00843057

246 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 2, APRIL 2000

Reactive Power Measurement Using theWavelet Transform

Weon-Ki Yoon and Michael J. Devaney, Member, IEEE

AbstractThis paper provides the theoretical basis for themeasurement of reactive and distortion powers from the wavelettransforms. The measurement of reactive power relies on theuse of broad-band phase-shift networks to create concurrentin-phase currents and quadrature voltages. The wavelet realpower computation resulting from these 90 phase-shift networksyields the reactive power associated with each wavelet frequencylevel or subband. The distortion power at each wavelet subbandis then derived from the real, reactive and apparent powers ofthe subband, where the apparent power is the product of the v; ielement pair's subband rms voltage and current. The advantage ofviewing the real and reactive powers in the wavelet domain is thatthe domain preserves both the frequency and time relationshipof these powers. In addition, the reactive power associated witheach wavelet subband is a signed quantity and thus has a directionassociated with it. This permits tracking the reactive power flowin each subband through the power system.

Index TermsDigital signal processing, phase shift networks,measurement, power, RMS, subband, wavelets.

I. INTRODUCTION

TRADITIONAL power measurements have been per-formed in both the time domain and, to a lesser extent, inthe frequency domain using the Fourier Transform approach.The time domain approach is the most efficient and mostaccurate when rms and real power as well as their dependentquantities such as reactive power and power factor are con-cerned. This is because the starting point for all digital methodsare the voltage and current waveforms concurrently sampledat uniform intervals over one or more cycles. The frequencydomain approach permits the determination of distortionand harmonic influences but suffers from the requirement ofperiodicity and the loss of temporal insight. Even with thesubstantial efficiencies provided by the class of Fast FourierTransform algorithms, it is the most computationally intensiveover any span of frequencies since its spectral results are equalintervaled in frequency.

The advantage of power measurements using the wavelettransform data of each voltage and current element pair is that itpreserves both the temporal and spectral relationship associatedwith the resulting powers. That is, it provides the distribution ofthe power and energy with respect to the individual frequencyoctaves associated with each level of the wavelet analysis.Instead of breaking the spectrum into a set of bands of uniform

Manuscript received May 26, 1999; revised November 18, 1999.The authors are with Digital Power Instrumentation Group, Department of

Electrical Engineering, University of Missouri, Columbia, MO 65211 USA(e-mail: [email protected]; [email protected]).

Publisher Item Identifier S 0018-9456(00)02426-8.

frequency width as the FFTs, it yields a smaller number of binswhich relate the rms, power, and energy in octaves. The spanof each bin has twice the bandwidth of the next lower bin. Eachof subbands represents that part of the original instantaneouspower occurring at that particular time and in that particularfrequency band [7].

For reactive power measurement, analog 90 phase-shift net-works were used in [2] and the outputs of the networks werequadrature voltages vquad

and in-phase currents iin-phase

. Com-pared to the analog networks, the digital phase-shift networksprovide greater accuracy because their numeric coefficients arenot changed by temperature or drift. There are three differentmethods to design the digital 90 phase-shift networks; 1) theequal-ripple; 2) the maximally-flat; and 3) the weighted leastsquare methods. The former two methods are based on stableanalog allpass filter designs [3], [4]. The analog allpass net-works are then transformed to digital allpass networks by thebilinear transform. These digital allpass networks are stable andyield the order of filters from the specified conditions. On thecontrary, the weighted least square method is used in the di-rect design of digital phase-shift networks without the bilineartransform [5]. Several specified frequency points are weightedand the phase results of the networks at those frequency pointsare very accurate. The disadvantage of the least square methodis that the resulting design is sometimes unstable. Therefore,the former two methods are more useful and convenient in thedesign of the phase-shift networks. These two methods will bestudied and their relative advantages and disadvantages identi-fied.

