COMPUTING and COMPUTATIONAL INTELLIGENCE
Numerical Simulations for Energy Calculation in Power Measurements
SORIN DAN GRIGORESCU
Faculty of Electrical Engineering
Politehnica University of Bucharest
Splaiul Independentei 313, s.6, Bucharest
ROMANIA
sorin.grigorescu@upb.ro
COSTIN CEPISCA
Faculty of Electrical Engineering
Politehnica University of Bucharest
Splaiul Independentei 313, s.6, Bucharest
ROMANIA
costin.cepisca@upb.ro
ION POTARNICHE
S.C. ICPE-Actel SA, Bucharest
ROMANIA
potarniche.ion@icpe-actel.ro
OCTAVIAN MIHAI GHITA
Faculty of Electrical Engineering
Politehnica University of Bucharest
Splaiul Independentei 313, s.6, Bucharest
ROMANIA
octavian.ghita@upb.ro
MIRCEA COVRIG
Faculty of Electrical Engineering
Politehnica University of Bucharest
Splaiul Independentei 313, s.6, Bucharest
ROMANIA
mircea.covrig@upb.ro
ELENA GRIGORESCU
S.C. DASON ELECTRO S.R.L.
Sos. Iancului 4, Bucharest
ROMANIA
elena_bortoi_grigorescu@yahoo.com
Abstract: – Measurements limitations of the reactive power flow in the power systems have represented, since the
early stage of the processes of high voltage networks design, construction and operation, an important issue for the
specialists. This paper analyses the most used competitive definitions for the reactive power, implemented in digital
energy meters, and compares the most usual measuring methods used in reactive energy measurement. Numerical
simulations of the measurement techniques provide an error based evaluation of performances.
Key -Words: Reactive power, Power harmonics, Energy measurements.
1 Introduction
The definition of the reactive power as a component of
the apparent power led, in time, to several debates
reported by the dedicated literature [1]-[8].
ISSN: 1790-5117
Experimental data indicate that the actual functional
behavior of the electric networks is always disturbed,
and thus the use of different data processing algorithms
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ISBN: 978-960-474-088-8
COMPUTING and COMPUTATIONAL INTELLIGENCE
difficult to implement within an electronic system at a
reasonable cost and accuracy. Various techniques try to
implement practical forms for this definition. This paper
analyses these solutions.
for the reactive power calculation led to different results.
This paper investigates the basic methods to measure
reactive power for energy meters and uses simulation to
estimate associated errors for reference tests and
extended tests for electric energy instrumentation.
3.1 The Hilbert transform method
This method, obeying equation (4), needs a dedicated
DSP, witch should process the Hilbert transformation in
real time, providing a constant displacement of π/2 for
each harmonic, without the attenuation of that amplitude
[3]. That feature makes Hilbert’s transformation an ideal
solution for Budeanu’s method implementation.
Reactive power is calculated using one signal (voltage
or current) phase shift on all harmonics, followed by the
same multiplication and mean value algorithms as for
active power.
The Hilbert transformation ensures the optimal accuracy
for the reactive power under non-sinusoidal regime,
depending on the deep of the associated FFT algorithm.
The block diagram of the system implementing Hilbert
transform is illustrated in figure 1.
2 Definitions for the Reactive Power
The dedicated literature abounds in references to
different definitions for the reactive power, but the
definitions do not maintain the physical aspect of the
non-sinusoidal regime. Among them, mostly used in
instrumentation design are the definitions formulated by
C.Budeanu and S.Fryze [1], [2], [3].
If active power may be calculated and accurately
measured in any circumstances of harmonic pollution
because it’s inherent definition:
P = ui =
1
nT
t1 + nT
∫ p(t )dt = UI cos ϕ
(1)
t1
Unfortunately this is not the case for reactive power.
2.1 Budeanu’s definition
This definition splits the apparent power in two
orthogonal components:
S = P ± jQ
(2)
The sign in equation (2) corresponds to the inductive
and capacitive reactive power, respectively.
