Small-signal Stability of Grid-connected Converter System in Renewable Energy Systems with Fractional-order Synchronous Reference Frame Phase-locked Loop

Peng Zhang; Wenjuan Du; Haifeng Wang

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Small-signal Stability of Grid-connected Converter System in Renewable Energy Systems with Fractional-order Synchronous Reference Frame Phase-locked Loop PDF

- ORCID：
Peng Zhang
✉
- ORCID：
Wenjuan Du (Member, IEEE)
✉
- ORCID：
Haifeng Wang (Senior Member, IEEE)
✉

School of Electrical Engineering, Sichuan University, Chengdu, China

Updated：2025-05-21

DOI：10.35833/MPCE.2024.000215

Abstract

Fractional-order control (FOC) has gained significant attention in power system applications due to their ability to enhance performance and increase stability margins. In grid-connected converter (GCC) systems, the synchronous reference frame phase-locked loop (SRF-PLL) plays a critical role in grid synchronization for renewable power generation. However, there is a notable research gap regarding the application of FOC to the SRF-PLL. This paper proposes a fractional-order SRF-PLL (FO-SRF-PLL) that incorporates FOC to accurately track the phase angle of the terminal voltage, thereby improving the efficiency of grid-connected control. The dynamic performance of the proposed FO-SRF-PLL is evaluated under varying grid conditions. A comprehensive analysis of the small-signal stability of the GCC system employing the FO-SRF-PLL is also presented, including derived small-signal stability conditions. The results demonstrate that the FO-SRF-PLL significantly enhances robustness against disturbances compared with the conventional SRF-PLL. Furthermore, the GCC system with the FO-SRF-PLL maintains stability even under weak grid conditions, showing superior stability performance over the SRF-PLL. Finally, both simulation and experimental results are provided to validate the analysis and conclusions presented in this paper.

Keywords

Small-signal stability; grid-connected converter (GCC) system; fractional-order control (FOC); fractional-order synchronous reference frame phase locked loop (FO-SRF-PLL).

I. Introduction

FRACTIONAL-ORDER elements (FOEs) and fractional-order control (FOC) have garnered significant interest in recent years within the electrical engineering community due to their enhanced flexibility and versatility in circuit design and applications [

1], [2]. Similarly, FOC has emerged as a promising approach for improving system performance and efficiency [3]. In renewable energy systems, FOEs and FOC provide a powerful framework for modeling system dynamics, which can be precisely described using fractional-order (FO) differential equations.

Recent advances have explored several novel applications of FOEs. For example, a high-power FO capacitor based on power converters has been proposed [

4], and FO models of filter inductors have been implemented within the control bandwidth of converters [5]. An FO modeling and analysis method for direct current (DC)-DC converters using FO inductors and capacitors was proposed in [6]. Additionally, an FO virtual capacitor aimed at active damping in multi-paralleled grid-connected current-source inverters has been introduced [7]. These innovations highlight the potential of FOEs to enhance the performance and efficiency of power electronic systems. In this paper, we use the FO model to describe the dynamics of transmission lines.

FOC, in particular, has gained significant attention as a control strategy for power and power electronic systems. Numerous studies have explored various FOC techniques, including FO proportional-integral-derivative (FOPID) control, FO sliding mode control (FOSMC), and FO terminal sliding mode control (FOTSMC). FOC has been successfully applied to enhance the robustness of multilevel converter integration into power grids [

8], improve power quality in grid-connected photovoltaic (PV) systems [9], and optimize output voltage tracking in DC-DC buck converters [10]. Furthermore, FOC has been utilized in passivity-based FOSMC [11] and robust fuzzy FOTSMC for grid-connected converter (GCC) systems [12]. It has also been applied in voltage and frequency control for microgrids [13], voltage control for high-voltage direct current (HVDC) transmission systems [14], robust controller design [15], and the experimental enhancement of fuzzy FOPID controllers for variable-speed wind energy conversion systems [16]. These studies demonstrate the broad applicability and advantages of FOC in power systems and power electronic applications.

Phase-locked loops (PLLs) are a core component of modern power systems and power electronics. The synchronous reference frame PLL (SRF-PLL) has become the standard for grid-connected renewable energy systems due to its efficiency, controllability, and adaptability. Several advanced PLL designs have been proposed to address power quality issues under abnormal grid conditions, such as enhanced PLL, dual second-order generalized integrator PLL, and double SRF-PLL [

17]. Comparative analyses have shown that while SRF-PLL may not be as robust under certain grid conditions as some of the more advanced PLLs, it remains the predominant choice for grid connection in renewable energy systems [18], [19].

The application of fractional calculus to PLLs has also shown promising results. In [

20], an FO analog PLL (FOAPLL) was introduced, demonstrating enhanced capture range and reduced lock time compared with the traditional integer-order analog PLL (IOAPLL). Subsequent studies [21], [22] analyzed the bandwidth, locking range, and transient response of FOAPLL, while [23] highlighted its superior performance in noisy environments. However, fractional calculus has yet to be applied to SRF-PLL in grid-connected control systems. Inspired by the FOAPLL model, this paper proposes an FO-SRF-PLL using FOPID control, aiming to improve the accuracy and performance of terminal voltage phase tracking.

