Calibration of power flow measurements

%Eirl!c

SYSTErnS REIFIRCH

Electric Power Systems Research 38 (1996) l-10

Calibration

of power flow measurements

Ahmad Fallaha I+, Edwin Cohen b, Ahmad Kayyali 2+>Alice Ming b b New Jersey

Institute

a Aleppo UnirersitJ’, Aleppo, Syria of Technology, 323 Martin Luther King Jr. Blud.. Newark,

NJ 07102,

G’SA

Received 29 March 1996: accepted 25 April 1996

Abstract Power systems are monitored by measuring the line power flows, bus voltage magnitudes and bus power injections. The real-time data are sent to various control centers and planning groups. So the reliability of a power system, the economy of its operation and the adequacy of its forecasting rcquirc accurate measurcmcnts. Yet these real-time remote measurements include errors caused by many factors, errors that affect the quality of state estimation solutions. A method is proposed to make up for the measurement errors and provide scale adjustments, that is, remotely calibrate real and reactive powers and voltage measurements. This approach requires information concerning the network configuration, the impedanceof all lines,and reliable data measurements at a few points of the system. The proposed algorithm consists of determining calibration constants for the

various measuredvaluesby usingalternatelybuspower balanceand line powerloss,starting from reliablepoints andpropagating the processthroughout the network. Finally, the measurements are corrected(calibrated)by meansof thosecalibration constants for all hours at which they were taken.

1. Introduction

State estimation plays a key role in providing inputs to the security control of an electric power system. These inputs are the online load flow calculation, node load forecasting, external network modeling, and limit checking of the power system [I]. Since the system operators perceive the state of the system through state estimation, it is important to feed the state estimator with measurementswhich are as accurate as possible from its standpoint. The meaning of accuracy in this caseis that data be free of systematic errors becausethe state estimator assumesthat the data are unbiased and are only subject to random errors of known standard deviation. Thus, only spurious errors are detected by the state estimator. Errors in measurements are broadly categorized as systematic errors and random errors. The systematic ’ Visiting Professor at New Jersey lnstltute of ‘fechnology supported by Aleppo University. ’ Visiting Professor at New Jersey Institute of Technology as Fullbright Scholar. 0378.7796i96/$15.00 8 1996 Elsevier Science S.A. All rights reserved PZI SO378-7796(96)01070-X

errors are those due to the inaccurate calibration of the measuring devices, such as zero errors and amplifier gain errors, while random errors are those that are unknown and impossible to rectify by means of service adjustments.

Systematic

errors

can be minimized

by a

well-administered maintenance plan involving field calibration. The economics of operation, however, sets a limit on the extent of such a plan, thus precluding perfect calibration at all times, no matter how well intended and designed.The techniques presented in this paper have as their goal to processthe measurementsin order to minimize the systematic errors, in essence performing remotely the ‘soft’ calibration of the measurements. This would certainly improve the performance of the state estimator. The study of remote measurementcalibration (RMC) is of interest. Many studies have been published in the last ten years. The work of Adibi and Stovall [2] presented the calibration as a correction based on measurementsof bus voltage magnitude and phaseangle. in addition to line real and reactive power flows. An improvement was produced later by Adibi and Kafka [3] where phase angle measurementwas dispensedwith

2

A. Fdlaha

et al. , Electric

Power

and bus summing was used for real and reactive power and kV equality at each bus. A nonlinear relationship was used between measured and calibrated values. Error analysis is described in many papers [4,5]. Errors are present in all measurements. They are combinations of random errors and systematic errors. The latter are caused by inaccurate modeling and scaling at the control center, nonlinearities of instruments in the measurement and the drift and deterioration of instruments with time, temperature and environment. These errors can be reduced by adjusting the measurements by zero-offset and gain coefficients, thus providing the degree of confidence in the results which enables closer and more effective control.

Systerm

Research

38 (1996)

1 -IO

lines between them. The impedance R + jX represents a line in the figure. The variables of interest for monitoring purposes are: (1) in each transmission line and transformer - the real P and reactive Q powers at each end (flow) (2) at each bus - voltage magnitude - P and Q powers These quantities are assumed to have been periodically measured, say every hour, over a certain period of time, say a day. 2.2. Mathematical

formulution

The linear model describing the relationship between the corrected and measured values of V, P, and Q can be written as follows:

2. Method of calibration The algorithm presented in the sequel for remote measurement calibration (RMC) requires reliable data measurement at a few points of the power system. This means that at those points field calibration and data transmission have to be checked much more frequently than what is considered routine for the rest of the system. The selection of the point locations depends on the system configuration and the minimum number found to produce acceptable results. The planner should then experiment, using the algorithm, with different sets of reliable points in order to arrive at what he considers to be an adequate set of reliable measurement points. The calibration at those points is then assumed to be perfect and in no need for correction. Measurements at those points are only subject to random errors. Using the measurements at the reliable points, the algorithm proceeds to correct (calibrate) the measurements at the other points of the system. The reliable points, therefore, serve as a reference for the power and voltage levels of the system without which no correction is possible. Indeed, reducing all measurements to nil satisfies perfectly all laws of conservation of energy. 2.1. Stutement of the problem The one-line diagram of a simple system is illustrated in Fig. 1. The system has two buses, K and L, with two