− for sinusoidal waveforms, the reactive power is:
Q = UI sin ϕ
(3)
− for non-sinusoidal waveforms, the reactive power is:
Fig.1. Block diagram of the system supporting Hilbert
algorithm.
∞
Q B = ∑ U h I h sin ϕ h
3.2 The power triangle method
(4)
The power triangle method is based on the hypothesis
that the three powers (apparent, active and reactive)
create a rectangular triangle. Estimating the active and
the apparent power, the reactive power is:
h =1
where h represents the order of the harmonic.
Q = S 2 − P2
(6)
This method gives excellent results under sinusoidal
regime, but the presence of the harmonics can induced
large errors. The power triangle method corresponds to
the Fryze’s definition.
2.2 Fryze’s definition
Originating in 1932 this definition preserves the reactive
power form in any waveform regime:
QF = S 2 − P 2
(5)
The difference between the values QB and QF depends
on the harmonic spectrum of the current and voltage and
the phase displacement between the harmonic
components.
3.3 The quarter period time delay method
This method copes with reactive power formulas
translating phase displacement in time domain delay
with T/4 (where T is the fundamental period). If N is the
number of samples corresponding to T/4 time delay, and
fs is the frequency of the sampling signal, then:
NTs = T / 4
(7)
3 Reactive Power Measurement
Methods
The theoretical definition, given by Budeanu in equation
(4), of the reactive power under non-sinusoidal regime is
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COMPUTING and COMPUTATIONAL INTELLIGENCE
and
f sampling = f line ⋅ 4 N
π/2 is obtained, for all frequencies, by a low-pass filter
(LPF), with a cut-off-frequency much lower than the
line frequency, for instance 2 Hz, as in the figure 4.
(8)
The time shifting versus phase method has two
alternatives:
− a T/4 delay, where T is considered constant for the
ideal frequency of the network; that can generate major
errors when time changing line frequency is present,
with a block diagram in figure 2. The reactive power Q
is calculated multiplying the digitized current signal
with the processed voltage signal, the result being
additionally processed by a digital mean value extractor
represented by a low path filter.
Fig.4. Block diagram of the system using one pole low
pass filter method.
The frequency characteristic of a low-pass filter with a
single pole, responsible for the equivalent π/2 phase
shift, is described by the transfer function:
H ( jω) = H ( jω) ⋅ e
Fig.2. Block diagram of the system using T/4 delay
algorithm.
jϕ H ( jω)
(10)
Both module and phase are depending of frequency:
ω >> ω 0 ⇒ H ( jω ) dB = −20 lg
− an adaptive T/4 delay, where T is now the measured
period of the network. The voltage and current signals
are digitized (ADC), then the voltage signal is delayed,
using a digital shift register to the quarter of the
measured network period, as depicted in figure 3.
ω >> ω 0 ⇒ ϕ H ( jω ) = −
π
ω
ω0
2
(11)
(12)
where: ω 0 - represents the cut-off frequency of the filter.
One can observe the frequency dependence of both
amplitude and the phase.
4 Results of Simulation
Table 1 presents the test conditions, voltage and current,
used to test the measurement performances of the five
reactive power measurement solutions. Table 2 presents
the errors obtained for different tests using notations: Hfor Hilbert transform, LPF- for low pass filter, PTpower triangle, CTD- compensated time delay, TD-time
delay. The traditional measurement methods, like Power
triangle and the Time delay, comply with international
standards but show limitations in the presence of
harmonics or line frequency variation [10], [11], [12].
One can observe that Hilbert method give the best
results, followed by the low pass filter method and then
power triangle method. So, different energy meters
implemented with different formulas can give
discrepancies measuring the same loads.
Simulations have been conducted in MATLAB for
phase and amplitude harmonics. Supplementary to the
basic tests from table 1, additional test was conducted
using a strong bought even and odd harmonic content. In
this particular case errors in the case of the power
Fig.3. Block diagram of the system using compensated
T/4 delay algorithm.
