Chapter 6
Models for Representation and
Summation of Random Harmonic
Currents
R. Langella, A. Testa
6.1
Introduction
The chapter deals with the probabilistic modeling of harmonic currents. After some definitions
about single current injection models, the vectorial summation of random harmonic currents
characterized by distributions independent on the time is considered. The basic methods
presented in literature for harmonic current summation modeling are descried: Convolution
Method, the Joint Density Method and Monte Carlo Simulations. Applications of the methods
to different case-studies give an idea of the accuracy of the models considered.
One aspect of assessing the harmonics tolerated by a power system is the estimation of the
statistic figures of harmonics arising from the various time-varying / probabilistic sources.
The assessment of harmonics is not exact or uniform, since there will be unpredictable
variations in either the non-linear sources and/or parameters of the system which affect the
summation.
The combination of a number of harmonic time-varying sources will generally lead to less than
the arithmetic sum of the maximum values due to uncertainty of magnitude and phase angle.
Hence the resulting summation is extremely difficult to estimate accurately.
Before introducing the methods that can be adopted to solve the summation problem, it is
necessary to introduce and discuss some basic concepts: random harmonic and interharmonic
vectors and harmonic summation principle and essential nomenclature.
6.2
Random Harmonic and Interharmonic Vectors
The analysis of interface among utilities and consumers needs representing the harmonic
currents injected from disturbing loads as random vectors. The random behavior of harmonic
1
currents is related to the parameter of influence stochastic nature such as active and reactive
powers, network configuration, non-linear load operational conditions, etc.
The harmonic current of order h injected by a non linear load into the network can be
represented as a vector I h , of amplitude Ih and phase h and of Cartesian components Xh and
Yh. In any case, the statistical characterization of I h requires the determination of the joint
statistics of a pair of real random variables (Ih,h) or (Xh,Yh). With reference to (Xh,Yh) and
omitting the subscript h, the functions to be considered are:
- the joint distribution FXY(x,y), also called joint cumulative probability function (jcpf), that is the
probability of the event Xx, Yy;
- the joint probability density function (jpdf), that is, by definition, the function:
f XY (x, y) 
 2 FXY (x, y)
;
xy
- the marginal distributions FX(x) and FY(y), also called cumulative probability functions (cpf’s);
- the marginal probability density functions (pdf’s) fX(x) and fY(y).
The graphs in 0 show an example of the 5th harmonic time variability magnitude and of its pdf.
0a reports the scatter plot of the vectors edges registered and 0b the magnitude pdf.
15
Yh
Probability
10
5
0
0.6
Xh
a)
0.65
0.7
Magnitude [A]
0.75
0.8
b)
Figure 6.1. 5th harmonic current from measurements: a) Scatter plot, b) Probability histogram of magnitude.
6.3
Harmonic Summation Principle and Essential Nomenclature
The basis for harmonic combination is the superposition principle. To apply the superposition
principle properly, a phasorial composition should be used.
2
Here, the basic definitions and nomenclature are introduced together with some practical
considerations about the computational problems.
6.3.1 Summation of Random Harmonic and Interharmonic Vectors
The problem of modeling the injection of the sum of numerous current vectors in the network
needs technical analysis and statistical elaboration of measurements for each load and also
knowledge and representation of the way in which random vectors combine during the time to
give a resultant.
The sum of N random harmonic vectors gives:
N
Ih 

k 1
N
I h, k 

k 1
N
X h, k  j
Yh,k  S h  jW h , I

k 1
h
 S h2  Wh2
,  h  tan 1 (
Wh
).
Sh
Great is the practical interest to obtain the pdf of Ih. The correct theoretical approach would be
based on the use of the 2N-dimensional joint probability density function
f Zh ( z h ) , with Z h  [ X h,1 , X h,2 ,..., X h, N , Yh,1 , Yh,2 ,..., Yh, N ] .
Equation 6.1
The distribution of Ih is given by:
FI (i h ) 
h

