Reactive Power Measurement Using the Wavelet Transform
Weon-Ki Yoon and Michael J. Devaney, Member, IEEE
Abstract—This paper provides the theoretical basis for the measurement of reactive and distortion powers from the wavelet transforms. The measurement of reactive power relies on the use of broad-band phase-shift networks to create concurrent in-phase currents and quadrature voltages. The wavelet real power computation resulting from these 90?phase-shift networks yields the reactive power associated with
each wavelet frequency level or subband. The distortion power at each wavelet subband is then derived
from the real, reactive and apparent powers of the subband, where the apparent power is the product of the v,i element pair's subband rms voltage and current. The advantage of viewing the real and reactive powers in the wavelet domain is that the domain preserves both the frequency and time relationship of these powers. In addition, the reactive power associated with each wavelet subband is a signed quantity and thus has a direction associated with it. This permits tracking the reactive power flow in each subband through the power system.
Index Terms——Digital signal processing, phase shift networks, measurement, power, RMS, subband,
TRADITIONAL power measurements have been performed in both the time domain and, to a lesser extent, in the frequency domain using the Fourier Transform approach. The time domain approach is the most efficient and
most accurate when rms and real power as well as their dependent quantities such as reactive power and power factor are concerned. This is because the starting point for all digital methods are the voltage and current waveforms concurrently sampled at uniform intervals over one or more cycles. The frequency domain approach permits the determination of distortion and harmonic influences but suffers from the requirement of periodicity and the loss of temporal insight. Even with the substantial efficiencies provided by the class of Fast Fourier Transform algorithms, it is the most computationally intensive over any span of frequencies since its spectral results are equal intervaled in frequency.
The advantage of power measurements using the wavelet transform data of each voltage and current element pair is that it preserves both the temporal and spectral relationship associated with the resulting powers. That is, it provides the distribution of the power and energy with respect to the individual frequency octaves associated with each level of the wavelet analysis. Instead of breaking the spectrum into a set of bands of uniform frequency width as the FFT’s, it yields a smaller number of
bins which relate the rms, power, and energy in octaves. The span of each bin has twice the bandwidth of the next lower bin. Each of subbands represents that part of the original instantaneous power occurring at that particular time and in that particular frequency band.【7】
For reactive power measurement, analog 90? phase-shift networks were used in  and the outputs of
the networks were quadrature voltages vquad and in-phase currents iin-phase. Compared to the analog
networks, the digital phase-shift networks provide greater accuracy because their numeric coefficients are not changed by temperature or drift. There are three different methods to design the digital 90?
phase-shift networks; 1) the equal-ripple; 2) the maximally-flat; and 3) the weighted least square methods. The former two methods are based on stable analog allpass filter designs , . The analog allpass networks are then transformed to digital allpass networks by the bilinear transform. These digital allpass networks are stable and yield the order of filters from the specified conditions. On the contrary, the weighted least square method is used in the direct design of digital phase-shift networks without the bilinear transform . Several specified frequency points are weighted and the phase
results of the networks at those frequency points are very accurate. The disadvantage of the least square method is that the resulting design is sometimes unstable. Therefore, the former two methods are more useful and convenient in the design of the phase-shift networks. These two methods will be studied and their relative advantages and disadvantages identified.
The wavelet transform and the digital phase-shift networks are applied to the proposed reactive power measurement. The vquad and iin-phase wavelet transforms are derived from a sequence of
concurrent vquad–iin-phase samples using a common orthonormal wavelet basis applied over each power system cycle. Since the individual subbands for vquad and iin-phase are registered in both time and
frequency, each associated vquad–iin-phase product subband represents the contribution of this band to
the total vquad–iin-phase element reactive power or cycle reactive energy. The summation of these signed
subband powers then results in the total reactive power for this vquad–iin-phase element pair.
II. POWER DEFINITIONS
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 power P is given as follows:
Reactive energy is defined as a “quantity measured by a perfect watt-hour meter which carries
the current of a single-phase circuit and a voltage equal in magnitude to the voltage across the single-phase but in quadrature therewith” . The voltage vt leads the voltage in quadrate vt-90?by 90?
at each frequency over its range. If vt-90?and it are periodic signals with period T, then reactive power
Q is given as follows:
Apparent power U for single-phase circuits is simply the product of the rms voltage V and the
rms current I. Phasor power S, Distortion power D and Fictitious power F are expressed in term of the apparent, real and reactive power and given as follows:
;;; CALCULATIONS OF POWERS USING THE WAVELET TRANSFORM
The equations of both the rms and the real power using the wavelet transform were proved in , . The following are extended forms for the digital signal application and the reactive power
Analog signals, it; vt and vt：90? , are periodic waves with a period T, and in; vn and vt：90?(n) are
N ?：1 for the period T. digitized signals of it; vt and vt：90? , respectively, with n = 0; 1; …; 2
Voltage in-quadrature vt：90? lags the voltage signal vt 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:
where cj0;k and c’j0;k are scaling coefficients of in and vn, respectively, at scaling level j0 and time k. dj;k
and d’j;k are wavelet coefficients of in and vn, respectively, at wavelet level j and time k.
