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On the Synchronization Stability of Converters

connected to Weak Resistive Grids

Line 1: Authors Name/s per 1st Affiliation

Line 2: Author’s Name/s per 1st Affiliation

Line 6: e-mail address if desired

Abstract—The paper proposes a simplified yet accurate converter

model for the analysis of the synchronization stability considering

the effect of weak resistive grids. A thorough comparison with

simulations obtained with detailed EMT models shows that the

proposed model captures precisely the synchronization

transients. Simulation results also indicate that the impact on

synchronization stability of the resistances of the grid on the

synchronization stability depends on the state of the converter

operation.

Index Terms— Synchronization stability, converter-interfaced

generator (CIG), phase-locked loop (PLL).

I. INTRODUCTION

With a migration to a higher renewable penetration grid, the

synchronous generators are gradually replaced by converter-

interfaced generation [1,2]. The dynamic response of a power

system with high renewable penetration is dominated by the

operation and control of these converters. Converter-interfaced

generators (CIGs) shall no longer disconnect from the grid

during the fault but have to maintain the power generation to

avoid further contingencies or even blackout due to the wide

loss of generation [3]. Thus, the ability of the low voltage ride-

through is critical for a stable CIGs operation. However, during

a severe grid fault, the converter may still suffer loss of

synchronization and become unstable even if the low voltage

ride through (LVRT) requirements are satisfied [4]. For

example, this has been highlighted by the British transmission

system operator (TSO), National Grid, which indicated that the

risk of the loss of the synchronization stability of the phase

locked-loop (PLL) based converters is rising during faults in a

weak grid [5]. In this context, an accurate model of the CIG

synchronization stability is a pre-condition to analyze the

mechanism of its grid synchronization and to robustly design

the CIG controller for the enhancement of its stability. In

particular, this paper considers the impact of network losses on

the synchronization stability, an aspect that has not been fully

studied so far.

The study of different types of PLLs has already been

thoroughly investigated, yet only considering its grid-

synchronization loop in a strong grid, for which the voltage at

the point of common point (PCC) is assumed to be fixed and

invariable to the grid power injection from the CIG [6].

However, if the grid is weak, i.e., lines have a non-negligible

inductive component, the PCC voltage is no longer stiff but

couples to the CIG grid power injection and results in the

possibility of synchronization instability. In order to analyze

this phenomenon, a static model identifying the equilibrium

point of the CIG operation with respect to the grid state was

presented in [7,8]. It shows that the allowable LVRT of the CIG

depends on the grid impedance and the reactive current

injection at the PCC. Since this model is static, the proposed

stability criteria are only necessary but not sufficient. The CIG

may lose synchronization stability during the transition to the

equilibrium point. To capture this transient response, a Quasi-

Static Large-Signal (QSLS) model was proposed in [9]. This

model clearly shows that the positive feedback from the self-

synchronization loop worsens the PLL dynamics. Based on this

model, reference [10] illustrated the equivalence of the PLL

dynamics to the synchronization mechanism of the

synchronous generator. Hence, the equivalent damping and

inertia of the PLL is identified. Referring to the stable region of

the SG, reference [11,12] used phasor analysis and numerical

approximations to estimate the PLL stable region. Reference

[13] compares these methods [9-13] and verifies that the QSLS

is more precise than other methods.

In the literature, a second-order QSLS model has been

widely utilized to assess the synchronization stability of multi-

converter systems [14] and unbalanced systems [15]. The

differential equations of this model take into account only the

PLL dynamics, since the converter is assumed to be an ideal

current source. However, in practice, CIGs are commonly

based on voltage-sourced converters, which is essentially a

voltage source, which thus, has transients on the current control.

Reference [16] proves that the dynamics of these currents can

worsen the synchronization stability and modifies the QSLS

model by including a current dynamics loop. However, this

model is based on the inductive grid impedance, while some of

the CIGs are implemented in the distribution system which has

a non-negligible resistive component in the impedance. The

grid resistance changes the power flow in the system, couples

the active and reactive power, and increases the system

damping, thus has an impact on the synchronization stability.

To better capture the CIGs synchronization transients in the

distribution system, this paper modifies the synchronization

stability model in [16] and analyzes the effect of resistive grid

impedance on the synchronization stability.

