Chapter 2 – (chapter 11)
Probabilistic Harmonic Indices
P. Caramia, G. Carpinelli, A. Russo, P. Verde, P. Varilone
2.1 Introduction
When the time-varying nature of harmonics is considered, the most adequate indices to
represent it have to be chosen.
The indices can refer either to a single customer point-of-common coupling or to a
segment of the distribution system or, more generally, to the utility’s entire distribution system.
While the single site index probabilistic treatment requires the collection of different
observations in time, for system indices it is needed to collect also observations at different sites.
These last indices – well known to characterize other aspects of service quality as sustained
interruptions – only recently have been considered for harmonic purposes [1]; the system indices
can serve as a metrics only and are of particular interest in the new liberalized market frame,
because they can be used as a benchmark against which the quality level of different distribution
systems, or of parts of the same distribution system, can be compared.
In the next, both single site and system probabilistic harmonic indices are analyzed and some
considerations about their statistical characteristics are presented, also on the basis of
measurements on actual distribution systems; particular attention is paid to ascertain the
statistical measures adequate to synthetically describe their probability distribution functions
(mean value, standard deviation, percentiles, etc.) and to be included in a harmonic standard or in
a power quality contract.
On the basis of the statistical characteristics of the harmonic index probability density
functions, it is shown that the expected value of peak factor on single site is a suitable statistical
measure to be selected for inclusion in Standards or in a power quality contract.
Furthermore, different system indices can be introduced in qualifying a distribution
system or a part of it, particularly in the frame of the new liberalized market. Proper selection
among those here proposed depends on the particular aim of the analysis.
2.2 Single Site Indices
The harmonic distortion can be described by several indices (individual harmonic
distortion, total harmonic distortion, the peak factor, and so on). In the next, without loss of
generality, we will refer to the total harmonic distortion (THD) and peak factor (kp); on the other
hand, they are very useful indices [2-6] and also easily transferable in the probabilistic field,
since only the marginal probability density function is generally required to fully describe their
random properties. In any case, considerations about the probabilistic behavior of individual
harmonic distortion are reported in [7,8].
The total harmonic distortion of Y (current or voltage) is given by:

