Synchronization Stability Enhancement of Grid-Following Converter Under Inductive Power Grid

The loss of synchronization (LOS) is a main issue for dynamic behavior of grid-following converter (GFC) under inductive power grid. This letter reveals the underlying mechanism of insufficient damping or even negative damping during grid disturbance, and analyzes how the initial speed of angle-difference between PLL output and grid voltage affects the synchronization stability. Then a synchronization stability enhancement scheme to add a positive damping and a reverse angular frequency during transient process is proposed in this letter.


I. INTRODUCTION
W ITH the rapid increase of renewable energy sources, the active and reactive power can be injected into the grid through the grid-following converter (GFC) [1]. Differing from the synchronous generator (SG), the synchronization dynamic behavior of GFC is highly affected by the phase-locked loop (PLL), which may cause the loss of synchronization (LOS) due to insufficient damping [2].
Some improved PLL structures during grid disturbance have been published previously, such as the frozen PLL [3], the adaptive parameter PLL [4] and the first-order PLL [5]. However, the frozen PLL lacks accurate phase angle detection [4]. The adaptive parameter PLL and the first-order PLL both focus on increasing the damping ratio. Nevertheless, modifying PLL parameters may cause some resonance issues [6]. The logic transitions and threshold values of first-order PLL are complicated Manuscript  to design [7]. And it is also worth analyzing other methods to further enhance the synchronization stability besides changing damping ratio. This letter analyzes the underlying mechanism of insufficient damping or even negative damping for GFC under inductive power grid. In addition, this letter notices that the initial speed of angle-difference between PLL output and grid voltage is another important link to affect synchronization stability. To this end, an improved synchronization stability enhancement scheme related to the damping coefficient and initial angle speed is presented in this letter. Fig. 1 illustrates the topology of GFC connected with weak grid. U pcc and I pcc represent the three-phase voltage and current at the point of common coupling (PCC). U gcp represents the voltage at the grid connection point (GCP). Z g = R g +jω g L g is the grid impedance. L f is the filter inductance. The dc-link voltage V dc is assumed to be constant in this letter.

II. LOS MECHANISM FOR GRID-FOLLOWING CONVERTER UNDER INDUCTIVE POWER GRID
The GFC relies on the PLL to implement the grid synchronization, and the block diagram of the conventional synchronous reference frame PLL (SRF-PLL) is also depicted in Fig. 1, where the K p and K i are the proportional integral gains, ω g = 100π rad/s is the fundamental angular frequency. The dynamics of LOS lies in the low-frequency range [5], while the bandwidth of current controller is usually high, thus the synchronization stability of the GFC is dominated by the dynamics of the PLL. 0885-8969 © 2023 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information. The PLL output angle θ pll is equal to PCC voltage angle θ pcc at the steady state, while θ pll ࣔθ pcc in the transient process. It can define the angle difference δ between PLL output and GCP voltage as shown in (1), and depict the voltage-angle curves as shown in Fig. 2, to characterize the phase-swing behavior of GFC. The steady-state and max-deviation of δ can be calculated in (2). Note that the LOS mechanism and synchronization stability enhancement under resistive dominated grid have been analyzed in [8]. For example, GFC may lose the synchronism under smaller grid voltage and larger grid resistance. This letter pays more attention on inductive dominated grid. When the grid impedance is pure inductive, i.e., R g = 0, the steady-state angle difference δ 0 is always positive.
When the operating point of the converter-grid interconnected system changes, for example, changes from the point a to point b, the angle difference δ will move to a new steady state at δ 0 , and the trajectory of δ can be seen as the purple solid curve in Fig. 2. Due to the inertial effect of integral loop, δ will continue to move. Due to the damping effect of proportional loop, δ will converge to δ 0 after several phase-swing periods [5]. However, when the angle overshoot exceeds the δ max , the GFC will lose synchronization stability as shown in the purple dotted curve.
It can be found that the GFC may occur LOS when the new steady-state angle difference δ 0 is larger than the initial one δ i under weak inductive grid. There are three cases with increased angle difference:

A. Compared With Synchronous Generator
The traditional swing equation of SG is as follow, where P T is mechanical power, P E is electromagnetic power, J and D are inertia and damping coefficient.
The voltage-angle curve of GFC is similar to the power-angle curve of SG. However, the PLL-based system has two additional characteristics, i.e., maybe exist negative damping and the initial angle speed. Fig. 3 depicts the phase portrait of (3) to clearly describe the influence of different damping coefficient D and initial angle speed δ i , where the P T = 394.36 W, P E = 563.38sinδ W, δ i = 0, δ 0 = 0.78 rad, δ max = 2.36 rad, J = 2.
According to Fig. 3(a), it can be found that: 1) when there is no damping, i.e., D = 0, the system is always in dynamic process; 2) when with the positive damping, i.e., D = 10, the phase portrait will converge to steady state at δ 0 ; 3) when with the negative damping, i.e., D = −10, the system will lose synchronization; 4) when with a smaller negative damping, i.e., D = −1, the phase portrait is diverged, and finally occur LOS after several swing periods.
According to Fig. 3(b), it can be found that: 1) when the initial angle speed increases, i.e., δ i = 0.75, the phase portrait is closer to δ max ; 2) when the initial angle speed further increases, i.e., δ i = 1.5, the phase portrait may exceed δ max and cause LOS; 3) when with the opposite initial angle speed, i.e., δ i = −1.5, the second half of trajectory is the same as δ i = 1.5.
Overall, it is desirable to have large positive damping and small absolute value of initial angle speed to enhance the synchronization stability.

B. Detailed Transient Model of Grid-Following Converter
According to the detailed transient model of GFC, this section will analyze the causes and effects of negative damping and initial angle speed. Taking Case 1 as an example, the GFC is controlled with the unity power factor, and the active current startups from 0 to the rated current I max .
r Assuming K p = 0, ω pll = ω g and performing differentiation of (1), the swing equation and initial angle speed are shown as, It can be found that when considering K i only, the initial angle speed is 0, and the GFC is similar to the SG without damping. If δ does not exceed δ max = π-sin −1 (ω g L g I max /U gcp ) in the first swing period, the system will be stable.
r Considering K p and K i , assuming ω pll = ω g , the swing equation and initial angle speed are shown as, It can be found that when considering K p , the system has positive damping when δ is between 0 and 90°, while has negative damping when δ is between 90°and 180°. In addition, the system has the initial angle speed which deteriorate the synchronization stability.
r Considering K p and K i , assuming ω pll ࣔω g , the swing equation and initial angle speed are shown as, Note that the reactance under synchronous reference frame is not a constant in transient process, i.e., ω pll L g = (ω g +δ )L g . This phenomenon is caused by the dynamic of PLL output, and this influence is more obvious with larger grid inductance L g . When the GFC decreases the active current after extremely severe grid voltage sags, this phenomenon can be ignored [9].
It can be found that there is a negative damping −L g I max , resulting in worse synchronization stability. The initial angle    speed is faster when 0 < K p L g I max < 1, then increases the risk of LOS. Increasing K p can increase the damping coefficient, but it will also enhance the initial angle speed to deteriorate the synchronization stability.  It is noticed that when enabling the proposed control, δ will not exceed δ max , and f pll will recover to 50 Hz eventually. Figs. 5(c)-7(c) depict the enlarged view of frequency response. Note that f pll is still 50 Hz in the initial stage due to the constraints of current dynamic, but it will rise to the calculated initial value after 0.5 ms. It is easy to notice that there is a reverse 5 Hz offset after the fault occurs, which can decrease the initial angle speed. And the slower rising speed of f pll is caused by the positive damping. Compared with the frozen PLL, the proposed control has more accurate phase-tracking ability during grid fault. Compared with the first-order PLL, the logic transitions of proposed control are simpler, since it only need to judge when the reverse angular frequency should be enabled.

IV. PROPOSED SYNCHRONIZATION STABILITY ENHANCEMENT SCHEME AND SIMULATION RESULTS
According to Fig. 4, the model considering the additional damping can be elaborated in (10). Fig. 8 depicts the synchronization stable range with parameter deviations, where the blue area indicates that the system is synchronization stable. A constant ω n is subtracted in δ i to simulate the additional reverse angular frequency. It can be noticed that the α has an upper limit as 16.2 rad/s when ω n = 0, indicating the system will lose synchronization stability with high cut-off frequency of high-pass filter. When α is 2π·2 rad/s, K h is necessary to set between 2.2 and 18.2. However, a too small cut-off frequency will affect the dynamic response at fundamental frequency.
The appropriate reverse angular frequency can increase the synchronization stable range and increase the upper limit of α, i.e., ω n = 2π·2 rad/s and ω n = 2π·5 rad/s. However, adding a too large reverse angular frequency may cause the reversed direction of the initial angle speed, which deteriorates the synchronization stability, i.e., ω n = 2π·10 rad/s. ⎧ ⎨ ⎩ (δ ) = K p (αx−U gcp cos δδ )+K i ((ω g +δ )L g I d −U gcp sin δ−x) V. CONCLUSION This letter compares the swing equation between SG and GFC, then explains that when GFC encounters some grid faults such as line tripping and voltage sags, the larger positive damping and smaller absolute value of initial angle speed can enhance the synchronization stability and avoid LOS. A synchronization stability enhancement scheme is presented to increase the positive damping and decrease the initial angle speed, which has the simple logic transitions.