Analysis of the DC Dynamics of VSC-HVDC Systems

Analysis of the DC Dynamics of VSC-HVDC
Systems Connected to Weak AC Grids Using a
Frequency Domain Approach
Gustavo Pinares
Division of Electric Power Engineering
Department of of Energy and Environment
Chalmers University of Technology
412 96 Gothenburg, Sweden
Email: [email protected]
Abstract—In this paper, an approach to evaluate the dynamic
performance of the dc side of a voltage-source-converter-based
HVDC (VSC-HVDC) system is proposed. The approach is based
on the analysis of subsystems which compose the HVDC system.
The VSC and the dc grid subsystems are defined in this paper
and their frequency responses under different conditions are
studied. An explicit expression for the transfer function of the
VSC subsystem connected to a strong ac grid is derived, and
through this, it is shown that the VSC subsystem frequency
response has a 180° phase shift at high frequencies and when
it injects power into the dc cable. The frequency response of
the dc grid subsystem shows a resonant behaviour due to the
RLC nature of the dc cable. In unstable cases, the resonance
becomes undamped due to negative conductance characteristic
of the VSC subsystem at the dc grid resonance frequency. The
transfer function of a VSC subsystem connected to a weak ac
grid is derived as well, and it is shown that the amplification of
the resonance originated in the dc cable is more considerable.
Keywords—VSC, HVDC, Eigenvalue Analysis, Frequency Domain Analysis, Passivity, VSC Admittance, VSC conductance, dcside resonance.
I.
I NTRODUCTION
HVDC systems are composed of complex elements such as
converters and cables whose interaction may lead to undesired
behaviour. For instance, cables have a resonant behaviour due
to their RLC characteristic and the resonance may be amplified
by the converter dynamic characteristics [1]. Typically, the dynamic evaluation of dc systems has been investigated through
eigenvalue analysis. An example of that is [2] where unstable
cases in the dc side of the system has been detected. Through
eigenvalue analysis, it has been found that the instability is
related to the dc-side resonance phenomenon and the operating
conditions. Moreover, the instability is attributed to the negative resistance characteristic introduced by the Voltage Source
Converter (VSC) which controls the active power. Other works
that have applied eigenvalue analysis are [3]–[5]. However,
their analysis focus more on the controller design rather than
the analysis of the dc-side stability in itself.
Although eigenvalue analysis is a powerful tool to analyze the stability of a system, it gives little insight on the
origin of the problem. One way to gain understanding on the
This work has been funded by the Chalmers Energy Initiative program
th
18 Power Systems Computation Conference
causes of the instability is to study the frequency response
of the elements which compose the system. For instance, [8]
introduces the immittance analysis in dc microgrids. In this
method, the system is divided in two subsystems, the load and
the source. The frequency response of the load admittance,
YL , and the source impedance, ZS , are derived, and the dc
microgrid is stable if the contour defined by YL ZS fulfills the
Nyquist stability criterion. A similar approach is presented in
[6], however, with the aim of investigating interactions between
ac systems and VSCs. It is found that, at low frequencies,
the VSC subsystem has a negative conductance characteristic,
which can amplify resonances originated in the ac side.
This work is the continuation of [9], [10], where the dc-side
dynamics of VSC-HVDC systems is investigated. The main
result from [9] is that the studied system becomes unstable
when the power flow through the HVDC system exceeds
certain value. The origin of the instability is investigated in
[10], where frequency domain analysis is applied. The VSC
subsystem and the dc grid are modeled as separate subsystems,
and it has been found that the frequency response of the
VSC subsystem has a 180° phase shift which amplifies the
resonance originated in the dc circuit. In [9] and [10], one
of the main assumptions is that the VSCs are connected to
strong ac systems. However, in this paper, the ac system is
considered weak, meaning that the phase-locked loop (PLL)
has to be considered in the model.
