180_101
TRANSCRIPT
Identification of Synchronous Generator Parameters Based on 3-Phase Sudden Short-Circuit Current
Chaoxian HAN, Xinzhen WU*, Ping MA Electrical Engineering Department
Qingdao University Qingdao, China
Abstract-A novel computation method for the identification of synchronous generator parameters based on simulated data of 3-phase sudden short-circuit current is described. The 3-phase sudden short-circuit current data is simulated by MATLAB. Considering optimization of multidimensional parameters and solution of nonlinear equations, the particle swarm optimization (PSO) algorithm and improved Euler method are employed. Since the identification process of direct axis parameters is identical with that of quadrature axis parameters for the cylindrical synchronous generator, only the direct axis parameters identification is given. The comparison between identified values and real values proves the correctness of the method.
Keywords-Synchronous Generator; parameters identification; 3-phase sudden short-circuit; particle swarm optimization
I. INTRODUCTION As power systems become more interconnected and
complicated, the analysis of dynamic performance of such systems becomes more important. Synchronous generators, heart of power systems, play a very significant role in stability of the systems, therefore the valid parameters of generator are required for a accurate stability analysis. However, because internal temperature varying, eddy, magnetic hysteresis and magnetic saturation of the generator [1-3] are unconsidered, the manufacture parameters cannot completely reflect the real parameters during online operating. To obtain the real parameters of running generators, identification of the synchronous generator parameters is needed. Parameters of the synchronous generator are usually obtained by 3-phase sudden short-circuit test, which is an appropriate means to identify generator parameters. Traditionally, by existing methods, the generator steady parameters and transient parameters are identified at the same time [4-5], and during the calculation process, the standstill frequency response (SSFR) data are mostly utilized [6-8]. This frequency domain method has been shown to be effective, but the effect of damping winding is not considered and the precise orientation of the direct and quadrature axis for large synchronous generators is fairly difficult.
A novel time domain method is used in this paper. During the identification of the parameters, the direct axis
synchronous reactance is firstly obtained by the data of steady state, and the steady state parameter is then used to identify transient parameters with optimization method. Owing to the decrease of the identified parameters, the identification of transient parameters becomes easier and more accurate than that in existing methods. The comparison between the results of the existing method and new method inform the advantage of the new method. Meantime, the particle swarm optimization algorithm, which has a better performance at time cost, is chosen.
II. GENERATOR MODEL Currently, several generator models, which are with
different orders, are used in identifying parameters of the generator. In general, the generator can be better described when a higher order model is used, at the same time, however, the dimension of parameters will increase, which means the identification becomes more time cost and less accurate. Considering the balance of time and accuracy, a standard six-order model with one damping winding in the direct axis and two damping winding in quadrature axis is chosen. The direct axis equivalent circuit and the quardrature axis equivalent circuit for a six-order model are shown in Fig. 1 and Fig. 2 respectively [9-10].
In the equivalent circuit of Fig.1 and Fig. 2, Rf and Xfl represent resistance and reactance of the field winding; uf and if field voltage and field current; RD and XDl resistance and reactance of the damping winding; Xad and Xl direct axis armature reaction reactance and armature leakage reactance;
aR fRlX
adXDR
DlX
di
du
fi
fu
flX
Figure 1. Direct axis equivalent circuit for a six-order model
* Corresponding Author
978-1-4577-0365-2/11/$26.00 ©2011 IEEE 959
qu
aR lX
aqXQR
QlX
qi
qR
qlX
Figure 2. Quadrature axis equivalent circuit for a six-order model
Ra stator resistance; ud and id direct axis voltage and direct axis current; RQ and XQl Q winding resistance and leakage reactance; Rq and Xql q winding resistance and leakage reactance; Xaq quadrature axis armature reaction reactance; uq and iq quadrature axis voltage and quadrature axis current.
By the equivalent circuit for generator, the electrical equation can be written as (1) and (2). The suffixes d and q indicate direct axis and quadrature axis parameters. Where the direct axis time constant Td0' and Td0" represent open-circuit transient time constant and open-circuit subtransient time constant; Eq' and Eq" quadrature axis transient electromotive force and quadrature axis subtransient electromotive force; Ef field electromotive force.
