## Impact of phase-locked loop on stability of active damped

## LCL-filter-based grid-connected inverters with capacitor

## voltage feedback

Xiaoqiang GUO1, Shichao LIU2, Xiaoyu WANG3

Abstract Active damped LCL-filter-based inverters have been widely used for grid-connected distributed generation (DG) systems. In weak grids, however, the phase-locked loop (PLL) dynamics may detrimentally affect the stability of grid-connected inverters due to interaction between the PLL and the controller. In order to solve the problem, the impact of PLL dynamics on small-signal stability is investigated for the active damped LCL-filtered grid-con-nected inverters with capacitor voltage feedback. The system closed-loop transfer function is established based on the Norton equivalent model by taking the PLL dynamics into account. Using an established model, the system stability boundary is identified from the viewpoint of PLL bandwidth and current regulator gain. The accuracy of the ranges of stability for the PLL bandwidth and current regulator gain is verified by both simulation and experi-mental results.

Keywords Phase-locked loop, Small-signal stability, Grid-connected inverter, Active damping, LCL filter, Transfer function

### 1 Introduction

LCL-filter-based grid-connected inverters have been widely discussed in [1–4]. One of the important issues is their stability. Ref. [5] proposed a robust passive damping method for LLCL-filter-based grid-connected inverters, so as to minimize the effect of harmonic voltages on the grid. Another interesting filter was proposed in [6] for optimiz-ing the system performance and stability. A recent review of passive filters for grid-connected converters was pre-sented in [7], and different passive damping methods were discussed. In order to avoid the losses resulting from pas-sive damping, capacitor-current feedback active damping was proposed in [8] to enhance the stability of LCL-type grid-connected inverters. Ref. [9] presented an interesting active damping method that can handle stability with only grid-current feedback control. In [10], a virtual RC damping method was presented with extended selective harmonic compensation. Note that these solutions are for voltage-source inverters. In practice, there are also current-source inverters [11–13]. An interesting active damping solution was presented in [14]. However, while the above methods are insightful for stability analysis, the impact of phase-locked loop (PLL) dynamics on the stability of grid-connected inverters has been neglected. The PLL is used to track the phase angle of the voltage at the point of common coupling (PCC) and thus to achieve grid synchronization for the inverter [15]. In case of high grid impedance, it has been found that the stability of a voltage source converter (VSC) can be affected by PLL parameters [16, 17]. In CrossCheck date: 23 May 2017

Received: 17 January 2017 / Accepted: 23 May 2017 / Published online: 8 July 2017

The Author(s) 2017. This article is an open access publication & Xiaoqiang GUO

[email protected] Shichao LIU [email protected] Xiaoyu WANG

1 _{Department of Electrical Engineering, Yanshan University,}

Qinhuangdao, China

2 _{School of Mechanical, Electronic, and Control Engineering,}

Beijing Jiaotong University, Beijing, China

3 _{Department of Electronics, Carleton University, Ottawa, ON}

K1S 5B6, Canada

VSC-based high voltage direct current (HVDC) transmis-sion systems, the PLL parameters have been found to limit the maximum power transfer capability [18]. In [19], the impact of short-circuit ratio and phase-locked-loop parameters on small-signal behavior is investigated, However, it only takes L-filter-based inverters into account.

In summary, these early works have discussed the influence of PLL parameters on the system stability. However, they only focused on grid-connected inverter systems with L or LC filters. In fact, an LCL filter is pre-ferred in grid-connected inverters, due to its better higher frequency attenuation and reduced physical size of the inductor [20]. In practice, the LCL filter usually suffers from resonance problems, which result in system instabil-ity. In order to deal with the problem, active damping controllers have been used [21].

