Improved Fractional-order Damping Method for Voltage-controlled DFIG System Under Weak Grid

Zhen Xie; Xiang Gao; Shuying Yang; Xing Zhang

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Improved Fractional-order Damping Method for Voltage-controlled DFIG System Under Weak Grid PDF

- ORCID：
Zhen Xie
✉
- ORCID：
Xiang Gao
✉
- ORCID：
Shuying Yang
✉
- ORCID：
Xing Zhang
✉

the School of Electrical Engineering and Automation, Hefei University of Technology, Hefei, China

Updated：2022-11-20

DOI：10.35833/MPCE.2020.000843

OUTLINE

Abstract

When a doubly-fed induction generator (DFIG) is connected to a weak grid, the coupling between the grid and the DFIG itself will increase, which will cause stability problems. It is difficult to maintain the tracking accuracy and robustness of the phase-locked loop (PLL) in the weak grid, and the risk of instability of the current-controlled DFIG (CC-DFIG) system will increase. In this paper, a new type of voltage-controlled DFIG (VC-DFIG) mode is adopted, which is a grid-forming structure that can independently support the voltage and frequency with a certain adaptability in the weak grid. A small-signal impedance model of the VC-DFIG system is also established. The impedance of DFIG inevitably generates coupling with the grid impedance in the weak grid, especially in parallel compensation grids, and results in resonance. On the basis of the VC-DFIG, impedance stability analysis is performed to study the influences of the control structure and short-circuit ratio. Then, a feedforward damping method is proposed to modify the impedance of the VC-DFIG system at resonance frequencies. The proposed fractional order damping is utilized, which can enhance the robustness and rapidity of resonance suppression under parameter fluctuations. Finally, the experimental results are presented to validate the effectiveness of the proposed control strategy.

Keywords

Weak grid; voltage-controlled doubly-fed induction generator (DFIG); resonance; feedforward damping; fractional-order damping

I. INTRODUCTION

IN the existing wind turbines connected to the electric power grid, the doubly-fed induction generator (DFIG) is extensively deployed due to its small converter rating, variable speed operation, and flexible regulation of active and reactive power [

1]-[6].

Several vector control strategies for rotor current such as direct torque control [

7] and direct power control [8] are proposed to enhance the performance of DFIG systems. A new model-free predictive current control is introduced to solve the uncertainty of the control parameters in [9]. The current control model with phase-locked loop (PLL) is still widely used in wind farms to transmit the desired active and reactive power to the grid through closed-loop rotor current control. The stability of the DFIG control system is closely related to the dynamic characteristics of PLL [10]-[13]. Nevertheless, it is difficult to guarantee the accuracy and robustness of PLL in weak grid due to the enhanced coupling between the current-controlled DFIG system (CC-DFIG) and the weak grid, which greatly increases the risk of system instability. In [11], an improved design method for PLL controller parameters has been proposed to reduce the negative effect of PLL on the current control in weak grid. The intermediate frequency resonance of 200-800 Hz caused by the PLL appears in the CC-DFIG system [13]. Therefore, the stability of CC-DFIG is negatively affected by the existence of PLL.

At present, with the increasing scale of wind power injected into the grid from remote wind farms, the DFIG systems are interconnected with weak grid and exhibit high impendence characteristics [

10], [11], [14], [15]. A new instability phenomenon that is different from that in a synchronous generator-dominated system is detected when a DFIG-based wind turbine is connected to an ultraweak alternating current (AC) grid [16].

Reference [

17] proposes a voltage-controlled virtual synchronous generator (VC-VSG) control structure. Compared with the sequence impedance of CC-VSG, the VC-VSG is more suitable for grid-connected inverter in an ultraweak grid from the perspective of system stability. In addition, there are other structures that enhance the stability in weak grid such as power synchronization control (PSC) [18] and grid-forming model (GFM) [19]-[22]. The voltage-controlled DFIG (VC-DFIG) system proposed in this paper is also such a structure, which uses a power synchronization method without PLL that emulates the characteristics of synchronous generators. The VC-DFIG can provide active power support under frequency disturbances, and automatically adjust the voltage through reactive power [23].

In the previous research related to VC-DFIG, the voltage and frequency regulators of the power synchronization itself [

23]-[25] and grid fault crossing [26] are concerned. Reference [23] analyzes the impact of P-f droop coefficient in the control of rotor-side converter (RSC) on synchronous stability. Reference [24] presents a virtual synchronous control for DFIG-based wind turbines to provide inertia contribution in particular when integrated into weak AC grid with low short-circuit ratio (SCR). In [25], a detailed analysis is carried out among different virtual inertia control schemes in terms of inertia response performance, weak grid operation stability, secondary frequency drop, and wind tower load analysis through the real-time digital simulator and pneumatic analysis software. For the low-voltage ride through methods, [26] presents an improved VSG strategy of DFIG to accelerate the decay of transient components and limit the rotor current during symmetrical voltage fault.

In the VC-DFIG system, the RSC is transformed into power synchronized method, but the grid-side converter (GSC) utilizes PLL for grid synchronization. The stability of VC-DFIG system in weak grid is reduced due to the influence of the GSC. Therefore, several potential resonance problems caused by the impedance coupling between the VC-DFIG system and the weak grid will decrease the stability of the system [

27].

In recent years, related researches focus on resolving the resonance problem of DFIG grid-connected systems [

27]-[30]. The harmonic linearization method for establishing the sequence impedance model of the CC-DFIG system connected to a grid is described in [27]. An impedance stability criterion is also proposed. When the CC-DFIG system is connected to a harmonic grid with parallel compensation, the frequency ranges of the resonance and harmonics may have intersections [28]. A variable frequency resistance and Chebyshev filter are therefore proposed. Reference [29] investigates the high-frequency resonance (HFR) mechanism of the CC-DFIG system and proposes an HFR damping method. Another way to solve the resonance problem is to improve the dynamic behavior of the PLL. In [30], the symmetrical PLL method is proposed to eliminate the frequency coupling terms caused by the asymmetric dynamics of conventional PLL method. The essence of these control strategies is to modify the CC-DFIG impedance to improve the stability of the system. Both RSC and GSC of the CC-DFIG system utilize PLL to synchronize the grid, so there are more problems in weak grid. For the VC-DFIG system, the problem is to solve the synchronization mode on the GSC. Therefore, the sequence impedance modeling of VC-DFIG is established by the harmonic linearization method. Moreover, the fastness and robustness of resonance suppression are seldom considered in the existing control strategies.