Despite the growing use of FOEs and FOC in power systems, their application to SRF-PLL for grid-connected renewable energy systems remains unexplored. This paper introduces the FO-SRF-PLL for grid-connected control systems and investigates the small-signal stability of GCC systems employing the FO-SRF-PLL. The contributions of this paper are described as follows.

1) The FO-SRF-PLL, utilizing FOPID control for terminal voltage phase tracking in GCC systems, is proposed. The FO-SRF-PLL outperforms its integer-order counterpart in terms of faster response, higher tracking accuracy, and quicker settling time under varying grid conditions.

2) A linearized model of GCC systems with FO-SRF-PLL is derived, along with stability criteria specific to these systems.

3) Small-signal stability conditions for GCC systems employing the FO-SRF-PLL are derived. Simulation and experimental results demonstrate that the FO-SRF-PLL significantly improves small-signal stability, especially in weak grid connections, compared with conventional SRF-PLL.

The remainder of this paper is organized as follows. Section II evaluates the performance of the proposed FO-SRF-PLL. Section III presents the linear model derivation of the GCC system with FO-SRF-PLL. The stability analysis is discussed in Section IV. Section V presents simulation and experimental results under varying grid conditions to validate the proposed FO-SRF-PLL. Finally, Section VI concludes this paper.

II. Performance Evaluation of FO-SRF-PLL

A. FO-SRF-PLL

The FO-SRF-PLL is developed by applying the FOPID controller proposed in [

24] to the conventional SRF-PLL. Figure 1 shows the structural block diagram of the FO-SRF-PLL, where the FO proportional-integral (FOPI) controller and FO integrator (FOI) are used in the FO-SRF-PLL. In Fig. 1,

V_{g a}

V_{g b}

, and

V_{g c}

are the input three-phase voltages;

V_{g d}

and

V_{g q}

are the d-axis and q-axis voltages obtained by coordinate transformation, respectively;

K_{p}

and

K_{i}

are the proportional gain and integral gain of the FOPI controller, respectively;

ω_{0}

is the nominal frequency; and

α

is the order of the FOPI controller and the FOI. The output of the FOI is the phase angle

θ_{p}

estimated by the FO-SRF-PLL. The conventional SRF-PLL can be seen as a special case of the FO-SRF-PLL with

α = 1

Fig. 1 Structural block diagram of FO-SRF-PLL.

Figure 2 shows the linear model of the proposed FO-SRF-PLL, where $V_{g 0}$ is the magnitude of the input voltage; $x_{p}$ is the output of the FOI in the FOPI controller; and $θ$ is the phase of the input voltage. The closed-loop transfer function of the estimated phase $θ_{p}$ to the actual phase $θ$ is:

\frac{θ_{p} (s)}{θ (s)} = \frac{V_{g 0} (K_{p} s^{α} + K_{i})}{s^{2 α} + V_{g 0} K_{p} s^{α} + V_{g 0} K_{i}}

(1)

Fig. 2 Linear model of FO-SRF-PLL.

In this paper, the subscript “0” is used to denote the steady-state value of the variable or variable vector.

B. Implementation of FO-SRF-PLL

The implementation of an FOPID controller typically requires a rational approximation. The Oustaloup filter algorithm (OFA) [

25] is commonly used for the continuous-time approximation in the numerical implementation of FO systems. Within the considered frequency band

(ω_{b}, ω_{h})

s^{α}

can be approximated by:

G (s) = K \prod_{k = 1}^{N} \frac{s + ω_{k}^{'}}{s + ω_{k}}

(2)

where N is the order of the filter; $K = ω_{h}^{α}$ ; $ω_{k}^{'} = ω_{b} ω_{u}^{(2 k - 1 - α) / N},$ $ω_{k} = ω_{u}^{2 α / N} ω_{k}^{'},$ and $ω_{u} = \sqrt[]{ω_{h} / ω_{b}}$ .

In the paper, the above FOA is used for realization of FOPID controllers, and the approximated FOPID transfer function is:

H (s) = K_{p} + K_{i} {(K \prod_{k = 1}^{N} \frac{s + ω_{k}^{'}}{s + ω_{k}})}_{- α}

(3)

With respect to the digital implementation, the Tustin discretization method is used for FO-SRF-PLL [

26].

C. Performance Evaluation of FO-SRF-PLL

The performance of the proposed FO-SRF-PLL is evaluated under varying grid conditions. To simulate realistic grid operations, the 9-bus test system described in [

18] is used, as shown in Fig. 3 (active and reactive power flows are in MW and Mvar, respectively). The system is modeled in MATLAB/Simulink, with bus 6 selected as the point of common coupling (PCC) for integrating the GCC system. The testing scheme and implementation details are provided in Table I.

Fig. 3 Power flow diagram of test system.