 Yh2
THDY 
h2
(1)
Y1
with Yh harmonic superimposed to the fundamental Y1.
The peak factor is defined as:
k pY 
Yp
(2)
Y p1n
where Yp is the peak value of the distorted waveform Y, and Yp1n the rated value of the
fundamental.
The peak factor can be separated into two aliquots kpsY and kphY, the first related to the
effects of fundamental variations and the second one related to the effects of harmonics
superimposed to fundamental Yp1; it results:
k pY 
Y p1 Y p
Yp1n Yp1
 k psY k phY
.
(3)
Both total harmonic distortion and peak value can be applied to voltage and current
distortions. The peak factor of voltage given by (2) and its aliquots given by (3) have been
considered in [5,9] for translation in the probabilistic field while the current indices have been
considered in [6].
The probability density functions (pdf’s) of the above mentioned indices exhibit different
statistical characteristics, as clearly revealed by the analysis of several on site measurements and
simulation results.
Firs of all, the pdf’s of all above indices can have several shapes and there is not a general
rule to choice the probability density function which best fits one of them; in fact, various basic
pdf’s (gaussian, uniform, linear increasing, and so on) have been tested to ascertain the best
approximating the harmonic index pdf behavior, but any of them can’t be generally applied.
From the analysis of several daily or weekly on-site measurements and of numerical
simulations on low voltage and medium voltage distribution systems, the following
considerations on their probabilistic behavior can be provided.
The pdf’s of the total harmonic distortion of the voltage (THDV) and current (THDI) can
have both monomodal and multimodal (usually bimodal) pattern. Moreover, mainly when the
harmonic data refer to a week, they are generally characterized by significant values of standard
deviations, due to the variability of harmonics in large ranges.
As an example, Figs. 1 a) and b) report the pdf’s of the THDV (a) and of the THDI (b)
obtained processing over one week recorded data referred to a medium voltage distribution
system.
Starting from the considerations about both the pdf shapes and statistical measures of
THDV and THDI, it follows that to impose reasonable probabilistic limits on these indices is
practically impossible. In fact, to directly set limits on the whole pdf’s is neither realistic nor a
reasonable solution, as a consequence of the obvious difficulties in the successive
implementation in the Standards. On the other hand, to impose limits on one or more statistical
measures (minimum value, maximum value, mean value, standard deviation, 95th percentile, and
so on) can be practical but can lead to unreasonable decisions. In fact, if we refer to the 95 th
percentile or maximum value, as proposed by the IEC 1000-3-6 [3] in its assessment procedure,
and we account for all the harmonic effects, unacceptable consequences can follow, since THDV
and THDI pdf’s with the same 95th percentile and maximum value can have very different effects
on electrical component behavior. As a very easy example, we can refer to the joule losses in a
conductor (in percent of the losses at fundamental frequency) associated to the two different
THDI pdf’s of Fig.2, characterized by the same 95th percentile and maximum. Assuming the
conductor resistance independent on the frequency, the joule loss expected values in the two
cases are equal to 0.095 (Fig.2a) and 0.035 (Fig.2b), that are far each other.
The pdf’s of the voltage peak factor (kpV) are characterized, in the most general case, by
very low values of standard deviations. In all the results reported in literature and in our
experience, the standard deviation shows values at least of very few percents, because the
voltage harmonics have values lower than fundamental voltage that ranges in no particularly
large intervals for system operation safety. Due to the very low values of standard deviations, it
is useless to talk about the shape of the pdf (monomodal or multimodal). A similar behavior has
been experimented with reference to the aliquot kphV, related to the effects of harmonics
superimposed to the fundamental.
As an example, Fig. 3 shows the pdf’s of the kpV (a) and of kphV (b) obtained processing over
one week recorded data referred to a medium voltage distribution system.
= 1.64 %
= 18.7 %
fTHDV
1.5
fTHDI
 = 0.3 = 0.07
CP95 = 0.4
f
kpV
=1.05
=0.80 %
40
30
1
20
0.5
10
0
1
1.5
2
THDV [%]
(a)
THDI [%]
(a)
fTHDI
= 6.70 %
= 24.1 %
0.2
fTHDI
0
1.02
f kphV
 =0.17  =0.08
CP95 = 0.4
0.1
40
0.05
20
0
0.98
0
10
THDI [%]
(b)
kpV
= 1.01
= 0.70 %
80
60
5
1.06
(a)
0.15
0
1.04
THDI [%]
(b)
1
1.02
kphV
(b)
Fig. 1 – Probability density
Fig. 2 – Different THDI
functions: a) THDV;
pdf’s with same 95th
b) THDI
percentile
and
Fig. 3 – Probability density
functions: a) kpV; b) kphV
maximum values
From the analysis of Fig.3b and from all the measurements on distribution systems we
analyzed, it was also noted that the voltage harmonics in several cases cause increasing of the
