IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006
2071
Steady-State Solution of a Voltage-Source
Converter with Full Closed-Loop Control
K. L. Lian and P. W. Lehn, Member, IEEE
Abstract—An iterative method based on a hybrid time/frequency-domain approach is proposed in this paper to solve for the
steady state of a pulse width modulated voltage-source converter
(VSC) with a -frame closed-loop controller. The method solves
the linear controller equations in the frequency domain, while
solving the VSC equations using time domain techniques. The
model is validated against time domain simulation results. It is
shown that the hybrid time/frequency-domain approach is both
highly efficient and accurate, and it provides a viable alterative
to brute-force time domain simulation for large signal harmonic
analysis of the VSC.
Index Terms—Broyden’s method, closed-loop control, harmonics, Jacobian, steady-state analysis, voltage-source converter
(VSC).
I. INTRODUCTION
EREGULATION of electric utilities, together with
increasing consumer loads have placed a level of unprecedented stress on power systems. Power electronic equipment
is increasingly being installed to meet the evolving needs of
modern power systems. At the generation level, back-to-back
converter circuits are needed as interfaces for distributed energy
sources; at the transmission level, converter based flexible ac
transmission systems (FACTS) controllers are employed to
improve system stability and at the distribution level converter
based Custom Power controllers are used to improve power
quality. While many benefits may be realized from large scale
introduction of power electronics into the power grid, these
come at the expense of generating harmonic distortion. It
is therefore essential to predict the harmonics generated by
voltage-source converters (VSCs) and to understand how these
harmonics will interact with the power system.
Brute-force time-domain simulation can provide accurate
harmonic analysis results if the simulation step is chosen
appropriately. However, the disadvantage of brute-force time
domain simulation is that it needs to go through an initialization
transient before reaching steady state where Fourier analysis
can be performed. Simulation times are especially long when
analyzing pulse-width-modulated (PWM) VSCs because the
simulation time step must be sufficiently small to capture
high frequency switching dynamics while the time interval
D
Manuscript received August 18, 2005; revised November 18, 2005. Paper no.
TPWRD-00482–2005.
The authors are with the Energy System Group of University of
Toronto, Toronto, ON M5S 3G4, Canada (e-mail: liank@ecf.utoronto.ca;
lehn@ecf.utoronto.ca).
Digital Object Identifier 10.1109/TPWRD.2006.877081
of simulation must be sufficiently large to provide the correct
fundamental frequency solution.
Alternative methods have been proposed to rapidly calculate the steady state operating point of various power electronic
circuits. These methods can be classified into three categories:
fast time domain methods [1]–[10], frequency-domain methods
[11]–[16], and hybrid time/frequency-domain methods [17].
To date only frequency-domain methods have been employed
for steady-state analysis of the closed-loop three-phase converter circuits. Wood employed linearization about a known operating point to obtain the harmonic domain admittance matrix
of the controlled thyristor bridge [15]. He later employed a similar technique to analyze a STATCOM with simple firing angle
control [16]. The main limitation of this approach is that it assumes the system harmonics do not influence the operating point
of the converter. For the thyristor bridge, this limitation is overcome in [14], where iteration is employed to find the steady-state
operating point. However, the iterative approach presented in
[14] cannot be directly applied to the VSC circuits because the
controllers of VSCs are typically operated in the synchronous
reference frame [18]. To date, no steady state model has yet
been developed for a conventional PWM VSC with a -frame
closed loop controller.
This paper presents an iterative large signal method for
steady state analysis of the PWM VSC with closed loop
-frame curcontrol. The converter controls consist of a
rent controller together with a dc bus voltage controller. The
proposed approach employs hybrid time/frequency-domain
modelling [17] in which the VSC is modeled using a fast time
domain approach and controllers are modeled in the frequency
domain. In contrast to frequency-domain techniques, the proposed approach is not limited by the Gibbs phenomenon [19],
thus far fewer harmonics need be calculated to accurately determine the steady state of the VSC. The simulation results of the
hybrid method are compared with those of PSCAD/EMTDC to
demonstrate the validity of the proposed method.