The contribution of this paper is a modelling approach
that helps on explaining dc-side instabilities in a VSC-HVDC
system. Moreover, a procedure to identify the conditions that
leads to instability is proposed. The procedure is based on the
analysis of the frequency response of two subsystems, named
in this paper as the VSC and the dc grid subsystems. The
VSC subsystem is the VSC which controls the direct-voltage
connected to an ac grid. The dc grid is the cable, modelled as a
Π-section, merged with the resistances that originate from the
linearization procedure (later explained in Section III-B). The
organization of this paper is as follows: Section II introduces
the system under study and some preliminary investigations
are carried out through the eigenvalue analysis. In Section
III, the models of the subsystems, i.e. the dc grid and the
VSC subsystem, are derived. With the models derived, the
analysis procedure is presented in Section IV. Finally, the main
conclusions of this work are presented in Section V.
Wroclaw, Poland – August 18-22, 2014
II.
P RELIMINARY STUDIES
reactor is described by
The system under study consist of two converters, VSC1 ,
which controls the direct-voltage, and VSC2 , which controls
the active power. The converters are interconnected through a
50 km cable as depicted in Fig. 1. The data assumed in this
paper is indicated in Table I.
Fig. 1.
0
uqc
=−kpc (iqref
f
+uqg
−
−
iqf )
ωg Lf iqf
− kic
t
Z
0
(iqref
− iqf )dt
f
(2b)
Parameter
Value
VSC rated power (Sbase )
600 MW
VSC rated alternating-voltage (Uacbase )
300 kV
VSC rated direct-voltage (Edcbase )
±300 kV
kpc = αLf , kic = αRf
AC inductance base (Lacbase )
0.477 H
DC impedance base (Zdcbase )
300 Ω
with α selected as 4 pu (200 Hz) as suggested in [6].
Considering (1), (2) and (3), the closed-loop system becomes a
first-order decoupled system, as shown in the Laplace domain
α dref q
α qref
i , if =
i .
(4)
idf =
s+α f
s+α f
150 Ω
DC inductance base (Ldcbase )
0.954 H
DC capacitance base (Cdcbase )
10.61 µF
Nominal frequency (fbase )
50 Hz
Reactor inductance, Lf
119.4 mH (0.25 pu)
Reactor resistance, Rf
0.375 Ω (0.0025 pu)
Converter capacitor, C
33 µF (3.14 pu)
Cable capacitance, C1
10.35 µF (0.97 pu)
Cable inductance, L12
9.45 mH (9.88·10-3 pu)
Cable resistance, R12
1.9 Ω (6.27·10-3 pu)
A. Control system description
In Fig. 2 the control system of VSC1 is sketched. The
core of the control system is the current controller, which is
implemented using the vector current control method. With this
method, the alternating three-phase quantities are transformed
into two-component dc quantities, in the so-called rotating
dq-frame, with the d axis aligned to the rotating vector ug .1
In order to perform the transformation, the angle θg and the
frequency ωg are estimated by the Phase-Locked Loop (PLL).
In the dq-frame, the dynamics of the current over the phase
Sketch of the VSC control system.
1 Underlined variables with not superscript denote two-component vectors
in a stationary reference frame (the so called αβ-frame). Underline variables
with the dq superscript denote two-component vectors in the dq-frame. For
instance z dq = z d + jz q .
th
(1b)
+udg + ωg Lf idf
S YSTEM DATA
AC impedance base (Zacbase )
Fig. 2.
(1a)
The current controller is usually implemented as a
Proportional-Integral (PI) controller with a feedforward of the
voltage ug and a current cross-coupling compensation. The
control law in the dq-frame is defined as
Z t
d
udc =−kpc (idref
−
i
)
−
k
(idref
− idf )dt
(2a)
ic
f
f
f
Two-terminal VSC system under study.
TABLE I.
udg
ud
− c
Lf
Lf
uqg
uq
− c.
Lf
Lf
didf
Rf
=− idf + ωg iqf +
dt
Lf
diqf
Rf q
=− if − ωg idf +
dt
Lf
18 Power Systems Computation Conference
where kpc and kic are selected as suggested in [11]
(3)
The Direct-Voltage Controller (DVC) is also a PI controller,
and the control law assumed in this paper is
Z t
ref
idref
=
k
(e
−
e
)
+
k
(eref
(5)
pe 1
1
ie
f
1 − e1 )dt.