peak voltage with respect to the value that would have been in presence of only the fundamental
component. In [9] it was instead experimented a lowering of the peak voltage.
Starting from the considerations about the kpV and kphV pdf shapes and their statistical
measures, it follows that to impose probabilistic limits on these indices is now possible and can
be a reasonable choice. In fact, the very low value of pdf standard deviations allow us to say that
it is enough to assign a limit on only the mean value of the pdf. Moreover, in practice only the
mean value of the voltage peak factor1 kpV seems to be suitable for inclusion in Standards or in
power quality contracts. The aliquot kphV does not allow to compute all the harmonic effects on
distribution system components, because to distinguish the effects between fundamental and
harmonics is a very difficult matter2.
As an example of setting limits on the mean value of peak factor, let’s consider the case
of low voltage industrial systems. The limits can be assigned making use of the diagram in Fig. 4
[5], where the normalized expected value of life is plotted versus the mean value of peak factor3;
each point of Fig. 4 is obtained averaging the expected values of normalized useful life for
different distribution system components (cables, transformers, capacitors and ac motors).
The pdf’s of the current peak factor4 (kpI) and of its aliquot kphI, related to the effects of
harmonics superimposed to the fundamental can have both monomodal and multimodal pattern,
2
E[L]/Ln
1.5
1
0.5
0
0.96
1
0.98
1.00
1.02
1.04
1.06
As shown in [4], the voltage peak has dominant effect in accelerating aging
of distribution system components in distorted regime and so the
E[Kp]
peak factor is the index most adequate to take into account the harmonic effects.
2
The aliquot kphV seems most appropriate for component designers as a valuable complementary decisional criterion in choosing among
different design solutions.
3
The expected
of normalised useful
life of
is thenormalized
ratio between theuseful
expected value
useful lifeexpected
and the nominal
one. of peak
Fig. 4value
– Expected
value
life ofversus
value
4
It is useful to remind that the current peak factor is a valuable index taken into account also in the Standards (for instance the IEC Std
for cables,
capacitors,
transformers
and ac motors
60947-2 [10], with factor
referenceaveraged
to the circuit-breakers
with electronic
over-current
protection).
once again usually bimodal. The variance of the kpI pdf is characterized, in the most general case,
by significant values of standard deviations; on the contrary, reduced variances of kphI pdf’s were
experimented.
As an example, Figs. 5 a) and b) show the pdf’s of the kpI (a) and of the kphI (b) obtained
processing over one week recorded data referred to an actual medium voltage distribution
system.
From the analysis of Fig.5 a and from all the measurements we analyzed on distribution systems,
it was also noted that the high variability of current peak factor kpI is due to the wide variation
range of the fundamental component, as clearly highlighted in Fig. 6, which reports pdf of the
aliquot kpsI corresponding to the same measurements of Fig. 5. This is a reasonable result, for the
very large variability of connected loads.
fk
= 0.53
= 29.0 %
psI
3
2
1
0
0.2
0.4
0.6
0.8
kpsI
Fig. 6 – Probability density function of kpsI
From the analysis of Fig.5b, it appears that the current harmonics cause decreasing of the
peak current with respect to the value that would have been in presence of only the fundamental
component. However, this is not a general rule, because in other on site measurements on
distribution network it was experimented an increasing of the peak current, in some cases
significant.
Finally, starting from the considerations about both the kpI pdf shapes and their statistical
measures, it follows that to impose reasonable probabilistic limits on this index is impossible, for
the same reasons evidenced in the case of THDV.
On the contrary, if the trend evidenced by our measurements were confirmed, to impose
probabilistic limits on kphI index would be possible and also a reasonable choice. In fact, the very
low value of pdf standard deviation allows to say, once again, that it is enough to assign a limit
only on the mean value. A possible criterion for assigning limits to the mean value of kphI could
be to set the fundamental current at rated value and derive the limits on the basis of the
component effects.
2.3 System Indices
As clearly pointed out in [1], the system indices serve as metrics only and are not
intended as an exact representation of the quality of service provided to each customer served
from the assessed system. They can be used as a benchmark against which quality levels of parts
of the distribution system and of different distribution systems can be compared.
The system indices here taken into account only refer to voltage distortion; they are:

the System Total Harmonic Distortion 95th percentile, STHD95;

the System Average Total Harmonic Distortion (Peak Factor) SATHD (SAkp5);

the System Average Excessive Total Harmonic Distortion (Peak Factor) Ratio Index,
SAETHDRITHD*, (SAEkpRIKP*).
The system indices referred to Total Harmonic Distortion have been defined in [1], the
ones concerning the Peak Factor are introduced in [6].
Let’s refer to a distribution system with N busbars. Even if the system indices can also
refer to small segments of the entire distribution system, such as a single feeder or a single
busbar, in the following, without loss of generality, we consider indices referred to the entire
distribution system.
The STHD95 index is defined as the 95th percentile value of a weighted distribution; this
weighted distribution is obtained collecting the 95th percentile values of the M individual index
distributions, each distribution obtained with the measurements recorded at a monitoring site.
The weights can be linked to the connected powers or the number of customer served from the
area the monitor data represents, and so on; in the next, without loss of generality, we assume as
weights the connected powers. Under these assumptions, the following relations compute
STHD95:
STHD95
 f t CP95THDs Ls


 0.95 ;
(4)
 0..95
(5)
 f t CP95THDs Ls

with CP95THDs calculated via the succeeding expression:
 fTHD,s X THD
CP 95 THD , s
s,i


 fTHD,s X THD

s,i


where:
s = monitoring site;
XTHDs,i = ith steady state THD measurement at the site s;
Ls= connected kVA served from the system segment at the monitoring site s;
fTHDs (XTHDs,i) = probability distribution function of the sampled THD values for monitoring
site s;
CP95THDs = 95th cumulative probability value of THD for the monitoring site s;
ft(CP95THDs) = probability distribution function of the individual monitoring site CP95THDs
values.
The system index STHD95 allows to summarize the measurements both temporally and
spatially handling measurements at M sites of the system in a defined time period, assumed as
significant for the characterization of system service conditions.
As an example, the value of STHD95 obtained processing over one week recorded data
referred to the same medium voltage distribution system considered in Section 4.2.2 results
equal to 2.2%.
The System Average Total Harmonic Distortion (Peak Factor) is based on the mean value
of the distributions rather than the CP95 value:
M
SATHD 
 Ls  (THD s )
s 1
LT
(6)
M
SAk p 
 Ls  (k p, s )
s 1
(7)
LT
being :
s)
p,s)
= mean value of sampled THD values for monitoring site s;
= mean value of sampled kp values for monitoring site s;
LT = total connected kVA served from the system.
SATHD and SAkp give average indications on the system voltage quality and, thanks to
the introduced weights, allow to assign different importance to the various sections of the entire
distribution system.
The values of SATHD and of SAkp,again computed on the same distribution system before
referred to, result equal to 1.16% and 1.05, respectively. It is worthwhile to note that SATHD is
far from STHD95 due to the not negligible value of standard deviation.
The System Average Excessive Total Harmonic Distortion, (Peak Factor) Ratio Index,
SAETHDRITHD* (SAEkpRIkp* ) relates to the number of steady-state measurements that exhibit a
THD (kp) value exceeding the THD* (kp*) threshold. It is computed, for each monitoring site of
the system, counting the measurements exceeding the THD* (kp*) value and normalizing this
number to the total number of the measurements effected at site s. They are defined by the
following relations:
M
SAETHDRI THD* 
s 1
M
SAEk p RI k p * 
 NTHD*s 


