Application of State Estimation Technique to Power System in Czechoslovakia

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APPLICATION OF STATE ESTIMATION TECHNIQUE TO POWER SYSTEM IN CZECHOSLOV AKIA B. Sadeck y

Abstract. The paper deals with the problem of state estimation, bad data detection and identification in the Czechoslovak power system. The weighted least squares estimation method has been applied and its program realization is described. The residual test has been used as the tool to bad data identification. An original algorithm for the analysia and the reduction of the set of suspicious measuraments is dascribed. Results from off-line tests performed on a network with 2B nodes are presented and the efficiency of the method is discussed. Keywords. Power system control, state estimation, bad data detection, weighted least squares method, residual test. INTROOUCTION The knowledge of instantaneous state of the controlled system is a necessary condition of its successful control. In a powar system the state is represented by values of variebles characterizing the operational mode , and by the positions of breakers, which allow to determine the topology of the system. Treated variables are the nodal voltages, generated and supplied power P,Q, power flows and currents in lines and transformers and other derived quantities. These quantitias are measured in the components of the power system and remotely transmitted to the control centre. In the course of meesuring and remote data transmission some failures and errors may occur. Violstions of quality end validity of the state information then follows and possible deterioration of control functions may take place. We in a) b) c) d)

can distinguiah four groups of arrors state information: random measurement errors model parameter errors gross measurement errors (bad data) model atructure errors , e.g. errors of network topology. 2115

Some variables in power system are not measured at all and therefore are not directly available to the control centre. Consequently one of important functions of control system is the processing of input informations aiming to the increase of their quality and velidity, and ensuring in this way the reliable and full data base of system state information. This function has been decompoaed into a number of tasks, which ara dapictad in Fig. 1. In the paper, only the methods for state eatimation and bad data detection and identification are fully deacribed. Tha other tasks ara charactarized without details. Computing nodal P,Q balances and comparing power flows at both ends of lines have been applied as plausibility check. Observsbility of the natwork in the caae of measurement loss i& tested by means of the observability parameters (Alvarez and Albertos, 1982), which are assigned to the individual maaaurementa as a by-product of the bad data identification using rN - test. Critical maasurementa are replaced by pseudomeasurements.

B. Sadecky

2 )) 6

METHOD OF STATE ESTIMATION In the state estimation problem the vector of complex nodal voltages is considered as the state vector of power system x •

(U 1 •••• Un'

f 2' ... f

n)

T

(1 )

Here Ui i8 the module of complex voltage in node i and ' i is the angle. Dimension of the state vector equals 2n-1. where n is number of nodes. One of the nodes (i.1) is chosen as reference node and O.

r1.

From existing methoda of state estimation the static WLS (weighted least squares) method has been chosen es the most suitable for Czechoslovak power system. regarding to its measurements system. This method makes it possible to employ any combination of different types of meesurements. The non linear model of measurements is applied : z • h(x) + v

(2)

where z is the vector of measurements (dim. m). v is the vector of random measurements errors. We suppose that random vector v is gaussian with zero mean and diagonal covariance matrix R. Values of the variances of individual meaaurements have been derived from the assum6d accuracy of measuring devices. For individual measurements (Pik.Qik flows. Pi,Qi injections. nodal voltages) the functions h(x) have the usual form. The method of weighted least squares minimizes the cost function: 1 H(x) • [z - h(X)]T R- [z - h(X)] (3) Solving of the problem proceeds iteratively. at each iteration the system of linear algebraic equations is formed and solved: T....

~ (Xk)R

-1

,.

,.

...

T,.

~(Xk)(xk+1-xk)·~ (Xk)R

-1 [

'" ~

z-h(Xk~

Here ~k denotes the state estimate at iteration k. (4)

is a ~acobian. The iteration process ends if all components of the difference ~k+1 - ~k are less in absolute value than a given & > O. Vector ~k is then applied

as the estimate of state vector. REALIZATION OF THE METHOD The state estimation program works according to the flowchart in Fig. 2. The Cholesky's method of triangularization is used for solving the system of linear algebraic equations (5)

Al:J"x • Y

Thia method utilizes the decomposition of the matriX A. A • LDLT

(6)

where L is s lower triangular matrix with 1 in diagonal. D is a diagonal matriX. The solution I:J.x is obtained by the back substitution in two steps: Lw • y

DL T A X

•

w

(7)