VCl = x1, + x2i vrn,

(1)

pci =

Y2iPmi

(2)

+Qmr

(3)

Qcz

=

Yli

zil+

+

V,,, P,, and Q,, are the calibrated (corrected) values of voltage, real power and reactive power, respectively, for measured values I’,,, P,, and Qml at measuring point i. (xlr, yil, zlr) and (x,;, yZir zZi) are the zero-offset and gain coefficients, respectively. What is most important now in this analysis is to find a relationship between calibrated values, measured values and network elements. Equality constraints are established which, when violated by the measurement data, give rise to errors which are to be minimized using basic nonlinear least-square techniques:

min z11 j=,,f [J;(x)l’

(4)

where H is the number of periodic (hourly) measurements. J;(X) is the ith component of the constraining function satisfying, broadly speaking, conservation of energy. The result must satisfy the reliable values assumed at some points and not be trivially zero. The reliable points have therefore the effect of setting the level at which conservation of energy is to be satisfied. It is worth noting that, from a practical point of view, P and Q are commonly measured as opposed to phase measurements or even current measurements. The system shown in Fig. 1 has seven measuring points: (1, 2, 3, 4) at bus K and (5, 6, 7) at bus L. Points 1 and 2 at bus K are selected as reliable in this illustration. 2.2.1. Power balance constraint

Fig.

1, One-line

diagram

of a simple

system

PC, + PC. + PC3 + PC4= 0

(5)

Q,, + Qc2 + Qc,+ Qc4= 0

(6)

v

Q

P

1 2 3 4 5 6 7 1 2 3 4 5 6 1 1 2 3 4 5 6 7

Table 1 Measurement

0.154 0.231 - 0.17 -0.15 0.167 0.155 -0.34 0.136 0.204 -0.15 -0.15 0.144 0.131 -0.3 0.99 0.99 0.962 0.953 0.952 0.94 0.941

1

data

0.182 0.373 - 0.20 -0.19 0.201 0.189 - 0.41 0.131 0.196 -0.15 -0.15 0.136 0.123 -0.28 0.98 0.98 0.952 0.944 0.942 0.93 0.931

2

0.338 0.507 -0.4 -0.38 0.39 0.376 -0.79 0.111 0.167 -0.12 -0.12 0.10 0.087 -0.21 0.97 0.97 0.942 0.934 0.931 0.92 0.921

3 0.414 0.621 -0.49 -0.47 0.482 0.467 -0.97 0.067 0.1 -0.07 -0.07 0.039 0.025 -0.09 0.99 0.99 0.962 0.953 0.957 0.945 0.946

4 0.224 0.336 -0.26 -0 .24 0.253 0.239 -0.51 0.129 0.193 -0.15 -0.14 0.131 0.199 -0.27 1.0 1.0 0.972 0.963 0.962 0.95 0.951

5 0.168 0.252 -0.19 -0.17 0.184 0.172 -0.37 0.179 0.269 -0.21 -0.21 0.194 0.183 -0.4 1.02 1.02 0.992 0.983 0.977 0.965 0.965

6 0.132 0.198 -0.14 -0.13 0.141 0.129 ~ 0.28 0.104 0.156 -0.11 -0.11 0.108 0.095 - 0.22 1.03 1.03 1 .oo 0.993 0.99 0.985 0.985

7 0.242 0.363 -0.28 -0.26 0.274 0.261 -0.55 0.133 0.199 -0.15 -0.15 0.136 0.123 - 0.28 1.04 1.04 1.01 1.0 1 .o 0.99 0.99

8 0.374 0.562 - 0.45 - 0.42 0.434 0.42 -0.88 0.148 0.222 -0.17 -0.17 0.139 0.126 -0.29 0.989 0.98 0.952 0.944 0.937 0.925 0.926

9 0.379 0.569 -0.45 -0.43 0.44 0.426 -0.89 0.145 0.218 -0.17 -0.16 0.135 0.123 -0.28 0.97 0.97 0.942 0.934 0.927 0.915 0.916

10 0.43 0.646 -0.52 - 0.49 0.502 0.486 - 1.0 0.163 0.245 -0.19 -0.19 0.149 0.137 -0.31 0.968 0.968 0.94 0.942 0.923 0.911 0.911

I1 0.286 0.429 -0.33 -0.32 0.327 0.313 -0.66 0.12 0.18 -0.13 -0.13 0.116 0.103 -0.24 0.99 0.99 0.962 0.953 0.952 0.94 0.941