 NTot , s 
LT
 Ls 
 N k p *s 


Tot
,
s


LT
 Ls  N
s 1
(8)
(9)
where:
NTHD*s (Nkp*s) = number of steady state measurements at monitoring site s that exhibit a THD (kp)
value which exceeds the specified threshold THD* (kp*);
NTot,s= total number of steady state measurements effected at monitoring site s over the assessed
period of time.
SAETHDRITHD* and SAEkpRIkp* are linked to the measure of the total portion of the time
that the system exceeds the specified THD* and kp* thresholds, respectively. Exact indication on
the total time the system exceeds the thresholds limits can be obtained only if the weights are not
introduced.
Table I shows the values of SAETHDRITHD* and SAEkpRIkp* computed on the same
distribution system already referred to and with two thresholds.
Table I: Values of SAETHDRITHD* and SAEkpRIKP*
SAETHDRI5%
0
SAETHDRI2%
0.016
SAEkpRI1.03
0.97
SAEkpRI1.05
0.41
Summarising, the introduced indices provide different global measures to qualify the
system in terms of service quality. The availability of several indices for a phenomenon can be
useful to give evidence to different aspects of the same disturbance. No general rule can be
established to suggest the most adequate index to be used; it will depend on which aspect is
judged more relevant.
For example, in the considered real distribution system, SATHD equal to 1.16 % indicates
not only the average level of distortion, but also takes into account the sites serving larger loads.
This can be more evident if we consider that the average value of THD on the same distribution
system, without any weights, is greater of about 25%.
Further indication on the same system and on the same measurements concerns the
highest levels of distortion. STHD95 value indicates the weighted 95th percentile of the 95th
percentiles of the system sites and SAETHDRITHD* values give more details about the weighted
relative frequencies of the records above specified threshold. Analogous considerations can be
developed concerning the system indices referred to the peak factor.
An alternative way to compute the aforementioned indices concerns the statistical
analysis of all the measurements effected at the M sites of the system. Assuming that centralized
handling all the measurements of the M sites is not hard, this method seems to be more
adequate, since any statistical measure of a pdf is estimated with an accuracy that depends on the
number of processed data. This extensive procedure gives indication on the global behavior of
the system; on the contrary the first recalled procedure furnishes the indication on the behavior
of the sites.
For example, with reference to THD, for each site s we compute the relative frequencies
of the measured THD and we weight them with the ratio Ls/LT. Starting from the weighted
relative frequencies at each site s, we evaluate the system weighted relative frequencies WFTHD
and, then, compute the mean value and the 95th percentile that coincide with the indices SATHD
and STHD95, respectively. The system weighted relative frequencies of THD can be also used to
evaluate the indices SAETHDRITHD*.
An analogous procedure can be implemented to calculate the indices SAkp and SAEkpRIkp*
referred to the peak factor.
Figs. 7 and 8 show the histograms of system weighted relative frequencies of THD and kp
(WFTHD and WFkp), respectively, with reference to the distribution system under consideration.
Table II shows the results in terms of the system indices obtained with the
aforementioned procedure.
.
WFkp
WFTHD
0.16
0.25
0.12
0.2
0.15
0.08
0.1
0.04
0.05
0.5
1
1.5
2
THD [%]
1.02
1.03
1.04
1.05
1.06
kp
Fig. 7 - Histogram of system weighted relative Fig. 8 - Histogram of system weighted relative
frequencies for THD
frequencies for kp
Table II: Values of system indices
STHD95
1.94 %
SATHD
1.16%
SAkp
1.05
SAETHDRI2%
0.016
SAEkpRI1.03
0.97
Comparing the results of the indices computed via the two procedures, it is important to
evidence that the obtained values of SATHD, SAETHDRITHD*, SAkp and SAEkpRIkp* coincide, as
expected.
On the contrary, the values of STHD95 are different. This is due to the fact that in the first
procedure, the weighted distribution cannot take into account the values greater than the 95 th
percentiles. These values are instead considered, for the second procedure, in the construction of
the system weighted relative frequency histogram, so that they can determine a lower value of
the 95th percentile
2.4 References
[1]
D.D. Sabin: “Analysis of Harmonic Measurements Data”, in Proc. 2002 IEEE Power
Engineering Society Summer Meeting, pp. 941 –945.
[2]
“Voltage Characteristics of Electricity Supplied by Public Distribution Systems” ,
Standards EN50160, 1999.
[3]
IEC 1000-3-6, IEC 1OC1O-3-6:Assessment of Emission Limits for Distorting Loads in
LV and HV Power Systems – 1996.
[4]
P. Caramia, G. Carpinelli, P. Verde, G. Mazzanti, A. Cavallini, G.C. Montanari: “An
Approach to Life Estimation of Electrical Plant Components in the Presence of Harmonic
Distortions”, Proc. of 9th Inter. Conf. on Harmonics and Quality of Power, Orlando
(USA), 2000, pp. 887-891.
[5]
P. Caramia, G. Carpinelli, A. Russo, P. Varilone, P. Verde: “ An Integrated Probabilistic
Harmonic Index”, in Proc. 2002 IEEE Power Engineering Society Winter Meeting, pp.
1084-1089.
[6]
P. Caramia, G. Carpinelli, A. Russo, P.Verde: “Some Considerations on Single Site and
System Probabilistic Harmonic Indices for Distribution Networks”, IEEE general
Meeting, Toronto, July 2003.
[7]
Sabin, D.D.; Brooks, D.L.; Sundaram, A.; “Indices for assessing harmonic distortion
from power quality measurements: definitions and benchmark data” Power Delivery,
IEEE Transactions on , Volume: 14 Issue: 2 , April 1999.
[8]
A.E. Emanuel, S.R. Kaprielian: An Improved Real-time Data Acquisition Method for
Estimation of the Thermal Effect of Harmonics – IEEE Trans. On Power Delivery, No. 2,
April 1994, pp. 668-674.
[9]
D. Gallo, R. Langella, A. Testa: “ On the Effects on MV/LV Component Expected Life of
Slow Voltage Variations and Harmonic Distortion”, in Proc. 2002 ICHQP, Rio de Janeiro
(Brasil), October 2002.
[10]
IEC Int. Standard 60947-2, Ed. 2.2: Low Voltage Switchgear and Controlgear – Part 2 –
Circuit-breakers, November 2001.