The program considers the sparsity of ~. A matrices. Only nonzero elements ara stored in the memory. In the course of triangularization of matrix A an increase of the number of non-zero elements occura. Neverthelass. many of them are very small and can be neglected. Elements with absolute value less then a chosen limit are omitted from matrix L. Sparsity of matrix L then conSiderably increases without marked influence on the accuracy. Nominal values of nodal volt ages Ui and zero angles fi were used as the estimata ~O at first iteration. Tha ~acobian ~ and the matrix A are computed only at tha baginning of the iterative procedure in no/'. minal values xO' The influence of this simplification on the results accuracy is negligible. the number of iterations has increased by 1-2 and the total computer run time has decreased considerably. BAD DATA DETECTION AND IDENTIFICATION The algorithm for detection and identification of gross measurement errors is indispensable to the full solution of powar system state estimation problem. Estimation procedure assumes random meesurement errors characterized by the covariance

Appli c atio n of Sta t e Es t imation Techniqu e

matrix R. but does not accept gross arrors beyond the model of random noise v. Bsd data detection is a statistical test giving answer to the quastion , whether in tha input informstion ere gross errors or not. The usual chi-square test of the cost function value has been used as detection test. If the detection is positive, the identificstion procedure must find corresponding bad measurements and provide correction of errors. For this purpose we use tha analysis of weighted rasiduals, which ara the random variablas

~r 11

(8)

Weighted residual r~i) is the difference between measured and estimated values of i-th component of measurament vector, divided by the squara root of the corresponding diagonal element of matrix R. It is known that statistical test (socalled rw-tast) can be used:

2 11 7

rement vector z, and a new estimation follows. In addition to the rw - test, slso more reliable (but time consuming) normelized residual test (r N - test) is exemined, simultaneously with the computation of the observability parameters. RESULTS OF TESTS The state estimation program with detection and identification of bad data has been tested on a digital computer simuletion of a power system. One part of the Czechoslovak power system was selected for theae tests, namely the 400 kV network. The estimated network consisted of 28 nodes and 31 linas. The number of measurements was different in individual examples. Two basic varianta are: Varient I - 176 meas., radundancy 3.2 Variant 11 - 139 meas., redundancy 2.53 Variant I corresponds to the maximal set of measurements , variant 11 takes into account about a 20 % failure of measurements.

(9)

Here N~/2 is the critical value of the gaussian distribution N(O,1). Meaaurements that do not fulfill the inequality (9), ere designated as suspicious. Due to the spreading effect we designate as suspicious not only the measuremente really containing gross errors, but alao some measurements in their naar neighbourhood. Therefore, it is necessary to analyse the set of suspicious measuraments and to determine in this set only those really containing gross errors. This analysis is depicted in Fig. 3. Suspicious measurements are at first ordered by the magnituda of I r~i)lend the resulting ordered list is then inspected in a descending order. The algorithm of Fig. 3 realizes in that way the elimination of dapendent errors from the list of suspicious data. It works under the assumption that the measurement with dependent error always has a smaller residual than the measurement with a real error from which the dependent error has originated. Aftar the analysis we obtain a smaller group of suspicious measurements. The whole group is then removed from the measu-

The measurement vector z was simuleted by the equation

z • h(x)

+

v + b

(10)

for the given topology of the estimated network. State vector x was obtained by the load flow solution from given nodal generations and demands of power. Vector of random measurement errors v was generated by the generator of pseudo random gaussian variables with zero mean and given variance G' 2 • In the course of simulations the group values of G" were used, namely (j"'. 10-15 MW for P, 8-10 MVAr for Q, 2 kV for U. Vector b in the equation (10) is the vector of gross measurement errors. Various examples differring one from snother by the number, alocation and magnitude of errors were tested. The magnitude of tha simulated errors was about 10 kV in U and 50-300 MW in P,Q. The estimation elgorithm revaaled very good stability and convergency properties. The number of iterations from nominal velues x till those satisfying the desired accuracy £'. 0.0001 (in relat-i ve values , i.e. corresponding to 0.04 kV) was

B. Sadecky

2 J J8

4-5. The computation of the ~acobian and the matrix A was performed only at the beginning of the iterative procedure in nominal valuea x , without significant decrease of accuracy. Ths astimation accuracy in individual examples is characterized by the magnitude of average estimation arror of state variables and individual typea of maasurad variables, see Tabla 1. The average estimation error in the group of variables was computed by the formula EST

6.

j

1

-Iij

(11)

EX i s where Azi is tha estimated va 1 ue, zi the exact value, nj is the number of values in the group. In addition to average estimation errors, also average measurement errors in the groupa are presented in Table 1 :

(12)

where zi is the meaaured value. Further, the maXimal estimation and measurement errors (MAX EST , MAXM ) in the groups are also given in Table 1. Examples in Table 1: A-without bad data, redundancy 3.2 B-without bad data, redundancy 2.53 C-with 13 gross meas.errors, redund. 3.2 o-after elimination of errors from example C, redundancy 3.0 We can see from Table 1 that the accuracy of monitored variables after estimation has considerably incraased in the situation without bad data. Tha estimation error is smaller than the maasurement arror. In the group of voltage measurements this decrease of error amounts as much as to 75 %, and in the other groups to 40-50 %. When comparing axamples A,B ~ find that the decrease of radundancy from 3.2 to 2.53 has had only a small influence on tha estimation accuracy.

