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An Adaptive IIR Phase Equalizer for Electronic Energy Meters
An Adaptive IIR Phase Equalizer for Electronic Energy Meters
S.

Jayasimha and M. S. Shastri, Signion Systems Ltd., Hyderabad, India


Abstract: International Standards for static
watt-hour meters require wideband phase
equalization of current/ voltage transformer
(CT/ VT). The proposed adaptive filter
structure equalizes CT/ VT (whose responses
are near-ideal, but whose transfer function
parameters are unknown) phase without
altering their magnitude responses. A
calibration procedure, together with its
performance measures, is presented.

1. Introduction

As electronic meters are superior to their
electromechanical predecessors in terms of
accuracy, billing, tamper-detection
capabilities, flexibility, manufacturability,
reliability and cost, their use in power system
networks is becoming widespread. The
International Electrotechnical Commission
(IEC) recommends two standards for static
watt-hour meters (based on maximum
percentage of error). IEC-687 [1] applies to
0.2% and 0.5% class meters while IEC-1036
[2] applies to 2% and 1% accuracy for watt-
hour measurement. The American National
Standards Institutes ANSI C12.16-1991 [3]
specifies S and A classes of accuracy for
watt-hour measurements. Accuracy
requirements on reactive and apparent
energies are generally specified by national
standards [4]. Draft standards also require the
meter to register line frequency energy and
total energy (line frequency energy plus
harmonic energy) separately.

Line voltage distortion (pollution) occurs
when users draw large harmonic currents.
Current measurement should be accurate
when up to 30% total harmonic distortion is
present [4]. Harmonics required for
conformance testing to BS5406 and
EN60555-2[5] (standards for disturbances in
power supplies caused by household
appliances, that specify current thresholds up
to the 31
st
harmonic) should also be
measured.

a . H a rm o n ic D isto rtio n
b . N o tc h in g


Figure 1. Quasi-stationary distortion in supply
voltage

Electronic energy meters are also expected to have
field programmable current transformer (CT)
and voltage transformer (VT) turns ratios and
input-output maps. Thus, when a meter is
connected to external CTs and VTs, it can
equalize VT/ CT phase and CT non-linearity
[4]. Residual phase error (after phase
equalization) between CT and PT paths, at
any power factor and multiples of line
frequency, should not exceed 1
°. An adaptive
procedure that equalizes CT and VT phase
over a range of frequencies is described.

2. Adaptive IIR structure for phase
equalization

Lumped linear component models for CTs
and PTs are shown in Figure 2 (after
translating secondary circuits to the primary),
where R
p
=primary resistance, L
p
=primary
inductance,
R
s
=primary equivalent of
secondary resistance, L
s
=primary equivalent
of secondary inductance, R
e
=primary
equivalent of load resistance, I
o
=no load
current, E
p
=primary induced voltage and E
s
=primary equivalent of secondary induced
voltage. In terms of these parameters, the
transfer functions are:

P s
V
V
n sL R
s L L
L
L L
s L R
R
R L
L
L R
R R
R
s
p
p
o
e
s
o
p
o
p
s
p
e
s
o
p
o
p
p
e
s
( )
(
) / { [ (
)
]
[
(
)(
)
]
(
)}
'
'
'
'
'
'
'
=
=
+
+
+
+
+
+
+
+
2
+


(1)

where,n
p
<1.

C s
V
I
n sL R
s L
L
R
R
s
p
c
o
e
o
s
s
e
( )
(
) (
)
'
'
'
'
=
=
+
+
+
,
n
c
>1(2).

where n
p
and n
c
are the secondary to primary
turns ratios of PT and CT. Due to
manufacturing tolerances, the parameters R
and L are known only approximately, and in
many cases, all that is known is that the CT/
PT transfer functions are close to unity. The
PT and CT transfer functions are suitably
discretized (the bilinear transform or the
impulse invariance methods [6]). Using the
bilinear transform, the discrete time transfer
functions are:

H z
b
b z
a z
c
c
c
c
( )
=
+
+
0
1
1
0
1
1


(3)

H z
b
b z
b z
a z
a z
p
op
p
p
p
p
( )
=
+
+
+
+
1
1
2
2
0
1
1
2
1

(4)


where estimates of a
c
s and b
c
s are refined by
the adaptive system of Figure 3. If H
p
(z)
follows C(s) and H
c
(z) follows P(s), phase is
equalized without significant amplitude
distortion (as C(s) and P(s) are all-pass) in the
desired frequency-range.
R
p
sL
p
sL
o
R
o
R
e
V
p
R
s
sL
s
V
s
E
s
=E
p
I</i>o
(a)
PT
Model

