Download Application for Implementing the p

6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
Application for Implementing the p-q Theory in a
Three Phase System
Călin MUREŞAN, Radu A. MUNTEAUNU, Florin DRAGĂN, Adrian O. MÂRZA, Bogdan ŢEBREAN
Electrical Engineering and Measurements Department
Technical University of Cluj-Napoca
Cluj-Napoca,Romania
[email protected]
The transformation matrix for voltage and currents is
detailed below by equation (1) and (2):
Abstract— The paper is part of the results of a study
conducted in the analysis of powers in multy-phase electric
systems. It presents the results obtained in applying the
instantaneous power theory (p-q theory) on a three phase electric
system. The block schematics of the application for computing
the powers, the currents and the non-active currents according to
the theory are showed. The results for two compensation
situations are presented. First case is compensating the
instantaneous reactive power and the second case is for
oscillating instantaneous active power and the instantaneous
reactive power. The application was developed in the LabVIEW
programing environment.
1
1 
 1


2   u1 (t ) 
2
 u0 (t ) 
 2
2
1
1 



u (t )   3  1  2  2  u2 (t )
u (t )





3
3  u3 (t ) 

 0

2
2 

1
1 
 1


2
2   i1 (t ) 
 i0 (t ) 
 2
2
1
1 



i (t )   3  1  2  2  i2 (t )
i (t )





3
3  i3 (t ) 

 0

2
2 

Keywords— instantaneous power theory, non-active power,
LabVIEW, Clarke transformation
I.
INTRODUCTION
In the year 1982 Akagi and coauthors introduce a new
concept in electric systems called the p-q theory. This theory
uses the Clarke transformation on a three phase systems (with
or without a zero wire), to cross from the classic coordination
systems (1,2,3) to an orthogonal axis system (αß0) as is
presented in figure 1. [1-7]
(1)
(2)
The Inverse Clarke Transformation is given by the equation
(3) and (4) and the graphical illustration is presented in figure
2. [11-13][16]:
This section details the concept of the instantaneous power
theory and the way to compute the non-active currents and
powers. [1-7][21][22].
 1

1
0 

 2
  u0 (t ) 
 u1 (t ) 
2
1
3 

 1
u (t ) 
u (t ) 




 2 
3 2
2
2 
u3 (t ) 
u (t )
1
1
3   




2
2 
 2
 1

1
0 

 2
  i0 (t ) 
 i1 (t ) 
 1
1
3 
2

i (t ) 


 i (t ) 
2 
2
2 
3 2
i3 (t ) 
i (t )
1
3   
 1



2
2 
 2
Fig.1 Clarke Transformation
192
(3)
(4)
6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
Q  q (t ) - Reactive power
p0 (t ) - Active power given by the zero components
II.
APPLICATION FOR THE COMPUTATION AND
COMPENSATION OF NONACTIVE POWERS
This section will present the application designed to
calculate and compensate the non-active powers, using the
instantaneous power theory (p-q theory). The voltage and
current signals were acquired using the NI-PCI-4472 data
acquisition board. The experimental system (figure 3) consists
of an asynchronous induction motor, a PC equipped with the
mentioned acquisition board and the LabVIEW environment.
Fig. 2 The transformation from Orthogonal αβ0 system into the 123
coordinates system
Starting from the definition presented the authors propose
the following equation to compute the electric powers.
0
0   i0 (t ) 
 p0 (t ) u0 (t )


 p (t )    0
u (t ) u (t )  i (t ) 

 
 q (t )   0
u  (t )  u (t ) i (t )

