IEEE Transactions on Energy Conversion, Vol. 14, No. 4, December 1999
1177
An Accurate Method of Neglecting Dynamic Saliency of Synchronous
Machines in Power Electronic Based Systems
S. D. Pekarek, Member
School of Electrical Engineering
University of Missouri-Rolla
Rolla, Missouri
and
Abstract - In this paper, motivations for neglecting dynamic
saliency in the detailed models of power electronic based systems
are presented. A new method of neglecting dynamic saliency is
then introduced in which approximate operational impedances are
derived from original dynamic salient models. New machine
parameters are obtained by fitting to the approximate impedance
curves, yielding model parameters in which dynamic saliency is
eliminated. An example synchronous machindconverter system is
provided that demonstrates the accuracy and increased simulation
efficiency resulting from this technique. Therein, errors resulting
from neglecting dynamic saliency are reduced from greater than
25% using traditional approximations tn less than 0.6%. In addition, simulations using the new approximate model are greater than
48% faster than simulations based upon the original dynamic
salient model.
I. INTRODUCTION
There are several approximations, models, and model structures that are used to represent the dynamics of the synchronous
machine. The complexity of the model chosen for a particular
analysis is a function of the size of the system being studied and
the detail requircd. In several applications, including the development of reduced-order models of large-scale power systems
[I I and average-value models of machine/converter systems [2],
an approximation is made in which the dynamic saliency of the
synchronous machine is ncglected. In large-scale transient analysis programs, neglecting the dynamic saliency provides a
means to reduce computational requirements by eliminating
time-dependent matrices [1]. In average-value analysis of
machine/converter systems, neglecting dynamic saliency greatly
reduces analytical derivations by eliminating time-varying commutating inductances [2].
Although several researchers have documented the use of
the approximation, the actual methods used by researchers
vary. In particular, some equipment manufacturers, particularly
in North America, assume equal subtransient reactances and calculate q-axis machine parameters based upon this assumption
[3]. In other cases, the system analyst neglects any differences,
or uses a weighted average of the q- and d-axis dynamic reactances within the respective machine and network equations
[1,2,41.
PE-453-EC-0-12-1998 A paper recommended and approved by the
IEEE Electric Machinery Committee of the iEEE Power Engineering
Society for publication in the IEEE Transactions on Energy
Conversion. Manuscript submitted July 30, 1998; made available for
printing December 17, 1998.
E. A. Walters, Student Member
School of Electrical and Computer Engineering
Purdue University
West Lafayette, Indiana
Several researchers have expressed concern over the use of
these approximations, particularly in machines in which the
ratio of the q- and d-axes dynamic reactances is large [2,5]. In
the derivation of average-value models of synchronous
machine/converter systems [2], errors of greater than 20% were
documented when the dynamic saliency of an example synchronous machine was neglected.
In this paper, it is first shown that neglecting dynamic
saliency can substantially reduce the computational ovcrhead
required in the detailed analysis of power electronic based generation systems. Such systems are typically used in shipboard
and aircraft power and propulsion systems and consist of one or
more generator/power electronic converters which supply power
to a network consisting of highly-regulated loads. A new
method of neglecting dynamic saliency is then derived. Using
this method, the original parameters of the machine, which can
be determined using any technique, are used to calculate operational impedances. From the original impedanccs, new approximate operational impedances are derived. The new curves
match the original impedances over a user-specified frequency
range, while having high-frequency asymptotes that are equal.
A refitting of the machine parameters to the new approximate
curves is performed in order to establish parameters that yield
equal dynamic reactances.
This technique requires little work from the system analyst,
and greatly reduces the errors resulting from neglecting dynamic
saliency. Example simulations using a machine with an original
dynamic reactance ratio of 1.9 are provided in which the errors
resulting from neglecting dynamic saliency are reduced from
over 25%, using traditional methods, to less than 0.6%. In addition, simulations using the new approximate model arc greater
than 48% faster than simulations based upon the original
dynamic salient model.