12

0.286 0.429 -0.33 -0.32 0.327 0.313 -0.66 0.12 0.18 -0.13 -0.13 0.116 0.103 -0.23 0.99 0.99 0.962 0.953 0.952 0.94 0.941

13

0.157 0.236 -0.17 -0.16 0.171 0.159 -0.34 0.137 0.206 -0.16 -0.15 0.145 0.133 -0.3 1.0 1.0 0.972 0.963 0.62 0.95 0.951

14

0.509 0.763 - 0.61 -0.59 0.596 0.579 -1.2 0.172 0.258 -0.2 -0.2 0.148 0.135 -0.31 0.97 0.97 0.942 0.934 0.922 0.911 0.912

15

4

4. Fullnhu

Table 2 Actual and calibrated

P and Q at locations

cv a/. ‘Electric

Power

Swemr

Reseur-th

38 (I 996) 1~ IO

3 and 3

Hour

1

2

3

4

5

6

7

8

9

IO

II

12

13

14

15

P d3 ;:

0.193 0.17 0.20

0.228 0.164 0.235

0.422 0.139 0.432

0.517 0.84 0.528

0.28 0.161 0.288

0.21 0.225 0.217

0.165 0.13 0.172

0.302 0.166 0.31

0.468 0.185 0.478

0.474 0.182 0.485

0.538 11.204 0.549

0.357 0.15 0.366

0.357 0.15 0.366

0.196 0.172 0.20

0.636 0.215 0.648

Qei

0.171 0.193 0.186

0.16 0.228 0.22 0.164 0.16

0.14 0.422 0.413 0.139 0.138

0.08 0.517 0.507 0.084 0.083

0.16 0.28 0.272 0.161 0.159

0.22 0.21 0.203 0.225 0.222

0.13 0.165 0.159 0.13 0.129

0.16 0.302 0.294 0.166 0.164

0.18 0.468 0.458 0.185 0.183

0.18 0.474 0.464 0.183 0.18

0.2 0.538 0.527 0.20 0.202

0.15 0.357 0.348 0.150 0.148

0 I5 0 357 0.348 0.150 0.148

0.17 0.196 0.189 0.172 0.170

0.21 0.636 0.623 0.215 0.213

p,‘l P,,

where P and Q are taken as positive if entering the bus and negative if leaving it. The vanishing summations are hardly possible, however, and can be rewritten in coefficient form as follows: PC, + PC2 +

Yl3

Qc,+ Qcc,+z,,+

+

Ydm3

+

~3Qm3 +

Yl4 214

+ 1’24Pm4= +A-

(7)

+ =sQmz,= c<,/<

C-3)

In general,

j=

I

,f,Qc&f+

j=R-1

,(=,,+=z,Qm,)

=

CA

(10)

R is the number of reliable points at bus K; M is the total number of measured points at the same bus; E,,~ and C+ are the errors resulting from the mismatches of the corrected powers at bus K. The second power constraint will be the difference between the measured and corrected values for each measured point. Its purpose is to prevent the corrected value from straying too far from the measured value. Eq. (9) may be minimized, for instance, by having yr, =y?, = 0 for some j. This is unacceptable. So, in addition, at bus K:

The errors for real power at bus K in the figure, using the power balance constraint, are then cPn-,6P3,and cP4. The power balance objective function to be minimized involves the power balance constraints described above for all hours of the period of measurements, for example 24 hours. The total number of measurements is MH over that period. The objective, then, at bus K is

(13j By way of example, the minimization in Eqs. (12) and (13) was carried out over I5 hours for bus K in the simple system of Fig. 1. The measurement data used are given in Table 1 in per unit for the seven measurement points. They were obtained from correct load flow solutions (called here actual values), modified systematically by means of chosen calibration constants. Those calibration constants were kept the same over the hours for each measurement location, but differed from one location to the other. The subscripts a and m are used in the tables to denote actual and measured, respectively. Since points 1 and 2 are reliable points, the aforementioned least-square error minimization produced ~7,~~Y,~. ~~~~ yZ4, z,~, z~, I~?, and --24. This led to corrected values for P and Q at points 3 and 4, as shown in Table 2. Once corrected. points 3 and 4 become reliable points. 2.2.2. Power loss constraint The relationship between the corrected and measured values can be enhanced if the power losses in and the voltage drop across the impedance of the line are accounted for. Since the calibrated values of P and Q at one end of line 1 have been calculated and the voltage at bus K is known, the current at both ends of the line can be calculated. Therefore the loss of power in the impedance due to the real and reactive line flows between points 3 and 5 (Fig. 1) is given by

(z,2+ I,’ )(R + X) =

P,j2 +

Qc3

Vd2

(12) The same equation power as

can be written

for the reactive

= 2h

+

Pc52

+

Qcs2

VCS2

(R + X) i

- IP,,ll + 211Q,,l- IQcsll

Eq. (14) can be rewritten

as

(14)

