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.
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 [
Recent advances have explored several novel applications of FOEs. For example, a high-power FO capacitor based on power converters has been proposed [
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 [
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 [
The application of fractional calculus to PLLs has also shown promising results. In [
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.
The FO-SRF-PLL is developed by applying the FOPID controller proposed in [

Fig. 1 Structural block diagram of FO-SRF-PLL.
(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.
The implementation of an FOPID controller typically requires a rational approximation. The Oustaloup filter algorithm (OFA) [
(2) |
where N is the order of the filter; ; and .
In the paper, the above FOA is used for realization of FOPID controllers, and the approximated FOPID transfer function is:
(3) |
With respect to the digital implementation, the Tustin discretization method is used for 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 [

Fig. 3 Power flow diagram of test system.
Testing scheme | Condition | Implementation detail |
---|---|---|
Conducted on test system | Under-voltage | An inductive load of Mvar connected to bus 6 at 60 ms |
Over-voltage | A capacitive load of Mvar connected to bus 6 at 60 ms | |
Load rise | Load at buses 6 and 9 increased by 100 Mvar at 60 ms | |
Utilizing a three-phase programmable voltage source | Phase jump | Phase jump of π/6 radians at 50 ms |
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 , q-axis voltage , and estimated phase under different test conditions are depicted in Figs.

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

Fig. 9 Configuration of GCC system in renewable energy systems.
The GCC system shown in
(4) |
In the paper, prefix refers to a small increment of the variable or variable vector.
(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
(6) |
(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:
(8) |
Thus, from (4) and (8), we can obtain:
(9) |
The linearization of (6) is:
(10) |
From (7), (9) and (10), we can obtain:
(11) |
The FO model of transmission lines is built as:
(12) |
Ignoring the dynamics of the grid, the linearization of (12) is:
(13) |
Or, equivalently, we can obtain:
(14) |
where ; and
Let and be the steady-state active and reactive power outputs of the GCC system, then we have:
(15) |
From (9), (11), (14), and (15), we can obtain:
(16) |
(17) |
Next, the objective is to analyze the stability of system (16).
In fractional calculus, the FO linear time-invariant (FO-LTI) system is represented as follows:
(18) |
where 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 [

Fig. 12 Stable and unstable regions of system in (18) with .
(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 [
(20) |
In this paper, the objective is to determine the parameters that render the FO-LTI system in (21) stable.
(21) |
where is given.
From theorem 1, the equivalent LTI system in terms of stability for the FO-LTI system (21) can be expressed as:
(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:
(23) |
(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).
(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.
(26) |
For , it can be proven that the GCC system employing SRF-PLL is stable if and only if conditions in (27) are satisfied.
(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 and in the FO-SRF-PLL.
The system shown in
(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 is utilized.
From (27) and (28), the stability conditions for the GCC system with the SRF-PLL are obtained.
(29) |
From (26) and (28), when the FO-SRF-PLL with is utilized, the GCC system is unstable if and only if:
(30) |
Three tests have been conducted to validate the theoretical analysis. The results regarding the impact of the PLL control parameters and on the small-signal stability of the system are presented below. Moreover, the impacts of the transmission line reactance and the steady-state active power are detailed in Supplementary Material C.
To evaluate the impact of , we have conducted the following test. The reactance of the transmission line is fixed to be , and . Thus, , and . is fixed to be (). From (29), the stability condition for the GCC system with SRF-PLL is:
(31) |
Supplementary Material D shows that there does not exist a value of that would lead to instability in the GCC system with the FO-SRF-PLL. The nonlinear simulation results are depicted in Figs.

Fig. 13 Results of nonlinear simulation with variation of in SRF-PLL. (a) . (b) . (c) .

Fig. 14 Results of nonlinear simulation with in FO-SRF-PLL. (a) . (b) . (c) .

Fig. 15 Results of nonlinear simulation with variation of in SRF-PLL. (a) . (b) . (c) .

Fig. 16 Results of nonlinear simulation with in FO-SRF-PLL.
It can be observed from Figs.
To evaluate the impact of , the following test is conducted. The system parameters are set to the same as before, and the value of is fixed to be . From (29), the stability condition for the GCC system with the SRF-PLL is:
(32) |
Similarly, there is no value of that would result in instability of the GCC system with the FO-SRF-PLL. The nonlinear simulation results are depicted in Figs.

Fig. 17 Results of nonlinear simulation with variation of in SRF-PLL. (a) . (b) . (c) .

