42
IEEE Transactions on Energy Conversion, Vol. 13, No. 1, March 1998
cient and Accurate Model for the Simulation
Synchronous MachindconverterSystems
H. J. Hegner, Member
Naval Surface Warfare Center
Annapolis, Maryland
S. D. Pekarek, Student Member
0.Wasynczuk, Senior Member
School of Electrical and Computer Engineering
Purdue University
West Ldayette, Iadiana
Abstract - A new synchronous machine model is presented which is
readily implemented in either circuit-based or differentialequation-based simulation programs. This model is well suited for the
simulation and analysis of synchronous machine - converter systems. It is based upon standard representations and no a p p r o h a tions are made in its derivation. However, the numerical
implementation is shown to be significantly more eEcient An
example is provided which demonstrates a 1700%increase in si"
lation speed with no observable lws in accuracy. The model
includes provisions for an arbitrary number of damper or rotor
windings and may be easily modified to represent synchronous or
induction machines with an arbitrary number of stator phases.
I. INTRODUCTION
Although there are a large number of techniques and computer programs available for the simulation of power circuits and
systems, many fall into one of two general categories. In circuitbased simulation languages such as SABER [l] or SPICE [2],
the circuit data (resistances, inductances, capacitances) are specified branch by branch. Topological information may be defined
graphically or by a branch-to-node incidence matrix which is
easily established from the equivalent circuit. Independent and
dependent sources and time varying-parameters are easily incorporated. In order to establish the time-domain response, the differential equations are discretized at the branch level providing
an algebraic equation relating branch voltages and currents at
any given instant of time to their past values. These difference
equations may be assembled numerically using nodal techniques
to form a set of coupled difference equations of the interconnected system.
In differential-equation-based languages such as MATLAB
[3] or ACSL [4],the system is described by its differential equations or transfer functions. The differential equations are converted to difference equations in accordance with the specific
integration algorithm selected (e.g. Euler, Runge Kutta, Gears,
etc.). The disadvantage of this approach is that in many converter-machine systems, it is difficult to derive the differential
equations for all the potential topological modes. However, an
algorithmic method of developing the state equations of complex power circuits and systems was recently set forth in [5].
PE-689-EC-0-04-1997 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 November 13, 1996; made available
for printing March 26, 1997
Using this method, circuits may be described by the pertinent
branch parameters and a branch-to-node incidence matrix as in
circuit-based languages. The composite system state equations
are then established algorithmically, and are subsequently
solved using any one of a number of well established techniques.
Regardless of the simulation approach selected, the model
structure and choice of state variables can have a significant
impact upon simulation speed and accuracy. In thi
synchronous machine model is presented which is shown to
have significant advantages relative to existing models. It is
readily implemented in either circuit-based languages or in differential-equation-based languages using the state model generation algorithm in [5]. This model is based upon s
representations and no approximations are made in its derivation. In fact, the solution of the corresponding state equations
yields identical results for the same inputs and initial conditions.
However, the numerical implementation is significantly more
efficient. In the example system sfudied, the given model gives
rise to a 1700%increase in simulation speed with no discernible
loss in accuracy.
II. AUTOMATED STATE MODEL GENERATION
Before introducing the new synchronous machine model, it
is useful to describe briefly the state model generation algorithm
set forth in [5]. In switched inductive circuits, the state equations
may be expressed explicitly as
pi,
=
[--Li1(r,+pLx)]ix-LilBbebr
(1)
T
T
where L, = BbLbrBb, rx = BbrbrBb,
and B, is the socalled basic loop matrix which is numerically established from
the branch-to-node incidence matrix. Also, rbr and L,, are
the branch resistance and inductance matrix, respectively, which
are readily established from branch data. The currents and voltages of all branches in the system may be obtained using
T.
ibr= Bbi,
and
Capacitive elements may be incorporated into the state
model as described in [5]. An alternative formulation may be
derived in which flux linkage, rather than current is used as the
state variable. Therein, the current vector is written in terms of
flux linkage as
i,
=
L,-1 h,
0885-8969/98/$10.00 0 1997 IEEE
(4)
43
and the resulting state equation written
d-axis rotor windings
-1
p h , = - r,L, h, - B b e b r
.\\
I
Lbs
' b r = [ ( r b r + P L b r ) B E - L b r B bTL ,-1 (r, + p L , , ] L i l h +
[I - L b , B bT L,-1 B J e b r
) stator windings
'kdN
(5)
The current and voltage of all branches in the system may be
obtained using
T -1
i b , = B b L , h,
(6)
and
q-axis rotor windings
(7)
Equation (5), (6), and (7) define the state model of the system
with flux linkages as the state variables and branch voltages and
currents as outputs.