A. Falluha Table 3 Actual and calibrated

P, Q and

L’ at locations

et al. ‘Electric,

Power

Systems

Re.reavch

18 (1996)

l-10

5 and 6

Hour 1

2

3

4

5

6

I

8

9

10

II

12

I3

14

15

P nS PC, Q,j Q,.5 1’a5 PI: V

0.192 0.194 0.163 0.168 0.97 0.975 0.192

0.227 0.228 0.155 0.16 0.96 0.227 0.965

0.419 0.424 0.118 0.122 0.95 0.956 0.419

0.513 0.519 0.056 0.057 0.975 0.979 0.513

0.279 0.282 0.15 0.155 0.98 0.279 0.984

0.209 0.212 0.215 0.220 0.995 0.998 0.209

0.164 0.167 0.126 0.131 I.01 0.164 1.01

0.300 0.302 0.155 0.16 1.02 0.300 I .02

0.464 0.467 0.158 0.162 0.955 0.464 0.961

0.47 0.476 0.154 0.159 0.945 0.952 0.47

0.533 0.538 0.169 0.174 0.94 0.533 0.497

0.355 0.359 0.134 0.139 0.97 0.975 0.355

0.355 0.359 0.134 0.139 0.97 0.355 0.975

0.195 0.197 0.165 0.17 0.98 0.195 0.984

0.629 0.636 0.167 0.171 0.94 0.629 0.947

P ,.6

0.19

0.22

0.41 I

0.504

0.272

0.203

0.161

0.293

0.455

0.462

0.523

0.347

0.347

0.191

0.621

E”” “Z,” l’c, VL

::;;; 0.97 0.974 0.975

0.153 0.155 0.96 0.65 0.965

0.118 0.117 0.95 0.956 0.956

0.055 0.056 0.975 0.979 0.979

0.149 0.15 0.98 0.984 0.984

0.225 0.213 0.995 0.998 0.998

0.126 0.125 1.01 1.01 1.01

0.155 0.153 1.02 1.02 1.02

0.156 0.158 0.955 0.96 0.961

0.154 0.152 0.945 0.95 1 0.951

0.167 0.19 0.94 0.946 0.947

0.133 0.134 0.97 0.974 0.975

0.133 0.134 0.97 0.974 0.975

0.163 0.165 0.98 0.984 0.84

0.165 0.167 0.94 0.946 0.947

1

pcx2+ L2 + CplS+ Y*5p~5)2+ tz15+ 3*5Q~5)2 (x,5 + x2j Vm512 v,3*

x(R+X) =&

(15)

L

This error is now supplemented with the voltage error obtained from the voltage drop constraint. 2.2.3. Voltage drop constraint The relationship between the voltage at bus i (which has been corrected) and the voltage at busj (which is to be corrected) is found (see Appendix A) to be V*qf +

I
+ vc,B, = O

The corresponding

(16)

error is given by

Ey = Vmj + I,,A, + vciB,

(17)

where

Minimization of Eq. (19) over H hourly measurements using the least-square method (Eq.(4)) produces the gain constants (x,,, y2,, zZi) and zero offsets (x,,, .?I j2 zl j ) of point j. This is done for every line that ends with a reliable/corrected point at one end and an uncorrected point at the other. This is the way that the algorithm reaches out to new buses where the power balance constraint is next applied. For the simple system of Fig. 1 the minimization procedure produces the gain constants (x2,, yZ5, Zig) and zero offsets (x,,, y15, zlj) of point 5. The same is also done for point 6 starting with the corrected values of point 4, thus exhausting all possible power-loss/voltage-drop phases of the algorithm at this stage. Table 3 shows these values. The algorithm proceeds with a power balance constraint phase for a bus displaying reliable/corrected values at a few, but not all, of its measurement points. For the simple system, that would be bus L, and P and Q are corrected at point 7. The results are shown in Table 4. V, is the average of the two corrected voltages, namely, Vc5 and V,,.

RPmj+ XQmj AJ = (P,i’ z--= V’ci’+ !I

+ t&y)‘:2 2.3. Solution algorithm

Qw2 )I,? VU

Pci Prni + Qc;Qm, Bi = (pci2 + Qci2 )I:* (P,’ + (2,’

)‘:2

For the line between points 3 and 5 in the simple system of Fig. 1, substituting A,, B,, and ZXj in Eq. (17) gives %5

=x,,+x25vm5+

(P,,’ + Qc3’Y A + Vc,B 5

5

(18)

V,;

The total error to be minimized E.I* = ELj2+ &,y2

then takes the form (19)

The measurement calibration algorithm is now described. It requires at the outset the specification of reliable points and the numbering of all the measurement points. Step 1. Determine the position of all reliable data measurements at each bus of the power system. Step 2. For every line terminated with a single reliable point, carry out a power-loss/voltage-drop constraint minimization using Eq. (19). The outcome of such a minimization gives the calibration constraints of the real power, the reactive power and the voltage at the other end of the line, i.e. the uncorrected point.