II. VSC IN A CLOSED LOOP
Fig. 1 shows a general grid connected VSC circuit, where
the dc current injection, may represent either a source or a
load current. Fig. 1 also shows a typical VSC controller [20].
referCurrent regulators are controlled in the synchronous
ence frame to exploit the tracking properties of proportional-integral (PI) regulators. The -axis current determines the active
power exchanged with the grid. A dc voltage regulator assigns
the -axis reference current. The -axis current determines the
reactive power exchanged with the grid. Its reference value may
be set to meet grid requirements. Without loss of generality, this
0885-8977/$20.00 © 2006 IEEE
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006
Fig. 1. Schematic diagram of a VSC with its internal controllers.
paper considers the VSC operated as a STATCOM, where the
.
dc current injection
As shown in Fig. 1, the paper will denote time domain variables using lower case. Upper case variable names will indicate
quantities that are in the frequency domain.
III. HYBRID METHODS
Semlyen and Medina [17] developed hybrid analysis methods
based on the realization that nonlinear and time varying components are best modeled using an iterative fast time domain
approach while linear components are best described in the
frequency domain. Thus the VSC, as a time-varying component, is best modeled in the time domain. In contrast, the PI
voltage and current controllers are linear elements most easily
specified by their transfer functions. This reasoning naturally
leads to the solution flow diagram shown in Fig. 2. However,
implementation of such a solution approach is not feasible because the PI controllers have a pole at zero frequency, resulting
in singularity and “blow-up” of the solution. Thus the key
to solving the steady state solution of the VSC using hybrid
techniques lies in resolving the singularity introduced by the
PI feedback controllers.
The calculation flow diagram of Fig. 3 shows how the system
is subdivided into three solution blocks. The first block consists of the pattern generator and the VSC, and is called the
“VSC block.” Though not shown in Fig. 3, the VSC block may
also contain feedforward or auxiliary controllers, provided that
they do not include integral terms. The second block is the ac
Fig. 2. Simple solution flow diagram based on the hybrid method. This
algorithm will not converge due to singularity in the controller transfer
functions at j! = 0.
“current control block” while the third is the dc bus “voltage
control block.” The calculation paths are chosen to pass backwards through these control blocks. This way integrators are replaced by differentiators, and the singularity at zero frequency
is eliminated.
Solution of the system proceeds as follows (see Fig. 3). The
and
(Fig. 1) and active
modulating signal harmonics
current reference
are initialized, and supplied to the blocks
has
that require them. The initial active current reference
two calculation paths. It is supplied to a voltage controller block
. It is also supplied to the current confor prediction of
troller blocks, where it contributes to the prediction of
and
. Similarly,
, and
are supplied to both the
VSC block, to calculate the ac current and dc voltage harmonics
,
, and
) and the current control block, to
(
contribute to the prediction of
, and
.
LIAN AND LEHN: STEADY-STATE SOLUTION OF A VSC WITH CLOSED-LOOP CONTROL
2073
Since an infinite bus is assumed, one can assume the forcing
functions are perfect sinusoids, which can be modeled as harmonic oscillators [21]. Therefore, (1) can be written as (2)
(2)
where
Fig. 3. Proposed calculation flow diagram for simulating a closed-loop VSC.
Initial values (white arrows) supplied to blocks produce predictions (black thick
arrows). Calculation paths and mismatch equations are denoted by black thin
arrows and question marks, respectively.
If all of the current and voltage harmonics predicted by the
VSC block are equal to those predicted by the control blocks,
then the initial modulating signal harmonics and active current reference harmonics are correct (i.e., the loop is effectively
closed); otherwise, the modulating signal harmonics and active
current reference harmonics must be iterated until convergence
is achieved.
and
,
,
model bus voltages
,
,
, respectively.
Equation (2) is homogenous and involves only the computation of the exponential matrix over each switching interval. To
further reduce the computation time, the VSC can be directly
modeled in the alpha–beta reference frame so that only one oscillator is needed [21]. Consequently the size of the state transition matrix of the homogeneous equation is reduced by half
IV. MODELS
This section briefly describes the fast time domain model of
the VSC, and the frequency-domain model of the controller.
A. VSC Model
(3)
where
The essence of the fast time domain method is to determine
and thereby yield the perithe correct initial system states
odic solution without going through system transients. In [3] an
“augmented matrix method” is introduced to determine the corby augmenting forcing functions to the state marect state
trices of linear switched circuits. The differential equation of a
VSC is given in (1)
and
(1)
The correct initial conditions can be solved by partitioning
the system matrix of (3) and performing a partial inversion [4].