The wavelet transform and the digital phase-shift networksare applied to the proposed reactive power measurement. Thev

quad

and iin-phase

wavelet transforms are derived from a se-quence of concurrent vquad

i

in-phasesamples using a common

orthonormal wavelet basis applied over each power systemcycle. Since the individual subbands for vquad

and iin-phase

are registered in both time and frequency, each associatedv

quad

i

in-phase

product subband represents the contribution ofthis band to the total vquad

i

in-phaseelement reactive power

or cycle reactive energy. The summation of these signedsubband powers then results in the total reactive power for thisv

quad

i

in-phaseelement pair.

II. POWER DEFINITIONS

Definitions of various types of powers are found in the IEEEStandard Dictionary of Electrical and Electronics Terms [IEEEStd. 100-88] [1].

00189456/00$10.00 2000 IEEE

YOON AND DEVANEY: REACTIVE POWER MEASUREMENT USING THE WAVELET TRANSFORM 247

Fig. 1. The characteristics of 90 phase-shift networks in a broad-band frequency range (solid line: Maximally-Flat, dotted line: Equal-Ripple).

If vt

and it

are periodic signals with period T , then real powerP is given as follows:

P =

1

T

Z

T

0

i

v

v

t

dt: (1)

Reactive energy is defined as a quantity measured by a per-fect watt-hour meter which carries the current of a single-phasecircuit and a voltage equal in magnitude to the voltage acrossthe single-phase but in quadrature therewith [2]. The voltagev

t

leads the voltage in quadrate vt90

by 90 at each frequencyover its range. If vt90

and it

are periodic signals with periodT , then reactive power Q is given as follows:

Q =

1

T

Z

T

0

i

t

v

t90

dt: (2)

Apparent power U for single-phase circuits is simply theproduct of the rms voltage V and the rms current I . Phasorpower S, Distortion power D and Fictitious power F areexpressed in term of the apparent, real and reactive power andgiven as follows:

S =

p

P

2

+ Q

2 (3)D =

p

U

2

S

2 (4)andF =

p

Q

2

+D

2

: (5)

III. CALCULATIONS OF POWERS USING THEWAVELET TRANSFORM

The equations of both the rms and the real power using thewavelet transform were proved in [6], [7]. The following are

extended forms for the digital signal application and the reactivepower calculation.

Analog signals, it

; v

t

and vt90

, are periodic waves with aperiod T , and in

; v

n

and vt90

(n) are digitized signals of it

; v

t

and vt90

, respectively, with n = 0; 1; ;2N 1 for theperiod T . Voltage in-quadrature vt90

lags the voltage signalv

t

by 90 at each frequency over its range.The rms values of current and voltage with respect to their

associated scaling and wavelet levels are given as follows:

I

rms

=

s

1

T

Z

T

0

i

2

t

dt

=

v

u

u

t

1

2

N

2

N

1

X

n=0

i

2

n

=

v

u

u

t

1

2

N

2

j

0

1

X

k=0

c

2

j

0

;k

+

N1

X

jj

0

1

2

N

2

j

1

X

k=0

d

2

j;k

=

v

u

u

t

I

2

j

0

+

N1

X

jj

0

I

2

j

(6)

and

V

rms

=

s

1

T

Z

T

0

v

2

t

dt

=

v

u

u

t

1

2

N

2

N

1

X

n=0

v

2

n

=

v

u

u

t

1

2

N

2

j

0

1

X

k=0

c

02

j

0

;k

+

N1

X

jj

0

1

2

N

2

j

1

X

k=0

d

02

j;k

=

v

u

u

t

V

2

j

0

+

N1

X

jj

0

V

2

j

; (7)

where cj

0

;k

and c0j

0

;k

are scaling coefficients of in

and vn

,

respectively, at scaling level j0

and time k. dj;k

and d0j;k

are


Fig. 2. Diagram of new power metering system using digital 90 phase-shift networks and the wavelet transforms.

wavelet coefficients of in

and vn

, respectively, at wavelet levelj and time k.

