A Novel PLL Structure for Dynamic Stability Improvement of DFIG-based Wind Energy Generation Systems During Asymmetric LVRT

Lei Guan; Jun Yao

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A Novel PLL Structure for Dynamic Stability Improvement of DFIG-based Wind Energy Generation Systems During Asymmetric LVRT PDF

- ORCID：
Lei Guan
✉
- ORCID：
Jun Yao
✉

the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, School of Electrical Engineering, Chongqing University, Chongqing 400044, China

Updated：2023-07-25

DOI：10.35833/MPCE.2022.000164

OUTLINE

Abstract

The dynamic coupling effect, which is introduced by the dual-sequence phase-locked loops (PLLs) used in doubly- fed induction generator (DFIG) based wind energy generation systems (WEGSs) during asymmetric low voltage ride-through (LVRT) in weak grid, needs attention. In order to study this new dynamic coupling effect, an equivalent two-degree-of-freedom (2-DOF) spring damper particle model is used in this paper to develop a small-signal model for the dual-sequence PLLs. The dynamic interaction between the positive-sequence (PS) and negative-sequence (NS) PLLs is unveiled. Moreover, the impact of the dynamic coupling between the dual-sequence PLLs on the dynamic stability during the steady-state stage of an asymmetric fault is analyzed. The analysis results show that the dynamic coupling between the dual-sequence PLLs will cause drift in the frequency and damping for the PS and NS PLL modes. This will change the instability modal of the system and introduce the risk of dynamic instability. Hence, the effectiveness of existing control strategies for enhancing the dynamic stability will be decreased. Finally, a novel PLL structure is designed to improve the dynamic stability of the system during the steady-state stage of an asymmetric fault. The effectiveness of the proposed strategy is verified by simulations and experiments.

Keywords

Doubly-fed induction generator (DFIG); phase-locked loop (PLL); dynamic stability; dynamic coupling; weak grid; asymmetric low-voltage ride through (LVRT).

I. Introduction

WITH the development of the renewable energy power generation industry, doubly-fed induction generator (DFIG) based wind energy generation systems (WEGSs) have been widely applied in the market in recent years [

1]. However, owing to the uneven distribution of wind resources, wind farms are usually installed far away from the power load, as for most offshore wind farm projects. As a consequence, the long-distance transmission lines connected to the grid would have a high impedance (weak grid characteristic) [2]. The relatively low short-circuit ratio (SCR) of the network, which is used to connect renewable energy plants such as photovoltaic and wind power plants, has led to frequent grid faults, especially asymmetric faults [3], [4].

Some existing studies on DFIG-based WEGSs under asymmetric grid faults focus on control schemes for the positive-sequence (PS) and negative-sequence (NS) components. References [

5]-[7] propose advanced control strategies for DFIG-based WEGSs under asymmetric voltage drop. Reference [8] designs an asymmetric low voltage ride-through (LVRT) strategy for DFIG-based WEGSs based on a model predictive control method for the current. However, these studies only investigate control strategies for DFIG-based WEGSs under asymmetric voltage sag and do not consider the inter-sequence coupling caused by the asymmetric faulty network under weak grid conditions and the small-signal stability of the system during asymmetric faults.

The faults in a power system comprise two transient stages (fault-initiation and fault-clearing) and one steady-state stage [

9]. Hence, the stability issues of grid-connected systems under weak grid faults can be divided into two parts: the large-signal stability (transient stability) during the transient stage and the small-signal stability (dynamic stability) during the steady-state stage [10], [11]. Some studies on the dynamic stability of DFIG-based WEGSs during fault ride-through (FRT) in weak grid have been carried out. References [11] and [12] investigate the dynamic stability of WEGSs connected to weak grid during symmetric LVRT. It is pointed out in [11] and [12] that the possibility of a small-signal instability in the DFIG-based WEGS will increase owing to the changes in the operating state and network structure during the steady-state stage of a fault in a weak grid. Reference [13] establishes a dynamic model for the power grid under asymmetric fault and reveals the coupling effect between the PS and NS networks, which has an effect on the dynamic stability. Because of the existence of the NS control system and NS network under an asymmetric fault, the operating state and network structure under an asymmetric fault are completely different from those under a symmetric fault. Reference [14] analyzes the dynamic stability of DFIG-based WEGSs during asymmetric LVRT, considering the inter-sequence dynamic coupling effect under asymmetric faults. Similarly, [15] studies the dynamic stability of DFIG-based WEGSs during asymmetric LVRT and proposes an algorithm that optimizes the current control reference to improve the dynamic stability.

However, the systems investigated in the above studies apply the single-sequence phase-locked loop (PLL) structure during asymmetric LVRT, which is different from the dual-sequence PLL structure adopted in [

16] and [17]. The PLL structure applied in the renewable energy generation systems under asymmetric grid conditions can be divided into two types: ① the single-sequence improved PLL structure, which uses the reversal of the output of PS PLL to control the NS components, e.g., decoupled double synchronous reference frame (DDSRF) PLL [7], notch PLL [15], [18], etc.; and ② the dual-sequence PLL structure, which obtains the angles and frequencies of the PS and NS reference voltages, respectively [16], [17]. The application of the dual-sequence PLL structure results in a new dynamic coupling effect, i.e., the dynamic coupling between the PS and NS PLLs, in the system during asymmetric LVRT. Owing to this new characteristic, the small-signal stability of the system during the asymmetric LVRT may change. In addition, the effectiveness of the control strategies for improving the dynamic stability proposed in [15] and [18] may be influenced during the asymmetric LVRT. At present, these issues have not been studied.

This study is carried out for filling the study gap about the influence of applying dual-sequence PLLs structure on the dynamic stability of power system. The main contributions of this study are as follows.

1) Based on the developed equivalent two-degree-of-freedom (2-DOF) spring damper particle model for dual-sequence PLLs, the new dynamic coupling between the PS and NS PLLs introduced by the dual-sequence PLL structure is revealed.

2) The impact of the dynamic coupling effect on the stability of the system and the proposed control strategy for enhancing the dynamic stability during asymmetric LVRT are analyzed. Moreover, the risk of dynamic instability caused by the dynamic coupling is demonstrated.

3) A novel PLL structure with an adjustable damping shock absorber added to the NS PLL is designed to improve the dynamic stability of the system when dual-sequence PLLs are applied during asymmetric LVRT. Simulations and experiments verify the effectiveness of this structure.

The rest of this paper is organized as follows. The model of dual-sequence PLLs is developed, and the dynamic coupling between the PS and NS PLLs is discussed in Section II. In Section III, the influence of the new dynamic coupling introduced by the dual-sequence PLLs on the dynamic stability is analyzed. In Section IV, a novel PLL structure that can improve the dynamic stability of systems adopting dual-sequence PLLs is designed. The results of simulations and experiments are presented in Sections V and VI, respectively. Finally, Section VII concludes this paper.

II. Model of Dual-sequence PLLs and Dynamic Coupling Between PS and NS PLLs

The typical control structure of DFIG-based WEGS during asymmetric LVRT is shown in Fig. 1.

Fig. 1 Typical control structure of DFIG-based WEGS during asymmetric LVRT.

During an asymmetric voltage dip, the system needs to inject the corresponding PS and NS reactive currents into the faulty grid in accordance with the severity of the fault [

7], [14], [15], [20]. When a grid fault is detected, the rotor-side converter (RSC) control system will cut off the outer loop of the power control and switch to use only the inner loop of the current control to regulate the rotor current. When an asymmetric drop at the point of common coupling (PCC) voltage is detected, the system will enable NS current control. The red box in Fig. 1 indicates the asymmetric FRT control strategy that needs to switch.

