Global Stability Analysis for Synchronous Reference Frame Phase-Locked Loops

This article analyzes the global stability of synchronous reference frame phase-locked loops (SRF-PLLs) from a large signal viewpoint. First, a large-signal model of SRF-PLL is accurately established, without applying any linearization method. Then, according to the phase portrait and Lyapunov argument, the global performance of SRF-PLL is discussed in the nonlinear frame. Compared with the small-signal analysis methods, the proposed analysis, not relying on the small-signal model and linearization method, provides a global discussion of the SRF-PLL performance. The contributions of this article are as follows. First, it is found that SRF-PLL has infinite equilibrium points, including stable points and saddle points. Second, it provides a way to divide the global region of SRF-PLL into many small regions. In each small region, the SRF-PLL only has one stable equilibrium point. And for any initial states $(\tilde{\theta }(t_0),\tilde{\omega }(t_0))$ in a small region, all states $(\tilde{\theta }(t),\tilde{\omega }(t)),t>t_0$ will remain in this small region, and SRF-PLL will converge to the unique stable equilibrium point of this small region. Third, by dividing the global region of SRF-PLL into many small regions, it is found that when the frequency of grids varies largely, the SRF-PLL will converge to a new equilibrium point that is far away from the original equilibrium point. It is the reason why the frequency convergence of SRF-PLL has many oscillations and SRF-PLL has a rather slow dynamic, when the frequency changes largely. The experimental results are provided to verify the proposed global stability analysis of SRF-PLL.

SRF-PLL estimates the phase and frequency of the grid voltage, and it provides the information to achieve grid synchronization of grid-tied converters. The structure of SRF-PLL is shown in Fig. 1, and it contains three parts, including the phase detector (PD), the loop filter (LF), and the voltage-controlled oscillator (VCO) [10]. In details, the output of the PD contains the phase error information. LF is a PI controller and it attenuates the high-frequency components from the PD output. VCO generates the estimated phase, which follows the actual phase of the grid voltage.
Recently, it is found that SRF-PLL causes an important effect on the stability of power grids, and several papers have discussed the relationship between SRF-PLL and grid stability [11]- [16]. Generally, most existing methods simplify SRF-PLL as a smallsignal model [10] and discuss the SRF-PLL stability in the linear frame. However, the analysis of SRF-PLL is only a local result, and these methods are only available for the case where the difference between the initial estimated frequencyω and grid frequency ω is sufficiently small. In addition, the SRF-PLL is used for grid synchronization in some varying-frequency power systems, such as more electric aircraft (MEA) grids. In the varying-frequency power grids, the frequency changes largely. For example, the frequency range in MEA grids is from 360 to 800 Hz. Hence, large frequency jumps should be discussed for the SRF-PLL in these applications [17], [18]. Unfortunately, these methods fail to give an accurate analysis for SRF-PLL in the varying-frequency power grids.
Furthermore, some literature, such as [19] and [20], provide a large-signal analysis of the SRF-PLL. The arguments in these methods imply that SRF-PLL converges to the equilibrium point (0, 0), globally. However, when the frequency changes largely, the experiment results of SRF-PLL show that the SRF-PLL converges to a different equilibrium point and suffers a slow dynamic. This experimental phenomenon cannot be explained by these methods.
In this article, a global performance analysis of SRF-PLL is provided in the nonlinear frame. First, a large-signal model is established according to the structure of SRF-PLL, which does not rely on the linearization. Then, the phase portrait and Lyapunov argument are proposed to discuss the global performance of the SRF-PLL. Experimental results further confirm the proposed theoretical analysis. The main contributions of this article are as follows.
2) It provides a way to divide the global region of the SRF-PLL into many small regions. In each small region, the SRF-PLL only has one stable equilibrium point. And for any initial states (θ(t 0 ),ω(t 0 )) in a small region, all states (θ(t),ω(t)), t > t 0 will still remain in this small region, and SRF-PLL will converge to the unique stable equilibrium point of this small region.
3) By dividing the global region of SRF-PLL into many small regions, it is found that when the frequency of grids varies largely, the SRF-PLL will converge to a new equilibrium point that is far away from the original equilibrium point. It is the reason why the estimated frequency of SRF-PLL has many oscillations and a rather long transient process, when the frequency changes largely.
The rest of this article is organized as follows. Section II briefly reviews the small-signal analysis method for SRF-PLL. The large-signal model of SRF-PLL is established in Section III. Following, global performance analysis of SRF-PLL is given in Section IV. Experimental tests are, then, provided in Section V. Finally, Section VI concludes this article.

