A Method of Utilsing NonSource Measurements for Harmonic State Estimation

Electric Power Systems Research 56 (2000) 231–241

A method of utilising non-source measurements for harmonic stateestimation

Van Long Pham a, Kit Po Wong a,*, Neville Watson b, Jos Arrillaga b

a Department of EE Engineering, AI and Power System Research Group, The Uni6ersity of Western Australia, Nedlands, WA 6907, Australiab Department of EE Engineering, Power System Research Group, The Uni6ersity of Canterbury, Christchurch, New Zealand

Received 6 October 1999; received in revised form 20 March 2000; accepted 22 March 2000

Abstract

A power system reduction method is described for use in harmonic state estimation (HSE). The new method uses harmonicmeasurements at non-harmonic-source buses, such as those of generators, without loads or linear loads, to estimate thesystem-wide harmonic levels. In general, non-harmonic-source buses are more numerous and accessible, and are also betterequipped with measurement instruments than those of remote and non-linear loads. The new solution provides optimalmeasurement locations thus reducing the monitoring costs attached to harmonic state estimation. The enhanced HSE algorithmhas been tested in the New Zealand power system with promising results. © 2000 Elsevier Science S.A. All rights reserved.

Keywords: Harmonic state estimation; Observability analysis; Harmonic

www.elsevier.com/locate/epsr

1. Introduction

To maintain high quality power supply, the systemharmonic levels must be measured and kept undercontrol. To assess the harmonic levels without the needto make full system harmonic measurements, a HSEalgorithm has been developed [1–4], capable of estimat-ing the system-wide harmonic levels from a few synchro-nised, partial and asymmetric harmonic measurements.The minimum number of these measurements and theirlocation are determined by observability analysis (OA)[5–8] and the location of the harmonic sources can beidentified from the estimated nodal harmonic currentinjections by the HSE algorithm.

The solvability of the HSE algorithm and the accuracyof the estimated harmonic levels depend on the numberand location of the harmonic measurements. Generally,harmonic-source buses are those with attached non-lin-ear loads while buses with generators, linear or non-loadsdo not generate harmonics. Thus, the original OA [6] andHSE algorithms were based on harmonic measurementsat only potential harmonic-source (suspicious) buses.

However, harmonic measurement at suspicious busescan be difficult to obtain, as these buses are ofteninaccessible or lack of monitoring equipment. More-over, the number of suspicious buses is relatively smalland exclusive measurement at these buses does notensure solvability.

Based on harmonic measurements at non-suspiciousbuses, this paper develops a new approach to theharmonic state estimation problem that ensures systemsolvability and adds estimation accuracy. A systemreduction method is developed to show that measure-ments at non-suspicious buses are as significant as thatof suspicious buses for system solvability. This methodis incorporated into the existing HSE and its OA al-gorithm is modified to identify an optimal set of har-monic measurements at non-suspicious buses.

Furthermore, the new HSE algorithm also makes useof the load demand information at the suspicious busesto improve the accuracy of the system harmonic nodalcurrent estimation. The application studies show thatinformation of load demand at these buses when incor-porating into the algorithm increases the accuracy of theHSE.* Corresponding author.

0378-7796/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved.PII: S 0 3 7 8 -7796 (00 )00126 -7

V.L. Pham et al. / Electric Power Systems Research 56 (2000) 231–241232

2. Harmonic state estimation problem

This section gives a summary of the formulation of theHSE problem. More detailed theory can be found in [1]and Refs. therein. In the HSE problem, partial harmonicmeasurements of nodal voltages, nodal injection currentsand line currents, and system data are used to generate abest estimate of the harmonic levels in the power system.Nodal voltages of the whole system are chosen to be statevariables as they are independent of each other. Whenthey are found, all the harmonic injection currents,branch currents, line currents and harmonic power flowscan be calculated. Measurement values are availablemeasured harmonic nodal voltages and injection currentsat selected suspicious nodes, and selected line currents ateach harmonic frequency.

For each harmonic order n, let VN, IN and IL be thevectors of system harmonic nodal voltages, injectioncurrents and line currents respectively, the system equa-tions are given by

IN=YNNVN (1)

IL=YLNVN (2)

where YNN is the system nodal admittance matrix andYLN is the system line-node admittance matrix. Thesematrices are obtained from the system configuration,which is represented as an oriented graph with incidentmatrices representing the system as individual elementsconnected in a manner of the physical network [1,2].

