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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012
Small-Signal Stability Analysis of Multi-Terminal
VSC-Based DC Transmission Systems
Giddani O. Kalcon, Grain P. Adam, Olimpo Anaya-Lara, Member, IEEE, Stephen Lo, and
Kjetil Uhlen, Member, IEEE
Abstract—A model suitable for small-signal stability analysis
and control design of multi-terminal dc networks is presented. A
generic test network that combines conventional synchronous and
offshore wind generation connected to shore via a dc network is
used to illustrate the design of enhanced voltage source converter
(VSC) controllers. The impact of VSC control parameters on
network stability is discussed and the overall network dynamic
performance assessed in the event of small and large perturbations. Time-domain simulations conducted in Matlab/Simulink
are used to validate the operational limits of the VSC controllers
obtained from the small-signal stability analysis.
Index Terms—DC transmission, offshore wind generation,
small-signal stability, voltage source converter.
HVDC
NOMENCLATURE
High-voltage direct current transmission.
HVAC
High-voltage alternating current transmission.
VSC
Voltage source converter.
LCC
Line-commutated converter.
MTDC
Multi-terminal direct current transmission.
PCC
Point of common coupling.
DFIG
Doubly-fed induction generator.
FRC-WT
Fully-rated converter wind turbine.
SSSA
Small-signal stability analysis.
H
I. INTRODUCTION
IGH-VOLTAGE dc (HVDC) transmission is emerging
as the prospective technology to address the challenges
associated with the integration of future offshore wind power
Manuscript received June 07, 2011; revised October 03, 2011, December 05,
2011, and February 09, 2012; accepted February 24, 2012. Date of publication
April 17, 2012; date of current version October 17, 2012. Paper no. TPWRS00467-2011.
G. O. Kalcon, G. P. Adam, and S. Lo are with the Institute for Energy and Environment, University of Strathclyde, Glasgow G1 1XW, U.K.
(e-mail: giddani@eee.strath.ac.uk; grain.adam@eee.strath.ac.uk; k.lo@eee.
strath.ac.uk).
O. Anaya-Lara is with the Institute for Energy and Environment,
University of Strathclyde, Glasgow G1 1XW, U.K., and also with the
Faculty of Engineering Science and Technology, Norwegian University
of Science and Technology, NTNU, 7491 Trondheim, Norway (e-mail:
olimpo.anaya-lara@eee.strath.ac.uk; olimpo.anaya-lara@ntnu.no).
K. Uhlen is with the Department of Electrical Power Engineering, Norwegian University of Science and Technology, NTNU, 7491 Trondheim, Norway
(e-mail: kjetil.uhlen@ntnu.no).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRS.2012.2190531
plants [1], [2]. Small-signal stability analyses (SSSA) have become important in the design stage of HVDC controllers to enhance their resilience to faults, and to improve their ability to
contribute to power network operation [3].
Stability studies of hybrid networks comprising HVDC and
HVAC transmission are discussed in [4] and [5]. In [4], the
authors investigate the potential interactions between multi-infeed LCC-HVDC converters and synchronous generators’ dynamics using SSSA. However, the LCC-HVDC controllers are
not modeled in detail (only current and extinction angle controllers are incorporated). In [5], a detailed linearized model of
a point-to-point LCC-HVDC is presented, and SSSA is conducted using a sampled data modeling approach. However, the
LCC-HVDC controllers and ac network are not represented in
detail.
In [6], small-signal stability analysis is used to design the
controls of a point-to-point LCC-HVDC connected in parallel
with an ac line to provide damping of sub-synchronous oscillations. The state-space model is derived in detail including the
dynamics of the network, the machine multi-mass shaft systems, and the HVDC system. The authors of [1] also presented
a linearized model for a hybrid system that includes comprehensive dynamic models for point-to-point LCC-HVDC, ac network, and synchronous generators [7]. The paper addresses the
possibility of using small-signal stability analysis to investigate
sub-synchronous oscillations damping in hybrid systems.