In the present paper, the impedance model of the VC-DFIG system is established, and its resonance is analyzed in weak grid. The main contributions of this paper are enumerated below.

1) The overall impedance of the VC-DFIG system is obtained through small-signal impedance modeling. Moreover, the impedance intersection between the VC-DFIG and the weak grid is studied by utilizing the impedance stability criterion.

2) Combined with the impedance stability criterion, the feedforward damping (FD) control strategy is adopted to solve the system resonance problems.

3) To enhance the stability of VC-DFIG in weak grid, fractional-order damping (FOD) which can reshape the DFIG system impedance is proposed. The design of the damping parameter of the proposed FOD is flexible in a wide range of frequency. Theoretical analysis and experimental results confirm that FOD has obvious advantages in terms of the robustness and rapidity of resonance suppression.

The remainder of this paper is organized as follows. Section II presents the modeling and analysis of the VC-DFIG, by using the impedance stability criterion. In Section III, the control strategy based on VC-DFIG is proposed, where two impedance shaping schemes based on FD and FOD, respectively, are utilized and compared to suppress system resonance. On the basis of the capability to suppress system resonance and reject parameter variation, the stability of the control strategy based on FOD is analyzed in Section IV. Section V discusses the experimental results and validates the theoretical analysis on the control strategy of FOD. The conclusions are provided in Section VI.

II. MODELING AND ANALYSIS OF VC-DFIG

The small-signal impedance of VC-DFIG system includes the RSC and GSC, which are connected in parallel to the point of common coupling (PCC). The topology and control diagram of the VC-DFIG are shown in Fig. 1 [

31]. The instructions of the voltage amplitude U_s and frequency

ω_{s}

are acquired from the power loop, in which the feedback signals of the active power P_e and reactive power Q_e are calculated using the stator voltage u_sabc and the rotor current i_sabc; i_{rd_ref} and i_{rq_ref} are the reference values of rotor current; u_sdq and u_rdq are the dq-axis values of stator and rotor voltage, respectively; i_rdq are the dq-axis values of rotor current; θ_s and θ_sl are the stator and slip angle, respectively; and

θ_{r}

is the rotor angle. The modulation strategy is space vector pulse width modulation (SVPWM).

Fig. 1 Topology and control diagram of VC-DFIG.

A. DFIG Mode and VC System

The equivalent circuit of DFIG is modeled in the motor convention, and the voltage and flux equations in a stationary coordinate system can be expressed as:

[\begin{matrix} u_{s} \\ u_{r} \end{matrix}] = \frac{d}{d t} [\begin{matrix} ψ_{s} \\ ψ_{r} \end{matrix}] - [\begin{matrix} - \frac{R_{s}}{σ L_{s}} - j ω_{s} & \frac{R_{s} L_{m}}{σ L_{s} L_{r}} \\ \frac{R_{r} L_{m}}{σ L_{s} L_{r}} & - \frac{R_{r}}{σ L_{r}} - j ω_{r} \end{matrix}] [\begin{matrix} ψ_{s} \\ ψ_{r} \end{matrix}]

(1)

[\begin{matrix} ψ_{s} \\ ψ_{r} \end{matrix}] = [\begin{matrix} L_{s} & L_{m} \\ L_{m} & L_{r} \end{matrix}] [\begin{matrix} i_{s} \\ i_{r} \end{matrix}]

(2)

where u_s and u_r are the stator and rotor voltage, respectively; $ψ_{s}$ and $ψ_{r}$ are the stator and rotor flux chains, respectively; R_s and L_s are the stator-side resistance and inductance, respectively; R_r and L_r are the rotor-side resistance and inductance, respectively; L_m is the mutual inductance; $σ$ is the magnetic flux leakage coefficient; and $ω_{s}$ and $ω_{r}$ are the angular velocities of the stator and rotor, respectively.

The power loop of VC-DFIG is defined as:

\{\begin{array}{l} ω_{s} = ω_{0} + K_{P} \frac{ω_{p}}{s + ω_{p}} (P_{r e f} - P_{e}) \\ U_{s} = U_{0} + K_{Q} (Q_{r e f} - Q_{e}) \end{array}

(3)

where P_ref and Q_ref are the reference values of active and reactive power, respectively; K_P and K_Q are the coefficients of active and reactive power, respectively; $ω_{0}$ and U₀ are the reference frequency and voltage, respectively; and $ω_{p}$ is the cut-off frequency of low-pass filter. The active power loop converts the active power instruction and the change in system frequency into an angle. The reactive power loop controls the reactive power and automatically regenerates the capacitive/inductive reactive power in accordance with the grid voltage deviation.

The control equation of the voltage and current loop are represented by (4) and (5), respectively.

[\begin{matrix} i_{r d} \\ i_{r q} \end{matrix}] = \frac{1}{L_{m} (s^{2} + ω_{s}^{2})} [\begin{matrix} s & - ω_{s} \\ ω_{s} & s \end{matrix}] [\begin{matrix} u_{s d} \\ u_{s q} \end{matrix}]

(4)

[\begin{matrix} u_{r d} \\ u_{r q} \end{matrix}] = [\begin{matrix} P I_{i} & 0 \\ 0 & P I_{i} \end{matrix}] [\begin{matrix} Δ i_{r d} \\ Δ i_{r q} \end{matrix}] + [\begin{matrix} ω_{s l} (\frac{L_{m}^{2}}{L_{s}} i_{m s} + σ L_{r} i_{r d}) \\ ω_{s l} σ L_{r} i_{r q} \end{matrix}]

(5)

where the subscripts d and q are the d-axis and q-axis components, respectively; i_ms is the exciting current of DFIG; and $ω_{s l}$ is the slip angular frequency. The voltage loop adopts a d-axis orientation. The rotor current loop uses the PI_i regulator to control the fundamental wave component of the current and adopts the feedforward compensation item to decouple the components of d-axis and q-axis.