TABLE I Testing Scheme and Implementation Details

Testing scheme	Condition	Implementation detail
Conducted on test system	Under-voltage	An inductive load of $+ j 150$ Mvar connected to bus 6 at 60 ms
	Over-voltage	A capacitive load of $- j 80$ Mvar connected to bus 6 at 60 ms
	Load rise	Load at buses 6 and 9 increased by 100 $M W + j 100$ Mvar at 60 ms
Utilizing a three-phase programmable voltage source	Phase jump	Phase jump of π/6 radians at 50 ms
Utilizing a three-phase programmable voltage source	Frequency step	Frequency step of 2 Hz at 50 ms

Table II provides the performance evaluation metrics of PLL, including locking time, overshoot/undershoot, and settling time. Locking time refers to the duration required from receiving the input signal to achieving signal lock. The PCC voltage $V_{a b c}$ , q-axis voltage $V_{q}$ , and estimated phase $θ$ under different test conditions are depicted in Figs. 4-8.

Fig. 4 Transient response to under-voltage conditions.

Fig. 5 Transient response to over-voltage conditions.

Fig. 6 Transient response to load rise conditions.

Fig. 7 Transient response to a phase jump.

Fig. 8 Transient response to a frequency step.

TABLE Ⅱ Performance Evaluation Metrics of PLL

Condition	PLL	Locking time (ms)	Overshoot/undershoot (%)	Settling time (ms)
Under-voltage	SRF-PLL	46.5	5.9	23.5
Under-voltage	FO-SRF-PLL	4.4	0.4	2.9
Over-voltage	SRF-PLL	46.5	100.0	30.0
Over-voltage	FO-SRF-PLL	4.4	88.8	5.4
Load rise	SRF-PLL	46.5	9.5	7.3
Load rise	FO-SRF-PLL	4.4	1.5	2.0
Phase jump	SRF-PLL	25.0	49.2	16.0
Phase jump	FO-SRF-PLL	1.2	47.1	1.0
Frequency step	SRF-PLL	25.0	61.0	16.0
Frequency step	FO-SRF-PLL	1.2	59.0	1.0

Table II and Figs. 4-8 show that the locking time of the SRF-PLL is significantly longer than that of the FO-SRF-PLL. Across various test scenarios, the locking time of the SRF-PLL is at least ten times longer than that of the FO-SRF-PLL. Furthermore, the GCC system using the FO-SRF-PLL exhibits a lower transient peak overshoot/undershoot and a quicker recovery response compared with the SRF-PLL.

In general, the FO-SRF-PLL demonstrates a significantly faster locking time, reduced undershoot/overshoot and shorter settling time compared with the SRF-PLL. This enhanced performance allows for quicker and more accurate tracking of the desired phase in the GCC system.

III. Linear Model Derivation of GCC System with FO-SRF-PLL

Figure 9 shows the configuration of the GCC system in renewable energy systems, where $V$ is the converter output voltage; $V_{g}$ is the PCC voltage; $I$ is the converter output current; $V_{b}$ is the grid voltage; $I_{d}$ and $I_{q}$ are the d- and q-axis components of the GCC output current, respectively; $I_{d r e f}$ and $I_{q r e f}$ are the reference values of the d- and q-axis currents of the current control loop, respectively; $θ_{p}$ is the phase angle estimated by the PLL; $L_{f}$ is the filter inductance; and $L_{g}$ is the transmission line inductance.

Fig. 9 Configuration of GCC system in renewable energy systems.

The GCC system shown in Fig. 9 is commonly used in PV or wind power systems, employing closed-loop control of the output current. The closed-loop control of the output current in the GCC system is realized in d-q coordinate. Moreover, in practical scenarios, the bandwidth of the inner loop of the current control is typically much greater than that of the PLL. Therefore, when assessing the stability of small disturbances associated with the PLL dynamics, it is reasonable to consider the output current of the GCC as equal to the current reference value provided by the inner loop of the current control. Consequently, the GCC can be represented as a constant current source, i.e., $I_{d 0} + j I_{q 0}$ . Hence, we can obtain:

Δ I_{d} + j Δ I_{q} = 0

(4)

In the paper, prefix $Δ$ refers to a small increment of the variable or variable vector.

Figure 10 shows the equivalent circuit and phasor diagram of the GCC system in the d-q coordinate, where $V_{g 0} ∠ 0$ is the PCC voltage; $X_{g}$ is the transmission line reactance; and $V_{b 0} ∠ - δ$ is the voltage of the alternating current (AC) grid. From Fig. 10, the voltage amplitude $V_{g 0}$ can be obtained as:

V_{g 0} = \sqrt[]{V_{b 0}^{2} - I_{d 0}^{2} X_{g}^{2}} - I_{q 0} X_{g}

(5)

Fig. 10 Equivalent circuit and phasor diagram of GCC system in d-q coordinate. (a) Equivalent circuit. (b) Phasor diagram.

