Decoupled State-Estimation in Energy Control Centres

Copyright

© IFAC

Automation and

Instrume ntation for Puwer Plan ts,

Ball ~,t1()re.

India. 1986

DECOUPLED STATE-ESTIMATION IN ENERGY CONTROL CENTRES P. B. Subramanyam, G. S. Lakshminarayana and S. Parameswaran Celllra l Pawn R esearch Jllslilule, Bangalore 560 012

obtained from various redundancy rates, by considering active and reactive redundancy seperately obtained by uniformly distributing the measurements over the system is presented.

ABSTRACT

S TA TE-ES TIMA TION (SE) is the process of assigning a value to an unknown system state-variable based on measurements from that system according to some cri teria. Usuall y, the process involves imperfect measurements i;)herent in the process and ti , >refore estimating the syst e m state is based on a statistical c rite ria that estimate s the true va lue of the state-variable so as to minimize or maximize the selected c riteria. A commonly use d and fa mi liar criteria is that of minimizing the sum of squares of the di ffe rences between the estimated and 'true' (i.e. measured) va lues.

Line flows and bus injection measurements required for stateestimation studies, have been obtained from load-flow computations. Error (noise) terms (0 = 0.01 and 0.00 J) are added to these measurements. The error vector being simulated using a random number ge nerator. Convergence tolerance of 0.00 I PU and 0.00 I are used for V and <5, respect ive ly. Results obtained for various combinations of measurements; and with different order of processing these measurements have been presented. Measurements are classified as:

Power System Static State-Estimation (PSSE) is a proven tec hnique for pro vi ding modern ene rg y cont rol cent res with reliabl e rea l-tim e dat a bases fo r secu rity monitoring and on-line control. The state-estima tion processes real-time raw measurements to obtain optimal est imat es of voltage magnitudes and angles at a ll buses, which determines the static state of the Pow e r System (PS).

Type I, 2 Type 3, 4 Type 5

Diffe rent cases analysed are: i) Case I ii) Case 2 iii) Case 3 iv) Case 4

The estimation pr oblem of PSSE is usually formulated as a Among state estimator s nonlinear least-sq uare probl em . based on the nor mal equat ions approach, t he weighted least square (WLS) est im ato r provides reliable estimates but is slow, mainly du e to its requirement for updating a nd factorizat ion of the full-matrix in eve ry it era tion. The decoupled state-est imator requir es less computat ion time per iteration and less storage requi rement than WLS.

I, 2, 3, 4 &. 5 I, 3, 4 and 5 I, 2, 3 and 4 3 and 5 Type

Type measurements Type measurements Type measurements measurements

It is important that at least few voltage measurements (NBUS/IO to NBUS/5) must be included while estimating the states. When line-flows only are used as measurements, solution takes large number of iterations to converge.

The concept of p-<5, Q-V decoupling t echn ique as applied in load flow studies is ex tende d t o st ate es timation t echniqu e also which redu ces th e storage req uiremen t and also the time for conve rge nce.

Comparative result s obtained from 'method of equations' and 'Data equations' have bee n presented.

Normal

Illustrative examples considered are IEEE-14 bus system and 5-bus sytem.

The most common app roach to the SE problem is to formulate it as a nonlinear probl em usually solved by a n iterative procedure based on success ive lin ea ri zation; in each it e rati on one has t o solve a linear least-sq ua res problem. Assoc iat ed with WLS est im at ion is a set of redundant equa ti ons are to be solved to yield the so lution. These equations are 'Normal Equations' a nd a re solve d using Cholesky factorization. The normal equations a re ve ry satisfacto ry with respec t to preserving sparsity, but they may give rise t o numerica l ill-condit ioning. Methods using orth ogonal tr ansformation s, suc h as those of Househo lde r ha ve fa r bette r stability properties.

1. INTRODUCTION

In Power System Planning, when evaluating the state of the system, usually it is possible to start with fixed, predetermined values and therefore the conventional deterministic Load-Flow (LF) calculation is adequate. But in real-time Power System Control the information about the system has to be obtained from a set of measurements. Two problems arise -- the inacc uracies pr esent in the measurements, and the usual redundancy in the numbe r of measurements. Conventional Load Flow algorithms turnout to be not suited to over come these problems. On the other hand there is no possibility to make use of redundan cy and , the effects of errors in measure ments are not taken into account -- this leads to large deviations between the deterministic ally calculated and the true state of the system. Hence, the application of State-Estimation algorithms which takes certain statistical feature s of measurement inacc uracies as a base and calculates in a mathematical/statistical way an estimate of the Power System.