The real and reactive powers with respect to their associatedscaling and wavelet levels are given as follows:

P =

1

T

Z

T

0

i

t

v

t

dt

=

1

2

N

2

N

1

X

n=0

i

n

v

n

=

1

2

N

2

j

0

1

X

k=0

c

j

0

;k

c

0

j

0

;k

+

N1

X

jj

0

1

2

N

2

j

1

X

k=0

d

j;k

d

0

j;k

= P

j

0

+

N1

X

jj

0

P

j

(8)

and

Q =

1

T

Z

T

0

i

t

v

t90

dt

=

1

2

N

2

N

1

X

n=0

i(n)v

t90

(n)

=

1

2

N

2

j

0

1

X

k=0

c

j

0

;k

c

00

j

0

;k

+

N1

X

jj

0

1

2

N

2

j

1

X

k=0

d

j;k

d

00

j;k

= Q

j

0

+

N1

X

jj

0

Q

j

(9)

where c00j

0

;k

and d00j;k

are scaling and wavelet coefficients ofv

t90

(n) at level j0

and j, respectively, and time k.

TABLE IPOINTS, FREQUENCY BANDS AND ODD

HARMONICS OF THE WAVELET LEVELS AT 128 (27) POINTS PER CYCLE.(NOTE: 2 IS THE SCALING LEVEL)

Table I shows wavelet and scaling levels with their associ-ated numbers of coefficients, the frequency ranges and theirharmonics when power system signals are sampled at 128 (27)points per the fundamental cycle (60 Hz).

IV. DIGITAL 90 PHASE-SHIFT NETWORKSIn reactive power [(9)], there are 90 phase differences

between voltage signals v(n) and vt90

(n) at each frequencyover its range. For the realization of the relationship betweenv(n) and vt90

(n), 90 phase-shift networks are generallyused and very effective. Input signal v(n) is fed to a digital 90phase-shift networks and the outputs are the in-phase outputv

i

(n) and the quadrature output vq

(n). The vi

(n) leads vq

(n)

by 90 at each frequency over its range. The following twoprocedures are based on the theory of constant phase-shiftnetworks in [3] and [4].


TABLE IIHARMONICS AND THEIR PHASES OF THE SIMULATED i(t) and v(t) IN THE

SCALING AND WAVELET LEVELS

TABLE IIITRUE VALUES OF RMS AND POWER MEASUREMENTS OF THE

SIMULATED POWER SIGNALS

TABLE IVRESULTS OF RMS & POWER MEASUREMENTS OF THE SIMULATED POWERSIGNALS USING IIR (L = 6) POLYPHASE WAVELET TRANSFORM AND 90

PHASE-SHIFT NETWORKS

A. Equal-Ripple MethodThe Jacobi elliptic functions together with the bilinear trans-

form are very useful to design a pair of allpass networks whosephase difference is the closest possible approximation to 90over an interval of frequencies.

Assume the two phases are 1

= 90

" and 2

= 90

+

", and " is very small nonnegative number. The procedure forapproximating 90 with an error of " in the frequency ranges!

a

! !

b

. is given as follows:1) Compute

=

tan(!

a

=2)

tan(!

b

=2)

(10)

and

1

=

tan(

1

=2)

tan(

2

=2)

=

1 tan("=2)

1 + tan("=2)

2

: (11)

2) Compute the order N 0 = (K0()K(1

)=K()K

0

(

1

)),

and force N 0 to be the next higher integer.WhereK();K0();K(1

), andK0(1

) are the com-plete elliptic integrals of the first kind. For example,K()and K0() are defined by

K() =

Z

=2

0

d

(1

2

sin

2

)

1=2

(12)

Fig. 3. Energy flow between buses 1 and 2 of EMTP simulation.

and

K

0

() = K(

p

1

2

): (13)

3) Compute poles of allpass filters as follows:

p

l

= tan

!

a

2

sn[(4l + 1)K

0

()=2N

0

;

0

]

cn[(4l + 1)K

0

()=2N

0

;

0

]

(14)

for l = 0; 1; ; N 01, and sn and cn are Jacobi ellipticfunctions.

