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Identification of a Dynamic Model for the State Estimation of Large Power Systems in Normal Operating Conditions
(;opyrig-hl
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I F;\( : 1
I'''LlIlIl'IlT
ESlilllalioll I~)H:). York. l lK. I ~'X :)
IDENTIFICATION OF A DYNAMIC MODEL FOR THE STATE ESTIMATION OF LARGE POWER SYSTEMS IN NORMAL OPERATING CONDITIONS A. K. Mahalanabis t)"thimi i~' lIp:ill(,l'rillg /)1'/Jftr llllt' lIl,
Tit"
PfllIIS\'il'flllifl Slrllt'
Ullil'l'nily.
lillil'l'nil\' i'flrk, /'.'1. U.i;A
possible remedy has recently been suggested by Kurzyn (1983) through the appliction of the hierarchical state estimation approach. In this paper an entirely different approach is suggested.
ABSTRACT The paper is concerned with the problem of identification of a dynamic model for the bus voltages of an interconnected power system. It is proposed to follow a two step procedure: In the first step, the given power injection data are processed using d.c. load flow techniques in order to generate approximate values of the voltage magnitudes and angles. In the second step, these approximate solutions are then processed to identify a simple state model. Key words: Identification; voltage dynamics; d.c. load flow; state model; parameter estimation .
rhe real time state estimation problem for a given power system is really the problem of estimating the bus voltage magnitudes and angle for all the important buses from the latest set of measurements. The voltage - power relations provide only a part of the information which is used because it is readily available. However, as is well known from the theory of state estimation of dynamic systems, it is very important to be able to seek and employ the other part of the information. This exists in the form of the dynamic equation for the state vector. Unfortunately, for the problem under discussion, it appears too difficult to derive this equation from theoretical considerations. It is therefore proposed to make use of a system identification approach in order to estimate the parameters of the missing state equation for a given power system . Since the state vector for the system consists of the voltage magnitudes and angles and since a virtual decoupling between these two sets of components is often justifiable (Horrisberger, Richard and Rossier, 1976), it is possible to simplify the identification problem by considering the voltage magnitude and angle dynamics separately. An outline of the paper is as follows.
INTRODUCT ION The problem of state estimation of an interconnected power system has been studied extensively during the last fifteen years (see, e.g., Srinivasan, Rao and Indulkar, 1983 and the references therein). With a couple of exceptions (Debs and Larson 1970, Srinivasan a nd Robichaud 1974), all of these studies have been concerned with the development of the so-called static state estimation algorithms on the basis of the steady state voltage - power relations. The weighted least square estimation algorithm, with some modifications introduced in recent years in order to improve its numerical behavior, has been mostly employed. The socalled dynamic state estimators proposed by Debs and Larson (1970) and by Srinivasan and Robichaud (1974) are also in fact not much different: the recursive form of the least square estimation algorithm has been employed in preference to the nonrecursive form employed by others.
A formal statment of the problem of identification considered in this work is first given in the next section . This is followed by a discussion of the proposed method for the generation of the required data. The processing of these data for the identification of the dynamic models of the two parts of the system state. A Simple example to illustrate the feasibility of the proposed method is also presented. Finally, the possible advantages of the use of the identified state model in dealing with the on line state estimation problem are indicated.
One consequence of the use of the voltage power relations alone while developing the estimation algorithms has been the need for employing a redundant set of measurements. This is essentially required by the need for the observability of the state vector (see, e.g., Clements, Krumpholz and Davis, 1983 and the references therein). This implies that, for the state estimation of a large scale power system, one needs to have extensive measurement instrumentation and telemetering. The large size of the data vector also implies that the amount of computer storage and operation will have to be large. This may be particularly unattractive in the real time estimation Situations, which is being increaSingly recommended for security monitoring. One
PROBLEM FORMULATION We consider a system with N interconnected buses so that the state vector x is the 2N - 1 dimensional vector (V 1 V2 .... VN 02 03 ... 0N)T where it is assumed that the angle of bus 1 serves as the reference for the angle measurements. It is assumed that the data for the active and reactive power injections at the various buses are available and are to be used for both system identification and subsequent state estimation. For the i th bus, the relations between the active and reactive powers Pi and Qi and the bus voltages are
15 VOL l-K
~H7
A. K. Mahalanabis
28tl N
I
P
j~1
N
I
Q
j=1
Ay (k) in order to find the missing dynamic
Y v v cos (8 ij i j
-
8
(1)
+
ij
Y v V sin ( 8 ij i j
-
8
+
)
(2)
ij
where Yij is the magnitude andij is the angle of the admittance between the i th and j th buses.