In this paper, the command values of the PS and NS reactive currents that should be injected by the DFIG WEGS during asymmetric LVRT are expressed as [

16]:

\{\begin{array}{l} I_{s q, r e f}^{p} = K_{+} (0.9 - U_{s d}^{p}) \\ I_{s q, r e f}^{n} = K_{-} (0.05 - U_{s d}^{n}) \end{array}

(1)

where $K_{+} = K_{-} = 1.5$ .

A. Small-signal Model of Dual-sequence PLLs

The dynamic stability during the steady-state stage of a fault in which the transient fault component has completely attenuated is the main focus of this paper. The crowbar on the rotor side or the chopper on the DC side has been deactivated during the steady-state stage of a fault. Therefore, a small-signal model is obtained by linearizing the mathematical model at the stable operating point during steady-state stage of a grid fault.

The control structure of dual synchronous reference frames (SRFs) is normally used to control the PS and NS components [

21], [22]. In order to obtain accurate PS and NS synchronization angles, [16] and [17] employ the dual-sequence PLL structure, as shown in Fig. 2.

Fig. 2 Diagram of dual-sequence PLL structure.

The PCC voltage is first decomposed into PS and NS reference voltages by the PS and NS separation modules, respectively. The PS and NS separation algorithm can be expressed as:

\{\begin{array}{l} [\begin{matrix} U_{s α}^{p} \\ U_{s β}^{p} \end{matrix}] = \frac{1}{2} [\begin{matrix} 1 & - q \\ q & 1 \end{matrix}] [\begin{matrix} U_{s α} \\ U_{s β} \end{matrix}] \\ [\begin{matrix} U_{s α}^{n} \\ U_{s β}^{n} \end{matrix}] = \frac{1}{2} [\begin{matrix} 1 & q \\ - q & 1 \end{matrix}] [\begin{matrix} U_{s α} \\ U_{s β} \end{matrix}] \end{array}

(2)

where $q = e^{- j π / 2} .$

Subsequently, the q-axis components of the PS and NS voltages, i.e., $U_{s q, p l l +}^{p}$ and $U_{s q, p l l -}^{n}$ , will enter the proportional-integral (PI) controller in independent PS and NS PLLs to obtain the angular velocities for PS and NS SRFs, and then obtain the PS and NS angles for the PS and NS SRFs through the integral operation, respectively.

The small-signal model for the dual-sequence PLLs is expressed as:

\{\begin{array}{l} Δ {\dot{θ}}_{p l l}^{p, n} = Δ ω_{p l l}^{p, n} \\ Δ {\dot{x}}_{1,2} = Δ U_{s q \pm}^{p, n} \\ Δ ω_{p l l}^{p, n} = K_{p 1,2} Δ U_{s q \pm}^{p, n} + K_{i 1,2} θ_{p l l}^{p, n} \end{array}

(3)

Due to the PLL dynamics, the system will have four rotating synchronous frames during asymmetric grid faults: dq^p, dqⁿ, $d q_{p l l}^{p}$ , and $d q_{p l l}^{n}$ . The relationship between these four rotating synchronous frames and the $α β$ frame is shown in Fig. 3, where the control vector F in $α β$ , dq^p, dqⁿ, $d q_{p l l}^{p}$ , and $d q_{p l l}^{n}$ coordinate frames are marked with different colors.

Fig. 3 Control vector F in αβ, dq^p, dqⁿ, $d q_{p l l}^{p}$ , and $d q_{p l l}^{n}$ coordinate frames.

Considering the PLL dynamics, we have:

\{\begin{array}{l} θ^{p} = θ_{p l l}^{p} + Δ θ_{p l l}^{p} \\ θ^{n} = θ_{p l l}^{n} + Δ θ_{p l l}^{n} \end{array}

(4)

The relationships of the control vector F in these four SRFs are expressed as:

\{\begin{matrix} Δ F_{g d q}^{p} = Δ F_{g d q, p l l}^{p} + j F_{g d q 0}^{p} Δ θ_{p l l}^{p} \\ Δ F_{g d q}^{n} = Δ F_{g d q, p l l}^{n} + j F_{g d q 0}^{n} Δ θ_{p l l}^{n} \end{matrix}

(5)

Figure 4 shows two typical asymmetric grid faults that will cause an asymmetric voltage drop, i.e., phase-to-ground and phase-to-phase short-circuit faults.

Fig. 4 Structure of power grid under two typical asymmetric grid faults.

Because the DFIG-based WEGS generally adopts the three-phase three-wire system to connect an AC grid, there is no zero-sequence (ZS) component path in the system [

13]. The dynamic stability of DFIG-based WEGSs that apply the dual-sequence PLL structure during the asymmetric LVRT is studied. Admittedly, the single-phase-to-ground and phase-to-phase short-circuit faults can also cause asymmetric voltage dips. However, the external characteristic of the faulty grid is a single- or two-phase voltage dip [13]. Therefore, regardless of the fault type, the control structure of the DFIG system remains unchanged, as shown in Fig. 1. In this paper, a typical two-phase-to-ground fault will be used as an example to model the asymmetric faulty grid. The impedance matrix of the fault branch is expressed as:

Z_{f} = (R_{f} + s L_{f}) [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}]

(6)

During asymmetric grid faults, there exist dynamic coupling phenomena in the PS and NS networks [

13]. According to [13], when the severity of the asymmetric fault is low, the inter-sequence coupling effect is weak, and the coupling impedance can be neglected. However, the inter-sequence coupling effect cannot be neglected when the severity of the asymmetric fault is high. The method for modeling the asymmetric faulty grid, whose small-signal model is able to be expressed in the PS and NS SRFs, can be expressed as [13]:

[\begin{matrix} Δ U_{s d q}^{p} \\ Δ U_{s d q}^{n} \end{matrix}] = \underset{Z_{g}}{\underset{⏟}{\underset{}{[\begin{matrix} Z_{g}^{p p} & Z_{g}^{p n} \\ Z_{g}^{n p} & Z_{g}^{n n} \end{matrix}]}}} [\begin{matrix} Δ I_{g d q}^{p} \\ Δ I_{g d q}^{n} \end{matrix}]

(7)

where $Z_{g}^{p p} = R_{g}^{p p} + j X_{g}^{p p} = R_{g}^{p p} + j ω_{p l l 0}^{p} L_{g}^{p p}$ and $Z_{g}^{n n} = R_{g}^{n n} + j X_{g}^{n n} = R_{g}^{n n} + j ω_{p l l 0}^{n} L_{g}^{n n}$ are the grid sequence impedances; and $Z_{g}^{p n} = R_{g}^{p n} + j X_{g}^{p n} = R_{g}^{p n} + j ω_{p l l 0}^{n} L_{g}^{p n}$ and $Z_{g}^{n p} = R_{g}^{n p} + j X_{g}^{n p} = R_{g}^{n p} + j ω_{p l l 0}^{p} L_{g}^{n p}$ are the inter-sequence coupling impedances of the faulty network. The output current dynamics of the DFIG-based WEGS can be expressed as:

\{\begin{matrix} Δ I_{g d q}^{p} = Δ I_{g d q, p l l}^{p} + j I_{g d q 0}^{p} Δ θ_{p l l}^{p} \\ Δ I_{g d q}^{n} = Δ I_{g d q, p l l}^{n} + j I_{g d q 0}^{n} Δ θ_{p l l}^{n} \end{matrix}

(8)

Combining (3)-(8), the small-signal model of dual-sequence PLLs is shown in Fig. 5.

