section 5 concludes the paper. Figure 2 shows the power system primary frequency regulation framework. For the convenience of a statement, all the variables are displayed in per-unit values. R is the frequency droop coefficient; T G refers to the speed governor time constant; F HP , T RH , and T CH are the reheat turbine time constants; ∆P ref_pu is the input-power reference variation; ∆P m_pu is the mechanical power variation; and ∆P l_pu is the load power variation. According to the well-known swing equation [8], the power grid frequency, ∆f g_pu , is given by:

Power System Frequency Response
where H M is the inertia constant of the synchronous machine (SM). Equation (1) indicates that the power grid frequency deviates from the nominal value if there is a power imbalance between the power generation and the load demand. More specifically, Figure 3 illustrates the typical frequency response when the power grid is subjected to a disturbance (taking a sudden load change as an example).
( ) where HM is the inertia constant of the synchronous machine (SM). Equation (1) indicates that the power grid frequency deviates from the nominal value if there is a power imbalance between the power generation and the load demand. More specifically, Figure 3 illustrates the typical frequency response when the power grid is subjected to a disturbance (taking a sudden load change as an example).  Due to the inertia of the SM, the power grid frequency does not immediately drop or increase with a high ramp rate. The instantaneous power imbalance is compensated by the mechanical energy of the rotor, resulting in a comparatively small variation in the system frequency. This process is known as the inertia response and usually takes less than 10 s. For a longer timescale, the speed governor control takes effect, and the frequency finally reaches a new equilibrium point. There are two important metrics that require attention, from the power system frequency stability perspective. One is the frequency nadir (fmin in Figure 3), and the other is the RoCoF value. When the frequency nadir goes beyond the normal operating range, proper measures, such as load shedding, should be taken, which causes a negative impact on electricity customers. In addition to this, the high RoCoF value may also cause a pole slip, trip the SM, and even lead to frequency instability. In this regard, many countries have set stringent requirements for the frequency nadir and the RoCoF.
To illustrate the impact of inertia on the frequency nadir and the RoCoF, Figure 4 shows the grid frequency response with different inertia values. It can be observed that the increase in the power system inertia reduces the RoCoF value and also improves the frequency nadir. Consequently, the power system frequency stability is enhanced. However, when conventional SMs and SGs are gradually replaced by inertia-less power converters, the total inertia of the power grid will decrease, giving ride to frequency stability ( ) where HM is the inertia constant of the synchronous machine (SM). Equation (1) indicates that the power grid frequency deviates from the nominal value if there is a power imbalance between the power generation and the load demand. More specifically, Figure 3 illustrates the typical frequency response when the power grid is subjected to a disturbance (taking a sudden load change as an example).  Due to the inertia of the SM, the power grid frequency does not immediately drop or increase with a high ramp rate. The instantaneous power imbalance is compensated by the mechanical energy of the rotor, resulting in a comparatively small variation in the system frequency. This process is known as the inertia response and usually takes less than 10 s. For a longer timescale, the speed governor control takes effect, and the frequency finally reaches a new equilibrium point. There are two important metrics that require attention, from the power system frequency stability perspective. One is the frequency nadir (fmin in Figure 3), and the other is the RoCoF value. When the frequency nadir goes beyond the normal operating range, proper measures, such as load shedding, should be taken, which causes a negative impact on electricity customers. In addition to this, the high RoCoF value may also cause a pole slip, trip the SM, and even lead to frequency instability. In this regard, many countries have set stringent requirements for the frequency nadir and the RoCoF.
To illustrate the impact of inertia on the frequency nadir and the RoCoF, Figure 4 shows the grid frequency response with different inertia values. It can be observed that the increase in the power system inertia reduces the RoCoF value and also improves the frequency nadir. Consequently, the power system frequency stability is enhanced. However, when conventional SMs and SGs are gradually replaced by inertia-less power converters, the total inertia of the power grid will decrease, giving ride to frequency stability Due to the inertia of the SM, the power grid frequency does not immediately drop or increase with a high ramp rate. The instantaneous power imbalance is compensated by the mechanical energy of the rotor, resulting in a comparatively small variation in the system frequency. This process is known as the inertia response and usually takes less than 10 s. For a longer timescale, the speed governor control takes effect, and the frequency finally reaches a new equilibrium point. There are two important metrics that require attention, from the power system frequency stability perspective. One is the frequency nadir (f min in Figure 3), and the other is the RoCoF value. When the frequency nadir goes beyond the normal operating range, proper measures, such as load shedding, should be taken, which causes a negative impact on electricity customers. In addition to this, the high RoCoF value may also cause a pole slip, trip the SM, and even lead to frequency instability. In this regard, many countries have set stringent requirements for the frequency nadir and the RoCoF.
To illustrate the impact of inertia on the frequency nadir and the RoCoF, Figure 4 shows the grid frequency response with different inertia values. It can be observed that the increase in the power system inertia reduces the RoCoF value and also improves the frequency nadir. Consequently, the power system frequency stability is enhanced. However, when conventional SMs and SGs are gradually replaced by inertia-less power converters, the total inertia of the power grid will decrease, giving ride to frequency stability concerns. Therefore, it is of importance to enable power converters to make an active contribution to the power grid by properly providing grid support, e.g., emulating inertia.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 concerns. Therefore, it is of importance to enable power converters to make an active tribution to the power grid by properly providing grid support, e.g., emulating iner