II. REVIEW OF THE CLASSICAL SMALL SIGNAL METHOD
In this section, the small-signal analysis method of SRF-PLL is briefly reviewed. For ideal power grids, the grid voltages v a , v b , and v c are where V , ω, and θ are the amplitude, frequency, and phase angle, respectively. Applying the abc-dq transformation whereθ is the estimation of θ, and the grid voltages (1) (6) is strongly nonlinear. To simplify the model of SRF-PLL, a linearization assumption is provided as (A1) : The initial estimation error of the phase angle θ −θ is sufficient small.
With this assumption, (6) is linearized as From (6) and (7), it is carried out Hence, when satisfying the assumption (A1), the SRF-PLL in Fig. 1 is linearized, as shown in Fig. 2. And the small-signal model of the SRF-PLL is established as [10] E θ (s) = s 2 s 2 + k p s + k i (9) where k p and k i are the parameters of PI.
According to the small-signal model, such as (9), the smallsignal analysis discusses the SRF-PLL performance. However, it is limited by the linearization condition (A1), and the analysis of SRF-PLL is only a local result. Meanwhile, it is difficult to confirm whether a frequency/phase jump is suitable for the small-signal analysis from the rigorous theoretical viewpoint. And the small-signal analysis method fails to give an accurate analysis of SRF-PLL in the varying-frequency power grids.

III. LARGE-SIGNAL MODEL FOR THE SRF-PLL
A small-signal model is based on the linearization method, and hence, it only describes the performance of SRF-PLL in a local region. To analyze the global performance of SRF-PLL, a large-signal model is established as follows. According to the LF and VCO parts in SRF-PLL, it is deduced thatθ where ω c is a constant value. It implies thaṫ Definingθ =θ − θ andω =ω − ω, and notinġ Submitting (6) and (14) into (13) leads tȯ This is the large-signal model for SRF-PLL. Compared with the small-signal model, the proposed large-signal model, not relying on linearization and (A1), describes the performance of SRF-PLL, globally. Observing (15), the large-signal model of SRF-PLL is strongly nonlinear. In the next section, the global performance of SRF-PLL is analyzed based on the proposed model (15).

IV. LARGE-SIGNAL ANALYSIS
For nonlinear systems, the phase portrait is a power tool to analyze stability, globally. In this section, the global performance of SRF-PLL is discussed according to the phase-portrait analysis. Based on the large-signal model (15), the phase portrait of SRF-PLL is shown in Fig. 3, with k p = 92 and k i = 4232. And the step-by-step way is provided in the Appendix to draw the Fig. 3. From Fig. 3, the following can be found.
2) In Fig. 3, the blue lines divide the global region of SRF-PLL into many small regions. The Appendix provides a way to draw these special lines. For each saddle point, there exist two special lines that converge to this saddle point, although the saddle points are unstable. And these special lines divide the global region of SRF-PLL into many small regions. In each small region, the SRF-PLL only has one stable equilibrium point. And for any initial states (θ(t 0 ),ω(t 0 )) in a small region, all states (θ(t),ω(t)), t > t 0 will remain in this small region, and SRF-PLL will converge to the unique stable equilibrium point of this small region.
3) When the frequency of grids varies largely, the states of SRF-PLL will belong to a new small region. Hence, SRF-PLL will converge to the unique stable equilibrium point of this new small region. This stable point is far away from the original equilibrium point, and the frequency convergence of SRF-PLL has many oscillations. In this case, the SRF-PLL has a rather long transient process.
In the following, a strict theoretical analysis is provided to discuss the global performance of SRF-PLL.
According to the definition of equilibrium points shown in the Appendix, the equilibrium points for the SRF-PLL (15) are the real roots of the equation Solving the algebraic equations (16) implies that the SRF-PLL has infinite equilibrium points and these points are located at where If n = 2kπ, k = 0, ±1, ±2, . . . , it yields cos(nπ) = 1. Thus, a direct calculation implies that the matrix A n has two eigenvalues with negative real part. This means that the equilibrium points (2kπ, 0), k = 0, ±1, ±2, . . . , are stable, and they are called stable points. In Fig. 3, points (0, 0) and (2π, 0) are stable points.
According to the definition of equilibrium points, it concludes that when the initial states of SRF-PLL are on stable points or saddle points, the states of SRF-PLL will remain at this equilibrium points for all future time.
According to the abovementioned analysis of saddle points, the SRF-PLL in the small region of saddle points is unstable. It implies that the initial states in this region will go far away the saddle points, while there still exists and only exists two special lines around each saddle point. When the initial states are on the special lines, SRF-PLL will converge to the saddle point [21]. In Fig. 3, the blue lines are these special lines.
Moreover, these special lines divide the global region of SRF-PLL into infinite small regions. In Fig. 3, Regions I and II are highlighted as an example of different regions. When the initial states fall in different regions, SRF-PLL will converge to different stable points (2kπ, 0), k = 0, ±1, ±2, . . .. In each small region, a strict Lyapunov analysis is proposed for SRF-PLL to show the stability as follows.
It is defined as the coordination transformatioñ From (15), it is expressed aṡ Consider the Lyapunov function It is easy to verify that U (θ 2k ,ω 2k ) is positive in the compact From (19) and (20), it is deduced thaṫ Hence, (21) is negative semidefinite, and SRF-PLL in this region is stable. In the following, the steady-state errors of SRF-PLL are analyzed by LaSalles theorem.
Hence, S = {(θ 2k ,ω 2k ) ∈ Ω|ω 2k = 0}. Let (θ 2k (t),ω 2k (t)) be a solution that belongs identically to S It results inθ 2k ≡ 0. Therefore, the only solution that can stay identically in S is the trivial solution (θ 2k (t),ω 2k (t)) ≡ 0. According to the LaSalle's theorem [21], it concludes that for any initial states (θ 2k (t 0 ),ω 2k (t 0 )) ∈ Ω, the solutions (θ 2k (t),ω 2k (t)) approach 0 as t → ∞. Hence, the estimations of SRF-PLL converge to the stable points (2kπ, 0), k = 0, ±1, ±2, . . . , as From the abovementioned analysis, the global region of SRF-PLL is divided into infinite stable small regions. In each small region, the SRF-PLL has an unique stable equilibrium point (2kπ, 0), k = 0, ±1, ±2, . . .. And for any initial states (θ(t 0 ),ω(t 0 )) in such a small region, all states (θ(t),ω(t)), t > t 0 will remain in this small region, and SRF-PLL will converge to the unique stable equilibrium point of this small region. Moreover, for any state (θ(t 0 ),ω(t 0 )) of the SRF-PLL in a small region, the distance between this state point and the boundary of the small region can be applied to evaluate the SRF-PLL performance. When the frequency/phase jump is larger than the distance, the states of SRF-PLL will jump to a new small region, and converge to a new equilibrium point. In particular, when the frequency jumps largely, the states of SRF-PLL belong to a new small region, and the equilibrium point of the new region is far away from the original one. In this case, although SRF-PLL converges to the new stable equilibrium point, the frequency convergence of SRF-PLL has many oscillations and suffers a long transient process. The experimental results provide the detail discussion of this phenomenon in the next section.