2N
dz h,i

i 1
f Z (zh )
h
,
Equation 6.2
being  the 2N-dimensional region of the hyperspace z where the constraint I h ( z )  ih , is
verified.
Once solved the integral form, it is trivial to obtain the pdf of Ih.
6.3.2 Basic Considerations
The vectorial summation of N random harmonic and interharmonic components, in a defined
scenario of space (loads, utility network characteristics, ...) and of time (the year, the annual
maximum load day, ...), is in principle very simple and comprehensive if the 2N-dimensional
jpdf of the 2N real random variables representing the N vectors involved is available for the
scenario at hand.
3
In practice, also having in mind only numerical approaches that discretize the hyperspace z,
assuming M discrete values for each coordinate in its definition interval, the following dramatic
computational problems arise:
-
the matroid to be utilized for representing the jpdf f Z given in 0 assumes dimension
D=2M2N (f.i. if M=N=10, that are very low values, then D=2x1020);
-
the solution of integral 0 is very time consuming;
-
the determination of the jpdf is prohibitive to obtain by both experimental analyses or
simulation approaches.
If the dependence amongst the different random vectors is ignored (hypothesis A) or accounted
for outside the summation stage, the problem reduces to consider N matroids, each of
dimension Di=2M2, to represent N bidimensional jpdf’s (M=N=10 gives D=NDi=2000).
Moreover, if the dependence amongst the pairs of real random variables utilized to represent
each vector is ignored (hypothesis B) or accounted for outside the summation stage, the
problem reduces to consider 2N matrices, each of dimension Di=2M, to represent 2N marginal
pdf’s (M=N=10 gives D=2NDi=400).
Different procedures have been engaged for approaching the summation of stationary random
vectors, in order to avoid the Monte-Carlo simulation burden or the dramatic computational
problems deriving from the theoretical formulation reported in nomenclature; wide reviews are
developed in [4], [5], [6], [8] and [11]. Among these procedures the first part of this section
summarizes those widest diffused.
The various methods proposed have been originated from appropriate application of
probability theory [18]. Most of them are based on analytical models founded on fully
developed convolutions or on the central limit theorem; differences consist on the more or less
restrictive hypotheses required by each of them (mainly the statistical dependence or
independence amongst the random variables representing the Cartesian coordinates of the
vectors and the number of components). All of the methods presented in the following are
founded on two general hypotheses:
1)
the random vectors are statistically independent;
2)
the distributions of harmonic vectors are independent on the time.
4
6.4
Basic Methods
The basic methods proposed in literature are the convolution method, the joint density method
and finally Monte Carlo simulations.
6.4.1 Convolution method
A first method [4] assumes that the resultant resolved components are statistically
independent. The method goes on as follows:
f Sh ( s h )  f X h ,1 ( xh,1 )  f X h , 2 ( xh,2 )  ...  f X h , N ( xh, N ) ,
fWh ( wh )  f Yh ,1 ( y h,1 )  f Yh , 2 ( y h,2 )  ...  f Yh , N ( y h, N ) ,
1
f S 2 ( s h2 ) 
h
2
1
fW 2 ( wh2 ) 
h
[ f Sh ( s h2 )  f Sh ( s h2 )] ,
s h2
2
wh2
[ fWh ( wh2 )  fWh ( wh2 )] ,
f S W ( s h2  wh2 )  f S (s h2 )  fW ( wh2 ) ,
2
h
2
h
2
h
2
h
f I (ih )  2ih f S W (ih2 ) ,
h
2
h
2
h
Equation 6.3
where * denotes convolution.
In general the method requires the knowledge of each vector resolved component pdf’s. A
simplification is possible when, for a large number N of harmonic vectors, the central limit
theorem is applicable to the summation of the Xh,k and of the Yh,k. In this case Sh and Wh pdf’s
approximate to two Gaussian distributions of known means and variances. However, the
relation 0 requires the statistical independence between S2h and W2h.
6.4.2 Joint density method
Another method [5] does not need the independence between Sh and Wh but it requires N so
high to make the central limit theorem applicable. Therefore, Sh and Wh are jointly normal with
their joint density given by:
5

f ShWh ( s h , wh ) 
e

2(1r 2 )
2 S  W 1  r 2
,
where r is the correlation coefficient and

(s   S ) 2
 S2

2r (s   S )( w  W )
 SW

(w  W ) 2
2
W
.
Equation 6.4
The density function of Ih is directly derived by the following relation:
2
f I (ih ) 
h
0 f S W (ih cosh , ih sin h )ih dh .
h
h
Equation 6.5
When r equals 0, Sh and Wh are independent because they are jointly normal.
In the same hypotheses, the density function of the phase  is derivable solving the following
integral form :