11.
BACKGROUND
Since the techniques that are described in this paper are
directly related to the model structure used to reprcsent synchronous machines in power electronic based system studies, it is
convenient to describe the model of the synchronous machine
that will be used herein. A generic Park's equivalent circuit of
a synchronous machine is shown in Fig. 1. It is assumed that an
arbitrary number of armortisseur windings, (denoted by M and
N ) are represented in the q- and A x e s of the machine and that
the rotor windings are referred to the stator windings by an
appropriate turns ratio. It should be noted that sevcral analysts
use additional leakage inductances in the d-axis of the machine
[3,6,7]. The physical meaning and/or mathematical necessity
of these terms remain matters of debate [8] and it is not the
intention of this paper to support or dispute their use. Rather,
the model structures and techniques described herein are applicable, with slight modification, to machines modeled with an
0885-8969/99/$10 1.00 0 1998 IEEE
vos = r&O$ +PLlSiOS
Herein, the dynamic inductances are defined as
L, =
Ld
4s
=
Lmq
+
Lmd
where
L,,
= L",q II LlkqlII ... II LikqM
Lmd = LmdII LlfdII LIkdl II ... II LikdN
The back emfs are defined as
i-qd
I..
,,
M
kiS
vi,?
Z<i(sJ
I
I
I
"2
L,,,~
j = i
N
I
"2
v ~ = - o ~ h ; i CLmdrkd'
( ~ ( h ", - h i , , L8ndrkd'.r
+ ~ ~ d . j (13
d-axis
Fig. 1 q-d equivalent circuit of synchronous machine.
'Ikdj
j = 1
arbitrary number of additional terms in the d-axis of the machine,
Although Park's circuit is an extremely useful representation of the synchronous machine, it is often difficult to represent
power electronic circuits in terms of transformed stator variables.
Therefore, in order to model the machine/electronic interface, a where
coupling between the transformed stator variables and the physical variables of the power electronic circuit must be developed.
=
In many power electronic based generation systems, it is more
convenient to represent the machine in terms of physical variables, rather than variables based upon Park's transformation.
"2
M
h, L,,
model sources,
can be
developed
Using physical
using simple
variables,
circuit
the elements
machine/system
such as voltage
inductors, switches, and resistors lo describe the circuit configuration, without the need for physical/transformed variable coupling equations.
Derivations of efficient physical-variable models are based
upon manipulation of machine equations in the rotor reference
frame. Transformation between the physical variables and transformed variables is accomplished using Park's transformation
j = 1
(2)
E(%)]
h, = L+&+
j = 1
as
= r'riabcsCP'Labcs(er)iobcsl
1
2
2
2
The equations describing Park's circuit can be manipulated into a
Corm in which the stator voltage equations are expressed as [9]
.I
v 9s
r = rsiqs t
r
o,Ldi:,T+ pLqi,,v + vq
"
"
+
v'bcs
where r,r is a diagonal matrix of stator resistances, and
,,
,,
-1
(15
'Ikdj
Applying the inverse of Park's transformation to (5)-(7),the sta
tor voltage equations are expressed in terms of physical variable:
Ls(')= LlstL,-Lb"s(')
(4)
-1
(14
N
(1) givenby
2
Likdj
La
,I
LM(.)= - z - L b c o s ( . )
and
L i = (Lmq L m d ) / 3
(16
~i~~
i:
1179
parameters, the solution of (25) must be performed at each integration time-step. In contrast, if L, is time-invariant, the inverse
(22)
The stator voltage equations given by (16), along with the
voltage equations
r.
of L, may be precomputed and the solution involves only a
matrix-vector multiply at each time-step. Thus, the disparity in
the number of operations required at each time-step between the
time-varying and time-invariant models, increase on the order of
3
, where n represents the number of states in the system. Addi(23)
tionally,
the time-varying inductances can produce time-varying
Llj
positive
eigenvalues [91 that can reduce the efficiency when
r.
ph. =
(24)
implementing a variable step-size integration algorithm [14].