V. EXPERIMENTAL RESULTS
In this section, SRF-PLL is implemented on a DSP 28335based platform, and the abovementioned analysis is verified by the following experimental results. The parameters of SRF-PLL are selected as k p = 92 and k i = 4232. The initial states of In Cases A and B, the initial states (θ(t 0 ),ω(t 0 )) are (0,20) and (0.5π, −20), respectively, and these initial states are all in Region I. From Figs. 4 and 5, it is observed that all states (θ(t),ω(t)), t > t 0 in Cases A and B remain in the Region I, and SRF-PLL converges to (0, 0), which is the unique stable equilibrium point of Region I. It confirms the conclusion in Section IV. Meanwhile, observing the phase portrait in Fig. 5, it is found that the frequency convergence has no oscillations, which implies that the dynamics in Cases A and B are fast according to the theoretical analysis in Section IV. It is verified by observing the transient time of Cases A and B, as shown in Fig. 4.

B. Part 2: The Initial States are in Region II
In this part, SRF-PLL is tested in Cases C and D, and the initial states of these cases are all in the Region II. Figs. 6 and  7 are the experimental results and the phase portrait of Cases C and D.
In Cases C and D, the initial states (θ(t 0 ),ω(t 0 )) are (2π, 20) and (2.5π, −20), respectively, and these initial states are in the Region II. From Figs. 6 and 7, it is found that all states (θ(t),ω(t)), t > t 0 in Cases C and D are in the Region II, and SRF-PLL converges to (2π, 0), which is the unique stable   Fig. 7, it is observed that the frequency convergence has no oscillations in this part, which implies that the dynamics in Cases C and D are fast according to the theoretical analysis in Section IV. It is further confirmed by Fig. 6.

C. Part 3: The Initial States are Closed but in Different Regions
In this part, SRF-PLL is implemented in Cases E and F. The initial states are closed but in different regions. Figs. 8 and 9 are the experimental results and the phase portrait of this part, respectively.
In Case E, the initial state (θ(t 0 ),ω(t 0 )) is (π − 0.01, 0), and it is in the Region I. In Case F, the initial state (θ(t 0 ),ω(t 0 )) is (π + 0.01, 0), and it is in the Region II. The two initial states of Cases E and F are closed but in the different small regions. From Fig. 8 and 9, it is observed that all states (θ(t),ω(t)), t > t 0 in Case E belong to the Region I, whereas all states (θ(t),ω(t)), t > t 0 in Case F remain in the Region II. SRF-PLL in Case E converges to the unique stable equilibrium point (0, 0) in the Region I, whereas SRF-PLL in Case F converges to the unique stable equilibrium point (2π, 0) in the Region II, although the initial states in Cases E and F are rather closed, and the distance between them is only 0.02 rad. The experimental results further confirm the theoretical analysis in Section IV. The phase portrait in Fig. 9 shows that the frequency convergence in Cases E and F has no oscillation. From the conclusion in the last section, the SRF-PLL dynamics in Cases E and F are fast, which is verified by the experimental results in Fig. 8.

