- Email: [email protected]
Identification of analog and topological measurement errors in generalized state estimation
Journal of Materials Processing Technology 161 (2005) 121–127
Identification of analog and topological measurement errors in generalized state estimation George N. Korres a,∗ , Peter J. Katsikas a , George E. Chatzarakis b a
School of Electrical and Computer Engineering, National Technical University, Athens, Greece b School of Pedagogical and Technological Education, Athens, Greece
Abstract This paper presents a method for analog and topological measurement error identification in generalized state estimation. Errors of analog measurements depend on measurement equipment accuracy. Errors of topological measurements (switching device statuses) result in erroneous network models. In generalized state estimation, analog and topological data are processed as a single interacting information set. The conventional measurement model is extended by incorporating the breaker statuses (open or closed) and the active and reactive power flows as continuous state variables. Depending on breaker statuses, additional pseudomeasurements are introduced. The method treats the analog and status measurements as regular measurements. Hypothesis testing for detection and identification of bad analog and topological measurements is used (J(ˆx) and rˆN , test respectively). Test results for the RTS substation system are reported. © 2004 Elsevier B.V. All rights reserved. Keywords: Generalized state estimation; Analog and topological measurement errors; Circuit breaker status; Bad data analysis
1. Introduction In the conventional state estimator (SE), the topology processor determines the bus/branch network model by using switching device statuses, and the analog data is processed next by the state estimator. The generalized state estimation treats the analog and topological data as a single interacting information set. Generalized state estimation models substations in detail (bus-section/switching device level), making it possible to detect and identify topology errors, which would appear as interacting bad data. Methods for topology error identification can be found in [1–9]. In [10–15] detailed physical-level modeling of substations is performed by incorporating the circuit breakers as zero impedance branches and their power flows as variables. The well known J(ˆx)-test is extended to validate hypothesis about the status of switching devices [16]. Method [17] overcomes the drawbacks of [16] by introducing the breaker statuses as additional state variables.
∗
This paper presents a unified approach for identification of errors in topological and analog measurements. The conventional state estimator is extended by incorporating the active and reactive power flows and the statuses of the circuit breakers as continuous state variables. Hypothesis testing is used to detect and identify bad analog and topological measurements. The proposed method is illustrated with the RTS substation system.
2. The conventional state estimation model In conventional state estimation the breaker statuses are processed by the topology processor. By merging bus sections joined by closed breakers into “buses”, the bus/branch model is built. For a N-bus system, the state vector x = [δT VT ]T , of dimension n = 2N − 1, consists of the N − 1 bus voltage angles δi , i = 2, 3, . . ., N, with respect to a reference bus, and the N bus voltage magnitudes Vi , i = 1, 2, . . ., N. The measurement model is: z = h(x) + v
Corresponding author. Tel.: +30 210 7723621; fax: +30 210 7723659. E-mail address: [email protected] (G.N. Korres).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.012
E(v) = 0,
(1a) E(vv ) = R T
(1b)
122
G.N. Korres et al. / Journal of Materials Processing Technology 161 (2005) 121–127
Fig. 1. Status modeling of circuit breaker k–l. Fig. 2. State variables assigned to circuit breaker k–l.
where z is a m × 1 measurement vector, v is a m × 1 noise vector, h(·) is a m × 1 vector of nonlinear functions and R is the diagonal measurement covariance matrix. The estimated state vector xˆ minimizes the WLS objective function
If we assume that status of breaker k–l is certain (skl = 1 for closed breaker, skl = 0 for open breaker), from (7), (9) and (8), (10) respectively we derive that:
J(x) = [z − h(x)]T R−1 [z − h(x)]
(1 − skl )Pkl = 0
(11)
skl δkl = skl (δk − δl ) = 0
(12)
(2)
it satisfies the optimality condition ∂J(ˆx) =0 ∂x and it is computed by an iterative scheme G(xk )xk = H T (xk )R−1 zk ,
Similarly, we derive that:
(3)
k = 0, 1, 2, . . .
(4)
(1 − skl )Qkl = 0
(13)
skl Vkl = skl (Vk − Vl ) = 0
(14)
where xk = xk+1 − xk , zk = z − h(xk ), H = ∂h/∂x is the m × n Jacobian matrix of h(x), G(xk ) = H T (xk )R−1 H(xk ) is the n × n gain matrix.
where Qkl is the true reactive power flow on CB k–l. Eqs. (11)–(14) are based on the hypothesis that status of breaker k–l is certain (skl = 1 or skl = 0). If (11–14) are not satisfied (0 < skl < 1) breaker status is uncertain.
