Download SECTION-1-Chapter 4

Chapter 4 – Electric Power
Definitions Under Probability
Conditions: The Radomness Power
Alexander E. Emanuel
5.1 - Introduction
Let’s assume an observation time T , hours or days, divided in  equal observation time
intervals T such that T   T . For each interval i , 1 i  , a measuring instrument
V /
I /
records voltage and current harmonic phasors hi hi and hi hi , ( h  1, 2, 3, where
h is the harmonic order), The rms voltage and current, Vi and I i , as well as symmetrical
components may be also computed and recorded. The intervals T are small enough to
consider these quantities as cyclic within each time interval or to assume the interval as a
period of a recurrent set of signals. This approach enables, for each interval i , the
computation and recording of apparent S i , active Pi , and different nonactive powers N i ,
Qi , D Ii , DVi (for the definitions of these powers see ref. [1]). Evidently each electrical
quantity will be characterized by a certain probability distribution for the observation time
T.
The main issue to be clarified on this approach is the formulation of correct definitions
that characterize the flow of electric energy for the entire duration T. As it will be explained,
the mean value is the representative quantity for each of the total apparent components,
regardless of the type of probability distribution. However, this is not true for the total
apparent power. A most significant fact is that if such probabilistic methods are used there
is necessary to include a new type of nonactive power, the randomness power DR .
5.2- Single-Phase Sinusoidal Case.
This is the simplest case and it will help prove the existence of the randomness power.
For each interval i there are the active and reactive powers
Pi  Vi Ii cos(i )
(1)
Qi  Vi Ii sin( i )
(2)
For the total observation time T the rms values of the voltage and current are
V 
1  2
1 2
Vi T 
Vi
T i 1
 i 1
I
1  2
1 2
 Ii T 
 Ii
T i 1
 i 1
(3)
(4)
yielding a total apparent power
S V I
(5)
The total active or real power for the duration T is
P
1 
1
 Pi T   Pi  P
T i 1
 i 1
(6)
i.e. the mean of all the measured active powers Pi .
The reactive power can be defined starting from the total lost energy W in the
supply line. Assuming the resistance of the line that supplies the monitored load to be r , the
total lost energy is
Pi 2  Qi2
(7)
W  r 
T
Vi 2
i 1
an has two distinct terms: one proportional with the squared active power and the

other proportional with the reactive powers squared. A power factor compensating capacitor
meant to minimize W will “deliver” the reactive power QC . The optimum value of QC can be
found by minimizing the function

F (QC )  
i 1
(Qi  QC ) 2
Vi 2
(8)
which leads to

Qi
V
2
1

(9)
Q  Q
1  i 1 i

2
i 1 Vi
This result helps define for the total reactive power when the powers are monitored as
QC 
i 1

i

statistics and holds true for any type of probability distribution.
A simple numerical example will help shade light on this approach. A random load has
its voltage, current and power recorded for five equal intervals, (Table I). This hypothetical load
is assumed to be a resistance that varies randomly.
The total rms voltage and current (3) and (4) are
V  99.92 V and I  124.90 A
giving an apparent power S  12,480.0 VA.
The active power (6) is P  11,579.6 W leading to a power factor PF  P / S  0. 928.
This result indicates that a certain type of nonactive power exists, otherwise the apparent and
the active powers will be equal.
Table I.
i
Vi (V)
1
100.0
2
99.5
3
99.8
4
100.1
5
100.2
I i (A)
100.0
180.0
160.0
80.0
60.0
Pi (W)
10,000.0
17,910.0
15,968.0
8,008.0
6,012
The nonactive power in question is the Randomness Power and in this example it has
the value
DR  S 2  P2  4,654.5
var
In the general case the substitution of (1), (2), (3) and (4) in (5) gives
S 2  V 2I 2 
1 
1  2
1  2
2  2
2
2
 Vi  I i 
 Vi  [ I i cos(i )] 
 Vi  [ I i sin( i )]
2
2
2
 i 1 i 1
 i 1 i 1
 i 1 i 1
(10)
By applying Lagrange’s identity
2
 2  2 
  1
 ai  bi    aibi   
i 1
 i 1
i 1


2
 (ambn  anbm )
m 1 n  m 1
to the squared apparent power (10), one finds
2
 1
1 

S  2  Vi I i cos(i )   
  i 1
m 1

2
2
 1
1 

V
I
sin(

)





i
i
i
 2  i 1
m 1

 
2
 P Q
2

 V
n  m 1
m

 V
n  m 1
m
I n cos( n )  Vn I m cos(m )  
I n sin(  n )  Vn I m sin(  m ) 
(11)
 D R2
where
DR 
 1