- hmd)+ v, ; j = fd, k d l , , , , kdN
J
J
Llj
Second, several commercial algorithmic languages,
such as
. .
where kmq and Am, are the magnetizing flux linkages, define Spice [lo] and Saber [ I l l , do not include a component representation of time-varying inductors. Therefore, the elimination of
the so-called physical-variable voltage-behind-reactance (PVBR) time-varying inductances in the PVBR model provides a
model of the synchronous machine. It is important to note that no to easily implement an efficient physical-variable model of the
approximations are made in its derivation. Neglecting numerical synchronous machine in such
error, the solution of the corresponding equations yields the Same
These incentives provide a motivation to develop an accutime-domain response of a full-order Park's circuit model upon
rate physical-variable synchronous machine model in which the
which it is based.
inductances are time-invariant.
p h . = -L(hj-k,,,,)+vj;j
'
= kql,
...kqM
111. NUMERICAL ADVANTAGES OF NEGLECTING DYNAMIC
SALIENCY
n
IV. TRADITIONAL METHODS OF NEGLECTING DYNAMIC
SALIENCY
Traditionally, a significant challenge in the development of
For machines in which at least one d-axis amortisseur windpower electronic based systems has been the derivation of ing is represented, equipment manufacturers often provide
detailed models from which a system can be analyzed and machine parameters that have equal dynamic reactances [3]. Hisdesigned. In particular, the use of power electronic devices torically, deriving models that have equal dynamic reactances has
makes it difficult for an individual to analytically derive the dif- been facilitated by the calculation procedures used to determine
ferentialequationsforallpotentialswitchingtopologiesofasystem.machine parameters. In particular, using traditional methods,
In order to alleviate the task of analytically deriving the equations parameters of the d-axis are calculated based upon open-circuit
of complex power systems, several algorithmic methods have and short-circuit tests. Similar tests to determine q-axis dynamic
been developed [10,11,12]. The syntax of the algorithms, as well values are impractical; therefore, approximations are often used.
as the equations that are formed and solved, vary. However, In particular, each axis is modeled with an identical number of
regardless of the method, the complexity and model structure of rotor windings and the q-axis dynamic reactance is assumed to be
components are critical factors in establishing computationally
equal to that of the d-axis, X, = X d . Values of the q-axis rotor
efficient system models.
The PVBR model described in Section I1 has been shown to Circuit parameters are then determined using these approximabe an efficient model structure for detailed analysis of power tions [3l. TO the authors' knowledge, the accuracy of neglecting
electronic based systems. Comparisons between model struc- dynamic saliency using this technique has never been analytically
tures of synchronous machines used in algorithmic languages established, although it has been questioned [51. Furthermore,
have demonstrated that the PVBR is significantly more efficient the test procedures used to obtain the q-axis parameters from
than standard physical-variable models [9] and models in which these methods has been opened to concern [3]. This technique
Park's circuit is used with associated coupling equations [13]. will not he explored in depth in this paper for two reasons, First,
Although efficient, there are significant incentives to eliminate from an analyst's Perspective, if Parameters are provided in this
the time-varying inductances present in the model, The incen- form, the accuracy of neglecting dynamic saliency only becomes
an issue if the responses predicted by the model are deemed inactives are two-fold.
First, significant numerical gains can be achieved if time- Curate. In such instances, the analyst will most likely need to pervarying inductances are eliminated. In particular, in differential form tests in which the q-axis machine paramcters can he
equation-based algorithms, a major computational step is the measured, such as a Standstill Frequency Response (SSFR) test.
manipulation of the system equations into an explicit state model Results Of these tests will most likely yield a model that does not
form. This manipulation, in which currents are state variables, have equal dynamic reactances, in which case the technique
requires the solution of a system of equations of the form
described in this paper can be applied. Second, with advances in
(25)
test
procedures to measure q-axis parameters, a more common
L,& = -(rx+pL,)ix+vx
approximation applied in power electronic based system analysis
for the derivative of the state vector, pi,. In ( 2 3 , rz , L x , and is to neglect dynamic saliency in the machine and system equa-
pL, are algorithmically generated matrices consisting of branch tions [21. The accuracy of neglecting dynamic saliency using
this technique has been analytically examined for machine/conresistances, inductances, and time-rates of change of inductances, verter systems, and serious
regarding the accuracy of
respectively. The size of these matrices are directly proportional the auuroximation have
raised
121,
_.