(V1
V2
(82 83 (P 1 P2
y
one can then obtain the following power flow equation (3)
y = h(x)
X~)T and h(x) is a nonlinear
wnere x =
function whose components are defined eqs (1) and (2).
During normal operation, the vector x is assumed to suffer from small perturbations around its nominal value Xo = (~T QT)T where ~ stands for an N vector of 1 's and 0 stands for an (N-1) vector O's. In the state estimation problem, it is required to note the perturbations lIy in the power data and then estimate the perturbations lIxv and lIXO in the two parts of the state vector x. The relations of interest are thus obtained from eq (3) through linearization: H (4) + u lIy = HVV 8v wh~re H stands for the Jacobian of the function h(x) and the subscripts indicate appropriate partitions of this matrix. The term u is added on the right hand side in order to indicate the errors of linear approximation and also to account for measurement errors. Often the matrices HvO and HOv are negligible and eq (4) may be written in the decoupled form (Horrisberger, Richard and Rossier 1976)
yp
(5a)
Hvv + u
(5b)
q
where the subscripts p and q are used to indicate the parts associate with the real and reactive power deviations respectively. In the usual static state estimation problem, eq (5) is used in order to obtain estimates of lIx v and lIXO from the instantaneous measurement perturbations lIYp and lIYq' In the real time estimation problem, it is recognized that these measurements are in fact time dependent so that eq (5) is rewritten as discrete time equations Hvv
lIxv(k)
+
up(k)
(6a)
H88
lI~
+
Uq(k)
(6b)
(k)
It is expected that once these models are identified, possibly in an off line manner, the state estimation problem can then be handled in the on line manner by making use the Kalman filtering algorithm or an equivalent. It is also expected that this will reduce the computational burden of the on line state estimation task. DATA GENERATION
Defining the vectors Xv Xo
mo~els for the state vectors Axv(k) and AXO(k).
where k = 1,2, ... is the discrete time index. As mentioned earlier, the current literature contains studies on the on line state estimation problem on the basis of the measurement relations (6). The problem studied in this work is that of processing the ,lata for IIYp(k) and
To solve the system identification problem, it is proposed to employ a stochastic realization algor i thm of the form suggested by Tse and Weinert (1975). This requires that noisy data of the values of the state vector be available. Since in the system under consideration, we do not have a direct measurement of the bus voltage magnitudes and angles, it would be necessary to generate these data by proceSSing the power injection data. Since only approximate values of the state variables are needed, it is proposed to make use of the d.c. load flow techniques to generate these values from the measured power data. In view of the large scale nature of the problem, it is proposed to process separately the active and reactive power injections in order to generate the data for the voltage angles and the magnitudes separately. Generation of the Angle Data Let us consider eq (1) and make the fOllowing assumptions: (i) The voltage magnitudes are approximately equal to uni ty (ii) The transmission lines have negligible resistences so that
It is then possible to express the injection at bus i as N
Pi = r
- Yij sin (8j - 8i)
j -1
Let us also assume that the difference between the angles 8j and 8i are relatively small so that eq (7) can be approximated as N
Pi = j
r
Yij
(e i - ej )
(8)
=1
Using this relation for i = 1,2, ... N, we can obtain the d.c. load flow equation P
= B 8
(9)
where P is the vector of the injections, 8 is the vector of the angles and the matrix B-has elements related to Yij. Assuming B nonsingular, we can use eq (9) for obtaining a solution for the angle vector (') for the given value of the power vector P. Since a number of approximations have been introducted while obtaining eq (9), it is apparent that the solution for (') obtained from the d.c. load flow is necessarily rather approximate. It may also be noted that since the angle 81 has been used as the reference, we need some adjustments in the choice of the elements of the first row of B (see, e.g., Wood and Wollenberg, 1984).
