- Email: [email protected]
Identification of Dynamic Equivalents for Large Power Systems
Copyright © IFAC Th~ory and Application of Digital Control N~w Ddhi. India 1982
IDENTIFICATION OF DYNAMIC EQUIVALENTS FOR LARGE POWER SYSTEMS S. Agrawala, S. I. Ahson and A. K. Sinha Department of Electrt'cal Engineert'ng, Indian Institute of Technology, New Delhl~ India
Abstract. The paper presents a new method for finding dynamic equivalent nodels for large inter-connected power systems using !«)senbrock' s rotating coordinate method of cptimisation. An integrated dynamic model consisting of a kn
The problem of obtaining dynamic equivalents for large interconnected power systems has been considered by many authors (Chang et al., 1970; Ibrahim, et al. 1976; Schlueter et al., 1978; Undril et al.,1971, 1971a; Yu et al., 1979). There are basically two main approaches of solving the problem: (i) Reducing the order of the entire power system through eigenmode analysis (Undril et al., 1971, 1971a) :ii) Grouping a number of synchronous machines by an equivalent machine based on the concept of coherency (Chang et.al., 1970) In the analysis of power system stability and decentralised stabilizer design, it is generally assumed that the power system under consideration is connected to an infinite bus of constant voltage and constant frequency. There have been same efforts in replacing the infinite bus by a dynamic equivalent of the unkn
time nodels. An algor i thm based on direct search method of RDsenbrock, 1960 is presented for evaluating the unkn
internal power system connected to a large external power system through a tieline is considered as shown in Fig.i. The internal power system is represented by a single synchronous generator with local load while the external unkn
X(k+1) =
X (k) +
B
u(k)
(1)
The out-put equation is given by y(k) = c
In the present paper an alternative method is proposed for obtaining dynamic equivalent of the external power system using discrete-
A(~)
(~)
X (k)
(2)
where k = 0, 1 ,2. • • •• is the sarrpling index X(k) is the vector consisting of all the 403
404
S. Agrawala, S. I. Ahson and A. K. Sinha
'states' of the combined system ex. is a vector or unKnown parameters of the dynamic equivalent, U(k) is the scalar 'input' and y(k) is the scalar 'output' of ti1e system. These are defined as follows: X(k)
=[Ll6i, Llwj
I
LlEFDi, .LlEqiJ LlOj, LlWi1LlEqd
where 66= 6w 6Fm = 6Eq '
perturbation perturbation perturbation perturbation voltage.
in in in in
rotor angle rotor speed field voltage generated internal
The subscripts i and j are used for the internal system and the dynamic equivalent, respectively.
ex. =
M
j'
D
j'
X
j'
X"
j'
T~ ]
where M j D.
inertia constant damping constant
Xj
synchronous reactance
X. '
transient reactance
T~
field time constant
]
]
]
)T
It is assumed that the measurements of in;; put torque 6 Tin and the internal bus voltage 6 Vt are possible. CA'le has to identify the dynamic equivalent rrodel using the measurements of the internal system only. A precise statement of the problem considered in this paper can now be given as follows: "Given a record of input U(k) and output y(k), k = 0,1,2, ••••••••• M, where M is a sufficiently large number, find the estimates of the unknown parameter vectoro: using the rrodel (1-2)" IDENTIFICATICN POCCEDURE
The problem stated above is transformed in to a function minimisation problem and the rrodel parameters are eu.aluated using a rrodel reference technique described-in Fig. 3. The function to be minimised is defined as M J(
9; )
L
[ y(k) - Ym(k))
2
••• (3)
k=1 where y(k)
&
SIMUIATICN RESULTS
U(k) =6Tin, 6Tin being the torque input to the synchronous generator. and y(k) = 6Ye 6Vt being the voltage at the internal bUs.