The Method of Data Equations which uses the Househo lde r orthogonal transformation is more efficient than the conventional method vi norm
Power System State-Estimation is a digital data processing algorithm for converting telemetered readings and topological information, including redundant measurements into reliable estimate of power system state-vector. The measurement and t e lemetering equipment in power system is inherently subjected to systematic and random errors. For minimising the e rror present in the measurements, the principle of The principle of 'estimation' 'ESTIMATION' is applied. serves well for this purpose. Once the states are known, all

In this paper the deco upled estimator based on tran sformation method is presented using diffe rent algorithms and the result s agree with the method of normal equat ions. Measurements are transfo rmed into new ' measu rements ' that are functions of the states and of the original measurements. In PSSE redundan cy rate is an import ant aspect.

Power injec tions Power flow s Voltage magnitudes

Estimator

39

40

P. B. Subramanyam, G. S. Lakshminarayana and S. Parameswaran

of the steady-state variables, including those, that are not telemetered or that are, for some reason missing can be readily calculated. The state-variable to be estimated are the voltage magnitudes and phase-angles at each of the N-buses of the system by representing available measurement equations. Since, phase angles are all relative to some fixed reference, the angle at the slack-bus is set equal to zero and the phase-angles at remaining buses are specified with reference to this. Therefore (2N - l) states are to be estimated. In literature various methods have been proposed for evaluating PSSE. The usual method employed to obtain optimal estimate from an over-determined system of equations is that of least-square or generalised inverse[20, 21]. State-Estimation based on these techniques have been proposed by number of authors[ I, 2, 3, 4]. However, the least-squares solution is only optimal when the measurement noise assumed is Gaussian. Basically most of these methods are based on Weighted Least-Square (WLS) approach. Since weighted sum of the squares of the residuals is minimised, these are termed as WLS estimators. When the noise (present inherently) in the measured data possesses certain statistical properties, the resulting estimates are known as unbiased and minimum variance estimates. The measurements in power system are nonlinear functions of the state variables. Due to non linear measurements, in the WLS approach an iterative process based on successive linearization is applied, so that reliable estimates can be obtained. The concept of state-estimation in power system using static model, was first suggested by Schweppe[l, 3]. The suitability of state-estimation techniques to real systems has been demonstrated by Larson et.a!.[4]. Schweppe and Messiello[14] considered a tracking state-estimator, assuming estimator updating time is less compared to the scanning time. Dopazo et.al.[5] proposed state-estimation algorithm based on lineflows only. Deb et.a!. [J 5] proposed sequential state-estimator. Debs and Lawson[13] proposed dynamic stateestimators. These algorithms are based on method of normal equations. The PSSE is formulated as a WLS problem. The problem is solved by the iterative normal equations method. The method of normal equations fails or leads to erroneous results in case of rank deficient or ill-conditioned situations. (Golub, 1965; Hanson and Lawson [22]). To over come the numerical instability Golub proposed the use of Householder orthogonal transformations to the estimation problem. The normal equations may occasionally become ill-conditioned . Alternative approaches based on orthogonal transformations that over come numerical ill-conditioning have been proposed to solve the WLS state-estimation. Rao and Lu[16] proposed method of data equations for sequential processing of measurements for estimation. Monticelli et.a!. [18] proposed a hybrid approach combining the sparsity of the normal equations method and the numerical robustness of the orthogonal transformation method. In this paper Singular Value Decomposition (SVD)[21,22] method is applied to solve state-estimation problem and compared with normal equations method. SVD method is efficient and specially suitable for large systems, based on transformation, which is less error prone and is more stable. The structure of the paper is as follows: Section 2 presents the possible combination of measurements for state-estimation and redundancy of measurements. In Section 3 the SVD algorithm is presented. Section 4 presents IEEE-14 bus and 5-bus system studies and compares results of SVD algorithm with normal equation method. The discussions and conclusions are then given in Sections 5 and 6 respectively. References and Appendices are given at the end. 2.