Fig. 5 Small-signal model of dual-sequence PLLs.

Owing to the inter-sequence coupling effect, there is dynamic interaction between the PS and NS PLLs. These interaction is not investigated in previous studies. From Fig. 5, we could have:

\{\begin{matrix} \begin{array}{l} Δ θ_{p l l}^{p} + K_{p 1} (X_{g}^{p p} I_{g q 0}^{p} + U_{g d 0}^{p}) Δ θ_{p l l}^{p} + K_{p 1} X_{g}^{p n} I_{g q 0}^{n} Δ θ_{p l l}^{n} + \\ K_{i 1} (X_{g}^{p p} I_{g q 0}^{p} + U_{g d 0}^{p}) Δ θ_{p l l}^{p} + K_{i 1} X_{g}^{p n} I_{g q 0}^{n} Δ θ_{p l l}^{n} = 0 \end{array} \\ \begin{array}{l} Δ θ_{p l l}^{n} + K_{p 2} (X_{g}^{n n} I_{g q 0}^{n} + U_{g d 0}^{n}) Δ θ_{p l l}^{n} + K_{p 2} X_{g}^{n p} I_{g q 0}^{p} Δ θ_{p l l}^{p} + \\ K_{i 2} (X_{g}^{n n} I_{g q 0}^{n} + U_{g d 0}^{n}) Δ θ_{p l l}^{n} + K_{i 2} X_{g}^{n p} I_{g q 0}^{p} Δ θ_{p l l}^{p} = 0 \end{array} \end{matrix}

(9)

As shown in (9), the interactively coupled dual-sequence PLLs are similar to two coupled harmonic oscillators. Therefore, the Newton’s equations of motion can be written for the system as:

\underset{M}{\underset{⏟}{\underset{}{[\begin{matrix} m_{p p} & m_{p n} \\ m_{n p} & m_{n n} \end{matrix}]}}} [\begin{matrix} {\ddot{x}}_{1} \\ {\ddot{x}}_{2} \end{matrix}] + \underset{D}{\underset{⏟}{\underset{}{[\begin{matrix} d_{p p} & d_{p n} \\ d_{n p} & d_{n n} \end{matrix}]}}} [\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] + \underset{K}{\underset{⏟}{\underset{}{[\begin{matrix} k_{p p} & k_{p n} \\ k_{n p} & k_{n n} \end{matrix}]}}} [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = 0

(10)

where M, D, and K are the mass, damping, and spring matrices of the system, respectively, and M is a unit matrix indicating that the masses of both blocks are 1. In the damping and spring matrices, the diagonal elements $d_{i i}$ and $k_{i i}$ represent the damping and spring coefficients of an independent single-sequence PLL, respectively, whereas the nondiagonal elements $d_{i j}$ and $k_{i j}$ ( $i, j = p, n,$ $i \neq j$ ) represent the damping and spring coupling terms between the dual-sequence PLLs, respectively.

The equivalent 2-DOF spring damper particle model with the force analysis of particles is shown in Fig. 6.

Fig. 6 Equivalent small-signal model of dual-sequence PLLs. (a) 2-DOF spring damper particle model. (b) Force analysis of particles.

The two blocks in Fig. 6 can represent the PS and NS PLL modes of a DFIG-based WEGS during the steady-state stage of an asymmetric fault. The vertical displacements x₁ and x₂ of the two blocks with masses $m_{p}$ and $m_{n}$ represent the angular deviations of the dual-sequence PLLs, i.e., $Δ θ_{p l l}^{p}$ and $Δ θ_{p l l}^{n}$ , respectively. The equivalent mathematic model is expressed as:

M = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]

(11)

D = [\begin{matrix} K_{p 1} (X_{g}^{p p} I_{g q 0}^{p} + U_{g d 0}^{p}) & K_{p 1} X_{g}^{p n} I_{g q 0}^{n} \\ K_{p 2} X_{g}^{n p} I_{g q 0}^{p} & K_{p 2} (X_{g}^{n n} I_{g q 0}^{n} + U_{g d 0}^{n}) \end{matrix}]

(12)

K = [\begin{matrix} K_{i 1} (X_{g}^{p p} I_{g q 0}^{p} + U_{g d 0}^{p}) & K_{i 1} X_{g}^{p n} I_{g q 0}^{n} \\ K_{i 2} X_{g}^{n p} I_{g q 0}^{p} & K_{i 2} (X_{g}^{n n} I_{g q 0}^{n} + U_{g d 0}^{n}) \end{matrix}]

(13)

There is a coupling effect between the two blocks, which will affect the damping and oscillation frequencies of these two modes.

B. New Dynamic Coupling Introduced by Dual-sequence PLLs

According to the established model, the application of dual-sequence PLLs has introduced a new dynamic coupling effect to the system (the dynamic coupling between the PS and NS PLLs). The influence of this new dynamic coupling effect on the system stability is still unclear.

In order to study the influence of this new dynamic coupling effect, a small-signal state-space equation is established for the DFIG-based WEGS during asymmetric LVRT, which includes the dynamics of the RSC current control loops (CCLs), grid-side converter (GSC) CCLs, and dual-sequence PLLs. During asymmetric faults, the small-signal model of a DFIG in the PS and NS dq frames is expressed as:

\{\begin{matrix} \begin{array}{l} Δ {\dot{I}}_{s d q +}^{p} = - \frac{R_{s}}{σ L_{s}} Δ I_{s d q +}^{p} - j (ω_{p l l 0}^{p} + \frac{ω_{r} L_{m}^{2}}{σ L_{s} L_{r}}) Δ I_{s d q +}^{p} + \frac{L_{m} R_{r}}{σ L_{s} L_{r}} Δ I_{r d q +}^{p} - \\ j \frac{L_{m} ω_{r}}{σ L_{s}} Δ I_{r d q +}^{p} + \frac{1}{σ L_{s}} Δ U_{s d q +}^{p} - \frac{L_{m}}{σ L_{s} L_{r}} Δ U_{r d q +}^{p} \end{array} \\ \begin{array}{l} Δ {\dot{I}}_{s d q -}^{n} = - \frac{R_{s}}{σ L_{s}} Δ I_{s d q -}^{n} - j (ω_{p l l 0}^{n} + \frac{ω_{r} L_{m}^{2}}{σ L_{s} L_{r}}) Δ I_{s d q -}^{n} + \frac{L_{m} R_{r}}{σ L_{s} L_{r}} Δ I_{r d q -}^{n} - \\ j \frac{L_{m} ω_{r}}{σ L_{s}} Δ I_{r d q -}^{n} + \frac{1}{σ L_{s}} Δ U_{s d q -}^{n} - \frac{L_{m}}{σ L_{s} L_{r}} Δ U_{r d q -}^{n} \end{array} \end{matrix}

(14)