.to the size of the system. If the matrix L, contains time-varying
Considering the PVBR model of the machine given in (16)-
1180
(24), the time-varying inductances are eliminated if the following
assumption is made, namely
I ,
I,
Ld-L
!?
Lb =
-o.
(26)
3
Determining mathematical hounds for errors resulting from this
approximation is a tedious, if not intractable, task. Therefore, to
better understand this approximation, it is helpful to transform the
stator equations (16) in which (26) is applied, to the rotor reference frame. Application of Park's transformation to the approximate stator equations yields
U
,
over the entire frequency range.
2
V. NEW METHOD OF NEGLECTING DYNAMIC SALIENCY
Observing that the traditional method of neglecting dynamic
saliency introduces errors over the entire frequency range, a new
method is derived that only introduces errors at and beyond a user
specified frequency, f, . Independent of how the synchronous
machine parameters are determined, the operational impedances
for the q- and d-axes can he expressed as
(29)
(27)
(30)
(28) where X4, X,, and zXy are determined from the machine param-
"d
Comparing the original and approximate models, it is seen that
the q- and d-axis dynamic reactances in (5)-(6) are replaced with
average values of the q- and d-axis dynamic reactances in (27)(28). This is interesting, since (27)-(28) are similar to representations of the stator voltage equations which raised concerns
regarding the accuracy of neglecting dynamic saliency in [Z].
To further compare the original and approximate models, it
is interesting to consider the models in the frequency domain. In
particular, by simulating a machine to a steady state, and setting
the rotor angular velocity and stator resistance to zero, perturbations to v i $ and v i s can he made and the resulting perturbations
eters as shown in 1151. From these expressions, the high-frequency asymptotes, which correspond to the dynamic reactances,
are defined as
x" =
4
x" --
(
2y15y2"'TqM
4 ZYlTYZ"'TQM
1
(31)
(32)
d(2Tdf:l"'TdN)
Df D1 " " D N
In general, xq and xd Will not be equal. Therefore, from (211,
the PVBR model will contain time-varying inductances. As
shown in the Section 111, significant advantages can be realized if
I
,
.I
measured. The resulting input impedance of the q- time-varying inductances are eliminated. Thus, the goal of this
technique is to fit machinc parameters to approximate impedance
and d-axes equivalent circuits, Zq(s) and z d ( s ) , can then he curves that have equal dynamic reactanceS while matching the
obtained numerically and used to establish the operational imped- original operational impedances for frequencies less than a user
ances X,(s) and X,(s) [E].The numerically calculated q- and specified rotor fit frequency, f,. This goal can be accomplished
d.axes operational impedances of the synchronous machine by altering the rotor windings in the original model or by adding
described in section
VI are shown in pig, 2, comparing
the a winding in either or both axes. From experience with this technique, either or both q- and d- axes may he manipulated with
identical results. Therefore, for simplicity, only the q-axis
Model
parameters will be modified herein.
Once the rotor fit frequency, f,,is sct by the user, the refitIn tqS
and i;,
01
10'~
J
io2
io-'
ioo
IO'
10%
io3
10'
ting of the impedances is a two step procedure. The first step is to
consider whether an additional winding is required. A winding
will need to he added if
T
1
QM
>-207lfr
(33)
Failure to add a winding- if (33)
. . is satisfied will produce a q-axis
impedance that will not match the original q-axis impedance to
'
z* 0
the specified frequency f,.
IO-' 10.'