0
If the current controller is assumed much faster than the
DVC, then, the characteristic polynomial of the closed-loop
system is of second-order, s2 + kpe C −1 s + kie C −1 , then,
according to [12] kpe and kie can be selected as
kpe = 2αe ξC, kie = αe2 C
(6)
where αe is the natural resonance frequency of the closed-loop
system, and ξ is the damping ratio. For the tests performed in
Section II-B, kpe is selected as 9.23 pu and kie as 1.23 pu.
B. Eigenvalue study
In [9], the system shown in Fig. 1, considering both VSCs
and the dc cable dynamics, is linearized and its state-space
model is derived. Considering that the VSCs are connected
to strong ac grids, the poles are calculated for power flows,
starting from −1 to +1 in steps of 0.1 pu (positive means
from VSC1 to VSC2 ). The pole placement is shown in the
left plot of Fig. 3, and it can be seen that they move towards
the unstable region as the power flow from VSC1 to VSC2
increases, becoming unstable when the power exceeds around
0.8 pu. In the right plot of Fig. 3, the poles are plotted for the
case when VSC1 is connected to an ac system with a ShortCircuit Ratio2 (SCR) of 5, with the PLL modelled as described
2 The SCR is defined as the ratio between the short-circuit power of the ac
system and the rated power of the converter, as defined in [13].
Wroclaw, Poland – August 18-22, 2014
later by (39). It can be seen that, in this case, the system
becomes unstable when the power exceeds around 0.5 pu. It
must be highlighted that a SCR of 5 represents a rather strong
ac system; however, the low limit of the power transference
is due to the fact that the gains of the DVC are high and that
no voltage support in the ac side has been considered. Fig.
·103
1
0
−1
−2
1
0
−1
−2
−3
−2
−1
0
real axis
1
idc1 = Pdc1 /e1 .
−4 −3 −2 −1
0
1
real axis
·102
2
3
·102
∆idc1 = −(Pdc10 /e210 )∆e1 + (1/e10 )∆Pdc1
∆idc2 = −(Pdc20 /e220 )∆e2
(9)
(10)
R10 = −e210 /Pdc10 , R20 = −e220 /Pdc20 .
(11)
where the subscript “0” means that the variable is a constant
whose value corresponds to the initial operating conditions.
Considering (9) and (10) the VSC-HVDC system can be
represented as shown in the Fig. 5, where the dc cable is
modelled as a Π-section and the resistances R10 and R20
originate from (9) and (10), respectively. They are
Fig. 3. Pole placement for power flows of −1(◦) to +1(4). Left: VSC1
connected to a strong ac grid. Right: VSC1 connected to a weak ac grid.
4 shows simulations results for the same two cases. In the
left plot, the power flow is increased from 0.79 pu to 0.80
pu in the VSC-HVDC system connected to strong ac sources
and it can be seen that instability occurs as predicted by the
eigenvalue analysis. Similarly, in the right plot, it can be seen
that instability takes place when the power flow increases from
0.38 pu to 0.40 pu, which is slightly smaller than the value
predicted by the eigenvalue analysis.
(8)
The linearization of (7) and (8), gives
·103
2
imaginary axis
imaginary axis
2
where Pdc20 is constant and e2 is the voltage of the dc node to
which VSC2 is connected (see Fig. 1). Similarly, the current
injected by VSC1 to the dc side is
Note that the resistances R10 and R20 can take negative
values and that they have opposite signs. In this paper, the
dc grid model is the result of merging the resistances the
resistances R10 and R20 with the Π-model of the cable, as
shown in Fig. 5. In the figure, the current injected by the VSC1
is ∆i∗dc1 , which, from (9), is
∆i∗dc1 = ∆Pdc1 /e10 .
(12)
e1 [pu]
From the analysis, it can be seen that, for the selected
values for kpe and kie , the power flow is limited even when
VSC1 is connected to a strong ac grid. The power limit is
further reduced when the ac system is assumed weak. The
origin of the instability is investigated using the frequency
domain approach in the following sections.