POWER SYSTEM MEASUREMENTS

The power system measurements are normally nonlinear functions of the states. [n order to apply SVD algorithm to PSSE the measurement equations need to be modified for

use by the algorithm. These measurements are telemetered to the central control station for the purpose of SE. Depending upon the way the data is received, and multiplexing facilities available, either all the measurements are available prior to the initiation of the estimator or the measurements are received sequentially for processing. Complex bus voltages at all buses constitute the statevariables of power system. These are to be estimated from the other variables that can be measured. For an N bus system with M line the following number of measurements can be obtained depending upon the meterplacement: SI.No.

~

No.of Measurements 2(N-l)

l.

Type I +Type2 at all buses

2.

Typel +Type2+Type5

2(N-l)+L

3.

Type3+Type4

4M

4.

Type3 +Type4 +Type5

4M+L

5.

Typel+Type2+Type3 +Type4

2M+2(N-l)

6.

Type 1+Type2 +Type3 +Type4 +Type5 2M+2(N-l)+L

7.

Type I +Type2 +Type3 +Type4 +Type5 4M +2(N-ll+L

2a. Redundancy of Measurements The complex voltages at all the buses in a power system are not available directly for measurement. These are to be ESTIMATED from other variables of the system, which can be measured (to be more precise, which can be metered). The power injections, power flow s and voltage magnitudes at certain buses of the system can be metered. Essential parameter of SE is the redundancy rate of the measurements. This is the ratio between the number of usable and the number of state-variables. This ratio should be greater than 2 and be obtained with measurements distributed uniformly over the system in order to be able to detect erroneous measurements. [n practice, the redundancy of the installed measurements should be little higher than 2, to take into account the various operation layouts used and to cover unavailabilities of transmission and telemetering equipment failures. Redundant measurements do contain useful information about the system. Though, increased redundancy improves the accuracy of the estimator, the solution may still be obtained with no redundancy. Redundancy rate for the installed measurements of the order of 2.75 has proved to be satisfactory. Since active and reac tive power are independent to a ce rtain degree, redundancy rate of 2.0 must be considered. An active redundancy rate is the ratio of the number of active power measurements to the number of nodes (N- D. Reactive redundancy rate is the ratio of the number of reactive power and voltage measurements to the number of nodes (N). 2b. Voltage Measurements Voltage measurements can be of a reasonable number (NBUS/lO to NBUS/5) and cor rect ly distributed over the network. These measurements furnish the general level of volt ages essential. 2c. Pseudo Measurements The state-estimation procedure fails completely when the availability of measureme nts goes on reducing because of telemetry failure or for any other reason. Under this condition, the network is said to be 'unobservable'. Large power system network will have missing data leading to an unobservable state. Under these conditions, procedure involving use of 'pseudo measurements' is incorporated, so

41

State-estimation in Energy Control Centres that the estimator continues to c1aculate the 'state'. Pseudo measurement, is used in the SE just as if it were an actual measured value. It is preferable to assign a large standard deviation (SO) to the pseudo measurements. So that, the large SO allows the estimator algorithm to treat the pseudo measurement as if it were a measurement from a very poor quality metering equipment.

n

Number of state variables,

P

Active power flow measurement,

KM : K Q KM QK: P

VK: 3.

STATE ESTIMATOR IN ENERGY CONTROL CENTRES

In modern energy control centres state-estimator plays a very important role [11, 12]. The system gets its information about the power system from remote terminal units that encode measurement transducer outputs and circuit breaker/switches status information into digital signals that are transmitted to the operations centre over communication circuits. This is needed as the breakers/switches in any sub-station can cause the network configuration to change. Also, the control centre can transmit control information such as lower/raise instructions to generators and open/close instructions to circuit breakers and switches. The electrical model of the power system and transmission system is sent to the SE program together with the analog measurements. The output of the SE consists of all bus voltage magnitudes and phase angles, transmission line flows, calculated from the bus voltage magnitude and phase angles and bus loads and generations. 3a. Oecoupled State-Estimator The concept of P-o, Q-V decoupling technique has been successfully applied in Load-Flow studies [18]. This is also extended to SE technique. This reduces the storage requirement and also the time for convergence. It can be recalled that the sensitivity relationship in a power system (PS) is that the phase angles are more sensi ti ve to the changes in real-powers. b) the voltage magnitudes are more sensitive to the changes in reactive power.

