A Study of Synchronous Machine Model Implementations in Matlab/Simulink
Simulations for New and Renewable Energy Systems
Z. Chen, F. Blaabjerg, F.Iov
Institute of Energy Technology, Aalborg University, Aalborg, DK-9220, Denmark
Abstract-A direct phase model of synchronous machines
implemented in MATLAB/SIMULINK is presented. The
effects of the machine saturation have been included. Simulation studies are performed under various conditions. It has
been demonstrated that the MATLAB/SIMULINK is an
effective tool to study the complex synchronous machine and
the implemented model could be used for studies of various
applications of synchronous machines including in renewable and DG generation systems.
I. INTRODUCTION
Synchronous machines have been widely used in
power systems, they are not only the main generation
units in large scale conventional power stations, but also
in small and remote stand alone systems. Various new
types of synchronous generators are being developed for
newly emerging renewable and distributed generation
(DG) systems, including multi-pole machine for wind
power conversion systems. These machines play a very
important role to achieve a high efficiency and reliable
power system with good power quality. A detailed and
accurate model is essential to investigate the performance
of a synchronous machine and its control strategies.
Various modelling and simulation techniques have
been reported to study the belabour of synchronous machines, dqO reference frame models of synchronous generators are widely used due to the simplicity, however,
these models have some limitations [1,2], for example in
studying unbalanced and nonlinear loading conditions. On
the other hand, a direct phase model is possible to provide
a more accurate solution [3] and has the potential to study
the performance of various new type of synchronous generators. Also, various conditions, such as sudden application and removal of balanced and unbalanced loads, connection of power electronic converters and symmetrical
and asymmetrical faults, can be easily investigated.
Currently Matlab/Simulink is a widely used simulation tool for dynamic systems. A wide range of components will be involved for modelling large dynamic systems, for example, power system, including prime movers, generators, transformers, power electronic converters.
Matlab/Simulink is an effective tool for such applications
[4].
This paper presents important aspects regarding the
implementations of a direct phase synchronous machine
model. It first briefs the synchronous machine model in
both abc frame and qdO frame, then discusses the saturation considerations, the saturation is considered in q-d
frame as usual way, however, an mathematical expression
is used to relate the current and inductance, then the in-
ductance may be directly found with the fluxes, which are
used as the state variables in the simulation model. The
model of synchronous machine has been implemented in
MATLAB/SIMULINK. The effects of nonlinearity of the
machine have been included in the model. Simulation
studies are performed under various conditions. It has
been demonstrated that the MATLAB/SIMULINK implemented model has the potential to be used for studies
of various applications of synchronous machines such as
in renewable and DG generation systems.
II. SYNCHRONOUS MACHINE MODELS
Both direct phase model and commonly used dqO
frame model that is used for considering the saturation
effects are briefed in this section.
A. Direct phase model
The six circuits of an idealized synchronous machine,
3 phase windings, a field winding and two equivalent
damper coils, are shown in Fig. 1.
kQ
O01 r
a
/
a
(o,
f
b,
kD
.ryy)n
C,
Fig. 1. Diagram of idealized synchronous machine.
The performance of a synchronous generator can be
described by the voltage equations in direct phase quantities for the three armature phases, the field and two
equivalent damper coils. The position of the rotor at any
instant is specified with reference to the axis of phase a by
the angle Or. In terms of flux linkage, the voltage equations for the six circuits can be expressed in phase frame
as:
[V]=[R].[i]+ d[A]
dt
where
1960
[V] = [Va
Vb
VC
[i]
2b
Pc
[ia
[R]= diagonal[r,
0
(1)
0
Vf 1
r
'Q'D D
f
rb
r,
rQ
rD
rfT
The flux is related to the current by:
(2)
[A I = IL I * [id
Where L is the machine inductance matrix, which is a
function of the rotor position and the saturation condition
of the machine. The general form of the inductance matrix
may be expressed as
Lss
L 2abc
labc 1
Lsr
L
(3)
2AQDf L _Lrs Lrr j LiQDf j
The stator inductance sub-matrix
Ls+Los L2scos 2Or
2I
C
3)
[LSS ] = LOS /2-L2S OS-L20S
CoS(20r-+2S2w 13)
+
/
2;T
cos(20,
2-L2S
13)
-LOS
[VqdO] [T][Vabc]
[ZqdO] [T][2abc,]
LID +LmD LfD
LDf
Lif + Lmf
Clearly the components of the stator inductance submatrix, the stator and rotor mutual inductance submatrixes are the functions of O, which is the angle difference between the reference frame and the magnetizing
axis of stator phase a. Therefore these elements are time
varying.