4) For the negative pl

, compute the digital coefficients ofin-phase allpass network,

z

1;l

=

1 + p

l

1 p

l

(15)

and for the positive pl

, compute the digital coefficients ofquadrature allpass network,

z

2;l

=

1 p

l

1 + p

l

: (16)

B. Maximally-Flat MethodThe procedure for approximating 90 with an error of " in the

frequency ranges !a

! !

b

is given as follows:1) Compute

=

tan(!

a

=2)

tan(!

b

=2)

(17)

and

W

0

=

p

tan(!

a

=2) tan(!

b

=2): (18)

2) Compute the order N 0 = (tanh1[tan(=4 "=2)])=tanh

1

p

) and force N 0 to be the next higher integer.3) Compute poles of allpass filters as follows:

p

l

= (1)

l+1

W

0

tan

(l + :5)

2N

0

for l = 0; 1; ; N 0 1: (19)

4) For the negative pl

, compute the digital coefficients ofin-phase allpass network,

z

1;l

=

1 + p

l

1 p

l

(20)


Fig. 4. Real energy of four cycles at each wavelet levels from 3 to 5.

and for the positive pl

, compute the digital coefficients ofquadrature allpass network,

z

2;l

=

1 p

l

1 + p

l

: (21)

C. Comparison of Equal-Ripple and Maximally-FlatQuadrature Phase-Shift Methods

If power system signals are sampled at 128 points per thefundamental cycle (60 Hz), the sampling frequency is equal to7680 Hz and by the Nyquist rate, the band limit of the signals is3840 Hz.

In Fig. 1, the equal-ripple method of 90 phase-shift networksis compared with the maximally-flat method where each of thenetworks have the same maximum allowable phase error, 0.5,and frequency range from 46.93 to 3626.7 Hz. In the case of theequal-ripple method, the total order N 0 of allpass filters is onlyten compared with 67 of the maximally-flat method.

When the frequency band is narrow, the result of the maxi-mally-flat method is much more accurate than that of the equal-ripple. But, as its frequency range is broadened, the result ofthe maximally-flat method with the same phase error is worseat both start and stop frequencies. The equal-ripple method hasequal ripple phase errors around 90, but the ripple error is thesame whether its frequency range is broad or narrow. So, ifthe power measurement is applied to the broad-band test, the

equal-ripple method is more effective than the maximally-flatmethod.

V. POWER MEASUREMENT STRATEGYFig. 2 illustrates the proposed power metering system based

on (6)(9). The signals v(n) and i(n) are sampled at 2N pointsper the fundamental cycle. The ii

(n) is the in-phase output ofcurrent i(n). The vq

(n) is the quadrature output of voltage v(n).Outputs of the wavelet transform blocks are wavelet coefficients(dx

N1;k

dx

2;k

) and scaling coefficients (cx2

;k

) at time kwhere x represents one of the four signals (v(n); i(n); ii

(n) andv

q

(n)). The wavelet levels are from 2 to N 1 and the scalinglevel is 2 as shown in Table I. VN1

V

2

and IN1

I

2

are therms results of voltage and current with respect to their associatedwavelet levels from N 1 to 2. V2

and I2

are the rms valuesof scaling level 2. PN1

P

2

and QN1

Q

2

are the real andreactive powers with respect to their associated wavelet levelsfrom N 1 to 2. P2

and Q2

are the real and reactive powersat scaling level 2.

VI. EVALUATION

Based on the proposed power measurement method, two datasets are examined under steady- state conditions: The first isderived from analytic signals; the second is data derived fromEMTP (Electro-Magnetic Transient Program) simulation of en-ergy flow between two buses. The evaluation of analytic signals


Fig. 5. Reactive energy of four cycles at each wavelet level from 3 to 5.

proves that proposed power measurements, based on the wavelettransform and 90 phase-shift networks, are highly accurate.

A. Power Calculations of Simulated Power SignalTest input signals, current i(t) and voltage v(t) have several

harmonics with their associated phases, respectively, as shownin Table II. The first harmonic is in scaling level 2, the fifthin wavelet level 3, the eleventh and thirteenth in wavelet level 4,the twenty-third in wavelet level 5, and the forty-fifth in waveletlevel 6. Every harmonic has the same rms value of 1. The fun-damental frequency is 60 Hz. These signals are sampled at 128(27) points per cycle.

