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CHAPTER 4
INSTANTANEOUS p-q THEORY BASED
HARMONIC ELIMINATION
4.1. INTRODUCTION TO p-q THEORY
In 1983, a new theory for the control of active filters in three-phase
power systems was proposed (Hirofumi Akagi et al. 2007). It is called
Generalized Theory of the Instantaneous Reactive Power in Three-Phase
Circuits or p-q Theory. It is initially developed for three-phase three wire
systems and three-phase four-wire systems for compensation of harmonic
pollution in the utility side. The neutral current compensation is also done by
this theory.
The compensation in the utility systems is done by the generation
of reference compensation current using the three phase p-q theory
(Peterson and Singh 2009). But in some cases of voltage imbalance in three
phase systems due to harmonic voltages, the sinusoidal reference current for
compensation is not generated for all the phases. This limitation in the three
phase systems is overcomed by the introduction of single phase p-q theory. It
uses simple calculations for deriving the reference current in all conditions of
imbalances due to the power defects. The designing by using this theory is
also efficient and flexible (Leszek and Czarnecki 2004). It can be used for
compensation of both single phase and three systems.
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4.2 INSTANTANEOUS p-q THEORY BASED ACTIVE
REGENERATIVE FILTER
The single phase p-q theory is used for compensation of harmonic
pollution in the utility. This is based on a set of instantaneous powers defined
in time domain and it is used for getting real time control in both transient and
steady state conditions. This single phase p-q theory has an ability to generate
the sinusoidal current in the utility side under voltage imbalance conditions
due to power quality issues. Thus reference compensation current is derived
by this theory which will compensate the distorted voltages and currents. This
reference current can be used for tracking the switching of converters and thus
harmonics are reduced in the power supply.
The instantaneous phase voltage and current are v(t) and i(t)
respectively. Using the instantaneous space vector concept, the v(t) can be
represented as follows,
))t(sin(j))t(cos(VVe)t(v ))t((j
)t(jV)t(V ir (4.1)
)t(Vr and )t(Vi are the amplitudes of real and imaginary part of )t(v
Similarly current can be given as
)t(jI)t(I)t(i ir (4.2)
)t(Ir and )t(Ii are the amplitudes of real and imaginary part of
)t(i
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)t(p1
, )t(q1
are the single phase instantaneous active and reactive
powers. They are given by the equation as,
)]t(I)t(V)t(I)t(V[)t(piirr1
(4.3)
)]t(I)t(V)t(I)t(V[)t(qriir1
(4.4)
The matrix formation from the above equation is indicated as
)t(I
)t(I
)t(V)t(V
)t(V)t(V
)t(q
)t(p
i
r
ri
ir
1
1 (4.5)
The single phase p-q theory uses Clarke transform -
-
-
coordinate plane system is given by the equation
i
i
vv
vv
q
p (4.6)
The power components can be expressed as
p~pivivp (4.7)
p - Mean value of the instantaneous real power. It corresponds to
the energy per unit time that is transferred from the power source to the load.
It is the only desired power component that is to be supplied by the power
source.
p~ -Alternating value of the instantaneous real power. It is the
energy per unit time that is exchanged between the power source and the load.
q~qivivq (4.8)
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q - Mean value of instantaneous imaginary power
q~ - Alternating value of instantaneous imaginary power
The instantaneous imaginary power (q) corresponds to the power
exchanged between the system phases and there is no transference or
exchange of energy between the power source and the load. It is the
undesirable power component and should be compensated.
For eliminating the harmonics components of active ( p~ ) and
reactive power ( q~ ), the reference compensation current is calculated by
)q~q(
p~
vv
vv
i
i1
c
c (4.9)
-
coordinate plane system is given by the equation as
)q(v)p~(vvv
1i
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*C,ref (4.10)
The instantaneous p-q theory based controller is implemented for
compensating the nonlinear loads (Vasco Soares et al. 2000). It continuously
tracks the changes of harmonics content in the utility and compensation is
provided. The implementation of this controller with ARHF is depicted in
Figure 4.1.