.A. Fullaha

6 Table 4 Actual and calibrated

P and Q at location

er al. : Electrk

Pmcer

Systems

38 (I 996)

Research

I- 10

7

Hour

pa7 PC7

QU Qc7

1

2

3

4

5

6

I

8

9

10

I1

12

13

14

15

0.384 0.381 0.327 0.327

0.453 0.45 0.311 0.31 I

0.839 0.835 0.236 0.237

1.02 1.02 0.112 0.112

0.558 0.554 0.301 0.301

0.418 0.415 0.431 0.43 1

0.329 0.326 0.253 0.253

0.601 0.598 0.31 0.31

0.928 0.925 0.317 0.317

0.941 0.937 0.309 0.309

1.06 1.06 0.338 0.338

0.71 0.706 0.269 0.269

0.71 0.706 0.269 0.269

0.3911 0.388 0.33 0.33

1.25 1.25 0.335 0.335

Those constants are used to correct (calibrate) the measured values at the uncorrected point using Eqs. (l)-(3). This step is performed for all similar lines before proceeding to Step 3. Step 3. Select the bus with the maximum number of reliable/calibrated points, say bus i. Carry out a power balance constraint error minimization (Eqs. (12) and (13)) involving all points at bus i over all hourly measurements. The outcome of this procedure gives the calibration constants which are used to correct (calibrate) the power and reactive power measurements, using Eqs. (l)-(3), at all but the reliable points associated with bus i. Step 4. Identify all lines connected to bus i which have an uncorrected point at their other end. Carry out a power-loss/voltage-drop constraint minimization, using Eq. (19) considering the previously corrected end as reliable. The outcome of such a minimization gives the calibration constraints of the real power, the reactive power and the voltage at the other end of the line, i.e. the uncorrected point. Those constants are used to correct (calibrate) the measured values at the uncorrected point using Eqs. (l)-(3). This step is performed for all similar lines before proceeding to Step 5. Step 5. If all measurement points have been corrected, stop. Otherwise, go to Step 3.

3. Numerical

3.1. Description

3.2. Measurement

datu

Load flow solutions were obtained for a variety of loading conditions in order to generate hourly measurements. These are referred to as actual values. Systematic errors of gain and zero offset were then introduced at all but a few points in order to generate the uncalibrated data. The few points which were left unchanged were considered reliable points. The modified data and the reliable point data became the measurements. At the same time, random errors were added to those measurements. Thus, the measurement data are derived from the actual data with both random and systematic errors. 3.3. Discussion

of the results

The proposed algorithm was applied extensively to the system of Fig. 2 and was found to be satisfactory, in the sense that the resulting calibrated values came out close to the actual values. Only a few of these cases

results

of the system

The proposed method was tested on a nine-bus system (26 kV/4 kV) which consists of a main generator at bus 1, nine transformers, nine lines and three reactors. The system also has four loads at buses 4, 5, 7, and 8, as well as capacitors or reactive power generators. The system is depicted in Fig. 2 which also shows 52 measurement points. These measurements include line real and reactive power flows and bus voltage. The power measurements consist of injection powers at some points, such as loads, and power flows at both ends of each line of the system.

Fig. 2. Test system

A. Fulluhu Table 5 Actual, measured Hour

A

M

Cl

G

c,

cl

Mr

cr

3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9

and corrected

Meaurement

power

et al. : Electric

Power

Systems

Research

38 (1996)

I- 10

P f x 10 ‘)

point

5

7

9

10

14

17

20

23

25

30

35

39

42

-3.86 -4.96 ~ 15.6 -3.57 -4.65 - IS.2 -3.91 - 5.0 - 15.6 -3.19 -4.90 - 15.6 - 3.89 -4.99 - 15.6 -3.82 -4.89 -15.3 -3.55 -4.63 - 15.0 -3.81 -4.89 - 15.4

- 1.28 -2.78 -5.87 - 0.902 -2.36 -5.36 - 1.25 -2.72 -5.73 -1.27 -2.16 -5.85 -1.3 -2.80 -5.89 -1.19 -2.66 -5.13 - 0.897 - 2.34 - 5.33 - 1.19 - 2.66 -5.67

-0.95 -2.21 -0.518 -0.753 -2.01 -5.0 -0.94Y -2.21 -5.18 -1.01 -2.27 - 5.25 -0.989 -2.23 -5.15 - 1.05 -2.32 -5.29 - 0.748 ~ 2.00 -4.98 - 1.05 -2.32 -5.30

2.52 3.18 8.05 2.174 2.82 7.59 2.53 3.185 8.01 2.53 3.18 8.01 2.529 3.185 8.019 2.53 3.184 8.016 2.163 2.809 7.51 2.53 3.18 7.98

