IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, JANUARY 2002
279
Time-Varying Harmonics: Part II—Harmonic
Summation and Propagation
Y. Baghzouz, R. F. Burch, A. Capasso, A. Cavallini, A. E. Emanuel, M. Halpin, R. Langella, G. Montanari,
K. J. Olejniczak, P. Ribeiro, S. Rios-Marcuello, F. Ruggiero, R. Thallam, A. Testa, and P. Verde
Abstract—This paper represents the second part of a two-part
article reviewing the state of the art of probabilistic aspects of harmonics in electric power systems. It includes tools for calculating
probabilities of rectangular and phasor components of individual
as well as multiple harmonic sources. A procedure for determining
the statistical distribution of voltages resulting from dispersed and
random current sources is reviewed. Some applications of statistical representation of harmonics are also discussed.
Index Terms—Statistical analysis, sum of random phasors, timevarying harmonics, voltage distortion.
I. INTRODUCTION
E
LECTRIC utilities have experienced an increase in the
level of harmonic currents and voltages on their electrical
delivery systems. This is primarily due to the widespread use of
power electronic devices found in residential, commercial, and
industrial loads.
The potential harmonic effects on power equipment and
system operation have become a concern for utilities. As a
consequence, harmonics can no longer be ignored in industrial
power systems since their ignorance may lead to problems
such as capacitor failure or transformer and neutral conductor
overheating.
Over the past two decades, much attention has been given
to deterministic harmonic analysis. Deterministic criteria, however, ignore the variability of nonlinear load operating conditions and resulting changes in harmonic currents injected by
these loads into the utility network.
Field measurements clearly indicate that voltage and current
harmonics are time-variant due to continual changes in load conditions, and to some extent in system configuration. A common
philosophy is to conduct a deterministic study based on the
worst case in order to provide a safety margin in system design and operation. But this often leads to overdesign and excessive costs. Consequently, statistical techniques for harmonic
analysis are more suitable, similar to other conventional studies
like probabilistic load flow and fault studies [1]. Such an analysis would calculate harmonic currents and voltages based not
simply on the expected average or maximum values, but would
also obtain the complete spectrum of all probable values together with their respective probabilities.
Manuscript received July 19, 1999; revised July 18, 2001. The authors are
member of the Probabilistic Aspects Task Force of the Harmonics Working
Group Subcommittee of the Transmission and Distribution Committee, Y. Baghzouz–Chair.
Y. Baghzouz, Task Force Chairman, is with the Electrical and Computer engineerign Department, University of Las Vegas, Las Vegas, NV 89154 USA.
Publisher Item Identifier S 0885-8977(02)00547-2.
Probabilistic harmonic analysis in real power networks is not
a simple task due to several factors including the following:
a) there exist a large number of different nonlinear loads that
generate harmonic currents which depend on the magnitude and
harmonic content of the voltage supply; b) load composition on
a feeder is constantly changing; c) there is a lack of data on
how different voltage waveforms affect the harmonic currents
of several electronic loads; and d) load modeling at harmonic
frequencies is a complex subject that is not fully understood.
The objective of the Task Force is to review and summarize
probabilistic aspects of harmonics in power systems. Due its
length, the subject is divided into two parts: Part I [2] reviews
problems associated with direct application of the fast Fourier
transform to compute harmonic levels of nonstationary distorted
waveforms, harmonic measurement, and various ways to describe recorded data in statistical form. Part II covers the summation of random harmonic phasors, the characteristics of harmonic voltages caused by dispersed nonlinear loads, and some
applications of these probabilities.
The paper is divided into three sections. The general representation of a harmonic phasor and the marginal distribution of its
real and imaginary components are given in Section II. Means
of deriving the probability characteristics of the sum of independent random phasors are also discussed in Section II. Section III
describes the linearized method for determining harmonic bus
voltages caused by distributed harmonic sources. Finally, some
applications to probabilistic harmonic indices are addressed in
Section IV.