Under weak grid conditions, the current regulator and the PLL dynamics of the active damped inverter may interact as the grid current passes the grid impedance, especially in case of a high bandwidth PLL. In fact, fast grid synchronization and PLL is important for operating the grid-connected inverters. Increasing PLL bandwidth is helpful for the fast and accurate grid synchronization. The PLL bandwidth should be large enough for the fast grid synchronization, and this is why diverse PLL designs have been proposed, as shown in [22–28]. However, when the PLL bandwidth is increased beyond a certain limit, the system may become unstable due to resonance between the inverter and the weak grid. Therefore, the interaction between fast PLL dynamics and current regulation deserves further investigation.

The objective of this paper is to contribute both ana-lytical and experimental verification on the interaction of PLL dynamics and current regulation in three-phase grid-connected active damped inverters with LCL filters under weak grid conditions. Using the Norton Theorem, a closed-loop transfer function is established to model the whole grid-connected inverter system including the PLL dynam-ics. By analyzing the poles of the closed-loop transfer function, the ranges of stability of the PLL bandwidth and the current regulator gain are obtained. Both a simulation and experimental results from a three-phase grid-connected inverter are provided to verify the effectiveness of the proposed model. The remainder of this paper is organized as follows. In Sect.2, a small-signal model is established for the three-phase grid-connected LCL-filter-based inver-ter with capacitor voltage feedback and active damping. In Sect.3, the small-signal stability of the inverter is dis-cussed in terms of the PLL dynamics and the current reg-ulator. In Sect.4, the analytical results are verified through simulations and experimental tests. Finally, conclusions are presented in Sect.5.

### 2 Inverter system model

Figure1 shows the structure of a typical three-phase grid-connected inverter system. S1* S6are the switches; L1 is the inverter-side filter inductance; C is the filter capacitance; L2is the grid-side filter inductance; Lgis the grid line inductance; VCa, VCb, VCc are the capacitor volt-ages in the grid frame ‘abc’; Iga, Igb, Igcare the grid cur-rents in the grid frame ‘abc’.

In the grid dq frame, h1is the PLL output phase angle; x1is the angular frequency; VC, Ig, IL, VPCC, Vg are the voltage vector at the filter capacitors, grid current vector, inverter-side filter current vector, voltage vector at the point of common coupling, and voltage vector of the grid-side voltage sources, respectively. In the inverter dq frame, the PLL output phase angle is denoted as h, which is almost the same as h1 in steady-state conditions. Other variables in the inverter dq frame are labeled with a superscript c, when referred in the grid dq frame. As current references are constant, in both dq frames, they are denoted as Iref.

Let VCs denote the capacitance voltage in the ab frame.
V_{C}¼ ejh1_{V}s

C when it is in the grid dq frame, and

VCc = ejhVCs in the inverter dq frame. Then, the following relationship is obtained: Vc C¼ ejDhVC Dh¼ h h1 ð1Þ

where Dh is the phase angle difference in these two dq frames and is approximately equal to zero when the grid-connected inverter operates in steady state. The PLL dynamics, however, make Dh depart from zero. Conse-quently, it is important to investigate the impact of the PLL on the dynamic stability of the grid-connected inverter under weak grid conditions.

*I*ga
DC
power
supply
*C*
*θ*
+
+
+
+ + + *II*gbgc
*V*Ca
*I*qref
*V*Cb
*V*Cc
*dq*
abc *dq*
abc
PI
S1
PWM
PCC
PLL
S3 S5
*L*1
*F*
*F*
+
+
+
*F*
*I*dref
*L*2
S4 S6 S2
*L*g
~
~
~

2.1 Current regulation with active damping

As shown in Fig.2, the current regulator for the inverter system includes a PI regulator and an active damping controller, whose transfer function is F(s).

Note that an active damped LCL filter increases the system complexity while increasing the stability of the inverter. Passive damping can also be used, but it incurs additional power losses. Therefore, in this paper, active damping is used due to avoid these power losses, and it is based on capacitor voltage feedback, where the capacitor voltage is differentiated by s and multiplied by a gain K before being fed back to the current regulator output. The transfer function F sð Þ is KCs.