-2.28 -3.18 -8.13 -2.28 -3.18 ~ 8.73 - 2.37 ~ 3.31 -9.1 ~ 2.29 -3.24 ~ 9.06 -2.28 -3.18 -8.73 -2.28 -3.18 -8.73 -2.28 -3.18 ~ 8.73 ~ 2.28 -3.18 -8.73

-3.05 - 3.67 -10.4 - 2.78 -3.39 -10.0 - 3.05 -3.66 ~ 10.3 -2.97 -3.58 - 10.3 - 3.05 -0.366 - 10.3 - 3.03 - 3.66 - 10.4 - 2.76 - 3.37 - 10.0 - 3.03 -3.66 -10.4

0.95 2.2 5.16 0.731 1.95 4.83 0.945 2.24 5.33 1.06 2.31 5.26 I .03 2.21 5.21 1.117 2.351 5.212 0.727 1.941 4.821 1.112 2.32 5.23

~ 1.07 -2.41 - 5.4 -0.854 -2.20 -5.2 -0.99 -2.37 - 5.45 -1.05 - 2.41 -5.45 - 1.07 - 2.41 -5.4 -1.13 -2.47 ~ 5.47 -0.849 -2.19 -5.18 -1.12 -2.48 -5.41

1.07 2.4 5.31 0.831 2.14 5.08 1.01 2.33 5.271 1.056 2.364 5.286 1.051 2.36 5.3 1.051 2.366 5.301 0.8267 2.138 5.070 1.07 2.34 5.16

3.04 3.14 10.37 3.85 3.5 10.06 2.93 3.64 10.3 3.038 3.142 10.4 2.93 3.64 10.27 2.29 3.64 10.4 2.195 3.488 10.03 3.036 3.742 10.39

2.32 2.6 8.8 2.019 2.9 8.28 2.36 3.31 9.1 2.29 3.25 9.08 2.27 3.18 8.72 2.278 3.184 8.714 2.007 2.887 8.254 2.309 3.18 8.75

0.73 1.34 2.98 0.532 1.14 2.79 0.73 1.35 3.03 0.725 1.34 2.99 0.474 1.37 3.047 0.0667 I .300 3.004 0.5289 1.139 2.780 0.7255 1.31 2.988

2.2 2.6 1.4 1.825 2.213 6.87 2.41 2.81 1.63 2.202 2.602 7.405 2.22 2.63 7.37 2.18 2.576 7.32 1.814 2.202 6.84 2.18 2.57 7.33

are reported here, at some of the points and for some of the hours. The cases differ in the number and location of the reliable points. Many cases were studied where measurements included systematic errors alone and others which comprised in addition random errors of Y/o, 3%, and 2%. Five cases are illustrated in Table 5 for real power, Table 6 for reactive power, and Table 7 for voltage. The measurement points are (5, 7, 9, 10, 14, 17, 20, 23, 25, 30, 35, 39, and 42). In the first column, A stands for actual, M for measured and C for calibrated, each comprising three rows corresponding to the third, sixth and ninth hours. Under C,-C, are four cases differing in reliable points and characterized by the existence of systematic errors alone. These are followed by a case including random errors with its measurement set M, followed by the calibrated points C, resulting from the algorithm. In case C, the reliable points are located at points 1, 2, 3, 8, 9, 27, 32, and 33. In case C, the reliable points are located at 1, 2, 3, 16, 19, 22, 32, 36, and 40. Inspection of these tables shows that case C, displays calibrated values of P, Q and V which are much closer to the actual values than the calibrated values observed in the first case, although both are satisfactory. With a judicious choice of reliable

points, one can select the most satisfactory locations resulting in calibrated values being very close to actual values. The reliable points of case C, are 1, 2, 3, 4, 15, 16, 18, 19, 27, 32, 33, 36, and 40; those of case C, are 1, 2, 3, 13, 14, 22, 32, 37, and 40. Finally, the last case, C,, with a random error of 5% and reliable points identical to those of case C, above, somehow produced the best results. Thus, the algorithm was capable of minimizing the effects of systematic errors, even in the presence of random errors.

4. Conclusions Experience with this technique of calibration that mitigates the effects of systematic errors has been quite satisfactory. In actual practice the system planner would conduct studies in a similar manner to the procedure tried in this paper in order to arrive at the best location and number of reliable points. In other words, a set of load flow solutions and systematic modification of their readings is all that is required for that purpose. Once a set of reliable points are chosen, the planner would require that those points in the system be calibrated more frequently and regularly than all the other measurement points.