Inertia Provision Principle
To achieve this objective, the output power of renewable energy resources (R needs to be coupled with the power grid frequency. Figure 5 illustrates the inertia p sion principle, where Hv is the virtual inertia provided by RESs, and ΔPi_pu is the ou power variation of the RES. From Figure 5, it can be seen that the power system frequency variation, Δfg given by: given that the RES active power variation is proportionally linked to the grid RoCoF (2) and (3), it can be obtained that: Comparing (4) with (1), the system inertia is equivalently increased from HM to Hv. To achieve this, the power converter of the wind turbine system should be propor ally linked with the grid RoCoF, as shown in (3). The corresponding controller design be discussed in Section 3. Therefore, the essential point is to realize the dynamics of E tion (3) through proper control of the interfaced power converters, which will be det in Section 3.

Inertia Provision Principle
To achieve this objective, the output power of renewable energy resources (RESs) needs to be coupled with the power grid frequency. Figure 5 illustrates the inertia provision principle, where H v is the virtual inertia provided by RESs, and ∆P i_pu is the output power variation of the RES.
concerns. Therefore, it is of importance to enable power converters to make an active contribution to the power grid by properly providing grid support, e.g., emulating inertia.

Inertia Provision Principle
To achieve this objective, the output power of renewable energy resources (RESs) needs to be coupled with the power grid frequency. Figure 5 illustrates the inertia provision principle, where Hv is the virtual inertia provided by RESs, and ΔPi_pu is the output power variation of the RES. From Figure 5, it can be seen that the power system frequency variation, Δfg_pu, is given by: given that the RES active power variation is proportionally linked to the grid RoCoF as: From (2) and (3), it can be obtained that: Comparing (4) with (1), the system inertia is equivalently increased from HM to HM + Hv. To achieve this, the power converter of the wind turbine system should be proportionally linked with the grid RoCoF, as shown in (3). The corresponding controller design will be discussed in Section 3. Therefore, the essential point is to realize the dynamics of Equation (3) through proper control of the interfaced power converters, which will be detailed in Section 3. From Figure 5, it can be seen that the power system frequency variation, ∆f g_pu , is given by: given that the RES active power variation is proportionally linked to the grid RoCoF as: From (2) and (3), it can be obtained that: Comparing (4) with (1), the system inertia is equivalently increased from H M to H M + H v . To achieve this, the power converter of the wind turbine system should be proportionally linked with the grid RoCoF, as shown in (3). The corresponding controller design will be discussed in Section 3. Therefore, the essential point is to realize the dynamics of Equation (3) through proper control of the interfaced power converters, which will be detailed in Section 3.