In bought methods the reactive power Q is calculated
multiplying the digitized current signal with the T/4
delayed voltage signal a quarter of a period delay, in
order to obtain a voltage signal with a π/2 phase
displacement for fundamental frequency:
T
1
Q=
i(t).u (t − T / 4)dt
T
∫
0
(9)
3.4 One pole low-pass filter method
This method is used by both static analogue and digital
energy meters. In this case the desired phase shift with
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COMPUTING and COMPUTATIONAL INTELLIGENCE
triangle methods become very large showing an
inappropriate method for highly distorted current signals
in the case of reactive energy measurements.
In figure 5 are illustrated the errors in table 2 for
different test and methods.
Table 2.
Table 1.
Type of Test
Current and Voltage
(Normalized to rms values)
Type of test
H
LPF
PT
CTD
TD
IEC1268 -Reference
Voltage and current
input: sin wave,
~0
~0
~0
~0
~0
~0
~0
~0
~0
5.6
PF = 0.87
~0
~0
~0
~0
-5.7
IEC1268 -Harmonic
Reference test + 10%
of the third harmonics
on the current signal
~0
~0
2
~0
~0
~0
-1.3
1.9
-3.9
-3.9
~0
2.83
390
3
3
f line ; PF = 0
IEC1268 Frequency variation
(±2%)
sin wave;
IEC1268 -Reference
Voltage and current
input: sin wave,
f line ; PF = 0
IEC1268 Frequency variation
(±2%)
sin wave; PF = 0.87
Reference test + 10%
of the third harmonics
on voltage input and
20% of the third
harmonics on current
input
( ϕ1 = ϕ 3 = 30 0 )
IEC1268 -Harmonic
Reference test + 10%
of the third
harmonics on the
current signal
Additional Test
Reference test + 10%
of the third
harmonics on
voltage input and
20% of the third
harmonics on current
input
( ϕ1 = ϕ 3 = 30 0 )
Additional Test
The negligible error values for the Hilbert transform
stand for the reference considered in the simulations
related to this method.
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COMPUTING and COMPUTATIONAL INTELLIGENCE
diagram of the circuit used in simulation in presented in
figure 8.
Fig.5. Errors distribution due different simulated
situations and methods.
4.1 Identification of the load
Data for the additional test in table 1 were collected
using a power harmonic analyzer on a real load.
Waveform reconstruction was performed in MATLAB
and the current waveform is illustrated in figure 6.
Fig. 8. Schematic of the equivalent load.
5. Conclusions
Results of simulation proved once again the fact that
among the alternative definitions proposed for reactive
power measurement of distorted waveforms, Budeanu’s
theory is still the most commonly and internationally
accepted definition, and Hilbert transform is the best
implementation of it.
Recently, the main effort is to reduce the difference
between active and apparent power, since the effect of
distortion on losses cannot be neglected. The emphasis
is shifting from a direct optimisation of the power factor,
at the point of common coupling, to a global
minimisation of the power loss throughout the system.
The important difference between reactive energy
measured with different methods demand a special
attention on different energy meters available on the
market, because in the pay chain of energy different,
various instruments may leave the energy budget
unclosed. To prevent this, one should request the meter
manufacturer to provide details about the reactive power
processing algorithm.
It is to notice that, given the presence of non-linear loads
in urban and rural households, the trend, increasing
worldwide, is to include in the electricity bill not only
the active energy but also the reactive one.
Fig.6. Actual voltage and current reconstruction.
Using basic knowledge on different industrial loads, the
power consumer was identified as a three phase rectifier
with filter [9].
Simulation of the load using the SPICE program
confirmed the similarity of the current waveforms in the
measured and modeled situations. Figure 7 illustrated
the SPICE simulated signals.
1
800V
2
40A
20A
400V
0A
References:
[1] Arrillaga, J, N. R. Watson, S. Chen, Power System
Quality Assessment, John Wiley & Sons, 2001.
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[3] Arseneau, R.; Baghzouz, Y.; Belanger, J.; Bowes,
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1
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I(V1)
Time
45ms
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COMPUTING and COMPUTATIONAL INTELLIGENCE
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