Small-signal stability analysis has also been used in [8] to
design the controllers of an LCC-HVDC connecting a wind
farm based on fixed-speed induction generators. The results reported contain very high-frequency components due to the interaction between the HVDC converter controller and the wind
farm network.
In [9], a modeling platform to analyze conventional electromechanical oscillations and high-frequency interactions in
hybrid networks, comprising an LCC-HVDC and the ac grid,
using small-signal stability analysis is proposed. The linearized
models of the dynamic devices and the network dynamics are
combined together using Kirchhoff’s laws. Then, the resultant
network dynamic models are combined with the admittance
matrix of the rest of the network, using current injection models.
The authors in [10] present the small-signal stability analysis
of ac/dc systems with a novel discrete-time representation of a
two-terminal LCC-HVDC based on multi-rate sampling. The
complete state-space model of the ac/dc system incorporates
suitable interfaces of the various subsystems involved. The
synchronous machine and ac network use a common dq-axes
reference frame. The ac and dc networks are interfaced using
current injection relationships.
0885-8950/$31.00 © 2012 IEEE
KALCON et al.: SMALL-SIGNAL STABILITY ANALYSIS OF MULTI-TERMINAL VSC-BASED DC TRANSMISSION SYSTEMS
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Fig. 1. Test system.
In this paper, the authors present a detailed state-space model
of a more elaborated 4-terminal VSC-MTDC system connecting
two offshore wind farms to an ac network. Small-signal stability
analysis is carried out to define the ranges for the gains of the
VSC controllers that ensure dynamic stability, and the results
are confirmed via time-domain simulations in Matlab/Simulink.
Also, a simple example to calculate the converter controller
gains using root-locus is provided in the Appendix.
The wind turbine generators are modeled as fixed-speed induction generators (FSIGs) to represent the worst-case scenario,
in terms of wind turbine controllability. However, the model
presented can also be used with variable-speed wind turbines
such as doubly-fed induction generators (DFIGs), or fully-rated
converter wind turbines (FRC-WTs), at the expense of increased
modeling complexity due to the power electronic converters
(and associated controllers) comprised in these type of wind
turbines.
II. GENERIC TEST NETWORK
Fig. 1 shows the network used in this research. It consists
of four VSC stations connecting two offshore wind farms to
the onshore grid (
and
). Each wind farm is rated
at 33 kV, 400 MVA. The dc transmission voltage is 300 kV
pole-to-pole (
-bipolar). The length of the dc link cables is 150 km, and the length of the auxiliary cables is 5 km.
The onshore grid comprises conventional thermal generation
aggregated and modeled by a synchronous generator, SG, with
ratings of 33 kV, 2400 MVA. Due to the asynchronous connection, the offshore wind farms and the onshore network are
treated as independent systems in the small-signal and transient
stability analyses [3].
III. SMALL-SIGNAL STABILITY MODEL DEVELOPMENT
A. Assessment of Small-Signal Stability
The most direct way to assess small-signal stability is via
eigenvalue analysis of a model of the power system [11]–[14].
In this case, the “small-signal” disturbances are considered
sufficiently small to permit the equations representing the
system to be linearized and expressed in state-space form.
Then, by calculating the eigenvalues of the linearized model,
the “small-signal” stability characteristics of the system can
be evaluated. The way in which system operating conditions
and controllers’ parameters influence dynamic performance
can be demonstrated by observing the influence on the loci of
the dominant eigenvalues, i.e., the eigenvalues having the most
significant influence on network dynamic performance.
The linearized model of the test system in Fig. 1 is expressed
in state-space form as [9], [15]
(1)
where is the state vector, is the input vector, is the state
matrix, and
is the input or control matrix. The eigenvalues
of the state matrix provide the necessary information about
the small-signal stability of the system. The participation factor
matrix formed from the left and right eigenvectors of matrix
gives information about the relationship between the states and
the modes.