R
p
sL
p
sL
o
R
o
R
e
i
p
R
s
sL
s
i
s
I</i>o
V
p
V
s
(b) CT Model

Figure 2. PT and CT Model


The model parameters are estimated during
the calibration process by applying a
wideband excitation to both the CT and VT
after initializing the digital IIR CT and VT
filter parameters (of Figure 3) to nominal
values. In many cases of interest, the PT
transfer function can be assumed to be
known. This is because a voltage
transformers input voltage varies only within
±20% of a nominal voltage (the line voltage),
while a current transformer should be
accurate from 0.5% to 120% of the rated
value. Thus, for a CT, a single model (using
lumped linear components) is not accurate at
all amplitudes
1
.
A Hilbert transform (with a passband
covering the range of frequencies of interest)
pair follows the IIR filter pair, P(z) and C(z),
the outputs of which are multiplied to obtain
an error signal which adjusts the parameters
of the IIR filter C(z) using a variation of the
output error method [7].The latter method,

1
When there is significant non-linearity in the CT,
phase calibration at a few current set points may be
required.
where the mean square error of the difference
between x
1
(n) and x
2
(n) is minimized, could
be used if one of the transfer functions is
close to ideal (a transfer function of unity). As
described in the next section, this assumption
can be relaxed if the variation on the output
error method described by Figure 3 is used.
Also, the method of Figure 3 equalizes phase
by only making minor perturbations to the
initial parameter values, rather than the (in
general) large perturbations needed for both
magnitude and phase equalization of the
output error method. The computational
requirements of Figure 3 (executed during
calibration only) is dominated by the Hilbert
transformer, while the IIR filters (executed
during normal operation as well) are of low
complexity.

In Figure 3, if P(z) is initialized to H
p
(z), and
C(z) adapts to H
c
(z), y
1
(n) and y
2
(n), are
orthogonal and the expected value of e(n) is
zero. However, if C(z) is different from H
c
(z),
the outputs y
1
(n) and y
2
(n) will not be
orthogonal, and e(n) can be used to adjust
coefficients of C(z) such that mean error
squared E e n
2
[ ( )], where E[ ] is the
expectation operator, is minimized. The
minimization procedure applies a correction
equal to a step-size multiplied by the negative
of a vector of partial derivatives to the
parameters of C(z).
w n
w n
n
(
)
( )
( )
+ =

1
µ

(5)
where w(n) is the coefficient vector at n
th

iteration and
µ
the step-size.
(n), the
gradient at n
th
iteration, is

=
( )
( )
[ ( )]
n
w n E e n
2


=
2
1
2
E e n
w n E y n y n
[ ( )]
( ) [ ( ) ( )]


=
2
1
2
E e n E y n w y n
[ ( )] [ ( )
{ ( )}]
(6)
as P(z) is not adapted by the adaptation
procedure. From figure(3), y
2
(n) is given by
the difference equation

y n
b x n p
b x n p
c
c
2
0
2
1
2
1
( )
(
)
(
)
= +



a y n
c
0
2
1
(
)
(7)
where p is the delay in the Hilbert Transform-
pair, y
2
(n) and x
2
(n) are the output and input
sequences respectively and b
0c
, b
1c
and a
0c
are
the coefficients to be updated by the
adaptation procedure. Taking the partial
derivatives of y
2
(n) with respect to the
coefficients
b
0c
(n),
b
1c
(n)and
a
0c
(n),
adaptation of k (in the present case, 2)
numerator coefficients is done as follows:
b n
b n
y n
ic
ic
( )
(
)
(
)
=
1
2
1
1
µ

E e n
X n
ic
[ (
)]
(
)
1
1
, 0
i<k (8)
where
(
)
X n
x n i
p
ic
(
)
=

1
1
2



= a
n
X n
l
l
c
ic
l
m
(
)
(
)
(
).
1
1
1
1

where m is the number of denominator
coefficients (in the present case, 1).

y
2
(n)
y
1
(n)
e(n)
x
1
(n)
x
2
(n)
Current channel
C(s)
P(s)
ADC
P(z)
C(z)
Hilbert
Delay
Voltage channel
Step-size
Avg


Figure 3. Phase estimation procedure.

The adaptation of m denominator coefficients
is as follows:

a n
a n
y n
jc
jc
( )
(
)
(
)
=
1
2
1
1
µ