The general estimation consists of estimating the statevector [X] based on a set of observations [Z] in the presence of an error vector [W]. Mathematical model describing the functional relations between [X], [Z] and [W] is expressed in the form of a set of non-linear equations which relates the measurement [Z] and the true state-vector [X].

Z

[F(X)] + [W]

Z

[Z A ZR] Measurement Vector

ZA = [P ZR =

KM

P ] Active Measurement Vector K

[QKM Q

K

VK 1 Reactive Measurement Vector

Reactive power injection measurement, and Voltage magnitude measurement.

The reader is referred measurement equations. is given in Appendix-Ill.

to Appendix-II for the detailed The relevant computer algorithm

The next section presents application of the above algorithm to some standard IEEE-14 and 5 bus examples. 3b. System Studies Illustrative example considered is IEEE-14 bus system and 5 bus system. Line flows and bus injections required for state-estimation studies (as true measurements) have been obtained from load-flow computations. Error or noise terms (a = 0.0 I and 0.00 I) are added to these measurements. Measurements are used with random errors, using random number generating algorithm, so as to be representative of values drawn from a set of numbers having a normal probability density function with zero mean and variance for each measurement. Convergence tolerance of 0.001 PU and 0.00 I radians are used for voltage magnitudes and phase angles respectively. 3c. Effect of mix of Measurements The measurements may be classified into 3 different groups: Active and Reactive injections at all buses (Type I and Type 2), Active and Reactive line-flows at both ends of all the lines (Type 3 and Type 4), and Voltage magnitudes at all the buses (Type 5).

A) B)

a)

Therefore, the error in measurement of real power should have insignificant effect on estimation of voltage magnitude. Similarly the error in reactive power measurement should cause insignificant error in voltage angle estimation.

Active power injection measurement, Reactive power flow measurement,

C)

The effect of using different types of measurements in obtaining state-estimator is presented in Table 1. 4.

SIMULATION RESULTS

Table I Bus No.

Case

Voltage Case III Case II

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

1.06 1.051 1.018 1.056 1.088 1.036 1.062 1.042 1.053 1.045 1.04 1.037 1.033 1.024

1.06 1.0509 1.0174 1.055 1.0869 1.0356 1.0611 1.0415 1.0521 1.0446 1.039 1.03615 1.03215 1.02396

1.06 1.04483 1.0095 1.0676 1.0885 1.0147 1.058 1.0175 1.0531 1.0481 1.05437 1.05318 1.0465 1.0302

Case IV 1.06 1.04 1.01 1.07 1.09 1.0155 1.0605 1.0183 1.055 1.050 1.0565 1.0551 1.0137 0.9947

Case I Type 1,3 Type 2,4,5 Case II Type 1,3 Type 2,4 Case III Type 1,3 Type 4,5 Case IV Type 3 Type 5

where F(X)

Non-linear function relating quantities to the states,

the

measured

Z

Measurement vector of size (M x I),

X

State vector, formed by (N- J) bus voltage angles (the reference bus angle is known) and N bus voltage magnitudes,

M

Number of measurements,

AIPP- D

Measurements considered from Bus K -to- Bus M and Bus M -to- Bus K Bus No.

UOU

NORM

1. 2. 3. 4. 5.

1.04823 1.03715 1.02141 1.0254 1.06

1.04828 1.03721 1.02147 1.02545 1.06

42

P. B. Subramanyam, G. S. Lakshminarayana and S. Parames\\'aran

Measurements: Injections + Power flows from Bus K -toBus M, Bus M -to- Bus K + Voltage (Total 38 Meas)

Measurement: All injections power flows at lines 1, 2 (Type 1, 3, 5 --- >9) and (Type 2,4 --- / 7)

Bus No.

UDU

NORM

Bus No.

UDU

NORM

I.

1.04902 1.03667 1.02308 1.01983 1.06230

1.04905 1.0367 1.0231 1.01987 1.06234

I.