B. Grid-Side VSC Converter Model
Fig. 2 shows the equivalent circuit of the grid-side converters
and
, which control the dc link voltage and the ac
voltage at buses
and
, respectively. The dynamic equations of these converters in the dq reference frame are (inverter
operation) [8], [13], [14]
(2a)
(2b)
(3)
where
and are the total resistance and inductance between
the VSC and the PCC; ,
are the voltages at the VSC terminals and PCC, respectively;
is the dc voltage; and is
the dc capacitor.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012
Fig. 2. One-phase of a VSC converter.
After linearization of (2) and (3), the linearized model of the
grid-side converter is
(4a)
(4b)
Fig. 3. Control system of the grid-side converters.
(5)
Fig. 3 shows the control system block diagram for both gridside converters
and
.
From Fig. 3, the reference currents
and
are obtained
from the dc voltage and ac voltage controllers as
(6)
(7)
,
,
, and
are the proportional and integral
where
gains of the dc voltage and ac voltage controllers, respectively.
The auxiliary variables
and
are used to represent the
integral parts of these controllers.
The voltages at buses
,
, and
(onshore grid), and
their linearized forms are expressed as
(8)
From Fig. 3, the VSC terminal voltage obtained from the current
controllers, including the feed-forward terms, is expressed in dq
coordinates as
(11a)
(11b)
where
and
are the active and reactive current components;
and
are the proportional and integral gains of
the current controller;
and
are auxiliary variables representing the integral parts of the controllers, where
and
.
After manipulation of the equations and change of variables,
the final linearized differential equations of
and
are
expressed as in (12) (the full matrix representation is provided
in the Appendix):
(12a)
(9)
The linearized forms of (6) and (7) are
(10a)
(10b)
(12b)
KALCON et al.: SMALL-SIGNAL STABILITY ANALYSIS OF MULTI-TERMINAL VSC-BASED DC TRANSMISSION SYSTEMS
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(12c)
to
introduced to represent
where the auxiliary variables
the integral parts of the dc voltage, ac voltage and current controllers are
Fig. 4. Control system of the wind farm-side converters.
(13a)
(13b)
(13c)
Fig. 4 shows the control system of the wind farm-side
converters.
Based on Fig. 4, the reference currents,
and
, obtained from the active power and ac voltage controllers are
(15a)
(15b)
(13d)
C. Wind Farm-Side VSC Converter Model
are the ac voltage controllers’ gains;
represents
where
the integral part of the ac voltage controllers. The final linear
representation of each wind farm-side converter is
The linearized model of the wind farm-side converters
and
in the dq coordinates are (rectifier operation)
(16a)
(14a)
(14b)
(14c)
(16b)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012
Fig. 5. Single-line diagram of the dc offshore network.
Fig. 6. Single-line diagram of the onshore network.
E. Synchronous Generator
The synchronous generator in the onshore grid, SG, is modeled with a seventh-order model, including excitation and turbine-governor control [14], [16].
F. Wind Farm Based on Fixed-Speed Induction Generator
(16c)
The offshore wind farms are assumed fixed-speed with fifthorder model induction generators. A detailed state-space model
including the static capacitors is given in the Appendix [17],
[18]. Variable-speed wind turbines such as DFIG or FRC-WT
can also be used, but at the expense of increased model complexity due to the power converters (and associated controllers),
incorporated in these wind turbine generator technologies.
G. Onshore Network
(16d)
The onshore network in Fig. 6 is modeled using the
impedance matrix (18) based on the matrix partitioning technique (load buses are neglected). The current and voltage in
each bus is referred to a common reference frame as described
in [16], [19]
(16e)
(16f)
(18)
where is the reduced impedance matrix. The voltage-current
relationship is
(19)
The matrix form of (16) is given in the Appendix.
The state-space representation of the onshore network is
D. DC Offshore Network
The dc network in Fig. 1 is represented by a set of quasisteady-state equations, which are linearized as follow (the dc
link voltages and currents are shown in Fig. 5). The converter
stations are based on simple two-level converter with common
dc link capacitors, which attenuate high-frequency harmonics
that may result from any transient in a similar manner as dc cable
series inductance do:
(20)
where
, and
and
are the currents and voltages in buses ,
. R and X are the impedance matrix components.