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Figure 4.1 Proposed ARHF with p-q theory based current controller
This controller determines the compensating current reference as in
equation and force the power converter (Leszek and Czarnecki 2007) to
synthesize accurately for having sinusoidal current at the source side. This
control strategy (Juraj Altus et al. 2005) makes the compensated current
proportional to the phase voltage, which resembles the waveform of pure
resistive load. It also eliminates the power oscillations (Joao Afonso et al.
2000). Thus p-q theory forms a very efficient basis for designing the
controllers of harmonic damping (Murat Kale and Engin Ozdemir 2005).
4.2.1 p-q theory based current wave shaper with High Pass Filter
For the same p-q theory based current controller implementation, a
High Pass Filter (HPF) is added as shown in the Figure 4.2.
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Figure 4.2 Proposed ARHF with p-q theory based current controller
and High Pass Filter
The main usage of this passive high pass filter is to improve the
damping performance of high order harmonics. So p-q theory based reactive
component of passive HPF is also derived in addition to the above reference
current derivation This derivation includes active, reactive and harmonics
component of nonlinear loads (Khadkikar et al. 2009).
For a single phase system with nonlinear load the load current can
be given as
)tnsin(2)t( nn,L1n
L ii (4.11)
Under normal circumstances the voltage at the PCC can be given as the
)tnsin(v2)t(v nPCCPCC (4.12)
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The HPF current can be represented as
)90tsin(i2)t(i n,hphp
(4.13)
From the above the instantaneous active power for nonlinear load
can be calculated as
)t(i).t(v)t(pLPCCL
LL
p~p (4.14)
And in the same way the instantaneous reactive power is given as
)t(i).t(v~)t(qLPCCL
LL
q~q (4.15)
Also the instantaneous reactive power of HPF is given as follows,
)t(i).t(v~(t)qhpPCChp
hphp
q~qLL
q~q (4.16)
where L
p , L
q and hp
p represent the constants and L
p~ , L
q~ and hp
p~
denote the variable component and )t(v~PCC
denotes the PCC voltage shifted
by 90º.
By obtaining the constant part in (4.13),(4.14),(4.15) the active
(iL,p),reactive (iL,q) and harmonics current (iL,h), components of nonlinear load
current and the reactive (ihp,q) components of passive HPF current can be
readily calculated as follows,
)t(uv
p2)t(i
PCC
L
P,L (4.17)
)90t(uv
q2)t(i
PCC
L
P,L (4.18)
)t(i)t(i)t(i)t(iP,LP,LLh,L (4.19)
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and
)90t(uv
q2)t(i
PCC
hp
q,hp (4.20)
where u(t) is the unit vector in phase with PCC voltage.
The generalized transfer function approach to harmonic filter
design is given below. The HPF impedance transfer function Hhp(s) can be
given in normalized form as
1S
Q
1)
S(
)1S
(S
A)S(Z)S(H
P
2
P
P
hphp (4.21)
hP
CA
1,
hPhP
0CL
1,
hP
hP
PL
R,
hP
hP
hPL
CRQ
w 0 p is the
pole resonant frequency and Q is the quality factor.
The passive HPF is tuned to 1.28 kHz of resonant frequency. The
inductor bypass resistor Rhp, is chosen to have desired high pass response and
The Q factor value is selected as 0.69 considering the required high pass
response over a wide frequency band.
The injected current transfer function Hcds(S) can be derived as
)S(Z)S(Z
)S(Z
)S(i
)S(i)S(H
Shp
hp
h
h,s
cds (4.22)
Thus the ARHF with this passive high pass filter combined to
mitigate the lower and higher order harmonics
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4.3 SIMULATION BY USING MATLAB
In the Indian context the following factors from load end can be
considered for the generations of higher-level harmonics. More use of solid
state power converters for industrial drives, use of arc and induction furnaces
for mini steel and non-ferrous metal plants, use of thyristors for locomotives,
especially for railways due to the massive electrification program and
extensive use of single phase electronics loads in domestic sectors are causes
for generation. Not only that the rapid use of energy conservation devices in
both domestic sector and industrial sector such as electronics chokes for tube
lights, electronics energy controllers for motors and electronics fan regulators
etc., also injects harmonics substantially.