II. SUM OF RANDOM HARMONIC PHASORS
It is a well-known fact that the sum of a number of harmonics
currents with some statistical variations generally leads to less
than the arithmetic sum of the maximum values. This section
addresses the probability characteristics of such a sum by first
reviewing the representation of a harmonic phasor in terms if
its joint probability density function (jpdf) and its rectangular
components in terms of their marginal probability density functions (pdf). In order to avoid complexity of conditional probabilities used to describe dependent harmonic phasors, only independent harmonic sources are studied in this paper. Furthermore, the following analysis applies to harmonics of any order
; consequently, the subscript is left out in all symbols to simplify notation.
A. Representation of Harmonic Phasors
A harmonic phasor can be described by either by its mag, or in terms of its rectangular components
nitude and phase
0885–8977/02$17.00 © 2002 IEEE
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, JANUARY 2002
. These four parameters are related to one another by
,
,
,
the following:
if
, and
if
. Eior polar parameters
ther rectangular parameters
can be used when evaluating phasors. The rectangular components or – projections are chosen because of the convenience
they offer when adding phasors.
Generally, and are dependent of each other and their
distribution is described by a joint probability density function
which is related to the joint cumulative distribution
by [3]
(1)
The characteristic measures of the above jpdf include five parameters: the mean values and standard deviations of and ,
, , ,
, and the joint second moment about the mean
values or covariance defined by
(2)
Fig. 1. Simple shapes in x–y plane.
The correlation coefficient which measures the dependence
between the – components is defined by
(3)
B. Marginal pdf of – Components
The marginal pdf of one of the projections of a phasor is calculated by integrating its jpdf over the other variable, e.g.,
(4)
, and are
Note that when the correlation coefficient
statistically independent and the jpdf simplifies to the product
of the marginal pdf of both variables
(5)
Some special cases where the jpdf is constant over simple curves
or surfaces are reviewed next.
Consider a phasor whose magnitude is fixed, while its phase
varies randomly with a uniform distribution between and ,
as depicted in Fig. 1(a). The joint pdf is
i.e.,
Fig. 2.
Pdf of x–y components of random phasor.
, in addition to phase variation as shown by the shaded area in
Fig. 1(b), the jpdf becomes
(6)
satisfying
for all
or is given by [4]
. Then the marginal pdf of
for
(8)
satisfying
for all
components becomes [5]
. Then the pdf of the –
(7)
(9)
elsewhere. The mean value and standard deand
, respectively.
viation of the – projections are and
If the magnitude of the phasor above also varies randomly
with a uniform distribution between and its maximum value
, and the mean and standard deviations of
Here,
, respectively. Both pdfs in (7) and
either or are and
(9) are plotted in Fig. 2.
BAGHZOUZ et al.: TIME-VARYING HARMONICS: PART II—HARMONIC SUMMATION AND PROPAGATION
Fig. 1(c) shows the case where there is zero correlation be), and the constant jpdf over a rectantween and (i.e.,
gular surface is
(10)
, and
. As indicated by
where
(5), the above expression is simply the product of
and
. It is clear that the
, and the
mean values of rectangular components are
standard deviation can easily be derived in terms of the extremes
values of and .
Finally, Fig. 1(d) shows the surface in the – plane where
there is no correlation between the magnitude and phase, and
the constant jpdf is calculated by
(11)
and
. The analytical exwhere
pressions of the marginal pdf is relatively complex, but can be
derived according to (4). The mean values of the – projections are found directly from the mean phasor value
Note that independence between magnitude and phase as in
Fig. 1(d) results in dependence between and . The reverse
is also valid as illustrated in Fig. 1(c) where independence between and results in dependence between magnitude and
phase.
required, but this is becoming a manageable computational
problem with the advent of faster and inexpensive computers.