In practice, the differential term may result in a high-frequency disturbance. To avoid this problem, a partially differentiation is used in the form s s

ssþ1, where s is a

time constant parameter. It is close to a differential element when s is very small. This gives the active damping con-troller a transfer function of the form F sð Þ ¼ K Cs

ssþ1.

The inverter output voltage is described as follows:
VPWMð Þ ¼s Irefð Þ Is cgð Þs
kpþ
ki
s
þ F sð ÞVc
Cð Þs
_{V}
dc
2

From Fig.2, the system variable relationships can be described by the following open-loop transfer functions [29]:

I_{L}cð Þ ¼ Hs 1ð ÞVs PWMð Þ þ Hs 2ð ÞVs cPCCð Þs

I_{g}cð Þ ¼ Hs 3ð ÞVs PWMð Þ þ Hs 4ð ÞVs cPCCð Þs

V_{C}cð Þ ¼ Hs 5ð ÞVs PWMð Þ þ Hs 6ð ÞVs cPCCð Þs

ð2Þ

where the transfer functions H1(s) * H6(s) are

H1ð Þ ¼s L2Cs2þ 1 L1L2Cs3þ Lð 1þ L2Þs ; H2ð Þ ¼ s 1 L1L2Cs3þ Lð 1þ L2Þs ; H3ð Þ ¼s 1 L1L2Cs3þ Lð 1þ L2Þs ; H4ð Þ ¼ s L1Cs2þ 1 L1L2Cs3þ Lð 1þ L2Þs ; H5ð Þ ¼s L2s L1L2Cs3þ Lð 1þ L2Þs ; H6ð Þ ¼s L1s L1L2Cs3þ Lð 1þ L2Þs :

Based on the above equations, the grid current can be obtained as:

I_{g}cð Þ ¼ Gs Tð ÞIs refð Þ Ys ið ÞVs cCð Þs ð3Þ

which in vector matrix form is:

Ic_{g}ð Þ ¼s gtð Þs 0
0 gtð Þs
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
GTð Þs
Irefð Þ s
yið Þs 0
0 yið Þs
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
Yið Þs
Vc_{C}ð Þs
ð4Þ
where gt(s) and yi(s) can be expressed as:

gt¼
Vdckpþk_{s}iH3H6 Vdckpþk_{s}iH4H5
2H6þ Vdc kpþk_{s}i
H3H6 Vdc kpþk_{s}i
H4H5
yi¼
VdcFðsÞH4H5 2H4 VdcFðsÞH3H6
2H6þ Vdckpþk_{s}iH3H6 Vdckpþk_{s}iH4H5
8
>
>
>
<
>
>
>
:
ð5Þ

The small-signal transfer function can be expressed as:

DIcg ¼ yið Þs 0 0 yið Þs |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Yið Þs DVcC ð6Þ

where DIc_{g} is a grid current disturbance and DVc_{C} is a
capacitor voltage disturbance in the inverter dq frame.

2.2 Small-signal model of PLL dynamics

Figure3 illustrates the PLL closed-loop control system used in the three-phase grid-connected inverter. A PI reg-ulator is used to enable the q-axis element of the capacitor voltage to follow the zero input reference Vq*, while VCac ,VCbc ,VCcc are the PLL inputs; x0 is the base phase angular velocity; f and h are the PLL output frequency and phase angle, respectively. The output of the PI regulator is

* I*gref

*PWM PI*

**V***G*vc

*(s)*

*F(s)*+ + +

_{+}+

*V*dc/2 c C

**V***I*g(

*s)*

*V*C(

*s)*c g

**I**Fig. 2 Structure of inverter current regulator

*αβ*
abc
*αβ*
*dq*
*θ*
*f*
Filter
Oscillator
PI +
+
*
*q* 0
*V =*
+
+
C*q*
c
*V*
C*d*
c
*V*
+
0
*ω* _{1/2π}
1
*s*
Ca
c
*V*
Cb
c
*V*
c
Cc
*V*
Fig. 3 PLL system
* V*PCC(

*s)*

*L1(*

**Z***s)*

*T(*

**G***ref (*

**s)I***s)*

*C(*

**V***s)*

*g(*

**I***s)*+

*g(*

**Z***s)*

*g(*

**V***s)*+

*eq(*

**Z***s)*

the instantaneous angular velocity disturbance Dx in the
following form:
Dx¼ kpPLLþ
kiPLL
s
|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
FPLLð Þs
Im V c_{C}
ð7Þ

where kpPLL and kiPLL are the proportional and integral gains of the PI regulator, respectively.