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⋅′′−′′=
′⋅′′+′′−′−′′−′=
′′⋅′′
′′−′′′−′′−−′−=
′⋅′
ddqq
qddddqq
qd
qqdd
ddqf
qd
iXEudtEd
TiXXEEdtEd
T
EEXXXXEE
dtEd
T
00
0
)(
)(
(1)
0
0 0
( )
( )
q qdq d d d
q q
d dq d d q q q q
d d q q
X XdET E E E
dt X X
dE dET E E X X i T
dt dtu E X i
′−′⎧ ′ ′ ′ ′′⋅ = − − −⎪ ′ ′′−⎪⎪ ′′ ′⎪ ′′ ′ ′′ ′ ′′ ′′⋅ = − + − + ⋅⎨⎪
′′ ′′= + ⋅⎪⎪⎪⎩
(2)
Compare the direct axis equivalent circuit to quardrature
axis equivalent circuit, it can be observe that these two equivalent circuits are nearly the uniform; the only different between them is the existence of the filed voltage in direct axis equivalent circuit. The format of the direct axis and quardrature axis state equivalents also prove this conclusion, as：
Ef=K·uf (3)
Where K is a constant coefficient. Because of the field windings is in direct axis equivalent circuit merely, in quardrature axis equivalent circuit, the field voltage uf is equal
to 0, thus, according to (3) the field electromotive force is equal to 0. Without the consideration of Ef, these two state equivalents are completely corresponding and the identification processes of direct axis and quardrature axis are uniform, thus, in this study, only the identification of direct axis parameters is discussed.
During the identification the initial values of these variables are given as follow:
Eq'=1 (p.u); Eq"=1 (p.u); Ef=K·uf=1 (p.u).
III. SHORT CIRCUIT CURRENT DATA
By reason of the stator current can be measured easily and precisely, the direct axis current, which is obtained from stator current by Park transform, is used as output of identification. The direct axis short circuit current data are yielded from (4).
d
mT
t
d
m
d
mT
t
d
m
d
md x
EexE
x
Eex
E
x
Ei dd +−′+′−″=′−″−
)()( (4)
Where Em is induction electromotive force in stator winding. The sampling time is selected to be 0.01 second, and
gives:
Em=1 (p.u); xd=2.2835 (p.u); xd' =0.2909 (p.u);
xd"=0.1774 (p.u)； Td0' =1.6153 (s)； Td0"=0.0465 (s).
Thus, the direct axis short circuit time constants are acquired:
Td' =(xd' /xd)·Td0' =(0.2909/2.2835)×1.6153=0.2058 (s),
Td"=(xd"/xd')·Td0"=(0.1774/0.2909)×0.0465=0.0284 (s).
The acquired direct axis short circuit current data are showed in Fig. 3.
0 0.5 1 1.5 20
1
2
3
4
5
6
Time (s)
Dire
ct c
urre
nt (A
)
Figure 3. The direct axis short circuit current
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IV. OPTIMIZATION METHOD AND PROCESS In the identification, particle swarm optimization (PSO),
which has a satisfying performance at both accuracy and time cost, is employed.
Particle swarm optimization is an optimization method based on iteration. Each particle, which represents the group of solutions, searches in the range following the best particles. Because of lacking crossover and mutation, which appear in the Genetic Algorithms and Evolutionary Strategies, the calculation time of partial swarm optimization is much less than that taken by other two optimization methods, in the same condition. During the iteration, the particles update their location and velocity by two extreme values. One of the values, which is the best value reached by the particle himself, called personal best, the other value, which is the best value reached by the whole swarm, called group best. During the optimization process, the particles update their position and velocity by (5).
⎪⎩
⎪⎨⎧
+=−⋅⋅+
−⋅⋅+⋅=
Vpresentpresentpresentgbestrc
presentpbestrcVwV)(
)(
22
11
(5)
Where V represent velocity, present represent the present position; pbest and gbest denote the personal best position and group best position; c1 and c2 represent learning factors; r1 and r2 are stochastic numbers between 0 and 1; w is a weighing factor.
There are four important factors that can affect the precision of optimization results: the number of particles, the time of iteration, the maximum velocity, and the weighting factor. In this process, the number of particles is selected to be 100; the time of iteration is estimated to be 1000, which is proved to be big enough to find the best solution; the maximum velocity is one thirtieth of the deviation between upper limit and lower limit; the weighing factor is linear decreasing as :
w=(wini-wend)·(G-g)/G+wend. (6)
In which wini and wend are the initial and least values of weighing factor respectively, G is the total time of iteration, g is the present time of iteration.
In the optimization process an object function is constructed to describe the fitness of particles, using the least square method. The function can be written as follow:
∑=
−=n
tsoldd IIJ
1
2_ )( (7)
Where Id is the simulated direct axis current data and Id_sol is the calculated current data, J the summation of squares of deviations between Id and d_sol .
