1. Introduction
The single-phase PLL represents a significant task in many engineering applications, including various types of grid-connected single-phase power converters. Namely, accurate and fast frequency and phase angle estimation of the grid voltage is required, which needs to operate with the PLL input signals contaminated by higher harmonics, voltage dips, and also frequency and phase angle variations. Consequently, several comprehensive single-phase PLL survey papers have been published [
1,
2,
3], in which major PLL design problems and issues are reviewed and presented.
1.1. Motivation
The main motivation behind the work outlined in this paper emerges from the possibility to improve significantly the estimation precision of existing FFPLL solutions. Also, the importance of the FFPLL solutions is based on several analyses, which show that implementation of frequency non-adaptive FFPLL solutions, when compared to conventionally used frequency adaptive PLL solutions, introduces significant improvements in resulting stability, maximum response speed, and phase-locked loop robustness of operation.
Namely, the main drawback of existing FFPLL solutions comprises the approximated estimated phase angle compensation, contrary to the proposed solution, which is based on the analytically accurate compensation factor.
1.2. Literature Review
Generally, the single-phase PLL solutions can be divided into two main groups–power PLL [
3] and PLL based on different orthogonal signal generators (OSG) [
1]. Power PLL algorithms represent simple and effective solutions, which, however, suffer from the significant double main frequency component, which results in the substantially reduced PLL response speeds that need to be tuned. Orthogonal signal generator based solutions, however, result in much higher response speed, and they can operate with a DC offset present at the PLL input.
There is a wide range of different OSG filters and techniques [
1] proposed in the literature, which is outside the scope of this paper. However, one of the most commonly used OSG algorithms, SOGI [
4], represents the basis of the FFPLL solution proposed in this paper. Namely, SOGI based applications are commonly used as adaptive resonant frequency filters fit for the OSG, which can also successfully be applied in the case when a DC offset is present at the PLL input [
4,
5]. However, the fact that the single-phase PLL closed-loop algorithm operates with an adaptive frequency OSG filter results in a non-linear PLL operation, which introduces difficulties in parameter tuning in order to enable stable and fast PLL operation.
Consequently, in order to avoid the nonlinear adaptive frequency SOGI filter application, single-phase PLLs are proposed using OSG with a fixed frequency tuned SOGI [
6,
7,
8,
9,
10]. In [
9], an OSG is implemented based on the fixed-frequency SOGI, with the accurate orthogonal voltage amplitude and phase angle corrections, which are necessary because of the estimation error caused by the fixed frequency SOGI tuning. However, in [
9] a complex input signal frequency estimation method is proposed, when compared to other different FFPLL solutions.
In [
8], a detailed analysis of the FFPLL structure and dynamics is presented, while in [
7] the original FFPLL structure is outlined. Also, in [
8] a modification of the basic FFPLL [
7] structure is proposed, with the increased PLL frequency and phase angle estimation speed. In [
10], the FFPLL based estimator is presented, used to separate the positive and negative sequence components in the non-symmetrical PLL input signals.
Regarding the application of the FFPLL algorithms in radio frequency (RF) and micro-wave applications, it is limited by the features of the employed control platform. Namely, in order to implement an FFPLL based on a digital signal processing (DSP) in an RF application, a specialized RFSoC platform [
11] could be employed. However, there is a possibility of an analogue FFPLL application [
12], which can overcome shortcomings of a DSP based solution.
1.3. Contribution and Paper Organization
In this paper, the modification of the original FFPLL [
7] structure is proposed, with the accurately calculated phase angle estimate correction value, as opposed to [
7] in which an approximation is employed. Namely, in conventional FFPLL applications [
7] phase compensation value is approximated for the estimated input signal frequency value close to fixed resonant frequency of the employed SOGI term, while the new proposed solution comprises an analytically calculated accurate phase compensation factor. In this way, accurate operation of the FFPLL for the much wider differences between the input signal frequency and SOGI term fixed resonant frequency value, which was not the case in conventional FFPLL solutions. Also, in this paper, the FFPLL structure is analyzed to operate with a DC offset present at the PLL input together with the PLL based on the input signal positive sequence separation, which is not the case in any of the existing FFPLL structures.
This paper comprises six sections. In
Section 2, the existing FFPLL structures are outlined and compared. In
Section 3, the improved FFPLL structure is proposed, including the modification, which enables FFPLL operation with the DC offset at the PLL input. In
Section 4 the simulation results are presented, while in
Section 5 the experimental tests are outlined.
Consequently, the problem statement comprises an effort to improve the phase angle estimation accuracy in the complete input signal frequency range for the existing FFPLL solutions, which are based on several analyses [
7,
8] dynamically superior to conventionally used frequency adaptive SOGI based PLLs.