2
2
 (Vm I n )  (Vn I m )  2VmVn I m I n cos(m  n )
m1 n  m1
(12)
is the expression of the randomness power (measured in var). In practical situations S is
computed from (3), (4) and (5), P and Q are computed from (6) and (9) and DR from (11), i.e.
from the basic formula
   Q 
DR  S 2  P
2
2
This approach enables the incorporation in DR of the randomness reactive power, a
minute component, otherwise neglected by the approximation (9).
5.3- Three-Phase, Sinusoidal and Unbalanced
The IEEE Std. 1459 – 2000 [1] recommends the use of effective apparent power
Se  3Ve I e
(13)
where Ve and I e are the effective voltage and current, respectively
Ve2  (V  ) 2  Vu2
(14)
I e2  ( I  ) 2  I u2
(15)
with V  , I  the positive-sequence voltage and current and the residual voltage and current
Vu , I u , caused by the load imbalance. According to IEEE Std. 1459 the imbalance components
Vu , I u are function of the negative- and zero-sequence components, i.e.
Vu2  (V  ) 2  (V 0 ) 2 / 2
and
I u2  ( I  ) 2  4 ( I 0 ) 2
From (13), (14) and (15) is found that the effective apparent power (13), can be separated in
two components:
Se2  9Ve2 I e2  ( S  ) 2  Su2
(16)
where S  is the positive-sequence apparent power, considered as the most important
component. The positive-sequence active and reactive powers which are the useful and
dominant terms,
S   ( P  )2  (Q )2 ;
P   3V  I  cos(  )
Q   3V  I  sin(   )
The second term is the unbalanced power (VA)
Su  (V  Iu ) 2  (Vu I  ) 2  (Vu Iu ) 2
(17)
The unbalanced power has three sub-terms; the third term Vu Iu contains the active and
nonactive powers associated with the negative- and the zero-sequence components that can be
easily separated from S u .
The case we are interested in is when the stored information recorded for the  equal times
intervals includes the voltages Vi , Vui , the currents Ii , Iui and the powers Pi and Qi . The
effective voltage and current for the total duration of time T are

1
Ve 
[(V

Ie 
1
i 1


) 2  Vui2 ]
(18)
) 2  I ui2 ]
(19)
i

[( I
i 1

i
thus, the total effective apparent power squared has four terms
   
 
2



2 
2  2
  Vi  Ii   Vi  Iui


9 i 1

i 1
i 1
i 1
2
Se  9Ve I e 

2
2
 
 2  
 2  2
 Vui  I ui
Vui  I i


i 1
i 1
i 1
i 1 
(20)
 
The first term is due to the positive-sequence components and just like in the previous case
leads to three components:
9 
 2
(S ) 
2
   
 2 
i 1

 Ii
i 1
2
2
 Vi
2

  P     Q     DR 
 




where
P 
Q 
1 

(21)
2
;
Pi  3Vi I i cos(i )
(22)

 Qi ;
Qi  3Vi I i sin( i )
(23)
 Pi
i 1
1 



2
2
 1   
9 
  
 

 
 Vi I i
 Vm I n Vn I m

 2 
m 1 n  m 1

i 1
are the total positive-sequence active and reactive powers and
DR 

9  1

2

m 1 n  m1
9  1
2



 2
[Vm I n Vn I m
]

  2
  2
   


 (Vm I n )  (Vn I m )  2Vm Vn I m I n cos(m  n )
m1 n  m1
(24)
is the positive-sequence randomness power
The remaining three terms in (20), correspond to the three terms in (17), leading to the
following expressions:
9

2


i 1
i 1
 
 (Vi  ) 2  ( I ui ) 2  S u'
2
'

 DuR
2
with
Su' 
9

2
3  
  Vi I 
  i 1 ui 


i 1
i 1
   D
 (Vui ) 2  ( I i ) 2  S u''
9
'
DuR

and
2

2
 1

m 1
n  m 1
 
( Vm I un  Vn I um ) 2

'' 2
uR
with
S u' ' 
3    
  Vui I i 
  i 1

''
DuR

and
9

2
 1

m 1
n  m 1
 1

m 1
n  m 1
 

( Vum
I n  Vun I m ) 2
and
9

2


i 1
i 1
 
 (Vui ) 2  ( I ui ) 2  S u'''
2
'''