IV. FORMULATION OF THE OVERALL LINEARIZED SYSTEM
(17)
The complete state-space representation of the test system
in Fig. 1 is formulated by combining the individual state-space
models of the wind farms, offshore and onshore converters, dc
network, onshore ac network, and synchronous generator, as
KALCON et al.: SMALL-SIGNAL STABILITY ANALYSIS OF MULTI-TERMINAL VSC-BASED DC TRANSMISSION SYSTEMS
shown by the block diagram in the Appendix. The dc currents
and voltages in (17) are used to link the grid-side converters to
the offshore converters. The synchronous generator and wind
farms are linked to the converters using nodal theory. The complete state-space matrix has a dimension of 56 56.
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TABLE I
EIGENVALUES OF TEST SYSTEM FOR THE BASE-CASE SCENARIO
V. SMALL-SIGNAL STABILITY ANALYSIS
The small-signal stability of the test network is assessed using
eigenvalue analysis. A base-case scenario is considered in order
to provide a yardstick against which the influence of VSC controllers and network loading can be judged. The power flow
results for the base-case scenario are shown in Fig. 1, and the
eigenvalues associated with this case are given in Table I.
As seen in Table I, all eigenvalues have negative real parts
indicating a stable operating condition for the base-case scenario. The eigenvalues that dominate the transient response of
the system are
and
. The participation factor matrix
indicates that the synchronous generator states have a dominant
effect on the complex pair
(with time constant of 26.3 s,
frequency of oscillation of 1.65 Hz, and 0.004 damping ratio).
Therefore, any attempt to improve network damping must take
these states into account. The participation factor matrix also
indicates that the VSC states influence greatly the eigenvalues
associated with super-synchronous oscillation modes
to
. Therefore, proper tuning of the VSC control parameters
may result in fast damping of these oscillation modes. In
addition, Table I shows that the complex pairs
and
(corresponding to fast transients, with frequencies of 3169 Hz,
time constant of 3.18 ms, and 0.016 damping ratio) are damped
out at a much faster rate. These modes are often caused by
super-synchronous oscillations due to the interaction between
adjacent converters, as reported in [20].
For example, modes
and
are associated with oscillations of the dc voltage of the two grid-side converters
and
and their effect on the direct-axis currents. Modes
and
are associated with the interaction between
converters
and
, and
and
through their
dc voltage and active current components. It is observed that
modes
and
represent the interaction between the
offshore converters and wind farms through their voltage control loops and reactive current components.
VI. IMPACT OF VSC CONTROLS ON SMALL-SIGNAL STABILITY
The transient behavior of interconnected ac/dc systems is
highly dependent on the characteristics of both synchronous
generators and VSC converters and their controllers. Large
synchronous generators have slow response during abnormal
conditions due to their relatively large inertia, while VSC
converters are fast-acting devices, which can respond within
tens of milliseconds and influence the transient behavior significantly. Hence, during a disturbance, the transient behavior
of the interconnected ac/dc system will mainly depend on the
ability of the VSC converter controllers to damp out network
oscillations, and to provide the necessary reactive power
during the fault, allowing sufficient time for the synchronous
machines to adjust their controllers to provide further support.
This section investigates the suitable range for different VSC
controllers’ gains that ensures network stability (time-domain
simulations are also used to validate the results). To this aim,
a line-to-ground fault with fault resistance
is
applied at bus
at
with 0.05 s duration.
A. Grid-Side VSC—Current Controller Effect
The effect of the proportional gain of the grid-side VSCs current controllers on system stability is investigated in this section.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012
EFFECT OF
OF THE
TABLE II
GRID-SIDE VSC CURRENT CONTROLLER
EFFECT OF
OF THE
TABLE III
GRID-SIDE VSC CURRENT CONTROLLER
Fig. 8. Active power output of
Fig. 7. Active power output of
for different values of
and
for different values of
and
.
TABLE IV
DC LINK VOLTAGE CONTROLLER
EFFECT OF VARYING
OF THE
EFFECT OF VARYING
OF THE
TABLE V
DC LINK VOLTAGE CONTROLLER
.