The d-q coordinate is aligned by continuously tracking the phase angle of PCC voltage using the PLL. From Fig. 2, the FO differential equations of FO-SRF-PLL can be obtained as:

\{\begin{array}{l} \frac{d^{α} x_{p}}{d t^{α}} = K_{i} V_{g 0} (θ - θ_{p}) \\ \frac{d^{α} θ_{p}}{d t^{α}} = K_{p} V_{g 0} (θ - θ_{p}) + x_{p} \end{array}

(6)

Figure 11 shows the relationship between the d-q coordinate and the common x-y coordinate. The PLL utilizes the tracked phase of the PCC voltage to determine the orientation of the d-axis in the common x-y coordinate.

Δ θ = [- \begin{matrix} \frac{V_{g y 0}}{V_{g 0}^{2}} & \frac{V_{g x 0}}{V_{g 0}^{2}} \end{matrix}] [\begin{array}{l} Δ V_{g x} \\ Δ V_{g y} \end{array}] = [- \begin{matrix} \frac{V_{g y 0}}{V_{g 0}^{2}} & \frac{V_{g x 0}}{V_{g 0}^{2}} \end{matrix}] Δ V_{g x y}

(7)

Fig. 11 Relationship between x-y and d-q coordinates.

where subscript x and y are used to indicate the x and y component of the variable or variable vector in the common x-y coordinate.

Then, we have:

\begin{array}{l} [\begin{array}{l} I_{x} \\ I_{y} \end{array}] = [\begin{matrix} c o s θ_{p} & - s i n θ_{p} \\ s i n θ_{p} & c o s θ_{p} \end{matrix}] [\begin{array}{l} I_{d} \\ I_{q} \end{array}] \\ [\begin{array}{l} I_{d} \\ I_{q} \end{array}] = [\begin{matrix} c o s θ_{p} & s i n θ_{p} \\ - s i n θ_{p} & c o s θ_{p} \end{matrix}] [\begin{array}{l} I_{x} \\ I_{y} \end{array}] \end{array}

(8)

Thus, from (4) and (8), we can obtain:

Δ I_{x y} = [\begin{matrix} Δ I_{x} \\ Δ I_{y} \end{matrix}] = [\begin{matrix} - I_{y 0} \\ I_{x 0} \end{matrix}] Δ θ_{p}

(9)

The linearization of (6) is:

\{\begin{matrix} \frac{d^{α} Δ x_{p}}{d t^{α}} = K_{i} V_{g 0} (Δ θ - Δ θ_{p}) \\ \frac{d^{α} Δ θ_{p}}{d t^{α}} = K_{p} V_{g 0} (Δ θ - Δ θ_{p}) + Δ x_{p} \end{matrix}

(10)

From (7), (9) and (10), we can obtain:

\begin{array}{l} \{\begin{array}{l} \frac{d^{α} Δ X}{d t^{α}} = A Δ X + B Δ V_{g x y} \\ Δ I_{x y} = C Δ X \\ Δ X = [\begin{matrix} Δ x_{p} \\ Δ θ_{p} \end{matrix}] \\ A = [\begin{matrix} 0 & - K_{i} V_{g 0} \\ 1 & - K_{p} V_{g 0} \end{matrix}] \\ B = [\begin{matrix} - \frac{K_{i} V_{g y 0}}{V_{g 0}} & \frac{K_{i} V_{g x 0}}{V_{g 0}} \\ - \frac{K_{p} V_{g y 0}}{V_{g 0}} & \frac{K_{p} V_{g x 0}}{V_{g 0}} \end{matrix}] \\ C = [\begin{matrix} 0 & - I_{y 0} \\ 0 & I_{x 0} \end{matrix}] \end{array} \end{array}

(11)

The FO model of transmission lines is built as:

\{\begin{matrix} \frac{X_{g}}{ω_{0}} \frac{d^{α} I_{x}}{d t^{α}} = V_{g x} - V_{b x} + X_{g} I_{y} \\ \frac{X_{g}}{ω_{0}} \frac{d^{α} I_{y}}{d t^{α}} = V_{g y} - V_{b y} - X_{g} I_{x} \end{matrix}

(12)

Ignoring the dynamics of the grid, the linearization of (12) is:

\{\begin{matrix} \frac{d^{α} Δ I_{x}}{d t^{α}} = \frac{ω_{0}}{X_{L}} Δ V_{g x} + Δ ω_{0} I_{y} \\ \frac{d^{α} Δ I_{y}}{d t^{α}} = \frac{ω_{0}}{X_{L}} Δ V_{g y} - Δ ω_{0} I_{x} \end{matrix}

(13)

Or, equivalently, we can obtain:

Δ V_{g x y} = X_{g} [\begin{matrix} \frac{s^{α}}{ω_{0}} & - 1 \\ 1 & \frac{s^{α}}{ω_{0}} \end{matrix}] Δ I_{x y} = (\frac{X_{g} s^{α}}{ω_{0}} U_{1} + X_{g} U_{2}) Δ I_{x y}

(14)

where $U_{1} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ ; and $U_{2} = [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] .$

