Implementation of Phasor Measurements in State Estimator at Sevillana de Electricidad

Juan M. Gonzalez Provost Ilya W. Slutsker, Senior Member, IEEE

Leonard A. Jaques, Member, IEEE

Siemens Empros Power Systems Control Sevillana de Electricidad GHERSA R&D Brooklyn Park, MN 55428-1540, USA Seville, Spain Cadiz, Spain

Jose Benaventa Sierra

Juan M. Montes Figueroa Sasan Mokhtari, Member, IEEE Miguel Baena Perez Fernando Gonzalez Gonzalez

Abstract: The paper describes the first implementation of phase angle measurements in an industrial state estimator. Transducers for measuring phase angles using satell ite clock synchronization were installed at Sevilfana de Electricidad and incorporated into the data acquisition subsystem of the EMS. The State Estimator was modified to support the processing of phasor telemetry. The enhancements to the state estimator algorithm are described in the paper. The modified state estimator was subjected to intensive testing prior to installation in the field. The results of all experiments as well as classification of phasor measurement effectiveness are reported in the paper. Phasor measurements were found to provide valuable information about the power network and improve the accuracy of the state e s t imat ion .

1. INTRODUCTION

State estimation plays a key part in the real-time monitoring and control of power systems. It provides estimated data to network analysis security and optimization applications as well as to power system dispatchers. The measurement set normally includes voltage magnitude, branch real and reactive flow, real and reactive injection, and ampere magnitude measurements. The prospects of using direct measurements of key state variables have long intrigued state estimator developers. Phase angle telemetry has only recently become available in power systems, primarily for relaying purposes, and their accuracy has not initially been adequate for state estimation. With an advent of satellite clock synchro- nization, phasor metering achieved a level of precision that made phasor telemetry a valuable source of measurement data.

Sevillana de Electricidad (CSE), a Spanish utility with the installed capacity of 5700 MW, has added phasor metering to its SCADA system. It has contracted Siemens Empros to

perform necessary modifications in the state estimator function to enable it to support processing of phasor measurements, and to conduct an extensive test program to establish the robustness, accuracy requirements and effectiveness of phasor measurements. In the course of the project, the state estimator was modified to process phasor measurements, and, upon successful fulfillment of the test program, the modified state estimator was installed at CSE and is currently operational.

The use of phasor measurements in state estimation, including the analysis of their impact on solution algorithms, observability analysis and the bad data identification, has already been addressed in [1,2]. This paper describes several additional enhancements to the observability processing and bad data identification components of the state estimator. It also reports the results of comprehensive tests that included viability testing, relative value analysis, and simulator testing.

The second chapter of the paper is devoted to phasor metering hardware and data acquisition issues. It is followed by section describing modifications to state estimator algorithms. Test results are presented in the fourth section. The final section contains conclusions.

2. PHASOR METERING AND DATA ACQUISITION

Phase angle transducers, designed by the Spanish company GHERSA R&D, were added to the station remote terminal units (RTU) of the data acquisition system and used to obtain phasor measurements at network buses. The phase transducer consists of analog and digital sections and the i80386-based main processing unit. The analog section contains input circuitry, ADC and DAC converters, signal multiplexers and conversion control circuitry. The digital section contains RAM memory, VME bus control circuitry, address selection switches, buffer memory and pulsed signal input stages. Using the GPS. interface, local clock is monitored against GPS UTC signals and clock synchronization and frequency steering are performed. The input signal is first clipped to +7V and sent to the ADC input. Upon detection of wave passage through -5V, the ADC starts conversion. The conversion data is then processed to obtain the time estimate of zero-crossing. The time of zero-crossing relative to the GPS reference is converted into a phase angle measurement. Phase data undergoes further filtering and processing to compute an estimate of voltage phase angle every 10 UTC seconds. The transducer's design ensures that the largest error

0-7803-2663-6195 $4.00 0 1995 IEEE 392

in clock synchronization does not exceed 2 ps . Measured angles are added to the RTU data collection and transmitted to the control center. The detailed description of phasor transducer's design and operation will be the subject of an upcoming paper.

Modifications in SCADA included creation of several new analog data formats, modification of existing scan group definitions, and expansion of the SCADA/SE interface to accommodate phasor telemetry.

A direct comparison of measured phase angle differences with the results of state estimator showed excellent correlation, with the standard deviation of measurement residuals better than 0.15O, confirming high accuracy of metering hardware.