1.04
1.04
1.02
1.02
1
1
0.98
0.98
0.96
0.5
1
1.5
2
0.85
0
0.5
1
1.5
B. DC grid model
2
0.85
0.7
0.7
0.55
0.55
0.4
0.4
0.25
0.25
0
0.5
1
1.5
Time [s]
2
0
0.5
1
1.5
Time [s]
2
Fig. 4. Simulations in the studied HVDC system. Left: VSCs connected to
strong ac grids. Right: VSCs connected to ac grids with an SCR of 5.
III.
F REQUENCY DOMAIN MODELLING
In [2], [10], it has been shown that the converter which
controls the active power, in this case VSC2 , can be assumed
to be a constant power device while still conserving the main
dynamic characteristics of the dc side. That is, the current
injected by VSC2 can be represented as simply
idc2 = Pdc20 /e2
18 Power Systems Computation Conference
A Single-Input Single-Output (SISO) model, with the current ∆i∗dc1 as the input and the voltage ∆e1 as the output, is
derived in this section. Solving the circuit enclosed by the “dc
grid subsystem” box in Fig. 5 (which is equivalent to find the
impedance seen from the point where ∆i∗dc1 is injected), the
following transfer function is obtained
G(s) =
−1
Ceq
n(s)
∆e1
(13)
=
∆i∗dc1
(s + ωc1 + ωc2 )(d(s) + ωc1 ωc2 ) + δ
where
A. Preliminary considerations
th
Linearized model of the VSC-HVDC system.
0.96
0
P1 [pu]
Fig. 5.
(7)
2
d(s) = s2 + ωrl s + 2ωlc
2
n(s) = s2 + (ωrl + ωc2 )s + ωlc
+ ωrl ωc2
δ
= −ωc1 ωc2 (ωc1 + ωc2 )
Ceq = C + C1 (VSC cap. plus cable equiv. cap.)
ωc1 = 1/(R10 Ceq ), ωc2 = 1/(R20 Ceq ),
2
ωlc
= 1/(L12 Ceq ), ωrl = R12 /L12 .
(14)
(15)
(16)
(17)
(18)
Wroclaw, Poland – August 18-22, 2014
Considering that the maximum value that |Pdc10 | and
|Pdc20 | can reach is 1 and also that R10 + R20 = R12 , it
can be shown that
R12
R12
1
− 3 ≤ δ ≤ 0, − 2 ≤ ωc1 +ωc1 ≤ 0, −
≤ ωc1 ωc2 ≤ 0
Ceq
Ceq
Ceq
which, numerically, from Table I, are
−0.9 · 10−4 ≤ δ ≤ 0
−1.3 · 10−3 ≤ ωc1 + ωc2 ≤ 0
−5.9 · 10−2 ≤ ωc1 ωc2 ≤ 0.
(19a)
(19b)
(19c)
and, considering the Fig. 2, the voltages udc and uqc are
udc = uds − (Rt + sLt )idf + ωg Lt iqf
uqc = uqs − (Rt + sLt )iqf − ωg Lt idf
where Lt = Lf + Ls , and Rt = Rf + Rs . Furthermore udq
s
is the source voltage in the converter dq-frame.3 It should be
4
noted that udq
s is actually constant in a synchronous dq-frame
sdq
since us is a fixed-frequency ac voltage source (then ∆us
is equal to zero). From Fig. 6, quantities in the synchronous
dq-frame and the converter dq-frame are related as
θg = θgs + ∆θg
The denominator of (13) is a third-degree polynomial,
a3 s3 + a2 s2 + a1 s + a0 , where the coefficient a0 is
2
a0 = (ωc1 + ωc2 )(2ωlc
+ ωc1 ωc2 ) + δ
(20)
and from (19), it can be shown that δ can be neglected since
it is much lower than a0 . Then, (13) is approximated as
G0 (s) =
−1
Ceq
n(s)
.