1.05799 1.03976 1.02514 1.0165 1.0594

1.05797 1.03974 1.02512 1.0164 1.05938

2. 3. 4. 5.

2. 3. 4. 5.

Solution converges with inclusion of minimum of ONE Voltage measurements

Measurements: Real flows from Bus K -to- Bus M Reactive: Bus M -to- Bus K + Voltage Bus No.

UDU

NORM

Bus No.

NORM

I.

1.03973 1.02745 1.0172 5 1.01001 1.05124

1.03962 1.02733 1.01713 1.00989 1.05113

1. 2. 3. 4. 5.

1.05137 1.03910 1.02554 1.0223 2 1.06469

2. 3. 4. 5.

Measurements: Power flows from Injections meas + Voltage

Bus

M -to- Bus

Bus No.

UDU

NORM

11. 2. 3.

1.04806 1.03615 1.026014 1.018405 ],05974

],04794 1.03604 ],02589 ],01828 ],05926

4.

5.

K +

Solution not converging without inclusion of voltage measurements Bus No.

NORM

].

],049166 ],47928 1.4586 ],45507 ],4998

2. 3. 4. 5.

The combination of (Type 1,2 + Type 3,4 + Type 5 and Type 3,4 + Type 5) converged in the same number of iterations. When a combinat ion of (Type 1,2 + Type 3,4) is tried, the equations have not converged . This leads to the hypothesis that the presence of bus injection measu rements without voltage magnitudes may lead to converge nce problems. When line-flows only are used as measurements the solution overshoots initially and takes a large number of iterations to converge. Order of Processing the Measurements In practice the availability of different types of measurements for estimation purpose will follow different sequence . The digital data processing algorithm must be capable of processing the measurements irrespe c tive of order, and give reliable estimator. The SVD algorithm serves well for this purpose. Results obtained by processing the same measurements in different order is tabulated in Table 2. Table 2

Line measurements from Bus M -to- Bus K Bus No.

UDU

NORM

1. 2. 3. 4. 5.

],04942 ],0429 ],03727 ],0333 ],0686

],04948 1.04296 ],03733 1.0333 1.0686

Measurements: Power flows from Bus K -to- Bus M + Injections + Voltage (Total 24 Meas) Bus No.

UDU

NORM

1. 2. 3. 4. 5.

1.05797 1.042 1.02495 1.02168 1.06576

1.05814 1.04217 1.02513 1.02186 1.066

Measurements: Real flows from Bus M -to- Bus K Reactive: Bus K - to- Bus M + Voltage Bus No.

UDU

NORM

1. 2. 3. 4. 5.

1.04790 1.03228 1.016012 ],01226 ],05632

1.04810 1.03243 ],01616 1.01242 1.05646

&5

Bus No . ].

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Type 1/5/3/4

Type 3/4/5/1

],06 1.044832 ],009515 ],067674 1.08853 1.01473 1.058967 1.017598 1. 053 134 1. 0482 1.05437 1.05318 1.04654 1. 030211

],06 ], 044832 ],009514 1.067675 1.088 53 1.014 73 1.058967 1.0 17598 1.053134 1. 0482 1.05437 1.05318 1.04654 1.030211

4a. Effect of Measurement Redundan c y In practice, high redundancy ratio is not realistic due to economic limitations. If the metering strategy is good enough, with a redundancy ratio between 1.2 to 1.6, the estimator can give an accurate estimate . (Assuming, that the system is not ill-conditioned). Fundamental assumption for successfu l performance of a state-estimation algorithm is that the measurement system should have a degree of redundancy greater than 1 (redundancy = no. of measu re ments/no. of states). Though the accu racy impr oves with higher redundancy it reflects on metering cost. Requirement of more sensible measurements and locat ions can therefore be decided considering the active and reactive redundancy rate s independently. Results of 5 bus system with various redundancy rates are tabulated in Table 3. Comparing different cases redundancy with 2.66 gives a reliable estimator.

State-estimation in Energy Control Centres

~ MEASUREMENTS REDUNDANCY-RATE Total Active Reactive Total Active Reactive 2.5 15 10 5 1.0 1.66 17 11 6 2.75 1.2 1.88 2.4 2.44 22 10 12 2.5 2.75 2.6 2.66 24 13 11 4.22 4.5 4.0 18 20 38

CASE I. 11. Ill. IV. V.