D. Part 4: The Initial State is Far Away From the Unique Stable Equilibrium Point of the Small Region
In this part, SRF-PLL is tested in Case G. Figs. 10 and 11 are the phase portrait and the experimental results of this part, respectively.
In Case G, the initial state (θ(t 0 ),ω(t 0 )) is (0, −45). The initial state is in Region VI and far away from the unique stable equilibrium point (−8π, 0) in Region VI. From Figs. 10 and 11, it is observed that all states (θ(t),ω(t)), t > t 0 in Case G belong to Region VI, and SRF-PLL converges to the unique stable equilibrium point (−8π, 0) in Region VI. It confirms the conclusion of Section IV. Meanwhile, from Fig. 10, it is found that the frequency convergence in Case G occurs oscillations. It will result in slow dynamics, according to the analysis in Section IV. Observing Fig. 11, the dynamic of the SRF-PLL is slow, which further confirms the conclusion in last section.
Remark: According to theoretical part, the states of SRF-PLL will remain at the saddle points for all future time when the states first stand on its saddle point. Meanwhile, there exists two special lines for each saddle point, and the SRF will converge to the saddle points when its initial states are on these special lines. However, due to the truncation errors of the experiment platform, the initial states of SRF-PLL are difficult to exactly stand on saddle points or special lines, and its experiments are hard to be provided in practical cases.

VI. CONCLUSION
In this article, the global performance of SRF-PLL was analyzed in the nonlinear framework. The large-signal model of SRF-PLL was accurately established, and phase portrait and Lyapunov argument was proposed to analyze the global stability of SRF-PLL. It was found that SRF-PLL had infinite equilibrium points, including stable points and saddle points. Furthermore, a way was provided to divide the global region of SRF-PLL into many small regions. Each small region only had a stable equilibrium point. And for any states in a small region, the states of SRF-PLL still belonged to this small region for all future time, and SRF-PLL will converge to the unique stable point of this small region. In addition, it was found that when the frequency of grids varied largely, the SRF-PLL converged to a new equilibrium point that was far away from the original equilibrium point. In that case, there were many oscillations for frequency dynamic, and the SRF-PLL had a rather long transient process. Experimental tests were also provided to verify the validity of the theoretical discovery.

APPENDIX
In this section, the related nonlinear system theory is introduced, including the definitions of phase portraits, equilibrium points, stable points, and saddle points. Then, the step-by-step way is provided to draw the phase portraits of SRF-PLL.
Consider a second-order nonlinear systeṁ Let x(t) = (x 1 (t), x 2 (t)) be the solution of (26) that starts at a certain initial state x(0) = (x 1 (0), x 2 (0)). The locus in the x 1 − x 2 plane of the solution x(t) for all t ≥ 0 is a curve that passes through the point x(0). The x 1 − x 2 plane is usually called the phase plane. The family of all solution curves in x 1 − x 2 plane is called the phase portrait of (26).
A point x = x * is said to be an equilibrium point of (26) if it has the property that whenever the state of the system starts at x * , it will remain at x * for all future time. The equilibrium points of (26) are the real roots of the equation Let p = (p 1 , p 2 ) be an equilibrium point of (26). It is defined that y 1 = x 1 − p 1 and y 2 = x 2 − p 2 . Under a sufficiently-small neighborhood the equilibrium point and the nonlinear system (26) is linearized as where When all the eigenvalues of the matrix A have negative real part, the equilibrium point p is called the stable point. When the matrix A has an eigenvalue with positive real part, and an eigenvalue with negative real part, the equilibrium point p is called the saddle point. In the following, a step-by-step way is shown how to obtain the phase portraits of SRF-PLL.
The lines by solving (34) and (35) previously converge to saddle points and divide the global region of SRF-PLL into many small regions. In such a small region, the SRF-PLL only has one stable equilibrium point. For any initial states (θ(t 0 ),ω(t 0 )) in a small region, all states (θ(t),ω(t)), t > t 0 in this case will remain in this small region, and SRF-PLL will converge to the unique stable equilibrium point of this small region, which is proven in Section IV. In addition, (34) and (35) are related to the SRF-PLL's parameters k p and k i . Hence, the shapes and the sizes of the small regions are related with the parameters of SRF-PLL.