# An Improved Phase-Locked-Loop Control with Alternative Damping Factors for VSC Connected to Weak AC System

Authors: Bin Yuan; Jianzhong Xu; Chengyong Zhao; Yijia Yuan
Journal: Journal of Control Science and Engineering
 (2016)

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## Abstract

The gains of phase-locked-loop (PLL) have significant impacts on the power transfer limits for the voltage source converter (VSC) connected to weak AC system. Therefore, in this paper, an improved PLL control, respectively, with alternative damping factors for rectifier and inverter is proposed. First, it is proved that the impedance angle of AC system has a great impact on the small-signal stability of the VSC system. With the same variation tendency of Thévenin equivalent resistance, the limits of power transmission are changing in opposite trends for rectifier and inverter. Second, the improved PLL with alternative damping factors is proposed based on the participation factor analysis. Third, the optimal damping factors of the improved PLL control for rectifier and inverter are calculated. Simulations and calculations validated the following three conclusions: (1) in rectifying operation, the equivalent system resistance has a negative impact on the stability of the system and this is not the case for inverting operation; (2) adding the alternative damping factors to PLL control shows similar results compared with changing the impedance angle of AC system; (3) the proposed optimal damping factors of PLL can effectively extend the power transfer limits under both rectifier and inverter modes.


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## Body

## 1. Introduction

Renewable energy resources are emerging as a future energy vector, and the voltage source converters (VSCs) are widely used to integrate such energies into power system [1–4]. The VSC-HVDC link connected to weak AC system with very low short circuit ratio (SCR < 2) will emerge quite often in the future [5–7]. However, the conventional vector-current control in d-q frame exhibits poor dynamic performance when applied to VSC connected to it. This brings a problem that the transmitted power cannot reach the ideal limitation for the unstable of small-signal model [8–17].There are three possible approaches to solve this problem. The first approach is shown in [8], which proposed an advanced vector-current control to decouple the d-q outer-loop control completely by optimizing the control parameters. However, the provided method is quite complicated and it is not suitable for frequent and rapid power changing.The second approach shown in [9] is adopting power synchronization control (PSC) as the main control strategy. PSC is similar with power angle control. This control strategy will not cause stable operating problems in extremely weak AC systems. However, it behaves in relatively low response speed due to the lack of the inner-loop current control and hence it cannot satisfactorily meet the requirement of the AC system.The last approach shown in [10] is changing the parameters of phase-locked-loop (PLL), especially the proportional gain in PI controller. It has been recognized that the challenging for VSCs operating in weak AC system is caused by the PLL dynamics. The response speed and small-signal stability are contradictory characteristics of the system. With a high proportional gain, the system response becomes quicker while the power transfer limitations decrease. Further, [16] reported that PLL has negative impact on the stability of VSC connected to weak AC system with reduced order model. However, quite few literatures have attempted to optimize PLL control system to enhance the stability of VSCs connected to weak AC system.This paper aims to propose an improved PLL control to extend the power transfer limitations. In the research of this work, it has been recognized that the impedance angle of weak AC system can also influence the power transfer limits. Meanwhile, an important observation is that for VSCs under different operation modes the equivalent resistance (related to the impedance angle directly) has opposite effects on the stability of VSCs. An advanced PLL with damping factor is proposed in this paper to enhance the power transfer limitations of grid-connected VSCs.The rest of this paper is organized as follows. Section2 presents the fundamental analysis of VSC connected to weak AC system. Section 3 studies the influence on power transfer limitations caused by impedance angle. Section 4 proposes an advanced PLL control system to enhance the stability of VSC connected to weak AC system. Section 5 verifies the proposed control by several case studies. And Section 6 concludes this paper.


## 2. Small-Signal Model of VSC Connected to Weak AC System

### 2.1. Benchmark Test Model

A two-level VSC is adopted in this paper as the topology. The test model is shown in Figure1. The weak AC system is represented by a Thévenin equivalent circuit and the equivalent impedance is Rg+jωLg. The DC side of the converter is represented by a DC voltage source. With the consideration of the current limitation of transformer, a capacitor is shunted at PCC to provide reactive power compensation. Lc is the leakage inductance of transformer and Rc is the resistance between PCC and converter.Figure 1
The benchmark test model of VSC connected to weak AC system.Vector-current control is selected as system control strategy and the control diagram is shown in Figure2(a). Active power (P) control and AC voltage (V) control are adopted. Direct and quadrature current reference signals for the inner-loop are generated from outer-loop control [18]. The simplified PLL model [19] is shown in Figure 2(b). With PI controller, the quadrature voltage at PCC point equals zero and the voltage phase angle can be accurately locked and measured.Figure 2
Control diagram of VSC connected to weak AC system.
(a)
Vector-current control model
(b)
The simplified model of PLL


### 2.2. System and Control Equations

The state-space model derived in this paper includes AC system and VSC controller shown in Figures1 and 2, respectively.