B. Impedance of VC-DFIG

The first work of this paper is the establishment of the impedance model of VC-DFIG, which provides a research basis for analyzing the stability of the voltage control system in weak grid. The inertia and frequency response characteristics of the active power controller and the voltage regulation characteristics of the reactive power controller are considered in the model. The DC voltage control is assumed to be stable, and the angle disturbance is considered.

In the VSG algorithm, the active controller simulates the inertia and primary frequency regulation characteristics of the synchronous generator, and the reactive controller simulates the primary voltage regulation characteristic of the synchronous generator. The output frequency of VSG is integrated to obtain the expression of the output angle of the voltage-controlled DFIG system.

\{\begin{array}{l} θ = T (s) (ω_{n} D + \frac{P_{r e f}}{ω_{n}} - \frac{P_{e}}{ω_{n}}) \\ T (s) = \frac{1}{s (J s + D)} \end{array}

(6)

where J is the virtual moment of inertia; D is the active damping coefficient; and $ω_{n}$ is the rated frequency of the grid.

\{\begin{array}{l} P_{e} = 1.5 (v_{α} i_{α} + v_{β} i_{β}) \\ Q_{e} = 1.5 (v_{β} i_{α} - v_{α} i_{β}) \end{array}

(7)

where the subscripts $α$ and $β$ represent the stator voltage and current in the $α β$ coordinate system. The output active power P_e and output reactive power Q_e of the voltage-controlled DFIG system can be calculated according to the instantaneous power theory. The power is calculated by the $α β$ coordinate system. The accuracy of power calculation is not achieved, but the angle influence can be avoided.

Assuming that positive- and negative-sequence small signal disturbances are added to the voltage and current in the time domain, the expression of active power in the frequency domain can be obtained by using the frequency domain convolution theorem. The superscript * in the formula indicates the conjugate of complex numbers in the frequency domain.

P_{e} [f] = \{\begin{array}{l} 3 (V_{1} I_{1}^{*} + V_{1}^{*} I_{1} + V_{p} I_{p}^{*} + V_{p}^{*} I_{p} + V_{n} I_{n}^{*} + V_{n}^{*} I_{n}) & f = 0 \\ 3 (V_{1}^{*} I_{p} + V_{p} I_{1}^{*}) & f = \pm (f_{p} - f_{1}) \\ 3 (V_{1} I_{n} + V_{n} I_{1}) & f = \pm (f_{n} + f_{1}) \\ 3 (V_{p} I_{n} + V_{n} I_{p}) & f = \pm (f_{p} + f_{n}) \end{array}

(8)

where $f = 0$ represents the disturbance response at the fundamental frequency; V₁ and I₁ are the steady-state operating voltage and current, respectively; $f = \pm (f_{p} - f_{1})$ defines the angular response to the positive voltage disturbance; V_p and I_p are the positive disturbances of voltage and current, respectively; $f = \pm (f_{n} + f_{1})$ represents the angular response to the negative voltage disturbance; $f = \pm (f_{p} + f_{n})$ represents the angular response to the positive and negative voltage disturbances; and V_n and I_n are the negative disturbances of voltage and current, respectively.

By substituting the above formula into the expression of the output angle of the voltage-controlled DFIG system, and ignoring the small-signal quadratic term appearing in the derivation, the expression in the frequency domain can be obtained.

θ_{s} [f] = \{\begin{array}{l} T (s) [ω_{n} D + \frac{P_{r e f}}{ω_{n}} - (V_{1} I_{1}^{*} + V_{1}^{*} I_{1}) \frac{3}{ω_{n}}] & f = 0 \\ - T (s) (V_{1}^{*} I_{p} + V_{p} I_{1}^{*}) \frac{3}{ω_{n}} & f = \pm (f_{p} - f_{1}) \\ - T (s) (V_{1} I_{n} + V_{n} I_{1}) \frac{3}{ω_{n}} & f = \pm (f_{n} + f_{1}) \end{array}

(9)

In combination with (4)-(9), the small-signal response of the VC-DFIG system is obtained by the signal angle disturbance of the power loop and the inner voltage current loop. Finally, the machine-RSC impedance Z_RSC is obtained.

Z_{R S C p} = - \frac{V_{p}}{I_{p}} = \frac{(R_{s} + L_{s} s) T_{2 p} (s) + V_{1}^{*} T_{1 p} (s) - L_{m} s σ_{s p} (s) / K_{e}}{I_{1}^{*} T_{1 p} (s) \pm T_{u p} (s \mp j ω_{1}) G_{i} (s \mp j ω_{1}) - T_{2 p} (s)}

(10)

T_{1 p} (s) = \frac{3}{ω_{n}} T (s \pm j ω_{1}) (\mp \frac{1}{2} j D_{r o} + \frac{1}{2} Q_{r o})

(11)

T_{2 p} (s) = \frac{K_{e}}{L_{m} s} (\frac{R_{r}}{K_{e}^{2}} + L_{r} s \frac{σ_{s p} (s)}{K_{e}^{2}} + G_{u} (s \mp j ω_{1}))