In addition, further augmentation of the system matrices may be
carried out to permit the harmonics of the state variables to be
analytically solved [3], [4]. Consequently, the augmented matrix
method can determine system harmonics without applying the
discrete Fourier transform (DFT) to simulation data sets—thus
avoiding limitations imposed by the Nyquist theorem.
where
B. Current Controller Model
,
switching functions.
in Fig. 1 and
,
,
are the
Fig. 4 shows the current controller of a VSC [18] in the syngives the steady
chronous reference frame, where
state response of the PI current controller at harmonic frequen, and
, the ac current harmonics at the
cies. For given
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006
Fig. 5. Block diagram of the dc voltage controller in the frequency domain.
Complete expressions for
,
,
,
,
,
,
are listed in the Appendix.
C. Voltage-Controller Model
Fig. 4. Block diagram of the current controllers in the frequency domain.
current controllers,
and found as in (4)
and
can be back calculated
Fig. 5 shows the outer dc bus voltage controller that assigns
, where
gives
the active current reference
the steady-state response of the PI voltage controller at harmonic
frequencies.
, the dc voltage harmonics are back calculated to
Given
be (8)
(8)
For each harmonic of interest, an equation in the form of (8)
is required, yielding the matrix relation (9)
(9)
Complete expressions for
in the Appendix.
,
, and
are listed
V. NUMERICAL ANALYSIS
(4)
Note that (4) must be evaluated at each harmonic of interest.
For multiple harmonics of interest one equation of the form of
(4) is required for each harmonic. In general, we may write this
in compact form as (5)
(5)
To form a mismatch equation between solutions of the current
controller block and the VSC block, (5) must be transformed
frame. This can be done by multiplying both sides
into the
of (5) by the connection matrix, C [22], [23]
(6)
are to be iniHowever, since the modulating signals
tialized in the -frame as indicated in Fig. 3, a more convenient
form of the equation is given by (7)
(7)
The steady-state solution of the closed-loop VSC is found
by using a hybrid time/frequency-domain approach based on a
Newton-type algorithm.
A. Flowchart
Fig. 6 shows the overall flow diagram of the proposed large
signal analysis of a closed-loop VSC. Note that the algorithm
initializes the modulating voltage signals in terms of positiveand negative-sequence quantities (
, and
), rather than
, and
as mentioned in the previous sections to simplify initialization and improve convergence under unbalanced
operation [23].
frame comSequence components are linearly related to
ponents through transform (10)
(10)
and
, together with
, are passed to the
where
VSC and controller blocks. In the VSC block, the modulating
signal harmonics are converted to the abc components and fed
to a PWM pattern generator. Any desirable modulating strategy
may be implemented by the pattern generator.
LIAN AND LEHN: STEADY-STATE SOLUTION OF A VSC WITH CLOSED-LOOP CONTROL
2075
Fig. 7. Simplified open-loop VSC model with the assumption of the switching
frequency being infinite.
The mismatch equations set the stage for a Newton-type it, and
erative method, which starts with an initial seed value
by (12)
generates the sequence,
(12)
where
.
The convergence and efficiency are dependent on the initial
, and the Jacobian, . Sections V-B and V-C will
values,
be devoted to the discussions on initialization and finding the
Jacobian matrix.
B. Initialization
Proper initialization is critical for Newton’s method. Poor initialization is not only detrimental to the speed of the simulation but can, in extreme cases, even lead to overall convergence
problems. A good seed value must therefore be found for the
first iteration. Given that the dominant harmonic produced by a
VSC is the positive sequence fundamental component, a simple
phasor solution, based on Fig. 7 is used for initialization. The
simplified model assumes infinite switching frequency and neglects both control action and converter losses.
The phasor analysis yields phase A modulation index
and firing angle
associated with the fundamental frequency
,
positive-sequence component for a given value of , ,
and
. All other harmonic components are initialized near
zero.
C. Jacobian Matrix
The Jacobian matrix needed for Newton-type iteration is
given by the partial derivative of the mismatch equations
.
with respect to the input vector
Consequently
Fig. 6. Proposed flowchart for hybrid method.
