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Estimating General Pollution Waveforms in

Non-Monitored Electrical System Buses

Jacques Szczupak,Fellow, IEEE1

Abstract-- This work studies the estimation of injected currentsin internal system buses. It generalizes previous studies on

harmonic pollution estimation, such as to allow estimating

general injected current waveforms. The electrical network is

first modeled in order to derive necessary and sufficient

conditions to permit single frequency component estimation. This

result is shown to be sufficient when applicable to the desired

spectrum region. Digital signal processing techniques are used as

a basis to return from frequency-domain back into time-domain,

recovering the unknown waveform. An illustrative case supports

the theory, indicating excellent match between injected and

recovered currents. Computational load is discussed and possible

reduction procedures are suggested.

Index Terms Quality, Signal Processing, Estimation,

Harmonics, Injections

I. INTRODUCTION

RESERVING energy quality is an important factor in

order to keep electrical utility customers satisfied.

However, to maintain minimum standards of quality is not

only expensive, but frequently also technically difficult to

achieve. Electrical pollution may have unknown time varying

sources, from load polluting customers to malfunctioning

equipments. Correctly monitoring quality is essential to limit

its effects by some type of control procedure; technically and,

sometimes, commercially acting on the pollution source.

It is common sense in the electrical sector to face the large

variety of electrical disturbances by assuming itsdecomposition in a collection of qualified signal distortions

such as harmonics, sags, swells, flickers etc. This tactic makes

it easier to understand and study causes and consequences of

the sets of signal perturbations. For instances, specific studies

and estimating techniques exist and are being developed for

each of the phenomena.

Estimating the pollution source is not a simple problem.

System dimension/complexity and the large variety of

possible pollution distortions make it a difficult task. It is

common practice to reduce problem complexity by studying

pollution estimation and source location, restricted to the

existence of just one of the previously mentioned phenomena.

Practical observations, however, indicate a commonsituation; the pollution involves more than one of the

previously mentioned types of perturbations. Not only this,

but sometimes the phenomena somehow interact, as for

instance is the case of larger sags associated with larger

harmonic energies.

J. Szczupak is with the Catholic University of Rio de Janeiro (PUC-RJ) and

Engenho Pesquisa, Desenvolvimento e Consultoria Ltda., Rio de Janeiro,

Brasil (e-mailjsz@engenho.com)

As a result, the energy quality estimation problem may getinvolved, even for the case of directly monitored electrical

signals.

Economical reasons limit the number of signal-monitored

buses, requiring pollution estimation for non-monitored buses.

This has been successfully done, for instances, in the

harmonic pollution case.

II. NETWORKMODEL

The power network is usually composed by linear and

nonlinear subsystems, making it difficult to find a general and

adequate model for representation. The network nonlinear

subsystems include operating equipments, such as saturated

power transformers, but may also result from nonlinear loads,

partly responsible for the network loss of quality. As a result,

the selected network model must deal with unknown, possibly

time-variant, types of non-linear behavior.

The selected model assumes the non-linear network to be

monitored by L access ports, delimiting its external frontiers.

In addition, extra monitoring ports are created by the

extraction of non-linear sub-systems from the network. This

extraction procedure may be understood as pulling the non-

linear device out of the network without destroying its

electrical connections. As such, new access ports are created,

L+1, , L+K, corresponding to the non-linear sub-system

ports, as indicated in Figure 1.

Figure 1 Transformer Equivalent Circuit

P

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The resulting network is still non-linear, since there might

be unknown internal non-linear loads. These internal loads are

modeled by proper injection of currents to the internal buses.

Figure 2 represents the final linear network model, after

extraction of the non-linear loads. The total number of ports is

now N, consisting of L external, K non-linear sub-systems

extraction related and (N-L-K) representing the non-linear

loads internal buses injections.

Fig. 2 Linear N-Port Network

In practice the model requires monitoring all first (L+K)

ports, such as the port vectors

[ ]t

1 2 K= V V ... V1V (1)

and

[ ]t

1 1 2 K = I I ... II (2)

are supposed to be known.

All monitored port voltages and currents are supposed

acquired with sampling frequency fT, covering from dc to the

Nyquist frequency fT/2 [1]. The data acquisitions are supposed

to be satellite synchronized.

The unknown port variables are described by the port vectors

[ ]t

2 K+1 K+2 N= V V ... VV (3)

and

[ ]t

2 K+1 K+2 N= I I ... II . (4)

The port admittance matrix [2] description was chosen for

the linear network , with no loss of generality. The resulting

N-port description is

1 11 12

2 21 22

1

2

=

Y YY Y

I V

I V, (5)

where the index 1 identifies monitored variables and index 2

stands for unknown.

III. NETWORKESTIMATION MODEL

If exists, it is possible to determine from (5)-1

12Y

-1

12 (= 2 1 11Y Y )1V I V (6)

and substituting its value on eq. (5) it follows

21 22 2Y Y= +2 1I V V . (7)

The existence of this type of matrix description and of

depends on the network topology and parameters and on

the relative port locations. Any choice for the Monitoring portlocations satisfying these conditions may be used.