The set of the system Eqs. (1) and (2) can be trans-formed into the following form, which relates the mea-surements to the state variables

Z(n)= f[X(n)]+o(n) (3)

In Eq. (3), X is the vector of nodal voltages, which are thestate variables; Z is the vector of measurements; and n isthe harmonic order. The function f(·) is related to thesystem topological configuration, system admittance ma-trix and placement of measurement points. o(n) is thevector of measurement errors produced from systemmeasurement noises due to transducer errors and prob-lems in data communications. The Gaussian noise modelis used such that

E [o(n)]=0, E [o(n)o(n)T]=R(n) (4)

where E(·) is an operator of expectation value and R is aweighted matrix.

The objective of the HSE algorithm is to estimateaccurately the system harmonic nodal voltages using Eq.(3). However, this equation is usually an under-deter-mined equation, which means that the number of nodalvoltages is larger than the number of measurements. Theresults obtained from solving this under-determinedproblem using state estimation techniques are usuallyinaccurate [1].

To obtain accurate results, it is required to transformthe under-determined problem into a completely-deter-mined one, where number of unknowns equals thenumber of measurements. This can be achieved usingmatrix-partitioning technique based on the harmonicinformation of the system and available harmonicmeasurements.

3. A new power system reduction procedure

A new procedure for system reduction enabling the useof measurements at non-suspicious nodes for HSE ispresented here. The procedure transforms the under-de-termined system equation in Eq. (3) into a completely-de-termined one taking in the non-source measurements.

The system node set N is partitioned into two subsetsof non-source (Vo, Lo) buses and suspicious (Vs, Is) buses,i.e.

VN=�Vo

Vs

n, IN=

�Io

Is

nEqs. (1) and (2) can be partitioned into

�Io

Is

n=�Yoo Yos

Yso Yss

n�Vo

Vs

n(5)

IL= [YLo YLs]�Vo

Vs

n(6)

From Eq. (5) and with Io=0, the vector of harmonicnodal voltages at non-suspicious nodes derived from

Vo= −Yoo−1YosVs (7)

Substitution of Vo into Eqs. (5) and (6) gives

Is=Y( ssVs

IL=Y( LsVs (8)

where

Y( ss=Yss−YsoYoo−1Yos

Y( Ls=YLs−YLoYoo−1Yos (9)

In Eq. (8), the number of unknowns in the HSEproblem has been reduced from VN to Vs. Because thenumber of non-source buses in a system is generally largerthan that of suspicious buses, the reduction of the numberof unknowns from all bus nodal voltages to that ofsuspicious buses is significant.

Next the reduced system set in Eq. (8) is partitionedbased on the measured and unmeasured values. Thenon-suspicious bus nodal voltage vector, Vo is subdividedinto two subsets of measured and unmeasured values Vom

and Vou, respectively. Similarly, the suspicious-bus nodalvoltage vector Vs is subdivided into Vsm and Vsu, the

V.L. Pham et al. / Electric Power Systems Research 56 (2000) 231–241 233

injection current vector Is into Ism and Isu, and the linecurrent vector IL into ILm and ILu. Eqs. (7) and (8) arethen partitioned into

�Vom

Vou

n=�Yomsm Yomsu

Yousm Yousu

n�Vsm

Vsu

n(10)

�Ism

Isu

n=�Ysmsm Ysmsu

Ysusm Ysusu

n�Vsm

Vsu

n(11)

�ILm

ILu

n=�YLmsm YLmsu

YLusm YLusu

n�Vsm

Vsu

n(12)

To take into account the measurement errors, letV%om, V%sm, I%sm and I%Lm be real measurement value vec-tors of Vom, Vsm, Ism and ILm with measurement noisevectors oom, osmv, osmi and oLm, respectively, then

V%om=Vom+oom

V%sm=Vsm+osmv

I%sm=Ism+osmi

I%Lm=ILm+oLm (13)

Substitution of Eq. (13) into Eqs. (10)–(12) gives

V( om=YomsuVsu+oom

I( sm=YsmsuVsu+osm

I( Lm=YLmsuVsu+oLm (14)

where

V( om=V%om−YomsmV%sm

I( sm=I%sm−YsmsmV%sm

I( Lm=I%Lm−YLmsmV%sm (15)

and

oom=oom−Yomsmosmv

osm=osmi−Ysmsmosmv

oLm=oLm−YLmsmosmv (16)

In the reduction process from Eq. (8) to Eq. (14), thenumber of unknowns has been reduced from number ofsuspicious nodes, Vs to the number of unmeasuredsuspicious nodes, Vsu. The reduced system Eq. (14), canbe written in the following form

Z=HX+E

where

Z=

ÆÃÃÃÈ

V( om

I( sm

I( Lm

ÇÃÃÃÉ

, H=

ÆÃÃÃÈ

Yomsu

Ysmsu

YLmsu

ÇÃÃÃÉ

X=Vsu, and E=

ÆÃÃÃÈ

oom

osm

oLm

ÇÃÃÃÉ

(17)

Overall, the above reduction process has reduced thenumber of unknowns in the HSE problem from allsystem nodal voltages, VN to unmeasured suspiciousnodal voltages, Vsu. Voltage measurements at non-sus-picious nodes, Vom have been incorporated into thereduced system Eq. (17).