10.'
loo
IO'
IO'
10'
io4
The second step is to determine the q-axis operational
frequency (Hz)
impedance time constants. If a rotor winding is added, numerator
Fig. 2 q- and d-axis operational impedances, original and traditional approximate and denominator time constants, 2Ty(M
+
and T Q ( M+ 11, must
models.
_"._
L--
,
~
impedances of the original and traditional approximate model
shows the frequency domain errors associated with the traditional
method of neglecting dynamic saliency. In Particular, in the traditional approximate model, the curves are both shifted by
he added to (29) to fully describe the higher order system, If a
winding is not added, TqM and T~~ must he modified to provide
equal dynamic reactances. In either case, only the numerator and
denominator time constants associated with the highest order
result of the equal dynamic reactances.
Iff, is large, a high frequency rotor time constant is introduced. However, the rotor terms required to produce accurate
results are generally slow compared to many stator time constants
in power electronic based systems. Therefore, the effect on simulation performance is minimal.
VI. COMPUTER STUDY
In order to illustrate the accuracy and numerical efficiency
resulting from the new approximation technique, the results of
two computer studies are described. The synchronous machine
= 0.08
rs = 5.15
'L,?
x mq
rkd = 0.024
= 1.00
xLkd= 0.334
XLfd = 0.137
XI,,,
= 0.330
r*
r*
= 0.062
kq2
kql
= 10.44
XTkq2 = 0.331
X m d = 1.71
rj.. = 1.11
r k q Z = 0.061
XTkql
= 0.146
* indicate refitted
parameter.
variable
.I
traditional
approximate model
new
approximate model
29.7 %
1.3 %
21.5 %
0.1 %
31.1 %
0.7 %
24.4 %
0.2 %
NAM, which is expected since the state model of the system is
relatively small in order. Therefore, the addition of a rotor state
has a more significant impact than would occur for larger-scale
systems. Although the CPU time required by the OM are not prohihitive for this study, as the size of the system increases, the
computations required to reform the state model at each time-step
increases. Therefore, the difference in efficiency between the
NAM and OM will significantly increase with the size of the svs-
'qs
.I
Ids
T.
$"d
4
- 2
.
so
-2
-4
10
Traditional Approximate
h
8 5
c2
'0
1
I
Traditional Approximate
6
6.05
6.1
6.15
6.2
6.25
6.3
time (s)
In Fig. 4 Responseof
NAM and OM are less than o,6 % for all system
contrast, the TAM response is significantly less accurate. Errors
for the TAM system variables are greater than 25%, which correspond to results published in 121. Close inspection of the
responses also reveals that the TAM does not accurately predict
the switching dynamics in either the steady-state or transient
response.
The solution of the NAM and TAM responses was accomplished by pre-computing the matrix L, at each change in topol-
ogy. Thereby, the solution of (25) only required a matrix-vector
multiply at each integration timc-step. In contrast, because of the
time-varying inductances of the OM model, a linear solve routine
was required to form the OM state model at each integration
time-step. The CPU times required to calculate the 0.2 s studies
on a 133-MHz Pentium PC, are given in Table 4. Therein, it is
seen that the time-invariant form of the PVBR is significantly
more efficient than the time-varying form. The NAM is on the
order of 48% faster, while the TAM is on the order of 58% faster.
gas-turbinegeneratorto application and removalafa3-phase
fault at stator terminals (From tov. - a-axis
. stator current. d-axis Stator current.
field current, and electromagnetic torque)
c
.
.
.
.
_
.
_
.
_
_
_
.
~
.
.
.
.
.
.
_
Synchronous Machinc
\,
i
_._..~....._........_
Fig. 5 Synchronous machinclconvener system
1183
5
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Approximate
.’cf5 d a d i t i o n a l Approximate
Model
variable
traditional
mode1
new
approximate
‘dc
28.2 %
0.5 %
lJ<l
25.1 %
0.5 %
las
28.3 %
0.6 %
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