3. Probabilistic modeling of circuit breakers
4. The generalized state estimation model
The status of a circuit breaker k−l can be defined by a discrete (zero–one) random variable Xkl , as in Fig. 1: The probability function of this random variable is: P(Xkl = 1) = skl 0 ≤ skl ≤ 1 (5) P(Xkl = 0) = 1 − skl
In generalized state estimation parts of the network are modeled at the bus-section/switching-device level. The measurement model (1) is extended, by taking into account the circuit breakers. For each circuit breaker k–l, the voltage angles and magnitudes) δk , Vk and δl , Vl , the active and reactive power flows Pkl , Qkl and the probability skl of being closed, are added as state variables (Fig. 2). For every circuit breaker k–l we add: • pseudo measurements based on breaker status
where skl is a continuous (non-integer) variable. The expected value and variance of Xkl are defined as [20]: E(Xkl ) = skl
(6a)
Var(Xkl ) = skl (l − skl )
(6b)
Let Pkl be the true active power flow on CB k–l. We define by Ykl {Θkl } a zero–one random variable and by E(Ykl ) {E(Θkl )} its expected value as follows: Ykl = Xkl Pkl ⇒ E(Ykl ) = E(Xkl )Pkl = skl Pkl
0 = δkl = δk − δl , 0 = Pkl ,
0 = skl δkl ,
0 = (1 − skl )Pkl ,
(8)
m skl
(16)
(17)
0 = (1 − skl )Qkl
(18)
• status measurements
(9)
If Xkl = 0 (open), δkl may have any value. If Xkl = 1 (closed) then δkl = 0. Therefore, Θkl always satisfies: Θkl = 0 ⇒ E(Θkl ) = 0
(if Xkl = 0)
0 = skl Vkl
(7)
If Xkl = 1 (closed), Pkl may have any value. If Xkl = 0 (open) then Pkl = 0. Therefore, Ykl always satisfies: Ykl = Pkl ⇒ E(Ykl ) = Pkl
0 = Qkl ,
(15)
• pseudo measurements based on (11–14)
Θkl = Xkl δkl ≡ Xkl (δk − δl ) ⇒ E(Θkl ) = E(Xkl )δkl = skl δkl
0 = Vkl = Vk − V(if l , Xkl = 1)
(10)
= skl + vskl
(19)
m skl
m (skl
where is the measured status of CB k–l = 0 for open m = 1 for closed breaker. A flow measurement breaker and skl in breaker k–l (Fig. 3) is written as follows: Pklm = Pkl + vPkl ,
Qm kl = Qkl + vQkl
(20)
A flow measurement in regular branch k–l is written as: Pklm = Pkl (δk , δl , Vk , Vl ) + vPkl
(21a)
G.N. Korres et al. / Journal of Materials Processing Technology 161 (2005) 121–127
123
where measurements are related to equations as follows
Fig. 3. Flow measurement on circuit breaker k–l.
ZA (mA × 1) ZB (mB × 1)
to (15), (22) and (23) with Bk = ∅ to (22), (23) with Lk = ∅, Bk = ∅
ZI (mI × 1)
to (16), (20), (22) and (23) with Lk = ∅
ZS (l × 1)
to (19)
hc (x, s)(2l × 1) to (17) hD (f, s)(2l × 1) to (18) Matrices MB (mB × 2l), MI (mI × 2l) are measurementto-breaker incident matrices and Il is the l × l identity matrix. The state vector yˆ minimizes the objective function:
Fig. 4. Injection measurement at node k.
Qm kl = Qkl (δk , δl , Vk , Vl ) + vQkl
(21b)
An injection measurement at node k (Fig. 4) is written as: Pkm
=
Pki (δk , δi , Vk , Vi ) +
i ∈ Lk
Qm k =
Pkj + vPk
(22)
j ∈ Bk
Qki (δk , δi , Vk , Vi ) +
i ∈ Lk
Qkj + vQk
J(y) = [z − g(y)]T R−1 [z − g(y)]
(28)
The following iterative scheme is used to compute yˆ : G(yk )yk = H T (yk )R−1 zk
k = 0, 1, 2, . . .
(29)
where yk = yk+1 − yk , zk = z − g(yk ), H = ∂g/∂y is the m × n Jacobian matrix of g(x):
(23)
j ∈ Bk
where Lk , Bk is the set of nodes connected to node k by regular branches and circuit breakers, respectively.
5. Generalized state estimation algorithm The nodal state vector x includes voltages at single buses and bus sections of explicitly modeled substations. We define an extended state vector y as follows: T
y = [xT f T sT ]
T
f = [P T QT ]
(24) (25)
where P, Q are the l × l (two occurrences) vectors of the unknown active and reactive flows (state variables) through circuit breakers and s is the l × l vector of CB statuses. The extended measurement model is written as: z = g(y) + v ≡ g(x, f, s) + v
(26)
and G(yk ) = H T (yk )R−1 H(yk ) is the n × n gain matrix. The partial derivatives in Jacobian (30) are computed with respect to variables δk , Vk , Pkl , Qkl , skl . The number of measurements is m = mA + mB + mI + 5l and the number of state variables is n = 2N − 1 + 3l. The detailed representation of switching devices increases significantly the number of state variables and requires the availability of enough measurements at the substation level, so that the system remains observable and redundancy high. However, the introduction of pseudomeasurements (15)–(18) and zero injections increases significantly the measurement redundancy. Pseudomeasurements (15–16) are given lower weights than the analog measurements. Breaker status measurements (19) and pseudomeasurements (17–18) are given weights comparable to analog measurements. Zero injection measurements of type (22–23) are given larger weights.
or
0 0 0 hA (x) vA z h (x) M 0 0 B B B vB zI 0 MI 0 vI 0 = + z 0 + 0 f + I s + v 0 S l S 0 0 0 0 hC (x, s) vC 0 0 0 0 hD (f, s) vD zA
(27)
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6. Observability analysis
Table 1 Bad measured values of CB power flows (in p.u.)