2. Fixed-Frequency PLL
In this section, the existing FFPPL solutions are presented and analyzed. Namely, the FFPLL is derived from the single-phase PLL with the adaptive frequency SOGI filter used for the orthogonal signal generation, which is outlined in
Figure 1.
In
Figure 1 U represents the PLL input,
and
orthogonal components generated by SOGI,
Ks the SOGI parameter,
ωest and
θest the estimated input signal frequency and phase angle,
Uα,
Uβ,
Ud, and
Uq the PLL input and auxiliary signals,
Kp and
Ki the PLL parameters, and
ωff the estimated frequency feed-forward value. The SOGI parameter value
Ks = 2 is commonly used, while the PLL parameters
Kp and
Ki are commonly designed by the symmetrical optimum technique [
2].
However, in [
8] a shortcoming of the adaptive frequency PLL in
Figure 1 is analyzed, caused by the nonlinear SOGI filter operation, which restricts the resulting PLL dynamics. Consequently, the FFPLL is proposed [
7,
8] in which the SOGI filter operates with a fixed frequency, resulting in linear orthogonal signal generation. Namely, the FFPLL is designed based on the following SOGI Equations (1) and (2), derived from the structure outlined in
Figure 1a. Namely, Equations (1) and (2) represent the basis for the derivation of the FFPLL operating equations, which are outlined in the following part of the paper.
As it was shown in [
1], for
s =
jωest is equal to
, while
is orthogonal with
. However, for the input signal frequency
ω ≠ ωest this is not the case, which is of special interest for the FFPLL applications.
Namely, in order to avoid the PLL operation with the adaptive SOGI filter, the SOGI rated frequency is fixed to the reference value
ωn (usually equal to 2π50 rad/s), with the corresponding PLL structure outlined in
Figure 2, which includes the fixed-frequency SOGI filter (FFSOGI) [
7,
8].
Figure 2 presents the FFPLL1 structure, where
ωn represents the rated SOGI frequency, while
θffpll represents the resulting estimated phase angle value. The main features of the FFPLL1 are outlined by analyzing the following Equations (3) and (4) of the FFSOGI output signals in
Figure 2.
In (3), Gα(s) represents the transfer function from the PLL input to the α axis output, while Gβ(s) represents the transfer function from the PLL input to the β axes output. By analyzing (3) and (4), it can be concluded that for s = jωest and are orthogonal, which enables the PLL to estimate successfully the PLL input signal frequency. However, there is a phase angle error Δφ (5) between and , which results in the erroneous PLL output phase angle estimate θest, which is consequently corrected.
Consequently, for
ωest ≅
ωn the (5) is in [
7,
8] approximated by
Finally, the resulting FFPLL1 phase angle estimate is equal to θffpll = θest − Δφ. However, the shortcoming of (6) is that it is not accurate for ωest, which differs significantly from ωn.
In order to avoid the aforementioned phase angle compensation (6), the FFPLL2 [
8] structure is proposed, outlined in
Figure 3, which according to [
8] statically and dynamically corresponds to the derivative element (DE) based single-phase PLL [
13].
In
Figure 3,
Vcos and
Vsin represent unity orthogonal signals, generated by the FFSOGI for the input signal cos(
θest). However, although FFPLL2 in
Figure 3 generates accurate frequency and phase angle estimates for
ωest ≠ ωn, it is more complex to implement.
In the next section, the improved FFPLL1 structure is proposed.
6. Conclusions
In this paper, an improved fixed frequency PLL is presented, with the proposed contribution comprising the modified algorithm for the compensation of the estimated phase angle value, which is typically required by an FFPLL. The analysis outlined in the paper shows that the novel FFPLL enables accurate phase angle estimation for any combination of the fixed FFSOGI resonant frequency value ωn and estimated input signal frequency ωest, while the conventional FFSOGI operates accurately only for ωn = ωest, with the phase angle estimation error increasing with the difference between ωn and ωest. Also, the corresponding FFPLL parameter tuning procedure is proposed and tested by a series of simulation and experimental tests. Three different FFPLL applications are examined: (i) single-phase PLL with no DC offset at the input, (ii) single-phase PLL designed to compensate a DC offset at the input, and (iii) PLL designed to separate a positive sequence component and to estimate its phase angle and frequency. For all three cases, corresponding simulation and experimental tests were performed, for three sets of FFPPL parameters, and for step variations of the input signal frequency, phase angle, and a DC offset. Both simulation and experimental tests verified the proposed method’s dynamic and static performance, with improved dynamic performance compared to the conventional adaptive filter based single-phase PLL applications. In a global context, by the method outlined in the paper, the existing frequency non-adaptive FFPLL algorithms (which are dynamically superior to conventional frequency adaptive PLL solutions) were further improved, by enabling analytically accurate phase angle estimation in the simplest FFPLL1 case.