1) True Values of RMS and Power Measurements: Table IIIshows the true values of the power measurements. Irms andV rms are rms values of current and voltage, respectively. U;P;and Q are apparent, real,and reactive powers with their associ-ated wavelet levels, respectively. S;D; and F are phasor, dis-tortion, and fictitious powers, respectively, with their associatedlevels.

2) RMS and Power Measurements Using the WaveletTransform: Tables III and IV illustrate the results for the truevalues and compared them to the others using the IIR (L = 6)polyphase wavelet transform. This IIR wavelet transform isintroduced in [6][8].

The results of total Irms; V rms, apparent, real, and fictitiouspowers are same in all cases. This proves that the proposed rmsand power calculation methods using the wavelet coefficients

are correct. For the computation of reactive power, equal-ripplemethod of 90 phase-shift networks is used with the phase error(" = 0:01

) and the frequency range 46.9333413.333 Hz.The errors of total reactive, phasor and distortion powers resultfrom the approximation of the equal-ripple method, but the er-rors are generally quite small.

In the IIR polyphase wavelet transform, a small amount ofleakage occurs at each wavelet level due to the roll-off charac-teristics of the low-pass and high-pass filter pairs. Compared tothe true values of power measurements at each level, the resultsof the application of the IIR (L = 6) polyphase wavelet trans-form are very accurate.

B. Energy Flow Analysis of EMTP DataFig. 3 is an example of energy flow between buses 1 and 2.

The source is located in BUS 1 and the load in BUS 2 consistsof R2

and L3

, as shown in the figure. The conductors betweenbuses 1 and 2 are 1 mile long and equal toR1

and L2

in a series.For the analysis of energy flow at each wavelet level, the sourceis included with several frequency components as follows:

V (n) = 133K

p

2Sin(260n)

+ 13:3K

p

2 Sin(2360n+ 90

)

+ 6:65K

p

2 Sin(2700n+ 45

)

+ 6:65K

p

2 Sin(21400n+ 180

):


With respect to V (n), the second term is the sixth harmonic,located in level 3 of Table I. The third and fourth terms arethe twelfth and the twenty-fourth harmonics with the same rmsvalue of 6.65K in levels 4 and 5, respectively.

Fig. 4 is the real energy bar graph of four cycles at eachwavelet level of buses 1 and 2, based on (8) and using the IIR(L = 6) polyphase wavelet transform. Fig. 5 is the reactive en-ergy bar graph of four cycles at each level of buses 1 and 2, basedon (9) and using the IIR (L = 6) polyphase wavelet transformand 90 phase-shift networks.

In these figures, energy of each level at BUS 1 is larger thanthat with its associated level at BUS 2, which means energy ofeach level flows from BUS 1 to BUS 2. The energy at level 3is larger than energies at level 4 and 5. As a result, the figuresillustrate the direction and amount of the flow of the real andreactive energies between buses 1 and 2 with their associatedwavelet levels.

VII. CONCLUSIONThe simulated signal test on various types of powers demon-

strated that the results from the IIR (L = 6) polyphase wavelettransform and the equal-ripple method of 90 phase-shift net-works were in good agreement with the reference for the totalpower measurements. As shown in Table III, the individual sub-band rms and power contributions of the IIR filter banks werevery accurate because of the IIR filter's sharper roll-off charac-teristics.

Energy flows between buses 1 and 2 were analyzed by EMTPdata under steady-state conditions. The real and reactive powerswith their associated wavelet levels are signed quantities andthus had directions associated with them. In Figs. 4 and 5, theproposed method's energies, computed at each level, are close tothe true powers of each level. This permits tracking the real andreactive energy flows at each wavelet level through the powersystem.