It can be interpreted from Fig.3 that, by integrating Dx ? x0, the following relationship is obtained, where FPLLð Þ is defined in (s 7): dh dt ¼ x0þ Dx ¼ x0þ FPLLð ÞIm Vs c C ð8Þ

Based on (1), the capacitor voltage can be written as

Vc_{C}¼ ejDhVC 1 jDhð Þ V C0þ DVC
¼ V0
Cþ DVC jDhVC0
) Im Vc
C
¼ Im DVf Cg DhVC0
ð9Þ

where VC0 is the steady-state capacitor voltage and DVCis the capacitor voltage disturbance. The capacitor voltage disturbance in the inverter dq frame DVCc is:

DVcC¼ DVC jDhVC0 ¼ DVC jVC0GPLLð ÞIm DVs f Cg

ð10Þ where GPLL(s) is given in the following equation:

Dh¼ FPLLð Þs sþ V0 CFPLLð Þs |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} GPLLð Þs Im DVf Cg ð11Þ -2000 -1500 -1000 -500 0 500 -6000 -4000 -2000 0 2000 4000 6000 Real axis Imaginary axis Real axis -500 0 500 -1000 -500 0 500 1000 Imaginary axis 8000 -8000 1500 -1000

**(a) Root locus**

**(b) Detail near the origin**

Fig. 5 Root locus of the system as PLL bandwidth changes

Real axis Imaginary axis -2000 -1500 -1000 -500 0 -6000 -4000 -2000 0 2000 4000 6000 Real axis Imaginary axis -400 -300 -200 -100 0 100 200 -600 -400 -200 0 200 400 600 8000 500 -8000 800 -800

**(a) Root locus**

**(b) Detail near the origin**

The vector matrix form of DVc_{C} is
DVcC¼
1 0
0 1 V0
CGPLLð Þs
DVC ð12Þ

Similarly, the grid current disturbance DIgc can be expressed as follows:

DIcg¼ DIg jDhI0g ¼ DIg jIg0GPLLð ÞIm DVs f Cg ð13Þ

where Ig0 is the steady-state value of the grid current and DIg is the current disturbance in the grid dq frame. The vector matrix form of (13) can be written as:

DIc g¼ DIg 0 0 0 IrefGPLLð Þs DVC ð14Þ

With (12), (13) and (14), the inverter input impedance
Zeqcan be obtained:
Zeq¼ DVC DIg
1
¼ 0 0
0 IrefGPLLð Þs
þ yið Þs 0
0 yið Þs
_{1} _{0}
0 1 V0
CGPLLð Þs
ð15Þ

Then with (14) and (15), the grid current can be expressed as

Igð Þ ¼ Gs Tð ÞIs refð Þ Zs eqð Þs1VCð Þs ð16Þ

The equivalent Norton model of the inverter system is shown in Fig.4.

In Fig.4, ZL1(s) denotes the grid-side filter inductance and Zg(s) denotes the grid line impedance:

ZL1ð Þ ¼s 1=L2s 0 0 1=L2s Zgð Þ ¼s Lgs 0 0 Lgs 8 > > > < > > > : ð17Þ

The final transfer function of the closed-loop system can be obtained from Fig.4 as follows:

G sð Þ ¼ VPCCð Þ Is ½ refð Þs1 ¼ GT ZL1þ Zg Zeq Zeqþ ZL1þ Zg þ Zeq Zeqþ ZL1þ Zg VgðIrefÞ1 ð18Þ

### 3 Small signal stability analysis

In general, a weak grid condition means the grid has a high impedance. In this section, a small signal stability analysis is carried out for the grid-connected inverter sys-tem with high grid impedance. More specifically, the ran-ges of the PLL bandwidth and the current regulator gain kp required to keep the inverter system stable are investigated. It should be noted that optimized design of the current regulator and the PLL gains is beyond the scope of this paper.