Bus No. I.

CASE

2. 3. 4. 5. CASE

11

I.

2. 3. 4. 5. CASE III

I.

2. 3. 4. 5. CASE

IV

I.

2. 3. 4. 5.

SVD

NORM

1.06 1.047 1.0249 1.02518 1.01726

1.06 1.05571 1.03581 1.03635 1.032

1.06 1.0462 1.0229 1.02275 1.01641

1.06 1.05578 1.036 1.0366 1.032

1.06 1.04666 1.0229 1.0228 1.01858

1.06 1.04865 1.02617 1.02595 1.020 I

1.06 1.04748 1.02476 1.02469 1.0186

1.06 1.04854 1.02615 1.0259 1.01955

5. DISCUSSION

43

information acquisition system and a data base, but also improvements may be acquired in the areas of improved economic operation, regulation etc. Since, SE is sensitive to modelling errors the results in real-time will be degraded, this sensitivity will provide a means for the detection and and identification of modelling errors. 6. CONCLUSIONS The SE based on least-squares approach minimizes weighted sum of the errors between measured and calculated system variables. The difference between the load-flow calculations and SE method is that, conventional LF calculations do not take into account, the inaccuracies and redundancy present in actual data. The SE uses the available telemetered measurements along with certain aprior information about the system to estimate the states. SSE is a digital data processing algorithm to obtain BEST estimate, which minimizes certain objective function, given a set of redundant measurements. It is important to obtain a reliable estimate even for much distributed measurement data, than an acc urate estimate of the state. State-estimator can detect the presence of bad-data and identify which data is in gross error. This basic property is the justification for including estimator in control centres. State-estimator as an integral part of the measurement system is one of the requirements for a computer based monitoring and control schemes of electric transmission systems using real time measurements. 7. ACKNOWLEDGEMENT The authors wish to thank Dr. M. Ramamoorty, Director General, Central Power Research Institute for his encouragement and guidance. 8. REFERENCES j) Scheppe, F .C. and Handschin, E., "Static state-estimation

The estimation problem of PSSE are usually formulated as a non-linear least-square problem. Among state-estimators based on the normal equation approach the WLS estimator exhibits excellent convergence reliability and filtering performance but fails to be satisfactorily fast, mainly due to its requirement for updating and factorization of the full-sized information matrix in each iteration. The decoupled estimators presented in this paper require less computation time per iteration and less computer storage than WLS. They however may suffer from slow convergence and biases either in heavy system loading conditions or in the presence of transmission lines with large R/X ratios. In the basic WLS algorithm, relinearization necessitates recalculation of all the large matrices. This may be accounted as a draw-back of the method, since it is difficult to predict how frequent relinearization is necessary. Based on the study carried out here the following general observations can be made: If line flow measurements are used, two different measurements can be obtained for each line connected to a particular bus. Bus injection measurements, however, provide a total of only two different measurements regardless of how many lines are connected to the bus. It is interesting to note that, voltage-magnitude measurement equation contains only one variable and therefore provides no coupling between two sets of equations (the measurement can always be grouped with the set that contains that particular voltage). Therefore, it is desirable to limit the number of buses at which measurements are taken. Line flow measurements appear to have economic advantages since they should have a higher information content to cost ratio. The 'line flow', 'tracking' and 'generalized' estimators are extensions of basic power flow methods. SE provides a suitable tool for system operations and will extend the operator's facility for observation. SE application not only provides a step toward security enhancement by providing a feasible means of complementing the