If the available measurements are adequate so that state vector yˆ can be computed by (29), the system is considered as observable. Eq. (29) are solvable if and only if the Jacobian matrix H (or gain matrix G) are of full column rank (t = null (G) = null (HT H) = null (H) = 0). The rank deficiency t of matrix H (or G) is equal to the number of zero pivots in the diagonal matrix D of the UT DU decomposition of G [18]. If the system is unobservable (t = 0), a minimal set of t additional nonredundant(critical) pseudo measurements should be added to make the network barely observable, without affecting the estimated states of the already observable part. A measurement is said to be critical if its suppression from the measurement set makes the network unobservable. It can be proven [17] that the rank deficiency (nullity) of matrix H(y0 ) at flat start is equal to rank deficiency of matrix F: HA 0 (32) F = H B MB
Value
P101–102
P304–305
Q204–209
Q1803–1806
Measured True
−0.3 −0.4
−7.6 −7.7
0.026 −0.074
0.459 0.359
0
MI
7. Bad data analysis Bad topological (breaker status) and analog measurements are detected and identified by statistical tests on J (ˆy) and 2 , normalized residuals rˆN , respectively [19]. If J(ˆy) > Xk,p 2 bad data is detected (threshold Xk,p is obtained from a X2 distribution with k = m − n degrees of freedom and (1 − p) false alarm probability). The normalized residuals are defined as: rˆN = (diag Pr )1/2 rˆ
(33)
where rˆ = z − g(ˆy)
(34) −1
Pr = Cov(r) = R − H(HR−1 H)
HT
(35)
Random vector rˆN has unit normal distribution. We adopt a detection threshold of 3, corresponding to 0.003 false alarm probability. If |ˆrN,i |max > 3, the ith measurement is flagged as bad. After a cycle of bad data removal, yˆ and rˆN reestimation, all bad data have been identified when |ˆrN,i |max < 3. At flat start, any circuit breaker k–l has δkl = Vkl = 0 and Pkl = Qkl = 0. As a consequence, all status measurements (19) become critical and if any of them is flagged as bad, it cannot be removed from the measurement set. However, we can give very small weights to a bad status measurement (19), which may become noncritical during subsequent iterations, since δkl = 0 and Pkl = 0, and may be corrected. If the proposed formulation is used for modeling the entire network, no topology preprocessing is necessary. The disadvantage of this approach is the increase in problem size. Instead, to cope with the dimensionality problem, state estimation can
Table 2 Bad measured values of breaker statuses (1→closed, 0→open) Status
S101–102
S1003–1006
S1501–1510
S1606–1607
True Measured
1 0
1 0
0 1
1 0
be executed using the regular bus/branch model for the entire network and if bad data is flagged, then only substations suspect of being affected by bad data will remodeled according to the proposed method.
8. Test results The proposed method is tested on the IEEE Reliability Test System [21,22] shown in Fig. 5. Each bus of the original RTS has been represented by its substation configuration. The network consists of 197 nodes, 85 branches (lines and transformers) and 168 breakers. In Fig. 5 the true breaker statuses are shown. One voltage measurement is assumed at angle reference node 1310. Only closed breakers are flow measured. Nodes adjacent to a conventional branch and one or two breakers are zero injections. By taking into account the pseudo measurements of type (15)–(18) and status measurements (19), the total number of measurements and the global measurement redundancy are 1725 and 1.93 respectively. Although the number of state variables is high, global redundancy is satisfactory and no new measurements are required. Normally-distributed random errors N (0, σ) are added to the exact load flow values to simulate measurement noise. The standard deviation is σ = 0.02 × P + 0.003 × FS for the power flow and injection measurements (P exact measurement value, FS: full scale measurement value) and σ = 0.005|V|p.u for the voltage magnitude measurements. Bad measurements were simulated by adding to the their true value an error of 10σ (Tables 1 and 2). Bad data identification is based on the rˆN test. After the first estimate yˆ is calculated the normalized residuals are computed (Table 3). The measurement with the largest normalized residual rˆN,i max > 3 is flagged as bad and is eliminated from the measurement set. A cycle of successive state estimation runs and bad data removal is established until Table 3 The four largest normalized residuals (first SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
CB-status CB-status CB-status CB-status
1606 1003 1606 0101
1607 1006 1607 0102
19.998 19.857 19.633 16.854
G.N. Korres et al. / Journal of Materials Processing Technology 161 (2005) 121–127
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Fig. 5. Single line diagram of IEEE RTS substation system. Table 7 The five largest normalized residuals (fifth SE run)
Table 4 The four largest normalized residuals (second SE run) Measurement type CB-status CB-status CB-status CB-status
Sending bus 1003 0101 0101 1501
Receiving bus 1006 0102 0102 1510
|rN,i |
Measurement type
Sending bus
Receiving bus
|rN,i |
19.997 16.854 16.756 9.680
P-flow Q-flow Q-injection Q-injection Q-flow
0304 0103 0103 0204 0204
0305 0107 – – 0209
7.137 4.921 4.921 4.920 4.920
Table 5 The four largest normalized residuals (third SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
CB-status CB-status P-flow P-injection
0101 1501 0304 0101
0102 1510 0305 –
16.854 9.680 7.227 5.514
Table 8 The five largest normalized residuals (sixth SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
Q-flow Q-flow P-flow P-flow P-injection
1803 2104 0101 0101 0101
1806 2105 0102 0110 –
4.908 4.887 4.061 4.061 4.061
Table 6 The five largest normalized residuals (fourth SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
CB-status P-flow Q-flow Q-injection Q-injection
1501 0304 0103 0103 0204
1510 0305 0107 – –
9.670 7.196 4.922 4.922 4.918
|ˆrN,i |max < 3 and all bad are identified (Tables 4–11). After the ninth state estimation run, all normalized residuals have |ˆrN,i | < 3, meaning that all bad data have been eliminated. After bad data elimination, the correct estimated flows and statuses of the circuit breakers, which had gross errors, are shown in Table 12.