3. STATE ESTIMATOR MODlFICATIONS

A. CSE State Estimator CSE SE employs the Enhanced 6i.vens method [3] of

solving a state estimation problem. Tlhe Enhanced Givens method combines a high execution speed and superior numerical properties. It is widely recognized as the most numerically stable technique of solving least square problems. Taking advantage of the solution engine's ability to process measurements with widely different weights, CSE SE uses a one pass solution scheme in which both the internal (observable) and external networks are solved simultaneously in a single solution. The observability is extended into the external network by creating forecasted injection and voltage pseudo measurements at unobservable buses. The one pass solution scheme allows fast estimator cycling by avoiding an excessive overhead of a two pass solution in which internal and external networks are solved separately. In addition, the highly weighted internal telemetry influences extemal solution close to the observable boundary pushing errors further into the extemal network and eliminating the boundary mismatch problem. The solution in the internal network is protected from contamination by the omnipresent errors in the forecasted measurements through the use of very high weighting factors.

CSE SE executes in two modes: partial and full. In partial execution mode, a solution is obtained ordy for portions of the network observable from telemetered measurements. In full execution mode, the solution is obtained for the whole network. Partial and full execution modes are complimentary and used intermittently in the on-line environment. The main purpose of the partial mode is to refresh estimator displays at higher cycling rates in between full solutions.

CSE SE uses the bad data identification method based on measurement compensation and linear residual calculation [5] , which has been shown to identify multiple interacting and conforming bad data (see test case 3 in [?)I). CSE SE employs an advanced hot start feature in which a solved state vector is projected into the future and is used as initial state vector in the next estimation cycle. The hot start feature substantially reduces the iteration burden of the estimator solution as will be seen in chapter 4.

B. Weights of Phasor Measurements The accuracy of phasor measurements depends on the clock

synchronization error in the metering hardware. Assuming 50 Hz base frequency, the standard deviation of the phasor metering error is computed as 0.006O per 1 ys of the clock synchronization error .

For a given clock accuracy, weights of phasor measurements, similar to other measurements, are determined as inverse standard deviations of their error distribution

(1) 0. =r <Ti'

C. Selection of Reference Bus Selection of reference bus becomes more complicated in the

presence of phasor measurements. For a given set of measurements, the choice of angle reference (zero or non-zero), uniquely determines the angle solution in the observable network. In other words, the choice of reference bus establishes the angle profile in the solved network.

At the same time, phase angles are measured with respect to some other reference that has no power system significance. If such phasor measurements are added to the measurement set as is, each one of them will contain a bias term equal to the angle shift between the references. For phasor measurements to be properly used in estimator solution, the reference bus must chosen among buses with phasor measurements and values of all remaining phasor measurements must be corrected by the value of the reference measurement

where ai, S j are the original and corrected values of phasor measurement at bus i, and tiref is the reference measurement.

Phasor measurements are, therefore, included into state estimation as phase angle difference measurements. It is obvious that, if error in uncorrected phasor measurement is ei = ~(o ,o?) , then e;. = N ( O , 2 0 ? ) , and weights of corrected measurements are obtained as

Gi =- I &<Ti

(3)

Reference bus in each observable island is selected among buses with phasor measurements in that island. The number of phasor measurements actually included in solution is always reduced by the number of phasor measurements at reference buses. Reference phasor measurements do not participate in estimator solution directly but are used to synchronize phasor and non-phasor measurements.

D. Observability Processing CSE SE uses the highly efficient symbolic Jacobian

reduction method of the observability analysis [4]. The symbolic method is a matrix rank computation technique which determines the observability of the network by processing a symbolic Jacobian matrix. The method is well suited for inclusion of various measurement types into observability analysis. In addition to power measurements,

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the method can process unpaired voltage and ampere measurements when determining the observability of the network. The symbolic method can identify both the coupled (combined P-6 and Q-V) observability as well as decoupled (separate P-6 and Q-V) observability.

The method has been enhanced to include phasor measurements in the coupled observability analysis by creating pseudopairs [4] consisting of phasor measurements and voltage magnitude or unpaired power measurements. Addition of phasor measurements may have a profound effect on the observability of the network.