289
Identification of a Dynamic Model It is thus possible process the discrete injections data Pi(k) in order to generate the approximate values of the state vector. Let this approximate value of 6XO(k) be denoted by ZO(k) so that we can write the relation
The matrix A is assumed to have the following canonical form
o o
o
o
A
( 10)
( 15)
where n(k) represents the unknown error. Using eqs(14) and (15). it is possible to show that the i th output component zi(k) can be related to the preceding components as follows:
Generation of the Magnitude Data Let us consider now eq (2) and make the following assumptions:
zi (k+l) = ail zl (k) + ai2 z2(k) + ... + aiiZi(k) + mi(k). i = 1.2 •... N
(i) The angles 0i. i - 2.2 •... N have the values determined above
(16)
where mi(k) is a noise term that depends on the linear combination of nl(k). n2(k) •... ni(k). We can see that the elements of the various rows of the matrix A may be estimated using the usual parameters estimation algorithms. This requires that we rewrite eq (16) in the form
(ii) The angles ~ij = 90 0 It is then possible to express the reactive injection at bus i as N
Qi
=
L j =1
Yij ViVj cos (0j - 0i)
(11 ) ( (1)
Let us also assume that Vi = 1 and cos (01 - 0i) _ MiJ. a known quantity. It is then poss ble to rewrlte eq (11) in the linear form
where we have defined the two vectors on the RHS as
( 12)
Q = C V
where Q is the vector of the reactive injectIons. V is the vector of voltage magnitudes and the matrix C has elements related to Yij and Mij' Again. assuming C to be nonsingular. we can use eq(12) in order to compute the vector V for any given value of the injection vector Q.- It is thus possible to process the discrete time data for the bus injections in order to generate the approximate values of the part 6x v (k) of the state vector. If we use the notation zv(k) for these approximate solutions. we get the relation
~i = (ail
Since we do not have any apriori information about the noise term. it is convenient to make use of the least squares estimation algorithm either in the batch processing form or in the recursive form (see. e.g .• Sinha and Kuszta (1983). For large N. it is of course better to employ the recursive form. Note. however. that the dimension N in most pratical cases may be kept relatively small by making use of the concept of 'external network equivalencing' (see. e.g .• Dopazo et al 1914).
(13 ) AN EXAMPLE IDENTIFICATION OF THE STATE DYNAMICS
°
For convenience. let us drop the subscripts and v and just consider the question of a suitable dynamic model for the state 6x(k) from the observed data for z(k). This is a stochastic realization problem and a number of approaches have been developed for solving this problem in recent years. One such solution has been proposed by Tse and Weinert who have shown that the problem is simplified to some extent if the state transition matrix is assumed to have block triangular form (Tse and Weinert (915). To simplify the problem further. we assume that the observability subindices of the components of z(k) are all equal to 1 so that the state transition matrix is in fact a triangular one. Note that even this is an improvement over the choice of this matrix as the identity matrix in the earlier papers on the dynamic state estimation of power systems. Following the argument above. let us assume that the desired dynamic model for the state vector is gi ven by
In order to illustrate the method. let us consider a simple 3 bus network for which the following constants are specified (all expressed in per unit on a 100 MVA base): PlO - .35pu; P20 = .65 pU; P3 0 = -1 pU; Y 12
=
j
0.4 pu;
Y - j 0.25 pU; Y - j 0.2 pu 23 13 Since bus #1 is the reference bus. we intend to solve for the angloes 82 and 83 only using the d.c. load flow equation. The matrix B needed for this is formed by noting the following relations: bij
=
-(
l/Y ij)
L wi th the summation being bii - j (l/Yij)' performed over all lines connected to bus i For the given admittance data. we get 1.5 -5 B
!\x(k+l) - AIIx(k) + w(k)
(14b)
while the output equation is z(k)
=
ilx(k) + n(k)
( 14a)
-5
9
A. K. Mahalanabis
290
Using eq (9), we then obtain the following relations between the angle vector 0 and the injection vector P: 0.212
0.118
0.118
0.176
9
It is possible to make use of this relation in order to compute the values of 9(k) for the given values of P(k). The table below gives a set of these values. TABLE P2(k) 0.6 0.65 0.7 0.55 0.5 0.75
P3(k) -0.95 -1 -1. 05 -0.9 -0.85 -1.1
92(k) 0.015 0.02 0.024 0.01 0.006 0.029
93(k) -0.096 -0.1 -0.102 -0.093 -0.091 -0.105
~
Clements, K. A., Krumpholz, G. R. and Davis, P. W., (1983) "Power system estimation with measurement deficiency: an observability measurement placement algorithm," paper No. 83 WM 058-5, IEEE PES Winter Meeting, New York, Jan. 30-Feb. 4, 1983. Debs, A. S. and Larson, R. E., (1970) "A dynamic estimator for tracking the state of a power system," IEEE Trans Power App & Syst, Vol PAS 89, pp 1670-1678. Dopazo, J. F. et al, (1977) "An external system equivalent model using real time measurements for system security evaluation," IeEE Trans Power App & Syst, Vol. PAS-96, pp. 431-446.