f  ( ) 
f
SW (i cos  , isin )idi
.
0
Equation 6.6
6.4.3 Monte Carlo method
Usually, Monte Carlo methods [17][19] are used to simulate a prescribed random behavior of
the network loads. That is, random number generators are used to assign a specific probability
distributions to certain parameters of the loads, thusly reflecting the random variations in the
loads operating condition. In this way, deterministic models of the load can then be used to
generate the random harmonic current injection. Subsequently, the statistics of the resulting
harmonic voltages are numerically determined, usually from the linear propagation of these
harmonic currents through the system impedances. The advantages of this approach is in the
possibility of simulating a wide variety of random load characteristics until the resulting
statistics agree with available field measurements. The disadvantage is that this method is
computationally intensive and time consuming since it is difficult to determine how to adjust
6
the load random models in order to produce desired results. That is, although the method is
flexible, the direct relation between the load probability models and the resulting harmonics is
not apparent. This means that if a given set of load random characteristics yields unsuccessful
results (simulated harmonics do not match available measurements), then limited insight is
gained from this simulation trial in terms of altering load models for more accurate results in
the next trial. Unless equipped with accurate information on the nature of the random loads, it
is extremely difficult to develop simulation models which generate adequate results.
6.5
Magnitude and phase distribution obtainable starting from the
Joint Density Method [16]
In the following some brief remarks on analytical distributions, firstly for the magnitude and
then for the phase are recalled. The integral forms (0 and 0) are not simply resolvable, due to
the complicated structure of fSW. Nevertheless, it is worth noting that if a priori it is possible to
assume r equal to zero, then S and W are independent, because they are jointly normal, and
the integral forms in (0 and 0) become more light to handle. In this case the integration of 0 and
0 can also be performed by opportune convolutions, as fully shown in [1][2][3], so obtaining a
formal but not effective simplification because in the most general case difficulties arise due to
numerical problems and to the sensitivity to the real axes partition choice.
The rotation aim is to determine an opportune rotation angle  of the original Cartesian
reference SW that gives a new Cartesian reference S’W’ in which rS’W’=0. The angle  can be
obtained as a function of variances and covariances of the original resolved components [3]:
1
2
  tan 1 2
2
 SW
.
 S2  W2
Equation 6.7
S and W result two normal and correlated r.v.’s changed by the rotation (0) into S’ and W’,
normal and uncorrelated, that is to say, also independent, so simplifying the integral forms (0
and 0) and also solving the theoretical problem of the original distribution concerning the
assumption of normal jpdf. The outcomes of the rotation are fully shown in [3], also referring to
summation methods not utilizing the bivariate normal distribution.
6.5.1 Magnitude
The aim is to have a comprehensive insight into the different fully analytical or empirical
solutions, in order to compare the regions of applicability, the performances and, mainly, the
usefulness in practical engineering problems in which sometimes only some statistic
parameters are available.
7
6.5.1.1 Closed form solutions holding under particular hypotheses
In the hypotheses of rSW = 0 and W = 0, the integration of 0 provides a complicated solution [6],
based on a series of products of Ik, modified Bessel function of integral order k :
 i2
i
f I (i) 
exp 
 4
 SW



 1
1 
1


 exp   S
  2  2 
 2  S
W 
 S





2
  ,


Equation 6.8
where

  I0[


i2 1
1
i2 1
1
(  1) k I k [ (

)] I 2k (i S ) .
(

)] I 0 (i S )  ... ..  2
2
2
2
2
2
4  S W
4  S W
 S2
S
k 1

Equation 6.9
In some special cases, the resultant magnitude density given by 0 turns into a simple form. First
of all, if S = W =  then the 0 becomes :
f I (i) 
  S2  i 2    S  i 
exp
 I0  2 ,

2 


2
2


   
i
Equation 6.10
the 0 is called the Rician probability density function.
Moreover, when it results Si 2, that is the argument of I0(.) is large, the following
approximation of 0 arises :
f I (i) 
 1i 
S
exp  
2
2