'THE
to the unknown parameter vector ex. to obtain the optimal estimates There are several optimisation algor:Lthms available in the literature both analytic and numerical. For canputational convenience it is proposed to use the lbsenbrock' s rotating co-ordinate method, lbsenbrock, 1960, which is a direct search algorithm for finding that minimises the cost function J (a ) • The canplete identification algorithm is described in the flow chart of Fig.4. The flow chart consists of 3-l00ps. The first ldop does the parameter updating in each direction and tests for 'success' and 'failure'. The second loop tests whether there has been at least one ' success' and one 'failure' in each direction. The iterations continue till this condition is achieved and then the convergence criterion is tested by finding the increment in each parameter and canparing it with certain small number E.. If this is satisfied, the iterations stop and the converged parameter values are obtained. otherwise the third loop is used for finding new co-ordinate directions. The canplete procedure is repeated till convergence occurs.
is the measured cutput of the real system Ym (k) is the rrodel output obtained on the basis of the parameter vector a and M is a sufficiently large number. This function has to be minimised with respect
The output sequence is generated by simulating a power system with the following assumed d3ta : Internal system P~
Mi=
= 0.9, ~t = 1.05, Pf = 0.9 (Lag), 5.0, D1 = 10.0, Xd = 1.0 Xd'
0.1, Xq
= 0.6,
Ti
=
0.05, Ki
= 50,
Ti'
= 0.9
(Lag)
7.8, G
weal Load
=
0.5,
Pf
(only for test
External pynamic equivalent data generation) 9.3,
Mj Xj'
26.0,
Dj
0.4 and Tj'
Tie - line:
~
Xj
0.6,
5.3
= 0.04,
~
= 0.5
The following input is used for simulation Tin
0.2 per unit for t
~
0.5 sec.
0.0 per unit for t
>
0.5 sec.
With the given data the system matrices A (ex. ), B (ex. ) and the output matrix c (ex. ) are -evaluated. The output sequence is then generated by simulating the nodel (1-2)with the above given values. The response curve is as shown in Fig. 5. For identification, the initial data are assumed as follows: Constants
a
= 3.0,
b
= -0.5
405
Identification of Dynamic Equivalents
Step-length in each direction (number of unknown parameters is taken as the number of directions) A 1 - 0.01, A 2 = 0.01, A 3 = 0.1, A 4=0.1 ,
TABLE
i1I.lmber of iterations M"J
andA5 = 0.5 co-ordinate directions
o o
0
0
0
0
o o
0
1
0
001 000
[s1
2 3 4
I
Dj
Xj
Xi
T
J
12.253
11 .31
0.3
0.45
5.94
14.665 14.44
62.92 63.30
0.2299 0.2294
14.44
63.56
0.2296
0.4312 0.4311 0.4311
5.9140 5.9139 5.9135
0
000 o (each column represent one direction) The initial values of parameters are assumed as Mj = 5.655, Dj = 18.85, Xj= 0.5, XJ"= 0.5 and Tj' = 6.0 With the response curve and the data as above the parameters are identified aocordingly to the algorithm given in Fig. 3. The identified parameters are given in tabular form (iteration wise) in table - I. The response of the model with the identified parameters is then plotted as shown in Fig. 5 and compared with the original simulated response. It can be seen that the two response curv~s have matched completly. CCNCLUDING REMARKS
Use of simple numerical method of optimisation, namely the Rosenbrock's rotating coordinate method has been shown possible for ~taining dynamic equivalent of a large 1nterconnected power system. The simulation results show that the parameter estimates have converged qery quickly in 3 to 4 iterations. Al though, the converged values are not exactly the same as the ones used in simulation, the model and the system resjXlf1se curves have matched completely. The method besides being computationally simple and I having better convergence properties, has the greatest advantage of being equally applicable to linear and nonlinear systems. Thus, it is not essential to linearize the power system models as has been done in this paper. ACKNCWLEDGEMENI'
The authors acknowledge with thanks the helpful discussions they had with Dr. Y.P.Singh of Elect.Engg.Dept. of I.I.T., New Delhi.
REFERENCES
Chang, A., and .M.M. Abidi(1970); Power System dynamic equivalents. IEEE Trans PAS,89, 1737-1744 Ibrahim, M.A.H., O.M. Mostafa, and A.H. El(1976). Dynamic equi vaJ.ents using operat1ng data and stockastic modeling. IEEE Trans. PAS,95, 1713-1722.