The differential equation to be solved can be expressed in terms of winding flux as
d
dt ([L ][i = [V] - [R ]. [i]
dt~~~~~~~~
d (h[ I = [L] '([V ]
[R ]. [ ]
dt
=
Pe
Vala
+
Vblb
where
[Vqd O ] [vsq Vsd
['qd O] [isq
sd
AO | [iqd 0]
[sq
isd
io
dA,q
= rsq +
dt
d
dt
Vsd --:rsisd +dAd_Os
dt
r
d2Q
d ([L [J )
(6)
0 = rQ iQ +
rDiDdt
+ VcIc
(9)
O
=
dt
(7)
ri' DD+dD
dt
o
v =
i
+
dt
A
The flux equations related to mutual inductance are:
2sq = LIS isq+ Lmq imq
sdK= LlISd +L md imd
o rm
° rm
VO j
Apply the above transformation to the voltage and
flux equations, the following set of equations may be obtained:
The voltage equations:
Vs
Both equation (5) or (6) can be used as basic d ifferential equations for simulation, the expression of e quation
(5) is simple. The output electromagnetic torque of the
generator and rotor motion equations can be calcullated in
usual way, such as
Te
[iqdO I= [T][Iabc,]
(5)
[A ]= [V ]- [R ]. [L ]'[ ]
Or in terms of current as
(4.a)
introduced briefly here. For synchronous machine study,
the dqO model is commonly referenced to rotating rotor
frame, and then only the stator parameters need to be
transformed, such as
=
0
ifY] and QDJ] = [ jD jf]T are the
flux linkages and currents of damper and field circuits.
The sub-matrices in the flux equation (3) are given in (4)
KD ]
LOS /2-L2SCoS(20r- 2wT13) LOS /2-L2S CoS(20r + 2W 13)
LIS +L S-L2S CoS(20r + 2w-T3)
LOS /2-L2S COS 20r
LOS /2-L2S cos 20r
LIS +L S-L2S CoS(20r- 2wT/ 3)]
The stator and rotor mutual inductance sub-matrix
I Lsq cOW,
Lsf sinO 1 (4.b)
Lsd sinO,
[Ls, ] = [Ls ] LsqCOS(O,- 2ff 3) Lsd sin(O,- 2ff 3) Lsf sin(O -2fT 3)
Lsq COS(0 + 21ff 3) Lsd sin(O, + 21fI 3) Lsf sin(O, + 21ff 3)j
The rotor inductance sub-matrix
0
0
LIQ + LmQ
(4.c)
[Lrr] 0=
Where
[/2abc] [= a Ab A,]T and [iabc]= [ia 'b i ]T are the 3
phase flux linkages and currents of the stator windings.
and
(10)
AQ = LQiQ + Lmqmq
dcor
2H
dt
C0b
b, co,9r
the rated angular speed and rotor angular s
(T +TmechO-rKd)
XD
(8)
L=DiD+Lmdimd
Af =Llf if +Lmdmd
where
electrical and mechanical torque,
Te, Tmech
Kd
damping constant.
imd
isd +iD +if and imq =isq + iQ (11)
Also the q and d-axis magnetizing flux may be respectively defined as
B. qdOframe model
Many proposed methods of considering electri cal machine saturation are based on dqO frame [5-10], and so
does this reported model. Therefore, the dqO frame' is also
1961
'mq
=
Lmqimq and 2md= Lmdlmd
(12)
After the transformation to rotor frame, the inductance
parameters are no longer the function of 0r, which simplifies the calculation significantly. Lmd and Lmq are mutual
inductances along d and q-axis, which can be saturated
and may be considered as functions of currents, imd and
T
III. SYNCHRONOUS MACHINE SATURATION CONSIDERATION
LLmd
Lmq 0
m
=
Lmq
L md
const
(13)
Then the anisotropic salient pole machine may be converted into an equivalent isotropic machine by means of
saliency ratio as
= nd+ Aq /m2 and in Imd +miq
So that it becomes possible to define a unique magnetizing inductance
Lm = Am / im
Then the open circuit d-axis magnetizing curve may
be used as function A. = f (im ) for saturation consideration. However, q- and d-axis magnetization may be considered separately by using (12).
Using (10), (11) and (12), the currents may be expressed as:
*
_ isd
md
sd
isq
iQ
T
Asq
mq
= AD
(14)
+
LIS
1
md
Lmd
=
+
1
LIQ
(2sd
( S+ 2 D+ 2
Lmq
Lis
LIQ
mq
Keeping Lmd, and Lmq in (17) as constants, LmdO and
LmqO, would result in a non-saturated magnetizing inductance machine model. However, Lmd and Lmq are varying
with operation condition and the variation may be significant in certain conditions, and therefore the variation of
Lmd and Lmq will be taken into account in the reported
model.