(12)

where $Z_{R S C p}$ is the posotive-sequence impedance of RSC; $T_{u p} (s + j ω_{1})$ is the positive impedance of the voltage loop; $T_{1 p} (s)$ is the positive impedance associated with the power loop and the steady-state operating point; $T_{2 p} (s)$ is the positive impedance of the DFIG and the current loop; $σ_{s p} (s) = (s - j ω_{r}) / s$ is the conversion relationship between the stator and rotor frequencies corresponding to the positive-sequence harmonic component, and similarly, $σ_{s n} (s)$ is the negative-sequence conversion relationship; K_e represents the conversion relation between the stator and rotor of DFIG; $G_{u} (s - j ω_{1})$ and $G_{i} (s - j ω_{1})$ are the voltage and current loop regulators, respectively; and D_ro and Q_ro are the steady-state operating point correlation variables. The negative-sequence impedance of RSC $Z_{R S C n}$ is obtained as:

Z_{R S C n} = - \frac{V_{n}}{I_{n}} = \frac{(R_{s} + L_{s} s) T_{2 n} (s) - V_{1} T_{1 n} (s) - L_{m} s σ_{s n} (s) / K_{e}}{I_{1} T_{1 n} (s) \mp T_{u n} (s \pm j ω_{1}) G_{i} (s \pm j ω_{1}) - T_{2 n} (s)}

(13)

T_{1 n} (s) = \frac{3}{ω_{n}} T (s \pm j ω_{1}) (\pm \frac{1}{2} j D_{r o} + \frac{1}{2} Q_{r o})

(14)

T_{2 n} (s) = \frac{K_{e}}{L_{m} s} (\frac{R_{r}}{K_{e}^{2}} + L_{r} s \frac{σ_{s n} (s)}{K_{e}^{2}} + G_{u} (s \pm j ω_{1}))

(15)

The GSC adopts an LC filter. The derivation of the corresponding model has been proposed in [

32], and will not be repeated in this paper. Formula (16) shows the total impedance expression of the VC-DFIG system, where Z_VC-DFIGp, Z_VC-DFIGn,

Z_{G S C p}

, and

Z_{G S C n}

are the positive- and negative-sequence impedances of the VC-DFIG and GSC systems, respectively.

\{\begin{matrix} Z_{V C - D F I G p} = Z_{G S C p} ∥ Z_{R S C p} \\ Z_{V C - D F I G n} = Z_{G S C n} ∥ Z_{R S C n} \end{matrix}

(16)

C. Impedance Responses of VC-DFIG

Based on (10) and (13), the positive- and negative-sequence impedances are close in terms of amplitude, but there is a phase difference of 180°. The negative-sequence impedance models of both GSC and RSC can be expressed as the complex conjugate of the corresponding positive-sequence impedance models at a negative frequency [

27]. Therefore, it is reasonable to analyze the stability of VC-DFIG in terms of positive-sequence impedance [27].

The method of impedance scan is widely reported in [

33] and [34]. The simulation model of VC-DFIG is implemented in MATLAB and includes both RSC and GSC. The simulation parameters are consistent with the experimental parameters. To run an impedance scan, a balanced small-signal voltage perturbation is superimposed to the grid voltage.

The perturbation is in either positive or negative sequence and has an arbitrary initial phase angle. Sweeping the frequency over the range of interest shows how the impedance changes with frequency. In this paper, the frequency of the injected harmonic source increases from 0 to 1000 Hz with an interval of 1 Hz, and the amplitude is set to be 5% of the rated value, so as to avoid the influence on the steady-state operation of the system.

Figure 2 shows the impedance Bode diagram of VC-DFIG, including the VC-DFIG system impedance Z_{VC_DFIG} obtained via theoretical derivation, and the measured impedance Z_scan. The result of theoretical derivation is approximately consistent with the impedance scanning validation, especially in high-frequency bands (above 200 Hz).

Fig. 2 Impedance Bode diagram of VC-DFIG system.

The impedance diagram of the VC-DFIG system and weak grid is shown in Fig. 3. The DFIG side can be regarded as the equivalent voltage source and series impedance Z_{VC_DFIG}. The grid-side impedance Z_g includes inductive impedance and parallel capacitive impedance. The capacitive impedance characteristic includes the shunt capacitive impedance of the LC filter of the GSC. The impedance responses are illustrated in Fig. 4.

Fig. 3 Impedance diagram of VC-DFIG system and weak grid.

Fig. 4 Impedance responses of VC-DFIG connected to weak grid.

In Fig. 4, Z_g isthe grid impedance including the equivalent capacitive part. The phase is -90° at high frequencies, which indicates the capacitive characteristics. Regardless of the value of SCR, an intersection always exists between the curves of Z_{VC_DFIG} and Z_g, and the 180° phase difference appears at the corresponding intersection. According to the impedance stability criterion, the system will resonate in the illustrated situation and the resonance frequency will decrease as the SCR decreases. For instance, when the SCR varies from 5 to 1, the resonance frequency decreases from 649 Hz to 410 Hz.

III. CONTROL STRATEGY BASED ON VC-DFIG

A. Improved Damping Method Based on FD

Figure 5 shows the control strategy of RSC with FD. The d-axis and q-axis components of the DFIG stator output current capture the high-frequency resonance component through high-pass filter, which can increase the damping of the DFIG system.

Fig. 5 Control strategy of RSC with FD.

The control strategy based on FD includes the designs of the high-pass filter $ω_{c} = 2 π f_{c u t}$ and the proportional gain K_fd. As shown in Fig. 4, when $S C R = 1$ , the resonance frequency is 410 Hz. Considering certain control margin requirements, f_cut is set to be 300 Hz as the cut-off frequency. By combing the FD-based control strategy and (5), the new rotor voltage control equations can be obtained.

