A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 17-25, 2007
A New Algorithm for Reactive Electric Power
Measurement
Adalet Abiyev1
Girne American University, Departmernt of Electrical Electronics Engineering,
Mersin 10, Turkey
Abstract
In this paper a Walsh function (WF) based new algorithm for reactive power
measurement is presented. The proposed algorithm allows the convenient
calculating process to obtain a reactive power from the entire instant power signal
without time delay between current and voltage signal. To test the validity of the
suggested approach the simulation tool has been developed by use of “Matlab 6.5”
software environment.
Keywords: Discrete Walsh function, active power, reactive power, phase shift
Introduction
The reactive power directly influences the power factor and, as a result, overloads
the connecting cables between the electrical energy sources and energy user
devices and plays a vital role in the stable operation of power systems (Fairney
1994).
The extension of the wavelet transform to the measurement of power components
(reactive power and active power) through the use of a broad-band quadrature
phase-shift networks is demonstrated in (Yoon & Devaney 2000). The proposed
wavelet-based power metering system requires the phase shift of the input voltage
signal. According to Purkayastha & Savoie (1990) the amplitude-pulse modulation
together with phase shift operation is used to measure reactive power in the
frequency range of from 50 to 70 Hz. An electronic shifter based on stochastic
signal processing for simple and cost-effective digital implementation of a reactive
power and energy meter has been proposed by Djokic et al. (2000). Toral et al.
(2001), suggest a computer algorithm for calculating reactive (quadrature) power.
In scientific papers the averaging of the value of the product of the current samples
and the voltage samples with shifting to the quarter one of the samples (current or
1
aabiyev@gau.edu.tr
17
Algorithm for Electric Power Measurement
voltage) relatively to another is used. The Fourier transformation including fast
Fourier transformation based digital or analogue filtering algorithms, used to
measure reactive power, involves quite complex computations.
In single-phase circuits an instant electric power (EP) can be evaluated by using
different methods depending on load type. In a simple case, when a source voltage,
u(t), and a current flowing through load, i(t), are the pure sinusoidal signals, the
instant EP is defined by Abijev (2006).
p (t ) = P − [P cos 2ω t + Q sin 2ω t ]
(1)
For applying a discrete WF for measuring EP components the instant power, p(t ) ,
is written in the discrete form as
⎡
⎛ 2π
⎞
⎛ 2π
⎞⎤
P(n) = P − ⎢ P cos⎜ 2
∆t ⋅ n ⎟ + Q sin⎜ 2
∆t ⋅ n ⎟⎥
⎝ T
⎠
⎝ T
⎠⎦
⎣
(2)
Where ∆t is the time interval between neighbourhood samples, ∆t =T/N, N is the
number of samples within observation period of T, n=0, 2, ...N-1. Considering
these equalities, final expression for power is written as
⎡
⎛ 4π ⎞
⎛ 4π ⎞⎤
P(n ) = P − ⎢ P cos⎜
⋅ n ⎟ + Q sin ⎜
⋅ n ⎟⎥
N
⎝
⎠
⎝ N
⎠⎦
⎣
(3)
Thus we have suitable expressions (Eq. 2 and Eq. 3) allowing for simultaneous
measuring of active and reactive components of the EP by use of WF.
Analogue Measurement Approach
This approach is based on analogue signal processing theory and implies the
integral of the product of instant power, p (t ) , and corresponding order of analogue
WF:
S ( m) =
18
1
nT
nT
∫ p(t )Wal (m, t )dt
0
(4)
A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 17-25, 2007
where m is an order of WF, n is the required integer number of averaging periods
T. The higher the n the lower the random error influence to the measurement
results.
When m=0, considering Eq. 2 we get from Eq. 4:
S ( 0) =
1
nT
nT
∫ {[ P − ( P cos 2ω t + Q sin 2ω t )]Wal (0, t )}dt
(5)
0
Since zero-order WF, Wal (0, t ) has only +1 value in the normalized period of T[7],
Eq.(5) results in average(active) power, P:
S ( 0) =
1
nT
nT
∫ Pdt = P
(6)
0
When m=3, considering Eq. 2 we get from Eq.4:
nT
S (3) =
1
{[ P − ( P cos 2ω t + Q sin 2ω t )]Wal (3, t )}dt
nT ∫0
(7)
Third-order WF, Wal (3, t ) is an odd function with normalized cycling period of T/2
and is orthogonal with the P cos 2ω t term as it should be (figure 1). So
1
nT
nT
∫ P cos 2ω tWal (3, t )dt = 0
0
The average power, P has constant value within a period of T, therefore:
1
nT
nT
∫ PWal (3, t )}dt = 0 . So, S (3) =
0
1
nT
nT
∫
Q sin 2ω tWal (3, t )}dt
(8)
0
As can be seen from figure 1, the product of Q sin 2ω t ⋅ Wal (3, t )}dt results in
rectification of the reactive component of the entire instant power signal, p(t ) . As
a result integral of Eq.(8) gives the average value of the reactive power, Q .
19
Algorithm for Electric Power Measurement
Figure 1. Graphical interpretation of the rectifying effect
Digital Measurement Approach
For digital measuring of the active and reactive power components we use discrete
expression of the WF (Abiyev & Aliyev2003, Abijev 2006).
Wal (i, β k ) = (− 1)
m ⎛
∑ ⎜⎝ ω m − k +1 ⊕ωm − k ⎞⎟⎠ β k
k =1
(9)
where i is order of WF in the WF system, i=0,1,2,…,N-1, β k is argument of WF
and defines the bit(digit) coefficients of β k represented in binary code,
β = (β1 , β 2 ...β k )2 , β k = 0,1 , ω m is the bit(digit) coefficients of ω m represented
in binary code, ω = (ω 0 , ω1 , ω 2 ...ω m )2 , ω m = 0,1 , m is a in binary representation
of highest-order WF serial number in the WF system. For example, if number of
samples, N=64 the dimension of WF system would also be 64. So from N = 2 m
we get m = 6.