2

1



2
i

,
 S

Equation 6.11
which is a Gaussian law except for the factor (i/S)1/2, and for this reason it is called “almost
Gaussian”. It is interesting to note that the previous relations can be utilized also when W0 by
the substitution of S with (S2+W2)1/2.
8
Finally, when S = W = 0 then the 0 becomes the Rayleigh distribution :
 i2 
.
f I (i) 
exp 
 2 2 
2


i
Equation 6.12
As well known, this is the case of random vectors with phases distributed in the whole interval
(0,2).
0 summarizes the methods for obtaining fI(i) from the bivariate normal distribution fSW(s,w) in
closed-form solution holding under particular hypotheses.
Table 6.1 methods for obtaining fi(i) from the bivariate normal distribution fsw(s,w): closed-form solution holding under particular
hypotheses
NAME
HYPOTESES
EXPRESSION
rSW = 0,
f I (i) 
Bessel
W = 0
 i2
exp 
 4
 SW

i
 1
1


 2  2
W
 S


  exp  1   S

 2   S


rSW = 0,
Rician
W = 0,
S = W = 
  S2  i 2 


  I  S  i 
f I (i) 
exp 
0


 2 
2
2 2 




i
rSW = 0,
“Almost
W = 0,
Gaussian”
S = W = ,
f I (i) 
 1i
S
exp  
2
2



2
1
Si 2
rSW = 0,
Rayleigh
W = S = 0,
S = W = 
9
 i2 

f I (i) 
exp 
 2 2 
2


i



2
i



S




2
 


6.5.1.2 Empirical solutions
Other methods seem to follow a not fully analytical demonstration for obtaining a closed form
for the solution of 0. Basically, they impose the equating of some moments of the actual f I with
the corresponding moments of an approximate distribution. In spite of all that, they obtain very
powerful and accurate results.
In the hypothesis of rSW=0, in [9] a 2 distribution is introduced so obtaining for the magnitude
density :
f I (i) 
2 (1 / 2) i ( 1)
 i2 
exp   ,
 / 2 ( / 2)
 2 
Equation 6.13
with the parameters expressed by the following relations :



2
2
2
2 2
S  W   S   W
2 2
4
2 S2 S2  2W
 W   S4   W

2 2
4
2 S2 S2  2W
 W   S4   W
.
2
2
 S2  W
  S2   W

,

Instead, also in the hypothesis of rSW  0, in [15] a method for assimilating the resultant
magnitude distribution with a generalized gamma distribution (GGD) is proposed :
f I (i) 
2  i ( 2 1)
( )  2


exp   (i /  ) 2 ,
Equation 6.14
where  is the Gamma function and the parameters  and  can be expressed in terms of the
bivariate normal distribution model five parameters :
10
   S2  W2   S2   W2 ,

4
2 2
4 S2 S2  4W
 W  2 S4  2 W4  C

,

2
2
C  4 rSW  S2 W
2 S W  rSW  S2 W
.
It is easy to demonstrate that the expression in 0 arises from the simplification of the more
general expression in (0 and 0). Parameters , ,  and  can be derived from a moment fitting
procedures as follows. The moment generating function of two jointly normal random variables
X and Y is:
M (t1 , t 2 )  exp[  X t1  Y t 2  ( x2t12  2r X  Y   Y2 t 22 ) / 2] ,
with their joint moment of order j+k given by:
t 0
 jk
.
m jk  E[ X j Y k ] 
M (t1 , t 2 ) 1
j k
t2  0
t1 t 2
On the other hand, the kth order moment of the GGD can be obtained by means of the
following expression
mk 
(r  k / 2) s k
( ) .
(r )
r
Equating the same order moments gives the GGD model parameters.
0 summarizes the methods for obtaining fI(i) from the bivariate normal distribution fSW(s,w)
giving “empirical” solutions.
Table 6.2 methods for obtaining Fi(I) from the bivariate normal distribution Fsw(S,W): empirical solution
11
NAME
HYPOTESES
2 procedure
rSW = 0
EXPRESSION
f I (i ) 
2 (1 / 2) i ( 1)
 i2 
exp 

 / 2 ( / 2)
 2 
2 procedure
f I ' (i' )  f I (i) 
+ rotation
2 (1 '/ 2) i'( '1)
 i'2 
exp 

 ' '/ 2 ( ' / 2)
 2 ' 
Generalized
f I (i) 
Gamma
2  i ( 2 1)
( )  2

exp   (i /  ) 2

Distribution
6.5.2 Phase distribution
Attention is paid also to the resultant phase distribution which until now has received little or
no importance in literature but which can considerably help in understanding the role played by
clusters of harmonic injections at different buses of the network.
The density function of the phase  is derivable solving the following integral form:

f  ( ) 
0 f SW (i cos, i sin  )idi .
Equation 6.15
The solutions require rSW = 0. Also in this case it is possible to take advantage of an axis
rotation. Nevertheless, it is necessary to consider that an angle  rotation does change the
phase density :
f  ( )  f ' (   ) .
6.5.2.1 Closed form solution
In the same hypotheses of the Rayleigh distribution for the resultant magnitude, the tan
distribution results a Cauchy so that:
f  ( ) 
1
,
2
Equation 6.16
12
that is the resultant is phase uniform in (0,2).
6.5.2.2 Almost analytical solution
In the hypothesis of rSW=0, the solution of the integral form 0 is obtained in Appendix of
[10]and it is given by :
f     K