#### 2.2.1. AC System Equations

The equations of AC system are(1)edeq-vdvq=Rgi1di1q+Lgddti1di1q+ωLg-i1qi1d,vdvq-vcdvcq=Rci2di2q+Lcddti2di2q+ωLc-i2qi2d,i1di1q-i2di2q=Cfddtvdvq+ωCf-vqvd.In this case,ed equals Emcos⁡δ and eq equals -Emsin⁡δ and δ is the angle to which PCC voltage V leads equivalent AC system voltage E.


#### 2.2.2. Control System Equations

Active power control and AC voltage control are adopted as system control.Pref and Vref are the references for active power and AC RMS voltage. Inner-loop references of direct current i2dref and quadrature current i2qref are calculated by outer-loop control:(2)kppPref-P+kip∫Pref-Pdt=i2dref,kpvVref-V+kiv∫Vref-Vdt=i2qref.The inner-loop control equations are shown in(3)kp1i2dref-i2d+ki1∫i2dref-i2ddt=Lcdi2ddt+Rci2d,kp2i2qref-i2q+ki2∫i2qref-i2qdt=Lcdi2qdt+Rci2q.The simplified PLL model is given by(4)ω=ω0+kpllvq+ckpll∫vqdt,θ=∫ωdt.In this paper,ω0 is specified as 100πrad/s.


#### 2.2.3. State-Space Model

The detailed derivation of the state-variable equations is given in the Appendix. Considering that the references of active power and AC voltage are not supposed to change for small-signal model, bothΔPref and ΔVref equal zero. The linearized state-space model is given by(5)ΔX˙=AΔX,in which ΔX=[Δi1d,Δi1q,Δi2d,Δi2q,Δvd,Δvq,Δx1,Δx2, Δx3,Δx4,Δθ,Δx5]T. A is a 12-order matrix (the former 6 variables are AC system variables and the others are control system variables). The detailed information of matrix A and the definitions of x1~x5 are shown in the Appendix.The operating point of VSC system and the power controller parameters are shown in Table1.Table 1
Parameters of VSC connected to weak AC system.
Parameter symbols
Value
Main circuit parameters
Equivalent AC source voltageE
1.0 pu, 50 Hz
Equivalent impedance of AC systemRg+jXg
0.547 +j0.048 pu, impedance angle 85°
Equivalent AC system SCR
1.83
Rated DC power rating
1.0 pu
PCC voltage
1.0 pu
Converter impedanceRc+jXc
0.003 +j0.15 pu
CapacitorCf
0.15 pu
Controller parameters
Power controller gains (kpp, kip)
(0.5, 50)
AC voltage controller gains (kpv, kiv)
(0.35, 30)
Innerid controller gains (kp1, ki1)
(1, 10)
Inneriq controller gains (kp2, ki2)
(1, 10)
PLL gains (kpll, c)
(50, 10)The eigenvalues of test model at the operating point ofPref=1.33 and Pref=1.50 are shown in Table 2.Table 2
Eigenvalues of the test model.
SCR = 1.37 (Pref = 1.33 pu)
SCR = 1.22 (Pref = 1.5 pu)
λ
1
-
1510
-
1509
λ
2,3
-
304
±
j
1120
-
366.9
±
j
1180.6
λ
4
-
1010
-
840.6
λ
5,6
-
46.3
±
j
629
-
78.4
±
j
631.3
λ
7
-
61.29
-
59.8
λ
8,9
−5.3 ± j23.1
5.2 ± j21.7
λ
10
-
11.18
-
10.8
λ
11
-
99.54
-
10
λ
12
-
99.92
-
10With the eigenvalues shown in Table2 (see the bold italic), the predominant poles [20] of matrix A are selected to be λ8,9. The root-locus of λ8,9 with active power changing is shown in Figure 3. And some of the values in Figure 3 are picked up and shown in Table 3.Table 3
The detailed information of predominant poles.
P
λ
8,9
Damping ratio
Oscillation frequency (Hz)
1.30 pu
-
9.86
±
j
24.08
0.379
3.83
1.33 pu
-
5.30
±
j
23.1
0
0.224
3.68
1.37 pu
-
2.80
±
j
22.00
0.126
3.50
1.40pu
0.22 ± j21.90
—
3.48
1.43 pu
1.51
±
j
21.71
—
3.46Figure 3
Root-locus of the predominant poles.Figure3 and Table 3 both show that, with the power rising, the small-signal stability of test model is getting worse and system becomes unstable when the active power reaches about 1.4 pu.