3 6 Y 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9

Hour

Table 6 Actual. measured

2.84 3.32 0.19 2.769 3.24 0.171 2.83 3.28 0.36 2.86 3.32 0.26 0.82 3.27 0.37 2.153 3.224 0.17 2.753 3.224 0.170 2.86 3.33 0.18

5

Measurement

and corrected

0.98 0.8 -2.64 0.95 0.774 -2.51 1.03 0.84 -2.67 0.971 0.77 -2.51 1.01 0.82 -2.7 1.02 0.83 -2.67 0.945 0.77 -2.56 1.002 0.82 -2.61

7

point

power

0.97 0.99 -1.5 0.96 0.98 - 1.491 0.97 0.99 - 1.5 0.96 I .06 - 1.6 0.99 1.02 -1.57 1.02 1 .O4 - 1.59 0.955 0.976 -1.48 1.001 1.03 -1.53

9

Q ( x IO-‘)

-0.74 -0.79 2.73 -0.714 -0.764 2.67 -0.730 -0.782 2.807 -0.731 -0.78 2.8 -0.73 -0.78 2.8 -0.73 -0.782 2.808 -0.711 -0.761 2.656 -0.73 -0.78 2.77

10 2.31 2.59 -0.25 2.31 2.59 -0.24 2.28 2.56 -0.18 2.26 2.54 -0.22 2.31 2.59 -0.25 2.26 2.551 ~ 0.337 2.31 2.59 -0.25 2.31 2.59 -0.25

14 0.21 0.37 -2.06 0.18Y 3.44 - 1.9s 0.203 0.366 - 2.06 0.198 0.36 -2.07 0.188 0.352 -2.1 0.184 0.348 -2.10 0.18 0.34 - 1.97 0.201 0.358 -2.01

17 -0.97 ~~ 1.0 1.48 -0.93 -0.959 1.42 -0.96 -0.99 1.5 -1.02 - 1.03 I.63 -0.98 ~ I .02 1.57 ~ 1.01 - 1.05 1.6 -0.92 -0.954 1.42 ~ 1.0 -1.03 1.5

20 ~ 0.97 0.93 - 1.9 - 0.934 0.89 ~ 1.84 -0.95 0.92 - 1.89 -0.97 0.93 - 1.9 -0.97 0.93 ~ 1.9 -0.973 0.931 - 1.93 ~ 0.928 0.89 ~ 1.83 -0.96 0.92 - 1.89

23 -0.99 - 1.0 1.53 -0.946 -0.95 1.46 -0.96 -0.97 1.6 -0.97 -0.98 1.6 -0.Y67 -0.Y77 1.601 - 0.967 - 0.977 1.6 -0.94 -0.95 1.465 -0.96 -0.97 1.6

25

--

-0.24 -0.41 I .64 - 0.201 --0.36 1.56 -0.24 -0.41 1.65 -0.24 -0.415 1.63 -0.226 - 0.404 1.67 -2.4 -0.415 1.63 -0.2 -0.365 1.55 - 0.236 - 0.409 1.605

30

-2.32 -2.6 0.18 -2.3 -2.54 0.158 -2.3 -2.58 0.13 -2.29 -2.56 0.148 -2.33 -2.61 0.128 -2.28 -2.57 0.224 -2.2 -2.5 0.15 -2.32 -2.61 0.16

35

-0.94 -- 1.08 0.73 -0.93 - 1.07 0.72 -0.975 -1.1 0.83 -0.9 -1.08 0.75 -0.97 -1.12 0.77 -0.96 -1.10 0.77 -0.92 - 1.06 0.7 -0.94 -1.08 0.72

39

-1.5 0.02 -1.40 -1.58 0.0187

-1.31 - 1.44 0.02 -1.3

- 1.55 0.02 - 1.39 ~ 1.56 0.01

~ 1.42 - 1.57 0.02 -I .41

- 1.56 0.0295

~ I.62 0.0 - 1.39

I 2

2 3 2 2 3 2 2 % 9

22 0 $

2 5

“a ‘.__ 5 s ;..

Table I Actual. measured Hour

A

M

c,

C?

c,

c4

M,.

c,

and corrected Measurement

3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9 3 6 9

I; point

5

I

Y

10

14

17

20

23

25

30

35

39

42

1.04 1.04 1.04 1.(I23 1.023 1.023 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.039 1.039 1.039 1.016 1.018 1.019 1.039 1.039 1.04

1.04 1.04 1.04 I.006 1.007 1.006 1.04 1.40 1.04 1.04 1.04 1.04 1.04 1.04 1.04 1.039 1.039 1.039 1.009 1.002 1.003 1.039 1.039 1.04

1.04 1.04 1.04 I.011 I.011 1.011 1.04 1.04 1.04 1.04 I .04 1.028 1.04 1.04 1.04 1.03Y 1.039 I .03Y 1.005 1.007 1.008 I.039 1.039 1.04

I.04 1.039 I .029 1.007 1.002 0.992 I.039 1.038 1.028 1.039 1.038 1.028 1.039 1.038 1.028 I.039 1.ois 1.0’8 0.99 0.997 0.988 1.038 1.038 I.029