The reactive phase shifting filter and the dyadic filters associ-ated with the concurrent voltage and current wavelet transformsrequire synchronously sampled data. However, if the voltageand current samples are acquired asynchronously with a suffi-ciently small inter-sample interval, simultaneous interpolation,in synchronism with the power system, would permit the realand reactive wavelet transform algorithms to be compute asshown.

This study demonstrated the extension of the wavelet trans-form to the measurement of reactive power through the use ofa broad-band quadrature phase-shift networks. The proposedwavelet-based power metering system of Fig. 2 was introducedfor computing the rms value of the voltage and current and thereal and reactive power with their associated wavelet levels,respectively, from the v-i wavelet transform pair and thequadrature v-i wavelet transform pair. In the proposed meteringsystem, powers at each wavelet level retained both the temporal

and spectral relationship associated with the powers from theproperty of wavelets.

REFERENCES[1] P. S. Filipski, Y. Baghzouz, and M. D. Cox, Discussion of power defi-

nitions contained in the IEEE dictionary, IEEE Trans. Power Delivery,vol. 9, pp. 12371244, July 1994.

[2] B. Djokic, P. Bosnjakovic, and M. Begovic, New method for reactivepower and energy measurement, IEEE Trans. Instrum. Meas., vol. 41,pp. 280285, Apr. 1992.

[3] J. E. Storer, Passive Network Synthesis. New York: McGraw-Hill,1957, pp. 298302.

[4] B. Gold and C. M. Rader, Digital Processing of Signals. New York:McGraw-Hill , 1969, pp. 9092.

[5] S. S. Kidambi, Weighted least-squares design of recursive allpassfilters, IEEE Trans. Signal Processing, vol. 44, pp. 15531557, June1996.

[6] W.-K. Yoon and M. J. Devaney, Power measurement based onthe wavelet transform, IEEE Trans. Instrum. Meas., vol. 47, pp.12051210, Oct. 1998.

[7] W.-K. Yoon, Power measurements via the wavelet transform, Ph.D.dissertation, Univ. Missouri, Columbia, Dec. 1998.

[8] A. N. A. Mark and J. T. Smith, Subband and Wavelet Transforms Designand Application. Norwood, MA: Kluwer, 1996.

[9] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM,1992.

Weon-Ki Yoon was born in Seoul, Korea. Hereceived the B.S. degree in electronic engineeringfrom Hanyang University at Seoul in 1986, and theM.S. and Ph.D. degrees in electrical engineeringfrom the University of Missouri, Columbia, in 1995and 1998, respectively.

He had industrial experiences with Dae-YoungElectronic Co., Korea, in designing analog anddigital telecommunication systems from 1986 to1989, and with LG Electronic Co., Korea, in satelliteTV receiver design from 1989 to 1991. He was a

Research Assistant on digital power metering at the University of Missourifrom 1996 to 1998. He is currently with Tadiran Microwave Networks as aSignal Processing Engineer.

Michael J. Devaney (S'60M'64) was born in St.Louis, MO. He received the B.S.E.E. degree fromthe University of Missouri, Rolla, in 1964 and theM.S. and Ph.D. degrees in electrical engineeringfrom the University of Missouri, Columbia, in 1967and 1971, respectively.

He worked for the Bendix Corporation (nowAllied-Signal) from 1964 to 1967, in automatedtest equipment design and joined the faculty of theElectrical and Computer Engineering Departmentof the University of Missouri, Columbia, in 1969,

where he is now an Associate Professor. From 1974 to 1979, he was anInvestigator at the university's John M. Dalton Research Center, where heworked on bio-telemetry and instrumentation for the study of micro-circulation.From 1980 to 1988, he was the Undergraduate Program Director for ComputerEngineering and in 1987 he became affiliated with the university's PowerElectronics Research Center and served as its Associate Director in 1989 and1990. He has published 12 journal articles, 29 conference papers, and for thepast eleven years, has been engaged in research in power metering and powerquality measurement supported by Square D.

Dr. Devaney was the Associate Editor of the IEEE Engineering in Medicineand Biology News from 1978 to 1979.

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