-1

12Y

It should be noticed that the network modeling is all

frequency domain based. This naturally leads to the harmonic

case estimation procedure, since it consists of applying the

derived equations for the fundamental and for each of the

harmonic frequencies [3-8]. In the case of general waveforms

this is no longer the situation, since the frequency spectrum is

no more discrete, but continuous. This more general injection

pollution problem is discussed next.

IV. GENERALWAVEFORMCURRENT INJECTION

Although general waveforms require a continuous

frequency spectrum description, it is possible to very closelyapproximate this spectrum by sampling it at a sufficiently

dense discrete number of points. This can be done by

uniformly sampling the voltages and currents on the jw-axis

(of the Laplace plane), satisfying the Nyquist frequency [1].

Therefore, eqs. (6) and (7) are successfully applied so as to

uniformly sample the values of the pair of vectors (V2, I2) . At

the end of this sampling evaluations, each of the elements of

V2 and I2 may now form a complex vector ordering its

frequency sampled values.

Clearly, it is not necessary to evaluate for positive and

negative frequencies, since one may use the symmetry and

antisymmetry properties for the corresponding spectrum realand imaginary parts. However, the final vectors must contain

all those values.

Applying the IFFT (Inverse Fast Fourier Transform)

operator to each of the frequency domain described vectors

results the equivalent time-domain pair of vectors (v2, i2)

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The proper choice of sampling frequency is strongly

dependent of the analysis objectives. In the case of harmonics

this is quite obvious, choosing a sampling frequency of at

least twice the highest harmonic frequency. For general

waveforms, the usual constraint is related to spikes or any

other fast varying situation to be registered. The faster the

transient, the higher the sampling frequency.

Window length is a second parameter to be considered. If a

continuous recovery is necessary, subsequent windows shouldhave their results concatenated by known procedures [1].

Larger window length imply heavier computation and delay in

applying the IFFT. The designer will have to balance between

larger window length with less concatenation to next window

effort or smaller windows (faster IFFT) with a larger number

of concatenation to next window operations. The decision

depends on the application.

V. ILLUSTRATIVECASE

The chosen illustrative case is based on the meshed six

buses system shown in Figure 3, slightly modified from the

original WSCC (Western System Coordinating Council)

unifilar diagram [10].

1,026 3,7

j0,0586j0,0625

0,0085 j0,072 0,0119 j0,1008

0,032

j0,161

0,01j0,088

0,017

j0,092

0,039

j0,17

1

2 8

5 6

4

7

9

10

11

12

13

14

15

j0,0576

-j4,399

-j5,9878

-j5,5844-j3,5274

-j3,878-j4,151

0,913

j0,311

j0,2707 j0,3287

0,6831,02SWING

3

1,016 0,7 1,032 2,0

0,996 -4,0 1,013 -3,7

1,026 -2,2

I1

I3I2

Fig. 3 Case Sudy Network

Computer synthesized current waveforms, besides the

usual 60 Hz currents indicated in the figure, were injected into

the buses. These currents are composed not only by

fundamental and harmonic components, but they also include

non-harmonic modulating sinusoids as well as exponentially

decreasing magnitude spikes of random nature.

The set of bus injected current spectrum magnitude is

shown in Figure 4. The legend identifies the corresponding

bus numbering. As it is possible to observe, it was supposed

buses 1, 2 and 3 are the external connections, while buses 5, 6

and 8 are those were non-linear loads could be connected. The

synthesized waveforms occupy practically all the frequency

spectrum.

Fig. 4 Magnitude of the Bus Injected Currents

The unknown vectors, (V2, I2), are estimated from the pair

of known vectors (V1, I1), following eqs. (6) and (7),

sweeping the frequency axis, up to the Nyquist frequency. In

the sequence, the estimated frequency-domain pair of vectors,

(V2, I2), is IFFT mapped back into time-domain (v2, i2). The

result offers an almost perfect match between original and

reconstructed waveforms.

This is illustrated in Figure 5 displaying bus 8 original and

estimated currents. It is possible from this figure to observe

the complexity of the recovered waveform.

Fig. 5 Injected Current in Bus 8

Figure 6 represents the detail of bus 8 current also

exhibiting almost perfect match between original and

recovered waveforms.

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Fig. 6 Injected Current in Bus 8 Detail

Figure 7 displays details of the bus 6 injected currents,

while figures 8 and 9 shows bus 5 currents, the whole windowand a detail, respectively. Purposely the waveforms are

severely distorted with respect to the desired sinusoidal

waveform. This was done in order to emphasize the methods

possibilities.

All the figures indicate a very close match between original

and recovered currents. It is possible to detect, mostly in the

details, almost unnoticeable mismatches.

Real life data will not fit as nicely as those, computer

synthesized, used in the illustrative case. There will exist

sampling time uncertainties as well as quantization effects,

from A/D conversion to arithmetic word length nonlinear

effects, altering the exact results. The author is now pursuing

this line of research.