Inertia Provision Control Strategy
As discussed previously, the control strategy of inertia provision can be mainly divided into two aspects. The first is the acquisition of the grid RoCoF value, whereas the second is to control the wind turbine system's active power variation according to Equation (3).

Acquistion of RoCoF
Conventionally, the grid RoCoF value is obtained by measuring the real-time grid frequency and calculating its derivation. Figure 6 is a schematic diagram of the conventional approach for RoCoF value calculation [33,34]. Initially, the grid voltage waveform is obtained, and its frequency, f g , can be detected through zero crossing or Fourier transformation. Then, a derivation block is implemented to calculate df g /dt, i.e., RoCoF. However, directly differentiating the frequency signal would result in noise amplification and therefore a compromise the RoCoF calculation performance. To address this problem, a measuring window (defined as the number of frequency-measuring periods over which the RoCoF is calculated) is usually employed. However, there is a tradeoff between RoCoF measurement accuracy and noise-rejection capability. If the length of the measuring window is considerable, the noise due to the derivation can be well attenuated. However, the measured RoCoF value will be differ more from the real value. If the filtering capability of the measuring window is weak, the measured RoCoF would be closer to the real RoCoF, whereas noise cannot be attenuated well. tion (3).

Acquistion of RoCoF
Conventionally, the grid RoCoF value is obtained by measuring the real-time grid frequency and calculating its derivation. Figure 6 is a schematic diagram of the conventional approach for RoCoF value calculation [33,34]. Initially, the grid voltage waveform is obtained, and its frequency, fg, can be detected through zero crossing or Fourier transformation. Then, a derivation block is implemented to calculate dfg/dt, i.e., RoCoF. However, directly differentiating the frequency signal would result in noise amplification and therefore a compromise the RoCoF calculation performance. To address this problem, a measuring window (defined as the number of frequency-measuring periods over which the RoCoF is calculated) is usually employed. However, there is a tradeoff between RoCoF measurement accuracy and noise-rejection capability. If the length of the measuring window is considerable, the noise due to the derivation can be well attenuated. However, the measured RoCoF value will be differ more from the real value. If the filtering capability of the measuring window is weak, the measured RoCoF would be closer to the real RoCoF, whereas noise cannot be attenuated well. To address this problem, a frequency-locked loop (FLL) is implemented in this paper for RoCoF calculation. In comparison with the conventional RoCoF calculation method the FLL-based approach avoids the use of a derivation block and therefore provides noisefree RoCoF acquisition. Figure 6 displays the control block diagram of the FLL-based RoCoF acquisition, where vg is the power grid voltage, kp-fll and ki-fll are the PI control parameters, and vfll-α and vfllβ are the orthogonal signals obtained from a second-order generalized integrator (SOGI) [35,36].
From Figure 7, the frequency-domain transfer functions from vg to vfll-α and from vg to vfllβ can be respectively derived as:  To address this problem, a frequency-locked loop (FLL) is implemented in this paper for RoCoF calculation. In comparison with the conventional RoCoF calculation method, the FLL-based approach avoids the use of a derivation block and therefore provides noise-free RoCoF acquisition. Figure 6 displays the control block diagram of the FLL-based RoCoF acquisition, where v g is the power grid voltage, k p-fll and k i-fll are the PI control parameters, and v fll-α and v fll-β are the orthogonal signals obtained from a second-order generalized integrator (SOGI) [35,36].
From Figure 7, the frequency-domain transfer functions from v g to v fll-α and from v g to v fll-β can be respectively derived as: At the fundamental frequency, ω0, it can be derived from (5) and (6) that: from which it is known that vfll-α is in-phase with the grid voltage, vg (with the same mag At the fundamental frequency, ω 0 , it can be derived from (5) and (6) that: ∠v fll−α (jω 0 ) = ∠v g (jω 0 ) = ∠v fll−β (jω 0 ) + π 2 . (8) from which it is known that v fll-α is in-phase with the grid voltage, v g (with the same magnitude), whereas v fll-β is orthogonal with v g (with the same magnitude). On this basis, the FLL dynamically adjusts ∆ω g through the integral controller, k i-fll /s. By doing so, the frequency input of the SOGI, i.e., ω 0 + ∆ω g , can track with the real grid frequency. It should be noted that there is an inherent integral controller before ω 0 + ∆ω. Therefore, the RoCoF can be obtained from the input of this integral controller, as illustrated in Figure 7.
To further attenuate harmonic distortions and also improve the performance of RoCoF calculation, a moving average filter (MAF) is incorporated into the control loop, the time window of which should be properly selected to be one fundamental period, i.e., 20 ms for the 50 Hz power system. In order to verify the efficacy of the RoCoF calculation, simulations were conducted, and the results are shown in Figure 8. In Figure 8, the RoCoF of the power grid voltage, v g , changes from 0 Hz/s to −0.1 Hz/s at t = 0.5 s. It can be observed from Figure 8 that the obtained RoCoF value can quickly track with the real RoCoF value within around 100 ms.
At the fundamental frequency, ω0, it can be derived from (5) and (6) that: from which it is known that vfll-α is in-phase with the grid voltage, vg (with the same magnitude), whereas vfll-β is orthogonal with vg (with the same magnitude). On this basis, the FLL dynamically adjusts Δωg through the integral controller, ki-fll/s. By doing so, the frequency input of the SOGI, i.e., ω0 + Δωg, can track with the real grid frequency. It should be noted that there is an inherent integral controller before ω0 + Δω. Therefore, the RoCoF can be obtained from the input of this integral controller, as illustrated in Figure 7. To further attenuate harmonic distortions and also improve the performance of RoCoF calculation, a moving average filter (MAF) is incorporated into the control loop, the time window of which should be properly selected to be one fundamental period, i.e., 20 ms for the 50 Hz power system. In order to verify the efficacy of the RoCoF calculation, simulations were conducted, and the results are shown in Figure 8. In Figure 8, the RoCoF of the power grid voltage, vg, changes from 0 Hz/s to −0.1 Hz/s at t = 0.5 s. It can be observed from Figure 8 that the obtained RoCoF value can quickly track with the real RoCoF value within around 100 ms.

