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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 2, APRIL 2000
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, wavelets.
I. INTRODUCTION
RADITIONAL 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
T
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 [2] 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 [3], [4]. 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 [5]. 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
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: weonyoon@hotmail.com; DevaneyM@missouri.edu).
Publisher Item Identifier S 0018-9456(00)02426-8.
Definitions of various types of powers are found in the IEEE
Standard Dictionary of Electrical and Electronics Terms [IEEE
Std. 100-88] [1].
0018–9456/00$10.00 © 2000 IEEE
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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).
P
If vt and it are periodic signals with period T , then real power
is given as follows:
P
=
Z
1
T
0
T
(1)
iv vt dt:
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” [2]. The voltage
vt leads the voltage in quadrate vt090 by 90 at each frequency
over its range. If vt090 and it are periodic signals with period
T , then reactive power Q is given as follows:
Q
=
1
T
Z
0
T
0
it vt
90
S
D
=
=
and
F
=
P2
U2
Q2
+
0
Q2
(3)
S2
=
=
=
(5)
THE
The equations of both the rms and the real power using the
wavelet transform were proved in [6], [7]. The following are
sZ
vuu X0
t
vuu X0
t
T
1
0
T
1
2N
2j0 1
k
2 +
0
and
Vrms
=
=
=
vu X0
t
X0 X0
+
j
1
N
2N
0
=0
i2
n
n
1
N
cj ;k
0
=0
Ij
2
2N 1
1
2
it dt =
j
(4)
+ D2 :
III. CALCULATIONS OF POWERS USING
WAVELET TRANSFORM
Irms
(2)
dt:
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:
p
p
p
extended forms for the digital signal application and the reactive
power calculation.
Analog signals, it ; vt and vt090 , are periodic waves with a
period T , and in ; vn and vt090 (n) are digitized signals of it ; vt
and vt090 , respectively, with n = 0; 1; 1 1 1 ; 2N 0 1 for the
period T . Voltage in-quadrature vt090 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:
0
j
1
2j 1
2N
k
=0
2
dj;k
2
(6)
Ij
j
vuu X0
sZ
t
vuu X0 X0 X0
0
0
t
vu X0 t
T
1
0
T
1
2N
2
vt dt
2j0 1
k
2 +
0
j0 ;k
=0
N
Vj
j
2N
N
2
c
0
2
Vj ;
1
+
j
1
2N 1
1
=
j0
=0
2
vn
n
1
2N
2j 1
k
=0
2
dj;k
(7)
j
where cj0 ;k and c0j0 ;k are scaling coefficients of in and vn ,
respectively, at scaling level j0 and time k . dj;k and d0j;k are
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 2, APRIL 2000
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 level
j and time k .
The real and reactive powers with respect to their associated
scaling and wavelet levels are given as follows:
P
Z
1
=
T
k
P0+
j
Q=
=
=
Z
1
T
T
2N
+
j
0
2N
0
j
2j 1
j;k
k
=0
d0
j;k
(8)
i v 090 dt =
t
j0 ;k j0 ;k
=0
N
1
j
X c c00
X0 Q
k
n
j
2j0 01
Q0+
j
=0
N 1
n
1
N
t
0
1
j0 ;k j0 ;k
=0
j
and
n
N
2j0 01
2N
=
t
X c c0
X0 P
0
1
=
t
Xiv
X0 X0 d
2N 01
1
i v dt =
2
T
1
j
X i nv0 n
X0 X0 d d00
( )
2N
N
+
2N 01
1
=0
n
1
1
2j
02
j
1
j;k
N
j
t
k
=0
j ;k
j;k
t
(9)
j
j;k
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
90 ( )
0
where c000 and d00 are scaling and wavelet coefficients
v 090 (n) at level j0 and j , respectively, and time k .
j
TABLE I
POINTS, FREQUENCY BANDS AND ODD
HARMONICS OF THE WAVELET LEVELS AT 128 (27 ) POINTS PER CYCLE.
(NOTE: 23 IS THE SCALING LEVEL)
of
In reactive power [(9)], there are 90 phase differences
between voltage signals v(n) and vt090 (n) at each frequency
over its range. For the realization of the relationship between
v(n) and vt090 (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 [3] and [4].