m. CIRCUIT-BASED SYNCHRONOUS MACHINE MODEL
For notational purposes, it is assumed that the synchronous
machine has three stator windings, one rotating field winding,
and an arbitrary number of short-circuited damper windings
along the q and d axes. Machines with other than three stator
windings require only a minor change of notation. The equivalent
circuit is depicted in Fig. 1. Therein, the orientation of each winding (depicted in Fig. 1 as inductors) portrays the physical dmction of the Corresponding magnetic axis.
The voltage equations may be expressed
Fig. 1 Coupled equivalent circuit of synchronous machm.
state model with voltages as inputs and winding currents as state
variables. This representation of the machine is subsequently
referred to as the coupled circuit (CC) model. The corresponding
equations may be solved numerically using either circuit-based or
differential-equation-based languages. In circuit-based languages, the nodes and branches of Fig. 1 are first labelled and/or
numbered, the branch parameters (resistances, self- and mutualinductances) are specified and topological information is defined
graphically or by a branch-to-node incidence list. In differentialequation based approaches, equations (8) and (11) are programmed using syntax specific to the selected language. In either
case, a variety of numerical algorithms (e.g. Runge-Kutta, Gears,
Euler) may be applied to establish the time-domain response.
IV. VOLTAGE BEBIND REACTANCE REPRESENTATION
where
Designating 8, as the electrical position of the rotor, the stator variables may be transformed to the rotor reference frame
using Park's transformation [SI.
<dos = K:(er)fabcs
Here, f can be
i , or h . The stator resistance matrix is
rs13 . The rotor resistance matrix rr is diagonal with entries
corresponding to the appropriate field or damper winding resisU,
(14)
tance. The flux linkage equations may be expressed
where f may be a voltage, current, or flux linkage, and
-
_I
Icos(Qri cos(€+-$)
Where Lss is the stator inductance matrix, L,, and L,, repre-
cos(Or+q
sent the mutual inductances between the stator and rotor windings, and L,, represents the rotor inductance matrix. The stator
-1
inductance matrix is of the form
2
(12)
i
The equations of the synchronous machine may be expressed in
the rotor reference frame as [6]
.r
Phis
(17)
%gs P h i ,
(18)
vis! = rs'slqs+ % G s
where, for example
La,,, = LlS+ LA - LgCOS28,
(13)
Expressions for the remaining stator and rotor self- and mutualinductances are given in [6] and will not be repeated here due to
space considerations.
Equations (8) and (11) are readily manipulated to form a
l2
.r
= r.2d.S VOS
UJ.
=
QOS
+Phos
= rJ. iJ. + p h j
+
4-
(19)
(20)
where j = kql, ...,k q M , fd, kdl, ..., k d N . Here, the super-
ables are expressed in the
this paper, rotor variables
to the stator by the appropriate turns
he stator and rotor flux linkages per sec-
Substituting (31) and (32
equations may be re
r
the
.r
r ~
s qs
.r
' d s = 's'ds-
U
qs
r
+
=
In the analysis of machin
an approximate voltage-b
(37) and (38) wherein it is
constant during fast switch
A.J = L Ui j + A m d ; j
= fd,kdl,
...kdN
(24)
where
p h i terms may be neglected. H
algebraically incorporated
which case an exact voltag
be obtained. This is achieved
oltage equations in a voltagebehind-reactance form, the magnetizing flux linkages are f i t
expressed in terms of rotor flux linkages. Solving (21)-(24) for
currents and substituting into (25) and (26) yields, after algebraic
manipulation,
Expressions for the derivat
obtained by manipulating the
(24). Using
1
i . = -(A.J
Lij J
M
1-1
,,.r
.r
r
uLs = rqLqs+ q-LJdtds + p L q i q s + e;
1
1
N
-1
w
.r
(30)
j=
(42)
for the rotor currents and substituting the resulting expressions
into (37) and (38) yields, after algebraic manipulation, an exact
voltage-behind-reactanceform of the stator voltage equations
where
r
hmd);j = fd, kdl, ...kdN
1
fi
.r
uLs = r i i & - o , . L ~+LP L~d~i d s + e ;
(43)
(44)
where
The stator flux linkage equations can then be expressed as
xis = L i i i , + h ;
(32)
r $ = r s +Lm%fd
-
where the double primes ("1 are used to denote subtransient
(46)
L;fd
quantities. The subtransient inductances, L i and Ld are given
by
and
,,
L; = L l s + L m 4
(33)
(34)
and the subtransientflux linkages by
(35)
These equations
applying the inverse
(44), which yields,
Y
d
45
where Am, and hmd are given by (27)-(28),define the so-called
voltage-behind-reactance (VBR) model of the synchronous
machine. It is important to note that no approximations were
made in its derivation. Neglecting numerical error, the solution of
the corresponding equations should yield the same time-domain
response as the CC model upon which it is based. Moreover, no
assumptions have been made in regard to the stator winding configuration. The windings may be connected in wye, delta, or the
individual windings may be supplied to isolated converter circuits. With only minor modification, the model may be extended
to include machines with an arbitrary number of stator phases.