The switching times
produced by the pattern generator are fed to the VSC model. The VSC model outputs cur, and
), and dc voltage harmonics,
rent harmonics (
. In the controller blocks,
, and
are converted
-components,
, and
via (10). These voltages,
to
are substituted into (7) and (9) to yield
together with
, and
and
.
Once
,
,
,
,
, and
are obtained, the mismatch equations can be found by (11).
(11)
(13)
Since the VSC block contains two nonlinear systems, the
PWM pattern generator and the converter itself, it is difficult
to obtain an analytical expression for the Jacobian of the entire
. In addition, one needs to re-derive the JaVSC block
cobian of the VSC block each time a different type of pattern
generator is used. An alternative is to compute the Jacobian numerically [6], [10], [17], [24] based on a sequential perturbation
of the input vector calculated at the nominal case. The Jacobian
, on the other hand, can be eiof the controller block
ther evaluated numerically or obtained analytically.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006
D. Broyden’s (Quasi-Newton) Method
The amount of computation required to carry out a NewtonRaphson solution is extensive. As mentioned, the Jacobian of
the VSC block is obtained numerically, which requires cycles of simulation for an
Jacobian [4]. In addition, the
Newton-Raphson method requires evaluation and inversion of
the Jacobian matrix each iteration as well as evaluation of the
mismatch equations in each iteration.
In 1965, Brodyden [25] introduced a Quasi-Newton’s method
in which only an approximate Jacobian matrix is needed to
start the iteration process. Refinement of the Jacobian matrix
is achieved by introducing correction terms in the iteration
loop. This avoids re-evaluating the partial derivatives for each
column and yields the inverse of the Jacobian matrix by simple
matrix-vector multiplications. The total arithmetic calculations
, rather than
as required by the original
required is
Newton’s method [26]. A simple algorithm for implementing
Broyden’s method [26] can be found in the Appendix.
VI. SIMULATION RESULTS
A. System Under Study
To validate the proposed method, its steady state solutions
are compared with those obtained by brute force time domain
simulation (PSCAD/EMTDC). The system under study is as
shown in Fig. 1. The block diagram of the current and voltage
controllers are the same as in Fig. 4 and Fig. 5. The parameters and PWM strategy are extracted from the IEEE benchmark
model of the D-STATCOM [18]. However, in order to show
the uncharacteristic harmonics more clearly for the unbalanced
case, the frequency modulation ratio is reduced from the original value of 27 to 9 (i.e. a switching frequency of 540 Hz), and
the value of the dc capacitance is reduced from the original value
. A simulation time step of 14
is chosen
of 4860 to 1000
for PSCAD/EMTDC to limit integration error.
Low order harmonics are expected to be heavily influenced
by control action. To demonstrate how harmonics are affected
by the controls, two different current controllers are employed.
The two current controllers differ by a factor of 10 in their gains.
Both have their zero placed to cancel the left-half plane pole of
[18]. The transfer
the linearized VSC model, which is at
functions of the slow and fast current controllers are
and (
,
respectively. The same slow outer dc voltage controller
is employed for both studies.
In the following sections the VSC is studied under first balanced and then unbalanced operating conditions.
B. Balanced Operation
The VSC is assumed to connect to a perfectly balanced threephase system, with system voltages given by
TABLE I
TOTAL NUMBER OF ITERATIONS AND CPU TIME
UNDER BALANCED OPERATION
1) Initialization and Efficiency: As mentioned in Section V-B, under balanced condition, phasor analysis is used
to initialize the positive sequence fundamental component
quantities; other harmonics are initialized close to zero. For the
,
,
and
system under study,
resulting in initial values of
and equal to
, equivalent to an initial positive-sequence
0.8911 and
.
fundamental component of
While the converter model is not inherently bandwidth limited, only a finite number of harmonics can be considered for
computation of the controller response. Given that measured
current and voltage signals are typically low pass filtered before entering the controllers, the modulating harmonics need
only be included up to the cut-off frequency of these filters. If
no feedback filters exist, as is the case in Fig. 1, the number of
modulating harmonics included is chosen by considering 1) the
low-pass nature of the power circuit itself and 2) the diminishing
amplitude of PWM harmonics with frequency.