Let $P_{0}$ and $Q_{0}$ be the steady-state active and reactive power outputs of the GCC system, then we have:

\{\begin{array}{l} P_{0} = I_{x 0} V_{g x 0} + I_{y 0} V_{g y 0} \\ Q_{0} = I_{x 0} V_{g y 0} - I_{y 0} V_{g x 0} \end{array}

(15)

From (9), (11), (14), and (15), we can obtain:

\frac{d^{α} Δ X}{d^{α}} = {(U_{1} - \frac{X_{g}}{ω_{0}} B C)}^{- 1} (A + X_{g} B U_{2} C) Δ X = A_{c} Δ X

(16)

\{\begin{array}{l} A_{c} = \frac{1}{d} [\begin{matrix} a & b \\ 1 & c \end{matrix}] \\ a = \frac{X_{g} K_{i} P_{0}}{ω_{0} V_{g 0}} \\ b = (1 - \frac{X_{g} K_{p} P_{0}}{ω_{0} V_{g 0}}) (\frac{X_{g} K_{i} Q_{0}}{V_{g 0}} - K_{i} V_{g 0}) + \\ \frac{X_{g} K_{i} P_{0}}{ω_{0} V_{g 0}} (\frac{X_{g} K_{p} Q_{0}}{V_{g 0}} - K_{p} V_{g 0}) \\ c = \frac{X_{g} K_{p} Q_{0}}{V_{g 0}} - K_{p} V_{g 0} \\ d = 1 - \frac{X_{g} K_{p} P_{0}}{ω_{0} V_{g 0}} \end{array}

(17)

Next, the objective is to analyze the stability of system (16).

IV. Small-signal Stability Analysis of GCC System with FO-SRF-PLL

In fractional calculus, the FO linear time-invariant (FO-LTI) system is represented as follows:

\{\begin{array}{l} D^{q} x = A x + B u \\ y = C x \end{array}

(18)

where $D^{q} = d^{q} / d t^{q}$ is the fractional differential operator; and q is the order of derivation.

It has been demonstrated that the system in (18) is stable when (19) is satisfied [

27], where

e i g (A)

denotes the eigenvalues of matrix A, and the argument of

e i g (A)

is denoted by

a r g (e i g (A))

. Figure 12 shows the stable and unstable regions of system in (18) with

0 < q \leq 1

Fig. 12 Stable and unstable regions of system in (18) with $0 < q \leq 1$ .

|a r g (e i g (A))| \frac{q π}{2}

(19)

The following theorem establishes a crucial condition for confining the eigenvalues of a matrix within specific sectors. It serves as the foundation for establishing a direct association between the stability of LTI systems and FO-LTI systems.

Theorem 1 [

28]: if and only if the LTI system in (20) is asymptotically stable, the system in (18) with

0 < q \leq 1

is unstable.

\dot{x} = - [\begin{matrix} A s i n (\frac{q π}{2}) & - A c o s (\frac{q π}{2}) \\ A c o s (\frac{q π}{2}) & A s i n (\frac{q π}{2}) \end{matrix}] x

(20)

In this paper, the objective is to determine the parameters that render the FO-LTI system in (21) stable.

D^{α} Δ X = A_{c} Δ X

(21)

where $0 < α \leq 1$ is given.

From theorem 1, the equivalent LTI system in terms of stability for the FO-LTI system (21) can be expressed as:

\{\begin{array}{l} \frac{d Δ X}{d t} = A_{e} Δ X \\ A_{e} = [\begin{matrix} - \frac{a s i n (α π / 2)}{d} & - \frac{b s i n (α π / 2)}{d} & \frac{a c o s (α π / 2)}{d} & \frac{b c o s (α π / 2)}{d} \\ - \frac{s i n (α π / 2)}{d} & - \frac{c s i n (α π / 2)}{d} & \frac{c o s (α π / 2)}{d} & \frac{c c o s (α π / 2)}{d} \\ - \frac{a c o s (α π / 2)}{d} & - \frac{b c o s (α π / 2)}{d} & - \frac{a s i n (α π / 2)}{d} & - \frac{b s i n (α π / 2)}{d} \\ - \frac{c o s (α π / 2)}{d} & - \frac{c c o s (α π / 2)}{d} & - \frac{s i n (α π / 2)}{d} & - \frac{c s i n (α π / 2)}{d} \end{matrix}] \end{array}

(22)

Therefore, the instability of the GCC system with the FO-SRF-PLL is equivalent to the stability of the system in (22). The system in (22) has the following characteristic polynomial:

P (s) = s^{4} + a_{3} s^{3} + a_{2} s^{2} + a_{1} s + a_{0} = 0

(23)

\{\begin{array}{l} a_{0} = {(a c - b)}^{2} / d^{4} \\ a_{1} = 2 s i n (α π / 2) (a + c) (a c - b) / d^{3} \\ a_{2} = (2 b - 4 b s i n^{2} (α π / 2) + a^{2} + c^{2} + 4 a c s i n^{2} (α π / 2)) / d^{2} \\ a_{3} = 2 s i n (α π / 2) (a + c) / d \end{array}

(24)

Supplementary Material A demonstrates that the instability conditions of the system described by (21) are given by (25) and further verifies the correctness of (25).