Fig. 2. Diagram of new power metering system using digital 90_ phase-shift networks and the wavelet transforms.
The real and reactive powers with respect to their associated scaling and wavelet levels are given as follows:
where c’’j0;k and d’’j;k are scaling and wavelet coefficients of vt-90?(n) at level j0 and j, respectively, and time k.
TABLE I POINTS, 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 associated numbers of coefficients, the
frequency ranges and their harmonics when power system signals are sampled at 128 (27) points per the fundamental cycle (60 Hz).
IV. DIGITAL 90_ PHASE-SHIFT NETWORKS
In reactive power [(9)], there are 90? phase differences between voltage signals v(n) and vt-90?(n) at
each frequency over its range. For the realization of the relationship between v(n) and vt-90?(n), 90?
phase-shift networks are generally used and very effective. Input signal v(n) is fed to a digital 90?
phase-shift networks and the outputs are the in-phase output vi(n) and the quadrature output vq(n). The
vi(n) leads vq(n) by 90? at each frequency over its range. The following two procedures are based on
the theory of constant phase-shift networks in  and .
HARMONICS AND THEIR PHASES OF THE SIMULATED i(t) and v(t) IN THE SCALING AND WAVELET LEVELS
TRUE VALUES OF RMS AND POWER MEASUREMENTS OF THE SIMULATED POWER SIGNALS
RESULTS OF RMS & POWER MEASUREMENTS OF THE SIMULATED POWER SIGNALS USING IIR (L = 6) POLYPHASE WAVELET
TRANSFORM AND 90?PHASE-SHIFT NETWORKS
A. Equal-Ripple Method
The Jacobi elliptic functions together with the bilinear transform are very useful to design a pair
of allpass networks whose phase difference is the closest possible approximation to 90?over an
interval of frequencies.
-ε andΦ2=90?+ε, and ε is very small nonnegative Assume the two phases are Φ1=90?
number. The procedure for approximating 90?with an error of ?ε in the frequency ranges ωa?
ωb?ωc is given as follows:
2) Compute the order N`= (K`(κ)K(κ)=K(κ)K`(κ1)), and force N` to be the next higher integer.
Where K(κ);K`(κ);K(κ1), and K`(κ1) are the complete elliptic integrals of the first kind. For
example,K(κ)and K`(κ) are defined by~
Fig. 3. Energy flow between buses 1 and 2 of EMTP simulation
3) Compute poles of allpass filters as follows:
for l = 0; 1; …;N`-1, and sn and cn are Jacobi elliptic functions. 4) For the negative pl , compute the digital coefficients of in-phase allpass network,
and for the positive pl , compute the digital coefficients of quadrature allpass network,
B. Maximally-Flat Method
The procedure for approximating 90?with an error of " in the frequency rangesωa?ω?ωb is given as follows:
-1-12) Compute the order N` = (tanh[tan(π/4 –ε/2)])= tanh?κ) and force N` to be the next
3) Compute poles of allpass filters as follows:
4) For the negative pl , compute the digital coefficients of in-phase allpass network,
Fig. 4. Real energy of four cycles at each wavelet levels from 3 to 5.
and for the positive pl , compute the digital coefficients of quadrature allpass network,
C. Comparison of Equal-Ripple and Maximally-Flat Quadrature Phase-Shift Methods
If power system signals are sampled at 128 points per the fundamental cycle (60 Hz), the sampling frequency is equal to 7680 Hz and by the Nyquist rate, the band limit of the signals is 3840 Hz. In Fig. 1, the equal-ripple method of 90? phase-shift networks is compared with the
maximally-flat method where each of the networks have the same maximum allowable phase error, 0.5?, and frequency range from 46.93 to 3626.7 Hz. In the case of the equal-ripple method, the total order N` of allpass filters is only ten compared with 67 of the maximally-flat method.
When the frequency band is narrow, the result of the maximally-flat method is much more accurate than that of the equalripple. But, as its frequency range is broadened, the result of the maximally-flat method with the same phase error is worse at both start and stop frequencies. The equal-ripple method has equal ripple phase errors around 90?, but the ripple error is the same whether
its frequency range is broad or narrow. So, if the power measurement is applied to the broad-band test, the equal-ripple method is more effective than the maximally-flat method.