TABLE I. SEARCH RANGE OF PARAMETERS
Direct Parameters
Synchronous Generator Parameters
Xd (p.u) Xd'(p.u) Xd"(p.u) Td0'(s) Td0"(s)
Real values 2.2835 0.2909 0.1774 1.6153 0.0465
Upper limit 3.4253 0.4364 0.2661 2.4230 0.0698
Lower limit 1.1418 0.1455 0.0887 0.8077 0.0233
In reality, due to the design values, which are approximates of real values, can be used as initial values for identification, a narrow research range is got. Meantime, since the direct axis synchronous reactance is pre-calculated, this steady parameter is determined to be a constant during optimization. Table � shows the search range of parameters.
V. IDENTIFICATION OF PARAMETERS
A. Calculation of steady parameter According to (4), after sudden short circuit occurs, the
direct axis current becomes constant with new steady state.
dmtd xEi =∞=
(8)
The direct axis synchronous reactance of the generator will be attained by solving (8). From the sudden short circuit current data, the constant direct axis current is known as 0.4379 (p.u). Hence the direct axis synchronous reactance of the generator is calculated:
xd=Em/id_constant=1/0.4379=2.2836 (p.u)
Because of the direct axis synchronous is calculated by an algebraic equation, the calculated value is very close to the real value.
B. Obtaining of transient parameters with optimization
The direct axis synchronous reactance of the generator, which is calculated in last section, is used as known value in the following optimization. The particle swarm optimization illuminated in section 3 is employed to search the best solution in the search range of parameters, and during the optimization, the improved Euler method is also selected to solve the system of high-order differential equation (1) simultaneously. The calculated direct axis current and simulated data are given in Fig. 4, which shows fairly exact identification.
According to Fig.4 the identified direct axis current and the simulated data are in good agreement, the correctness of the method employed in this working is verified.
In order to show the deviations between the simulated current and currents calculated by these two methods clearly, relative errors of the results are computed and showed in Fig. 5.
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0 0.5 1 1.5 20
2
4
6
Time (s)
Dire
ct c
urre
nt (A
)
simulated currentnew methodexisting method
Figure 4. Simulated current and calculated current
0 0.5 1 1.5 2-2
-1.5
-1
-0.5
0
0.5
1
Time (s)
Rel
ativ
e er
rors
(%)
new metnodexisting method
Figure 5. The relative errors of result
TABLE II. THE IDENTIFICATION RESULT OF NEW METHOD
Direct Parameters
Synchronous Generator Parameters
Xd (p.u) Xd'(p.u) Xd"(p.u) Td0'(s) Td0"(s)
Real values 2.2835 0.2909 0.1774 1.6153 0.0465 Identification
values 2.2836 0.3050 0.1801 1.7430 0.0438
Relative errors (%) 0.004 4.85 1.53 7.90 -5.89
TABLE III. THE IDENTIFICATION RESULT OF EXISTING METHOD
Direct Parameters
Synchronous Generator Parameters
Xd (p.u) Xd'(p.u) Xd"(p.u) Td0'(s) Td0"(s)
Real values 2.2835 0.2909 0.1774 1.6153 0.0465 Identification
values 2.2781 0.2977 0.1761 1.7341 0.0384
Relative errors (%) -0.24 2.35 -0.76 7.35 -17.41
According to Fig.5, the relative errors of the new method is less than that of the traditional method, therefore, the new identification method have a better accuracy. Table � and Table Ⅲ illustrate the identified results of the new method and traditional method respectively, which also prove the advantage of the new method.
VI. CONCLUSION
A new method for identification of synchronous generator parameters is presented in this paper. The short circuit current data that utilized for identification are yielded by equations. The direct axis and quadrature axis equivalent circuits for a six-order model and the corresponding state equation are described. Owing to the satisfactory performance both in precision and time cost, particle swarm optimization is employed.
During the identification process of synchronous generator parameters, using 3-phase sudden short-circuit current data, the highly accurate steady parameter is calculated by steady data firstly. In this new method, the pre-calculated steady parameter reduces the dimension of the identified variables, thus, the optimization process of transient parameters becomes more accuracy and time-saving. The comparison between the results of the traditional method and new method inform the advantage of the new method. The good agreement of the identified current and simulated current prove the validity and accuracy of this method.
The errors of the identified parameters are caused by the following reasons: the solution of the nonlinear function is approximate and the particle swarm optimization method may reach locally optimal solution instead of the global optimal solution.
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