When implementing the VBR model, only the stator
branches and nodes are included when defining the circuit topology. The rotor voltage equations are expressed explicitly in state
model form with rotor flux linkages as state variables. The subtransient voltages represent outputs of the rotor model and are
incorporated in the stator circuit as dependent sources. The stator
branch currents are transformed into the rotor reference frame
and represent inputs to the rotor state model. A circuit/blockdiagram of the VBR model is given in Fig. 2
In circuit-based approaches, it is necessary to solve an ndimensional set of linear equations at each time step where n corresponds to the number of nodes. In differential-equation-based
approaches using the state model generation algorithm, it is necessary to solve a k-dimensional set of linear equations where
k = b-n+l.Here,bisthenumberofbranchesannisthe
number of nodes. The elimination of rotor branches and nodes
reduces the dimensionality of the linear equations that need to be
solved at each time step in either approach.
The resistance and inductance matrices are given by
-,
r
1
n
n
La = Lmq
+
xkz&-I
Lmd
;'I-'
I
l
l
3
.
It is interesting to note that the stator voltage equations are
complicated by the addition of the time varying "mutual" resistances contained in (51). However, these are easily incorporated
by introducing off-diagonal terms in the branch resistance matrix
rbr.
The stator voltage equations given by (49), along with the
rotor voltage equations
r.
p h . = -L(hj-hmd)+Vj; j = fd, k d l , ...k d N
Llj
_ .
,
U
Fig. 2 Voltage-behind-reactancemachine model.
V. EIGENSYSTEM ANALYSIS
Although the CC and VBR models are equivalent in the
sense that the solutions are identical for the same inputs and initial conditions, the eigenstmcturesmay be substantially different.
For example, if the CC model is implemented with currents as
state variables, the state model (1) can be expressed symbolically
as
p i , = A{@,) i, + Bier) ebr
(63)
where
(62)
A(er) = -Lil(O,)[r, +pL,(e,)l
(64)
46
B@,)
=
-Lil(Q,)
(65)
Also, ebr is a vector of dependent and independent branch voltages which are assumed to be inputs in this analysis. If the CC
model is expressed with flux linkages as the state variables, the
state equations may be expressed
p h , = 40.1 h,
where
+ Bhebr
(66)
Ah(@,)= -IzL;?e,)
(67)
(68)
Although (63) and (66) are algebraically related, the eigenstructures are substantially different. To show this, it is convenient to assume that the rotor of the synchronous machine is held
fixed at an arbitrary position ern. Here, the subscript denotes
where
pL(8,) = 2wFBsin(28,)
If the rotor is f i e d at an arbitrary position, the inductance is constant and the resulting eigenvalue [-r/L(B,@)I is negative.
However, if the rotor speed is such that 2 0 r L B > r , then for
some rotor positions the resulting eigenvalue will be positive.
The presence of positive real eigenvalues can lead to a significant error growth when integrating the state equations numerically. Generally, a smaller time step is needed in order to
maintain a specified truncation error. A detailed explanation of
the relationship between eigenstructure and time step is contained
in [9]. This analysis suggests that flux linkages represent a better
choice of state variables. In the VBR model with (5) used to represent the stator dynamics, the rotor and stator flux linkages are
state variables.
that w, = 0 . With the rotor fixed, the corresponding state matrices A$o,fo) and A,(0,@) are timeinvariant. Moreover, they
are similar since
q1
Ai @,fd=
(e,fo)A,( 6,fO )L,( % f O )
(69)
Therefore, at zero speed the two models have identical eigenvalues. Since the machine is stationary, the eigenvalues are negative
regardless of the selected rotor position.