It has been found that the range of
that ensures system stability over the entire operating range is between (0.6–35) with
best responses obtained with
. In Table II, three different gains for the proportional gain (
, 1, and 10)
are investigated. In this case, the pair
has an oscillation frequency of 211.3 Hz with damping time of 0.002 s and
0.345 damping ratio when
compared to 224 Hz with
damping time of 0.025 s and 0.028 damping ratio when
.
The system is unstable when
.
Table III shows the effect of the current controller integral
gains on system stability. The small-signal stability analysis indicates that the system remains stable for any value
,
with best responses obtained with
. For example, the
pair
has an oscillation frequency of 3192 Hz with damping
time of 0.02 s and 0.0025 damping ratio when
,
compared to 5048 Hz with damping time of 0.02 s and 0.0015
damping ratio when
. The system is unstable if
is
less than 10.
The time-domain simulation in Fig. 7 validates the results obtained from the small-signal stability analysis when the line-to-
ground fault is applied at bus . From the participation factor
matrix, it was found that the variations of
influence the states
associated with the active power control loops, while variations
in
affect those associated with the reactive power control
loops, in both converters.
B. DC Link Voltage Controller Effect
It is found that the dc voltage controller integral gains that
ensure stable operation lay in the range
.
Table IV shows the oscillation frequency and damping
time for selected eigenvalues for different values of
. The best time-domain response is
achieved with the integral gain
, where lower oscillation frequencies and fast damping time are observed (see
Fig. 8).
Table V shows the effect of the dc voltage controller proportional gain on system stability. It is established that large
decreases the damping time. For example, when
, the
damping time for the super frequency oscillations modes
is 0.025 s with 0.028 damping ratio, while the damping time is
0.01 s with 0.011 damping ratio for
and 0.063 s with
KALCON et al.: SMALL-SIGNAL STABILITY ANALYSIS OF MULTI-TERMINAL VSC-BASED DC TRANSMISSION SYSTEMS
Fig. 9. Active power output of
EFFECT OF CHANGING
EFFECT OF VARYING
for different values of
and
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.
TABLE VI
OF GRID-SIDE CURRENT CONTROLLER
TABLE VII
OF THE AC VOLTAGE CONTROLLER
0.009 damping ratio with
. These results are also confirmed by time-domain responses shown in Fig. 8.
C. AC Voltage Controller Effect
The small-signal stability analysis shows that the ac voltage
has a wide operational range
controller proportional gain
that ensures stable system as shown in Table VI. The best response is obtained with
. It is noticeable that the pair
has an oscillation frequency of 224 Hz, 0.079 s damping
time, and 0.009 damping ratio when
compared to
220 Hz, 0.038 s, and 0.019 damping ratio for
. Fig. 9
shows the time-domain simulation that validates the results obtained via small-signal stability analysis.
Similarly, the best guess for the ac voltage controller
integral gain,
, for stable operation ranges between
. The best damping response is achieved
with
. With this gain, the pair
oscillation
frequency is 227 Hz, 0.02 s damping time, and 0.35 damping
ratio compared to 225 Hz and the 0.025 s damping time and
0.028 damping ratio with
. The system becomes
unstable with values of
as shown in Table VII.
The ac voltage controller integral gain corresponding to the
Fig. 10. Active power at
and
. (B) Active power at
power at
during three-phase fault at
.
. (a) Active
system breakpoint (the transition from stable to unstable) lays
between 640 and 650. The shaded cells of Table VII indicate
eigenvalues
and
have positive real parts (instability).
For further validation of the VSC gain limits obtained based
on small-signal stability analysis, and to investigate the VSCs
response during three-phase faults, a solid three-phase fault is
applied at bus , at time
with a fault duration of 5 cycles (for 50 Hz). This scenario allows the robustness of VSCs
controllers designs based on small-signal stability analysis to
be assessed. Fig. 10 shows the power waveform at
and .
It can be seen that the system remains stable and returns to
the pre-fault operating condition after the fault is cleared. This
demonstrates the validity of the analysis presented in this paper.