DFIG System Control
Based on the acquired RoCoF value, the DFIG system can be controlled to provide the expected active power support (followed by inertia support). To achieve this target, the GSC is responsible for maintaining the DC-link voltage, v dc , and regulating the reactive power, Q g , injected to the grid. The RSC controls the induction machine.

GSC Control Scheme
Under the synchronous reference frame of the GSC, the relationship between the output voltage and output current is given by: where v d and v q are the GSC output voltage and current, respectively; i d and i q are the GSC output and current, respectively; v gd and v gq are the grid voltages; and R g and L g are the grid resistance and inductance, respectively. The active and reactive power injected to the grid can be expressed as: A phase-locked loop (PLL) is usually implemented to obtain the phase angle information for the abc/dq transformation. The PLL dynamically regulates the phase angle such that v q = 0. In this regard, the active power, P g , is determined by the d-axis current, i d , whereas the reactive power, Q g , is determined by the q-axis current, i q . On this basis, the current references for the GSC are given by: (14) where k p1,2 and k i1,2 are the proportional and integral control gains, respectively; v dc_ref is the reference DC-link voltage; and Q ref is the reference reactive power. Figure 9 shows the control block diagram of the GSC, where the cross-coupling terms in Equations (8) and (9) have been decoupled through the current feedforward control, and k p3 and k i3 are the proportional and integral control gains, respectively, for the inner-loop current control.
where vd and vq are the GSC output voltage and current, respectively; id and iq are the GSC output and current, respectively; vgd and vgq are the grid voltages; and Rg and Lg are the grid resistance and inductance, respectively. The active and reactive power injected to the grid can be expressed as: A phase-locked loop (PLL) is usually implemented to obtain the phase angle information for the abc/dq transformation. The PLL dynamically regulates the phase angle such that vq = 0. In this regard, the active power, Pg, is determined by the d-axis current, id, whereas the reactive power, Qg, is determined by the q-axis current, iq. On this basis, the current references for the GSC are given by: where kp1,2 and ki1,2 are the proportional and integral control gains, respectively; vdc_ref is the reference DC-link voltage; and Qref is the reference reactive power. Figure 9 shows the control block diagram of the GSC, where the cross-coupling terms in Equations (8) and (9) have been decoupled through the current feedforward control, and kp3 and ki3 are the proportional and integral control gains, respectively, for the inner-loop current control.