VI. COMPUTER STUDY
In order to illustrate the advantages of the new model, an
experimental system consisting of a 3.7 kW synchronous
machine connected to a line-commutator converter was simulated. The system studied is the same as that described in [5]. A
circuit diagram of this system is shown in Fig. 3. The synchronous machine parameters, as determined by standstill frequency
response testing, are summarized in Table 1.
If the machine is allowed to rotate, the matrices Aje,) and
Ah(e),
at any rotor position e,, may be expressed as
-4'(Or0)rX-Li1
(6,0)~L,1(
)e,
~$0~=
0)
(77)
Table 1: Synchronous machine parameters.
Ir, = 382 mQ
IL,, = 1.12 mH
IL,,
= 24.9 mH
(70)
(71)
Comparing (71) with (67) it can be seen that if
,e,
A~(O,,) = A,@,f,)
=
then
(rkd3= 1.58 Q
(72)
Thus, the real parts of the eigenvalues of A,(e,) are negative
regardless of speed or rotor position.
The same cannot be said for 4 (0,). Comparing (70) with
(64)it can be seen that if
,e, =
= 4.52 m~ Irkq3= 447 mQ
I
L,,
= 39.3 mH (Llkol= 4.21 mH (Llka2= 3.5 mH
then
~p,,)
= ~ p -L;l(e,,)pL;l(e,,)
,~)
(73)
Although the first term on the right of (73) was shown to have
eigenvalues that are negative, the addition of the second term,
which is a function of the changing inductances, can shift the
eigenvalues of 4 (0,) to the right-half plane. This can be shown
In the computer studies, it is assumed that the system is initially operating in the steady state with a base load resistance of
21 Q connected to the dc output terminals. A second load resistance of 4.04 Q is then connected in parallel with the original
load. The CC and VBR models were implemented in ACSL [4]
using a simple example.
using the state model generation algorithm. The simulated
In a single-phase reluctance machine the voltage equations
may be expressed as
response using the CC model is depicted in Figs. 4-5. The experimentally measured response is shown in Fig. 6 [5]. As shown, the
measured and simulated responses are in excellent agreement. In
calculating the response depicted in Figs. 4-5, Gear's algorithm
U = ri+p[L(O,)i]
where the winding self-inductance is given by
~ ( 0 ,=
) L,,+ L~ - L ~ C O S ( ~ ~ , )
(74)
(75)
with LA > LB ,the resulting state equation can be written in terms
was used with a maximum and minimum time step of 1 x
and 1 x
sec, respectively. The local truncation error, which
-4
of currents as
1
LWU
(76)
is used to determine the actual time step, was set to 1 x 10 for
all state variables. When a change in topology (change in diode
conduction state) is sensed, the time step is reduced and the pre-
I
'
b
VP
l
h
t) f)n 12
b 19
n16
b23
n20
nl
-
-
vbn
b 10
i
b8
vca
47
i.
U
n3
--,
== vdc
n4
Fig. 3 Example system.
*O
1
2o 1
0 '
0.75 1
0.751
0.0
I
40ms
I
'
Fig. 4 Simulated dc and field currents.
2o 1
-20 J
50 7
-50'
Fig. 5 Simulated ac current and vdtage.
ceding calculation is repeated so as to limit the uncertainty in
switching time to less than the minimum time step. Following
each topological change, the time step is set to its minimum value
and the integration algorithm is reset. Subsequent time steps are
adjusted in accordance with Gear's criteria [ 101.
Fig. 6 Measured dc and field currents.
For comparison purposes, the same study was repeated using
the VBR model. The results are indiscernible from those depicted
in Figs. 4-5. However, the computer time is significantly less.
The CC model required 212 sec and the VBR model required
12.3 sec of CPU time with both models implemented on a 133MHz Pentium-based personal computer. The average time step
selected by Gear's algorithm was 300% larger in the VBR model
with no observable difference in response. The further savings is
attributed to the reduction in the number of nodes and branches in
the VBR representation. The CC model includes 24 branches and
20 nodes whereas the VBR model includes 10 branches and 6
nodes.