VII. CONCLUSION
A detailed mathematical model for small-signal stability
analysis of VSC-based multi-terminal dc transmission systems
has been presented. The approach taken was to divide the
system into smaller subsystems representing each of them by a
state-space model. The individual state-space models were then
integrated into a single model to give the overall representation
of the network. The mathematical model developed was used
to investigate the small-signal stability performance of the
hybrid network utilizing the eigenvalues and the participating
factors matrix. The limits for the VSC controllers’ gains were
established and validated using time-domain simulations under
small perturbations. It was observed that using the small-signal
stability model, it was possible to design improved controllers
for the VSCs of the multi-terminal dc network, which ensure
stable network operation and enhanced dynamic performance.
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Fig. 11. Methodology used to obtain the complete state-space model of the test system.
The modeling approach and analysis presented can be extended
to larger systems with an arbitrary number of converters, synchronous machines, and wind farms.
is considered constant to reduce
turbine. In this paper,
system complexity.
D. Synchronous Machine State-Space Model [14]
APPENDIX
The complete state-space representation of the test system
in Fig. 1 is obtained by combining the individual state-space
models as shown in Fig. 11. The dc currents and voltages in
(17) are used to link the grid-side converters to the offshore-side
converters, while the synchronous generator and wind farms are
linked to the converters using current injection theory. The linearized models of individual subsystems are expressed in the
form
in the following subsections.
A. Grid-Side Converters State-Space Model
See the equation at the bottom of the next page.
represents vector
is the matrix
matrix of state variables;
that contains the interfacing variables that relate the onshore ac
network and the dc network.
B. Offshore Wind Farm-Side Converters State-Space Model
See the first equation at the bottom of page 1828.
is
the
matrix
that contains the offshore converter state variables;
represents the vector matrix of
interfacing variables that relate the converter to the offshore ac
network and the dc network.
C. Fixed-Speed Induction Generator [17], [18]
See the second equation at the bottom of page 1828.
is a vector
matrix of the state variable of the fixed-speed induction generator;
and
are interfacing variables between the
fixed speed induction generator and the offshore wind farm ac
network; and
is also an interfacing variable that relates
generator mechanical input torque to mechanical output of the
See
the
equation
at
the
bottom of page 1829.
is the matrix that
and
are interfacing variables
contains state variables;
that relate the synchronous generator to the onshore ac network;
and
and
also represent interfacing variables related
to the synchronous machine controllers, mainly turbine and
excitation systems.
E. Converter Control System Design and Gains Selection
The converter controller’s gains limits are first defined
using eigenvalues analysis, and then gains which provide
the best network dynamic performance are selected within
these limits. The proposed approach uses the overall system
linearized model (which involves 56 eigenvalues), making
the use of root-locus for control design very complex
(if possible at all). However, for demonstration purposes,
in this Appendix, the control system of each converter
station (using only the converter linearized model and its
associated controllers) is designed using root-locus based on
equations and transfer functions obtained from the linearized
model of the converter (assuming a two-level voltage source
converter):
Current controller:
Based on Fig. 2, the linearized model of the converter ac side
is
(E1.1)
KALCON et al.: SMALL-SIGNAL STABILITY ANALYSIS OF MULTI-TERMINAL VSC-BASED DC TRANSMISSION SYSTEMS
1827
(E2.4)
(E1.2)
and
.
and
are obtained from the proportional and integral (PI) controllers as
, and
, where
and
. After substitution in
(E1.1) and (E1.2), and algebraic manipulations the following
equations are obtained:
where
After Laplace manipulation of the state-space equations
(E2.1)–(E2.4), the following transfer function is obtained:
(E3)
(E2.2)
With the parameters used in the paper:
and
, the gains obtained from the root-locus analysis are
,
,
,
and maximum
overshoot of 2.6% (these gains put the system closed-loop poles
at
and a zero at
). These gains do not
provide a satisfactory performance over all operating conditions
when the converters are connected to the system under investigation. The final gains obtained based on eigenvalue analysis of
the overall system, when all possible interactions are taken into
account, are
,
.