#### 2.2.4. Participation Factor Analysis

Participation factor can be utilized to analyze the relationship between predominant poles and state-variables [21–23]. The participation factor of test model is shown in Table 4.Table 4
The calculated participation factors of test model.
State-variable
Participation factor
State-variable
Participation factor
X
1
0.0277
X
7
0.1904
X
2
0.0582
X
8
0.1773
X
3
0.0032
X
9
0.0014
X
4
0.0087
X
10
0.0037
X
5
0.0098
X
11
0.3639
X
6
0.0040
X
12
0.1517From Table4, it can be discovered that the outer-loop and PLL control diagrams are likely to have more impacts on the stability of VSC connected to weak AC system. Reference [8] has proposed the outer-loop control approach, and this paper will mainly focus on the PLL improvements.


## 2.1. Benchmark Test Model

A two-level VSC is adopted in this paper as the topology. The test model is shown in Figure1. The weak AC system is represented by a Thévenin equivalent circuit and the equivalent impedance is Rg+jωLg. The DC side of the converter is represented by a DC voltage source. With the consideration of the current limitation of transformer, a capacitor is shunted at PCC to provide reactive power compensation. Lc is the leakage inductance of transformer and Rc is the resistance between PCC and converter.Figure 1
The benchmark test model of VSC connected to weak AC system.Vector-current control is selected as system control strategy and the control diagram is shown in Figure2(a). Active power (P) control and AC voltage (V) control are adopted. Direct and quadrature current reference signals for the inner-loop are generated from outer-loop control [18]. The simplified PLL model [19] is shown in Figure 2(b). With PI controller, the quadrature voltage at PCC point equals zero and the voltage phase angle can be accurately locked and measured.Figure 2
Control diagram of VSC connected to weak AC system.
(a)
Vector-current control model
(b)
The simplified model of PLL


## 2.2. System and Control Equations

The state-space model derived in this paper includes AC system and VSC controller shown in Figures1 and 2, respectively.


### 2.2.1. AC System Equations

The equations of AC system are(1)edeq-vdvq=Rgi1di1q+Lgddti1di1q+ωLg-i1qi1d,vdvq-vcdvcq=Rci2di2q+Lcddti2di2q+ωLc-i2qi2d,i1di1q-i2di2q=Cfddtvdvq+ωCf-vqvd.In this case,ed equals Emcos⁡δ and eq equals -Emsin⁡δ and δ is the angle to which PCC voltage V leads equivalent AC system voltage E.


### 2.2.2. Control System Equations

Active power control and AC voltage control are adopted as system control.Pref and Vref are the references for active power and AC RMS voltage. Inner-loop references of direct current i2dref and quadrature current i2qref are calculated by outer-loop control:(2)kppPref-P+kip∫Pref-Pdt=i2dref,kpvVref-V+kiv∫Vref-Vdt=i2qref.The inner-loop control equations are shown in(3)kp1i2dref-i2d+ki1∫i2dref-i2ddt=Lcdi2ddt+Rci2d,kp2i2qref-i2q+ki2∫i2qref-i2qdt=Lcdi2qdt+Rci2q.The simplified PLL model is given by(4)ω=ω0+kpllvq+ckpll∫vqdt,θ=∫ωdt.In this paper,ω0 is specified as 100πrad/s.


### 2.2.3. State-Space Model

The detailed derivation of the state-variable equations is given in the Appendix. Considering that the references of active power and AC voltage are not supposed to change for small-signal model, bothΔPref and ΔVref equal zero. The linearized state-space model is given by(5)ΔX˙=AΔX,in which ΔX=[Δi1d,Δi1q,Δi2d,Δi2q,Δvd,Δvq,Δx1,Δx2, Δx3,Δx4,Δθ,Δx5]T. A is a 12-order matrix (the former 6 variables are AC system variables and the others are control system variables). The detailed information of matrix A and the definitions of x1~x5 are shown in the Appendix.The operating point of VSC system and the power controller parameters are shown in Table1.Table 1
Parameters of VSC connected to weak AC system.
Parameter symbols
Value
Main circuit parameters
Equivalent AC source voltageE
1.0 pu, 50 Hz
Equivalent impedance of AC systemRg+jXg
0.547 +j0.048 pu, impedance angle 85°
Equivalent AC system SCR
1.83
Rated DC power rating
1.0 pu
PCC voltage
1.0 pu
Converter impedanceRc+jXc
0.003 +j0.15 pu
CapacitorCf
0.15 pu
Controller parameters
Power controller gains (kpp, kip)
(0.5, 50)
AC voltage controller gains (kpv, kiv)
(0.35, 30)
Innerid controller gains (kp1, ki1)
(1, 10)
Inneriq controller gains (kp2, ki2)
(1, 10)
PLL gains (kpll, c)
(50, 10)The eigenvalues of test model at the operating point ofPref=1.33 and Pref=1.50 are shown in Table 2.Table 2
Eigenvalues of the test model.
SCR = 1.37 (Pref = 1.33 pu)
SCR = 1.22 (Pref = 1.5 pu)
λ
1
-
1510
-
1509
λ
2,3
-
304
±
j
1120
-
366.9
±
j
1180.6
λ
4
-
1010
-
840.6
λ
5,6
-
46.3
±
j
629
-
78.4
±
j
631.3
λ
7
-
61.29
-
59.8
λ
8,9
−5.3 ± j23.1
5.2 ± j21.7
λ
10
-
11.18
-
10.8
λ
11
-
99.54
-
10
λ
12
-
99.92
-
10With the eigenvalues shown in Table2 (see the bold italic), the predominant poles [20] of matrix A are selected to be λ8,9. The root-locus of λ8,9 with active power changing is shown in Figure 3. And some of the values in Figure 3 are picked up and shown in Table 3.Table 3
The detailed information of predominant poles.
P
λ
8,9
Damping ratio
Oscillation frequency (Hz)
1.30 pu
-
9.86
±
j
24.08
0.379
3.83
1.33 pu
-
5.30
±
j
23.1
0
0.224
3.68
1.37 pu
-
2.80
±
j
22.00
0.126
3.50
1.40pu
0.22 ± j21.90
—
3.48
1.43 pu
1.51
±
j
21.71
—
3.46Figure 3
Root-locus of the predominant poles.Figure3 and Table 3 both show that, with the power rising, the small-signal stability of test model is getting worse and system becomes unstable when the active power reaches about 1.4 pu.