\{\begin{matrix} \begin{array}{l} Δ {\dot{I}}_{r d q +}^{p} = \frac{L_{m} R_{s}}{σ L_{r} L_{s}} Δ I_{s d q +}^{p} + j \frac{L_{m} ω_{r}}{σ L_{r}} Δ I_{s d q +}^{+} - j (ω_{p l l 0}^{p} - \frac{ω_{r}}{σ}) Δ I_{r d q +}^{p} - \\ \frac{R_{r}}{σ L_{r}} Δ I_{r d q +}^{p} - j I_{r d q 0 +}^{p} Δ ω_{p l l}^{p} - \frac{L_{m}}{σ L_{s} L_{r}} Δ U_{s d q +}^{p} + \frac{1}{σ L_{r}} Δ U_{r d q +}^{p} \end{array} \\ \begin{array}{l} Δ {\dot{I}}_{r d q -}^{n} = \frac{L_{m} R_{s}}{σ L_{r} L_{s}} Δ I_{s d q -}^{n} + j \frac{L_{m} ω_{r}}{σ L_{r}} Δ I_{s d q -}^{n} - j (ω_{p l l 0}^{n} - \frac{ω_{r}}{σ}) Δ I_{r d q -}^{n} - \\ \frac{R_{r}}{σ L_{r}} Δ I_{r d q -}^{n} - j I_{r d q 0 -}^{n} Δ ω_{p l l}^{n} - \frac{L_{m}}{σ L_{s} L_{r}} Δ U_{s d q -}^{n} + \frac{1}{σ L_{r}} Δ U_{r d q -}^{n} \end{array} \end{matrix}

(15)

where L_m is the mutual inductance; and $σ = 1 - L_{m}^{2} / (L_{s} L_{r})$ . The small-signal model of the RSC control system in the PS and NS dq frames is defined as:

\{\begin{array}{l} Δ {\dot{x}}_{3,4} = - Δ I_{r d q +}^{p} \\ \begin{array}{l} Δ U_{r d q +}^{p} = R_{r} Δ I_{r d q +}^{p} - K_{p 3,4} Δ I_{r d q +}^{p} + K_{i 3,4} Δ x_{3,4} + j (ω_{p l l 0}^{p} - ω_{r}) \cdot \\ σ L_{r} Δ I_{r d q +}^{p} + j σ L_{r} I_{r d q 0 +}^{p} Δ ω_{p l l}^{p} + \frac{L_{m}}{L_{s}} (Δ U_{s d q +}^{p} - R_{s} Δ I_{s d q +}^{p} - \\ j ω_{r} L_{s} Δ I_{s d q +}^{p}) \end{array} \\ Δ {\dot{x}}_{5,6} = - Δ I_{r d q -}^{n} \\ \begin{array}{l} Δ U_{r d q +}^{n} = R_{r} Δ I_{r d q -}^{n} - K_{p 5,6} Δ I_{r d q -}^{n} + K_{i 5,6} Δ x_{5,6} + j (ω_{p l l 0}^{n} - ω_{r}) \cdot \\ σ L_{r} Δ I_{r d q -}^{n} + j σ L_{r} I_{r d q 0 -}^{n} Δ ω_{p l l}^{n} + \frac{L_{m}}{L_{s}} (Δ U_{s d q -}^{n} - R_{s} Δ I_{s d q -}^{n} - \\ j ω_{r} L_{s} Δ I_{s d q -}^{n}) \end{array} \end{array}

(16)

Similarly, the small-signal model of the GSC and its control system in the PS and NS dq frames can be obtained as:

\{\begin{matrix} \begin{array}{l} Δ {\dot{I}}_{c d q +}^{p} = - \frac{R_{c}}{L_{c}} Δ I_{c d q +}^{p} - j ω_{p l l 0}^{p} Δ I_{c d q +}^{p} - \\ j I_{c d q 0 +}^{p} Δ ω_{p l l 0}^{p} - \frac{1}{L_{c}} Δ V_{c d q +}^{p} + \frac{1}{L_{c}} Δ U_{s d q +}^{p} \end{array} \\ \begin{array}{l} \overset{}{Δ {\dot{I}}_{c d q -}^{n}} = - \frac{R_{c}}{L_{c}} Δ I_{c d q -}^{n} - j ω_{p l l 0 -} Δ I_{c d q -}^{n} - \\ j Δ ω_{p l l 0}^{n} I_{c d q 0 -}^{n} - \frac{1}{L_{c}} Δ V_{c d q -}^{n} + \frac{1}{L_{c}} Δ U_{s d q -}^{n} \end{array} \end{matrix}

(17)

\{\begin{array}{l} Δ {\dot{x}}_{7,8} = Δ I_{c d, r e f +}^{p} - Δ I_{c d q +}^{p} \\ \begin{array}{l} Δ V_{c d q +}^{p} = (K_{p 7,8} Δ I_{c d q +}^{p} - R_{c} Δ I_{c d q +}^{p}) - K_{i 7,8} Δ x_{7,8} - j ω_{p l l 0}^{p} L_{c} Δ I_{c d q +}^{p} - \\ j Δ ω_{p l l 0}^{p} L_{c} I_{c d q 0 +}^{p} + Δ U_{s d q +}^{p} \end{array} \\ Δ {\dot{x}}_{9,10} = - Δ I_{c d q -}^{n} \\ \begin{array}{l} Δ V_{c d q +}^{n} = (K_{p 9,10} Δ I_{c d q +}^{n} - R_{c} Δ I_{c d q +}^{n}) - K_{i 9,10} Δ x_{9,10} - \\ j ω_{p l l 0}^{n} L_{c} Δ I_{c d q +}^{n} - j Δ ω_{p l l 0}^{n} L_{c} I_{c d q 0 +}^{n} + Δ U_{s d q +}^{n} \end{array} \end{array}

(18)

\{\begin{array}{l} Δ {\dot{x}}_{11} = - Δ U_{d c} \\ Δ I_{c d, r e f +}^{p} = - K_{p 11} Δ U_{d c} + K_{i 11} Δ x_{11} \end{array}

(19)

By combining the above equations and the model of the faulty grid, the small-signal model of a DFIG during the steady-state stage of a weak-grid asymmetric fault can be obtained. By solving the small-signal state-space equation, the dominant oscillation model of the DFIG-based WEGS and its corresponding participation factors can be studied during asymmetric LVRT. In addition, the severity of the asymmetric faults is measured by the voltage unbalance factor (VUF) defined as $δ = | U_{s}^{n} | / | U_{g}^{p} |$ .

When $δ = 0.63$ and $S C R = 2.4$ , there is a pair of poles entering the right half of the complex plane. The dominant oscillation mode of the system and its analysis results are summarized in Table I.

Table II Eigenvalue Analysis of Dominant Poles Under Different Coupling Conditions

Condition	Eigenvalue λ_3,4	Damping ratio
Without considering coupling between CCLs and NS PLL	$1.32 \pm j 147.1$	-0.009
Without considering coupling between dual-sequence PLLs	$- 6.10 \pm j 156.1$	0.039

Table Ⅰ Dominant Oscillation Mode of System and Its Analysis Results

Eigenvalue λ_3,4	Damping ratio	Frequency (Hz)	Dominated state variable	Participation factor
$1.45 \pm j 145.5$	-0.01	23.16	$Δ θ_{p l l}^{n}$	0.47
			$Δ θ_{p l l}^{p}$	0.22
			$Δ I_{s d q +}^{p}$	0.02, 0.02
			$Δ I_{s d q -}^{n}$	0.05, 0.05
			$Δ I_{r d q +}^{p}$	0.02, 0.02
			$Δ I_{r d q -}^{n}$	0.05, 0.05

From the analysis, the dynamic stability of a DFIG-based WEGS adopting the dual-sequence PLL structure is dominated by the NS PLL during asymmetric LVRT. In contrast, the PS PLL affects the dynamic stability of the system by the dynamic coupling between the dual-sequence PLLs during asymmetric LVRT.