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Table 9 The five largest normalized residuals (seventh SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
P-flow P-injection P-flow Q-injection Q-flow
0101 0101 0101 0204 0204
0102 – 0110 – 0209
4.261 4.031 3.945 3.864 3.864
Table 10 The five largest normalized residuals (eighth SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
Q-flow Q-injection Q-flow Q-injection P-flow
0204 0204 0103 0103 1102
0209 – 0107 – 1103
4.258 4.230 4.227 4.227 2.230
Table 11 The five largest normalized residuals (ninth SE run) Measurement type P-flow P-flow P-injection Q-injection Q-flow
Sending bus
Receiving bus
1102 1006 1008 0803 0803
1103 1008 – – 0804
|rN,i | 2.232 2.142 2.076 1.944 1.944
Table 12 Estimated values after bad data elimination CB flow
Estimated value
CB status
Estimated value
P101–102 P304–305 Q204–209 Q1803–1806
−0.407 −7.702 −0.075 0.356
S101–102 S1003–1006 S1501–1510 S1606–1607
1.000 1.000 1.9 × 10−5 1.000
9. Conclusions In this paper a new state estimation algorithm is presented, for identification of interacting errors in analog and topological measurements of generalized state estimation. Substations of interest are modeled in detail, by extending the conventional WLS state estimator and incorporating the active and reactive power flows and the statuses of circuit breakers as state variables. Depending on breaker statuses, additional pseudomeasurements are introduced. Analog and status measurements are treated as regular measurements. Observability analysis is carried out for analog and topological measurements. Hypothesis tests, based on index J(ˆx) and normalized residuals rˆN , are performed to detect and identify bad analog and topological measurements. The proposed method is illustrated with the RTS substation system.
References [1] M.R. Irving, M.J. Sterling, Substation Data Validation, in: IEE Proceedings, pt C, vol. 129 (3), 1982, pp. 119–122.
[2] F.F. Wu, W.H. Liu, Detection of topological errors by state estimation, Paper 88 WM 216-4, IEEE/PES Winter Meeting 1988. [3] K.A. Clements, P.W. Davis, Detection and identification of topology errors in electric power systems, IEEE Trans. Power Syst. 3 (4) (1988) 176–183. [4] N. Singh, H. Glavitsch, Detection and identification of topological errors in on-line power system analysis, IEEE Trans. Power Syst. 6 (1) (1991) 324–331. [5] A. Simoes Costa, J.A. Leao, Identification of topology errors in power system state estimation, IEEE Trans. Power Syst. 8 (4) (1993) 1531–1538. [6] H. Singh, F.L. Alvarado, Network topology determination using least absolute value state estimation, IEEE Trans. Power Syst. 10 (3) (1995) 1159–1165. [7] K.A. Clements, A.S. Costa, Topology error identification using normalized lagrange multipliers, IEEE Trans. Power Syst. 13 (2) (1998) 347–353. [8] J.C.S. Souza, A.M. Leite da Silva, A.P. Alves da Silva, On line topology determination and bad data suppression in power system operation using artificial neural networks, IEEE Trans. Power Syst. 13 (3) (1998) 796–803. [9] L. Mili, G. Steeno, F. Dobraca, D. French, A robust estimation method for topology error identification, IEEE Trans. Power Syst. 14 (4) (1999) 1469–1476. [10] A. Monticelli, A. Garcia, Modeling zero impedance branches in power system state estimation, IEEE Trans. Power Syst. 6 (1991) 1561–1570. [11] A. Monticelli, The impact of modeling short circuit branches in state estimation, IEEE Trans. Power Syst. 8 (1) (1993) 364– 370. [12] A. Monticelli, Modeling circuit breakers in weighted least squares state estimation, IEEE Trans. Power Syst. 8 (3) (1993) 1143– 1149. [13] O. Alsac, N. Vempati, B. Stott, A. Monticelli, Generalized state estimation, IEEE Trans. Power Syst. 13 (3) (1998) 1069–1075. [14] A. Monticelli, in: Proceedings of the IEEE, vol. 88 (2), Electric power system state estimation (2000) 262–282. [15] A. Abur, H. Kim, M. Celik, Identifying the unknown circuit breaker statuses in power networks, IEEE Trans. Power Syst. 10 (4) (1995) 2029–2035. [16] A. Monticelli, Testing equality constraints hypothesis in weighted least squares state estimators, IEEE Trans. Power Syst. 15 (3) (2000) 950–954. [17] G.N. Korres, P.J. Katsikas, Identification of circuit breaker statuses in WLS state estimator, IEEE Trans. Power Syst. 17 (3) (2002) 818–825. [18] G.C. Contaxis, G.N. Korres, A reduced model for power system observability analysis and restoration, IEEE Trans. Power Syst. 3 (4) (1988) 1411–1417. [19] A. Monticelli, A. Garcia, Reliable bad data processing for real-time state estimation, IEEE Trans. Power Apparatus Syst. PAS 102 (1983) 1126–1139. [20] A. Papoulis, Probability, Random Variables and Stochastic Processes, Mc Graw-Hill, 1965. [21] IEEE Committee Report, IEEE Reliability Test System, IEEE Trans. Power Apparatus Syst. PAS-98 (10) (1979) 2047–2054. [22] R. Billinton, P.K. Vohra, S. Kumar, Effect of station originated outages in a composite system adequacy evaluation of the IEEE reliability test system, IEEE Trans. Power Apparatus Syst. PAS 104 (10) (1985) 2649–2656. George N. Korres received the Dipl.Eng. and the PhD degrees in electrical engineering from the National Technical University of Athens, Greece, in 1984 and 1988, respectively. Currently, he is an Associate Professor in the School of Electrical and Computer Engineering at the National Technical University of Athens, Greece. He is member of IEEE and CIGRE.