### 2.2.4. Participation Factor Analysis

Participation factor can be utilized to analyze the relationship between predominant poles and state-variables [21–23]. The participation factor of test model is shown in Table 4.Table 4
The calculated participation factors of test model.
State-variable
Participation factor
State-variable
Participation factor
X
1
0.0277
X
7
0.1904
X
2
0.0582
X
8
0.1773
X
3
0.0032
X
9
0.0014
X
4
0.0087
X
10
0.0037
X
5
0.0098
X
11
0.3639
X
6
0.0040
X
12
0.1517From Table4, it can be discovered that the outer-loop and PLL control diagrams are likely to have more impacts on the stability of VSC connected to weak AC system. Reference [8] has proposed the outer-loop control approach, and this paper will mainly focus on the PLL improvements.


## 2.2.1. AC System Equations

The equations of AC system are(1)edeq-vdvq=Rgi1di1q+Lgddti1di1q+ωLg-i1qi1d,vdvq-vcdvcq=Rci2di2q+Lcddti2di2q+ωLc-i2qi2d,i1di1q-i2di2q=Cfddtvdvq+ωCf-vqvd.In this case,ed equals Emcos⁡δ and eq equals -Emsin⁡δ and δ is the angle to which PCC voltage V leads equivalent AC system voltage E.


## 2.2.2. Control System Equations

Active power control and AC voltage control are adopted as system control.Pref and Vref are the references for active power and AC RMS voltage. Inner-loop references of direct current i2dref and quadrature current i2qref are calculated by outer-loop control:(2)kppPref-P+kip∫Pref-Pdt=i2dref,kpvVref-V+kiv∫Vref-Vdt=i2qref.The inner-loop control equations are shown in(3)kp1i2dref-i2d+ki1∫i2dref-i2ddt=Lcdi2ddt+Rci2d,kp2i2qref-i2q+ki2∫i2qref-i2qdt=Lcdi2qdt+Rci2q.The simplified PLL model is given by(4)ω=ω0+kpllvq+ckpll∫vqdt,θ=∫ωdt.In this paper,ω0 is specified as 100πrad/s.