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YOON AND DEVANEY: REACTIVE POWER MEASUREMENT USING THE WAVELET TRANSFORM
249
TABLE II
HARMONICS AND THEIR PHASES OF THE SIMULATED i(t) and v (t) IN THE
SCALING AND WAVELET LEVELS
TABLE III
TRUE VALUES
AND POWER MEASUREMENTS OF THE
SIMULATED POWER SIGNALS
OF
RMS
Fig. 3. Energy flow between buses 1 and 2 of EMTP simulation.
and
p
K 0 () = K ( 1 0 2 ):
(13)
3) Compute poles of allpass filters as follows:
!a sn[(4l + 1)K 0 ()=2N 0; 0 ]
(14)
2
cn[(4l + 1)K 0 ()=2N 0; 0]
for l = 0; 1; 1 1 1 ; N 0 0 1, and sn and cn are Jacobi elliptic
pl = 0 tan
TABLE IV
RESULTS OF RMS & POWER MEASUREMENTS OF THE SIMULATED POWER
SIGNALS USING IIR (L = 6) POLYPHASE WAVELET TRANSFORM AND 90
PHASE-SHIFT NETWORKS
functions.
4) For the negative pl , compute the digital coefficients of
in-phase allpass network,
z1;l =
1 + pl
1
0 pl
(15)
and for the positive pl , compute the digital coefficients of
quadrature allpass network,
z2;l =
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.
Assume the two phases are 1 = 90 0 " and 2 = 90 +
", and " is very small nonnegative number. The procedure for
approximating 90 with an error of 6" in the frequency ranges
!a ! !b . is given as follows:
1) Compute
=
tan(!a =2)
tan(!b =2)
tan(1 =2)
tan(2 =2)
=
1
0 tan("=2) 2 :
1 + tan("=2)
(10)
(11)
2) Compute the order N 0 = (K 0 ()K (1 )=K ()K 0 (1 )),
and force N 0 to be the next higher integer.
Where K (); K 0(); K (1 ), and K 0 (1 ) are the complete elliptic integrals of the first kind. For example, K ()
and K 0 () are defined by
K () =
Z
0
=2
d
2
2
(1 0 sin )1=2
0 pl :
(16)
1 + pl
B. Maximally-Flat Method
The procedure for approximating 90 with an error of " in the
frequency ranges !a ! !b is given as follows:
1) Compute
=
tan(!a =2)
(17)
tan(!b =2)
and
and
1 =
1
(12)
W0 =
p
tan(!a =2) tan(!b =2):
(18)
2) Computepthe order N 0 = (tanh01[tan(=4 0 "=2)])=
01 ) and force N 0 to be the next higher integer.
tanh
3) Compute poles of allpass filters as follows:
pl = (01)l+1 W0 tan
for l
(l + :5)
2N
= 0; 1;
0
1 1 1 ; N 0 0 1:
(19)
4) For the negative pl , compute the digital coefficients of
in-phase allpass network,
z1;l =
1 + pl
1
0 pl
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(20)
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 2, APRIL 2000
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,
equal-ripple method is more effective than the maximally-flat
method.
0
V. POWER MEASUREMENT STRATEGY
z2;l
=
1
pl
1 + pl
:
(21)
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 0 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
Fig. 2 illustrates the proposed power metering system based
on (6)–(9). The signals v (n) and i(n) are sampled at 2N 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 01;k –dx2;k ) and scaling coefficients (cx23 ;k ) at time k
where x represents one of the four signals (v(n); i(n); ii(n) and
vq (n)). The wavelet levels are from 2 to N 0 1 and the scaling
level is 23 as shown in Table I. VN 01 –V2 and IN 01 –I2 are the
rms results of voltage and current with respect to their associated
wavelet levels from N 0 1 to 2. V23 and I23 are the rms values
of scaling level 23 . PN 01 –P2 and QN 01 –Q2 are the real and
reactive powers with respect to their associated wavelet levels
from N 0 1 to 2. P23 and Q23 are the real and reactive powers
at scaling level 23 .
VI. EVALUATION
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
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YOON AND DEVANEY: REACTIVE POWER MEASUREMENT USING THE WAVELET TRANSFORM
251
Fig. 5. Reactive energy of four cycles at each wavelet level from 3 to 5.
proves that proposed power measurements, based on the wavelet
transform and 90 phase-shift networks, are highly accurate.
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 23 , 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 frequency is 60 Hz. These signals are sampled at 128
(27 ) points per cycle.
1) True Values of RMS and Power Measurements: Table III
shows the true values of the power measurements. I rms and
V rms 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 [6]–[8].