G.N. Korres et al. / Journal of Materials Processing Technology 161 (2005) 121–127 Peter J. Katsikas received the Dipl. Eng. degree from the National Technical University of Athens, Greece, in 1998. He is currently pursuing the Ph.D. degree in the School of Electrical and Computer Engineering at the National Technical University of Athens.
127
George E. Chatzarakis received the Dipl. Eng. and the PhD degrees in electrical engineering from the National Technical University of Athens, Greece in 1986 and 1990, respectively. Currently, he is an Associate Professor in the School of Pedagogical and Technological Education (A.S.PE.T.E) in Athens, Greece. He is member of IEEE.
Identification of analog and topological measurement errors in generalized state estimation George N. Korres a,∗ , Peter J. Katsikas a , George E. Chatzarakis b a
School of Electrical and Computer Engineering, National Technical University, Athens, Greece b School of Pedagogical and Technological Education, Athens, Greece
Abstract This paper presents a method for analog and topological measurement error identification in generalized state estimation. Errors of analog measurements depend on measurement equipment accuracy. Errors of topological measurements (switching device statuses) result in erroneous network models. In generalized state estimation, analog and topological data are processed as a single interacting information set. The conventional measurement model is extended by incorporating the breaker statuses (open or closed) and the active and reactive power flows as continuous state variables. Depending on breaker statuses, additional pseudomeasurements are introduced. The method treats the analog and status measurements as regular measurements. Hypothesis testing for detection and identification of bad analog and topological measurements is used (J(ˆx) and rˆN , test respectively). Test results for the RTS substation system are reported. © 2004 Elsevier B.V. All rights reserved. Keywords: Generalized state estimation; Analog and topological measurement errors; Circuit breaker status; Bad data analysis
1. Introduction In the conventional state estimator (SE), the topology processor determines the bus/branch network model by using switching device statuses, and the analog data is processed next by the state estimator. The generalized state estimation treats the analog and topological data as a single interacting information set. Generalized state estimation models substations in detail (bus-section/switching device level), making it possible to detect and identify topology errors, which would appear as interacting bad data. Methods for topology error identification can be found in [1–9]. In [10–15] detailed physical-level modeling of substations is performed by incorporating the circuit breakers as zero impedance branches and their power flows as variables. The well known J(ˆx)-test is extended to validate hypothesis about the status of switching devices [16]. Method [17] overcomes the drawbacks of [16] by introducing the breaker statuses as additional state variables.
∗
This paper presents a unified approach for identification of errors in topological and analog measurements. The conventional state estimator is extended by incorporating the active and reactive power flows and the statuses of the circuit breakers as continuous state variables. Hypothesis testing is used to detect and identify bad analog and topological measurements. The proposed method is illustrated with the RTS substation system.
2. The conventional state estimation model In conventional state estimation the breaker statuses are processed by the topology processor. By merging bus sections joined by closed breakers into “buses”, the bus/branch model is built. For a N-bus system, the state vector x = [δT VT ]T , of dimension n = 2N − 1, consists of the N − 1 bus voltage angles δi , i = 2, 3, . . ., N, with respect to a reference bus, and the N bus voltage magnitudes Vi , i = 1, 2, . . ., N. The measurement model is: z = h(x) + v
Corresponding author. Tel.: +30 210 7723621; fax: +30 210 7723659. E-mail address: [email protected] (G.N. Korres).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.012
E(v) = 0,
(1a) E(vv ) = R T
(1b)
122
G.N. Korres et al. / Journal of Materials Processing Technology 161 (2005) 121–127
Fig. 1. Status modeling of circuit breaker k–l. Fig. 2. State variables assigned to circuit breaker k–l.