3.1 Impact of PLL bandwidth

When the current regulator gain kp= 0.1 and the PLL regulator gain kpPLLis varied within the range from 2 to 22,

Table 1 System parameters

Variable Value Per unit Variable Value Per unit

Vdc 200 V 1 p.u. Vg 80 V 1 p.u.

L2 2.5 mH 0.05 Lg 6.5 mH 0.12

L1 3 mH 0.06 C 9.4 lF 21

Table 2 Control parameters

Variable Value Variable Value

f0 50 Hz fswitch 10 kHz
Ts 50 ls Igref 5 A
ki 10 K 0.11
s 10-5 kp 0.1
*t (s)*
-50
0
50
*I*g
(A)
5
1
.
0
0
1
.
0
5
0
.
0
-50
0
50
*I*g
(A
)
*t (s)*
5
1
.
0
0
1
.
0
5
0
.
0

**(a)** Introde the active damping function

**(b)** Remove the active damping function

*I*ga * _{I}*
gb

*I*gc

*I*ga *I*gb

*I*gc

Fig. 7 Simulated input current waveform in the grid-connected inverter

the root loci of the closed-loop transfer function (18) are shown in Fig.5. The arrows indicate the direction of increasing PLL bandwidth, and the intersection with the vertical axis is at kpPLL= 12. As can be seen from the root loci, the poles of the closed-loop system are all in the left half side of the complex plane, and the system is stable, when kpPLLis within the range from 2 to 12 (equally, the PLL bandwidth is less than 90 Hz). It should be noted that when the current regulator gain is fixed, the system tends to

become less stable as the PLL bandwidth increases. Therefore, under weak grid conditions, a relatively small PLL bandwidth should be chosen to ensure system stability for the inverter.

3.2 Impact of current regulator gain

When the PLL bandwidth is fixed at 50 Hz and the current regulator gain kpis varied within the range from 0 to 0.1, the root loci of the closed-loop transfer function (18) are shown in Fig.6. The arrows indicate the direction of decreasing gain, and the intersection with the vertical axis is at kp= 0.04. As can be seen from this figure, the poles of the closed-loop system are all in the left half side of the complex plane, and the system is stable when kpis greater than 0.04. It can be observed that, when the PLL bandwidth is constant, the system tends to be less stable as the current regulator gain decreases.

### 4 Simulation and experimental verifications

Based on the results of the small-signal stability analy-sis, it can be seen that the PLL dynamics may significantly affect the inverter system stability under weak grid con-ditions. In this section, time-domain simulations and experimental tests are conducted to verify the theoretical results.Parameters of a three-phase grid-connected inverter system with the structure shown in Fig.1 are listed in Tables1 and2.

4.1 Simulation results

In order to verify the effectiveness of the active damping controller, the following two test cases were simulated with sampling period Ts = 50 ls and with integral gain of the

PI-type current regulator ki= 10:

Case 1 The inverter is initially only equipped with a PI-type current regulator, then the active damping function is added at t = 0.05 s.

Case 2 The inverter is initially installed with a PI-type current regulator and the active damping function, then the damping function is removed at t = 0.1 s.

As shown in Fig.7, the system becomes stable as soon as the active damping function is added, and it becomes unstable after the active damping function is removed.