dc voltage controller:
From Fig. 2 and assuming a lossless converter, the converter
dc-side dynamics can be expresses as
(E2.3)
(E4)
(E2.1)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012
Using Taylor series with the higher-order terms neglected, the
linearized form of (E4) is obtained as
(E5)
Let
and
KALCON et al.: SMALL-SIGNAL STABILITY ANALYSIS OF MULTI-TERMINAL VSC-BASED DC TRANSMISSION SYSTEMS
and the variable
be obtained from the dc voltage controller
as
and
,
then the linearized form of the differential equations that described the dc side, including dc voltage controller are
(E6.1)
(E6.2)
After Laplace manipulation of equations (E6.1) and (E6.2), the
transfer function for the dc voltage controller is
(E7)
Selection of the dc voltage controller gains can be accomplished
in a similar way as that for the current controller using root-locus
or frequency-domain techniques. Normally, the converter load
angle (the angle of the converter terminal voltage relative to the
voltage at the point of common coupling) is sufficiently small
as the total impedance between the converter terminals and the
point of common coupling must be kept small in order not to
1829
compromise the available dc voltage for reactive power compensation,
and similarly
.
Therefore the reference current
is obtained:
, where
and
are
normalized by
.
In the control system design, the authors rely on feed-forward
terms of the current controller, which are introduced during the
decoupling of
from
to improve system disturbance rejection. However, the controllers’ gains obtained from such designs are always subject to change when the converter is operated in a complex power system. For this reason, the gains obtained from the control design are used only as a starting point;
and the final values of the gains are selected as those that may
produce the best performance taking into account all the system
interactions. Gain fine-tuning is also employed in an attempt
to establish the influence of voltage source converter gains and
controllers on the overall network performance.
ACKNOWLEDGMENT
The authors would like to thank NOWITECH for facilitating
the interaction between the researchers and institutions involved
in the preparation of this research paper. Dr. O. Anaya-Lara
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 4, NOVEMBER 2012
would like to thank Det Norske Veritas (DNV) for the sponsorship provided for his Visiting Professorship at NTNU, Trondheim, Norway.
REFERENCES
[1] P. Bresesti et al., “HVDC connection of offshore wind farms to the
transmission system,” IEEE Trans. Energy Convers., vol. 22, pp.
37–43, 2007.
[2] H. Li and Z. Chen, “Overview of different wind generator systems and
their comparisons,” IET Renew. Power Gen., vol. 2, pp. 123–138, 2008.
[3] C. Osauskas and A. Wood, “Small-signal dynamic modeling of HVDC
systems,” IEEE Trans. Power Del., vol. 18, pp. 220–225, 2003.
[4] C. Karawita and U. D. Annakkage, “Multi-infeed HVDC interaction
studies using small-signal stability assessment,” IEEE Trans. Power
Del., vol. 24, pp. 910–918, 2009.
[5] X. Yang and C. Chen, “HVDC dynamic modelling for small signal
analysis,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol. 151, pp.
740–746, 2004.
[6] K. Dong-Joon et al., “A practical approach to HVDC system control
for damping subsynchronous oscillation using the novel eigenvalue
analysis program,” IEEE Trans. Power Syst., vol. 22, pp. 1926–1934,
2007.
[7] C. Karawita and U. D. Annakkage, “HVDC-generator-turbine torsional
interaction studies using a linearized model with dynamic network representation,” in Proc. Int. Conf. Power Systems Transients (IPST2009),
Kyoto, Japan, Jun. 3–6, 2009.
[8] W. Li et al., “Power-flow control and stability enhancement of four
parallel-operated offshore wind farms using a line-commutated HVDC
link,” IEEE Trans. Power Del., vol. 25, pp. 1190–1202, 2010.
[9] C. Karawita and U. D. Annakkage, “A hybrid network model for small
signal stability analysis of power systems,” IEEE Trans. Power Syst.,
vol. 25, pp. 443–451, 2010.
[10] R. K. Pandey, “Stability analysis of AC/DC system with multirate discrete-time HVDC converter model,” IEEE Trans. Power Del., vol. 23,
pp. 311–318, 2008.