Fig. 18 Results of nonlinear simulation with 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 and in the GCC system with SRF-PLL, the stability of the system remains unaffected by the values of and 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.
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 HIL experimental platform.
Parameter | Value | Parameter | Value |
---|---|---|---|
DC-side voltage | 200 V | OFA parameters | |
Filter inductor | 5.25 mH | PLL proportional gain | 0.01-50 |
Grid impedance | 3-15 mH | PLL integral gain | 300-5000 |
Grid phase voltage | 78 V | Current loop proportional coefficient | 0.15 |
Switching frequency | 10 kHz | Current loop integral coefficient | 60 |
Sampling frequency | 10 kHz | Current reference of d-axis | 10-30 A |
Order of Oustaloup filter N | 5 | Current reference of d-axis | 0 |
The performance of the FO-SRF-PLL is evaluated under a phase jump condition. The transient response to a phase jump of radians at 100 ms is shown in

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.
Type | Locking time (ms) | Overshoot (%) | Settling time (ms) |
---|---|---|---|
SRF-PLL | 55.2 | 5800 | 38 |
FO-SRF-PLL | 5.6 | 870 | 6 |
The experimental waveforms of PCC voltage and active power of the GCC system when changes are given in the test ( , , ). When increases and decreases, the experimental results of the GCC system with the SRF-PLL and the FO-SRF-PLL are shown in Figs.

Fig. 21 Experimental results of GCC system when increases. (a) SRF-PLL. (b) FO-SRF-PLL.

Fig. 22 Experimental results of GCC system when 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 are presented in this test, with . The results of the GCC system with the SRF-PLL and the FO-SRF-PLL are shown in

Fig. 23 Experimental results of GCC system when increases. (a) SRF-PLL. (b) FO-SRF-PLL.
The experimental results of PCC voltage and output active power of the GCC system when changes are given in this test, with and . The results of the GCC system with the SRF-PLL and the FO-SRF-PLL are shown in

Fig. 24 Experimental results of GCC system when changing given current. (a) SRF-PLL. (b) FO-SRF-PLL.
The experimental results under varying with A,, and are shown in