\{\begin{array}{l} u_{r d} = (i_{r d}^{*} - i_{r d} - i_{s d} G_{F D} (s)) G_{i} (s) \\ u_{r q} = (i_{r q}^{*} - i_{r q} - i_{s q} G_{F D} (s)) G_{i} (s) \end{array}

(17)

By substituting (17) to (10), the final VC-DFIG impedance Z_FD is established. The increased FD term $G_{F D} = s K_{F D} / (s + ω_{c})$ is included in the equivalent impedance of the voltage loop. Compared with (10), the addition of FD only affects T_up. This phenomenon can be mathematically expressed as:

T_{u p}^{F D} (s \mp j ω_{1}) = j G_{u} (s \mp j ω_{1}) - G_{F D} (s \mp j ω_{1})

(18)

As shown in Fig. 6, Z_FD is the resulting impedance after VC-DFIG adopts FD, and Z_g is set to be $S C R = 2$ . After reshaping the system impedance by FD, the intersection point between the system and grid impedance curves shifts. The black arrows indicate the phase difference at the impedance intersection. As a result, the impedance phase angle difference at the intersection of the impedance curves is lower than 180°. As shown in Fig. 5, the control strategy of FD can initially avoid system resonance, but cannot guarantee sufficient phase margin. Therefore, the high-pass filter parameters must be redesigned to ensure the stability of the DFIG system.

Fig. 6 Impedance of VC-DFIG system after reshaping by FD.

B. Improved Damping Method Based on FOD

To reach the above requirements, an improved damping method based on a fractional-order filter, which possesses flexibility in the order of the transfer functions, is designed in [

35]. The transfer function of FOD is depicted in Fig. 7.

G_{F O D} = K_{F O D} \frac{s^{α} R / L}{s^{α + β} + s^{α} R / L + 1 / (L C)}

(19)

Fig. 7 Bode diagram of fractional-order filter.

Figure 7 shows the amplitude and frequency response characteristics of the fractional-order filter at different orders ( $α$ , $β$ ). The solid red line in Fig. 7 represents the amplitude frequency response of the integer-order ( $α$ , $β = 1$ ) bandpass filter, where the low- and high-frequency stopband attenuations are 20 and -20 dB/decade, respectively. The attenuation ratios of the low- and high-frequency bands can be adjusted independently by adjusting $α$ and $β$ separately. In terms of phase characteristics, when $α + β > 2$ , the phase in the high-frequency band is 270°. When $α + β < 2$ , the phase is 90°, which suggests that the impedance is capacitive. Considering the flexibility of impedance reconstruction, fractional filters have more advantages than traditional filters. In this paper, $α$ and $β$ are set to be 1 and 0.8, respectively, to guarantee a high bandwidth and a minimal effect on the fundamental frequency, while enhancing the extraction effect on the resonance frequency band.

The FOD control branch is added to the VC-DFIG voltage control loop, as shown in Fig. 8. The part of FOD that affects the VC-DFIG impedance is only in the voltage loop. The positive impedance of the voltage loop with FOD can be expressed as:

T_{u p}^{F O D} (s \mp j ω_{1}) = j G_{u} (s \mp j ω_{1}) - G_{F O D} (s \mp j ω_{1})

(20)

Fig. 8 Control strategy of VC-DFIG with FOD control branch.

Figure 9 presents the impedance comparison of VC-DFIG after reshaping by FOD and FD. In this scenario, the grid impedance is still set as $S C R = 2$ . Z_FOD denotes the impedance after adopting FOD (green line), and Z_FD (red line) indicates the impedance of VC-DFIG with FD. As illustrated in the figure, the phase margin at 495 Hz is 8.6°. The original system resonance point indicates that the system is on the verge of instability. After reshaping by FOD, the resonance point moves to 470 Hz while the phase margin increases by 61°. Compared with the control strategy based on FD, the stability margin of the DFIG system with the FOD is increased.

Fig. 9 Impedance comparison of VC-DFIG after reshaping by FOD and FD.

C. Parameter Design Based on FOD

Figure 10 shows the Overall block diagram of voltage and current loop based on FOD, where the control loop and the DFIG model are depicted in black and red, respectively. Especially, the block of FOD control branch is drawn with a green line.

Fig. 10 Overall block diagram of voltage and current loop based on FOD.

To design the FOD, the transfer function from the stator voltage to the stator current is established using (18)-(20).

G_{u s_o p e n} = G_{n o F O D} + L_{m} G_{F D} G_{i} G_{p} Z_{g r i d} / L_{s}

(21)

\begin{array}{l} G_{n o F O D} = 1 + G_{i} G_{p} + L_{m} G_{u} G_{i} G_{p} G_{g r i d} / L_{s} + \dots + \\ L_{m} G_{1} G_{2} G_{p} G_{g r i d} / L_{s} + G_{1} G_{g r i d} / L_{s} + G_{1} Z_{g r i d} G_{i} G_{p} / L_{s} \end{array}

(22)

where G_{us_open} is the open-loop transfer function of VC-DFIG. The information of system stability can be easily obtained through an open-loop Bode diagram (Fig. 11(a)). $G_{n o F O D}$ is a part of the transfer function that is not affected by the increased FOD. In Fig. 11(a), with the increase of K_FOD, the gain of G_{us_open} at the resonance frequency gradually decreases, and the ability to resist weak grid disturbances and suppress system resonance is enhanced. This result is also reflected in Fig. 9. However, blindly increasing K_FOD does not guarantee better results. To effectively design the parameter range of K_FOD, the closed-loop transfer function of the VC-DFIG system is obtained as:

G_{u s_c l o s e} = \frac{U_{s d q}}{I_{s d q}} = \frac{- G_{g r i d} (1 + G_{u} G_{p})}{G_{n o F O D} + L_{m} G_{F O D} G_{i} G_{p} Z_{g r i d} / L_{s}}

(23)

Fig. 11 Bode diagram of transfer function. (a) G_{us_close}. (b) G_{us_open}.

Figure 11(b) shows the closed-loop Bode diagram that visually presents the input-output response characteristics of the VC-DFIG, which can be closely related to the time-domain response. For the transfer function of a closed-loop system, the phase margin of a generally designed system in engineering practice is 45°-70°. Combined with the Bode diagram of G_{us_close}, when K_FOD is 0.001, 0.01, 0.1, and 0.2, the system stability margin is 40°, 65°, 70°, and 82°, respectively. Considering the suppression effect, K_FOD is set to be 0.01-0.02.