The reduced system Eq. (17) has a unique solution oris completely determined when the number of harmonicmeasurements, which are at either non-suspicious orsuspicious nodes, equals to that of the unmeasuredsuspicious-bus nodal voltages VSU. This, therefore,shows that harmonic voltage measurements at non-sus-picious buses are as significant as that of suspiciousbuses for ensuring the solvability of the reduced systemequation.

4. A new system measurement matrix for OA

To ensure the solvability of the system Eq. (17) andthe accuracy of the results, the minimum number ofmeasurement and the optimal measurement locationsfirst need to be determined using OA techniques. Thesymbolic OA method developed in [6] shows that thesystem observability can be determined from the systemmeasurement matrix, which relates the suspicious-busnodal voltages, Vs to the measurement values. Thesystem measurement matrix in [6] however, does notinclude harmonic nodal voltage measurements at non-suspicious buses. In this section, a new system measure-ment matrix is obtained which include non-suspiciousbus harmonic nodal voltage measurements.

From Eqs. (10)–(12), the equation set, which relatesVs to the measurement values are given by

ÆÃÃÃÈ

Vom

Ism

ILm

ÇÃÃÃÉ

=

ÆÃÃÃÈ

Yomsm Yomsu

Ysmsm Ysmsu

YLmsm YLusm

ÇÃÃÃÉ

�Vsm

Vsu

n(18)

Thus, the new symbolic OA algorithm uses measure-ments at suspicious and non-source buses to obtain anoptimal set of measurement locations, which ensuresthe system solvability. The existing OA algorithm canalso be modified to search for more convenient non-source bus measurement locations as optimal harmonicmeasurement locations for the HSE purpose.

5. Modelling of suspicious loads for HSE

Because accurate loading level of potential-harmonic-source loads is not clear before the harmonic


information is obtained, in [1], the load branch atthe suspicious buses are removed when forming thesystem admittance matrix. To make use availableinformation of load demand at suspicious buses, in thispaper the suspicious-bus loads are modelled to in-clude two components: a linear load and a harmonicsource. The loading level of the linear-load componentis given by the load demand at the bus. The injectionlevel of harmonic-source component is unknown andis estimated by the HSE algorithm. The linear-load-component admittance sub-matrices at thesuspicious buses are incorporated into the systembranch admittance matrix when it is formed for theHSE. The load demand information will be the trueloading value if the suspicious-bus load is in fact alinear load. However, if the load is a harmonic sourceload then the load demand information may not be thetrue value and some error is expected.

6. A state estimation algorithm for harmonic estimation

A number of state estimation techniques [9,10] canbe used for determining the most accurate solution forEq. (17).

A weighted-least-squared estimation algorithm is se-lected here given that the harmonic statistical charac-teristic in a power system is unknown.

To obtain the best estimate of X in Eq. (17),

Z=HX+E

let X% be a known vector of old state estimate beforethe measurement Z is made, and M is a known posi-tive matrix given by

M=E [(X%−X)(X%−X)T (19)

where E [·] is the expectation value, then a weightedleast-squares estimator minimises J(X) where

J(X)= (Z−HX)TR−1(Z−HX)

+ (X−X%)TM−1(X−X%) (20)

To find vector X( , which minimises J let

(J(X)((X)

)X=X(

=0 (21)

then

X=X( %+PHTR−1(Z−HX%) (22)

where P is an estimate error covariance matrix and isgiven by

P=E[(X( −X%)(X( −X%)T]= (M−1+HTR−1H)−1 (23)If X% is unknown then set

X%=0 and M=� (24)

The solution X( is given by

X( = (HTR−1H)−1HTR−1Z (25)

and

P= (HTR−1H)−1 (26)

If no prior estimate of the harmonic content in thepower system is available and all the harmonic mea-surement instruments being used are of the same type,then R is a unit matrix and

X( = (HTH)−1HTZ (27)

P= (HTH)−1 (28)

Eq. (27) gives the best estimate of the suspicious-busunmeasured harmonic nodal voltages. The algorithmalso calculates system-wide harmonic nodal voltages,harmonic nodal current injections, line currents andharmonic power throughout the power system.