B. Saturation considerations
In general, a magnetizing characteristic can be obtained by testing the electrical machine. A typical magnetizing characteristic of an electrical machine is shown in
Fig. 2.
Magnetizing cret(p.)
Fig. 2. Magnetization curve
The magnetizing inductance characteristic may be represented as [9]:
for i < i(t)
(1 8.a)
L =L 0
1 +
(1 8.b)
for im > im(sat)
LimO
a(im m(sat) )
A resultant magnetizing curve with (18) is shown in
Fig. 3.
md
.1
1.6
LID
_f
(17)
r) Lddl
LIS LID L1i Lmd
L
Amq
(qmq 2 + 2"Q) L ql
s
L
ZQ -Zmq
1
and
'm=
imd
A. Flux and current relationship
In the discussed model, the leakage flux saturation and
cross saturation are ignored, only main flux saturation is
considered.
In some studies [6,7], it is assumed that a saliency factor m denoting parameters and variables associated with
the magnetizing flux is constant as in (13), i.e. it is assumed that d- and q- axis are saturated to the same degree.
1
Lmq
imq, respectively.
(16)
1
qE
.-,.
.,
md
I
Then the magnetizing flux linkage components can be
expressed with d-q stator and rotor flux linkages and the
current variables are eliminated, as
AKd = (d+
+
LIS LID LI1
) )Lq
= (L sq~ + LA)
A"mq Amq
LIS
where
L dY -
LIS
I
0
1
1
LID
Lc
0
2
4
6
magnetizing cret(p.)
a
10
Fig. 3. Magnetizing inductance
(15)
L IQ
1
L,d
)Ldy
12
Equation (18) gives the relation of magnetizing current and inductance. The relationship between the fluxes,
magnetizing current, imd and imq, and inductance, Lmd and
Lmq may be established by substituting equation (16) into
(17)
± 1 ±
1md 1-+-+-))
=(sd + AD + ) (19.a)
L1S LID Lif
L1S LIDLD
1962
L((L+
1
2)
(I + L,,q
LLIs ±Q
i"q/(1±
L-)) =('Ad
LIQ
LIs LIQ
I
LIia14X_H
(19.b)
l9b
( l(n'Xt- + 'mq(,at) IU + Lmq(L + L J
L L)
Q
= LI,bothLID un-20.b
used for
a L(l9.a)
and )~
(l9.b) may
Equations
q
Li,be LlQj
Equations (19.a) and (19.b) may be used for both unIf the q-axis magnetizing curve is unknown, the as-
saturated and saturated conditions. For saturation condition, (1 8.b) can be rewritten as
'md
1mdO
ax ( Lmd
)m(
1) + md(sat)
for imd > imd(sat)
sumption of (13) may be used.
IV. MODEL IMPLEMENTATION IN MATLAB/SIMULINK
The model discussed above has been implemented in
MATLAB/SIMULINK environment. Equation (5) is used
as the basic differential equation for the implemented
model, i.e. the fluxes of the six circuits shown in Fig. 1 are
the state variables.
In order to simplify the implementation, the main
computation part has been written in a function format.
The simulation model is shown in Fig. 4, where a Matlab
function is used.
Then equation (l9.a) can be expressed as
1
F+
ii
(Ld+ALAf)
lL,.dO _-)
md
a
L
d
sL ID LI j IIS I
Equation (20.a) is a second order equation of the magnetizing inductance Lmd, therefore the saturated Lmd can be
directly solved out from a set of values of d-q axis flux
variables.
Similarly, the q-axis equation can be obtained as
,
P
J
M-
+
.
0 Y 1h, >
I
-4
.
-1
I
M.km!
.--
I~
Fig. 4. Synchronous machine simulation model
V. SIMULATION STUDY
A synchronous machine system has been simulated
with the developed abc direct phase model, the same parameters as given in Synchronous machine projects 1 of
reference [5] are used.
The machine is driven by a mechanical torque as
shown in Fig. 5 (a), it can be seen that the machine first
operates as a generator with a leading angle and change to
a motor mode at simulation time of 3 seconds due to the
change of the direction of the mechanical torque. The ac
system voltages, machine stator phase and rotor circuit
currents are given in Fig. 5 (b), (c) and (d). Also the variation of magnetizing reactance of the machine is shown in
Fig. 5 (e).
Fig. 6 gives the results of the machine under the same
driving condition as that of Fig. 5, but a three phase volt-
age drop (to 7000 of the rated voltage) occurs at 8 seconds
and lasts for 300 ms. The machine speed, angle, voltages
and currents are presented.
1963
2I
Tmechi
:3-1
.
El
0
I .....