Ab~ad
Rosenbrock, H.H. (1960). An autanatic method for finding the greatest or least value of a function. Computer journal, 3, 175 - 184. SchJ11€ter, R.A., H.Akhtar, and R.r·tHr (1978). An RMS coherency measure: a basis for unifi-
cation of coherency and model analysis model aggregation techniques. IEEE PES SllITDTler ~ ting, A78 533-2. undril, J.M., and A.E. TUrner (1971). Construction of power system electromechanical equivalents by model analysis. IEEE Trans. PAS, 90, 2049 - 2059. undril, J.M., J.A. Gassaza, E.M. Gulachenski and LK. Kirchmayer (1971a). Electranecha-' nical equivalents for use in power system stability studies. IEEE Trans. PAS 90 20602071. ' , YU, Yao-nan, M.A. El-Sharkawi, and M.D.Wvong (1979). Estimation of unknown large power system dynamics. IEEE Trans. PAS, 98 279289. '
S. Agrawala, S. I. Ahson and A. K. Sinha
Unknown external equivalent
Known 'Internal system
Fig , 2
Block d 'l agram of combined
power system
u(k)
•
Combined system model
GIB
i
.,
Internaloc, Unknown or known I external sys tem I sys tem I
I
Fig , 1 An internal power system connected to an external large power sys tem
unction m',n ', m',sation 01 or'lthm
F'lg , 3 Model reference technique for parameter identification _ _ _ _ Gener ated
-2
2xlO
-2 lxl0
f -lxlO
•
••
outpu t sequence
Response with identified parameters
Identificat ion o f Dynamic Equiva lents
Select constants a,b Select initial parameter
~
407
B
Choose step length A 1 ' - - - - - ->, N O1cx:>se co-ordinate duections sl' s2' · ··s Read N, M Read y (k), k = 1, 2, •••••••••• M} Calculate
Ym(k~Jk=1,2,
Calculate J B = Initialise NF(J) Ns(J)
,M
k~l
••••••••• M
y(k)- Ym(k)
2
=[0] J=l ,2 •••••••• N
Calculate Y(k), k=l, ••••••••• M M ,2 Calculate J = 1=1 (y (k) -Ym(k), i
yes
(sucess)
A
.s.1
1
no (fails)
,---_ _L-_---,
Ai
no
-bAt NF(i) = 1 =
no
yes
L - - - - - - 4 Fihd New Direction using Gram-
schmidt orthogonalisation
Fig. 4
Flow chart for parameter identification algorithm
IDENTIFICATION OF DYNAMIC EQUIVALENTS FOR LARGE POWER SYSTEMS S. Agrawala, S. I. Ahson and A. K. Sinha Department of Electrt'cal Engineert'ng, Indian Institute of Technology, New Delhl~ India
Abstract. The paper presents a new method for finding dynamic equivalent nodels for large inter-connected power systems using !«)senbrock' s rotating coordinate method of cptimisation. An integrated dynamic model consisting of a kn
The problem of obtaining dynamic equivalents for large interconnected power systems has been considered by many authors (Chang et al., 1970; Ibrahim, et al. 1976; Schlueter et al., 1978; Undril et al.,1971, 1971a; Yu et al., 1979). There are basically two main approaches of solving the problem: (i) Reducing the order of the entire power system through eigenmode analysis (Undril et al., 1971, 1971a) :ii) Grouping a number of synchronous machines by an equivalent machine based on the concept of coherency (Chang et.al., 1970) In the analysis of power system stability and decentralised stabilizer design, it is generally assumed that the power system under consideration is connected to an infinite bus of constant voltage and constant frequency. There have been same efforts in replacing the infinite bus by a dynamic equivalent of the unkn
time nodels. An algor i thm based on direct search method of RDsenbrock, 1960 is presented for evaluating the unkn
internal power system connected to a large external power system through a tieline is considered as shown in Fig.i. The internal power system is represented by a single synchronous generator with local load while the external unkn
X(k+1) =
X (k) +
B
u(k)
(1)
The out-put equation is given by y(k) = c
In the present paper an alternative method is proposed for obtaining dynamic equivalent of the external power system using discrete-
A(~)
(~)
X (k)
(2)
where k = 0, 1 ,2. • • •• is the sarrpling index X(k) is the vector consisting of all the 403
404
S. Agrawala, S. I. Ahson and A. K. Sinha
'states' of the combined system ex. is a vector or unKnown parameters of the dynamic equivalent, U(k) is the scalar 'input' and y(k) is the scalar 'output' of ti1e system. These are defined as follows: X(k)
=[Ll6i, Llwj
I
LlEFDi, .LlEqiJ LlOj, LlWi1LlEqd
where 66= 6w 6Fm = 6Eq '
perturbation perturbation perturbation perturbation voltage.