where z is a m × 1 measurement vector, v is a m × 1 noise vector, h(·) is a m × 1 vector of nonlinear functions and R is the diagonal measurement covariance matrix. The estimated state vector xˆ minimizes the WLS objective function
If we assume that status of breaker k–l is certain (skl = 1 for closed breaker, skl = 0 for open breaker), from (7), (9) and (8), (10) respectively we derive that:
J(x) = [z − h(x)]T R−1 [z − h(x)]
(1 − skl )Pkl = 0
(11)
skl δkl = skl (δk − δl ) = 0
(12)
(2)
it satisfies the optimality condition ∂J(ˆx) =0 ∂x and it is computed by an iterative scheme G(xk )xk = H T (xk )R−1 zk ,
Similarly, we derive that:
(3)
k = 0, 1, 2, . . .
(4)
(1 − skl )Qkl = 0
(13)
skl Vkl = skl (Vk − Vl ) = 0
(14)
where xk = xk+1 − xk , zk = z − h(xk ), H = ∂h/∂x is the m × n Jacobian matrix of h(x), G(xk ) = H T (xk )R−1 H(xk ) is the n × n gain matrix.
where Qkl is the true reactive power flow on CB k–l. Eqs. (11)–(14) are based on the hypothesis that status of breaker k–l is certain (skl = 1 or skl = 0). If (11–14) are not satisfied (0 < skl < 1) breaker status is uncertain.
3. Probabilistic modeling of circuit breakers
4. The generalized state estimation model
The status of a circuit breaker k−l can be defined by a discrete (zero–one) random variable Xkl , as in Fig. 1: The probability function of this random variable is: P(Xkl = 1) = skl 0 ≤ skl ≤ 1 (5) P(Xkl = 0) = 1 − skl
In generalized state estimation parts of the network are modeled at the bus-section/switching-device level. The measurement model (1) is extended, by taking into account the circuit breakers. For each circuit breaker k–l, the voltage angles and magnitudes) δk , Vk and δl , Vl , the active and reactive power flows Pkl , Qkl and the probability skl of being closed, are added as state variables (Fig. 2). For every circuit breaker k–l we add: • pseudo measurements based on breaker status
where skl is a continuous (non-integer) variable. The expected value and variance of Xkl are defined as [20]: E(Xkl ) = skl
(6a)
Var(Xkl ) = skl (l − skl )
(6b)
Let Pkl be the true active power flow on CB k–l. We define by Ykl {Θkl } a zero–one random variable and by E(Ykl ) {E(Θkl )} its expected value as follows: Ykl = Xkl Pkl ⇒ E(Ykl ) = E(Xkl )Pkl = skl Pkl
0 = δkl = δk − δl , 0 = Pkl ,
0 = skl δkl ,
0 = (1 − skl )Pkl ,
(8)
m skl
(16)
(17)
0 = (1 − skl )Qkl
(18)
• status measurements
(9)
If Xkl = 0 (open), δkl may have any value. If Xkl = 1 (closed) then δkl = 0. Therefore, Θkl always satisfies: Θkl = 0 ⇒ E(Θkl ) = 0
(if Xkl = 0)
0 = skl Vkl
(7)
If Xkl = 1 (closed), Pkl may have any value. If Xkl = 0 (open) then Pkl = 0. Therefore, Ykl always satisfies: Ykl = Pkl ⇒ E(Ykl ) = Pkl
0 = Qkl ,
(15)
• pseudo measurements based on (11–14)
Θkl = Xkl δkl ≡ Xkl (δk − δl ) ⇒ E(Θkl ) = E(Xkl )δkl = skl δkl
0 = Vkl = Vk − V(if l , Xkl = 1)
(10)
= skl + vskl
(19)
m skl
m (skl
where is the measured status of CB k–l = 0 for open m = 1 for closed breaker. A flow measurement breaker and skl in breaker k–l (Fig. 3) is written as follows: Pklm = Pkl + vPkl ,
Qm kl = Qkl + vQkl
(20)
A flow measurement in regular branch k–l is written as: Pklm = Pkl (δk , δl , Vk , Vl ) + vPkl
(21a)
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where measurements are related to equations as follows
Fig. 3. Flow measurement on circuit breaker k–l.
ZA (mA × 1) ZB (mB × 1)
to (15), (22) and (23) with Bk = ∅ to (22), (23) with Lk = ∅, Bk = ∅
ZI (mI × 1)
to (16), (20), (22) and (23) with Lk = ∅
ZS (l × 1)
to (19)
hc (x, s)(2l × 1) to (17) hD (f, s)(2l × 1) to (18) Matrices MB (mB × 2l), MI (mI × 2l) are measurementto-breaker incident matrices and Il is the l × l identity matrix. The state vector yˆ minimizes the objective function:
Fig. 4. Injection measurement at node k.
Qm kl = Qkl (δk , δl , Vk , Vl ) + vQkl
(21b)
An injection measurement at node k (Fig. 4) is written as: Pkm
=
Pki (δk , δi , Vk , Vi ) +
i ∈ Lk
Qm k =
Pkj + vPk
(22)
j ∈ Bk
Qki (δk , δi , Vk , Vi ) +
i ∈ Lk
Qkj + vQk
J(y) = [z − g(y)]T R−1 [z − g(y)]
(28)
The following iterative scheme is used to compute yˆ : G(yk )yk = H T (yk )R−1 zk
k = 0, 1, 2, . . .