(s + ωc1 + ωc2 )(d(s) + ωc1 ωc2 )
(22)
and the poles of (22) are
λ1 = −(ωc1 + ωc2 )
λ2,3
√
ωrl
=−
± j 2ωlc
2
(23a)
s
2
ωrl
2 −1
8ωlc
(23b)
where λ1 is an unstable pole and, therefore, Bode diagrams
cannot be used to study the system described by (22). However, the minimum bandwidth of the feedback control system
required to stabilize (22) is small since the unstable pole λ1 is
small, compared to λ2,3 . Then, as shown in [10], (22) can be
further approximated by the marginally stable transfer function
e 0 (s) =
G
−1
Ceq
n(s)
s × d(s)
(24)
which can be analyzed with Bode diagrams to study the
stability of the VSC-HVDC system. It should be noted that this
approximation is valid for this particular case. For other cable
parameters the approximation procedure must be re-evaluated.
C. Model of the VSC subsystem connected to a weak ac grid
If the VSC is assumed lossless, the active power at the ac
side of VSC1 is equal to the power at the dc side. That is
Pdc1 = udc idf + uqc iqf
(25)
which in terms of small deviations is
∆Pdc1 = udc0 ∆idf + uqc0 ∆iqf + idf0 ∆udc + iqf0 ∆uqc ,
(26)
therefore, (12) can be expressed as
∆i∗dc1 =
th
1
ud ∆id + uqc0 ∆iqf + idf0 ∆udc + iqf0 ∆uqc (27)
e10 c0 f
18 Power Systems Computation Conference
dq
sdq −j∆θg
z =z e
∆z dq = ∆z sdq − jz sdq
0 ∆θg
(29a)
(29b)
(29c)
where z is any voltage or current from the ac side. It should
(21)
2
Furthermore, the numerical value of 2ωlc
is 49.1 pu which
is much greater than ωc1 ωc2 . Therefore, (21) can be further
approximated as
−1
n(s)
Ceq
e
G(s)
=
(s + ωc1 + ωc2 )d(s)
(28a)
(28b)
Fig. 6.
sdq
udq
represented in the dq and the sdq frame.
g and ug
be also noted that, initially, θg is equal to θgs . Then, (28) can
be linearized and expressed in the converter dq-frame as
∆udc = uqs0 ∆θg − (Rt + sLt )∆idf + ωg Lt ∆iqf
+iqf0 Lt ∆ωg
(30a)
∆uqc =−uds0 ∆θg − (Rt + sLt )∆iqf − ωg Lt ∆idf
−idf0 Lt ∆ωg
(30b)
and, in steady state udc0 and uqc0 are
udc0 = uds0 − Rt idf0 + ωg0 Lt iqf0
uqc0 = uqs0 − Rt iqf0 − ωg0 Lt idf0 .
(31a)
(31b)
It should be highlighted that if VSC1 is connected to an
dq
infinite ac source, Ls and Rs are zero and udq
s is equal to ug
dq
and it is constant, then, ∆ug is zero. Continuing with the nonstrong ac source case, (30) and (31) are put into (27), and the
following is the current injected to the dc side by the VSC1 .
iq Lt
idf0 Lt
(s + zd )∆idf − f0 (s + zq )∆iqf
e10
e10
uqs0 idf0 − uds0 iqf0
+
∆θg
(32)
e10
∆i∗dc1 =−
where zd and zq are
zd = 2
Rt
ud
Rt
uq
− d s0 , zq = 2
− q s0 .
Lt
Lt
if0 Lt
if0 Lt
(33)
3 In the converter dq-frame, the transformation is performed with the angle
θg estimated by the PLL.
4 The synchronous dq-frame rotates with the constant frequency ω , and
g0
the transformation is performed with the angle θgs which can be defined as:
θgs = θg0 + ωg0 t. Quantities in the synchronous dq-frame are represented as
z sdq = z sd + jz sq .