Considering different coupling effects, an eigenvalue analysis of the dominant poles is summarized in Table II. Combining the analyses in Tables I and II, the introduction of the dual-sequence PLL structure has added a new dynamic coupling effect to the system during asymmetric LVRT, which is the main factor affecting the dynamic stability and changes the influence mode of the dominant poles of the system during asymmetric LVRT.

Because of the influence of this new dynamic coupling introduced by dual-sequence PLLs, the improved control strategies proposed in the existing studies may be less effective for WEGSs that apply the dual-sequence PLL structure. Therefore, this paper will focus on this influence on the dynamic stability of the system and strategies for addressing it to improve the stability of the system.

III. Influence of New Dynamic Coupling Introduced by Dual-sequence PLLs

The damping ratios $ξ_{e i}$ and natural oscillation frequencies (eigenfrequencies) $ω_{e i}$ for the decoupled PLL modes can be calculated using (20).

\{\begin{matrix} ξ_{e i} ω_{e i} = d_{i i} / 2 m_{i i} \\ ω_{e i} = \sqrt[]{K_{i i} / m_{i i}} \end{matrix}

(20)

According to [

24], the system is stable if its damping matrix is positive definite.

It is assumed that the 2-DOF system is viscous proportional damping. Substituting its natural exponential solution $x_{1,2} = X_{1,2} e^{s t}$ into (10) and dividing both sides by the nonzero term e^s^t, we have:

(M s^{2} + K) [\begin{matrix} x_{1} \\ x_{2} \end{matrix}] = 0

(21)

According to Cramer’s Rule, when the coefficient determinant of an n-variable homogeneous linear equation system composed of n equations is nonzero, the homogeneous equation system has only trivial solutions, i.e., $x_{1} = x_{2} = 0$ . Thus, if (21) has a nontrivial solution, the determinant of the coefficient matrix must be zero.

|M s^{2} + K| = 0

(22)

Substituting $s = j ω$ into (22), the solution to (22) can be obtained as:

\{\begin{matrix} ω_{c p} = \sqrt[]{\frac{k_{p p} + k_{n n} + \sqrt[]{{(k_{p p} - k_{n n})}^{2} + 4 k_{p n} k_{n p}}}{2}} \\ ω_{c n} = \sqrt[]{\frac{k_{p p} + k_{n n} - \sqrt[]{{(k_{p p} - k_{n n})}^{2} + 4 k_{p n} k_{n p}}}{2}} \end{matrix}

(23)

where $ω_{c p}$ and $ω_{c n}$ are the approximate modal frequencies of the PS and NS PLLs considering the dynamic coupling between the dual-sequence PLLs, respectively. Intuitively, $ω_{c p} > ω_{e p} > ω_{e n} > ω_{c n}$ owing to k_pnk_np.

In accordance with (23), the dynamic coupling between the dual-sequence PLLs increases the frequency of the PS PLL mode and decreases the frequency of the NS PLL mode. In addition, the magnitude of the frequency shift is related to k_pnk_np.

From (10), the dynamic coupling between the dual-sequence PLLs also affects the damping of the system. Using (10), we can obtain:

\{\begin{matrix} (s^{2} + d_{p p} s + k_{p p}) x_{1} + (d_{p n} s + k_{p n}) x_{2} = 0 \\ (s^{2} + d_{n n} s + k_{n n}) x_{2} + (d_{n p} s + k_{n p}) x_{1} = 0 \end{matrix}

(24)

To find a nontrivial solution, we have:

\{\begin{matrix} (s^{2} + d_{p p} s + k_{p p}) - \frac{(d_{p n} s + k_{p n}) (d_{n p} s + k_{n p})}{s^{2} + d_{n n} s + k_{n n}} = 0 \\ (s^{2} + d_{n n} s + k_{n n}) - \frac{(d_{p n} s + k_{p n}) (d_{n p} s + k_{n p})}{s^{2} + d_{p p} s + k_{p p}} = 0 \end{matrix}

(25)

G_{p} (s) = s^{2} + d_{p p} s + k_{p p}

(26)

G_{n} (s) = s^{2} + d_{n n} s + k_{n n}

(27)

G_{p n} (s) = - \frac{(d_{p n} s + k_{p n}) (d_{n p} s + k_{n p})}{s^{2} + d_{n n} s + k_{n n}}

(28)

G_{n p} (s) = - \frac{(d_{p n} s + k_{p n}) (d_{n p} s + k_{n p})}{s^{2} + d_{p p} s + k_{p p}}

(29)

In (25), the PS and NS PLLs are equated to coupled complex torque models that have symmetric structures. For example, the model of NS PLL can be depicted as shown in Fig. 7.

Fig. 7 Model of NS PLL. (a) Spring damper particle model of coupled NS PLL. (b) Equivalent complex torque model of NS control subsystem.

According to complex torque theory, the magnitude of the imaginary part of G_np(s) can be used to measure the influence of the dynamic coupling between the dual-sequence PLLs on the damping of the NS PLL. Substituting $s = j ω_{c n}$ into G_np(s), the damping offset $Δ D_{n}$ for the NS PLL mode can be obtained as:

Δ D_{n} = k_{p n} k_{n p} \frac{(ω_{c n}^{2} - ω_{e p}^{2}) τ_{n} + ω_{c n}^{2} τ_{p} (1 - ω_{e p}^{2} τ_{n} τ_{p})}{{(ω_{c n}^{2} - ω_{e p}^{2})}^{2} + τ_{n}^{2} ω_{e n}^{4} ω_{c n}^{2}}

(30)

where $τ_{p} = k_{p p}^{p} / k_{p i}^{p}$ ; and $τ_{n} = k_{p p}^{n} / k_{p i}^{n}$ .

It can be observed from (30) that $Δ D_{n} < 0$ because $ω_{c n} < ω_{e p}$ , i.e., the dynamic coupling between the dual-sequence PLLs will provide negative damping of the NS PLL mode. Owing to the symmetry of the PS and NS PLLs, the dynamic coupling between the dual-sequence PLLs will provide positive damping of the PS PLL mode. The damping offset of the PS PLL mode caused by dynamic coupling is expressed as:

Δ D_{p} = k_{p n} k_{n p} \frac{(ω_{c p}^{2} - ω_{e n}^{2}) τ_{p} + ω_{c p}^{2} τ_{n} (ω_{e n}^{2} τ_{p} τ_{n} - 1)}{{(ω_{c p}^{2} - ω_{e n}^{2})}^{2} + τ_{p}^{2} ω_{e p}^{4} ω_{c p}^{2}}

(31)

It can be observed from (31) that $Δ D_{p} > 0$ because $ω_{c p} > ω_{e n}$ .

Similar to the frequency offset, the magnitude of the damping offset is related to k_pnk_np. The magnitude of k_pnk_np is related to the coupling impedances $X_{g}^{p n}$ and $X_{g}^{n p}$ and the PS and NS reactive currents $I_{g q 0}^{p}$ and $I_{g q 0}^{n}$ . The more serious the asymmetrical faults, the more obvious the inter-sequence coupling impedance [

13]. Moreover, the more severe the asymmetrical fault, the larger the reactive currents injected into the faulty grid. Hence, the magnitudes of the frequency and damping deviations are related to the severity of the asymmetric fault.

The above small-signal state-space model will be used for a modal analysis to obtain the dynamic stability of the dominant poles of the system during the steady-state stage of an asymmetric fault: the poles dominated by the PS and NS PLL modes. Figure 8 and Table III present the movement trajectories and an eigenvalue analysis of the dominant poles, respectively, as the severity of the asymmetric fault increases with or without consideration of the dynamic coupling between the dual-sequence PLLs. The decoupled and coupled dominant poles of the PS and NS PLLs are obtained by modifying the coefficients related to the coupled terms in the state-space equation.