D. Fractional-order Approximation

The final procedure is to realize the FOD function. Related literatures introduce several methods [

36], where higher-order approximation can provide higher accuracy of the magnitude. Therefore, this method is adopted in this paper. The approximation of the irrational fractional-order operator

s^{λ}

into a rational transfer function is obtained through the following steps.

1) Decompose each factor $s^{λ}$ ( $s^{α}$ , $s^{α + β}$ ) into s^m and s^v, where $m = |γ|$ ; and $ν = γ - m$ . Then, replace s^v in the filter model.

2) Process the numerator and denominator separately and determine a higher-order transfer function representation from the pseudo polynomial.

3) Apply a minimal implementation of the obtained high-priced integer-order transfer function to obtain the simplest transfer function model.

4) Consider the implementation of the physical platform and implement it discretely.

In Fig. 12, the solid green line represents the ideal Bode diagram of the fractional-order filter, whereas the blue line denotes the higher-order approximation process of FOD. The magnitude and phase responses of the higher-order approximated FOD can accurately imitate the original one. After the discretization, the red line is obtained, which indicates that the fractional-order approximation has little effect on the full-band performance. Moreover, the approximation can maintain good characteristics within the frequency band range of 1-2000 Hz.

Fig. 12 Fractional-order implementation.

Ⅳ. STABILITY ANALYSIS OF CONTROL STRATEGY BASED ON FOD

A. Motor Parameter Deviations

Considering the skin effect as well as the temperature rise, the parameters of the generator might change in practical applications. Therefore, the performance of the control strategy with generator parameter deviations must be investigated.

Figure 13 shows the impedance of VC-DFIG with FOD under the mutual inductance parameter deviations.

Fig. 13 Impedance of VC-DFIG with FOD under mutual inductance parameter deviations.

When the mutual inductance parameter varies by 20% below the fundamental frequency, the impedance curve based on FOD changes inconspicuously. No obvious difference is observed in the changes above the fundamental frequency. The maximum amplitude change is 5 dB, and the phase frequency curve has no obvious change. In other words, the ability of the proposed FOD to suppress resonance is not affected by the deviations of the motor mutual inductance parameters.

B. Uncertainty of Grid Parameters

In the case of weak grid, the long-distance transmission line impedance accounts for most of the entire grid impedance. According to the impedance stability criterion, the actual grid impedance is closely related to the safe and reliable operation of the system, the accuracy of the parameter estimation, and the reliability of the control strategy under parameter fluctuations.

In Fig. 14, setting the reference impedance of the grid as $S C R = 2$ , the three solid blue line Z_g represents the grid equivalent impedance, in which the equivalent inductance parameters are changed to 80%, 100%, and 120%, respectively. The red solid line Z_FD represents the impedance curve of control strategy by FD, and the green solid line Z_FOD indicates the impedance curve that adopts the FOD-based strategy. The results show that the resonance frequency is high when the inductance of the grid impedance increases. At the same time, the phase margins of Z_FD and Z_FOD slightly decrease, which is an unfavorable phenomenon for the system stability. Compared with the FD-based scheme, the FOD-based strategy can still provide sufficient phase margin.

Fig. 14 Resonance suppression under grid parameter uncertainty.

C. Dynamic Performance Comparison

The dynamic performance is an important index for evaluating resonance suppression strategies. The dynamic performance of the strategies based on FD and FOD are analyzed and compared in Fig. 15. The results show that the closed-loop poles and zeros of the VC-DFIG are all on the left half plane, thereby suggesting that the system is a stable high-order system.

Fig. 15 Pole-zero diagram of different values of K_FOD and K_FD. (a) K_FOD. (b) K_FD.

The pole-zero diagram of VC-DFIG with FOD is shown in Fig. 15(a), where K_FOD varies from 0.003 to 0.03. The closed-loop zero point is not affected by the variation in K_FOD, whereas the closed-loop non-dominant pole moves to the imaginary axis, and eventually reduces damping and accelerates the response speed of the VC-DFIG system. The result is in accordance with the dynamic performance analysis of higher-order systems. Although a compromise K_FOD value is selected to balance the performances of resonance suppression, the system maintains a satisfactory speed at the maximum K_FOD of 0.02. Compared with the pole-zero diagram of the DFIG system based on the impedance of FD in Fig. 15(b), when K_FD changes from 1 to 10, the closed-loop non-dominant pole of the system moves away from the imaginary axis, and thus, reduces the response speed of the VC-DFIG system. After comprehensive comparisons, the findings reveal that the improved damping strategy based on FOD has greater advantages in terms of resisting parameter variations and rapidly suppressing resonance.

Ⅴ. EXPERIMENTAL VERIFICATION

An 11 kW DFIG-based experimental platform is constructed to validate the control strategy of the VC-DFIG and the stability performance in weak grid. The schematic diagram of the experimental platform is presented in Fig. 16. The experimental system parameters are listed in Table I. An INOVANCE MD320 inverter is used to drive a three-phase squirrel cage asynchronous motor (SCAM). The control strategy of RSC and GSC is implemented on two independent DSP TMS320F28335, and the experimental waveforms are collected using YOKOGAWA DL750 scopes.

Fig. 16 Schematic diagram of experimental platform.

TABLE I PARAMETER OF DFIG

Parameter	Value	Parameter	Value
Rated power (kW)	11	Stator rated voltage (V)	195
Rotor resistance (Ω)	0.2858	Stator resistance (Ω)	0.2983
Stator inductance (H)	0.068923	Rotor inductance (H)	0.069381
Mutual inductance (H)	0.0676	Polar	2
Voltage loop parameter	$k_{p} = 0.5$ , $k_{i} = 0.025$	Current loop parameter	$k_{p} = 2$ , $k_{i} = 0.1$

Figure 17 shows the resonant state of the VC-DFIG system and FFT analysis. The change in the running state may induce the damping of the system at the analysis of inherent resonance point. When $S C R = 3.5$ , according to the experiment data, the resonance point on the stator side is 540 Hz, and the resonance peak is 118.3% of the fundamental wave.