Although the eigenvalues vary with time, it is interesting to
compare them at a specific instant of time. The eigenvalues associated with the two models are summarized in Table 2 for
t = 0.02. While the CC model has two eigenvalues with positive real parts, all of the eigenvalues of the VBR model have negative real parts, In fact, if the eigenvalues are calculated at other
time instants, the CC model consistently has eigenvalues with
positive real parts, while the eigenvalues of the VBR model stay
in the left-half plane. Consequently, a larger integration time step
may be used in the VBR model. This, coupred with the reduced
dimensionality of the linear equations that must be solved at each
time step, results in a much faster simulation.
48
Bble 2: System Eigenvalues using CC and VBR Models
t
I
326
-46
-279
-802
- 1,377
-4,718
-13,442
-218,060
I
I
I
-52
-205
-370
-798
-1,176
-13,315
-214,489
[47 Mitchell and Gauthier Associates, Advanced Continuous
Simulation Language Reference Manual,” Concord, MA,
1993.
IS] 0. Wasynczuk and S. D.Sudhoff, “Automated State Model
Generation Algorithm for Power Circuits and Systems,” Paper 96 WM 259-2 PWRC presented at the IEEE Power Engineering Society Winter Meeting, Baltimore, MD, Jan. 21-25,
1996.
Electric Machinery,
nous Machine Systems ”
version, Vol. 8, No. 1,
scataway, N5, 1995.
, ‘Analysis and Averageutated Converter-Synchroions on Energy Con-
Vn. SUMMARY
A new voltage-behind-reactance model of the synchronous
machine is derived. Therein, the rotor dynamic equations are
expressed explicitly in state model form while the stator equations are described in circuit form. The model is computationally
more efficient than existing machine models without making any
approximations, it is flexible with regard to the stator winding
configuration, and it is compatible with circuit-based and differential-equation-based simulation approaches.
age-Value Model of Line-CommutatedConverter - Synchronous Machine Systems,” IEEE Transactions on Energy
Conversion, Vol. 8, No. 3, pp. 404-410, September 1993.
[9] L. 0. Chua, P-M Lin, Computer Aided Analysis of Electronic Circuits: Algorithms and Computational Techniques, Prentice-Hall, Englewood Cliffs, New Jersey, 1975.
[lo] C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice Hall, NJ, 1971.
VIII. ACKKNOWLEDGEMENTS
This work was supported in part by P. C . Krause and Associates under contract “24-93-C-4180
with Naval Surface Warfare Center and in part by University of South Carolina under
Grant No. N00014-96-1-0926 with Office of Naval Research.
Steven D. Pekarek was bom in Cicero, Illinois on December 22, 1968. He
received the B.S.E.E, M.S.EE, degrees from Purdue University in 1991, 1993
respectively, and is currently pursuing his Ph.D. His interests include electric
machines and automatic control.
E.REFERENCES
Analogy, Inc., “Introduction to the Saber Simulator,” Beaverton, Oregon, 1991.
L. W. Nagel and D. 0. Pederson, “Simulation Program with
Integrated Circuit Emphasis,” University of California Electronic Research Laboratory, Memorandum EAL-M382,
April 1973.
The Mathworks, Inc. “Simulink Dynamic System Simulation Software - Users Manual,” Natick, Massachusetts, 1993.
01% W a s p a u k ( M 76, S M 88) was bom in Chicago, Illinois on June 26,1954.
He received the B.S.E.E. degree from Bradley University in 1976 an the M.S. and
Ph.D. degrees from Purdue University in 1977 and 1979, respectively. Since
1979, he has been at Purdue where he is presently a Professor of Electrical and
Computer Engineering. Dr. Wasynczuk is a member of Eta Kappa Nu, Tau Beta
Pi and Phi Kappa Phi and is a Senior member of the IEEE Power Engineenng
Society.
H. J. Hegner (S’ 88, M’89) received the B.S.E.E. degree from the Virginia Polytechnic Institute and State University in 1983 and the M.S.E.E. degree from Purdue University in 1992. He is currently a member of the US. Navy Advanced
Surface Machinery Programs in which he serves as a team leader of the DC Zonal
Electric System Program. For the past 16 years, he has specialized in electrical
systems and components for US.Navy shipboard systems