in electric power system", Proc. IEEE, July 1974, Vol.62, No.7, pp.972-982. 2) Handschin, L, "Real-time data process ing in electric power system". 3) Scheppe, F .C. et.al., "Power system static state-estimator", Parts 1,2 and 3; IEEE Trans. on PAS, Jan . 1970, pp.120-136. 4) Larson, R.E. et.al., "Static state-estimation in power systems", Parts I and 2; IEEE Trans. on PAS, Mar. 1970, pp.345-353. 5) Dopa zo, J. F. et.al., "Bus voltage and injection measurements in the aep state-estimator", PICA Proceedings, 1979 6) Gracia, A., Monticelli, A. and Abreu, P., "Fast-decoupled state-estimation and bad-data process", IEEE Trans. on PAS, Sept./Oct. 1979, Vol.98, pp.1645-1652. 7) Simoes, A. and Quintana, V.H., "A robust numerical technique for power system state-estimation", IEEE Trans. on PAS, Feb. 1981, Vo1.100, pp.691-698. 8) Simoea-Costa, A. and Quintana, V.H., "An orthogonal row· processing algorithm for power system sequential stateestimation", IEEE Trans. on PAS, Aug. 1981, VoI.IOO, pp.3791-3800. 9) Wang, J.W. and Qunitana, V.H ., "A decoupled orthogonal row processing algorithm for power system state-estimation", IEEE Trans. on PAS, Aug. 1984, pp.2337-2344. 10) Larson , R.E. et.al., "State-estimation in power systems", Parts I and 2; IEEE Trans. on PAS, Mar. 1970, pp.345-353. I J) Dy-Liacco, T.E., "Real time computer control of power systems", Proc. IEEE, Vol.62, No.7, pp.884-89 I. 12) Dy-Liacco, T.E., "State estimation in control centres", Int. Jr. Electric Power and Energy Systems, Vol.5, No.4, pp.218-221, Oct. 1983.

44

P . B . Subramanyam. G. S. Lakshminarayana and S. Parameswaran

13) Debs, A.S. and Larson, R.E., "A dynamic estimator for tracking the state of a power system", IEEE Trans. on PAS-90, pp.1025-1033. 14)Scheppe, F.e. and Masiello, R.D., "A tracking stateestimator", IEEE Trans. on PAS-90, pp.1025-1033.

In the above equations, VK and 0Km are magnitude and phase-angle at node K.

15) Debs, A.S., Larson, R.E. and Hajdu, L.P., "On line sequential state-estimation in power system", Proc. of 4th PSSC, France, 1972, pp.3.37.

P Km and QKm are the real and reactive power flows in the line joining nodes K and m.

16) Rao, N.D. and Lu, K.V., "An efficient algorithm for power system state-estimation", Proc. of Symposium on Computer Application in Large-Scale Power Systems, New Delhi, Sept. 1979, Vol.l, pp.81-88.

°

17) Stott, B. and Alsac, 0., "Fast decoupled load flow", IEEE Trans. on PAS-93, May-June 1974, pp.859-867. 18) Monticelli, A., Murari, C.A.F. and Wu, F.F., "A hybrid state-estimator: Solving normal equations by orthogonal transformations", IEEE Trans. on PAS-104, Dec. 1986, No.12, pp.3460- 3468. 19) Tony F. Chan, "An improved algorithm for computing the singular value decomposition", ACM Trans. on Math. Software, Mar. 1982, Vol.8, pp.72-83. 20) Nash (1969), Compact numerical algorithm for computing the singular value decomposition. 21) George, E. Forsythe, Michael A. Malcolm and Cleve B. Moler (1977), Computer methods for mathematical computations; Prentice Hall Series in Automatic Computation.

the bus

voltage

Km = IiK - om = the phase angle difference between nodes K and m

"K

is the set of branches connected to node K.

BKm

are

the

transfer

conductance

and

G and Km susceptance of

branch Km. By making substitutions for: 0Km + sin 0Km - Ii Km

sin 0Km

°

cos Km

1 + cos 0Km - 1

sin 0Km

0Km -

coso Km

=

1-

(O~m / 6)

(O~m

/ 2)

in equations (1 to 4) and rearranging the terms

22) Lawson, e.L. and Hanson, R.J., Solving least-squares problems; Prentice Hall, New Jersey. APPENDIX I SINGULAR VALUE DECOMPOSITION (S V D)

The over-determined system of equations of the form Ax = b are solved using singular value decomposition (SVD). SVD plays a useful role in analysing square, invertible matrices and its full power is realised in the analysis of non-square possibly rank-deficient matrices which arise in linear leastsquare problems. The objective of the algorithm is to find orthogonal U and V matrices, so that UT AV = E is diagonal. The individual components of E are "';j " I~ ~~. By proper choice of U and V It IS pOSSIble to make most of the cr ij zero. SVD of an (m x n) real matrix A is any factorization of the form A = U E V T, where U is (m x m) and V is (n x n) orthogonal matrices. E is (m x n) diagonal matrix, with cr .. = 0, if i f- j and cr .. = rJ. ~ 0 1)

11

I

Applying the SVD to the system equations Ax = b, where A is (m x n) real matriv., b is an m-vector and x is an n-vector; we get (U l: VT) X = b and hence l: z = d (diagonal equations), where z = VT x and d x vector is evaluated .