\{\begin{array}{l} d > 0 \\ a + c > 0 \\ a c - b > 0 \\ {(a + c)}^{2} - 4 (a c - b) c o s^{2} (α π / 2) > 0 α \in (0, 1] \end{array}

(25)

Supplementary Material B demonstrates that the GCC system described by (16) is unstable if and only if conditions in (26) are satisfied. The small-signal instability conditions in (26) can be utilized to evaluate the stability of the GCC system.

\{\begin{array}{l} X_{g} < \frac{V_{g 0} ω_{0}}{K_{p} P_{0}} \\ X_{g} > \frac{K_{p} V_{g 0}^{2} ω_{0}}{K_{i} P_{0} + K_{p} Q_{0} ω_{0}} \\ {(K_{p} V_{g 0} - \frac{K_{p} Q_{0} X_{g}}{V_{g 0}} - \frac{K_{i} P_{0} X_{g}}{V_{g 0} ω_{0}})}^{2} - \\ \frac{4 K_{i} c o s^{2} (α π / 2) (V_{g 0}^{2} - Q_{0} X_{g}) (V_{g 0} ω_{0} - K_{p} P_{0} X_{g})}{V_{g 0}^{2} ω_{0}} > 0 \end{array}

(26)

For $α = 1$ , it can be proven that the GCC system employing SRF-PLL is stable if and only if conditions in (27) are satisfied.

\{\begin{array}{l} X_{g} < \frac{V_{g 0} ω_{0}}{K_{p} P_{0}} \\ X_{g} < \frac{K_{p} V_{g 0}^{2} ω_{0}}{K_{i} P_{0} + K_{p} Q_{0} ω_{0}} \end{array}

(27)

It can be observed from (26) that the stability of the system is influenced by the operating conditions of the GCC system as well as the values of $K_{p}$ and $K_{i}$ in the FO-SRF-PLL.

V. Simulation and Experimental Results

A. Simulation Results

The system shown in Fig. 9 has been built in MATLAB/Simulink to verify the accuracy of the theoretical analysis in this paper. Assume that the system operates with a power factor of 1 and the nominal voltage is 1 p.u., which implies $Q_{0} = 0$ , i.e., $I_{q 0} = 0$ . For ease of calculation, let $V_{b 0} = \sqrt[]{10} / 3 \approx 1.054$ , and $α = 0.5$ . Thus, we can obtain:

V_{g 0} = \sqrt[]{10 - 9 I_{d 0}^{2} X_{g}^{2}} / 3

(28)

To facilitate a comparative analysis of the stability of GCC the system with the SRF-PLL and FO-SRF-PLL, the following stability conditions are provided for the system when SRF-PLL is employed, as well as the instability conditions when FO-SRF-PLL with $α = 0.5$ is utilized.

From (27) and (28), the stability conditions for the GCC system with the SRF-PLL are obtained.

\{\begin{array}{l} X_{g} < \frac{ω_{0}}{K_{p} I_{d 0}} \\ X_{g}^{2} < \frac{10 K_{p}^{2} ω_{0}^{2}}{9 K_{p}^{2} ω_{0}^{2} I_{d 0}^{2} + K_{i}^{2}} \end{array}

(29)

From (26) and (28), when the FO-SRF-PLL with $α = 0.5$ is utilized, the GCC system is unstable if and only if:

\{\begin{array}{l} X_{g} < \frac{ω_{0}}{K_{p} I_{d 0}} \\ X_{g}^{2} > \frac{10 K_{p}^{2} ω_{0}^{2}}{9 K_{p}^{2} ω_{0}^{2} I_{d 0}^{2} + K_{i}^{2}} \\ X_{g}^{2} > 10 - {\{\frac{- K_{i}}{K_{i}^{2} - K_{p}^{2} ω_{0}^{2}} [3 ω_{0}^{2} - \sqrt[]{9 ω_{0}^{4} + 10 (K_{i}^{2} - K_{p}^{2} ω_{0}^{2})}]\}}^{2} \end{array}

(30)

Three tests have been conducted to validate the theoretical analysis. The results regarding the impact of the PLL control parameters $K_{p}$ and $K_{i}$ on the small-signal stability of the system are presented below. Moreover, the impacts of the transmission line reactance $X_{g}$ and the steady-state active power $P_{0}$ are detailed in Supplementary Material C.