in in in in
rotor angle rotor speed field voltage generated internal
The subscripts i and j are used for the internal system and the dynamic equivalent, respectively.
ex. =
M
j'
D
j'
X
j'
X"
j'
T~ ]
where M j D.
inertia constant damping constant
Xj
synchronous reactance
X. '
transient reactance
T~
field time constant
]
]
]
)T
It is assumed that the measurements of in;; put torque 6 Tin and the internal bus voltage 6 Vt are possible. CA'le has to identify the dynamic equivalent rrodel using the measurements of the internal system only. A precise statement of the problem considered in this paper can now be given as follows: "Given a record of input U(k) and output y(k), k = 0,1,2, ••••••••• M, where M is a sufficiently large number, find the estimates of the unknown parameter vectoro: using the rrodel (1-2)" IDENTIFICATICN POCCEDURE
The problem stated above is transformed in to a function minimisation problem and the rrodel parameters are eu.aluated using a rrodel reference technique described-in Fig. 3. The function to be minimised is defined as M J(
9; )
L
[ y(k) - Ym(k))
2
••• (3)
k=1 where y(k)
&
SIMUIATICN RESULTS
U(k) =6Tin, 6Tin being the torque input to the synchronous generator. and y(k) = 6Ye 6Vt being the voltage at the internal bUs.
'THE
to the unknown parameter vector ex. to obtain the optimal estimates There are several optimisation algor:Lthms available in the literature both analytic and numerical. For canputational convenience it is proposed to use the lbsenbrock' s rotating co-ordinate method, lbsenbrock, 1960, which is a direct search algorithm for finding that minimises the cost function J (a ) • The canplete identification algorithm is described in the flow chart of Fig.4. The flow chart consists of 3-l00ps. The first ldop does the parameter updating in each direction and tests for 'success' and 'failure'. The second loop tests whether there has been at least one ' success' and one 'failure' in each direction. The iterations continue till this condition is achieved and then the convergence criterion is tested by finding the increment in each parameter and canparing it with certain small number E.. If this is satisfied, the iterations stop and the converged parameter values are obtained. otherwise the third loop is used for finding new co-ordinate directions. The canplete procedure is repeated till convergence occurs.
is the measured cutput of the real system Ym (k) is the rrodel output obtained on the basis of the parameter vector a and M is a sufficiently large number. This function has to be minimised with respect
The output sequence is generated by simulating a power system with the following assumed d3ta : Internal system P~
Mi=
= 0.9, ~t = 1.05, Pf = 0.9 (Lag), 5.0, D1 = 10.0, Xd = 1.0 Xd'
0.1, Xq
= 0.6,
Ti
=
0.05, Ki
= 50,
Ti'
= 0.9
(Lag)
7.8, G
weal Load
=
0.5,
Pf
(only for test
External pynamic equivalent data generation) 9.3,
Mj Xj'
26.0,
Dj
0.4 and Tj'
Tie - line:
~
Xj
0.6,
5.3
= 0.04,
~
= 0.5
The following input is used for simulation Tin
0.2 per unit for t
~
0.5 sec.
0.0 per unit for t
>
0.5 sec.