## 2.2.3. State-Space Model

The detailed derivation of the state-variable equations is given in the Appendix. Considering that the references of active power and AC voltage are not supposed to change for small-signal model, bothΔPref and ΔVref equal zero. The linearized state-space model is given by(5)ΔX˙=AΔX,in which ΔX=[Δi1d,Δi1q,Δi2d,Δi2q,Δvd,Δvq,Δx1,Δx2, Δx3,Δx4,Δθ,Δx5]T. A is a 12-order matrix (the former 6 variables are AC system variables and the others are control system variables). The detailed information of matrix A and the definitions of x1~x5 are shown in the Appendix.The operating point of VSC system and the power controller parameters are shown in Table1.Table 1
Parameters of VSC connected to weak AC system.
Parameter symbols
Value
Main circuit parameters
Equivalent AC source voltageE
1.0 pu, 50 Hz
Equivalent impedance of AC systemRg+jXg
0.547 +j0.048 pu, impedance angle 85°
Equivalent AC system SCR
1.83
Rated DC power rating
1.0 pu
PCC voltage
1.0 pu
Converter impedanceRc+jXc
0.003 +j0.15 pu
CapacitorCf
0.15 pu
Controller parameters
Power controller gains (kpp, kip)
(0.5, 50)
AC voltage controller gains (kpv, kiv)
(0.35, 30)
Innerid controller gains (kp1, ki1)
(1, 10)
Inneriq controller gains (kp2, ki2)
(1, 10)
PLL gains (kpll, c)
(50, 10)The eigenvalues of test model at the operating point ofPref=1.33 and Pref=1.50 are shown in Table 2.Table 2
Eigenvalues of the test model.
SCR = 1.37 (Pref = 1.33 pu)
SCR = 1.22 (Pref = 1.5 pu)
λ
1
-
1510
-
1509
λ
2,3
-
304
±
j
1120
-
366.9
±
j
1180.6
λ
4
-
1010
-
840.6
λ
5,6
-
46.3
±
j
629
-
78.4
±
j
631.3
λ
7
-
61.29
-
59.8
λ
8,9
−5.3 ± j23.1
5.2 ± j21.7
λ
10
-
11.18
-
10.8
λ
11
-
99.54
-
10
λ
12
-
99.92
-
10With the eigenvalues shown in Table2 (see the bold italic), the predominant poles [20] of matrix A are selected to be λ8,9. The root-locus of λ8,9 with active power changing is shown in Figure 3. And some of the values in Figure 3 are picked up and shown in Table 3.Table 3
The detailed information of predominant poles.
P
λ
8,9
Damping ratio
Oscillation frequency (Hz)
1.30 pu
-
9.86
±
j
24.08
0.379
3.83
1.33 pu
-
5.30
±
j
23.1
0
0.224
3.68
1.37 pu
-
2.80
±
j
22.00
0.126
3.50
1.40pu
0.22 ± j21.90
—
3.48
1.43 pu
1.51
±
j
21.71
—
3.46Figure 3
Root-locus of the predominant poles.Figure3 and Table 3 both show that, with the power rising, the small-signal stability of test model is getting worse and system becomes unstable when the active power reaches about 1.4 pu.


## 2.2.4. Participation Factor Analysis

Participation factor can be utilized to analyze the relationship between predominant poles and state-variables [21–23]. The participation factor of test model is shown in Table 4.Table 4
The calculated participation factors of test model.
State-variable
Participation factor
State-variable
Participation factor
X
1
0.0277
X
7
0.1904
X
2
0.0582
X
8
0.1773
X
3
0.0032
X
9
0.0014
X
4
0.0087
X
10
0.0037
X
5
0.0098
X
11
0.3639
X
6
0.0040
X
12
0.1517From Table4, it can be discovered that the outer-loop and PLL control diagrams are likely to have more impacts on the stability of VSC connected to weak AC system. Reference [8] has proposed the outer-loop control approach, and this paper will mainly focus on the PLL improvements.


## 3. The Impacts of System Impedance Angle on Power Transfer Limitations

This section will analyze the impact of the system impedance angle on VSC working in either rectifier or inverter modes regarding the power transfer limitations.The active power transmission at PCC can be calculated using(6)P=VRg2+Xg2-EXgsin⁡δ+ERgcos⁡δ-VRg.The power angle curves for different impedance angles(φ) of AC system are shown in Figure 4. It can be found that φ has a great influence on power transfer limits. The small-signal stability of VSC system with different φ will be analyzed in the subsections below.Figure 4
Power angle curves in different impedance angles.


### 3.1. Rectifying Operation

With the analysis of state-space matrixA, it is concluded that the equivalent resistance of AC system has a negative impact on VSC operating as a rectifier. The root-locus of predominant poles with φ changing is shown in Figure 5(a). It shows that, for VSC operating in rectifier mode, lower resistance of AC system will enhance the small-signal stability of VSC system.Figure 5
Root-locus of predominant poles withφ change.
(a)
Rectifier
(b)
Inverter


### 3.2. Inverting Operation

Again, with the analysis of state-space matrixA, an opposite conclusion can be drawn that, for VSC working in inverter mode, lower resistance of AC system will weaken the VSC system stability, as shown in Figure 5(b).The power transfer limitations of system with differentφ are expressed in Table 5.Table 5
Calculation result of power transfer limitations with differentφ.
φ
MaxP (rectifier)
MaxP (inverter)
80°
1.284
1.533
81°
1.302
1.524
82°
1.323
1.521
83°
1.358
1.518
84°
1.383
1.510
85°
1.400
1.505It can be concluded that lower resistance will enlarge the stable margin of VSC working at rectifier mode and will reduce the stable margin for inverter mode.


## 3.1. Rectifying Operation

With the analysis of state-space matrixA, it is concluded that the equivalent resistance of AC system has a negative impact on VSC operating as a rectifier. The root-locus of predominant poles with φ changing is shown in Figure 5(a). It shows that, for VSC operating in rectifier mode, lower resistance of AC system will enhance the small-signal stability of VSC system.Figure 5
Root-locus of predominant poles withφ change.
(a)
Rectifier
(b)
Inverter


## 3.2. Inverting Operation

Again, with the analysis of state-space matrixA, an opposite conclusion can be drawn that, for VSC working in inverter mode, lower resistance of AC system will weaken the VSC system stability, as shown in Figure 5(b).The power transfer limitations of system with differentφ are expressed in Table 5.Table 5
Calculation result of power transfer limitations with differentφ.
φ
MaxP (rectifier)
MaxP (inverter)
80°
1.284
1.533
81°
1.302
1.524
82°
1.323
1.521
83°
1.358
1.518
84°
1.383
1.510
85°
1.400
1.505It can be concluded that lower resistance will enlarge the stable margin of VSC working at rectifier mode and will reduce the stable margin for inverter mode.


