A Study of Synchronous Machine Model Implementations in Matlab/Simulink. Simulations for New and Renewable Energy Systems. Z. Chen, F. Blaabjerg, F.Iov.
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
[i]
[ia
Vb
2b
VC
Pc
[R]= diagonal[r,
0
(1) 0
Vf 1
'Q'D D f
rb
r,
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
Lsr
L
1
(3)
2AQDf L _Lrs Lrr j LiQDf j The stator inductance sub-matrix
Where [/2abc] [= a Ab A,]T and [iabc]= [ia 'b i ]T are the phase flux linkages and currents of the stator windings.
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 ]
Ls+Los L2scos 2Or
[LSS ] =
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 2I cos +L 20r + / 2;T S-L2S CoS(20r- 2wT/ 3)] cos(20, LOS /2-L2S LIS 2-L2S 13) -LOS
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
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) =
[VqdO]
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
dt
[A ]= [V ]- [R ]. [L ]'[ Or in terms of current as
][i
([L
=
Vala
Pe
°
[Vqd O ] [vsq Vsd
+
Vblb
dt b,
=
rsq
C0b
co,9r
(T
dt
+
[sq
isd
io
d
Vsd --:rsisd +dAd_Os
r
dt
d2Q []
d ([L
dt
[J
)
(6)
+ VcIc
+TmechO-rKd)
(9)
rQrDiDdt iQ +
0=
O =
(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
AQ = LQiQ 2H
AO | [iqd 0]
dt
and dcor
sd
dA,q
o rm
rm
['qd O] [isq
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 =
where
[V] - [R ]. [i]
dt~~~~~~~~ d (h[ I = [L] '([V ] [R ]. dt
Te
[iqdO I= [T][Iabc,]
(5)
]
[T][Vabc]
[ZqdO] [T][2abc,]
LID +LmD LfD LDf Lif + Lmf
0
d
(4.a)
C 3) /2-L2S OS-L20S CoS(20r-+2S2w 13)
[Lrr] 0=
3
XD
(8)
+L
md imd
(10)
+ Lmqmq
L=DiD+Lmdimd
Af =Llf if +Lmdmd
the rated angular speed and rotor angular s
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: *
sd
_ isd
md T
Asq
isq
mq
=
AD
(14)
+
LIS
=
+
1
LIQ
(2sd ( S+ 2 D+ 2
(17)
r) Lddl
LIS LID L1i Lmd L Amq (qmq 2 + 2"Q) L ql s
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
+
LimO a(im
(1 8.b)
im(sat)
for im >
m(sat) )
A resultant magnetizing curve with (18) is shown in Fig. 3.
md
LID
_f
1 md
Lmd
L
ZQ -Zmq
iQ
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
.1
1.6
.-,.
.,
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 L1S LID Lif L1S
1962
+
AD
+
LIDLD
) (19.a)
I
L((L+ 1 (I + L,,q i"q/(1± L-)) LIs LIQ
2) =('Ad LLIs ±Q LIQ
LIia14X_H
(19.b) l9b
( l(n'Xt- + 'mq(,at) IU + Lmq(L + L
Equations (19.a) and (19.b) may be used for both saturated and saturated conditions. For saturation condition, (1 8.b) can be rewritten as 'md
1mdO
ax ( Lmd
1) +
)m(
md(sat)
for imd > imd(sat)
Q
the as-
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.
P
J
M-
+
sumption of (13) may be 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
,
J L L)
= LI,bothLID un-20.b used for a L(l9.a) and )~ (l9.b) may Equations q Li,be LlQj unIf the q-axis magnetizing curve is unknown,
.
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
2
6
7
8
:31
Torque
0 4'- ----
Z5 -2
Tmech Telec
-41
0
9
2
4
6
10
12
14
10
12
14
10
12
14
Machine speed
1.1
1
8
-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
II
0
-
E.
1
.
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
21
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
14
=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
-s-,
--
Torquuje
--L .s-
10
12
Telechl 14
10
12
14
10
12
14
-2
_4 1
-4o
4
6
Machine speed
1.11I :.. 6
1.05
E.
L
0.95 0.91 0
-2,
Ag
E.
1
.
0
-a I
4
2
6
8
Machine angle
2
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
[1] P. Subramaniam, and O.P. Malik, 'Digital simulation of synchronous generator in direct-phase quantities', Proc. IEE, 1971, 118, (l),pp. 153-160. [2] I.R. Smith, and L.A. Snider, 'Prediction of transient performance of isolated saturated synchronous generator', Proc. IEE, 1972,119, (9), pp. 1309-1318. [3] M.A. Abdel-Halim, C.D. Manning, "Direct phase modelling of synchronous generators", IEE Proceedings-Electric Power Applications, Volume: 137, Issue: 4, July 1990, Pages:239 - 247. [4] F. yov, F.Blaabjerg, A.D. Hansen, Z. Chen, "Comparative study of different implementations for induction machine model in Matlab/Simulink for wind turbine simulations", Computers in Power Electronics, 2002. Proceedings. 2002 IEEE Workshop on, 3-4 June 2002 pp. 58 -63. [5] Chee-Mun Ong, "Dynamic Simulation of Electric Machinery: Using Matlab/Simulink", Prentice Hall, September 1997. [6] G.Xie, R.S.Ramshaw, "Nonlinear model of synchronous machines with saliency" IEEE Trans. Energy Conversion., vol. 1, no. 3, 1986, pp. 198-204. [7] E. Levi, "Saturation modeling in d-q axis models of salient pole synchronous machines"; Energy Conversion, IEEE Transactions on Vol. 14, Issue 1, March 1999 Page(s):44- 50. [8] L. Chedot, G. Friedrich, "A cross saturation model for interior permanent magnet synchronous machine. Application to a startergenerator", Industry Applications Conference, 2004. 39th IAS Annual Meeting. Conference Record of the 2004 IEEE Volume 1, 3-7 Oct. 2004 Page(s). [9] Z. Chen, A. C. Williamson, "Simulation Study of A Double Three Phase Electric Machine", International Conference On Electric Machine ICEM'98, 1998, Vol.1, pp 215-220. [10]Z. Chen, A. C. Williamson, "Simulation Study of Multiple Three Phase Electric Machine", Chinese International Conference On Electric Machine CICEM'99, Xi'an, China, August 1999, vol. 2. pp 1014-1017.
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