With the given data the system matrices A (ex. ), B (ex. ) and the output matrix c (ex. ) are -evaluated. The output sequence is then generated by simulating the nodel (1-2)with the above given values. The response curve is as shown in Fig. 5. For identification, the initial data are assumed as follows: Constants
a
= 3.0,
b
= -0.5
405
Identification of Dynamic Equivalents
Step-length in each direction (number of unknown parameters is taken as the number of directions) A 1 - 0.01, A 2 = 0.01, A 3 = 0.1, A 4=0.1 ,
TABLE
i1I.lmber of iterations M"J
andA5 = 0.5 co-ordinate directions
o o
0
0
0
0
o o
0
1
0
001 000
[s1
2 3 4
I
Dj
Xj
Xi
T
J
12.253
11 .31
0.3
0.45
5.94
14.665 14.44
62.92 63.30
0.2299 0.2294
14.44
63.56
0.2296
0.4312 0.4311 0.4311
5.9140 5.9139 5.9135
0
000 o (each column represent one direction) The initial values of parameters are assumed as Mj = 5.655, Dj = 18.85, Xj= 0.5, XJ"= 0.5 and Tj' = 6.0 With the response curve and the data as above the parameters are identified aocordingly to the algorithm given in Fig. 3. The identified parameters are given in tabular form (iteration wise) in table - I. The response of the model with the identified parameters is then plotted as shown in Fig. 5 and compared with the original simulated response. It can be seen that the two response curv~s have matched completly. CCNCLUDING REMARKS
Use of simple numerical method of optimisation, namely the Rosenbrock's rotating coordinate method has been shown possible for ~taining dynamic equivalent of a large 1nterconnected power system. The simulation results show that the parameter estimates have converged qery quickly in 3 to 4 iterations. Al though, the converged values are not exactly the same as the ones used in simulation, the model and the system resjXlf1se curves have matched completely. The method besides being computationally simple and I having better convergence properties, has the greatest advantage of being equally applicable to linear and nonlinear systems. Thus, it is not essential to linearize the power system models as has been done in this paper. ACKNCWLEDGEMENI'
The authors acknowledge with thanks the helpful discussions they had with Dr. Y.P.Singh of Elect.Engg.Dept. of I.I.T., New Delhi.
REFERENCES
Chang, A., and .M.M. Abidi(1970); Power System dynamic equivalents. IEEE Trans PAS,89, 1737-1744 Ibrahim, M.A.H., O.M. Mostafa, and A.H. El(1976). Dynamic equi vaJ.ents using operat1ng data and stockastic modeling. IEEE Trans. PAS,95, 1713-1722.
Ab~ad
Rosenbrock, H.H. (1960). An autanatic method for finding the greatest or least value of a function. Computer journal, 3, 175 - 184. SchJ11€ter, R.A., H.Akhtar, and R.r·tHr (1978). An RMS coherency measure: a basis for unifi-
cation of coherency and model analysis model aggregation techniques. IEEE PES SllITDTler ~ ting, A78 533-2. undril, J.M., and A.E. TUrner (1971). Construction of power system electromechanical equivalents by model analysis. IEEE Trans. PAS, 90, 2049 - 2059. undril, J.M., J.A. Gassaza, E.M. Gulachenski and LK. Kirchmayer (1971a). Electranecha-' nical equivalents for use in power system stability studies. IEEE Trans. PAS 90 20602071. ' , YU, Yao-nan, M.A. El-Sharkawi, and M.D.Wvong (1979). Estimation of unknown large power system dynamics. IEEE Trans. PAS, 98 279289. '
S. Agrawala, S. I. Ahson and A. K. Sinha
Unknown external equivalent
Known 'Internal system
Fig , 2
Block d 'l agram of combined
power system
u(k)
•
Combined system model
GIB
i
.,
Internaloc, Unknown or known I external sys tem I sys tem I
I
Fig , 1 An internal power system connected to an external large power sys tem
unction m',n ', m',sation 01 or'lthm
F'lg , 3 Model reference technique for parameter identification _ _ _ _ Gener ated
-2
2xlO
-2 lxl0
f -lxlO
•
••
outpu t sequence
Response with identified parameters
Identificat ion o f Dynamic Equiva lents
Select constants a,b Select initial parameter
~
407
B
Choose step length A 1 ' - - - - - ->, N O1cx:>se co-ordinate duections sl' s2' · ··s Read N, M Read y (k), k = 1, 2, •••••••••• M} Calculate
Ym(k~Jk=1,2,
Calculate J B = Initialise NF(J) Ns(J)
,M
k~l
••••••••• M
y(k)- Ym(k)
2
=[0] J=l ,2 •••••••• N
Calculate Y(k), k=l, ••••••••• M M ,2 Calculate J = 1=1 (y (k) -Ym(k), i
yes
(sucess)
A
.s.1
1
no (fails)
,---_ _L-_---,
Ai
no
-bAt NF(i) = 1 =
no
yes
L - - - - - - 4 Fihd New Direction using Gram-
schmidt orthogonalisation
Fig. 4
Flow chart for parameter identification algorithm