RSC Control Scheme
Similarly, the dynamic equations of the induction machine can be expressed as: Figure 9. Control block diagram of the GSC.

RSC Control Scheme
Similarly, the dynamic equations of the induction machine can be expressed as: The stator and rotor flux linkages are given by: where v dr and v qr are the rotor voltage, v ds and v qs are the stator voltage, λ dr and λ qr are the rotor flux, λ ds and λ qs are the stator flux, ω e is the angular frequency of the reference frame, ω r is the angular frequency of the rotor, L lr is the self-inductance, and L m is the mutual inductance. Electromagnetic torque can be expressed as: By aligning the rotor flux vector with the d-axis of the synchronous reference frame, λ qr equals zero. In this case, the electromagnetic torque is proportional to the q-axis stator current, i qs , whereas the d-axis stator current is responsible for producing the flux. Figure 10 shows the control block diagram of the RSC, where the indirect field-oriented control is implemented for the rotor flux orientation [37] and T ref is the reference electromagnetic torque.
where vdr and vqr are the rotor voltage, vds and vqs are the stator voltage, λdr and λqr are the rotor flux, λds and λqs are the stator flux, ωe is the angular frequency of the reference frame, ωr is the angular frequency of the rotor, Llr is the self-inductance, and Lm is the mutual inductance. Electromagnetic torque can be expressed as: qs dr ds qr T qs dr ds qr r By aligning the rotor flux vector with the d-axis of the synchronous reference frame, λqr equals zero. In this case, the electromagnetic torque is proportional to the q-axis stator current, iqs, whereas the d-axis stator current is responsible for producing the flux. Figure  10 shows the control block diagram of the RSC, where the indirect field-oriented control is implemented for the rotor flux orientation [37] and Tref is the reference electromagnetic torque. The determination of the reference electromagnetic torque Tref is displayed in Figure  11. It can be observed that the virtual inertia control, as well as the speed recovery control, are simultaneously incorporated into the control loop. ωr_ref represents the desired rotor speed that achieves the maximum power harvest, Pset is the active power setpoint as determined by ωr_ref, and Hv is the virtual inertia coefficient. It should be mentioned that the The determination of the reference electromagnetic torque T ref is displayed in Figure 11. It can be observed that the virtual inertia control, as well as the speed recovery control, are simultaneously incorporated into the control loop. ω r_ref represents the desired rotor speed that achieves the maximum power harvest, P set is the active power setpoint as determined by ω r_ref , and H v is the virtual inertia coefficient. It should be mentioned that the PI control gain, k p5 and k i5 , should be properly designed so that the virtual inertia control response time is much less than 10 s (the typical power system inertia response time). In addition, the PI control gain, k p6 and k i6 , need to be designed so that the speed recovery control response time is much more than 10 s. In this regard, the conflict between two control loops can be avoided. The objectives of inertia provision and speed recovery can be realized under a multi-timescale approach.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 10 of 15 PI control gain, kp5 and ki5, should be properly designed so that the virtual inertia control response time is much less than 10 s (the typical power system inertia response time). In addition, the PI control gain, kp6 and ki6, need to be designed so that the speed recovery control response time is much more than 10 s. In this regard, the conflict between two control loops can be avoided. The objectives of inertia provision and speed recovery can be realized under a multi-timescale approach.