 DuR
2
with
S u' ' ' 

3
  Vui I ui 
  i 1

'''
DuR

and
9
2
 
( Vum I n  Vun I m ) 2
Thus the final expression of the total effective apparent power for the duration
T   T is
S e2
where
2
2
2
  P     Q     S u   DR2




 
(25)
S   S   S   S 
2
'
u
u
2
''
u
2
'''
u
2
and
 2  DuR' 2  DR'' 2  DuR''' 2
DR2  DR
5.4- Three-Phase Systems with Nonsinusoidal and Unbalanced Conditions
This is the general case. For each time interval i, the effective voltage and current are
recorded, moreover the fundamental component is separated from the total harmonics
component, i.e.
2
Vei2  Ve21i  VeHi
2
2
I ei
 I e21i  I eHi
and
where
Ve1 and I e1 are the effective fundamental voltage and current and VeHi and I eHi
are given by the expressions
VeHi 

2
Vehi
I eHi 
and
h 1
The total effective apparent power squared is

  Ve1i
2


2
9
9
i 1
Se2  9Ve I e 
 Vei  I ei 
2
2
 i 1
 
i 1

   
   
 2
 I ehi
h 1
2  2


 I eHi 
i 1
i 1



 (26)
2
 2  
 2  2 
VeHi  I e1i  VeHi  I eHi

i 1
i 1
i 1
i 1
2 

 I e1i
i 1
2

  Ve1i
 
The first term is due to the contributions of the fundamental voltage and current and has
exactly the same terms as (25),
 2  

 i1Ie1i 
 2 i 1
9 
 Ve1i
2
2
2
2


  P1    Q1    S1u   D12R
 




The second term is due to the interaction between the fundamental voltages and the
harmonic currents
2
 2  2
2


 I eHi  DI   DIR
2
 i 1
i 1
9 
 Ve1i
with
DI 
1


 3Ve1i I eHi
and
i 1
D IR 
3  1


  V
m 1 n  m 1
I
e1m eHn
 Ve1n I eHm 
2
being the current distortion power and the randomness current distortion power, respectively.
The third term is due to the interaction between the harmonic voltages and the fundamental
currents

9

2
VeHi2
i 1

I
i 1
2
e1i
 
 DV
2
2
 DVR
with
DV 
1 
 3VeHi I e1i
and
DVR 
3  1

  V

I
 VeHn I e1m 
I
 VeHn I eHm 
2
 i 1
m 1 n  m 1
being the voltage distortion power and the randomness voltage distortion.
The fourth term is due to the interaction between the harmonic voltages and the harmonic
currents

9

2
VeHi2
i 1

I
i 1
2
eHi
 
 SH
2
eHm e1n
2
 D HR
with
SH 
1

 3V

i 1
eHi
I eHi
and
DHR 
3  1

  V

m 1 n  m 1
eHm eHn
2
The total harmonic apparent power S H can be further separated in the harmonic active
power PH and the nonactive harmonic power N H .
Finally, for the general case the apparent power squared can be resolved using the
expression
Se2
       
2
2
2
2


  S1   S1u  DI  DV 2 S H  DR2
 
where
2
2
2
DR  D12R  DIR
 DVR
 DHR
is the total randomness power.
(27)
5.5- Conclusions
It is common practice to observe electrical quantities by means of statistics. A large time
interval is divided in a finite number of small observation times and for each observation time
the electrical quantities of interest are measured and recorded thus leading to probability
densities for each one of the monitored quantities.
The present study proves that for the active and nonactive powers the equivalent value
for the large time interval is always the mean value. For the apparent power however, this
property does not apply. The apparent power is determined from the product of the equivalent
or effective rms voltages and currents. Under these circumstances the apparent power
resolution has to include a new type of nonactive power, named randomness power. The
randomness power is to be considered only when the apparent power is an equivalent value for
a load or a cluster of load that are monitored using a statistical approach.
Whenever randomness is present the subharmonics can be detected. If the
subharmonics and their associated powers are included, i.e. if the approach to the
measurement for the total duration T is deterministic, then the randomness power is nil.
5.6- References
[1] IEEE Std. 1459 – 2000, “Definitions for the Measurement of Electric Power Quantities Under
Sinusoidal, Nonsinusoidal Balanced or Unbalanced Conditions,” Trial-Use June 2000, Full-Use
August 2002.