= UT b.

(V K Vm BKm)V K - (V K Vm BKm ) Vm 2 P Km + VKG Km - VK Vm G Km

From these equations

2 3 + VK Vm [G Km (oKm/ 2) + BKm (oKm/ 6)]

SVD method is more useful when least-squares solution has to be updated as new data is available. APPENDIX II

Expressions for real and rea can be written as:

ve power injections at bus K

2 P K = VK G KK + VK ~Ea Vm (G Km cos IiKm+B Km sinli Km ) K 2 ) Q K = -VK BKK + VKrii Vm (G sin Ii -B cos Ii Km Km Km Km EelK

a KK oK - ~E

The expressions for real and reactive line-flows in the line joining nodes K and m are given by:

Reactive Power Injection Measurement:

A-3-6

Equations (8) to (11) describe the power measurements in the required form as given by: Real Power Injection Measurement:

<1<

a Km om + V = P K + P~ + P~

45

State-estimation in Energy Control Centres Real-Power Flow Measurements:

Reactive Power-Flow Measurement:

Step 3 :

Read - No. of iterations, EPSI, EPS2, No. of measurements, Line No. from bus to bus, meas type measurement.

Step 4 :

Initialize state-estimator.

Step 5 :

Form coefficient matrices, A and B.

Step 6 :

Apply SVD on matrix A and obtain UA, VA and sigmal matrix. Apply SVD on matrix B and obtain UB, VB and sigma2 matrix.

Step 7 :

Start iteration count, ITN=O.

Step 8 :

Modify the measurements measurement vector.

Step 9 :

Solve for phase angles using UA, VA and sigmal and modified measurement vector. Solve for volt ages using UB, VB and sigma2 and modified measurement vector.

Step 10:

Test for convergence: If (iTN.GT.ITNMAX) I convergence occured. Pr int values and stop.

Step 11 :

If (iTN.GT.ITNMAX) YES

Voltage Measurement:

V~ where the coefficients are given by:

form

the

new

Solution not converged. Stop.

The modifications P ' , p " , Q ' , 0", P'Km' p " ,Q ' ,Q " K "K Km Km Km K K are given by:

P~

and

Step 12:

Increment iteration count, ITN

Step 13:

Go back to Step 8

ITN + I

P~ METHOD

Q~

V

E

B

Steps I to 5 are as in SVD algorithm.

K m~ cK

- 1:

mECt

V

m

[G

UDU From normal equations method.

(6

Step 6 :

Formulate [A] [A] ; [B] [B]

Km Km

[A] [ZP] ; [B] [ZQ] matrices.

K

V G Km - V Vm G Km K K

Step 7 :

Formulate Decoupled Real and Reactive matrices: [A A] [A ZP]; [B B] [B ZQ]

Step 8 :

a) b)

Step 9 :

a) b)

Factorise [A A] into UDU Factorisation. Factorise [B B] into UDU Factor isa t ion.

equivalent PI of the tr.Jine. The above equations are suitably arranged to apply SVD algorithm. APPENDIX III COMPUTER-ALGORITHM STEPS

METHOD REF. AUTHOR

SVD Solving least-squares problems Lawson, C.L. and Hanson, R.J.

Step I :

Read the system data. No. of buses, No. of lines and Line parameters.

Step 2 :

Formulate your matrix.

Step 10:

using Cholesky

Solve for phase angles using [UDU] = [A] [ZP] Solve for volt ages using [UDU] = [B] [ZQ]

(Solution BcK is the charging susceptance of the K-th leg of the

using Cholesky

uses

forward

Test for convergence.

substitution

procedure).