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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