## 4. Improved PLL Control for VSC Connected to Weak AC System

In Section3, a conclusion can be drawn that the damping characteristic of AC network (impedance angle) has a great influence on the system stability. With the analysis of participation factor, it is acknowledged that PLL control also has a great impact on the stability of VSC connected to weak AC system. Therefore, an improved PLL control suitable for VSC connected to weak AC system is proposed as follows, which is the main contribution of this paper.Considering that the active power is proportional to the direct-axis current, a supplementary damping control with state-variablei2d is added in PLL control system (D is the damping factor). And the new PLL equations are shown in (7), in which “+” is for rectifying operation and “−” is for inverting operation. Figure 6 shows the improved PLL control diagram:(7)ω=ω0+kpllvq+ckpll∫vqdt,θ=∫ω±Di2dref-i2ddt.Figure 6
Improved PLL control for VSC connected to weak AC system.The small-signal model is changed and the state-variable matrixA with the improved PLL is shown in the Appendix. Figure 7 shows the root-locus of predominant poles of the modified A.Figure 7
Root-locus and participation factor of system with advanced PLL control.
(a)
Root-locus of VSC withP=1.4 pu
(b)
Root-locus of VSC with different transfer power
(c)
Participation factor analysisFrom Figure7, it can be seen that, with the rising of damping factor D in a certain range, the small-signal stability of VSC system is enhanced. After a critical value, the stability is reduced with the rising of D. The root-locus of VSC system with P rising in different damping factors has also verified the effectiveness of the improved PLL controller. Figure 7(c) shows that, with a proper damping factor, the relationship between PLL and dominant poles is weak. It also means that the most effective parts are changing from PLL to the outer-loop controller, which is the purpose of the improved PLL control strategy. There should be an optimal value for the damping factor D. When impedance angle of AC system is 85°, the optimal value of D is approximately 1100 for rectifying operation and 300 for inverting operation.


## 5. Case Studies

### 5.1. Validation of the Small-Signal Model

Table6 shows the base values of the system parameters of benchmark test model. The accuracy of the small-signal model including the oscillation frequency and damping ratio will be verified in this section.Table 6
Parameters of test model.
Parameters
Value
Dc voltage
±200 kV
Rated active power
300 MW
Equivalent AC source voltage
230 kV
PCC voltage
230 kV
Transformer
230 kV/204.12 kVFigure8 shows the simulation results of the system in which the impedance angle is 85°. The active power, PCC voltage, and D- and Q-axis current are shown in Figures 8(a)~8(d) respectively, with the active power being changing from 360 MW to 370 MW.Figure 8
Simulation results of VSC system (impedance angle is 85°).
(a)
Active power
(b)
PCC voltage
(c)
D-axis current
(d)
Q-axis currentIt can be seen from Figure8 that the maximum error value between EMT model and small-signal model of active power is 0.8%. The maximum value of PCC voltage is 0.15%. The maximum error value of D-axis current and Q-axis current both is 0.27%. Therefore, the correctness of the VSC system small-signal model is satisfactorily validated.


### 5.2. Validations of Impacts of Impedance Angle on Power Transfer Limits

The theory on the impact caused by impedance angle on stability of VSC connected to weak AC system which is proposed in Section3 is verified in this section.


#### 5.2.1. Rectifying Operation

Figure9 shows the maximum power transmission of system with different impedance angles.Figure 9
Maximum power transmission with different impedance angles (rectifying operation).
(a)
φ
=
8
0
°
(b)
φ
=
8
5
°It can be seen from the figure that the maximum active power is 385 MW whenφ equals 80°. Meanwhile, the maximum active power is 420 MW when φ equals 85°. Simulation results show that, with the rising of impedance angle, the VSC system operating in rectifier mode gets larger small-signal stable margin.