E. State Estimator Solution State estimator solution requires little changes to

incorporate processing of phasor measurements. The main modification is addition of new Jacobian matrix rows for participating phasor measurements. The only non-zero entry in each such row is a partial derivative of a phasor measurement with respect to the angle variable of its bus

a& - --1. -.- (4) dbi

F. Bad Data Ident@cation Gross errors in phasor measurements, as in other

measurements, can be identified using usual bad data identification methods. The exceptions are errors in reference measurements. Reference phasor measurements do not explicitly participate in state estimation and no residuals are available for such measurements. Conventional identification methods are based on processing of measurement residuals and can not be used to detect errors in reference measurements. At the same time, errors in reference measurements can have much more negative impact on solution than errors in other measurements. An error in reference measurement introduces a bias term in each phasor measurement corrected by it, distorting estimator solution, and often leading to good phasor measurements being erroneously rejected as anomalies.

A special method to identify errors in reference phasor measurements was developed for CSE SE. The method allows the computation of estimated errors in all phasor measurements, including reference measurements. Not only this approach enables identification of bad data in reference measurements, but it also allows a reliable selection of new reference measurements, as ones with smallest error component.

It can be shown (see Appendix A) that the error component in a reference measurement eR can be approximately computed as

r c P

where P is the set of phasor measurements in a given observable island, ri are the residuals of phasor measurements,

and oii are the diagonal terms of the residual sensitivity matrix of phasor measurements

w = I - H ( H ' R - ~ H ) H ' R - ' . ( 6 )

Here H is the measurement Jacobian matrix and R-' is the matrix of measurement weights.

approximately obtained as An error in each non-reference measurement is

r . ei =_-L+eR.

wii

If error component in reference measurement exceeds the identification threshold, the reference measurement is declared anomalous and replaced with another phasor measurement. A search for errors in reference measurements is performed as the first step in the bad data identification process. If no errors are found, then normal bad data identification can proceed as usual. If, however, gross errors are detected in any of reference measurements, a new reference measurement is chosen for the corresponding island and the state estimator solution is reexecuted.

4. TEST RESULTS

Prior to installation at CSE, the modified state estimator was subjected to intensive testing. The purpose of tests was to verify the correctness of solution algorithms, evaluate impact of phasor metering accuracy on the quality of estimator solution, and to establish a relative value of phasor measurements vis-a-vis other measurements. In all, nine groups of tests were conducted, the results of which are reported below.

All tests were performed on a 100-bus test network that was similar to CSE network in size (CSE has 98 modeled buses) and structure. In addition to bus voltages and angles, the state vector included LTC and phase-shifting transformer tap position variables. The measurement set consisted of 170 power measurements (including unpaired measurements), 4 ampere magnitude, 114 voltage magnitude, and 21 tap position measurements. 23 phasor measurements were added to approximate the size and placement of phasor telemetry at CSE.

Weights of measurements were computed assuming that metering in all voltage levels was performed using ANSI class 0.6 instruments [6].

The first group of tests analyzed the impact of phasor measurements on the quality of estimator solution. With Gaussian noise in non-phasor measurements, perfect phasor measurements of various accuracy were added to the measurement set, and the state estimator solution was obtained. The impact of phasor measurements was evaluated by computing the estimation error index E ( i ) , the measure of distance between measurement estimates and their true values

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substantially reducing the iteration load of the estimator solution, as will be further discussed below.

Here z y , zyt, oi2 are, respectively, the true value, estimated value. and variance of measurement i.

It is clear that addition of perfect measurements brings estimates of other measurements closer to their true values, which is reflected in the reduction of the estimation error index. A typical behavior of the estimai:ion error index as a function of phasor measurement accuracy (i.e. weights of phasor measurements) is plotted in Fig. 1.

I 900 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 , 1

0.1 0 . 2 0.3 Q.4. 0.5 0.6 STD of Phasor Metering Error (deg)

Figure 2. Effect of phasor measurements accuracy on the quality of state estimator solution.

As can be seen from Fig. 1, phasor measurements have little effect on estimator solution until their precision approaches 0.3O. The accuracy should be better than 0.12O (i.e. clock synchronization error of less than 20 ps) for phasor measurements to have a noticeable effect the quality of estimator solution.

The purpose of the next group of tests was to analyze how errors in phasor measurements degrade the accuracy of state estimation. Errors ranging -200 to 200 were added to phasor telemetry while the remaining measurements were kept error- free. Tests was repeated for several phasor metering accuracy levels. The results are shown in Fig. 2. As expected, errors in phasor measurements with highest precision caused the largest increase in the estimation error indlsx.