To evaluate the impact of the PLL bandwidth, with a fixed current regulator gain kp= 0.1, the simulations were conducted with PLL bandwidth of 30 Hz and 125 Hz. The Fig. 8 Simulated dynamics of Igand VPCCwhen PLL bandwidth is

30 Hz

Fig. 9 Simulated dynamics of Igand VPCCwhen PLL bandwidth is

resulting dynamics of grid current Igand voltage at PCC VPCC are shown in Figs.8 and 9 respectively. In Fig.8, when the PLL bandwidth is 30 Hz, the inverter is

stable and the oscillations of Ig and VPCC are small. However, in Fig.9, when the PLL bandwidth is increased to 125 Hz, the dynamics of both Igand VPCCdiverge and the system becomes unstable. Based on these simulation results, for a given current regulator gain, it can be seen that the stability margin of the inverter system is bigger when the PLL bandwidth is smaller. This conclusion ver-ifies the results of the small-signal analysis.

To evaluate the effect of the current regulator gain, keeping the PLL bandwidth fixed at 50 Hz, the simulations Fig. 10 Simulated dynamics of Igand VPCCwhen kp= 0.1

Fig. 11 Simulated dynamics of Igand VPCCwhen kp= 0.02

DC
power
supply
*C*
PCC
*L*1 *L*2
*L*g

Fig. 12 Schematic diagram of the experimental setup

*i*
(5 A/div),
*V*
(100 V/div)
Time (10 ms/div)
PCC
*V*
La
*i*
ga
*i*
**(a) Case 1**
*i*
(5 A
/div)
*, V*
(100 V
/div)
Time (10 ms/div)
PCC
*V*
La
*i*
ga
*i*
**(b) Case 2**

Fig. 13 Experimental waveforms for the grid-connected inverter

Fig. 14 Experimental waveforms when the PLL bandwidth is 30 Hz and the current regulator gain is kp= 0.1

were conducted with kp= 0.1 and kp= 0.02. The resulting dynamics of grid current Igand voltage at PCC VPCCare shown in Figs.10and11respectively. The grid-connected inverter system is stable when kp= 0.1, as shown in Fig.10. However, the dynamics of both Ig and VPCC diverge and the grid-connected inverter system becomes unstable when kp= 0.02, as shown in Fig.11. The con-clusion again verifies the results of the small-signal anal-ysis. These observations also indicate that the inverter system may become unstable when the PLL bandwidth is beyond its stable range, even when the current regulation gain is within its stable range. Therefore, it is important to take the PLL dynamics into account.

4.2 Experimental results

A 3 kVA experimental platform system was built to further verify the impacts of the PLL dynamics and the current regulator. The schematic diagram of the experi-mental setup is shown in Fig.12.

The inverter is controlled by a 32-bit fixed-point 150 MHz TMS320F2812 DSP. The system parameters are the same as those simulated above, listed in Tables1and2. The effectiveness of the active damping controller was verified using two test cases similar to those simulated, as follows:

Case 1 The inverter system is initially only equipped with a PI type current regulator, then the active damping function is added at t = 0.023 s.

Case 2 The inverter system is initially installed with a PI type current regulator and the active damping function, then the damping function is removed at t = 0.25 s.

As shown in Fig.13, the system becomes stable as soon as the active damping function is added, and it becomes unstable after the damping function is removed. These

results verify the effectiveness of active damping regarding the resonance which results from the LCL filters.

The interaction between the PLL dynamics and the current regulator was also experimentally tested. First of all, the current regulator gain was fixed as kp= 0.1, and experimental tests were carried out with the PLL band-width set to 30 Hz and 125 Hz. The results are shown in Figs. 14 and 15 respectively. It can be observed that the waveforms of VPCC, Ig, and iL are stable when the PLL bandwidth is 30 Hz. However, when the PLL bandwidth is

Fig. 15 Experimental waveforms when the PLL bandwidth is 125 Hz and the current regulator gain is kp= 0.1

Fig. 16 Experimental waveforms when changing the PLL bandwidth

Fig. 17 Experimental waveforms when the current regulator gain kp= 0.1 and the PLL bandwidth is 50 Hz

125 Hz, VPCC, Ig, and iL start diverging and operate unstably. This is consistent with the theoretical analysis and simulation results that, for a fixed current regulator gain, the system is more stable with a smaller PLL band-width. The same good agreement with theory was observed

when the PLL bandwidth was changed online, as shown in Fig.16.