#### 5.2.2. Inverting Operation

Figure10 shows the maximum power transmission of system with different impedance angles.Figure 10
Maximum power transmission with different impedance angles (inverting operation).
(a)
φ
=
85
°
(b)
φ
=
80
°However, the simulation results shown in Figure10 are opposite with the system working in rectifying operation. The maximum active power is 460 MW when φ equals 80°, and this value decreased to 450 MW when φ equals 85°. With the rising of impedance angle, the VSC system operating in inverter mode is getting smaller stable margin and even becoming unstable.Table7 shows the power transfer limitations with different impedance angles.Table 7
Limits of power transmission with differentφ.
φ
MaxP
MaxP
(rectifying operation)
(inverting operation)
80°
385 MW (1.283 pu)
460 MW (1.533 pu)
81°
390 MW (1.300 pu)
458 MW (1.527 pu)
82°
397 MW (1.323 pu)
456 MW (1.520 pu)
83°
408 MW (1.360 pu)
455 MW (1.517 pu)
84°
415 MW (1.383 pu)
452 MW (1.507 pu)
85°
420 MW (1.400 pu)
450 MW (1.500 pu)The conclusion drawn from Section3 is validated by Table 7. Meanwhile, it can be seen that the system operating at rectifier mode is more severe than inverter mode, which means that the stable margin for VSC working in rectifier mode is narrower than that of inverter mode.


### 5.3. Validations of the Proposed PLL Control

Figure11 shows the simulation results of maximum power transmission with the proposed improved PLL control.Figure 11
The simulation results of system with proposed PLL control.
(a)
Rectifying operation
(b)
Inverting operationIt can be obtained that, with the proposed PLL control strategy, the limits of power transmission are improved. The test model adopts 85° as the AC system impedance angle. Compared with 420 MW, the maximum active power at rectifier is enhanced to 460 MW. Meanwhile, the limits of power transmission at inverter are enhanced from 450 MW to 480 MW. The optimal value of damping factorD calculated at rectifying operation is 1100 and is 300 at inverting operation. Figure 12 shows the relationship between VSC power transfer limitations and damping factors in the proposed PLL control.Figure 12
The relationship betweenD and limits of power transmission.
(a)
Rectifying operation
(b)
Inverting operationIt can be seen from Figure12 that the optimal values of damping factors are approximately equal to the calculated results (1100 and 300, resp.). Figure 13 shows the simulation results of the relationship between system impedance angles and power transfer limitations with D being equal to zero and the optimal values (i.e., 1100 and 300, resp.).Figure 13
The power transfer limitations with the advanced PLL control.
(a)
Rectifying operation
(b)
Inverting operationNote that the power transfer limitation increments are shown in Figure13 using arrows. The validity of the advanced PLL control strategy and the previous impedance angle analysis results are simultaneously validated in Figure 13.


## 5.1. Validation of the Small-Signal Model

Table6 shows the base values of the system parameters of benchmark test model. The accuracy of the small-signal model including the oscillation frequency and damping ratio will be verified in this section.Table 6
Parameters of test model.
Parameters
Value
Dc voltage
±200 kV
Rated active power
300 MW
Equivalent AC source voltage
230 kV
PCC voltage
230 kV
Transformer
230 kV/204.12 kVFigure8 shows the simulation results of the system in which the impedance angle is 85°. The active power, PCC voltage, and D- and Q-axis current are shown in Figures 8(a)~8(d) respectively, with the active power being changing from 360 MW to 370 MW.Figure 8
Simulation results of VSC system (impedance angle is 85°).
(a)
Active power
(b)
PCC voltage
(c)
D-axis current
(d)
Q-axis currentIt can be seen from Figure8 that the maximum error value between EMT model and small-signal model of active power is 0.8%. The maximum value of PCC voltage is 0.15%. The maximum error value of D-axis current and Q-axis current both is 0.27%. Therefore, the correctness of the VSC system small-signal model is satisfactorily validated.


## 5.2. Validations of Impacts of Impedance Angle on Power Transfer Limits

The theory on the impact caused by impedance angle on stability of VSC connected to weak AC system which is proposed in Section3 is verified in this section.


### 5.2.1. Rectifying Operation

Figure9 shows the maximum power transmission of system with different impedance angles.Figure 9
Maximum power transmission with different impedance angles (rectifying operation).
(a)
φ
=
8
0
°
(b)
φ
=
8
5
°It can be seen from the figure that the maximum active power is 385 MW whenφ equals 80°. Meanwhile, the maximum active power is 420 MW when φ equals 85°. Simulation results show that, with the rising of impedance angle, the VSC system operating in rectifier mode gets larger small-signal stable margin.


### 5.2.2. Inverting Operation

Figure10 shows the maximum power transmission of system with different impedance angles.Figure 10
Maximum power transmission with different impedance angles (inverting operation).
(a)
φ
=
85
°
(b)
φ
=
80
°However, the simulation results shown in Figure10 are opposite with the system working in rectifying operation. The maximum active power is 460 MW when φ equals 80°, and this value decreased to 450 MW when φ equals 85°. With the rising of impedance angle, the VSC system operating in inverter mode is getting smaller stable margin and even becoming unstable.Table7 shows the power transfer limitations with different impedance angles.Table 7
Limits of power transmission with differentφ.
φ
MaxP
MaxP
(rectifying operation)
(inverting operation)
80°
385 MW (1.283 pu)
460 MW (1.533 pu)
81°
390 MW (1.300 pu)
458 MW (1.527 pu)
82°
397 MW (1.323 pu)
456 MW (1.520 pu)
83°
408 MW (1.360 pu)
455 MW (1.517 pu)
84°
415 MW (1.383 pu)
452 MW (1.507 pu)
85°
420 MW (1.400 pu)
450 MW (1.500 pu)The conclusion drawn from Section3 is validated by Table 7. Meanwhile, it can be seen that the system operating at rectifier mode is more severe than inverter mode, which means that the stable margin for VSC working in rectifier mode is narrower than that of inverter mode.