The remainder of the paper is organized as follows: Section

II analyzes the effect of the grid states on the synchronization

stability; Section III develops a model of the synchronization

stability analysis considering the current transients and

analyzes the effect of the grid resistance. Section IV verifies the

accuracy of the model and shows the effect of the grid

resistance on the synchronization stability in different situations

while Section V draws the conclusion.

Junru Chen1, Muyang Liu1, Terence O’Donnell2 and Federico Milano2

1Xinjiang University, Urumqi, China

2University College Dublin, Dublin, Ireland

junru.chen@xju.edu.cn

II. EFFECT OF GRID STATES ON SYNCHRONIZATION

STABILITY

The grid-feeding converter is widely used in renewable

generation, where is most commonly t controlled to behave like

a current source. The synchronization of this converter relies on

the PLL, which tracks the phase of the PCC voltage as shown

in Fig. 1, where is grid voltage and is grid impedance. The

control of the typical synchronous reference frame PLL (SRF-

PLL) as shown in Fig. 2 aims to force the detected phase from

the PLL to track the phase of the fundamental component

of PCC voltage , as follows:

Equation (1) describes the q-axis component of the PCC

voltage as detected by the PLL. When the phase is locked, i.e.

, then the q-axis component of the PCC voltage at

the fundamental frequency (1) should be zero. is thus

the condition for a successful converter synchronization.

Fig. 1. Grid-Feeding Converter system structure

The analysis of (1) is the key to understand the

synchronization stability. However, in the power system, only

the grid voltage is known and the PCC voltage is a

consequence of the power flow through the grid impedance,

which mainly depends on the state of the grid. Considering this,

the modeling of the synchronization stability using (1) has to be

extended correspondingly. This section models and analyzes

the effect of the grid state on the synchronization stability.

A. Synchronization Stability in a Strong Grid

In a strong grid with a negligible grid impedance, i.e.

, the PCC voltage always equals the grid voltage. Assuming

the is the reference angle, i.e. , then, (1)

can be rewritten as follows:

In this case, the synchronization stability (1) is solely

depending on the grid-synchronization loop of the PLL with

no influence from the grid state, i.e. Fig. 2. A proper H(s) can

ensure a solid synchronization. Hence, for the strong grid, the

research concerning synchronization stability focuses on the

design of the PLL controller [17].

Fig. 2. PLL structure

B. Synchronization Stability in Weak Grid

In the case where the grid impedance is nonnegligible, i.e.

weak grid, then the PCC voltage changes with the power flow.

For a simple computation, assuming that is the reference

angle, . Then, (1) can be rewritten as follows:

where and

is the reference value of the

d-axis component of the current, and

assumes perfect

current tracking. In comparison with (2), besides the PLL grid-

synchronization loop

, the synchronization

stability (3) in a weak grid has an additional self-

synchronization loop represented by the term , which

acts as a positive feedback in the overall synchronization loop,

which thus worsens the synchronization stability as shown in

Fig. 3. In the literature, grid-feeding converters are commonly

assumed to be ideal current sources,

. Hence, the effect

of the grid impedance is seen as a proportional gain and the total

order of this model remains the same as when applied in the

strong grid.

Fig. 3. Quasi-static large-signal model of the PLL

C. Synchronization Stability in a Distribution System

Small-capacity CIGs are generally connected to the

distributed system, where line impedances have a non-

negligible resistance . Based on (3), the synchronization

stability can be written as (4).

where . In general, CIGs are supposed to

compensate the reactive power to the grid during the LVRT, in

which case the value of is negative.

The following are relevant remarks based on (4):

• If , the phase angle is positive.

The capacitive current through the grid resistance

could partially cancel the positive effect of the

term on

, thus widen the CIG

operational range and increase the static voltage

stability. This conclusion has been well verified by the

previous literature [8].

• If , the phase angle is negative.

A further increase in will enlarge the phase

negatively thus degrade the converter stability.

• If

, no equilibrium point

exists. The converter is unstable.