Fig. 17 Resonant state of VC-DFIG system and FFT analysis. (a) Resonant state. (b) FFT analysis.

Figures 18(a) and 17(b) illustrate the comparison of the suppression effect after resonance occurs in the VC-DFIG system between the FD and FOD. Both suppression strategies exhibit good suppression effect. When the control strategies based on FD and FOD are activated, the former resonance suppression time is 150 ms, and the latter, which is nearly 47% faster, is roughly 80 ms. The experimental results show that the suppression strategy based on FOD has a faster suppression effect than that based on FD. These results are consistent with the theoretical derivation.

Fig. 18 Suppression effect based on FD and FOD. (a) Stator current and voltage. (b) Rotor current and active power.

Figure 19 shows the suppression effect with different parameter values of resonance suppression based on FOD. When K_FOD is extremely small, the suppression time is prolonged after the occurrence of the system resonance. When K_FOD is 0.003, the resonance suppression time is approximately 250 ms, and the suppression process increases by 250%. When K_FOD is 0.3, the phenomenon of overcompensation occurs, and a small resonance peak appears near the system resonance point. In conclusion, the improved damping method based on FOD delivers a fast and highly stable resonance suppression effect when the parameter is selected appropriately.

Fig. 19 Suppression effect with different parameter values based on FOD. (a) $K_{F D} = 0.003$ . (b) $K_{F D} = 0.3$ . (c) FFT analysis with $K_{F D} = 0.3$ .

VI. Conclusion

In this paper, the impedance model of a VC-DFIG system is established through the harmonic linearization method. Subsequently, the stability analysis of the VC-DFIG system is conducted on the basis of the derived impedance model. The results suggest that the increased coupling between the VC-DFIG system and the weak grid leads to instability. To address this issue, an improved damping control strategy based on FD and FOD, which can suppress the resonance by reducing the phase difference between the DFIG system and the weak grid at the potential resonance frequency, are proposed. The FOD strategy demonstrates the satisfactory performance in enhancing the dynamic and robustness of the VC-DFIG system for resonance suppression. The results of the VC-DFIG experiment platform verify that the proposed strategy has a robust dynamics and steady-state performance in enhancing stability.

REFERENCES

C. Wu, D. Zhou, and F. Blaabjerg, “Direct power magnitude control of DFIG-DC system without orientation control,” IEEE Transactions on Industrial Electronics, vol. 68, no. 2, pp. 1365-1373, Feb. 2021. [Baidu Scholar]

X. Xi, H. Geng, G. Yang et al., “Two-level damping control for DFIG-based wind farm providing synthetic inertial service,” IEEE Transactions on Industry Applications, vol. 54, no. 2, pp. 1712-1723, Mar.-Apr. 2018. [Baidu Scholar]

D. Zhu, X. Zou, L. Deng et al., “Inductance-emulating control for DFIG-based wind turbine to ride-through grid faults,” IEEE Transactions on Power Electronics, vol. 32, no. 11, pp. 8514-8525, Nov. 2017. [Baidu Scholar]

C. Zhou, Z. Wang, P. Ju et al., “High-voltage ride through strategy for DFIG considering converter blocking of HVDC system,” Journal of Modern Power Systems and Clean Energy, vol. 8, no. 3, pp. 491-498, May 2020. [Baidu Scholar]

Y. Chang, I. Kocar, J. Hu et al., “Coordinated control of DFIG converters to comply with reactive current requirements in emerging grid codes,” Journal of Modern Power Systems and Clean Energy, vol. 10, no. 2, pp. 502-514, Mar. 2022. [Baidu Scholar]

P. Chen, C. Qi, and X. Chen, “Virtual inertia estimation method of DFIG-based wind farm with additional frequency control,” Journal of Modern Power Systems and Clean Energy, vol. 9, no. 5, pp. 1076-1087, Sept. 2021. [Baidu Scholar]

M. R. Agha Kashkooli, S. M. Madani, and T. A. Lipo, “Improved direct torque control for a DFIG under symmetrical voltage dip with transient flux damping,” IEEE Transactions on Industrial Electronics, vol. 67, no. 1, pp. 28-37, Jan. 2020. [Baidu Scholar]

D. Sun and X. Wang, “Low-complexity model predictive direct power control for DFIG under both balanced and unbalanced grid conditions,” IEEE Transactions on Industrial Electronics, vol. 63, no. 8, pp. 5186-5196, Aug. 2016. [Baidu Scholar]

Y. Zhang, T. Jiang, and J. Jiao, “Model-free predictive current control of DFIG based on an extended state observer under unbalanced and distorted grid,” IEEE Transactions on Power Electronics, vol. 35, no. 8, pp. 8130-8139, Aug. 2020. [Baidu Scholar]

X. Zhang, D. Xia, Z. Fu et al., “An improved feedforward control method considering PLL dynamics to improve weak grid stability of grid-connected inverters,” IEEE Transactions on Industry Applications, vol. 54, no. 5, pp. 5143-5151, Sept.-Oct. 2018. [Baidu Scholar]

D. Zhu, S. Zhou, X. Zou et al., “Improved design of PLL controller for LCL-type grid-connected converter in weak grid,” IEEE Transactions on Power Electronics, vol. 35, no. 5, pp. 4715-4727, May 2020. [Baidu Scholar]

Y. Shen, J. Ma, L. Wang et al., “Study on DFIG dissipation energy model and low-frequency oscillation mechanism considering the effect of PLL,” IEEE Transactions on Power Electronics, vol. 35, no. 4, pp. 3348-3364, Apr. 2020. [Baidu Scholar]

Y. Song and F. Blaabjerg, “Analysis of middle frequency resonance in DFIG system considering phase-locked loop,” IEEE Transactions on Power Electronics, vol. 33, no. 1, pp. 343-356, Jan. 2018. [Baidu Scholar]