Based on the open loop analysis [27], [28], and engineering
are very small
experience, harmonic magnitudes after
for the studied system and have negligible effect on the converter
switching times. Furthermore, for balanced operation, there is
harmonics because only
no need to include all of the
characteristic harmonics will be nonzero [27]. Therefore, the
number of harmonics to be initialized is substantially reduced.
Thus, the following harmonics are initialized.
, 5, 7, 11, 13, 17, 19, 23, 25, 29,
• Modulating signal
and 31.
, 2, 4, 6, 8, 10, 12, 14, 16,
• Active current reference
18, 20, 22, 24, 26, 28, 30, and 32.
Table I shows the total number of iterations and CPU time
for the hybrid method versus PSCAD/EMTDC (on a 550-MHz,
Pentium III, Windows 2000 workstation). As Table I shows, a
significant number of iterations and simulation time are greatly
saved by the hybrid method.
2) Current and Voltage Harmonics: Once the closed loop
modulating signal harmonics are obtained, these signals are
passed to the PWM pattern generator to determine the steady
state switching times. Applying the switching times to the
the open loop augmented VSC model yield the ac current
and dc voltage harmonics. Note that although, the modulating
(i.e., 31st)
harmonics are only specified up to the
harmonic, the ac current and dc voltage harmonics may be
calculated up to much higher frequencies. The assumption is
merely that harmonics above the 31st do not propagate through
the control. Once correct switching times are determined, the
ac current and dc voltage harmonics may be found to arbitrarily
high frequency [4].
Figs. 8 and 9 show the ac current space vector
harmonic spectrum due to the slow and fast current controllers
LIAN AND LEHN: STEADY-STATE SOLUTION OF A VSC WITH CLOSED-LOOP CONTROL
Fig. 8. AC current space vector complex spectrum under balanced operation
).
(due to C
Fig. 9. AC current space vector complex spectrum under balanced operation
).
(due to C
under balanced operation. The nonzero ac current harmonics
shown in the figures (from left to right) are [ 53 47 41 35
29 23 17 11 5 1 7 13 19 25 31 37 43 49 55].
Figs. 10 and 11 show the dc voltage harmonic spectrum due
to the slow and fast current controllers, respectively, under balanced operation. Note that since dc voltage harmonics roll off
very rapidly, the magnitude of dc voltage harmonics are plotted
using a log scale to allow comparison of small high-frequency
components.
C. Unbalanced Operation
A 10% negative sequence voltage is superimposed on the
system voltage to study operation under unbalanced conditions.
The system voltage is given by
2077
Fig. 10. DC voltage spectrum under balanced operation (due to C
).
Fig. 11. DC voltage spectrum under balanced operation (due to C
).
Note that a 10% bus voltage imbalance is selected for visualization purpose only, and that such a high level of imbalance would
not ordinarily occur in a power system.
1) Initialization and Efficiency: For unbalanced operation,
one expects uncharacteristic harmonics to appear in the spectrum of the modulating signals. The harmonics of interest must
be either predicted a priori—a challenging task—or all odd harmonics up to the maximum harmonic of interest must be included. For the sample case, all odd harmonics up to
are all considered. Initialization of these harmonics is carried
out as follows.
, 5, 7, 11, 13, 17, 19, 23,
• Modulating signal: for
25, 29, and 31, the initialized values are set to those converged to under balanced operation. Other harmonics are
.
arbitrarily initialized to
• Active current reference: all even harmonics are initialized
to those converged to under balanced operation.
Table II shows the total number of iterations and CPU time
for the hybrid method versus PSCAD/EMTDC. Again, a great
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006
TABLE II
TOTAL NUMBER OF ITERATIONS AND CPU TIME
UNDER UNBALANCED OPERATION
Fig. 14. DC voltage spectrum under unbalanced operation (due to C
).
Fig. 15. DC voltage spectrum under unbalanced operation (due to C
).
Fig. 12. AC current space vector complex spectrum under unbalanced operation (due to C
).
using a log scale to allow comparison of small high-frequency
components.
VII. DISCUSSIONS
Fig. 13. AC current space vector complex spectrum under unbalanced operation (due to C
).
number of iterations and simulation time are saved by use of the
hybrid method.
2) Current and Voltage Harmonics: Figs. 12 and 13 show
harmonic spectrum due
the ac current space vector
to the slow and fast current controllers under unbalanced operation. All odd harmonics are shown.