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A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 17-25, 2007
Now we can write general expression for digitally measuring of the EP by use of
equalities of Trautman (1975) and Abiyev (2006):
m
S (i ) =
∑ (ω m− k +1⊕ω m− k )β k
1
∑ P (n)(−1) k =1
N n =0
N −1
(10)
For measuring the reactive component of the EP we use the third- order WF,
Wal (3, β k ) . For the third-order Walsh function ω = 3 therefore only ω 6 = 1 and
ω 5 = 1 . The remaining bit coefficients of ω m , m = 1,2,3,4 are equal to the zero:
3 = (0000011)2 . In this case the third-order WF is
W (3, β 2 ) = (-1) ( ω 5 ⊕ ω 4 ) β 2 = (-1) (1 ⊕ 0 ) β 2 = ( − 1) β 2
(11)
The argument, β 2 changes depending on normalized time of T=0.02s as shown in
figure 2.
Figure 2.
Time representation of final three bit coefficients of the binary
representation of β k
21
Algorithm for Electric Power Measurement
Figure 3 depicts the β k and third-order discrete WF, W (3, β 2 ) = (−1) β 2 .
So for the third-order component of the EP from Eq.(10) and Eq.(13) we obtain the
equality
S (3) =
1
N
N −1
∑ P(n)(−1) β
(12)
2
n =0
Figure 3. Time representation of β k and third-order discrete WF
Considering Eq.(3) we have
S (3) =
1
N
N −1
⎛
⎡
⎛ 4π
∑ ⎜⎜ P − ⎢⎣ P cos⎜⎝ N
n=0 ⎝
⎞
⎛ 4π ⎞⎤ ⎞⎟
⋅ n ⎟ + Q sin ⎜
⋅ n ⎟⎥ (−1) β 2
⎠
⎝ N ⎠⎦ ⎟⎠
The next terms of this sum are equal to zero:
1
N
N −1
∑
n =0
P (−1) β 2 = 0 and S (3) =
1
N
N −1
⎛ 4π
∑ P cos⎜⎝ N
n =0
Considering these, Eq. (13) becomes(see figure 3):
22
⎞
⋅ n ⎟(−1) β 2 = 0
⎠
(13)
A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 17-25, 2007
S (3) =
N / 2 −1
3 N / 4 −1
N −1
1 ⎡ N / 4 −1
⎛ 4π ⎞
⎛ 4π ⎞
⎛ 4π ⎞
⎛ 4π ⎞ ⎤
Q sin ⎜
n⎟ −
Q sin ⎜
n⎟ +
Q sin ⎜
n⎟ −
Q sin ⎜
n ⎟⎥
⎢
N ⎣ n=0
⎝ N ⎠ n= N / 4
⎝ N ⎠ N /2
⎝ N ⎠ n =3 N / 4
⎝ N ⎠⎦
∑
∑
∑
∑
(14)
The analysis of intervals indicated in the figure 3 shows that since the function of
sin (4πn / N ) has negative values at the intervals of [N/4, N/2-1] and [3N/4, N-1],
then the Eq. (14) results in:
S (3) =
1
N
N −1
⎛ 4π ⎞
∑ Q sin⎜⎝ N n ⎟⎠
(15)
n=0
⎛ 4π ⎞
This expression defines the average value of the signal of Q sin ⎜
n ⎟ and is
⎝ N ⎠
proportional to the average reactive value of the EP in the investigated circuit.
During experimental studying the input voltage, u (t ) , and the current, i (t ) , signals
were taken as
u (t ) = U m sin(ω t ) and i (t ) = I m sin(ω t − ϕ ) ,
where I m =2A, U m =4V, ω = 2πf , f = 50 is the linear frequency in Hz, ω =314 is
frequency in rad/sec, ϕ -phase shift between the voltage, u (t ) and the current, i (t )
signals.
During experimental studies the phase shift, ϕ between the voltage, u (t ) and the
current, i (t ) signals has been varied in the interval of ϕ = 0 − 90° .
The signal proportional to the instant value of the power, p(t) which is applied to
the first nputs of the pair of multipliers is represented in figure 4 and is written as
p(t) = 8 sin(314 t ) ∗ sin(314 t − ϕ )
The time representation of the signals S0(t) and S3(t) are shown in the figure 4. The
essential advantages of the proposed method for the measuring of the reactive
power have been verified by experimental studies. One of these advantages is that,
in contrast to the known existing methods, the proposed method does not require
the time delay of the current signal to the π / 2 with respect to the voltage signal.
The time delaying process requires the corresponding hardwire which may result in
the additional measurement error.
23
Algorithm for Electric Power Measurement
Figure 4. Component output signals versus time representation
Conclusion
The evaluation and measurement of components of EP with application of a Walsh
function simplifies the volume of computing operations on some order in
comparison with sets of algorithms based on decomposition of signals on
harmonics (trigonometric components). Measuring of the reactive EP Walsh
functions results in certain advantages:
- during the processing of the signals on the base of Walsh functions the timeshifting of the signals acts on the structure of the signals. This influence becomes
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A. Abiyev, GAU J. Soc. & Appl. Sci., 2(4), 17-25, 2007
useful during the evaluation of the power components allowing the obtaining of
extra knowledge concerning the phase-shifts on the harmonics of the input signals;
- during digital signal processing the sample values of the signals multiplication by
Walsh functions is replaced by summing of the samples with the corresponding
+1or -1 signs.
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