P. Sun, J. Yao, R. Liu et al., “Virtual capacitance control for improving dynamic stability of the DFIG-based wind turbines during a symmetrical fault in a weak AC grid,” IEEE Transactions on Industrial Electronics, vol. 68, no. 1, pp. 333-346, Jan. 2021. [Baidu Scholar]

J. Fang, H. Deng, and S. M. Goetz, “Grid impedance estimation through grid-forming power converters,” IEEE Transactions on Power Electronics, vol. 36, no. 2, pp. 2094-2104, Feb. 2021. [Baidu Scholar]

W. Tang, J. Hu, Y. Chang et al., “Modeling of DFIG-based wind turbine for power system transient response analysis in rotor speed control timescale,” IEEE Transactions on Power Systems, vol. 33, no. 6, pp. 6795-6805, Nov. 2018. [Baidu Scholar]

W. Wu, Y. Chen, L. Zhou et al., “Sequence impedance modeling and stability comparative analysis of voltage-controlled VSGs and current-controlled VSGs,” IEEE Transactions on Industrial Electronics, vol. 66, no. 8, pp. 6460-6472, Aug. 2019. [Baidu Scholar]

H. Wu and X. Wang, “Design-oriented transient stability analysis of grid-connected converters with power synchronization control,” IEEE Transactions on Industrial Electronics, vol. 66, no. 8, pp. 6473-6482, Aug. 2019. [Baidu Scholar]

B. K. Poolla, D. Groß, and F. Dörfler, “Placement and implementation of grid-forming and grid-following virtual inertia and fast frequency response,” IEEE Transactions on Power Systems, vol. 34, no. 4, pp. 3035-3046, Jul. 2019. [Baidu Scholar]

W. Du, Z. Chen, K. P. Schneider et al., “A comparative study of two widely used grid-forming droop controls on microgrid small-signal stability,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 963-975, Jun. 2020. [Baidu Scholar]

R. Rosso, S. Engelken, and M. Liserre, “Robust stability investigation of the interactions among grid-forming and grid-following converters,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 991-1003, Jun. 2020. [Baidu Scholar]

D. Pan, X. Wang, F. Liu et al., “Transient stability of voltage-source converters with grid-forming control: a design-oriented study,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 1019-1033, Jun. 2020. [Baidu Scholar]

L. Huang, H. Xin, L. Zhang et al., “Synchronization and frequency regulation of DFIG-based wind turbine generators with synchronized control,” IEEE Transactions on Energy Conversion, vol. 32, no. 3, pp. 1251-1262, Sept. 2017. [Baidu Scholar]

S. Wang, J. Hu, and X. Yuan, “Virtual synchronous control for grid-connected DFIG-based wind turbines,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 3, no. 4, pp. 932-944, 2015. [Baidu Scholar]

H. Shao, X. Cai, D. Zhou et al., “Analysis and test evaluation of VSG and virtual inertia control method for wind turbine,” High Voltage Engineering, vol. 46, no. 5, pp. 1528-1537, May 2020. [Baidu Scholar]

H. Nian and Y. Jiao, “Improved virtual synchronous generator control of DFIG to ride-through symmetrical voltage fault,” IEEE Transactions on Energy Conversion, vol. 35, no. 2, pp. 672-683, Jun. 2020. [Baidu Scholar]

I. Vieto and J. Sun, “Refined small-signal sequence impedance models of type-III wind turbines,” in Proceedings of 2018 IEEE Energy Conversion Congress and Exposition (ECCE), Portland, USA, Sept. 2018, pp. 2242-2249. [Baidu Scholar]

H. Nian and B. Pang, “Stability and power quality enhancement strategy for DFIG system connected to harmonic grid with parallel compensation,” IEEE Transactions on Energy Conversion, vol. 34, no. 2, pp. 1010-1022, Jun. 2019. [Baidu Scholar]

B. Pang, C. Wu, H. Nian et al., “Damping method of high-frequency resonance for stator current controlled DFIG system under parallel compensation grid,” IEEE Transactions on Power Electronics, vol. 35, no. 10, pp. 10260-10270, Oct. 2020. [Baidu Scholar]

D. Yan, X. Wang, F. Liu et al., “Symmetrical PLL for SISO impedance modeling and enhanced stability in weak grids,” IEEE Transactions on Power Electronics, vol. 35, no. 2, pp. 1473-1483, Feb. 2020. [Baidu Scholar]

Z. Xie, H. Meng, X. Zhang et al., “Virtual synchronous control strategy of DFIG-based wind turbines based on stator virtual impedance,” Automation of Electric Power Systems, vol. 42, no. 9, pp. 157-163, May 2018． [Baidu Scholar]

M. Cespedes and J. Sun, “Impedance modeling and analysis of grid-connected voltage-source converters,” IEEE Transactions on Power Electronics, vol. 29, no. 3, pp. 1254-1261, Mar. 2014. [Baidu Scholar]

I. Vieto and J. Sun, “Sequence impedance modeling and analysis of type-III wind turbines,” IEEE Transactions on Energy Conversion, vol. 33, no. 2, pp. 537-545, Jun. 2018. [Baidu Scholar]

H. Li, K. Wang, Y. Hu et al., “Impedance modeling and stability analysis of virtual synchronous control based on doubly-fed wind generation systems,” Proceedings of the CSEE, vol. 39, no. 12, pp. 3434-3443, Jun. 2019. [Baidu Scholar]

A. Adhikary, S. Sen and K. Biswas, “Practical realization of tunable fractional order parallel resonator and fractional order filters,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 63, no. 8, pp. 1142-1151, Aug. 2016. [Baidu Scholar]

H. K. Kwan and A. Jiang, “FIR, Allpass, and IIR variable fractional delay digital filter design,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 56, no. 9, pp. 2064-2074, Sept. 2009. [Baidu Scholar]

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