Fig. 25 Experimental results of GCC system when 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.
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.
References
I. Petras, Fractional-order Nonlinear Systems. Berlin: Springer Verlag, 2011, pp. 47-52. [Baidu Scholar]
M. S. Sarafraz and M. S. Tavazoei, “Passive realization of fractional-order impedances by a fractional element and RLC components: conditions and procedure,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 64, no. 3, pp. 585-595, Mar. 2017 [Baidu Scholar]
D. Pullaguram, S. Mishra, N. Senroy et al., “Design and tuning of robust fractional order controller for autonomous microgrid VSC system,” IEEE Transactions on Industry Applications, vol. 54, no. 1, pp. 91-101, Feb. 2018. [Baidu Scholar]
Y. Jiang and B. Zhang, “High-power fractional-order capacitor with [Baidu Scholar]
based on power converter,” IEEE Transactions on Industrial Electronics, vol. 65, no. 4, pp. 3157-3164, Apr. 2018. [Baidu Scholar]
S. F. Chou, X. Wang, and F. Blaabjerg, “A fractional-order model of filter inductors within converter control bandwidth,” in Proceedings of 2018 IEEE Electronic Power Grid (eGrid), Charleston, USA, Nov. 2018, pp. 1-6. [Baidu Scholar]
X. Chen, Y. Chen, B. Zhang et al., “A modeling and analysis method for fractional-order DC-DC converters,” IEEE Transactions on Power Electronics, vol. 32, no. 9, pp. 7034-7044, Sept. 2017. [Baidu Scholar]
M. A. Azghandi, S. M. Barakati, and A. Yazdani, “Passivity-based design of a fractional-order virtual capacitor for active damping of multi-paralleled grid-connected current-source inverters,” IEEE Transactions on Power Electronics, vol. 37, no. 7, pp. 7809-7818, Jul. 2022. [Baidu Scholar]
A. Zafari, M. Mehrasa, S. Bacha et al., “A robust fractional-order control technique for stable performance of multilevel converter-based grid-tied DG units,” IEEE Transactions on Industrial Electronics, vol. 69, no. 10, pp. 10192-10201, Oct. 2022. [Baidu Scholar]
M. Badoni, A. Singh, S. Pandey et al., “Fractional-order Notch filter for grid-connected solar PV system with power quality improvement,” IEEE Transactions on Industrial Electronics, vol. 69, no. 1, pp. 429-439, Jan. 2022. [Baidu Scholar]
B. Babes, S. Mekhilef, A. Boutaghane et al., “Fuzzy approximation-based fractional-order nonsingular terminal sliding mode controller for DC-DC buck converters,” IEEE Transactions on Industrial Electronics, vol. 37, no. 3, pp. 2749-2760, Mar. 2022. [Baidu Scholar]
B. Long, W. Mao, P. Lu et al., “Passivity fractional-order sliding-mode control of grid-connected converter with LCL filter,” IEEE Transactions on Industrial Electronics, vol. 38, no. 6, pp. 6969-6982, Jun. 2023. [Baidu Scholar]
B. Long, P. Lu, K. Chong et al., “Robust fuzzy-fractional-order nonsingular terminal sliding-mode control of LCL-type grid-connected converters,” IEEE Transactions on Industrial Electronics, vol. 69, no. 6, pp. 5854-5866, Jun. 2022. [Baidu Scholar]
M. K. Behera and L. C. Saikia, “An improved voltage and frequency islanded microgrid using BPF based droop control and optimal third harmonic injection PWM scheme,” IEEE Transactions on Industry Applications, vol. 58, no. 2, pp. 2483-2496, Apr. 2022. [Baidu Scholar]
R. Shah, R. Preece, and M. Barnes, “The impact of voltage regulation of VSC-HVDC on power system stability,” IEEE Transactions on Energy Conversion, vol. 33, no. 4, pp. 1614-1627, Dec. 2018. [Baidu Scholar]
S. Ghasemi, A. Tabesh, and J. Askari-Marnani, “Application of fractional calculus theory to robust controller design for wind turbine generators,” IEEE Transactions on Energy Conversion, vol. 29, no. 3, pp. 780-787, Sept. 2014. [Baidu Scholar]
A. Beddar, H. Bouzekri, B. Babes et al., “Experimental enhancement of fuzzy fractional order [Baidu Scholar]
controller of grid connected variable speed wind energy conversion system,” Energy Conversion and Management, vol. 123, no. 1, pp. 569-580, Sept. 2016. [Baidu Scholar]
M. E. Meral and D. Çelík, “A comprehensive survey on control strategies of distributed generation power systems under normal and abnormal conditions,” Annual Reviews in Control, vol. 47, pp. 112-132, Dec. 2018. [Baidu Scholar]
V. Khatana and R. Bhimasingu, “Review on three-phase PLLs for grid integration of renewable energy sources,” in Proceedings of 2017 14th IEEE India Council International Conference (INDICON), Roorkee, India, Dec. 2017, pp. 1-6. [Baidu Scholar]
S. Golestan, J. M. Guerrero, and J. C. Vasquez, “Three-phase PLLs: a review of recent advances,” IEEE Transactions on Power Electronics, vol. 32, no. 3, pp.1894-1907, May 2016. [Baidu Scholar]
R. El-Khazali and W. Ahmad, “Fractional-order phase-locked loop,” in Proceedings of 2007 9th International Symposium on Signal Processing and Its Applications, Sharjah, United Arab Emirates, Feb. 2007, pp. 1-4. [Baidu Scholar]
M. C. Tripathy, D. Mondal, K. Biswas et al., “Design and performance study of phase-locked loop using fractional-order loop filter,” International Journal of Circuit Theory and Applications, vol. 43, no. 6, pp. 776-792, Jun. 2015. [Baidu Scholar]
B. T. Krishna, “Fractional calculus-based analysis of phase locked loop,” in Proceedings of 2012 International Conference on Signal Processing and Communications (SPCOM), Bangalore, India, Aug. 2012, pp. 1-5. [Baidu Scholar]
R. El-Khazali, W. Ahmad, and Z. A. Memon, “Noise performance of fractional-order phase-locked loop,” in Proceedings of 2007 IEEE International Conference on Signal Processing and Communications, Dubai, United Arab Emirates, Nov. 2007, pp. 572-575. [Baidu Scholar]
I. Podlubny, “Fractional-order systems and [Baidu Scholar]
-controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208-214, Jan. 1999. [Baidu Scholar]
A. Oustaloup, F. Levron, B. Mathieu et al., “Frequency-band complex noninteger differentiator: characterization and synthesis,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no.1, pp. 25-39, Jan. 2000. [Baidu Scholar]
C. A. Monje, Y. Chen, B. M. Vinagre et al., Fractional-Order Systems and Controls. London: Springer, 2010, pp. 193-198. [Baidu Scholar]
D. Matignon, “Stability result on fractional differential equations with applications to control processing,” Computational Engineering in Systems Applications, vol. 2, no. 1, pp. 963-968, Jul. 1996. [Baidu Scholar]
M. S. Tavazoei and M. Haeri, “A note on the stability of fractional order systems,” Mathematics and Computers in Simulation, vol. 79, no. 5, pp. 1566-1576, Jan. 2009. [Baidu Scholar]