V. POWER MEASUREMENT STRATEGY
Fig. 2 illustrates the proposed power metering system based on (6)–(9). The signals v(n) and i(n)
Nare sampled at 2 points per the fundamental cycle. The ii(n) is the in-phase output of current i(n). The
vq(n) is the quadrature output of voltage v(n). Outputs of the wavelet transform blocks are wavelet coefficients(dxN-1;k–dx;k) and scaling coefficients (cx2_;k) at time k where x represents one of the four 2
signals (v(n); i(n); ii(n) and vq(n)). The wavelet levels are from 2 to N -1 and the scaling level is 2*as
shown in Table I. VN-1–V2 and IN-1–I2 are the rms results of voltage and current with respect to their associated wavelet levels from N -1 to 2. V2*and I2* are the rms values of scaling level 2*. PN-1–P2 and
QN-1–Q2 are the real and reactive powers with respect to their associated wavelet levels from N -1 to 2.
P2*and Q2*are the real and reactive powers at scaling level 2*.
Based on the proposed power measurement method, two data sets are examined under steady- state conditions: The first is derived from analytic signals; the second is data derived from EMTP (Electro-Magnetic Transient Program) simulation of energy flow between two buses. The evaluation of
analytic signals proves that proposed power measurements, based on the wavelet transform and
90? phase-shift networks, are highly accurate.
Fig. 5. Reactive energy of four cycles at each wavelet level from 3 to 5.
A. Power Calculations of Simulated Power Signal
Test input signals, current i(t) and voltage v(t) have several harmonics with their associated
phases, respectively, as shown in Table II. The first harmonic is in scaling level 2，, the fifth
in wavelet level 3, the eleventh and thirteenth in wavelet level 4, the twenty-third in wavelet level 5,
and the forty-fifth in wavelet level 6. Every harmonic has the same rms value of 1. The fundamental
7frequency is 60 Hz. These signals are sampled at 128(2) points per cycle；
1) True Values of RMS and Power Measurements: Table III shows the true values of the power
measurements. Irms and Vrms are rms values of current and voltage, respectively. U; P;and Q are
apparent, real,and reactive powers with their associated wavelet levels, respectively. S;D; and F
are phasor, distortion, and fictitious powers, respectively, with their associated levels.
2) RMS and Power Measurements Using the Wavelet Transform: Tables III and IV illustrate
the results for the true values and compared them to the others using the IIR (L = 6) polyphase
wavelet transform. This IIR wavelet transform is introduced in –
The results of total Irms; V rms, apparent, real, and fictitious powers are same in all cases.
This proves that the proposed rms and power calculation methods using the wavelet coefficients are correct. For the computation of reactive power, equal-ripple method of 90?phase-shift networks is
used with the phase error (ε= ?0.01?) and the frequency range 46.933–3413.333 Hz. The errors of
total reactive, phasor and distortion powers result from the approximation of the equal-ripple method, but the errors are generally quite small. In the IIR polyphase wavelet transform, a small amount of leakage occurs at each wavelet level due to the roll-off characteristics of the low-pass and high-pass filter pairs. Compared to the true values of power measurements at each level, the results of the application of the IIR (L = 6) polyphase wavelet transform are very accurate.
B. Energy Flow Analysis of EMTP Data
Fig. 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 consists of R2 and L3, as shown in the figure. The conductors between buses 1 and 2 are 1 mile long and equal to R1 and L2 in a series. For the analysis of energy flow at each wavelet level, the source is included with several frequency components as follows:
With respect to V(n), the second term is the sixth harmonic,located in level 3 of Table I. The third and fourth terms are the twelfth and the twenty-fourth harmonics with the same rms value of 6.65K in levels 4 and 5, respectively.
Fig. 4 is the real energy bar graph of four cycles at each wavelet level of buses 1 and 2, based on (8) and using the IIR (L = 6) polyphase wavelet transform. Fig. 5 is the reactive energy bar graph of four cycles at each level of buses 1 and 2, based on (9) and using the IIR (L = 6) polyphase wavelet
transform and 90? phase-shift networks.
In these figures, energy of each level at BUS 1 is larger than that with its associated level at BUS 2, which means energy of each level flows from BUS 1 to BUS 2. The energy at level 3 is larger than energies at level 4 and 5. As a result, the figures illustrate the direction and amount of the flow of the real and reactive energies between buses 1 and 2 with their associated wavelet levels.
The simulated signal test on various types of powers demonstrated that the results from the IIR (L = 6) polyphase wavelet transform and the equal-ripple method of 90? phase-shift networks were in
good agreement with the reference for the total power measurements. As shown in Table III, the individual subband rms and power contributions of the IIR filter banks were very accurate because of the IIR filter's sharper roll-off characteristics.
Energy flows between buses 1 and 2 were analyzed by EMTP data under steady-state conditions. The real and reactive powers with their associated wavelet levels are signed quantities and thus had directions associated with them. In Figs. 4 and 5, the proposed method's energies, computed at each