Wroclaw, Poland – August 18-22, 2014
Entering (4) into (32) and assuming that ∆iqref
is zero, it
f
becomes
αid Lt s + zd
Qs0
∆i∗dc1 = − f0
∆idref
+
∆θg (34)
f
e10
s+α
e10
where Qs0 = (uqs0 idf0 − uds0 iqf0 ). VSC1 controls the directvoltage e1 with the control law (5), which can be entered into
(34) in terms of small deviations. Considering kie = 0, the
following is obtained
∆i∗dc1
where
Qs0
= Fc (s)∆u +
∆θg
e10
αid Lt kpe
Fc (s) = − f0
e10
s + zd
s+α
(35)
.
(36)
If the ac system is infinite, the angle variation ∆θg is zero
since the ug has a fixed frequency. Then, the VSC subsystem
transfer function for infinite ac sources, Fis , is
∆i∗dc1
αidf0 Lf kpe s + zdis
Fis (s) =
=−
(37)
∆u
e10
s+α
where ∆u is (∆eref
1 − ∆e1 ) and
zdis = 2
Rf
−
Lf
(38)
.
0
with kpl and kil selected as 0.2 pu and 0.01 pu, respectively.
In terms of small deviations, in the Laplace domain (39) is
kpl s + kil
∆uqg = Fpll (s)∆uqg .
s2
(40)
Considering Kirchoff law and that ∆ωg = sθg , then
∆uqg = − (uds0 + sidf0 Ls )∆θg − (Rs + sLs )∆iqf
(41)
− ωg0 Ls ∆idf .
Combining (40) and (41), the following is expression is
obtained
(Rs + sLs )Fpll (s)
∆iq
1 + (uds0 + sidf0 Ls )Fpll (s) f
ωg0 Ls Fpll (s)
−
∆id .
1 + (uds0 + sidf0 Ls )Fpll (s) f
∆θg = −
th
(42)
∆iqref
f
= 0, (42) becomes
e10
Fθ (s)∆u.
Qs0
(43)
Considering (4) and (5), and that
∆θg =
Fθ (s) = −
18 Power Systems Computation Conference
ωg0 Qs0 Ls kpe Fpll (s)
.
e10 (1 + (uds0 + sidf0 Ls )Fpll (s))(s + α)
(44)
Finally, using (43) into (35), the transfer function F for the
weak ac grid case is
F (s) = Fc (s) + Fθ (s).
(45)
Then, the system shown in Fig. 5 can be represented by the
SISO feedback system shown in Fig. 7, where F is the VSC
subsystem transfer function mentioned in the previous sections,
e 0 is the approximated dc grid transfer function. In the
and G
following section, the characteristic of the derived transfer
functions are studied.
Fig. 7.
Block diagram of the simplified VSC-HVDC system.
IV.
udg0
idf0 Lf
Continuing with (35), ∆θg has to be expressed in terms of
∆idf and ∆iqf in order to derive a transfer function similar to
(37). The angle θg and the frequency ωg come from the PLL
block, which is defined as
Z t
ωg = kpl uqg + kil
uqg dt
(39a)
0
Z t
θg =
ωg dt
(39b)
∆θg =
where
F REQUENCY DOMAIN ANALYSIS
The stability of the system can be studied considering the
e 0 , and the Nyquist stability
passivity properties of F and G
criterion. According to [15], a feedback system as the one
e 0 , are
shown in Fig. 7 is stable if both subsystems, F and G
passive. However, if any of the subsystems is non-passive, the
SISO feedback system is not necessarily unstable. In such a
case, the stability of the system can be determined through the
e 0 should not
Nyquist criterion. That means that the contour F G
encircle clock-wise the point −1 of the real axis for the system
to be stable.
e 0 for power flows
Fig. 8 shows the frequency response of G
of −1 and +1. It also shows the frequency response of the
transfer function of the dc cable, G0 . The transfer function
G0 is obtained when the resistances R10 and R20 are not
considered as part of the model dc grid model, and it is
G0 (s) =
2
−1 2
(s + ωrl s + ωlc
)
Ceq
.
2
2
s(s + ωrl s + 2ωlc )
(46)
e 0 , are passive subsystems since they are stable
G0 and G
and their phase angle is always less than 90° (see Fig. 8).