Figs. 14 and 15 show dc voltage harmonic spectrum due to
the slow and fast current controllers, respectively. All even harmonics are shown. Note that similar to the case of balanced
operation, the magnitude of dc voltage harmonics are plotted
As can be noted from Figs. 8–15, excellent agreement exists
between the proposed hybrid technique and the brute-force time
domain (PSCAD/EMTDC) simulation results.
and
For balanced or unbalanced operation, both
successfully regulate the positive sequence fundamental
current to 30 A. However, the ac current spectra for the two
controllers differ at other frequencies. This is true even under
the case of balanced operation.
Comparing Figs. 8 and 9, one sees that harmonics at 47,
13, and 55 differ. Note that
suppresses these three
does not. These difharmonics to almost zero, while
ferences are sufficiently large to produce different 60-Hz operating points. Table III lists phase A modulation index
and the firing angle
of the fundamental component of the
modulating signal produced by PSCAD/EMTDC and those converged to by the hybrid method. As can be noted, the firing angle
LIAN AND LEHN: STEADY-STATE SOLUTION OF A VSC WITH CLOSED-LOOP CONTROL
TABLE III
OPERATION POINTS OF THE FUNDAMENTAL COMPONENT OF PHASE A
MODULATION SIGNAL UNDER BALANCED OPERATION
issued by the fast controller is about half of that issued by the
slow controller.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
2079
and
between PSCAD/EMTDC and
Discrepancies of
the hybrid method result from the fact that ideal bidirectional
switches are employed in the hybrid method whereas switches
in PSCAD/EMTDC are not perfectly ideal. Minor phase discrepancies are also attributable to delays between control and
circuit solutions in PSCAD/EMTDC.
Under unbalanced operation, the ac current harmonic spectra
for the two controllers differ very significantly; as evident from
Fig. 12 and Fig. 13. For instance, the negative sequence fundawhile it is
mental current component is about 26A for
is employed. Table IV shows how the
only 13.2A when
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 21, NO. 4, OCTOBER 2006
3)
TABLE IV
OPERATION POINTS OF THE FUNDAMENTAL COMPONENT OF PHASE A
MODULATION SIGNAL UNDER UNBALANCED OPERATION
where
;
, and
4)
5) set
6) end.
;
;
B. Matrices in Section IV
differing harmonic distortion levels impact the 60-Hz operating
point of the converter. The table lists the modulation indices and
firing angles of phase A, B, and C for both the hybrid method
and PSCAD/EMTDC simulations.
Figs. 10, 11, 14, and 15 show that the dc component of the
is successfully regulated to its reference
capacitor voltage
value of 230 V in all cases. As seen by comparing Figs. 10 and
11, for balanced operation, the ac current controller gain has
little impact on dc voltage harmonics.
In contrast, for unbalanced operation Figs. 14 and 15 show that
a design of the ac current controller has a significant influence on
dc voltage harmonics. Particularly noticeable discrepancies exist
in the second, fourth, and eighteenth harmonic voltages.
VIII. CONCLUSION
An efficient hybrid time/frequency-domain method has been
presented for steady-state analysis of the VSC with closed-loop
control. The method identifies the exact timing of converter
switching events, thereby avoiding limitations imposed by the
Gibbs phenomenon. Only harmonics that influence the switching
times are included in the iterative portion of the solution algorithm, thus minimizing computational burden. Once the timing
of the switching events is accurately known, an arbitrary number
of VSC harmonics may be calculated analytically.
Accuracy of the model is demonstrated through comparison with simulation results obtained from PSCAD/EMTDC.
In comparison to simulation, the hybrid time/frequency-domain method obtains the steady-state solution using roughly
one-tenth the computation time.
The simulation test cases considered also demonstrate the importance of including the effects of control action when calculating the harmonic steady state of the VSC. Control action is
shown to play a particularly important role in the case of unbalanced operation, where active power exchange at frequencies
other than the positive sequence fundamental can be significant.
APPENDIX
A. Broyden’s Method
Step 1) initialization (i.e.,
) to obtain
;
Step 2) set
Step 3) while
1) begin;
2) evaluate
;
and
);
In the equation at the bottom of the previous page,
,
,
where
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
.
Note that
.
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K. L. Lian, photograph and biography not available at the time of publication.
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