Using these data, it is also possible to estimate the state transition matrix using eq (16). In this case, the matrix A has only three unknown elements. These are computed as: all = 0.93; a21 = -0.14; a22
REFERENCES
17.41
For these computations, we used the following data: 0.6
Horrisberger, H. P., Richard, J. C. and Rossier, C, (1976), "A fast decoupled static state estimator for electric poer systems," IEEE Trans Power App & Syst., Vol. PAS 95, pp. 208-215. Sinha, N. K. and Kuszta, B. (1983), "Modeling and Identification of Dynamic Systems," Van Nostrand, N. Y. Tse, E. and Weinert, H. L., (1975), "Structure determination and parameter identification mul ti var iable stochastic linear systems," IEEE Trans Autom Contr, Vol. AC-20, pp. 603-613.
-0.95 These estimates may be considered as the initial data for an on-line algorithm which may be based on the recursive version of the estimation algorithm. CONCLUDING REMARKS An approach for the identification of the voltage dynamics of a power system operating in the steady state has been proposed on the basis of the data generated from the d.c. load flow analysis. This form of load flow can be performed for a large network without too much of computations. However, in view of the approximate nature of the solution, this has been modeled as the result of additive corruption of the correct solution. This model has been combined with a simple canonical form of the state transition matrix in order to develop a least squares algorithm for the parameter estimation. Once the state transition matrix has been estimated, it should be possible to use the following model equations for solving estimation problem: x(k+l )
A x(k) + w(k)
y(k) = H x(k) + u(k) With this model, it is expected that the need for redundant measurements will be eliminated since the pair (A, H) is expected to be observable. Investigations for test systems are currently in progress along this line.
Wood, A. J. and Wollenberg, B. F., (1984), "Power Generation, Operation and Control," John Wiley & Sons, N.Y.
©
I F;\( : 1
I'''LlIlIl'IlT
ESlilllalioll I~)H:). York. l lK. I ~'X :)
IDENTIFICATION OF A DYNAMIC MODEL FOR THE STATE ESTIMATION OF LARGE POWER SYSTEMS IN NORMAL OPERATING CONDITIONS A. K. Mahalanabis t)"thimi i~' lIp:ill(,l'rillg /)1'/Jftr llllt' lIl,
Tit"
PfllIIS\'il'flllifl Slrllt'
Ullil'l'nily.
lillil'l'nil\' i'flrk, /'.'1. U.i;A
possible remedy has recently been suggested by Kurzyn (1983) through the appliction of the hierarchical state estimation approach. In this paper an entirely different approach is suggested.
ABSTRACT The paper is concerned with the problem of identification of a dynamic model for the bus voltages of an interconnected power system. It is proposed to follow a two step procedure: In the first step, the given power injection data are processed using d.c. load flow techniques in order to generate approximate values of the voltage magnitudes and angles. In the second step, these approximate solutions are then processed to identify a simple state model. Key words: Identification; voltage dynamics; d.c. load flow; state model; parameter estimation .
rhe real time state estimation problem for a given power system is really the problem of estimating the bus voltage magnitudes and angle for all the important buses from the latest set of measurements. The voltage - power relations provide only a part of the information which is used because it is readily available. However, as is well known from the theory of state estimation of dynamic systems, it is very important to be able to seek and employ the other part of the information. This exists in the form of the dynamic equation for the state vector. Unfortunately, for the problem under discussion, it appears too difficult to derive this equation from theoretical considerations. It is therefore proposed to make use of a system identification approach in order to estimate the parameters of the missing state equation for a given power system . Since the state vector for the system consists of the voltage magnitudes and angles and since a virtual decoupling between these two sets of components is often justifiable (Horrisberger, Richard and Rossier, 1976), it is possible to simplify the identification problem by considering the voltage magnitude and angle dynamics separately. An outline of the paper is as follows.