(29)
where yk = yk+1 − yk , zk = z − g(yk ), H = ∂g/∂y is the m × n Jacobian matrix of g(x):
(23)
j ∈ Bk
where Lk , Bk is the set of nodes connected to node k by regular branches and circuit breakers, respectively.
5. Generalized state estimation algorithm The nodal state vector x includes voltages at single buses and bus sections of explicitly modeled substations. We define an extended state vector y as follows: T
y = [xT f T sT ]
T
f = [P T QT ]
(24) (25)
where P, Q are the l × l (two occurrences) vectors of the unknown active and reactive flows (state variables) through circuit breakers and s is the l × l vector of CB statuses. The extended measurement model is written as: z = g(y) + v ≡ g(x, f, s) + v
(26)
and G(yk ) = H T (yk )R−1 H(yk ) is the n × n gain matrix. The partial derivatives in Jacobian (30) are computed with respect to variables δk , Vk , Pkl , Qkl , skl . The number of measurements is m = mA + mB + mI + 5l and the number of state variables is n = 2N − 1 + 3l. The detailed representation of switching devices increases significantly the number of state variables and requires the availability of enough measurements at the substation level, so that the system remains observable and redundancy high. However, the introduction of pseudomeasurements (15)–(18) and zero injections increases significantly the measurement redundancy. Pseudomeasurements (15–16) are given lower weights than the analog measurements. Breaker status measurements (19) and pseudomeasurements (17–18) are given weights comparable to analog measurements. Zero injection measurements of type (22–23) are given larger weights.
or
0 0 0 hA (x) vA z h (x) M 0 0 B B B vB zI 0 MI 0 vI 0 = + z 0 + 0 f + I s + v 0 S l S 0 0 0 0 hC (x, s) vC 0 0 0 0 hD (f, s) vD zA
(27)
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6. Observability analysis
Table 1 Bad measured values of CB power flows (in p.u.)
If the available measurements are adequate so that state vector yˆ can be computed by (29), the system is considered as observable. Eq. (29) are solvable if and only if the Jacobian matrix H (or gain matrix G) are of full column rank (t = null (G) = null (HT H) = null (H) = 0). The rank deficiency t of matrix H (or G) is equal to the number of zero pivots in the diagonal matrix D of the UT DU decomposition of G [18]. If the system is unobservable (t = 0), a minimal set of t additional nonredundant(critical) pseudo measurements should be added to make the network barely observable, without affecting the estimated states of the already observable part. A measurement is said to be critical if its suppression from the measurement set makes the network unobservable. It can be proven [17] that the rank deficiency (nullity) of matrix H(y0 ) at flat start is equal to rank deficiency of matrix F: HA 0 (32) F = H B MB
Value
P101–102
P304–305
Q204–209
Q1803–1806
Measured True
−0.3 −0.4
−7.6 −7.7
0.026 −0.074
0.459 0.359
0
MI
7. Bad data analysis Bad topological (breaker status) and analog measurements are detected and identified by statistical tests on J (ˆy) and 2 , normalized residuals rˆN , respectively [19]. If J(ˆy) > Xk,p 2 bad data is detected (threshold Xk,p is obtained from a X2 distribution with k = m − n degrees of freedom and (1 − p) false alarm probability). The normalized residuals are defined as: rˆN = (diag Pr )1/2 rˆ
(33)
where rˆ = z − g(ˆy)
(34) −1
Pr = Cov(r) = R − H(HR−1 H)
HT
(35)
Random vector rˆN has unit normal distribution. We adopt a detection threshold of 3, corresponding to 0.003 false alarm probability. If |ˆrN,i |max > 3, the ith measurement is flagged as bad. After a cycle of bad data removal, yˆ and rˆN reestimation, all bad data have been identified when |ˆrN,i |max < 3. At flat start, any circuit breaker k–l has δkl = Vkl = 0 and Pkl = Qkl = 0. As a consequence, all status measurements (19) become critical and if any of them is flagged as bad, it cannot be removed from the measurement set. However, we can give very small weights to a bad status measurement (19), which may become noncritical during subsequent iterations, since δkl = 0 and Pkl = 0, and may be corrected. If the proposed formulation is used for modeling the entire network, no topology preprocessing is necessary. The disadvantage of this approach is the increase in problem size. Instead, to cope with the dimensionality problem, state estimation can
Table 2 Bad measured values of breaker statuses (1→closed, 0→open) Status
S101–102
S1003–1006
S1501–1510
S1606–1607
True Measured
1 0
1 0
0 1
1 0
be executed using the regular bus/branch model for the entire network and if bad data is flagged, then only substations suspect of being affected by bad data will remodeled according to the proposed method.