The results of total I rms; 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
(" = 60:01 ) 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:
p
V (n) = 133K 2 Sin(260n)
p
+ 13:3K 2 Sin(2 360n + 90 )
p
+ 6:65K 2 Sin(2 700n + 45 )
p
+ 6:65K 2 Sin(2 1400n + 180 ):
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 49, NO. 2, APRIL 2000
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.
VII. CONCLUSION
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 level, are close to
the true powers of each level. This permits tracking the real and
reactive energy flows at each wavelet level through the power
system.
The reactive phase shifting filter and the dyadic filters associated with the concurrent voltage and current wavelet transforms
require synchronously sampled data. However, if the voltage
and current samples are acquired asynchronously with a sufficiently small inter-sample interval, simultaneous interpolation,
in synchronism with the power system, would permit the real
and reactive wavelet transform algorithms to be compute as
shown.
This study demonstrated the extension of the wavelet transform to the measurement of reactive power through the use of
a broad-band quadrature phase-shift networks. The proposed
wavelet-based power metering system of Fig. 2 was introduced
for computing the rms value of the voltage and current and the
real and reactive power with their associated wavelet levels,
respectively, from the v -i wavelet transform pair and the
quadrature v -i wavelet transform pair. In the proposed metering
system, powers at each wavelet level retained both the temporal
and spectral relationship associated with the powers from the
property of wavelets.
REFERENCES
[1] P. S. Filipski, Y. Baghzouz, and M. D. Cox, “Discussion of power definitions contained in the IEEE dictionary,” IEEE Trans. Power Delivery,
vol. 9, pp. 1237–1244, July 1994.
[2] B. Djokic, P. Bosnjakovic, and M. Begovic, “New method for reactive
power and energy measurement,” IEEE Trans. Instrum. Meas., vol. 41,
pp. 280–285, Apr. 1992.
[3] J. E. Storer, Passive Network Synthesis. New York: McGraw-Hill,
1957, pp. 298–302.
[4] B. Gold and C. M. Rader, Digital Processing of Signals. New York:
McGraw-Hill , 1969, pp. 90–92.
[5] S. S. Kidambi, “Weighted least-squares design of recursive allpass
filters,” IEEE Trans. Signal Processing, vol. 44, pp. 1553–1557, June
1996.
[6] W.-K. Yoon and M. J. Devaney, “Power measurement based on
the wavelet transform,” IEEE Trans. Instrum. Meas., vol. 47, pp.
1205–1210, 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 Design
and Application. Norwood, MA: Kluwer, 1996.
[9] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM,
1992.
Weon-Ki Yoon was born in Seoul, Korea. He
received the B.S. degree in electronic engineering
from Hanyang University at Seoul in 1986, and the
M.S. and Ph.D. degrees in electrical engineering
from the University of Missouri, Columbia, in 1995
and 1998, respectively.
He had industrial experiences with Dae-Young
Electronic Co., Korea, in designing analog and
digital telecommunication systems from 1986 to
1989, and with LG Electronic Co., Korea, in satellite
TV receiver design from 1989 to 1991. He was a
Research Assistant on digital power metering at the University of Missouri
from 1996 to 1998. He is currently with Tadiran Microwave Networks as a
Signal Processing Engineer.
Michael J. Devaney (S'60–M'64) was born in St.
Louis, MO. He received the B.S.E.E. degree from
the University of Missouri, Rolla, in 1964 and the
M.S. and Ph.D. degrees in electrical engineering
from the University of Missouri, Columbia, in 1967
and 1971, respectively.
He worked for the Bendix Corporation (now
Allied-Signal) from 1964 to 1967, in automated
test equipment design and joined the faculty of the
Electrical and Computer Engineering Department
of the University of Missouri, Columbia, in 1969,
where he is now an Associate Professor. From 1974 to 1979, he was an
Investigator at the university's John M. Dalton Research Center, where he
worked on bio-telemetry and instrumentation for the study of micro-circulation.
From 1980 to 1988, he was the Undergraduate Program Director for Computer
Engineering and in 1987 he became affiliated with the university's Power
Electronics Research Center and served as its Associate Director in 1989 and
1990. He has published 12 journal articles, 29 conference papers, and for the
past eleven years, has been engaged in research in power metering and power
quality measurement supported by Square D.
Dr. Devaney was the Associate Editor of the IEEE Engineering in Medicine
and Biology News from 1978 to 1979.
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