Simulation Results
To verify the theoretical analysis that the rotor speed recovery bandwidth will affect the inertia emulation effect and the effectiveness of the proposed multi-timescale controller, simulations were conducted in the PLECS (piece-wise linear simulation of electrical circuits) environment. The DFIG wind power generation system (schematic diagram Figure 11. Control block diagrams for virtual inertia and speed recovery.

Simulation Results
To verify the theoretical analysis that the rotor speed recovery bandwidth will affect the inertia emulation effect and the effectiveness of the proposed multi-timescale controller, simulations were conducted in the PLECS (piece-wise linear simulation of electrical circuits) environment. The DFIG wind power generation system (schematic diagram shown in Figure 1) is connected to a practical power grid represented by an SM (primary frequency control framework illustrated in Figure 2). The power grid primary frequency regulation framework is illustrated in Figure 5. The main circuit and control parameters are listed in Table 1.  Figure 12 shows the simulation results when the DFIG generation system does not provide inertia support to the grid. At t = 5 s, a 5% step load change happens, and the power system frequency drops. As shown in Figure 12, the frequency nadir is 49.73 Hz, and the maximum RoCoF is −0.2 Hz/s. It can also be observed in Figure 12 that the wind turbine rotor speed, ω r , is decoupled from the grid frequency, as well as the RoCoF value (i.e., there is almost no change in ω r during the frequency event). This indicates that the mechanical energy stored in the rotor was not released to support the primary frequency response of the power system. Figure 13 shows the simulation results when the DFIG system provides inertia support. The speed recovery control bandwidth is designed to be 0.03 Hz according to the multitimescale control concept discussed in Section 3. Similarly, a 5% step load change happens when t = 5 s. It is observed that the wind turbine rotor speed is reduced during the inertia response period (5 s < t < 15 s). As a result, the rotor mechanical energy is released to support the grid frequency and limit the RoCoF. The frequency nadir is improved to 49.76 Hz, and the maximum RoCoF is reduced to −0.16 Hz/s. For a larger timescale (15 s < t < 40 s), the speed recovery control gradually takes effect, and the rotor speed comes back to the desired value of 300 rad/s. Note that the DC-link voltage is also under good control and remains close to the nominal value. Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 15 mechanical energy stored in the rotor was not released to support the primary frequency response of the power system.  Figure 13 shows the simulation results when the DFIG system provides inertia support. The speed recovery control bandwidth is designed to be 0.03 Hz according to the multi-timescale control concept discussed in Section 3. Similarly, a 5% step load change happens when t = 5 s. It is observed that the wind turbine rotor speed is reduced during the inertia response period (5 s < t < 15 s). As a result, the rotor mechanical energy is released to support the grid frequency and limit the RoCoF. The frequency nadir is improved to 49.76 Hz, and the maximum RoCoF is reduced to −0.16 Hz/s. For a larger timescale (15 s < t < 40 s), the speed recovery control gradually takes effect, and the rotor speed comes back to the desired value of 300 rad/s. Note that the DC-link voltage is also under good control and remains close to the nominal value. For the next scenario, the speed recovery control bandwidth was intentionally adjusted to 0.1 Hz, which is close to that of the inertia response. Figure 14 displays the corresponding simulation results. Although the wind turbine rotor speed is decreased at the start of the frequency event, the coupling between the inertia control and the speed recovery control makes ωr recover to the expected value within around 10 s. As a consequence, the inertia-provision effect is compromised. The frequency nadir is 49.73 Hz, and the maximum RoCoF is −0.17 Hz/s, which is not as satisfactory as the results of the previous case. For the next scenario, the speed recovery control bandwidth was intentionally adjusted to 0.1 Hz, which is close to that of the inertia response. Figure 14 displays the corresponding simulation results. Although the wind turbine rotor speed is decreased at the start of the frequency event, the coupling between the inertia control and the speed