Impact of phase-locked loop on stability of active dampedLCL-filter-based grid-connected inverters with capacitorvoltage 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

gxq@ysu.edu.cn

Shichao LIU

scliu@bjtu.edu.cn

Xiaoyu WANG

xiaoyuw@doe.carleton.ca

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

123

J. Mod. Power Syst. Clean Energy (2017) 5(4):574–583

DOI 10.1007/s40565-017-0302-3

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

Figure 1 shows the structure of a typical three-phase

grid-connected inverter system. S1 * S6 are the switches;

L1 is the inverter-side filter inductance; C is the filter

capacitance; L2 is the grid-side filter inductance; Lg is the

grid line inductance; VCa, VCb, VCc are the capacitor volt-

ages in the grid frame ‘abc’; Iga, Igb, Igc are the grid cur-

rents in the grid frame ‘abc’.

In the grid dq frame, h1 is the PLL output phase angle;

x1 is 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.

VC ¼ e�jh1VsC when it is in the grid dq frame, and

VCc = ejhVC

s in the inverter dq frame. Then, the following

relationship is obtained:

VcC ¼ e�jDhVC

Dh ¼ 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.

Iga

DCpower supply

C

θ+

+

++ +

+ IgbIgc

VCa

Iqref

VCbVCc

dqabc dq

abcPI

S1

PWM

PCC

PLL

S3 S5

L1

FF+

++

F

Idref

L2

S4 S6 S2

Lg~~~

Fig. 1 Structure of a three-phase grid-connected inverter system

Impact of phase-locked loop on stability of active damped LCL-filter-based grid-connected… 575

123

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 � sssþ1

, 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 Csssþ1

.

The inverter output voltage is described as follows:

VPWM sð Þ ¼ Iref sð Þ � Icg sð Þ� �

kp þki

s

� �þ F sð ÞVc

C sð Þ� �

Vdc

2

From Fig. 2, the system variable relationships can be

described by the following open-loop transfer functions

[29]:

IcL sð Þ ¼ H1 sð ÞVPWM sð Þ þ H2 sð ÞVcPCC sð Þ

Icg sð Þ ¼ H3 sð ÞVPWM sð Þ þ H4 sð ÞVcPCC sð Þ

VcC sð Þ ¼ H5 sð ÞVPWM sð Þ þ H6 sð ÞVc

PCC sð Þð2Þ

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

H1 sð Þ ¼ L2Cs2 þ 1

L1L2Cs3 þ L1 þ L2ð Þs ; H2 sð Þ ¼ � 1

L1L2Cs3 þ L1 þ L2ð Þs ;

H3 sð Þ ¼ 1

L1L2Cs3 þ L1 þ L2ð Þs ; H4 sð Þ ¼ � L1Cs2 þ 1

L1L2Cs3 þ L1 þ L2ð Þs ;

H5 sð Þ ¼ L2s

L1L2Cs3 þ L1 þ L2ð Þs ; H6 sð Þ ¼ L1s

L1L2Cs3 þ L1 þ L2ð Þs :

Based on the above equations, the grid current can be

obtained as:

Icg sð Þ ¼ GT sð ÞIref sð Þ � Yi sð ÞVcC sð Þ ð3Þ

which in vector matrix form is:

Icg sð Þ ¼ gt sð Þ 0

0 gt sð Þ

� �|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

GT sð Þ

Iref sð Þ � yi sð Þ 0

0 yi sð Þ

� �|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

Yi sð Þ

VcC sð Þ

ð4Þ

where gt(s) and yi(s) can be expressed as:

gt ¼Vdc kp þ ki

s

�H3H6 � Vdc kp þ ki

s

�H4H5

2H6 þ Vdc kp þ kis

�H3H6 � Vdc kp þ ki

s

�H4H5

yi ¼VdcFðsÞH4H5 � 2H4 � VdcFðsÞH3H6

2H6 þ Vdc kp þ kis

�H3H6 � Vdc kp þ ki

s

�H4H5

8>>><>>>:

ð5Þ

The small-signal transfer function can be expressed as:

DIcg ¼ � yi sð Þ 0

0 yi sð Þ

� �|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

Yi sð Þ

DVcC ð6Þ

where DIcg is a grid current disturbance and DVcC is a

capacitor voltage disturbance in the inverter dq frame.