Moreover, it can be seen that the three cases match closely
to the dc cable transfer function, G0 , which means that the
frequency spectrum of the actual impedance seen from the
point where VSC1 is connected can be utilized for the analysis.
The fact that the dc grid transfer function is passive implies
that VSC subsystem transfer function is non-passive in the
unstable cases. This is analyzed next.
A. Study of the VSC connected to a strong ac system
Equation (37) shows that the transfer function of a VSC
connected to a strong ac system has a zero, −zdis , which
depends on the operating point. Neglecting the resistance Rf , it
can be seen that the zero is located in the left-half plane (LHP)
of the s-plane when the current idf0 is negative (the current if is
Wroclaw, Poland – August 18-22, 2014
Ph. [deg.]
10−2
ω = 7 pu
10−1
100
101
102
45
0
−45
−90
10−1
100
101
102
Frequency [pu]
e0 (s) for a
Fig. 8. Frequency response of: Solid gray: G0 (s). Dashed: G
e0 (s) for a power flow of +1.
power flow of −1. Dotted G
positive if it follows the convention adopted in Fig. 2), while
it is located in the right-half plane (RHP) when idf0 is positive.
As mentioned in [16], RHP zeros impose a limitation on the
bandwidth that the closed-loop system can achieve. Moreover,
in this particular case, the fact that there is a RHP zero for
positive power flow implies that Fis is non-passive for any
positive power flow. This can be seen in Fig. 9 where the phase
of Fis decreases towards −180° as the frequency increases.
The opposite is also true, i.e. Fis is passive for negative power
flow. Again, Fig. 9 shows that the phase angle of Fis is around
zero when the power flow is −1.
Mag. [pu]
When the power flow is positive, Fis is a passive subsystem
for low frequencies, while it becomes non-passive for high
frequencies. Physically this means that the energy of a low
frequency dc-side resonance is dissipated by the converter and,
therefore, the system is stable. However, at high frequencies,
due to the 180° phase shift of the VSC subsystem transfer
function, the dc-side resonance might be amplified instead. As
mentioned earlier, the fact that the converter is non-passive is
not enough to claim that the feedback system is unstable. For
8
6
Ph. [deg.]
10−2
10−1
100
101
102
103
0
−90
−180
10−2
10−1
100
101
Frequency [pu]
102
103
Fig. 9. Frequency response of F for three different power flows. Solid gray:
Power flow equal to −1. Solid black: Power flow = +1. Dashed: Power flow
= +0.8. Dotted: Power flow = +0.5.
a power flow of 0.5 pu, Fis is non-passive, but the system
is stable as shown in Section II. It is only after the power
exceeds 0.8 pu that the system turns unstable. The instability
can be explained by looking at the magnitude of Fis plotted
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18 Power Systems Computation Conference
in Fig. 9 which increases when the power flow increases. This
e 0 increase in such a way that
makes the magnitude of Fis G
the Nyquist stability criterion is not fulfilled anymore when
the power exceeds 0.8 pu. From this analysis, Fis can be seen
as the VSC admittance whose conductance becomes negative
at high frequencies and positive power flows. In addition, the
VSC conductance increases as the power flow increases. From
(37), it can be seen that the proportional gain of the DVC
plays a roll also in the stability of the system. If kpe is high,
the VSC admittance increases and the SISO system becomes
more prone to instability. That is the reason why in [9] the
instability did not take place when kpe was low.
B. Impact of the SCR on the converter system transfer function
The reason why the power is further limited when VSC1
is connected to a weak ac grid can be investigated by studying
the frequency responses of F shown in Fig. 10 for three
different SCRs. It can be seen that the phase of F is around the
same for the three cases. Moreover, the figure shows that the
decrease of the SCR increases the VSC admittance magnitude,
increasing the risk of amplifying the dc-side resonances. In
Mag. pu]
peak=0.2 pu
10−1
20
15
10
10−2
Ph. [deg.]