7. Application examples

7.1. Test system

The developed algorithm is tested using the 220 kVinterconnected transmission network in the SouthIsland of New Zealand. The real system data andsystem modelling are obtained from references [11–13].Three-phase system modelling is applied to take intoaccount imbalances and couplings among phases atdifferent harmonic frequencies. Fig. 1 shows the three-phase diagram of the test network. The system includeseight transmission lines represented by equivalent pmodels, the three synchronous generators are modelledas shunt branches connected at their terminal nodesand the five transformers are connected in star-delta.There are a total of 27 nodes, 111 branches and 87lines.

Three loads are connected at bus bars R11B, I33A,T220. The actual harmonic sources are six and 12-pulse rectifiers at two of these buses. In Fig. 1, thesesuspicious harmonic source injections are representedby question marks and the linear loading levels are thegiven load demand at these buses. The remaining sys-tem buses are non-source buses. Because three-phasesystem is used, each bus includes three nodes. There-fore, there are a total of 19 non-source nodes and ninesuspicious nodes in the test system.

7.2. Optimal measurement placements

The suspicious-source buses, R110, I33A and T220geographically far from the utility processing station,are not ideal measurement locations. Moreover, the


generators at M14A and M14B and their step-up trans-formers are identical and the harmonic nodal voltagesat the bus bars M14A and M14B are the same. Mea-surements at only one of these two buses are thereforenecessary.

To obtain a unique solution for this system, theminimum required number of harmonic measurementsis 9. These can be measurements of nodal voltagesand/or line currents at the identified optimal locations.The symbolic OA algorithm is applied to determine thesystem observability from a set of measurement loca-tions. The set of measurement locations used for thistest is shown in Fig. 1. These measurements are partial,asymmetrical and distant from the suspicious-sourcebuses.

7.3. Test results

It is very expensive and time consuming and some-times even impossible to obtain the harmonic informa-tion for the whole test system by measurements for thevalidation purpose. System harmonic information fromthe harmonic penetration software program is consid-ered an accurate approximation of that of the measure-ments. Therefore the HSE results are validated hereusing harmonic penetration simulation results.

Vigorous testing using the real system for differentharmonic levels and harmonic source locations hasbeen carried out to ensure the robustness of the newalgorithm. This section shows the results of one of thetests. For comparison, both results obtained from the

Fig. 1. South Island of New Zealand power system. ", measurement points.


Fig. 2. Nodal voltage magnitude of harmonic simulation.

Fig. 3. Nodal voltage magnitude of harmonic state estimation.

Fig. 4. Nodal voltage phase angle of harmonic simulation.

developed HSE algorithm and the harmonic penetra-tion program are presented.

To show the improvement in the accuracy of theHSE results, first the linear load levels at the suspiciousbuses are set to zero. Figs. 2–5 show the harmonic

nodal voltage magnitudes and phases obtained by simu-lation and the state estimation methods, respectively. Inthe figures, each system bus includes three grid lines,each of which represents one node in the bus. Types ofharmonics can be readily identified from the harmonic


nodal voltage magnitudes. The results show that theestimated harmonic voltages are identical to that of thesimulation.

Figs. 6–9 show the nodal injection current magni-tudes and phases obtained by simulation and the HSE

methods respectively. The errors of the estimated cur-rents are shown in Figs. 10 and 11. Next the loadinglevels at the suspicious buses are set to their true valuesand the harmonic nodal currents are calculated fromthe nodal voltages using the HSE algorithm. The results

Fig. 5. Nodal voltage phase angle of harmonic state estimation.

Fig. 6. Nodal injection current magnitude of harmonic simulation.

Fig. 7. Nodal injection current magnitude of harmonic estimation for zero suspicious load demand case.


Fig. 8. Nodal injection current phase of harmonic simulation.

Fig. 9. Nodal injection current phase of harmonic estimation for zero suspicious load demand case.

Fig. 10. Nodal injection current magnitude difference for zero suspicious load demand case.

are shown in Figs. 12 and 13. The estimated nodalcurrents are now identical to that of the simulationresults as seen from the zero difference shown in Figs.14 and 15. In practice however, only an estimate of the

load demand at source buses is available and someerrors are expected from the HSE results.