8. Test results The proposed method is tested on the IEEE Reliability Test System [21,22] shown in Fig. 5. Each bus of the original RTS has been represented by its substation configuration. The network consists of 197 nodes, 85 branches (lines and transformers) and 168 breakers. In Fig. 5 the true breaker statuses are shown. One voltage measurement is assumed at angle reference node 1310. Only closed breakers are flow measured. Nodes adjacent to a conventional branch and one or two breakers are zero injections. By taking into account the pseudo measurements of type (15)–(18) and status measurements (19), the total number of measurements and the global measurement redundancy are 1725 and 1.93 respectively. Although the number of state variables is high, global redundancy is satisfactory and no new measurements are required. Normally-distributed random errors N (0, σ) are added to the exact load flow values to simulate measurement noise. The standard deviation is σ = 0.02 × P + 0.003 × FS for the power flow and injection measurements (P exact measurement value, FS: full scale measurement value) and σ = 0.005|V|p.u for the voltage magnitude measurements. Bad measurements were simulated by adding to the their true value an error of 10σ (Tables 1 and 2). Bad data identification is based on the rˆN test. After the first estimate yˆ is calculated the normalized residuals are computed (Table 3). The measurement with the largest normalized residual rˆN,i max > 3 is flagged as bad and is eliminated from the measurement set. A cycle of successive state estimation runs and bad data removal is established until Table 3 The four largest normalized residuals (first SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
CB-status CB-status CB-status CB-status
1606 1003 1606 0101
1607 1006 1607 0102
19.998 19.857 19.633 16.854
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Fig. 5. Single line diagram of IEEE RTS substation system. Table 7 The five largest normalized residuals (fifth SE run)
Table 4 The four largest normalized residuals (second SE run) Measurement type CB-status CB-status CB-status CB-status
Sending bus 1003 0101 0101 1501
Receiving bus 1006 0102 0102 1510
|rN,i |
Measurement type
Sending bus
Receiving bus
|rN,i |
19.997 16.854 16.756 9.680
P-flow Q-flow Q-injection Q-injection Q-flow
0304 0103 0103 0204 0204
0305 0107 – – 0209
7.137 4.921 4.921 4.920 4.920
Table 5 The four largest normalized residuals (third SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
CB-status CB-status P-flow P-injection
0101 1501 0304 0101
0102 1510 0305 –
16.854 9.680 7.227 5.514
Table 8 The five largest normalized residuals (sixth SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
Q-flow Q-flow P-flow P-flow P-injection
1803 2104 0101 0101 0101
1806 2105 0102 0110 –
4.908 4.887 4.061 4.061 4.061
Table 6 The five largest normalized residuals (fourth SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
CB-status P-flow Q-flow Q-injection Q-injection
1501 0304 0103 0103 0204
1510 0305 0107 – –
9.670 7.196 4.922 4.922 4.918
|ˆrN,i |max < 3 and all bad are identified (Tables 4–11). After the ninth state estimation run, all normalized residuals have |ˆrN,i | < 3, meaning that all bad data have been eliminated. After bad data elimination, the correct estimated flows and statuses of the circuit breakers, which had gross errors, are shown in Table 12.
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Table 9 The five largest normalized residuals (seventh SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
P-flow P-injection P-flow Q-injection Q-flow
0101 0101 0101 0204 0204
0102 – 0110 – 0209
4.261 4.031 3.945 3.864 3.864
Table 10 The five largest normalized residuals (eighth SE run) Measurement type
Sending bus
Receiving bus
|rN,i |
Q-flow Q-injection Q-flow Q-injection P-flow
0204 0204 0103 0103 1102
0209 – 0107 – 1103
4.258 4.230 4.227 4.227 2.230
Table 11 The five largest normalized residuals (ninth SE run) Measurement type P-flow P-flow P-injection Q-injection Q-flow
Sending bus
Receiving bus
1102 1006 1008 0803 0803
1103 1008 – – 0804
|rN,i | 2.232 2.142 2.076 1.944 1.944
Table 12 Estimated values after bad data elimination CB flow
Estimated value
CB status
Estimated value
P101–102 P304–305 Q204–209 Q1803–1806
−0.407 −7.702 −0.075 0.356
S101–102 S1003–1006 S1501–1510 S1606–1607
1.000 1.000 1.9 × 10−5 1.000
9. Conclusions In this paper a new state estimation algorithm is presented, for identification of interacting errors in analog and topological measurements of generalized state estimation. Substations of interest are modeled in detail, by extending the conventional WLS state estimator and incorporating the active and reactive power flows and the statuses of circuit breakers as state variables. Depending on breaker statuses, additional pseudomeasurements are introduced. Analog and status measurements are treated as regular measurements. Observability analysis is carried out for analog and topological measurements. Hypothesis tests, based on index J(ˆx) and normalized residuals rˆN , are performed to detect and identify bad analog and topological measurements. The proposed method is illustrated with the RTS substation system.
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G.N. Korres et al. / Journal of Materials Processing Technology 161 (2005) 121–127 Peter J. Katsikas received the Dipl. Eng. degree from the National Technical University of Athens, Greece, in 1998. He is currently pursuing the Ph.D. degree in the School of Electrical and Computer Engineering at the National Technical University of Athens.
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George E. Chatzarakis received the Dipl. Eng. and the PhD degrees in electrical engineering from the National Technical University of Athens, Greece in 1986 and 1990, respectively. Currently, he is an Associate Professor in the School of Pedagogical and Technological Education (A.S.PE.T.E) in Athens, Greece. He is member of IEEE.