To evaluate the impact of $K_{p}$ , we have conducted the following test. The reactance of the transmission line is fixed to be $X_{g} = 1$ , and $I_{d 0} = 1 / 3$ . Thus, $V_{g 0} = 1$ , and $P_{0} =$ $V_{g 0} I_{d 0} = 1 / 3$ . $K_{i}$ is fixed to be $ω_{0}^{2}$ ( $ω_{0} = 2 π f = 120 π$ ). From (29), the stability condition for the GCC system with SRF-PLL is:

125.66 \approx 40 π = \frac{1}{3} ω_{0} < K_{p} < 3 ω_{0} = 360 π \approx 1130.97

(31)

Supplementary Material D shows that there does not exist a value of $K_{p}$ that would lead to instability in the GCC system with the FO-SRF-PLL. The nonlinear simulation results are depicted in Figs. 13-16. For all these tests, $P$ corresponds to the active power output from the GCC. At 0.1 s of simulation, the active power output from the GCC in the example system increases by 10% for 100 ms.

Fig. 13 Results of nonlinear simulation with variation of $K_{p}$ in SRF-PLL. (a) $P$ . (b) $θ$ . (c) $V_{a b c}$ .

Fig. 14 Results of nonlinear simulation with $K_{p} = 0.1$ in FO-SRF-PLL. (a) $P$ . (b) $θ$ . (c) $V_{a b c}$ .

Fig. 15 Results of nonlinear simulation with variation of $K_{p}$ in SRF-PLL. (a) $P$ . (b) $θ$ . (c) $V_{a b c}$ .

Fig. 16 Results of nonlinear simulation with $K_{p} = 100 ω_{0}$ in FO-SRF-PLL.

It can be observed from Figs. 13-16 that when the value of $K_{p}$ exceeds the upper and lower limits obtained in (31), the GCC system with SRF-PLL becomes unstable, whereas the system with FO-SRF-PLL remains unaffected by the changes of $K_{p}$ .

To evaluate the impact of $K_{i}$ , the following test is conducted. The system parameters are set to the same as before, and the value of $K_{p}$ is fixed to be $ω_{0} / 3$ . From (29), the stability condition for the GCC system with the SRF-PLL is:

K_{i} ω_{0}^{2} \approx 1.42 \times 10^{5}

(32)

Similarly, there is no value of $K_{i}$ that would result in instability of the GCC system with the FO-SRF-PLL. The nonlinear simulation results are depicted in Figs. 17 and 18. Clearly, while the GCC system with the SRF-PLL loses stability due to the value of $K_{i}$ exceeding the range obtained in (32), the system with the FO-SRF-PLL remains stable.

Fig. 17 Results of nonlinear simulation with variation of $K_{i}$ in SRF-PLL. (a) $P$ . (b) $θ$ . (c) $V_{a b c}$ .

Fig. 18 Results of nonlinear simulation with $K_{i} = 50 ω_{0}^{2}$ in FO-SRF-PLL.

This test effectively validates the accuracy of the instability conditions by varying the control parameters of the PLL. Furthermore, it can be observed that when the system stability is constrained by the values of $K_{p}$ and $K_{i}$ in the GCC system with SRF-PLL, the stability of the system remains unaffected by the values of $K_{p}$ and $K_{i}$ in the GCC system with FO-SRF-PLL. This finding highlights the robustness of the proposed FO-SRF-PLL in maintaining system stability under varying control parameters.

B. Experimental Results

This subsection aims to evaluate the performance of the proposed FO-SRF-PLL and to verify the correctness of the theoretical analysis through experimental results. A hardware-in-the-loop (HIL) experimental platform is established, as shown in Fig. 19, and the main parameters are detailed in Table III.

Fig. 19 HIL experimental platform.

TABLE Ⅲ Main Experimental Parameters

Parameter	Value	Parameter	Value
DC-side voltage $V_{d c}$	200 V	OFA parameters $(ω_{b}, ω_{h})$	$(10^{- 2}, 10^{5})$
Filter inductor $L_{f}$	5.25 mH	PLL proportional gain $K_{p}$	0.01-50
Grid impedance $L_{g}$	3-15 mH	PLL integral gain $K_{i}$	300-5000
Grid phase voltage $V_{b}$	78 V	Current loop proportional coefficient $K_{c p}$	0.15
Switching frequency $f_{w}$	10 kHz	Current loop integral coefficient $K_{c i}$	60
Sampling frequency $f_{s}$	10 kHz	Current reference of d-axis $I_{d 0}$	10-30 A
Order of Oustaloup filter N	5	Current reference of d-axis $I_{q 0}$	0

1)　Performance Evaluation of FO-SRF-PLL

The performance of the FO-SRF-PLL is evaluated under a phase jump condition. The transient response to a phase jump of $π / 3$ radians at 100 ms is shown in Fig. 20. The PCC voltage response, q-axis voltage response, and phase angle tracking output results using the SRF-PLL and FO-SRF-PLL under the phase jump condition are shown in Fig. 20(a) and 20(b), respectively. Table IV presents the performance evaluation metrics, including locking time, overshoot, and settling time. Experimental results show that the FO-SRF-PLL exhibits faster locking time, reduced overshoot, and shorter settling time.

Fig. 20 PCC voltage, q-axis voltage, and phase angle tracking output results under phase angle jump condition. (a) SRF-PLL. (b) FO-SRF-PLL.