2.2 Small-signal model of PLL dynamics

Figure 3 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 ,VCb

c ,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

Igref VPWMPI Gvc(s)

F(s)+

++ ++ /2Vdc

cCV Ig(s)VC(s)

cgI

Fig. 2 Structure of inverter current regulator

αβ

abc

αβ

dq

θ

fFilter

Oscillator

PI +

+

*q 0V =

++

CqcV

CdcV

+

0ω 1/2π

1s

CacV

CbcVcCcV

Fig. 3 PLL system

VPCC(s)ZL1(s)

GT(s)Iref (s)

VC(s)

Ig(s)

+

Zg(s)

Vg(s)

+

Zeq(s)

Fig. 4 Equivalent Norton model of inverter system

576 Xiaoqiang GUO et al.

123

the instantaneous angular velocity disturbance Dx in the

following form:

Dx ¼ kpPLL þkiPLL

s

� �|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

FPLL sð Þ

Im VcC

� 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

FPLL sð Þ is defined in (7):

dhdt

¼ x0 þ Dx ¼ x0 þ FPLL sð ÞIm VcC

� ð8Þ

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

VcC ¼ e�jDhVC � 1� jDhð Þ V0

C þ DVC

�¼ V0

C þ DVC � jDhV0C

) Im VcC

� ¼ Im DVCf g � DhV0

C

ð9Þ

where VC0 is the steady-state capacitor voltage and DVC is

the capacitor voltage disturbance. The capacitor voltage

disturbance in the inverter dq frame DVCc is:

DVcC ¼ DVC � jDhV0

C ¼ DVC � jV0CGPLL sð ÞIm DVCf g

ð10Þ

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

Dh ¼ FPLL sð Þsþ V0

CFPLL sð Þ|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}GPLL sð Þ

Im DVCf g ð11Þ

-2000 -1500 -1000 -500 0 500

-6000

-4000

-2000

0

2000

4000

6000

Real axis

Imag

inar

y ax

is

Real axis-500 0 500

-1000

-500

0

500

1000

Imag

inar

y ax

is8000

-8000

1500

-1000

(a) Root locus

(b) Detail near the origin

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

Real axis

Imag

inar

y ax

is

-2000 -1500 -1000 -500 0

-6000

-4000

-2000

0

2000

4000

6000

Real axis

Imag

inar

y ax

is

-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

Fig. 6 Root locus of the system as current regulator gain changes

Impact of phase-locked loop on stability of active damped LCL-filter-based grid-connected… 577

123

The vector matrix form of DVcC is

DVcC ¼ 1 0

0 1� V0CGPLL sð Þ

� �DVC ð12Þ

Similarly, the grid current disturbance DIgc can be

expressed as follows:

DIcg ¼ DIg � jDhI0g ¼ DIg � jI0gGPLL sð ÞIm DVCf g 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:

DIcg ¼ DIg �0 0

0 IrefGPLL sð Þ

� �DVC ð14Þ

With (12), (13) and (14), the inverter input impedance

Zeq can be obtained:

Zeq ¼ �DVC DIg ��1¼ �

0 0

0 IrefGPLL sð Þ

� �

þyi sð Þ 0

0 yi sð Þ

� �1 0

0 1� V0CGPLL sð Þ

� � ð15Þ

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

expressed as

Ig sð Þ ¼ GT sð ÞIref sð Þ � Zeq sð Þ�1VC sð Þ 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:

ZL1 sð Þ ¼1=L2s 0

0 1=L2s

� �

Zg sð Þ ¼Lgs 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 VPCC sð Þ Iref sð Þ½ ��1

¼ GT

ZL1 þ Zg

�Zeq

Zeq þ ZL1 þ Zg

þ Zeq

Zeq þ ZL1 þ Zg

Vg Irefð Þ�1

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 kprequired 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 kpPLL is 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

)

51.001.050.0-50

0

50

I g(A

)

t (s)51.001.050.0

(a) Introde the active damping function

(b) Remove the active damping function

Iga Igb Igc

Iga IgbIgc

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

inverter

578 Xiaoqiang GUO et al.

123

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 kpPLL is 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 kp is 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 kp is 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

Tables 1 and 2.

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 Ig and VPCC when PLL bandwidth is

30 Hz

Fig. 9 Simulated dynamics of Ig and VPCC when PLL bandwidth is

125 Hz

Impact of phase-locked loop on stability of active damped LCL-filter-based grid-connected… 579

123

resulting dynamics of grid current Ig and 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 Ig and VPCC diverge 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 Ig and VPCC when kp = 0.1

Fig. 11 Simulated dynamics of Ig and VPCC when kp = 0.02

DCpowersupply

C

PCC

L1 L2

Lg

Fig. 12 Schematic diagram of the experimental setup

i(5

A/d

iv), V

(100

V/d

iv)

Time (10 ms/div)

PCCV

Lai

gai

(a) Case 1

i(5

A/d

iv) , V

( 100

V/ d

iv)

Time (10 ms/div)

PCCV

Lai

gai

(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

580 Xiaoqiang GUO et al.

123

were conducted with kp = 0.1 and kp = 0.02. The resulting

dynamics of grid current Ig and voltage at PCC VPCC are

shown in Figs. 10 and 11 respectively. 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 Tables 1 and 2.

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

Impact of phase-locked loop on stability of active damped LCL-filter-based grid-connected… 581

123

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. 17 and 18 respectively.

It can be observed from Fig. 17 that 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

Creative Commons Attribution 4.0 International License (http://

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.

gai

Lai

PCCV

i(5

A/d

iv), V

(100

V/d

iv)

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

582 Xiaoqiang GUO et al.

123

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Xiaoqiang GUO received 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 WANG received the B.Sc. and M.Sc. degrees from TsinghuaUniversity, 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.

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