INTRODUCT ION The problem of state estimation of an interconnected power system has been studied extensively during the last fifteen years (see, e.g., Srinivasan, Rao and Indulkar, 1983 and the references therein). With a couple of exceptions (Debs and Larson 1970, Srinivasan a nd Robichaud 1974), all of these studies have been concerned with the development of the so-called static state estimation algorithms on the basis of the steady state voltage - power relations. The weighted least square estimation algorithm, with some modifications introduced in recent years in order to improve its numerical behavior, has been mostly employed. The socalled dynamic state estimators proposed by Debs and Larson (1970) and by Srinivasan and Robichaud (1974) are also in fact not much different: the recursive form of the least square estimation algorithm has been employed in preference to the nonrecursive form employed by others.
A formal statment of the problem of identification considered in this work is first given in the next section . This is followed by a discussion of the proposed method for the generation of the required data. The processing of these data for the identification of the dynamic models of the two parts of the system state. A Simple example to illustrate the feasibility of the proposed method is also presented. Finally, the possible advantages of the use of the identified state model in dealing with the on line state estimation problem are indicated.
One consequence of the use of the voltage power relations alone while developing the estimation algorithms has been the need for employing a redundant set of measurements. This is essentially required by the need for the observability of the state vector (see, e.g., Clements, Krumpholz and Davis, 1983 and the references therein). This implies that, for the state estimation of a large scale power system, one needs to have extensive measurement instrumentation and telemetering. The large size of the data vector also implies that the amount of computer storage and operation will have to be large. This may be particularly unattractive in the real time estimation Situations, which is being increaSingly recommended for security monitoring. One
PROBLEM FORMULATION We consider a system with N interconnected buses so that the state vector x is the 2N - 1 dimensional vector (V 1 V2 .... VN 02 03 ... 0N)T where it is assumed that the angle of bus 1 serves as the reference for the angle measurements. It is assumed that the data for the active and reactive power injections at the various buses are available and are to be used for both system identification and subsequent state estimation. For the i th bus, the relations between the active and reactive powers Pi and Qi and the bus voltages are
15 VOL l-K
~H7
A. K. Mahalanabis
28tl N
I
P
j~1
N
I
Q
j=1
Ay (k) in order to find the missing dynamic
Y v v cos (8 ij i j
-
8
(1)
+
ij
Y v V sin ( 8 ij i j
-
8
+
)
(2)
ij
where Yij is the magnitude and
(V1
V2
(82 83 (P 1 P2
y
one can then obtain the following power flow equation (3)
y = h(x)
X~)T and h(x) is a nonlinear
wnere x =
function whose components are defined eqs (1) and (2).
During normal operation, the vector x is assumed to suffer from small perturbations around its nominal value Xo = (~T QT)T where ~ stands for an N vector of 1 's and 0 stands for an (N-1) vector O's. In the state estimation problem, it is required to note the perturbations lIy in the power data and then estimate the perturbations lIxv and lIXO in the two parts of the state vector x. The relations of interest are thus obtained from eq (3) through linearization: H (4) + u lIy = HVV 8v wh~re H stands for the Jacobian of the function h(x) and the subscripts indicate appropriate partitions of this matrix. The term u is added on the right hand side in order to indicate the errors of linear approximation and also to account for measurement errors. Often the matrices HvO and HOv are negligible and eq (4) may be written in the decoupled form (Horrisberger, Richard and Rossier 1976)
yp
(5a)
Hvv + u
(5b)
q
where the subscripts p and q are used to indicate the parts associate with the real and reactive power deviations respectively. In the usual static state estimation problem, eq (5) is used in order to obtain estimates of lIx v and lIXO from the instantaneous measurement perturbations lIYp and lIYq' In the real time estimation problem, it is recognized that these measurements are in fact time dependent so that eq (5) is rewritten as discrete time equations Hvv
lIxv(k)
+
up(k)
(6a)
H88
lI~
+
Uq(k)
(6b)
(k)
It is expected that once these models are identified, possibly in an off line manner, the state estimation problem can then be handled in the on line manner by making use the Kalman filtering algorithm or an equivalent. It is also expected that this will reduce the computational burden of the on line state estimation task. DATA GENERATION
Defining the vectors Xv Xo
mo~els for the state vectors Axv(k) and AXO(k).