Next group of tests explored the benefits of using measured phase angles for state vector initialization in an attempt to reduce the number of estimator iterations. Tests were conducted under both flat and hot stmt conditions. Little improvement in iteration count was observed in flat start initialization, as phasor measurements covered only about a quarter of network buses and could not significantly alter an unrealistic angle profile associated with the flat start. Similarly, no reduction in iteration count was observed when using measured angles in hot start initiakzation. The hot start algorithm already obtains relatively accurate initial bus angles,

x 6000 % fi 5000

4000 r;i .3 8 3000

h

c)

._ E" 2000

w" 1000 e

0

-20 -10 0 10 20 Error in Phasor Measurements (0)

Figure 2 . Impact of errors in phasor measurements on the quality of state estimator solution.

The impact of phasor measurement placement was a subject of another group of tests. Nine different configurations were studied and no changes in estimator convergence characteristics were detected.

The next group of tests analyzed the ability of state estimator solution to withstand large errors in phasor measurements. Flat start was used in all tests and the number of estimator iterations and factorizations were monitored. CSE SE uses a pseudo-Newton form of solution in which no factorization is performed unless, either a change occurred in the network or measurement topology, or solution was not obtained in a predetermined number of iterations. As seen in Fig. 3, CSE SE displayed remarkable resiliency in both partial and full execution modes, withstanding errors in excess of21000o. Positive errors under 5000 resulted in a more distressful flow pattem for the estimator solution than the one produced by negative errors under 5000 and required an additional iteration for convergence.

7 d Iterations

- - -

-1000 -500 0 500 1000 Error in Phasor Measurements (0)

Figure 3. Number of iteration and refactorizations in state estimator solution as function of error level in phasor

measurements.

The choice of reference bus was the subject of the next group of tests. The selection of either internal or external

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reference bus was found to have no impact on convergence properties of estimator solution.

Clearly, addition of phasor measurements, just as addition of other measurements, has a positive effect on estimator's ability to detect anomalies. It improves local redundancy, helps to eliminate critical measurements, breaks down bad data groups, and removes other obstacles to a successful bad data analysis. While it is clear that phasor measurements improve the bad data identification in principle, the purpose of the next group of tests was to establish an accuracy level at which their contribution becomes significant. Normalized residuals of phasor measurements were computed for various error and metering accuracy levels. A typical behavior of a normalized residual of a phasor measurement is shown in Fig. 4. As we can see, even small errors in phasor measurements lead to sizable normalized residuals, thus improving the probability of identification, once metering precision is better than 0.06O.

Another series of tests was devoted to the sensitivity analysis of phasor measurements. Terms of the residual sensitivity matrix W (see equation (6)), determine the level of interaction between measurements in state estimation and, ultimately, influence the outcomes of solution and bad data identification. Assuming a linear measurement model [5] , the ratio of W terms in the column of the phasor measurement j is equal to the ratio of changes in residuals of measurements i and j in response to a disturbance in the phasor measurement value.

w .) (9) p - Y - Ari

ojj Arj

0.0 0.1 0.2 0.3 0.4 0.5 0.6 STD of Phasor Metering Error (deg)

Figure 4. Effect of metering precision on normalized residuals of a phasor measurement.

For a measurement to be viable in state estimation, it must possess sufficient sensitivity with respect to other measurements. An example of sensitivity ratio p behavior as function of phasor metering accuracy is presented in Fig. 5. Sensitivity ratios are shown for two power measurements in the immediate vicinity of a phasor measurement. It is clear

that when metering accuracy is low, even large errors in phasor measurements have little influence on the residuals of other measurements, making presence of phasor measurements irrelevant. As metering accuracy increases, phasor measurements become viable in estimator solution, asserting stronger influence on the residuals of surrounding telemetry.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 STD of Phasor Metering Error (deg)

Figure 5. Sensitivity ratio as function of phasor metering accuracy.

The next group of tests was devoted to analysis of the relative value of phasor measurements vis-a-vis other measurements. For each phasor measurement, based on sensitivity analysis, a comparable power measurement in its immediate vicinity was identified and placed on what was referred to as the control set. Both phasor measurement and the control set were removed and the estimator solution was obtained with the Gaussian noise added to remaining measurements. Reintroduction of noise-free phasor measurements and control set measurements led to a reduction of the estimation error index. Based on whether phasor measurements had larger reduction in the index, they were judged to be more or less effective than the alternative power measurements. Tests were repeated for a several control sets and wide range of phasor metering accuracy. An example of ratio of improvement in the estimation error index for one of the control sets is shown in Fig. 6. While the precise breakpoint in effectiveness will differ for various measurement structures and weight profiles, it is safe to say, based on results of the tests, that phasor metering accuracy should be less than 0 . lo for the phasor measurements to be more effective than power measurements in state estimation. It should be noted that the cost of a phasor transducers is comparable to that of other power measurements.