III. EFFECT OF THE CURRENT TRANSIENTS ON THE

SYNCHRONIZATION STABILITY

In the previous literature, the grid-feeding converter is

assumed to be an ideal current source. The analysis of the

synchronization stability only considers the PLL dynamics but

neglects the current controller dynamics. This is because

generally the time constant of the PLL is around 50-100 ms

while that of the current controller is 0.5-5 ms [18]. However,

in reality, the grid-feeding converter is a voltage source

converter, for which the terminal voltage at the instant of the

fault remains fixed resulting in an excessive fault current.

Although the current controller can suppress this fault current

within a few millisecond, the fault current may be much higher

than the reference which enlarges the positive feedback

, resulting in a larger during the transient even

converging into an unstable region. The converter may be

unstable even if its QSLS model indicates stability, due to the

effect of the current transients. Reference [16] added the loop

of the current transients into the model (4) thus making its

stability criteria become both sufficient and necessary. Based

on the advanced model in [16], this section extends to consider

the effect of the grid resistance on the synchronization stability.

A. Model of Current Transients On Synchronizatio Stability

Defining as the transient current change, i.e.

and

and substituting these into (4) gives:

Compared to (4), the transient current change includes

another two loops for the transients of the and in the

model. Now, the synchronization stability couples to both

active and reactive current. The d-axis voltage at the PCC is

given by:

During the fault, the PCC voltage changes along with the

state of the fault, while the converter terminal voltage is the

consequence of the current controller actions. The transient

current is attributed to the voltage difference between the PCC

and converter terminal voltage dropped across the filter:

where the converter terminal voltage is and in line with the

current it has a static and a transient component, i.e.

; . The static component is

attributed to the reference current:

where

is the pre-fault

phase. The transient component is attributed to the transient

current change, or the current error,

and

, and arises only after the current controller acts :

Equations (5-12) represent the model of the synchronization

stability of the grid-feeding converter. Fig. 4 shows the model

structure. In comparison with the QSLS model, this model

includes additional loops for the current transients and thus

elevates the order of the model to be 4th order.

Fig. 4. Model of synchronization stability analysis considering the current

transients and grid resistance

B. Analysis of Grid Losses on Synchronization Transients

The effect of the grid resistance on the QSLS

synchronization transients, i.e. (4) has been described in Section

II-C. In addition, the grid resistance also affects the current

transients as indicated in (5~10), which will have an impact on

the synchronization stability.

At the instant of the fault , the converter terminal

voltage remains invariant as described by (9,10), approximately

at the nominal value. At the occurrence of the fault, the grid

voltage sags from

to

, thus instantly lowering the

PCC voltage. The voltage difference between the converter

terminal and grid at this moment is significant, thus, resulting

in a peak current. Substituting (9,10) and the grid voltage

change

into (5-10) gives the resulting transient current

change at the peak:

If the transients from the impedance (

) are neglected,

(13) and (14) indicate that the peak current is solely related to

the filter inductance. The d-axis current increases at the instant

of the fault whatever the initial state of the system, which

worsens the synchronization transients when while

improving it when as indicated before in (4). On the

other hand, the q-axis current change and the initial phase

present a negative correlation. As analyzed in Section II-C, the

negative increase in q-axis current enhances the

synchronization stability, thus, the transient q-axis current

benefits the stability. Note, since the initial phase

normally is small, the current transients on d-axis is more

significant than that on q-axis.

IV. SIMULATION RESULTS

A real-time Electromagnetic Transients (EMT) simulation

in Matlab/Simulink is used to validate the above analysis and

specifically to verify following aspects:

• The accuracy of the proposed model for the

synchronization stability analysis in comparison to the

QSLS model;

• The impact of the current transients on the synchronization

stability;

• The impact of the grid resistance on the synchronization

stability.

The proposed model and QSLS model are built in

Matlab/Simulink using only math blocks. The initial system is

at nominal with 50 Hz,10 kV. The PI controller of the PLL has

gains 0.022/0.392. A voltage sag occurs in the grid at 5 s from

nominal to 0.32 pu.

A. Model Validation and Accuracy

Two cases are considered in this section to validate the

accuracy of the proposed model: Case 1 has a small current

transients with 0.1 ms current controller time constant, i.e.

; and Case 2 has a larger

current transients with 0.5 ms current controller time constant,

i.e. . In both cases,

. Figure 5 shows

the transients of the PLL phase and the converter output

current in both d-axis and q-axis after the fault occurrence.