Secondly, the PLL bandwidth was fixed at 50 Hz, and experimental tests were carried out with the current regu-lator gain set to kp= 0.1 and kp= 0.02. The results are shown in Figs.17and18respectively.

It can be observed from Fig. 17that the waveforms of VPCC, Ig, and iL are stable when kp= 0.1. However, the system becomes unstable when kp= 0.02, with VPCC, Ig, and iL diverging and showing irregular behavior. This is consistent with the theoretical analysis and simulation results that, when the PLL bandwidth is fixed, the system is more stable for a larger current regulator gain. When the current regulator gain was changed online, as shown in Fig.19, the same good agreement with theory was observed.

### 5 Conclusion

In this paper, the interaction of phase-locked loop (PLL) dynamics and the current regulator on system stability has been investigated for active damped LCL-filter-based grid-connected inverters with capacitor voltage feedback under weak grid conditions. To efficiently model the system including the PLL dynamics, an equivalent Norton model was applied to obtain the closed-loop transfer function of the whole system. By investigating the poles of the transfer function, it has been found that the system tends to be instable as the PLL bandwidth increases with a fixed cur-rent regulator gain. Also, when the PLL bandwidth is fixed, the system tends to be instable as the current regulator gain decreases. The ranges of stability of the PLL bandwidth and the current regulator gain have been identified. The effectiveness of the designed active damping control method has been verified by both simulations and experi-mental results using a three-phase grid-connected inverter system. Future research can include stability analysis with a low sampling frequency under different control and operating modes.

Acknowledgements This work was supported by Science Founda-tion for Distinguished Young Scholars of Hebei Province (No. E2016203133) and Hundred Excellent Innovation Talents Support Program of Hebei Province (No. SLRC2017059).

Open Access This article is distributed under the terms of the
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creativecommons.org/licenses/by/4.0/), which permits unrestricted
use, distribution, and reproduction in any medium, provided you give
appropriate credit to the original author(s) and the source, provide a
link to the Creative Commons license, and indicate if changes were
made.
ga
*i*
La
*i*
PCC
*V*
*i*
(5 A/div
),
*V*
(100 V/div)
Time (10 ms/div)

Fig. 18 Experimental waveforms when the current regulator gain kp= 0.02 and the PLL bandwidth is 50 Hz

Fig. 19 Experimental waveforms when changing the current regula-tor gain kp

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Xiaoqiang GUOreceived the B.S. and Ph.D. degrees in Electrical Engineering from Yanshan University, Qinhuangdao, China, in 2003 and 2009, respectively. He has been a Postdoctoral Fellow with the Laboratory for Electrical Drive Applications and Research (LEDAR), Ryerson University, Toronto, ON, Canada. He is currently an associate professor with the Department of Electrical Engineering, Yanshan University, China. His current research interests include high-power converters and ac drives, electric vehicle charging station, and renewable energy power conversion systems.

Shichao LIU received the B.Sc. and M.Sc. degrees from Harbin Engineering University, Harbin, China, in 2007 and 2010, respec-tively, and the Ph.D. degree from Carleton University, Ottawa, ON, Canada, in 2014. His research interests include networked control systems, energy management, and optimization of microgrids. Xiaoyu WANGreceived the B.Sc. and M.Sc. degrees from Tsinghua University, Beijing, China, in 2000 and 2003, respectively, and the Ph.D. degree from the University of Alberta, Edmonton, AB, Canada, in 2008. He is currently an Associate Professor with the Department of Electronics, Faculty of Engineering and Design, Carleton Univer-sity, Ottawa, ON, Canada. His research interests include the integration of distributed energy resources and power quality.