In the final group of tests CSE SE was subjected to stress testing using the Real Time Sequence Simulator (RTSST), a variation of the PSM subsystem of the Operator Training Simulator. Using the RTSST, the estimator cycling behavior can be analyzed under a variety of loading and topological conditions. Test scenario consisted of 1500 estimator executions and represented an "obstacle course" including some

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of the more adverse conditions CSE SIE was expected to encounter in the field, such as load pick up and drop off, variety of equipment outages including loss of largest generator, largest load and outage of line with largest flow, network splits, loss of observability, modeling and topology errors. Tests were executed with arid without phasor measurements in both partial and full modes. The results of CSE execution in full mode are summarized in Table 1.

Synchroni- zation clock

accuracy (PS) > 30

20 - 30

0 .3 c

2 2.5 Y

8 5 2.0

E E 1.5 CI

STD of phasor Cla~ssification metering error

(deg) - > 0.54 Ineffectivi-. No impact

on estimator solution.

Little impact on

- 0.36 - 0.54 Somewhat effective.

Phasor measurements are more effective than power measurements

(ratio > 1.0)

are less effective than 0

power measurements

w

Y 0 . 5 4

.3 * d

0.0 0.1 0.2 0.3 0.4 0.5 0.6 STD of Phasor Metering Error r(deg)

Figure 6. Effectiveness of phasor measurements as function of metering accuracy.

TABLE 1 SIMULATOR TESTING SUMMARY (1:SQI EXECUTIONS)

I Iterations I Factorizations I I Total I Ave I T'otal I Ave I

With phasor I measurements I 4373 I 2.9 I 1047

I Without phasor 1 4414 I 2.9 I 1048 measurements

TABLE 2 PHASOR MEASUREMENT EFFECTIVENEISS CLASSIFICATION

superior ts3 that of othe measurements.

The purpose of simulator testing was not to analyze the effect of phasor measurements, which is difficult to assess due to

randomness of noise in the measurements, but to establish the viability of state estimator with a phasor measurement feature under near-field conditions prior to actual installation at the customer site. Note, that even under these adverse conditions, the average number of SE iteration is less than three, due to the hot start feature.

5. CONCLUSIONS

Phasor measurements can play a significant part in state estimation. Their contribution greatly depends on the precision of phasor metering. Based on results of the tests, four categories of phasor measurements effectiveness were identified and summarized in Table 2.

Modern satellite clock synchronization technology is expected to provide phasor metering accuracy better than 0. lo, and enable phasor telemetry to become an important source of data for state estimation.

It is important to realize that phasor measurements complement but do not replace other measurements. While an exclusive use of voltage angle and magnitude measurements does result in a simplified formulation of the state estimation problem, the remaining non-phasor measurements are not any less important and must not be ignored. Exclusion of non- anomalous measurements, for any reason, would deprive the state estimator of valuable measurement data. All available telemetry must always be utilized in state estimation to ensure optimal local redundancy and high accuracy of solution and bad data identification.

6. ACKNOWLEDGMENT

The authors gratefully acknowledge the sponsorship of OCIDE under project PIE No. 132-212.

References

J. S. Thorp, A. G. Phadke, K. J. Karimi. "Real Time Voltage-Phasor Measurements for Static State Estimation". IEEE Trans. on PAS, Vol. PAS-104, No. 11, August, 1985.

A. G. Phadke, J. S. Thorp, K. J. Karimi. "State Estimation with Phasor Measurements". IEEE Transactions on Power Systems, February, 1986.

N. Vempati, I. W. Slutsker, W. F. Tinney. "Enhancements to Givens Rotations for Power System State Estimation". IEEE Transactions on Power Systems, May, 1991.

I . W. Slutsker, J. M. Scudder. "Network Observability Analysis Through Measurement Jacobian Matrix Reduction". IEEE Transactions on Power Systems, May, 1987.

I. W. Slutsker. "Bad Data Identification in Power System State Estimation Based on Measurement Compensation and Linear Residual Calculation," IEEE Transactions on Power Systems, February, 1989.

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[6] M. M. Adibi, R. J. Kafka. "Minimizat ion of Uncertainties in Analog Measurements for Use in State Estimation," IEEE Transactions on Power Systems, August, 1990.