The proposed model can accurately capture the PLL

behaviour during the transient in comparison with the result

from the EMT model. Since it neglects current transients, the

QSLS model presents the same response for both cases. In the

case of , a larger current transient make the

PLL lose synchronization and the converter becomes unstable

after the fault. This is because the transient current in d-axis

surges to above 100 A and causes the phase to move into the

unstable region. After that, the current oscillates at the

saturation frequency of the PLL.

B. Effect of

This scenario considers same parameters of the converter as

those utilized in Case 1 of Section IV.A. Figure 6 shows the

results for different grid resistances but ensuring that

, i.e., 0 Ω, 5 Ω and 20 Ω. The negative reactive current

through the grid resistance adds a negative feedback into the

synchronization transients and results in a phase reduction in

both transient and steady state. The inclusion of the grid

resistance enhances the synchronization stability. Moreover,

the grid resistance does not impact on the peak of the current as

shown in Fig. 6 (b) at 5 s.

Fig. 5. Model Validation of synchronization stability assessment for two cases,

Case 1: small current transient; Case 2: larger current transient.

Fig. 6. Effect of on synchronization stability.

C. Effect of

This scenario considers same parameters of the converter

parameters as those considered in Section IV.B except for the

current references, which are set to

in

order to ensure that . Figure 7 shows the

results obtained with various values of the grid resistance. The

initial phase becomes negative and the imposing of a negative

feedback into the synchronization transients at both transients

and steady-state due to the inclusion of the capacitance current

through the grid resistance increases the phase negatively and

reduces the stability margin.

5 5.2 5.4 5.6 5.8 6

0

1

2

3

Phase (rad)

Time (s)

EMT1

Proposed1

EMT2

Proposed2

QSLS

5 5.2 5.4 5.6 5.8 6

80

100

120

id (A)

Time (s)

EMT1

Proposed1

EMT2

Proposed2

QSLS

5 5.2 5.4 5.6 5.8 6

-70

-60

-50

-40

-30

iq (A)

Time (s)

EMT1

Proposed1

EMT2

Proposed2

QSLS

(a) PLL phase transients

(b) Current transients on d-axis

(c) Current transients on q-axis

5 5.2 5.4 5.6 5.8 6

0

1

2

Phase (rad)

Time (s)

rg=0

rg=5

rg=20

5 5.2 5.4 5.6 5.8 6

81

82

83

84

85

id (A)

Time (s)

rg=0

rg=5

rg=20

(a) PLL phase transients

(b) Current transients on d-axis

Fig. 6. Effect of on synchronization phase transients.

D. Effect of current transients at

In this case, we repeat the tests carried out in Section IV.A

but use the current references in Section IV.C to impose the

condition . The grid fault now drops to 0.2 pu

at 5 s. Figure 7 shows that, compared to the EMT model, the

proposed model properly captures the synchronization

transients, whereas the QSLS model fails. For

, a larger current transient enhances the transient

stability and even make an unstable system stable. This

conclusion is opposite to that in a situation of

. The grid resistance adds damping into the system thus

slows down the phase change in comparison with Fig. 5 (a).

This gives an extra time to clear the fault.

Fig. 7. Effect of current transients in the situation of

V. CONCLUSIONS

The paper proposes a simple yet accurate model to assess

the synchronization stability of converters connected to weak

resistive grids. The proposed model properly takes into account

the transient behavior of the currents of the converter and

approximates well the transient response of the fully-fledged

EMT converter model. Simulation results indicate that the

impact of the resistance of the lines to which the converter is

connected is not negligible. A noteworthy conclusion of this

work is that these resistances have a significant role in the

synchronization stability. In particular, the transient behavior of

the converter currents worsens if and

improves if . Future work will focus on the

design of a converter current control that makes the

synchronization stability independent from the impedance of

the grid.

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5 5.2 5.4 5.6 5.8 6

-0.8

-0.6

-0.4

-0.2

0

Phase (rad)

Time (s)

rg=0

rg=5

rg=20

5 5.5 6 6.5 7 7.5 8

-3

-2

-1

0

Phase (rad)

Time (s)

EMT1

Proposed1

EMT2

Proposed2

QSLS