Mag. [pu]
100
10−1
100
101
102
103
10−1
100
101
Frequency [pu]
102
103
0
−90
−180
10−2
Fig. 10. Frequency response of F for a power flow equal to +1 pu and
three different SCRs. Solid: SCR infinite. Dashed: SCR = 5. Dotted: SCR =
2. Solid gray: SCR = 2, power flow = +0.8 pu.
other words, the weaker the ac system, the more negative the
VSC conductance is. In Fig. 10, one case with an SCR of 2
and power flow of +0.8 pu is plotted in order to illustrate
that, similarly to the strong ac system case, the magnitude of
F increases with the power flow increase.
Finally, Nyquist contours are shown in Fig. 11. In the left
plot, it can be seen that effect of decreasing the power flow is
that the nyquist plot intersects the real axis at a point higher
than −1, improving the gain margin of the system. The same
effect is observed when decreasing the DVC gain, kpe .
C. Recommendations for the dc-side stability analysis
From the analysis performed in this section, the following
is recommended:
1)
Identify the resonance frequency and the resonance
peak of the dc side. Commercial tools that calculates
the harmonic impedance of electrical networks can
be used for this purpose.
Wroclaw, Poland – August 18-22, 2014
0.5
0
0
imaginary axis
imaginary axis
0.5
−0.5
−1
−1.5
−1.5
−1
−0.5
real axis
0
0.5
ACKNOWLEDGMENT
The author thanks Prof. Lina Bertling-Tjernberg, Dr. Le
Anh Tuan and Prof. Claes Breitholtz for their support during
the first stage of the project. The valuable comments to this
manuscript provided by Associate Prof. Massimo Bongiorno
and Dr. Abdel-Aty Edris are also greatly appreciated.
−0.5
−1
−1.5
−1.5
R EFERENCES
−1
−0.5
real axis
0
0.5
Fig. 11. Nyquist plots. Left: Solid, kpe = 9.23 and 0.5 pu power flow;
dashed, kpe = 9.23 and 0.4 pu power flow. Right: kpe = 9.23 and 0.5 pu
power flow; dashed, kpe = 4.61 and 0.5 pu power flow.
[1]
[2]
[3]
2)
3)
Determine if the VSC conductance is negative at the
resonance frequency. Determine also if the Nyquist
stability criterion is fulfilled.
If the system is unstable, investigate if the magnitude
of the converter admittance can be decreased by
modifying the controller structure.
It should be highlighted that the VSC admittance is defined
as follow:
∆i∗dc
Y (s) =
(47)
∆u
so, for the analysis, the current ∆idc is not used to obtain
the VSC admittance. The use of ∆i∗dc is possible because it
has been shown in III-B and in [10] that the resistances R10
and R20 , which originate from the linearization of the powers
Pdc1 and Pdc2 , can be merged with the dc cable model, and,
actually, disregarded as shown in the simplification procedure
of the dc grid model.
[4]
[5]
[6]
[7]
[8]
[9]
V.
C ONCLUSION
In this paper, the dc-side dynamics of the system has been
studied using a frequency domain approach. A VSC-HVDC
system has been modeled as a SISO feedback system composed of two subsystems, the dc grid and the VSC subsystem.
It has been shown that the dc grid is a passive subsystem which
means that it is not the source of the instability. However,
the dc grid has a resonance phenomenon which has to be
considered in the analysis. The frequency response of the VSC
subsystem, called in this paper the VSC admittance, has a
non-passive characteristic at high frequencies when the power
flow is positive. If the dc-side resonance occurs at frequencies
where VSC conductance is negative, there is a risk that the
resonance becomes unstable. With the assumed control system,
it has been shown that the magnitude of the VSC admittance
depends on the power flow and the SCR of the ac grid to which
the VSC is connected. The control system also influences on
the magnitude of the VSC admittance.
After the analysis, a procedure to evaluate the stability of
a VSC-HVDC system has been recommended. Although in
this paper the vector current control method has been used to
implement the control system, the procedure is not restricted
to only this method, as long as the VSC admittance is derived
following the definition indicated in (47).
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18 Power Systems Computation Conference
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[11]
[12]
[13]
[14]
[15]
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Wroclaw, Poland – August 18-22, 2014

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