Fig. 7 Injected Current in Bus 6 Detail

Fig. 8 Injected Current in Bus 5

Fig. 9 Injected Current in Bus 5 - Detail

VI. COMPUTATIONALCOMPLEXITY

Although simple, the method involves a tremendous

computational load. This load is strongly dependent on the

size of the network under analysis and on the number of times

one should obtain , one for each sampled frequency.-1

12Y

It is possible to reduce this computational load by

restricting the sampling rate, but this is not always possible, as

previously discussed. A second approach is to obtain at a

reduced number of points, generating the remaining samples

by interpolation procedures. Multirate digital signal methods

are available for this purpose [1].

-1

12Y

This last approach was used in the illustrative case

resulting a reduction of at least an order of magnitude in the

computational load. This result is not always possible to

generalize.

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It is strongly connected to the network under examination

and a good hint on the result might come from the

interpretation of in eq. (6). It acts as the impedance

matrix mapping port 1 currents into voltages in port 2. For

instance for the chosen ports, in the used network, this almost

reduces in low frequencies to series RL sub networks. As a

result the impedance elements in have almost constant

real part, while the imaginary part varies linearly withfrequency. A simple linear interpolation procedure would

work as well as the more sophisticated used. This simple case

will probably be found repeatedly due to the very nature of

power networks.

-1

12Y

-1

12Y

VII. CONCLUSIONS

A new method is presented in order to estimate general

pollution injection in the electric energy network. The

approach is general, not being restricted to radial topology

power networks.

The original network is modified by the introduction of

extra monitoring ports, simplifying the network description

and allowing establishing a one to one relation betweenunknown injections and monitoring port measurements. A

mathematical description of the transformed network, leads to

a close form solution to the problem.

It is shown, for the general type of waveform injection case

that one may sequentially operate in frequency and time

domains alternating from one to the other. The approach leads

to very accurate signal reconstructions, as illustrated with a

meshed type of network. The injected pollution was not

limited to harmonics, but included modulating signals and

random generated spikes. A digital signal multirate approach

is suggested in order to reduce computational complexity in

large networks. The idea being to evaluate a reduced number

of inverse matrices and digitally interpolate the missingvalues.

VIII. REFERENCES

[1] S.K.MITRA, Digital Signal Processing A Computer-Based Approach,

McGraw-Hill, 1998

[2] N. BALABANIAN and T. A. BICKART, Network Theory, John Wiley

and Sons, 1969.

[3] A. C. PARSONS, W. M. GRADY, E. J. POWERS and J. C. SOWARD,

"A Direction Finder for Power Quality Disturbances Based upon

Disturbance Power and Energy", Proceedings of the 8th International

Conference on Harmonics and Quality of Power, pp. 693 to 699, Greece,

October 1998.

[4] A. TESHOME, "Harmonic source and type identification in a radial

distribution system," in Proc. 1991 IEEE Industry Applications Society

Annual Meeting, pp. 1605-1609.

[5] H. MA and A. A. GIRGIS, "Identification and tracking of harmonic

sources in a power system using a kalman filter," IEEE Transactions on

Power Delivery, vol. 11, no. 3, pp. 1659-1665, July 1996.

[6] V. L. PHAM, K. P. WONG, N. R. WATSON, and J. ARRILLAGA, "A

method of utilizing non-source measurements for harmonic state

estimation," ELSEVIER Electric Power Systems Research, vol. 56, pp.

231-241, May 2000.

[7] J. ARRILLAGA, M. H. J. BOLLEN, and N. R. WATSON, "Power

quality following deregulation," (invited paper) Proceedings of the

IEEE, vol. 88, no. 2, pp. 246-261, Feb. 2000.

[8] J. SZCZUPAK and G.P.B.CASTRO,Estimao e Localizao de

Fontes Poluidoras de Harmnicos, XVII SNPTEE, Brazil, 2003.

[9] J. SZCZUPAK, R. A. F. CHACON, P. A. M-S. DAVID, "Real TimePrecise Harmonic Estimation", Proceedings of the 13th Power System

Computation Conference, PSCC, vol.: II, pp. 1063 to 1069, Throndheim,

Norway, 1999.

[10] M. A. PAI e P. W. SAUER, Power System Dynamics and Stability,

Prentice Hall, 1998.

[11] S. HAYKIN, Adaptive Filter Theory, Prentice Hall, Upper Side River,

NJ, third edition, 1996.

IX. BIOGRAPHY

Jacques Szczupak (Fellow, IEEE) is an Electrical Engineer, graduated in

1964 (UFRJ), M.Sc. in 1967 (UFRJ) and Ph.D. in 1975 (University of

California). He was a professor in COPPE/UFRJ, leader of the Signal

Processing group (CEPEL) and is now a full professor at PUC-RIO. He is the

technical director of Engenho. Dr. Szczupak participated in different

committees and working groups, was editor of many technical magazines,

founded the IEEE Circuits and Systems Rio de Janeiro Chapter and was

Circuits and Systems Region IX President. His areas of interest include

Instrumentation, Signal Processing, Energy, Quality and Simulation.