2   exp  2 / 4 1  erf  / 2 
4A
,
Equation 6.17
where the auxiliary variable , which is a function in , is defined by means of the following
relation
  05
. B/ A,
Equation 6.18
and the quantities K, A and B are valuable as shown in Appendix of [16].
It is easy to demonstrate that the expression in 0 arises from the simplification of the more
general expression in 0.
6.6
Case studies
6.6.1 Numerical case-studies
All the above described procedures were tested in different case-studies, also performing
Monte-Carlo simulations to obtain a reference.
Case-studies of Rayleigh and Rician magnitude distributions applicability, not reported here for
the sake of brevity, were performed obtaining very good results also applying 2 and GGD
procedures. Instead, the cases fully reported in the following refer :

case 1A to completely verified Almost Gaussian applicability conditions, while case 1B and
1C move toward Almost Gaussian non-applicability conditions starting from 1A ;

case 2 to a scenario where the bivariate distribution is characterized by an elliptic
symmetry; since the principle axes of the ellipse are not parallel to the Cartesian axes, the
13
correlation coefficient is different from zero (case 2A) but it becomes equal to zero (case
2B) after a /3 angle rotation.
In 0 the parameter values utilized are reported for the original bivariate distribution, for the
GGD and the 2 distribution ; for the Monte-Carlo simulation a minimum of 15000 samples has
been considered in order to directly solve integral forms of 0 and 0.
Table 6.3 CASE STUDIES PARAMETERS
Bivariate Normal Distribution
2 Distribution
GGD
Cartesian
Reference

S
S
W
W
rSW




1A
SW
-
20.0
1.7
0.0
1.7
0.0
34.7
20.2
69.4
2.4
1B
SW
-
10.0
1.7
0.0
1.7
0.0
9.3
10.3
18.5
2.4
1C
SW
-
2.5
1.7
0.0
1.7
0.0
1.4
3.5
2.8
2.1
2A
SW
-
3.0
1.0
5.2
1.3
0.6
4.6
6.2
-
-
2B
S’W’
/3
6.0
1.5
0.0
0.7
0.0
4.6
6.2
9.2
2.1
Case
Study
From 0 to 0, the scatter plot, the magnitude and phase pdfs, for the respective cases, are
reported.
0 shows that Gaussian and Almost Gaussian give results very similar in 1A, similar in 1B and
appreciably wide apart in 1C. Concerning the Almost Gaussian it is worth to underline as the
right tail better approximates solution, that is to say that higher percentiles are better
estimated, due to the better verification of the condition Si  2, especially when this
condition is not verified for all the possible values.
It is worth to note as 2 and GGD procedure results are very close to the actual (Monte-Carlo) in
all of the cases considered, performing as well as the closed forms, when applicable. Moreover,
2 distribution and GGD procedures have been tested in more general scenarios as those
reported in [7], [12] not necessary characterized by scatter plot symmetry, in any case giving
good results.
14
0 .3 5
B
C
A
0 .3
Probability
P r o b a b ilit y
C B A
0 .2 5
0
0 .2
0
0 .1 5
0 .1
0 .0 5
0
0
5
1 0
1 5
M a g n it u d e
2 0
2 5
3 0
Magnitude
5
5
5
4.5
4.5
4.5
4
4
3.5
3.5
3
C
2.5
2
3
B
2.5
2
1.5
1.5
Probability
4
3.5
Probability
Probability
Figure 6.2. Case studies 1A,1B and 1C, current magnitude pdf, obtained by Monte Carlo Simulation (+), 2 procedure (o),
GGD model (), Almost Gaussian model () and Gaussian assumption (*).
3
2.5
2
1.5
1
1
1
0.5
0.5
0.5
0
-3
-2
-1
0
Phase [rad]
1
2
3
0
A
-3
-2
-1
0
Phase [rad]
1
2
3
0
-3
-2
-1
0
Phase [rad]
1
2
Figure 6.3. Case studies 1A,1B and 1C, current phase pdf. obtained by Monte Carlo Simulation (+), and by the 0 (o).
15
3
0 .3 5