Fig. 8 Movement trajectories of dominant poles of PLLs as δ varies from 0.41 to 0.63.

TABLE III Eigenvalue Analysis of Dominant Poles of PLLs

δ	Dynamic coupling	Dominant mode	Eigenvalue	Damping ratio	Frequency (Hz)
0.55	Decoupled	PS PLL	$- 7.7 \pm j 192.7$	0.040	30.67
	Coupled	PS PLL	$- 13.8 \pm j 203.3$	0.068	32.35
	Decoupled	NS PLL	$- 5.950 \pm j 154.2$	0.038	24.54
	Coupled	NS PLL	$0.105 \pm j 146.6$	-0.001	23.30
0.63	Decoupled	PS PLL	$- 7.0 \pm j 183.2$	0.038	29.16
	Coupled	PS PLL	$- 14.5 \pm j 197.1$	0.070	31.37
	Decoupled	NS PLL	$- 6.1 \pm j 156.1$	0.039	24.84
	Coupled	NS PLL	$1.45 \pm j 145.5$	-0.010	23.16

As can be observed from Fig. 8 and Table III, the dynamic coupling between the dual-sequence PLLs will repel the coupled PS and NS PLL modes, the frequency of the PS PLL mode will increase, and the frequency of the NS PLL mode will decrease. At the same time, the damping of the PS PLL mode will increase, and the damping of the NS PLL mode will decrease. Because of this, the dominant mode of the dynamic stability of the system during asymmetric LVRT changes from the PS PLL mode to the NS PLL mode, in contrast to systems that adopt the single-sequence PLL structure in [

15], [18]. The damping of the NS PLL mode will become negative as the severity of the asymmetric fault increases owing to the dynamic coupling between the dual-sequence PLLs, which is not beneficial for the stability of the system.

From the perspective of vibration mechanics, greater dynamic coupling between the dual-sequence PLLs will increase the nondiagonal elements of D, and D will change from a positive-definite matrix to a non-positive-definite one, which will affect the dynamic stability of the system during the steady-state stage of a fault with asymmetric LVRT.

IV. Design of Novel PLL Structure for Improving Dynamic Stability

A. Impact of Dual-sequence PLLs on Application of Improved Control Strategies

According to the above analysis, the adoption of the dual-sequence PLL structure has changed the form of the dynamic stability of the system during the steady-state stage of a fault when there is an asymmetric voltage dip. On one hand, the dynamic coupling between the dual-sequence PLLs has introduced negative and positive damping to the NS and PS PLL modes, respectively. On the other hand, the dynamic coupling between the dual-sequence PLLs becomes stronger as the severity of the asymmetric fault increases. Consequently, this dynamic coupling effect due to the dual-sequence PLL structure changes the dominant mode of the instability of the system.

There are two control strategies proposed in the existing studies to enhance the dynamic stability during the steady-state stage of asymmetric faults: the algorithm that optimizes the current control reference in [

15] and the compensated control strategy in [18]. However, the control strategies for improving the dynamic stability during asymmetric LVRT proposed in the existing studies are all based on a system that applies the single-sequence PLL structure. The algorithm in [15] for DFIG-based wind turbines shows that the WEGS can properly inject the PS and NS active currents into the grid during asymmetric LVRT, which can improve the dynamic stability of the system. Nevertheless, this algorithm is mathematically based on the dynamic coupling between the PS PCC voltage and the PS/NS currents. After applying the dual-sequence PLL structure, the instability mode of the system changes from the PS PLL mode to the NS PLL mode. Therefore, the algorithm in [15] is not suitable for a system with the dual-sequence PLL structure.

Reference [

18] designs a control strategy to improve the dynamic stability of a grid-connected system during asymmetric LVRT by compensating for the negative effect introduced by the dynamic coupling between the PLL and CCLs. Figure 9 shows the control structure of the compensated control strategy in [18], where H_fbv±, H_fbu±, and H_fbi± are the compensated coefficients, whose specific values are selected according to the discussion in [18]. However, in the systems applying the dual-sequence PLL structure, the new dynamic coupling effect introduced by the dual-sequence PLLs is not taken into account in this compensated control strategy, which is an important factor affecting the dynamic stability of the system during asymmetric LVRT. Therefore, the effectiveness of this compensated control strategy for enhancing the dynamic stability during the steady-state stage of an asymmetric fault will decrease.

Fig. 9 Control structure of compensated control strategy in [

18].

Figure 10 shows the motion trajectories of the dominant poles of the PS and NS PLLs as the severity of the asymmetric fault increases after applying the compensated control strategy in [

18].

Fig. 10 Motion trajectories of dominant poles of PS and NS PLLs when compensated control strategy in [

18] is adopted as δ varies from 0.41 to 0.63.

From Fig. 10, the system can maintain dynamic stability after applying the compensated control strategy in [

18] when the new dynamic coupling effect introduced by the dual-sequence PLL structure is not considered regardless of the severity of the asymmetric fault. However, the effectiveness of the compensated control strategy towards improving the dynamic stability gradually decreases as the severity of the asymmetric fault increases considering the dynamic coupling effect between dual-sequence PLLs. When the severity of the asymmetric fault increases to a certain value, a small-signal instability will occur in the system during asymmetric LVRT.

B. Design of Adjustable Damping Shock Absorber for Dual-sequence PLLs

When the dual-sequence PLL structure is applied, the improvement in the dynamic stability using the existing improved control strategies during asymmetric LVRT will be reduced or even invalidated. Hence, a more effective control strategy needs to be designed to improve the dynamic stability of systems that apply the dual-sequence PLL structure during asymmetric LVRT.

The negative damping of the NS PLL mode is the main cause of the dynamic instability of the system during the steady-state stage of asymmetric faults. Therefore, a method for enhancing the damping of the NS PLL mode is the key to enhancing the dynamic stability during the steady-state stage of an asymmetric voltage dip.

This paper proposes an improved NS PLL structure that negatively feeds the deviation between $ω_{p l l}^{n}$ and $ω_{g}^{n}$ back to the q-axis input voltage of the NS PLL through the PI controller. The control structure of improved NS PLL is shown in Fig. 11.

Fig. 11 Control structure of improved NS PLL.

After adopting this improved NS PLL structure, the damping ratio of the NS PLL mode during asymmetric LVRT is revised as:

ξ^{*} = \frac{K_{p 2} U_{s d 0}^{n} + K_{p 2} X_{g}^{n n} I_{g q 0}^{n} + K_{p 2} k_{c i} + k_{c p} K_{i 2} + I m (G_{n p})}{2 \sqrt[]{|K_{i 2} (1 + K_{p 2} k_{c p}) (U_{s d 0}^{n} + X_{g}^{n n} I_{g q 0}^{n} + k_{c i} + R e (G_{n p}) / K_{i 2})|}}

(32)

From (32), the actual damping of the NS PLL mode during asymmetric LVRT can be adjusted according to the additional PI controller parameters k_cp and k_ci. Figure 12 shows how the damping ratio changes as k_cp and k_ci change. The damping ratio of the NS PLL mode will increase as k_cp and k_ci become larger.

Fig. 12 Damping ratio of NS PLL mode as k_cp and k_ci change.