Telec
0
-1
2
0
1
2
2
(a)
3
4
5
6
7
8
9
3
4
5
time (seconds)
6
7
8
9
Machine torques, speed and angle
-
system voltages
21
l
c
1
2
0
1
2
3
4
5
Mechanical and Electrical Torque
2
6
7
8
:31
0
4'- ----
Z5 -2
Tmech
Telec
-41
0
9
2
4
6
8
10
12
14
10
12
14
10
12
14
Machine speed
1.1
1
-j5l 1 .05
E.
II
c
1
-L
-277
0
1
-L
-L
2
-1
-L
-1
11
-L
',-.
1
.
0.95
0.9
-1
0
2
4
6
Machine angle
2
,
z1
-
1
.
0
E.
II
0
-511
Iz
-5 -1
a5
0
1
2
3
4
5
time (seconds)
6
7
-V-\-
.2
0
2
9
8
4
(b) AC system phase voltages
6
8
time (seconds)
(a) Machine torques, speed and angle
Stator currents
system voltages
f2,1
5__
-5_
-10
0
1
3
2
6
5
8
7
-2~~~~~~~~~~~~~~~~~~~~~~1
9
10
5_
o1
-5_
-1 0
0
1
3
2
6
5
8
7
9
0
2
4
6
8
10
12
14
2
4
6
8
10
12
14
12
11
L10
L12
'L
14
10
12
1
10
-1
-2
o
-5_
_1 10
1
2
3
4
5
time (seconds)
6
8
7
2
9
~
~
~
~
(c) Machine stator three phase currents
~
~~~~~~ttrcret
(b) AC system phase voltages
Rotor currents
0.5,
~
10
-:31
le7
C)
5
0
EL
0
Z
-0.5
'W -
-5
0
1
2
3
4
5
6
7
8
9
-10
0.5
0
2
I1
2
4
I
I
I
10
I
105
-El
0*
.2
-0.5
-51
0
1
2
3
4
5
6
7
8
9
-1 0
l
o
10
-:31
~
~
~
~
~
~
~
~
~
~
~
-5_
ie(scns
~ ~ d 2ahn roo-icitcret
,e7
-10
0
2
4
time (seconds)
(c) Machine stator three phase currents
Rotor currents
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
21 =8
1
O
1
2
3
4
51
6
7
8
9
(e) q-and d- axis mutual reactance
Fig. 5. Simulation of normal operation of a synchronous machine
4
3
The results presented in Fig. 7 are under the same
conditions as that of Fig. 6, except that the voltage drop is
a single phase (to 30°0 of the rated voltage) and occurs on
phase "a".
1964
1
tidM
n
(seconds)
Fig. 6. Simulation of the synchronous machine under a three phase voltage drop to 70% of the rated voltage
VI. CONCLUSIONS
Mechanical and Electrical Torquuje
-s-,
--
--L
.s-
10
12
Telechl
14
10
12
14
10
12
14
-2
_4 1
-4o
4
6
Machine speed
1.11I
:.. 1.05
6
E.
L 0.95
0.91
0
-2,
Ag
E.
.
-a
I
4
2
6
8
Machine angle
2
1
0
V -1
a5
-21'o
2
4
6
8
time (seconds)
(a) Machine torques, speed and angle
system voltages
2,
0
o
2
4
6
10
8
12
REFERENCES
14
cLo
0
2
4
6
0
2
4
6
8
10
12
14
10
12
14
cLo
8
time (seconds)
(b) AC system phase voltages
Stator currents
10
10o
2
4
6
8
10
12
14
2
4
6
8
10
12
14
2
4
6
8
10
12
14
10
-5 _
10o
10
-5 _
10o
(c)
time (seconds)
Machine stator three phase currents
Rotor currents
1__
-1
0
2
4
6
8
10
This paper presents a synchronous machine model in
abc frame implemented in MATLAB/SIMULINK. The
saturation considerations have been considered, an mathematical expression is used to represent the saturation effects and implemented in the model. Simulation studies
are performed under various operation conditions. It has
been demonstrated that the MATLAB/SIMULINK is a
powerful tool to implement the complex synchronous
machine model.
The direct phase model can easily integrated into other
power system model for simulating various loading or
fault, balanced or unbalanced conditions. The implemented model could also be used for studies of various
applications of synchronous machines including renewable and DG generation systems.
12
14
10
12
14
10
12
14
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[9] Z. Chen, A. C. Williamson, "Simulation Study of A Double Three
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1__
0
-1__
0
2
4
6
o
2
4
6
8
4
3
1
8
time (seconds)
(d) Machine rotor circuit currents
Fig. 7. Simulation of the synchronous machine under a single phase "a"
voltage drop to 30% of the rated voltage
1965