However, the estimated nodal injection currents atthe suspicious buses for both the two extreme suspi-


cious-bus loading cases show that harmonic sources canbe identified at T220 and I33A while the load at R11Bis a purely passive load. The types of the identified

harmonics also show that the source at T220 is of atwelve-pulse converter while the one at I33A is of asix-pulse converter.

Fig. 11. Nodal injection current phase difference for zero suspicious load demand case.

Fig. 12. Nodal injection current magnitude difference for accurate suspicious load demand case.

Fig. 13. Nodal injection current phase angle of harmonic state estimation for accurate suspicious load demand case.


Fig. 14. Nodal injection current magnitude of harmonic state estimation for accurate suspicious load demand case.

Fig. 15. Nodal injection current phase difference for accurate suspicious load demand case.

8. Conclusions

A new power system reduction procedure has beendeveloped and incorporated into the HSE algorithm toaccurately estimate the system-wide harmonic levelsusing measurements at easily accessible non-sourcebuses in a power system such as generators, non-loadand linear-load buses. The application of the new al-gorithm to a real system has shown that completeharmonic information including system-wide harmoniclevels, location and type of harmonic sources can beobtained from a few measurements taken at non-sourcebuses geographically far from the harmonic sources.

This new development allows optimal harmonic mea-surement locations to be selected and increases the HSEsolvability for large and complicated systems. It reducesthe cost and time in obtaining harmonic measurementdata for harmonic state estimation purposes.

The new algorithm has the potential application forharmonic monitoring in a deregulation environment.The new algorithm enables individual organisations,

who manage different parts of the network to estimateharmonic levels and identify harmonic sources in otherparts of the network using only harmonic measure-ments obtained from their own part of the system.

Acknowledgements

The authors wish to express their gratitude to theAustralian Research Council and Western Power fortheir support. The authors thank Drs S. Chen andMatair for their valuable comments and discussions.

References

[1] Z.P. Du, J. Arrillaga, N. Watson, Continuous harmonic stateestimation of power systems, IEE Proc. Gen. Transm. Distrib.143 (4) (1996) 329–336.

[2] G.T. Heydt, Identification of harmonic sources by a state estima-tion technique, IEEE Trans. Power Delivery 4 (1) (1989) 569–576.


[3] Z.P. Du, J. Arrillaga, N. Watson, S. Chen, Implementation ofharmonic state estimation, Proceeding of the 8th InternationalConference on Harmonics and Quality of Power, Athens,Greece, Paper Number ICHP 0 18, October 1998.

[4] L.L. Lai, W.J. Chan, C.T. Tse, T.P. So, Real-time frequency andharmonic evaluation using artificial neural networks, IEEETrans. Power Delivery 14 (1) (1999) 52–59.

[5] I.W. Slutsker, J.M. Scudder, Network observability analysisthrough measurement Jacobian matrix reduction, IEEE Trans.Power Syst. 2 (2) (1987) 331–338.

[6] Z.P. Du, J. Arrillaga, N. Watson, A new symbolic method ofobservability analysis for harmonic state estimation of powersystems, IEEE Proceedings of the International Conference onElectrical Engineering, vol. 1, Beijing, People’s Republic ofChina, 12–15 August, 1996, pp. 431–435.

[7] J.E. Farach, W.M. Grady, A. Arapostathis, Optimal harmonicsensor placement in fundamental network topologies, IEE Proc.Gen. Transm. Distrib. 143 (6) (1996) 608–612.

[8] J.E. Farach, W.M. Grady, A. Arapostathis, An optimal proce-dure for placing sensors and estimating the locations of har-monic sources in power systems, IEEE Trans. Power Delivery 8(3) (1993) 1303–1310.

[9] W.H. Press, et al., Numerical Recipes in C, the Art of ScientificComputing, 2nd edn., Cambridge University Press, Cambridge,1992.

[10] J.H. Mathews, Numerical Methods for Mathematics, Science,and Engineering, 2nd edn., Prentice Hall, Englewood Cliffs, NJ,1992.

[11] Z.P. Du, Harmonic state estimation of power systems, MEthesis, University of Canterbury, New Zealand, 1994.

[12] A.J.V. Miller, M.B. Dewe, Multi-channel continuous har-monic analysis in real time, IEEE Trans. PD-7 (4) (1992) 1813–1819.

[13] Z.R. Du, J. Arrillaga, N. Watson, S. Chen, Identification ofharmonic sources of power systems using state estimation, IEEProc. Gen. Transm. Distrib. Part C 146 (1) (1999) 7–12.

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A Method of Utilsing NonSource Measurements for Harmonic State Estimation