## 5.2.1. Rectifying Operation

Figure9 shows the maximum power transmission of system with different impedance angles.Figure 9
Maximum power transmission with different impedance angles (rectifying operation).
(a)
φ
=
8
0
°
(b)
φ
=
8
5
°It can be seen from the figure that the maximum active power is 385 MW whenφ equals 80°. Meanwhile, the maximum active power is 420 MW when φ equals 85°. Simulation results show that, with the rising of impedance angle, the VSC system operating in rectifier mode gets larger small-signal stable margin.


## 5.2.2. Inverting Operation

Figure10 shows the maximum power transmission of system with different impedance angles.Figure 10
Maximum power transmission with different impedance angles (inverting operation).
(a)
φ
=
85
°
(b)
φ
=
80
°However, the simulation results shown in Figure10 are opposite with the system working in rectifying operation. The maximum active power is 460 MW when φ equals 80°, and this value decreased to 450 MW when φ equals 85°. With the rising of impedance angle, the VSC system operating in inverter mode is getting smaller stable margin and even becoming unstable.Table7 shows the power transfer limitations with different impedance angles.Table 7
Limits of power transmission with differentφ.
φ
MaxP
MaxP
(rectifying operation)
(inverting operation)
80°
385 MW (1.283 pu)
460 MW (1.533 pu)
81°
390 MW (1.300 pu)
458 MW (1.527 pu)
82°
397 MW (1.323 pu)
456 MW (1.520 pu)
83°
408 MW (1.360 pu)
455 MW (1.517 pu)
84°
415 MW (1.383 pu)
452 MW (1.507 pu)
85°
420 MW (1.400 pu)
450 MW (1.500 pu)The conclusion drawn from Section3 is validated by Table 7. Meanwhile, it can be seen that the system operating at rectifier mode is more severe than inverter mode, which means that the stable margin for VSC working in rectifier mode is narrower than that of inverter mode.


## 5.3. Validations of the Proposed PLL Control

Figure11 shows the simulation results of maximum power transmission with the proposed improved PLL control.Figure 11
The simulation results of system with proposed PLL control.
(a)
Rectifying operation
(b)
Inverting operationIt can be obtained that, with the proposed PLL control strategy, the limits of power transmission are improved. The test model adopts 85° as the AC system impedance angle. Compared with 420 MW, the maximum active power at rectifier is enhanced to 460 MW. Meanwhile, the limits of power transmission at inverter are enhanced from 450 MW to 480 MW. The optimal value of damping factorD calculated at rectifying operation is 1100 and is 300 at inverting operation. Figure 12 shows the relationship between VSC power transfer limitations and damping factors in the proposed PLL control.Figure 12
The relationship betweenD and limits of power transmission.
(a)
Rectifying operation
(b)
Inverting operationIt can be seen from Figure12 that the optimal values of damping factors are approximately equal to the calculated results (1100 and 300, resp.). Figure 13 shows the simulation results of the relationship between system impedance angles and power transfer limitations with D being equal to zero and the optimal values (i.e., 1100 and 300, resp.).Figure 13
The power transfer limitations with the advanced PLL control.
(a)
Rectifying operation
(b)
Inverting operationNote that the power transfer limitation increments are shown in Figure13 using arrows. The validity of the advanced PLL control strategy and the previous impedance angle analysis results are simultaneously validated in Figure 13.


## 6. Conclusions

In this paper, based on small-signal analysis and state-space matrix, the power transfer limitations of VSC linked with weak AC system are analyzed. Besides, the dynamic model with vector-current control is developed to evaluate the improved PLL control. The following conclusions are drawn:(i)
With the rising of active power at PCC, the stability of VSC system is getting worse and even unstable. The stable margin of VSC for rectifying operation is narrower than inverting situation. With the analysis of participation factor, the most sensitive parts to the stability margin are outer-loop control and PLL control.(ii)
The impedance angle of AC system has a great impact on the stable margin of VSC. For rectifying operation, with high impedance angle, the stability of system is enhanced; and this is not the case for inverting operation. The damping characteristic of AC system can change the stable margin of VSC as well.(iii)
The proposed PLL control appends the damping factorD into the existing PLL diagram to change the damping characteristic of VSC system. With the calculated optimal damping factors, the stable margins for VSC working in either rectifier or inverter modes are significantly enhanced.



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*Source: 9537342-2016-03-27.xml*