where k = 1,2, ... is the discrete time index. As mentioned earlier, the current literature contains studies on the on line state estimation problem on the basis of the measurement relations (6). The problem studied in this work is that of processing the ,lata for IIYp(k) and
To solve the system identification problem, it is proposed to employ a stochastic realization algor i thm of the form suggested by Tse and Weinert (1975). This requires that noisy data of the values of the state vector be available. Since in the system under consideration, we do not have a direct measurement of the bus voltage magnitudes and angles, it would be necessary to generate these data by proceSSing the power injection data. Since only approximate values of the state variables are needed, it is proposed to make use of the d.c. load flow techniques to generate these values from the measured power data. In view of the large scale nature of the problem, it is proposed to process separately the active and reactive power injections in order to generate the data for the voltage angles and the magnitudes separately. Generation of the Angle Data Let us consider eq (1) and make the fOllowing assumptions: (i) The voltage magnitudes are approximately equal to uni ty (ii) The transmission lines have negligible resistences so that
It is then possible to express the injection at bus i as N
Pi = r
- Yij sin (8j - 8i)
j -1
Let us also assume that the difference between the angles 8j and 8i are relatively small so that eq (7) can be approximated as N
Pi = j
r
Yij
(e i - ej )
(8)
=1
Using this relation for i = 1,2, ... N, we can obtain the d.c. load flow equation P
= B 8
(9)
where P is the vector of the injections, 8 is the vector of the angles and the matrix B-has elements related to Yij. Assuming B nonsingular, we can use eq (9) for obtaining a solution for the angle vector (') for the given value of the power vector P. Since a number of approximations have been introducted while obtaining eq (9), it is apparent that the solution for (') obtained from the d.c. load flow is necessarily rather approximate. It may also be noted that since the angle 81 has been used as the reference, we need some adjustments in the choice of the elements of the first row of B (see, e.g., Wood and Wollenberg, 1984).
289
Identification of a Dynamic Model It is thus possible process the discrete injections data Pi(k) in order to generate the approximate values of the state vector. Let this approximate value of 6XO(k) be denoted by ZO(k) so that we can write the relation
The matrix A is assumed to have the following canonical form
o o
o
o
A
( 10)
( 15)
where n(k) represents the unknown error. Using eqs(14) and (15). it is possible to show that the i th output component zi(k) can be related to the preceding components as follows:
Generation of the Magnitude Data Let us consider now eq (2) and make the following assumptions:
zi (k+l) = ail zl (k) + ai2 z2(k) + ... + aiiZi(k) + mi(k). i = 1.2 •... N
(i) The angles 0i. i - 2.2 •... N have the values determined above
(16)
where mi(k) is a noise term that depends on the linear combination of nl(k). n2(k) •... ni(k). We can see that the elements of the various rows of the matrix A may be estimated using the usual parameters estimation algorithms. This requires that we rewrite eq (16) in the form
(ii) The angles ~ij = 90 0 It is then possible to express the reactive injection at bus i as N
Qi
=
L j =1
Yij ViVj cos (0j - 0i)
(11 ) ( (1)
Let us also assume that Vi = 1 and cos (01 - 0i) _ MiJ. a known quantity. It is then poss ble to rewrlte eq (11) in the linear form
where we have defined the two vectors on the RHS as
( 12)
Q = C V
where Q is the vector of the reactive injectIons. V is the vector of voltage magnitudes and the matrix C has elements related to Yij and Mij' Again. assuming C to be nonsingular. we can use eq(12) in order to compute the vector V for any given value of the injection vector Q.- It is thus possible to process the discrete time data for the bus injections in order to generate the approximate values of the part 6x v (k) of the state vector. If we use the notation zv(k) for these approximate solutions. we get the relation
~i = (ail
Since we do not have any apriori information about the noise term. it is convenient to make use of the least squares estimation algorithm either in the batch processing form or in the recursive form (see. e.g .• Sinha and Kuszta (1983). For large N. it is of course better to employ the recursive form. Note. however. that the dimension N in most pratical cases may be kept relatively small by making use of the concept of 'external network equivalencing' (see. e.g .• Dopazo et al 1914).