Figure 13 shows the motion trajectories of the dominant poles of the dual-sequence PLLs as the severity of the asymmetric fault increases when the proposed improved NS PLL is applied to the dual-sequence PLL structure. With the proposed novel PLL structure, the system can maintain the dynamic stability during the steady-state stage of an asymmetric voltage dip, regardless of the severity of the asymmetric fault.

Fig. 13 Motion trajectories of dominant poles of dual-sequence PLLs after adopting improved PLL structure as δ varies from 0.41 to 0.63.

The improved control strategy proposed in this paper is physically equivalent to the adjustable damping shock absorber widely used in engineering mechanics (such as vehicle engineering), as shown in Fig. 14.

Fig. 14 Adjustable damping shock absorber applied to an automobile. (a) Coupling between vehicle body and tire. (b) Equivalent 2-DOF system in vehicle. (c) DFIG-based WEGS with proposed novel PLL structure.

While driving a vehicle, the small-signal disturbances are inevitable owing to uneven ground. The tires and vehicle body are equivalent to a coupled 2-DOF system. The adjustable damping shock absorber can adjust the coupled damping between the tires and the vehicle body according to the magnitude of the disturbance, thereby reducing the vibration of the vehicle body and improving the driving performance and the comfort of passengers [

26].

When an asymmetric voltage dip occurs in the grid, the DFIG-based WEGS activates the NS control system and injects NS current into the faulty grid. The PS and NS PLL modes of the system during the steady-state stage of an asymmetric fault are similar to a coupled 2-DOF mass damping spring system, similar to the vehicle body and tires. The severity of the asymmetric fault is analogous to the magnitude of the disturbance on the road while driving a vehicle. The coupled damping and coupling springs between the PS and NS PLL modes are related to the magnitude of the disturbance (severity of the fault). When the system enters the steady-state stage of an asymmetric fault after the transient process ends, the proposed adjustable damping shock absorber (the novel PLL structure) is activated. The parameters of the adjustable damper are set according to the magnitude of the disturbance (severity of the asymmetric fault).

When a fault is detected, the fault type is first determined. The fault type, fault location, and fault resistance can be obtained through the fault detection technology, which has been studied in [

28]-[30]. Since the fault diagnosis techniques and fault location methods in [28]-[30] are not the focus of this paper, they will not be studied in detail because of space limitations. The proposed novel NS PLL structure will be activated if the grid fault is determined to be an asymmetric voltage dip. The coefficients k_cp and k_ci in the proposed novel NS PLL structure are calculated using (32). Combining the fault impedance and system parameters, the original damping ratio

ξ

of the NS PLL mode is obtained from (32). Subsequently, k_cp and k_ci are changed to obtain the target damping ratio

ξ^{*}

. Hence, the dynamic stability of the system during the steady-state stage of an asymmetric voltage dip can be improved.

V. Simulation Validation

The above analysis and the effectiveness of the novel PLL structure for improving the dynamic stability during asymmetric LVRT are verified by simulation using MATLAB/Simulink. In the simulation model, the DFIG-based WEGS is connected to the power grid through step-up transformers. An asymmetric ground fault is triggered at the fault point.

The simulation results as the severity of the asymmetric fault changes without applying any improved control are shown in Figs. 15 and 16 when $δ = 0.55$ and $δ = 0.63$ , respectively, where phases a, b, and c are represented with red, green, and blue lines, respectively.

Fig. 15 Simulation results when $δ = 0.55$ without any improved control. (a) $U_{s a b c}$ . (b) $I_{s a b c}$ . (c) $I_{r a b c}$ . (d) $I_{c a b c}$ . (e) $U_{d c}$ . (d) THD.

Fig. 16 Simulation results when δ=0.63 without any improved control. (a) $U_{s a b c}$ . (b) $I_{s a b c}$ . (c) $I_{r a b c}$ . (d) $I_{c a b c}$ . (e) $U_{d c}$ . (d) THD.

A small-signal instability phenomenon occurs during the steady-state stage of an asymmetric voltage dip without any improved control strategy. The stator-side current I_sabc and voltage $U_{s a b c}$ , rotor-side current I_rabc, DC-side voltage U_dc, and GSC current I_cabc all have oscillations. The total harmonic distortion (THD) of U_sa during asymmetric LVRT are 17.73% and 25.54% when $δ = 0.55$ and $δ = 0.63$ , respectively. Comparing the simulation results in Figs. 15 and 16, it can be observed that an increase in the severity of the asymmetric fault will worsen the dynamic stability during asymmetric LVRT by aggravating the dynamic coupling effect between the dual-sequence PLLs.

Figures 17 and 18 show the simulation results when $δ = 0.55$ after applying the compensated control strategy proposed in [

18] and the proposed novel PLL structure, respectively. After adopting these two improved control strategies, the small-signal instability during the steady-state stage of the asymmetric voltage dip is avoided, the dynamic stability of the system is enhanced, and the THD of U_sa during asymmetric LVRT is reduced from 17.73% to 2.41% and 2.31%, respectively. From the simulation results, both improved control strategies can improve the small-signal stability of the system during asymmetric LVRT when the severity of the asymmetric fault is low (

δ < 0.55

), which justifies the conclusion of the analysis in Fig. 10.

Fig. 17 Simulation results when $δ = 0.55$ with compensated control strategy in [

18]. (a)

U_{s a b c}

. (b)

I_{s a b c}

. (c)

I_{c a b c}

. (d)

I_{r a b c}

. (e)

U_{d c}

. (f) THD.

Fig. 18 Simulation results when $δ = 0.55$ with novel PLL structure proposed in this paper. (a) $U_{s a b c}$ . (b) $I_{s a b c}$ . (c) $I_{r a b c}$ . (d) $I_{c a b c}$ . (e) $U_{d c}$ . (f) THD.

Figures 19 and 20 show the simulation results when $δ = 0.63$ after applying the two improved control strategies. As the severity of the asymmetric fault increases, the improvement in the dynamic stability of the system during asymmetric LVRT with the compensated control strategy in [

18] is limited, and the small-signal instability will still occur. The THD of U_sa is 19.88%, which also justifies the conclusion of the analysis in Fig. 10. In contrast, the small-signal instability during asymmetric LVRT is avoided after applying the proposed novel PLL structure, and the dynamic stability of the system is enhanced. The THD of U_sa is reduced from 25.54% to 2.52%. This justifies the conclusion of the analysis in Fig. 13.

Fig. 19 Simulation results when $δ = 0.63$ with compensated control strategy in [

18]. (a)

U_{s a b c}

. (b)

I_{s a b c}

. (c)

I_{r a b c}

. (d)

I_{c a b c}

. (e)

U_{d c}

. (f) THD.

Fig. 20 Simulation results when $δ = 0.63$ with novel PLL structure proposed in this paper. (a) $U_{s a b c}$ . (b) $I_{s a b c}$ . (c) $I_{r a b c}$ . (d) $I_{c a b c}$ . (e) $U_{d c}$ . (f) THD.

According to the analysis results in Fig. 8, the dynamic coupling effect between the dual-sequence PLLs is low when the severity of the asymmetric fault is low. At this time, the system can also remain small-signal stability during the steady-state stage of an asymmetric fault ( $δ < 0.51$ ) without applying any improved control. During a single-phase voltage dip, the maximum value of $δ$ is 0.5 (the single-phase voltage drops to zero). Therefore, the system can remain small-signal stability during a single-phase voltage dip without any improved control strategy.