TABLE Ⅳ Performance Evaluation Metrics Under Phase Angle Jump Condition

Type	Locking time (ms)	Overshoot (%)	Settling time (ms)
SRF-PLL	55.2	5800	38
FO-SRF-PLL	5.6	870	6

2)　Verification of Small-signal Stability Analysis

The experimental waveforms of PCC voltage and active power of the GCC system when $K_{p}$ changes are given in the test ( $I_{d 0} = 10$ $A$ , $L_{g} = 7.5$ $m H$ , $K_{i} = 300$ ). When $K_{p}$ increases and decreases, the experimental results of the GCC system with the SRF-PLL and the FO-SRF-PLL are shown in Figs. 21 and 22, respectively. The active power output increases by 10% within 150 to 160 ms. From the experimental results, it is clear that when $K_{p}$ increases from 10 to 15 or decreases from 10 to 0.1, the GCC system with the SRF-PLL loses stability. In contrast, the GCC system with the FO-SRF-PLL remains stable even when $K_{p}$ increases from 10 to 50 or decreases from 10 to 0.01. While the GCC system using the SRF-PLL loses stability due to the change of $K_{p}$ , the GCC system with the FO-SRF-PLL can remain stable.

Fig. 21 Experimental results of GCC system when $K_{p}$ increases. (a) SRF-PLL. (b) FO-SRF-PLL.

Fig. 22 Experimental results of GCC system when $K_{p}$ decreases. (a) SRF-PLL. (b) FO-SRF-PLL.

The experimental waveforms of PCC voltage and output active power of the GCC system during variations in $K_{i}$ are presented in this test, with $I_{d 0} = 10$ $A,$ $L_{g} = 7.5$ $m H,$ $K_{p} = 10$ . The results of the GCC system with the SRF-PLL and the FO-SRF-PLL are shown in Fig. 23. It is evident from the results that as $K_{i}$ increases from 300 to 3000, the GCC system with the SRF-PLL loses stability, while the GCC system with the FO-SRF-PLL remains stable even when $K_{i}$ increases from 300 to 5000. Compared with the SRF-PLL, the GCC system with FO-SRF-PLL has a lower risk of instability when $K_{i}$ changes.

Fig. 23 Experimental results of GCC system when $K_{i}$ increases. (a) SRF-PLL. (b) FO-SRF-PLL.

The experimental results of PCC voltage and output active power of the GCC system when $I_{d 0}$ changes are given in this test, with $L_{g} = 7.5$ $m H,$ $K_{p} = 10,$ and $K_{i} = 300$ . The results of the GCC system with the SRF-PLL and the FO-SRF-PLL are shown in Fig. 24. It can be observed from the results that when $I_{d 0}$ changes, i.e., $P_{0}$ changes, the GCC system with SRF-PLL loses stability, whereas the GCC system with FO-SRF-PLL remains stable. Compared with the SRF-PLL, the stability of the GCC system with FO-SRF-PLL is less affected by changes in $P_{0}$ .

Fig. 24 Experimental results of GCC system when changing given current. (a) SRF-PLL. (b) FO-SRF-PLL.

The experimental results under varying $L_{g}$ with $I_{d 0} = 10$ A, $K_{p} = 10$ , and $K_{i} = 300$ are shown in Fig. 25. It can be observed that as $L_{g}$ increases, the instability risk of the GCC system with SRF-PLL increases, while the stability of the GCC system using FO-SRF-PLL is not affected.

Fig. 25 Experimental results of GCC system when $L_{g}$ is changed. (a) SRF-PLL. (b) FO-SRF-PLL.

In summary, the performance of the FO-SRF-PLL proposed in this paper is significantly superior to that of the SRF-PLL. The locking time and settling time are considerably shorter, and the maximum error value during step changes is reduced. Furthermore, the GCC system using SRF-PLL tends to lose stability under small disturbances when the control and operating parameters change, whereas the GCC system with FO-SRF-PLL remains stable. The experimental results verify the effectiveness of the proposed FO-SRF-PLL.

VI. Conclusion

This paper introduces an FO-SRF-PLL for accurate phase angle tracking of the terminal voltage in GCC systems. The stability conditions of GCC system with FO-SRF-PLL are derived and analyzed. Through simulation and experimental results, several useful conclusions are drawn as follows.

1) The utilization of the FO-SRF-PLL demonstrates reduced undershoot, faster response, and shorter settling time. These improved performances enable quicker and more precise tracking of the desired phase in GCC system.

2) GCC system with the SRF-PLL may become unstable due to changes in the PLL control parameters. However, the stability of GCC system with FO-SRF-PLL is less likely to be affected by variations in control parameters.

3) As the steady-state active power and the transmission line reactance increase, GCC system with FO-SRF-PLL exhibits greater stability compared with that with the SRF-PLL. Notably, even with extremely weak grid connections, GCC system with the FO-SRF-PLL remains stable.

Further research and investigation are required to explore the potential applications of FOC and FOEs in renewable energy systems and their impact on system stability. This paper serves as an initial exploration, and future work should focus on expanding the understanding of these techniques and their implementation.

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