[11] O. Anaya-Lara et al., “Influence of windfarms on power system dynamic and transient stability,” Wind Eng., vol. 30, no. 2, pp. 107–127,
2006.
[12] P. Kundur et al., “Definition and classification of power system stability IEEE/CIGRE joint task force on stability terms and definitions,”
IEEE Trans. Power Syst., vol. 19, pp. 1387–1401, 2004.
[13] D. Y. Wong et al., “Eigenvalue analysis of very large power systems,”
IEEE Trans. Power Syst., vol. 3, pp. 472–480, 1988.
[14] P. M. Anderson and A. A. Fouad, Power System Control and Stability. New York: Wiley, 2007.
[15] Y. Shao and Y. Tang, “Voltage stability analysis of multi-infeed HVDC
systems using small-signal stability assessment,” in Proc. 2010 IEEE
PES Transmission and Distribution Conf. Expo., 2010, pp. 1–6.
[16] P. Kundur, Power System Stability and Control, ser. Electric Power Research Institute, Power System Engineering Series. New York: McGraw-Hill, 1994.
[17] A. Radunskaya et al., “A dynamic analysis of the stability of a network of induction generators,” IEEE Trans. Power Syst., vol. 23, pp.
657–663, 2008.
[18] S. Tohidi et al., “Influence of model simplifications and parameters on
dynamic performance of grid connected fixed speed wind turbines,” in
Proc. 2010 XIX Int. Conf. Electrical Machines (ICEM), 2010, pp. 1–7.
[19] M. A. A. Alyan et al., “Assessment of steady state stability of a multimachine power system with and without a superconducting machine,”
IEEE Trans. Power Syst., vol. 5, pp. 730–736, 1990.
[20] IEEE Guide for Planning DC Links Terminating at AC Locations
Having Low Short-Circuit Capacities, IEEE Std. 1204-1997, 1997.
Giddani O. Kalcon received the B.Eng degree (with honors) in power system
and machines from Sudan University of Science and Technology (SUST), Khartoum, Sudan, in 2001, and the M.Sc degree in electrical power system from
SUST. He is currently pursuing the Ph.D. degree at the University of Strathclyde, Glasgow, U.K.
His research interest include wind power integration and HVDC system.
Grain P. Adam received the first-class B.Sc. and M.Sc. degrees in electrical
machines and power systems from Sudan University of Science and Technology, Khartoum, Sudan, in 1998 and 2002, respectively, and the Ph.D. degree
in power electronics from Strathclyde University, Glasgow, U.K., in 2007.
He is currently with the Department of Electronic and Electrical Engineering,
Strathclyde University, and his research interests are multilevel inverters, electrical machines, and power systems control and stability.
Olimpo Anaya-Lara (M’98) received the B.Eng. and M.Sc. degrees from Instituto Tecnológico de Morelia, Morelia, México, and the Ph.D. degree from
University of Glasgow, Glasgow, U.K., in 1990, 1998 and 2003, respectively.
His industrial experience includes periods with Nissan Mexicana, Toluca,
Mexico, and CSG Consultants, Coatzacoalcos, México. Currently, he is a Senior Lecturer at the University of Strathclyde in Glasgow. His research interests
include wind generation, power electronics, and stability of mixed generation
power systems.
Stephen Lo received the M.Sc. and Ph.D. degrees from the University of Manchester Institute of Science and Technology, Manchester, U.K.
He is a Professor at Strathclyde University, Glasgow, U.K. His research interests include power systems analysis, planning, operation, monitoring, and
control, including application of expert systems and artificial neural networks,
transmission and distribution management systems, and privatization issues.
Kjetil Uhlen (M’95) received the M.Sc. degree and the Ph.D. degree in control
engineering from the Norwegian Institute of Science and Technology (NTNU),
Trondheim, Norway, in 1986 and 1994, respectively.
He is currently a Professor of Electrical Engineering at NTNU and leads the
NOWITECH work package on wind power integration. His main technical interests are operation and control of electric power systems.