0 .3
Probability
Probability
0 .2 5
0 .2
0 .1 5
0 .1
0 .0 5
0
0
2
4
6
8
10
12
M a g n itu d e [A]
Magnitude
Figure 6.4. Case-study 2B, current magnitude pdf, obtained by Monte-Carlo simulation (+), GGD model (),2 procedure (o)
and by relation (13) () after a /3 angle rotation.
3.5
3
Probability
2.5
2
1.5
1
0.5
0
0.2
0.4
0.6
0.8
1
Phase [rad]
1.2
1.4
1.6
Figure 6.5. Case-study 2B current phase pdf. obtained by Monte Carlo Simulation (+), and by the formula (21) (o), after a /3
angle rotation.
16
6.6.2 Real cases
In order to have an insight into the applicability of the proposed methods to harmonics
measured in medium voltage distribution networks, measurement results were processed for
comparing estimated probability density functions with sample histograms of measured
harmonic currents/voltages.
The measurements consist of 90 readings collected during a working day morning, in fall 1997,
over 3 medium voltage lines: line #1 supplying a large plant, line #2 a cluster of small industrial
laboratories and line #3 prevailingly residential loads. Measurements were collected
simultaneously, between 11.00 a.m. and 11.40 a.m., storing a spectrum each half a minute,
approximately. The observation interval is relatively short, but not too short for practical
purposes for, e.g., estimating the network behavior during a time interval deemed as the most
critical in the working cycle. Even if the 37 minutes are not a long time, data can exhibit nonstationary due to the particular time of the day when some consistent modification are likely to
occur in the production processes and in the way residential customers utilize electric energy.
#1
#3
#2
Figure 6.6. line #1 supplying a large plant, line #2 a cluster of small industrial laboratories and line #3 prevailingly residential
loads
Here emphasis is given only to lowest order harmonic, that is, the 5th. The graphs in 0 and 0
show the 5th harmonic magnitude histograms along the pdf estimated by 0 and 0, starting from
measured S, W, S2, W2 and rSW, and utilizing axis rotation when necessary. 0 reports also the
time behavior of the line #3 current. 0 shows the resultant of the line currents, evaluated by
adding up the contributions from each line. Harmonic busbar voltage (0) is considered too.
Then, the Almost Gaussian has been applied to the case of the voltage, by utilizing the
17
substitution of S with (S2+W2)1/2 and the assumption =(S2+W2)1/2. Finally, the phase of line
#2 current is analyzed reporting in 0 the histograms along the pdf estimated by 0. The scatter
plot of the recorded vectors in the complex plane are enclosed in all the graphs.
8
7
5
Probability
Probability
6
4
3
2
1
0
0 .7
0 .7 5
0 .8
0 .8 5
0 .9
M a g n itu d e [A ]
0 .9 5
1
1 .0 5
Magnitude [A]
Figure 6.7. Probability of 5th harmonic current magnitude estimated from sample and Gaussian assumption (continuous line).
Data are relative to line #2, feeding a cluster of industrial customers.
18
16
14
Probability
Probability
12
10
8
6
4
2
0
0 .4
0 .4 5
0 .5
M
a
g
n
itu d e [A]
[A]
Magnitude
0 .5 5
0 .6
Figure 6.8. Probability of 5th harmonic current magnitude estimated from sample histogram and Gaussian assumption
(continuous line). Data are relative to line #3, feeding a cluster of residential customers.
0.65
0.6
Line #3
5th Harmonic current
0.55
0.5
[A]
0.45
0.4
0.35
0.3
11.00
11.20
Time [hours]
Figure 6.9. 5th harmonic current in line #3, feeding a cluster of residential customers.
19
11.40
5
4 .5
4
Probability
3 .5
3
2 .5
2
1 .5
1
0 .5
0
1 .4
1 .5
1 .6
1 .7
1 .8
1 .9
2
2 .1
M a g n itu d e [A ]
Figure 6.10. Probability of 5th harmonic current magnitude estimated from sample histogram and Gaussian assumption
(continuous line). Data are relative to the sum of all lines.
0 .0 5
0 .0 4 5
0 .0 4
Probability
0 .0 3 5
0 .0 3
0 .0 2 5
0 .0 2
0 .0 1 5
0 .0 1
0 .0 0 5
0
60
70
80
90
100
110
120
M a g n itu d e [V]
Figure 6.11. Probability of 5th harmonic voltage magnitude estimated from sample (+), 2 procedure (o), GGD model () and
Almost Gaussian model ().
20
4
3 .5
3
Probability
2 .5
2
1 .5
1
0 .5
0
-1 .4
-1 .2
-1
-0 .8
-0 .6
Ph a se [ra d ]
-0 .4
-0 .2
0
Figure 6.12. Probability of 5th harmonic current phase estimated from sample histogram and Gaussian assumption
(continuous line). Data are relative to line #2, feeding a cluster of industrial customers.
A critical observation reveals that:

both line #1 and line #2 are described fairly well resorting to Gaussian modeling;

line #3 results show, even in the limited observation period, a multimodal behavior;

all of the scatter plots show elliptic shapes, suggesting that the hypothesis of Gaussian
distribution is verified for the resolved components; in fact, elliptic scatter plot will still be
displayed by vectors whose resolved components are distributed according to f(Q), being
f some positive function with unitary integral and Q the quadratic form in 0; the line #3
non-stationary can be observed either in the scatter plot (two groups of data can be
detected) and in the time domain; an increasing trend in harmonic injection can be
commonly detected and associated to residential activities and it is not a surprise,
therefore, that the discussed algorithms do not fit accurately the considered process;

increasing trends can be detected, even if in a less significant way, in all of the 5 th
harmonic currents measured in the 3 lines; it is interesting to observe that the
assumption of independence between loads would support the belief that, adding all the
harmonic currents, the resultant resolved components jpdf would be more close to the
21
normal jpdf than that of the contributions. In reality, trends seem to add-up so that the
estimates of the current flowing into the supply transformer are less accurately fitted by
the analytical equations here described. For this purpose, 0 and 0 report the magnitude
pdf for both the current resultant and the voltage measured at the supply busbar for both
quantities, estimates are not accurate as it would be expected owing to theoretical
considerations;

the Almost Gaussian performs very well when it can take advantage of the high mean and
low variance values ;

the line current phase estimation by the almost analytical solution gives an excellent
performance.
For higher order harmonics, a tendency to conform more easily to the Gaussian model has been
detected. These harmonics show, as anticipated, the tendency to display a single mode and are,
therefore, more likely to be described by the probability density functions discussed in the
presentation.
6.7
Conclusions
This section has shown opportune rotations of the Cartesian axes in which the random vectors
to be summed are represented given favorable consequences, reducing to zero the correlation
between the resolved components, that may be in actual cases relevant mainly for low order
harmonics. Moreover, the rotation application has been extended for dealing with the
summation of random vectors also in presence of low dependence among them.
Afterwards, Gaussian modeling of harmonic vectors in power systems has been dealt with the
aim of validating and/or proposing methods for resultant vectors magnitude and phase
distribution evaluation. Among the tested methods it is possible to distinguish those analytical
from the empirical. Analytical approaches must satisfy certain assumptions and therefore are
not suited for every possible field situation. Empirical solutions do not pass through any
rigorous analytical validation, but they perform rather well in a broad band of simulated
situations. Even if prevalent attention has been dedicated to magnitude distribution, as it is
normally done in literature, also phase distributions have been derived by an analytical
procedure.
Besides presenting the application of all these methods to simulated situations, the section has
also focused on a practical use of these distributions. For this purpose, harmonic measurements
performed on a node of the Italian medium voltage network have been processed. The results
22
have shown that most of the recorded quantities can be successfully described by the empirical
procedures for the magnitude distribution estimation. Moreover, the proposed relationship for
the phase distribution evaluation has shown to perform satisfactorily. Furthermore, when some
assumptions are met, analytical solutions can be successfully employed.
The results obtained lend hope that when a Gaussian modeling of harmonic vectors is
considered not far from reality, the possibility of performing and processing measurements in a
simple way is given. In fact, provided that in the considered observation interval the signals are
stationary, the useful information about a single harmonic vector can be derived by the
collection of only 5 real numbers: the means of the resolved components and the (symmetric)
covariance matrix elements. This can be of great usefulness being necessary for the application
of the standards to cumulate the efforts of numerous different stationary intervals to obtain
statistics extended to the whole reference time to be considered.
6.8
References
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23
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24