recovery control makes ω r recover to the expected value within around 10 s. As a consequence, the inertia-provision effect is compromised. The frequency nadir is 49.73 Hz, and the maximum RoCoF is −0.17 Hz/s, which is not as satisfactory as the results of the previous case. For the next scenario, the speed recovery control bandwidth was intentional justed to 0.1 Hz, which is close to that of the inertia response. Figure 14 displays th responding simulation results. Although the wind turbine rotor speed is decreased start of the frequency event, the coupling between the inertia control and the speed r ery control makes ωr recover to the expected value within around 10 s. As a consequ the inertia-provision effect is compromised. The frequency nadir is 49.73 Hz, and the imum RoCoF is −0.17 Hz/s, which is not as satisfactory as the results of the previous Finally, the speed recovery control bandwidth was designed as 1 Hz, which is faster than the inertia response. Figure 15 displays the simulation results. It is obs that the wind turbine rotor speed quickly recovers to 300 rad/s, even when it is sub to the load change disturbance. Therefore, the mechanical energy of the turbine roto not effectively contribute to power grid frequency regulation. The scenario is sim Figure 12, with a frequency nadir of 49.73 Hz and a maximum RoCoF of −0.2 Hz/s. Finally, the speed recovery control bandwidth was designed as 1 Hz, which is much faster than the inertia response. Figure 15 displays the simulation results. It is observed that the wind turbine rotor speed quickly recovers to 300 rad/s, even when it is subjected to the load change disturbance. Therefore, the mechanical energy of the turbine rotor does not effectively contribute to power grid frequency regulation. The scenario is similar to Figure 12, with a frequency nadir of 49.73 Hz and a maximum RoCoF of −0.2 Hz/s. For the next scenario, the speed recovery control bandwidth was intentionall justed to 0.1 Hz, which is close to that of the inertia response. Figure 14 displays the responding simulation results. Although the wind turbine rotor speed is decreased a start of the frequency event, the coupling between the inertia control and the speed r ery control makes ωr recover to the expected value within around 10 s. As a consequ the inertia-provision effect is compromised. The frequency nadir is 49.73 Hz, and the imum RoCoF is −0.17 Hz/s, which is not as satisfactory as the results of the previous Finally, the speed recovery control bandwidth was designed as 1 Hz, which is m faster than the inertia response. Figure 15 displays the simulation results. It is obse that the wind turbine rotor speed quickly recovers to 300 rad/s, even when it is subj to the load change disturbance. Therefore, the mechanical energy of the turbine rotor not effectively contribute to power grid frequency regulation. The scenario is simi Figure 12, with a frequency nadir of 49.73 Hz and a maximum RoCoF of −0.2 Hz/s. To summarize, Table 2 compares the simulation results. It is found that the maxim frequency error and maximum RoCoF for Case 2 are comparatively small. At the s time, the maximum rotor speed and DC-link voltage error for Case 2 are comparati To summarize, Table 2 compares the simulation results. It is found that the maximum frequency error and maximum RoCoF for Case 2 are comparatively small. At the same time, the maximum rotor speed and DC-link voltage error for Case 2 are comparatively large. The DFIG for Case 2 satisfies the multi-timescale control principle proposed in this paper, whereas other cases do not. As a result, the mechanical energy stored in the rotor can be effectively released to support frequency regulation. This explains why the rotor speed error for Case 2 is larger and the frequency error/RoCoF are smaller.

Conclusions
In this paper, we analyzed the impacts of wind turbine speed recovery control on the inertia-provision effect. It was revealed through analysis that the emulated inertia is different from the physical inertia if the speed recovery control bandwidth is close to or higher than the inertia control bandwidth. On this basis, a multi-timescale control scheme was developed in this paper to decouple the speed recovery control and the inertia control so that the control conflict is avoided. Extensive simulations were conducted to verify the feasibility of the proposed method, and a 23% reduction in the power grid RoCoF and 11% reduction in the maximum frequency deviation were achieved through the proposed multi-timescale DFIG control.