(13 ) AN EXAMPLE IDENTIFICATION OF THE STATE DYNAMICS
°
For convenience. let us drop the subscripts and v and just consider the question of a suitable dynamic model for the state 6x(k) from the observed data for z(k). This is a stochastic realization problem and a number of approaches have been developed for solving this problem in recent years. One such solution has been proposed by Tse and Weinert who have shown that the problem is simplified to some extent if the state transition matrix is assumed to have block triangular form (Tse and Weinert (915). To simplify the problem further. we assume that the observability subindices of the components of z(k) are all equal to 1 so that the state transition matrix is in fact a triangular one. Note that even this is an improvement over the choice of this matrix as the identity matrix in the earlier papers on the dynamic state estimation of power systems. Following the argument above. let us assume that the desired dynamic model for the state vector is gi ven by
In order to illustrate the method. let us consider a simple 3 bus network for which the following constants are specified (all expressed in per unit on a 100 MVA base): PlO - .35pu; P20 = .65 pU; P3 0 = -1 pU; Y 12
=
j
0.4 pu;
Y - j 0.25 pU; Y - j 0.2 pu 23 13 Since bus #1 is the reference bus. we intend to solve for the angloes 82 and 83 only using the d.c. load flow equation. The matrix B needed for this is formed by noting the following relations: bij
=
-(
l/Y ij)
L wi th the summation being bii - j (l/Yij)' performed over all lines connected to bus i For the given admittance data. we get 1.5 -5 B
!\x(k+l) - AIIx(k) + w(k)
(14b)
while the output equation is z(k)
=
ilx(k) + n(k)
( 14a)
-5
9
A. K. Mahalanabis
290
Using eq (9), we then obtain the following relations between the angle vector 0 and the injection vector P: 0.212
0.118
0.118
0.176
9
It is possible to make use of this relation in order to compute the values of 9(k) for the given values of P(k). The table below gives a set of these values. TABLE P2(k) 0.6 0.65 0.7 0.55 0.5 0.75
P3(k) -0.95 -1 -1. 05 -0.9 -0.85 -1.1
92(k) 0.015 0.02 0.024 0.01 0.006 0.029
93(k) -0.096 -0.1 -0.102 -0.093 -0.091 -0.105
~
Clements, K. A., Krumpholz, G. R. and Davis, P. W., (1983) "Power system estimation with measurement deficiency: an observability measurement placement algorithm," paper No. 83 WM 058-5, IEEE PES Winter Meeting, New York, Jan. 30-Feb. 4, 1983. Debs, A. S. and Larson, R. E., (1970) "A dynamic estimator for tracking the state of a power system," IEEE Trans Power App & Syst, Vol PAS 89, pp 1670-1678. Dopazo, J. F. et al, (1977) "An external system equivalent model using real time measurements for system security evaluation," IeEE Trans Power App & Syst, Vol. PAS-96, pp. 431-446.
Using these data, it is also possible to estimate the state transition matrix using eq (16). In this case, the matrix A has only three unknown elements. These are computed as: all = 0.93; a21 = -0.14; a22
REFERENCES
17.41
For these computations, we used the following data: 0.6
Horrisberger, H. P., Richard, J. C. and Rossier, C, (1976), "A fast decoupled static state estimator for electric poer systems," IEEE Trans Power App & Syst., Vol. PAS 95, pp. 208-215. Sinha, N. K. and Kuszta, B. (1983), "Modeling and Identification of Dynamic Systems," Van Nostrand, N. Y. Tse, E. and Weinert, H. L., (1975), "Structure determination and parameter identification mul ti var iable stochastic linear systems," IEEE Trans Autom Contr, Vol. AC-20, pp. 603-613.
-0.95 These estimates may be considered as the initial data for an on-line algorithm which may be based on the recursive version of the estimation algorithm. CONCLUDING REMARKS An approach for the identification of the voltage dynamics of a power system operating in the steady state has been proposed on the basis of the data generated from the d.c. load flow analysis. This form of load flow can be performed for a large network without too much of computations. However, in view of the approximate nature of the solution, this has been modeled as the result of additive corruption of the correct solution. This model has been combined with a simple canonical form of the state transition matrix in order to develop a least squares algorithm for the parameter estimation. Once the state transition matrix has been estimated, it should be possible to use the following model equations for solving estimation problem: x(k+l )
A x(k) + w(k)
y(k) = H x(k) + u(k) With this model, it is expected that the need for redundant measurements will be eliminated since the pair (A, H) is expected to be observable. Investigations for test systems are currently in progress along this line.
Wood, A. J. and Wollenberg, B. F., (1984), "Power Generation, Operation and Control," John Wiley & Sons, N.Y.