(5)
Fig. 3 Experimental system
The application makes a synchronous acquisition of the
three phase signals and computes the non-active powers. After
that it computes the non-active currents and removes them
from the initial currents thus compensating the non-active
powers.
Where:
p 0 (t )  u 0 (t )  i0 (t ) - Is the zero power component,
p(t )  u (t )  i (t )  u (t )  i (t ) - is the real power (active)
The notation (u ) and (i) were given to the voltage and
current vector in instantaneous time, and (u[n]) , (i[n]) in
discrete time. The active and non-active currents vectors were
given the notation (i a [n]) , (i n [n]) . A vector (x) in
q(t )  u (t )  i (t )  u (t )  i (t ) - is the imaginary power
(reactive)
The relation between the conventional power theory and the
power defined in the p-q theory (instantaneous power theory),
is better understood if we separate the new powers into average
components p(t ) , q (t ) , p0 (t ) and oscillating the components
~
p(t ) , q~(t ) , ~
p0 (t ) [1-7]:
instantaneous time, and ( x[n]) in discrete time, is described by
the following: [18-20],[24-27]:
( x)  ( x1 (t ), x2 (t ), x3 (t ))
( x[n])  ( x1[n], x2 [n], x3[n])
p (t )  p (t )  ~
p (t )
Imaginary power : q(t )  q (t )  q~ (t )
Real power :
Zero Squence : p0 (t )  p0 (t )  ~
p0 (t )
power
In figure 4 the schematics for the compensation algorithms
are presented [18-20],[24-27]:
(6)
avarage oscilating
power
power
The input data are the voltage (u[n]) and current (i[n])
vectors. The output data are the active (i a [n]) , and non-active
(i n [n]) vectors [18-20],[23-27].
In equation (6) the average powers represent the average
powers from the convention theory (Budeanu concept) and will
be noted as such:
The computation of the non-active currents is done
according to the desired non-active powers [21-22]. Thus we
can distinguish four cases:
P  p (t ) - Active power (without the zero components)
from the conventional theory
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6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
Fig. 4 The schematics for calculation of the active and non-active currents [19]
p (t ) and reactive
A. The compensation of oscillating active ~
~
q (t ) powers
in [n]  inQ [n] 
The algorithms for computing the non-active currents for
compensating the oscillating components is the following
[7][19]:
in~p [n] 
inq~ [ n] 
inq~ [n] 
u [n]
u2 [n]  u 2 [n]
C. The compensation of the reactive instantaneous power q(t )
The calculation method for this case is given by the
equation (10): [7][19]
u [n]
~
p[n]
u [n]  u2 [n]
2
u  [ n]
u2 [ n]  u 2 [ n]
u [n]
u2 [n]  u2 [n]
u  [ n]
in [n]  inq [n] 
q~[ n]
u  [ n]
2
u (t )  u2 (t )
q[n]
 u [ n]
in [n]  inq [n]  2  2 q[n]
u [n]  u [n]
(7)
q~[n]
Q
(10)
~
p[n]
p (t ) and
D. The compensation of the active oscillating power ~
the reactive instantaneous power q(t )
The non-active currents on the two axes are:
The algorithms for computing the currents that need to
be eliminated for this case is presented below, [7][19]:
in~p [n] 
2
u [n]  u2 [n]
in [n]  in~p [ n]  inq~ [ n]
in [ n] 
in~p [n] 
(8)
in~p [ n]  inq~ [ n]
inq [ n] 
B. The compensation of average reactive power Q
in [n]  inQ [n] 
u  [ n]
2
u (t )  u 2 (t )
Q
(9)
194
u [n]
2
2
u [n]  u [n]
u [ n]
2
2
u [ n]  u [n]
~
p[n]
q[ n]
(11.a)
6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
inq [n] 
in~p [n] 
u [n]
q[n]
u2 [n]  u2 [n]
u  [ n]
u2 [n]  u2 [n]
(11.b)
~
p[n]
The currents of the two axies are:
in [n]  in~p [n]  inq [n]
(12)
in [n]  in~p [n]  inq [n]
III.
Fig. 7 Instantaneous total power and average power
In figure 7 the total instantaneous power is presented. The
presence of the oscillating is remarked. [19]
RESULTS
The results for applying the computation and compensation
algorithms for cases in paragraphs 2.3 and 2.4 are displayed. In
figure 5 the initial voltages and currents acquired from the
asynchronous motor are displayed. In figure 6 the initial
parameters for the acquired signals are presented. [19].
The Clarke transformation has been applied and the results
are presented in figure 8 for voltages and 9 for currents.
Fig. 8 The voltages in the αβ0 system
Fig. 9 The currents in the αβ0 system
The active, reactive and zero powers (instantaneous and
average) are presented in figure 10-12.
fig. 5 The initial voltages and currents
Fig. 6 Initial phases, voltage and current phase displacement and total
harmonic distortion factor
Fig. 10 Instantaneous and average active power
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6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
The resulting powers after the compensation are:
Fig. 11 Instantaneous and average reactive power
Fig. 14 Instantaneous active power
Fig. 12 Instantaneous and average zero power
Fig. 15 Instantaneous reactive power
A. Compensating the instantaneous reactive power
Compensating the instantaneous reactive power implies the
elimination of the non-active currents (in1, in 2 , in3 ) .
B. Compensation of the oscillating active power and the
instantaneous reactive power
In this case the oscillating active power and the
instantaneous reactive power were eliminated. This is the ideal
situation, in terms of power compensation. The phase
displacement between voltage and current must be zero. The
results (active currents) are presented in the figure 16 [19]:
The active resulting currents are presented in figure 13
(ia1, ia 2 , ia 3 ) [19]:
Fig. 16 Active currents and voltages ( ~
p (t ) and q (t ) compensation)
Fig. 13 Active currents and voltages ( q (t ) compensation)
196
6TH INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS2015, 18-21 MAY 2015, CLUJ-NAPOCA, ROMANIA
The algorithm for computing the non-active currents were
presented in the discrete form required in the LabVIEW
environment.
Form the four cases presented (section 2) in the theory only
the results of applying two of them were shown. The analysis
was made on the three phase motor running with no consumer.
From the results presented in section 3, it can be observed
the efficiency and the utility of the theory and application. In
the first case the instantaneous reactive power was fully
compensated. As it was shown in sub-section 3.1, it had a zero
value. Instead the instantaneous active power presented an
oscillating component which can cause vibration problems in
the motor. This component was compensated in the second
case (sub-section 3.2).
Fig. 17 Currents Total Harmonic Distortion and phase displacement between
voltage and currents (at the fundamental frequency)
In the following figures the resulting powers are
represented.
The acquisition of the voltages and currents waveforms was
done also with the help of the LabVIEW programing
environment and the NI-PCI-4472 data acquisition board.
REFERENCES
[1]
Fig. 18 Instantaneous active power
[2]
[3]
[4]
[5]
[6]
Fig. 19 Instantaneous reactive power
[7]
From figure 17 we can observe that the phase displacement
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(the phase displacement was computed only on the
fundamental frequency); witch implies a reactive power equal
to zero.
[8]
[9]
Figure 18 and 19 shows that the instantaneous reactive
power is zero and the instantaneous active power has no
oscillating component.
[10]
IV.
CONCLUSIONS
[11]
The paper presented the Instantaneous Power Theory (p-q
theory), with the appropriate equation for computing the
powers and the non-active currents required for compensation.
[12]
An application for applying the theory was presented. It
was developed in the LabVIEW programming environment.
The presented application is part of a larger program design for
the analysis of non-sinusoidal three phase systems.
[13]
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