The influence of gross measurement errora on the estimation resulta and the degree of error spreading was alao analysed on some examples. One of them is presented as C in Table 1. The accuracy of estimeted values considerably decreesed in the presence of bad data. The average estimation error of stste variables increased about three times. The error detection was ppsitive (value of cost function. 1241.5) and the identification rw - test determined the set of 20 suspicious measurements with the largest weighted residuals. The degree of error spreading was small, at most influencing the neighbouring node, The analysis of the set of suspicious values by algorithm of Fig. 3 gave then 11 erroneous measurements which were removed from vector z. These 11 values ware all the measurements really containing gross error. Two erroneous valuea were not identified and remained in the vector z. The bad data identification proved to be sufficiently effective also in other examples. The resulting estimation errors after the removing of the 11 erroneous values are alao prasented in Table 1 lexample 0/. They are nearly the same as in the example without bad data (A). In all simulated examples only one detection and residual identificetion with following new estimation was sufficient for the elimination of the absolute majority of bad data and attaining of desired estimation accuracy. CONCLUSION The simulated tests of the state estimation program with bad data detection and identification confirmed the efficiency and sufficient speed for its implementation in operational conditions of power system in Czechoslovakia. At present, this progrsm is modified for the implementation in the Czechoslovak dispatch control centre to the estimation of the 400 kV network.

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Application of State Estimation Technique

TABLE 1

Average and Maximal Errors of

for power system state estimation.

Estimates and Messurements

IEEE Trans. Power Apparatu. Systems, PAS-94 , 329-337.

Var. Variables

Schweppe, F.C., J.Wildes and D.Rom (1970).

group

Power system static state estimation,

state U i state ~i meas. Ui

A

B

1.54

3.91

meas. Pi 13.65 meas. Q 7.67 i meas. Pij 7.12 meas. Qij 5.10

50.72 26.85 30.53

Part 1,11,111. IEEE Trans. Power Apps-

0.28

ratus Sy.tems, PAS-89 , 120-135

25.57

4.36 4.55

8.85 12.54

2.95

10.50

state Ui

0.31

1.13

state

0.18

0.28

21.85

ifi

1.25

1.47

3.91

0.30

0.62

meas. Pi 14.04 mess. Q 7.06 i meas. P 7.52 ij meas. Qij 5.11

50.72

9.56

37.56

19.91

4.74

9.61

30.53

5.71

23.55

21.85

3.54

9.84

0.92

2.11

0.31 0.91

0.51 2.11

250.7 11.56 144.3 17.80

49.97

194.2

i

state U i state fi mess. Ui

3.06 10.63

meas. Pi 22.17 meas. Q 22.59 i meas. P 10.5 ij meas. Qij 12.3

Oats acquiaition from power system

Plausibility check

Network topology determination

57.66

8.46

29.64

150.8 11.07

44.83

0.26

1.07

state Ui state f'?i meas. U i meas. Pi meas. Q i meas. P ij meas. Qij

D

1.25

0.37 7.28

meas. U

C

0.37 0.16

0.13

0.25

0.25

1.07

7.52

25.35

4.58

11.22

4.70

15.09

2.96

10.09

Observability test

Specifications of models

State estimation

Bad data detection

no In the table the values are in kV (U), dagrees (f), MW (P), MVAr (Q). Bad date identification and correction

REFERENCES Alvarez, C. and Albertos,P. (1982). Online observability determination as a further result of state estimation

Computing of nonmeasured qusntities, storage into data base

algorithms. IEEE Trans. Power apparatus Systems, PAS-l0l, 767-774. Broussolle, F. (1978). State estimation in power systems: detecting bad data through the sparse inverse matrix method. IEEE Trans. Power Apparatus Sys~,

PAS-97 , 678-682.

Handschin, F.,F.C.Schweppe, J.Kohlas and A. Fiechter (1975). Bad data analysis

IWC4-0*

Fig. 1

Power system state determination

2 120

B. Sadecky

Computing of estimates Zs h(xs ) Jacobian J •

;~

I x.~

T -1 ,.. Vector Y_J R (z-zs)

Linear system solutio ~

A • AX

Y

New estimates

"x N

Fig. 2

"

.. Xs + 6)(

State estimation progrem

no

END

Go to next maas.

Fig. 3

Analysis of the list of suspicious measurements