This is due to the semiconductors used in those solid state
electronics. This type of loads draws current in a short pulse only during the
peak of the sine wave. Thus resulting in the high frequency harmonics which
are superimposed onto the fundamental 50Hz frequency. Thus the uneven
switching of the converters increase the harmonic pollution in the supply
systems and these types of loads are called as nonlinear loads.
The uneven switching of the nonlinear converter loads injects the
harmonics and thus causes distortions in the electric power supply. The
current and voltage waveform distortion mainly depends upon the size of the
nonlinear loads and its source impedance. The amount of distortion increases
as the percentage of nonlinearity increases. So the uneven switching of
converters with their distortions is modelled using MATLAB. The converters
like single phase full bridge rectifier and single phase half bridge rectifier are
used as loads. The harmonics produced by these loads are controlled by single
phase H bridge regenerative rectifier which gives four quadrant
characteristics.
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In this modelling, the single phase full bridge and half bridge
rectifier is considered as nonlinear load 1(NL1) and nonlinear load 2(NL2).
The simulation circuits for nonlinear load 1 and 2 (Joorabian et al. 2011) are
given in Figure 4.3 to 4.6. The p-q theory based current controller is
implemented for both loads. Figure 4.3 gives the simulated circuit of NL1
without high pass filter and Figure 4.4 gives the circuit of NL1 with high pass
filter. In the same way simulation is also done for NL2 without and with HPF
(Figure 4.5 and 4.6). Figures 4.7 and 4.8 give the affected source current due
to NL1& NL2.The distorted source due to the uneven switching of NL2 is
given in Figure 4.8. Figure 4.9 and 4.10 show the source current waveforms
without any compensation controllers. The total current harmonic distortion
(THDi) for both loads are given as,
NL1 is 89.54% and
NL2 is 190.55%.
Figure 4.3 Simulated Nonlinear Load 1 with p-q theory based
controller & without HPF
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Figure 4.4 Simulated Nonlinear Load 1 with p-q theory based
controller & with HPF
Figure 4.5 Simulated Nonlinear Load 2 with p-q theory based
controller & without HPF
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Figure 4.6 Simulated Nonlinear Load 2 with p-q theory based
controller & with HPF
The two cycles of the source current are taken for analysis. The FFT
harmonic profiles are also shown for that selected signal. Figure 4.11 and 4.12
gives the two cycle FFT window and its profile due to the addition of NL1 with
source. The FFT window of the utility current due to NL2 is given in Figures
4.13 and 4.14. The results after the use of the active regenerative harmonic
filter with p-q theory based current compensator without HPF is given in
Figures 4.15 and 4.17 for both loads. After the usage of HPF further harmonic
mitigation is done for both loads and it is shown in Figures 4.16 and 4.18.
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Figure 4.7 Source current due to Nonlinear Load 1
Figure 4.8 Source current due to Nonlinear Load 2
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Figure 4.9 FFT analysis of uncompensated Nonlinear Load 1
Figure 4.10 FFT analysis of uncompensated Nonlinear Load 2
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Figure 4.11 Two cycle FFT window (NL1 source current)
Figure 4.12 FFT profile (NL1 source current) THDi=89.54%
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Figure 4.13 Two cycle FFT window (NL2 source current)
Figure 4.14 FFT profile (NL2 source current) THDi=190.55%
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Figure 4.15 Harmonics spectrum for NL1 with p-q theory based ARHF
& without HPF
Figure 4.16 Harmonics spectrum for NL1 with p-q theory based ARHF
& HPF
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Figure 4.17 Harmonic profile of NL2 with ARHF and without HPF
Figure 4.18 Harmonic profile of NL2with ARHF and with HPF
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The overall performance of regenerative filter incorporating p-q
theory based current controllers and HPF is given as
THDi (NL1) = 5.81%
THDi(NL2) =19.84%
4.4 SUMMARY
The single phase p-q theory for harmonic filtering is dealt in this
chapter. The power compensation leading to harmonic limitation derived from
p-q theory is analysed. In order to eliminate the higher order harmonics an
additional filter is also implemented. The usage of ARHF with the
application of p-q theory is explained with the simulation results. Thus the
harmonic cancellation by using the single phase p-q theory has different
functionalities as
Elimination of power oscillations
Improvement of power factor
Elimination of current harmonics
Harmonic damping.