1.04 I.039 1.029 1.04 1.039 1.029 1.039 I.038 I.028 1.039 1.038 1.028 1.039 I .038 1.028 1.039 1.038 I.028 1.04 1.039 I.029 1.038 1.038 1.029

1.04 1.039 1.029 1.023 1.002 1.012 1.039 1.038 1.028 1.039 1.038 1.032 1.039 1.038 1.028 1.039 1.038 1.028 1.016 1.017 I .008 I.038 1.038 1.029

1.04 1.039 1.035 0.995 0.994 O.Y9S 1.036 1.035 1.031 1.036 1.036 1.032 1.038 1.038 1.033 1.038 1.037 I .034 0.98 1 0.989 0.986 1.037 1.036 1.033

1.04 I.039 1.035 1.023 I.022 1.018 I.037 1.036 1.03 1.037 1.036 1.004 1.038 1.038 1.033 1.038 1.037 I .034 I.016 1.017 1.014 1.037 1.036 1.033

1.05 I.047 1.009 0.99 0.987 0.949 I .048 1.048 1.007 1.046 1.043 1.004 1.048 1.045 1.007 1.048 1.045 1.007 0.98 0.982 0.94 1.048 1.044 1.007

1.05 1.05 1.029 1.004 1.005 0.984 1.046 I .045 1.023 1.046 1.046 1.024 1.046 1.045 I.024 1.045 1.045 1.022 0.99 1.00 0.98 1.045 1.044 1.023

1.04 1.039 I .02 I .023 1.022 1.003 1.037 1.037 1.016 1.038 1.037 1.018 1.037 1.037 1.108 1.038 1.037 1.018 I.016 1.017 0.99 1.037 1.038 1.02

1.05 1.05 1.022 1.021 I.021 0.994 1.046 1.046 1.019 1.047 1.047 1.021 1.047 1.047 1.021 I.048 1.048 I.021 1.015 1.016 0.99 I .048 1.048 1.022

1.05 1.05 1.031 I.19 1.019 1 .oo 1.047 1.046 1.026 1.046 1.046 1.025 1.047 1.047 1.02 I.045 1.045 1.009 1.013 1.01 0.997 1.046 I .048 1.02

-

Appendix A

646)

The following equations are derived with reference to Fig. 1 for the sake of clarity, but the results are general in nature. Consider line 2 between bus K and bus L: v4 L (14, v,io, V,+(Z4L$4)(R+jX)-

cos0,= -

6

cos 0,

- Q4Qh V, P,'+Qe' v,

P4P,

for Substituting (A I ) gives

V, + I,R cos (c/4- 14X sin ti4 - cos 0, = 0

(Al)

P4 +jQ4 = - V414L(f14 - $A

(A21

+jQ, = VJ4L(- ti4)

(.43)

f’,

sin 8, - 9 6

(A7)

Solving Eqs. (A6) and (A7) for cos O,,

ISLJb4 V,iO,=O

Using the real part since it is more consequential the voltage magnitudes,

Q, = - q

v, + I&,

(A@

from Eqs. (A4), (A5), and (AS) into Eq.

+ V$, = 0

649)

where

Eq. (A3) produces I-- = I =

P, + jQ6 = VJ4 cm $4 - j VJ, sin i, cos $b4== PC,

(A4)

64

sin *4 = -=

I,

Q6 64

”

4

U’,,’

+ Q,4’ P2 V cl

Pd’m, + Qc4Qms = (P,,” + Q,.I )“2(P,,,;+ Q,;)‘:2

(A5) References

Using Eqs. (A4) and (A5) in Eq. (A2) produces P4 = - V,I, cos(0, - ti4)

[I] A. Abur absolute

and M.K. Celik. A fast algorithm for the weighted value state estimation, IEEE PES Surmner Meer.,

least Min-

A. Fallaha

et al. j Electric

Potiser Sqstenzs Research

neapohs, MN, USA, 1990, Paper No. 90 SM 273-3 PWRS: IEEE Trans. Power Syst., (Feb.) (1991) l-8. [2] M.M. Adibi and J.P. Stovall, On Estimation of uncertainties in analog measurements, IEEE Trans. Power Cyst., (Nov.) (1990) 1222-1230. [3] M.M. Adibi and R.J. Kafka, Minimization of uncertainties in analog measurements for use in state estimation, IEEE PES Winter

38 (1996)

1~ 10

Meet., IVew York, USA, 1990, Paper No. 90 WP 235-2 PWRS; IEEE Trans. Power Syst., (Aug.) (1990) 902-910. [4] D.M. Falcao and SM. Assis. Linear programming stateestimation error analysis and gross error identification, IEEE Trans. Power Syst.. 3 (1988) 809-815. [5] J.R. Tyler, An Introduction to Error Analysis, University Science Books, Denver, CO, 1982.