Ilya W. Slutsker (SM) received his M.S. degrees in electrical engineering, from Moscow Institute of Energy in 1975 and from University of Wisconsin in 1982. From 1975 to 1984 he was employed by the Scientific Research Institute of Heavy Industry, Moscow, U.S.S.R. and later Wisconsin Electric Power Company where he specialized in EMS applications. In 1984, he joined Siemens Empros Power Systems Control, where he presently works in the power system application research and development area. Mr. Slutsker is a recipient of the 1993 IEEE W. R. G. Baker award for best electrical engineering paper. He is a registered professional engineer in the State of Wisconsin.

Sasan Mokhtari (M) is native of Teheran, Iran. He received his BSEE, MSEE, and PhD degrees from the University of Missouri, Columbia, Missouri. Since graduation in 1984, he has been with Siemens Empros Power Systems Control where he currently is the Manager of Advanced Applications Research and Development.

Leonard A. Jaques (M) was born in Socorro, NM USA, on June 13, 1963. He received his B.S and M.S. degrees from New Mexico State University in 1987 and 1988, respectively. In 1988, he joined Siemens Empros Power Systems Control Division as Electrical Power System Analyst. He presently works in the Network Analysis Department.

Juan M. Gonzales Provost was born in Seville, Spain on April 12, 1959. He received the M.S. degree in Electrical Engineering from Esquela Superior de Ingenieros Industriales de Sevilla in 1982. From 1982 to 1986 he was the I&C chief engineer for Valdecaballeros Nucler Power Plant. In 1987 he joined the EMS project team at CSE where he is currently the software leader on the Empower Spectrum EMS project.

Miguel Baena Perez was born in Seville, Spain on January 22, 1967. He received the B.S. degree in Physics from the University of Seville in 1990. The same year he joined CSE where he is currently working at the dispatch center specializing in operation and maintenance of Empros ARTECS and Empower Spectrum network applications.

Jose Benaventa Sierra was born in Jerez, Cadiz, Spain in 1938. He graduated from the Spanish Naval Academy in 1961 as Navy Officer. He received M.S. degrees in mathematics and physics from the Spanish Royal Naval Observatory and the M.S. degree in engineering geoscience from the University of California at Berkeley. From 1978 to 1988 he worked at Spanish Royal Observatory as secretary, chief of Time and Frequency section and vice-director. In 1988 he joined GHERSA R&D where he is currently the R&D technical director.

Fernando Gonzalez Gonzalez was born in Bilbao, Vizcaya, Spain in 1942. In 1965 he graduated from Spanish Naval Academy as Navy Officer. He received the M.S. degree in electronic systems engineering from ENSTA, Paris, France. He has served in Spanish Navy specializing in the antisubmarine

warfare and air navigation, becoming the chief of the avionics division in 1981. In 1992 he joined GHERSA R&D where he is currently involved in several projects in areas of electrical power management and time and frequency applications in power industry.

Juan Manuel Montes Figueroa was born in Cadiz, Spain in 1964. He received the B.S. degree in electronics from the University of Cadiz in 1988. The same year he joined GHERSA R&D where he has worked on several R&D projects in areas of electrical power management and time and frequency applications in power industry.

APPEhDMA APPROXIMATE COMPUTATION OF ERROR IN REFERENCE

PHASOR MEASUREMENT

A residual of a non-referenced phasor measurement i, assuming the linear measurement model [ 5 ] , can be expressed as the following sensitivity relationship

where P is the set of phasor measurements and N is the set of non-phasor measurements in a given observable island.

Assuming the existence of a gross error in the reference measurement, gaussian noise in measurements and the dominance of the diagonal sensitivity terms, the last component in (10) will be much larger than the other two.

ri = wii(ei - e,)

Performing an algebraic summation for all phasor measurements in an island we get

i c P i c P i c P i c P

Since errors in phasor measurements are gaussian, the second component in (12) will clearly dominate, yielding an estimate of the reference measurement error

i c P

C w i i i c P

Note that even a gross error in a non-reference phasor measurement will not significantly affect the accuracy of (13). Such an error will mostly contribute to the residual of its own measurement but not the others. An increase in the algebraic sum of phasor residuals will be small. Further divided by the sum of diagonal sensitivity terms, the effect on e, will be insignificant. In contrast, even a small error in a reference measurement will produce a sizable increase in the algebraic sum of phasor residuals, allowing a reasonably accurate computation of that error using (13).

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