Figures 21 and 22 shows the simulation results for a phase-to-phase short-circuit fault. When the improved control strategy is not applied, there is a small-signal instability in the system during the steady-state stage of the asymmetric fault. After applying the proposed novel PLL structure, this small-signal instability is avoided, and the dynamic stability of the system is improved. The THD of U_sa is reduced from 21.22% to 2.9%.

Fig. 21 Simulation results under phase-to-phase short-circuit fault without improved control strategy. (a) $U_{s a b c}$ . (b) $I_{s a b c}$ . (c) $I_{r a b c}$ . (d) $I_{c a b c}$ . (e) $U_{d c}$ . (f) THD.

Fig. 22 Simulation results under phase-to-phase short-circuit fault with proposed novel PLL structure. (a) $U_{s a b c}$ . (b) $I_{s a b c}$ . (c) $I_{r a b c}$ . (d) $I_{c a b c}$ . (e) $U_{d c}$ . (f) THD.

The control strategies for improving the dynamic stability during asymmetric LVRT proposed in the existing studies have limited effect for the systems adopting dual-sequence PLLs. In particular, as the severity of the asymmetric fault increases, the enhancement in the dynamic stability provided by the existing improved control strategies will decrease or become invalidated. In contrast, the novel PLL structure proposed in this paper can improve the dynamic stability of the systems adopting dual-sequence PLLs during asymmetric LVRT, regardless of the severity of the asymmetric fault.

VI. Experimental Validation

In this paper, the experimental platform shown in Fig. 23 is used to further verify the analyses presented in this paper and the effectiveness of the proposed novel PLL structure. The DFIG is driven by a DC motor, and the impedance of the transmission lines is simulated by an air-core reactor. Asymmetric voltage dips are simulated by a Chroma 61830 grid simulator. Appendix A presents the parameters of the experimental platform in detail.

Fig. 23 Structure of experimental platform.

Figure 24 shows the experimental results for a two-phase asymmetric voltage dip when $δ = 0.55$ . From the experimental results, a small-signal instability occurs in the system during the steady-state stage of the asymmetric fault without applying the improved control strategy. After applying the proposed novel PLL structure, this small-signal instability is avoided; moreover, the THD of the stator voltage decreases from 16.51% to 4.82%, and the stability is improved.

Fig. 24 Experimental results under a two-phase asymmetrical voltage dip when $δ = 0.55$ . (a) $U_{d c}$ , $U_{s a b c}$ , $I_{r a b c}$ , $I_{s a b c}$ without improved control strategy. (b) $U_{d c}$ , $U_{s a b c}$ , $I_{r a b c}$ , $I_{s a b c}$ with novel PLL structure. (c) FFT analysis of U_sab without improved control strategy. (d) FFT analysis of U_sab with novel PLL structure.

Figure 25 shows the experimental results when $δ = 0.63$ . As the severity of the asymmetric fault increases, a small-signal instability will occur, and the oscillation will increase during asymmetric LVRT. After applying the proposed novel PLL structure, the stability of the system is significantly improved, and the THD of U_sab is reduced from 23.09% to 4.89%.

Fig. 25 Experiment results under an asymmetric voltage dip when $δ = 0.63$ . (a) $U_{d c}$ , $U_{s a b c}$ , $I_{r a b c}$ , $I_{s a b c}$ without improved control strategy. (b) $U_{d c}$ , $U_{s a b c}$ , $I_{r a b c}$ , $I_{s a b c}$ with novel PLL structure. (c) FFT analysis of U_sab without improved control strategy. (d) FFT analysis of U_sab with novel PLL structure.

A comparison of the experimental results in Fig. 24(a) and 25(a) verifies the conclusion in Section III, i.e., an increase in the severity of the asymmetric fault will not be beneficial to the dynamic stability of the system during asymmetric LVRT because the effect of the dynamic coupling between the dual-sequence PLLs is aggravated. The simulation and experimental results demonstrate the effectiveness of the proposed new PLL structure.

VII. Conclusion

The influence of the new dynamic coupling effect introduced by dual-sequence PLLs (the dynamic coupling between the PS and NS PLLs) on the dynamic stability of DFIG-based WEGSs during the steady-state stage of an asymmetric fault is studied. The conclusions are as follows.

1) The dynamic coupling between the dual-sequence PLLs will increase the frequency of the PS PLL mode and decrease the frequency of the NS PLL mode. Moreover, the damping of the PS PLL mode will increase, and the damping of the NS PLL mode will decrease. Further, the dynamic coupling between the dual-sequence PLLs will increase as the severity of the asymmetric fault increases, and the damping offset will increase. Hence, this dynamic coupling will change the mode of instability and introduce a risk of dynamic instability to the system.

2) The dynamic coupling introduced by the dual-sequence PLLs will decrease the effectiveness of the existing control strategies for enhancing the dynamic stability. An adjustable damping shock absorber from the field of engineering mechanics was used in this paper to design a novel PLL structure to enhance the dynamic stability of DFIG-based WEGSs that adopt dual-sequence PLLs during the steady-state stage of an asymmetric fault. Simulations and experiments verify the limitations of the existing control strategies and the effectiveness of the proposed novel PLL structure towards improving the dynamic stability during the steady-state stage of an asymmetric fault.

Nomenclature

Symbol	——	Definition
A.	——	Parameters and Variables
ω, θ	——	Angular velocity and angle
I, U	——	Current and voltage
P_g, Q_g	——	Active power and reactive power
R, L, C	——	Resistance, inductance, and capacitance
K_p1,2, K_i1,2	——	Proportional and integral coefficients in positive-sequence (PS) and negative-sequence (NS) phase-locked PLLs
K_p3,4, K_i3,4	——	Proportional and integral coefficients in PS current control loop (CCL) of rotor-side converter (RSC)
K_p5,6, K_i5,6	——	Proportional and integral coefficients in NS CCL in RSC
K_p7,8, K_i7,8	——	Proportional and integral coefficients in PS CCL of grid-side converter (GSC)
K_p9,10, K_i9,10	——	Proportional and integral coefficients in NS CCL in GSC
K_p11, K_i11	——	Proportional and integral coefficients in DC voltage control loop
U_dc	——	DC-side voltage
Z_g, Z_g1, Z_g2	——	Grid impedances
B.	——	Subscripts and Superscripts
α, β	——	Components in αβ coordinates
+, -	——	PS and NS components
0	——	Initial value
a, b, c	——	Phases a, b, and c
d, q	——	Components in synchronous dq coordinates
f	——	Components in faulted branch
pll	——	PLL-detected components
ref	——	Reference value
s, r, c, g	——	Stator, rotor, GSC, and grid sides
p, n	——	Components in PS and NS synchronous reference frames (SRFs)

Appendix

Appendix A

Table AI Simulation System Parameters

Parameter	Value	Parameter	Value
P_base	2 MW	U_base	690 V
U_dc	1200 V	R_s, L_s	0.02, 2.9 p.u.
R_r, L_r	0.03, 2.95 p.u.	L_m	2.8 p.u.
NS PLL bandwidth	25 Hz	PS PLL bandwidth	29 Hz
R_g, L_g	0.1, 0.5 p.u.

Table AII Experimental System Parameters

Parameter	Value	Parameter	Value
P_base	3 kW	U_base	220 V
U_dc	400 V	R_s, L_s	0.07, 1.3 p.u.
R_r, L_r	0.1, 1.35 p.u.	L_m	1.26 p.u.
R_g, L_g	0.1, 0.42 p.u.	Synchronous speed	1500 r/min
PS PLL bandwidth	29 Hz	NS PLL bandwidth	25 Hz

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