In case when the number of phasors to be added is sufficiently
large and no phasor is dominant over the others, an accurate solution can be found by using one of the most important theorems
in probability theory which pertains to the limiting distribution
of the sum of independent random variables. This is known as
the central limit theorem which states that the pdf of the sum
approaches a normal distribution regardless of the distributions
of the individual variables, as long as the number of variables is
sufficiently large and none is dominant. If the above conditions
are satisfied, the pdf of the sum of the – projections in (12) is
approximated by
(16)
where the mean value and variance of this sum are respectively
and
.
given by
D. PDF of Magnitude of Sum of Random Phasors
The jpdf of the sum of independent phasors, each of which
is described by its jpdf in (1), is obtained by applying convolution of bivariate functions by extending (14) to two-variable
functions. Once more, such integrations are complicated and
one often resorts to Monte Carlo simulation. If a relatively large
number of phasors are to be added and none of the phasors is
dominant, then the central limit theorem can be applied. In such
a case, the resulting jpdf is a normal bivariate distribution given
by
C. Pdf of Sum of Projections of Independent Phasors
Now consider a sum of
frequency
(17)
independent phasors of the same
where
(12)
The pdf of
and
281
are obtained by convolution
(18)
(19)
(13)
where
(20)
(14)
(21)
Another analytical solution method is to first find the Fourier
of each pdf
, and then take the inverse
transform
of the product of these functions, i.e.
The pdf of the magnitude the sum can be derived by transto polar coordinates
forming the rectangular components
(15)
Both the convolution and Fourier transform methods are
known to be difficult to evaluate analytically and numerically.
An alternative and most widely used solution is the Monte
Carlo simulation method which is simply a repeated process
of generating deterministic solutions to the problem, with each
solution corresponding to a set of deterministic values of the
random variables. Generally, thousands of simulations are
(22)
Approximate solutions to the above equation are derived under
various special cases in [6] and [7].
The magnitude of the total phasor can also be estimated form
the marginal pdf of the sum of the and projections calculated earlier [8]–[13], and the accuracy depends on the number
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, JANUARY 2002
Fig. 3. Pdf of magnitude of sum of random phasors with fixed magnitude and
random phase.
Fig. 4. Pdf of magnitude of sum of random phasors with random magnitude
and phase.
of phasors to be summed and the correlation between the rectangular components of each phasor. In case of low correlation
and a sufficiently large number of phasors, the resulting rectangular components can be considered independent and each
having a Gaussian distribution. The distribution of the magnitude of the total phasor can then be derived analytically. Further, in special cases where the mean value of – projections
is and equal variance , then the pdf of the magnitude of the
sum is a Rayleigh distribution [3]
(23)
The mean value and standard deviation of the above distribution
and
, respectively.
are
To illustrate the above statements, the magnitude of a sum of
identical and independent random phasors is considered. Each
of the four uniform joint probability distributions with – propu,
jections shown in Fig. 1 is investigated (with
,
pu,
pu,
pu,
,
). Let Cases a)–d) correspond to Fig. 1(a)–(d). The
and
resulting pdfs of the magnitude of the sum of 3 and 10 phasors
are shown in Figs. 3–6, respectively. The following is noted.
• Cases a) and b): Rayleigh distribution can be used since
both and have zero mean and the same variance. However, since is highly dependent on in individual vectors in Case a), a larger number is required for accurate
Rayleigh representation as shown in Fig. 3. On the other
hand, the – components are highly (but not totally) independent in Case b); thus requiring only few phasors for
a valid Rayleigh approximation as illustrated in Fig. 4.
• Cases c) and d): In these cases, Rayleigh distribution is
not expected even for a large number of phasors because
of the fact that the mean values of the – are nonzero
and their variances are unequal. The resulting distribution
is expected to be Gaussian even for a small number of
Fig. 5. Pdf of magnitude of sum of random phasors with independent x–y
components.
phasors due to total independence in Case c) (see Fig. 5)
and high independence in Case d) (see Fig. 6).
E. Practical Considerations
In practice, analytical expressions describing joint distributions of harmonics are often unavailable due to their
complexity. Unlike the simple distributions considered earlier,
actual distributions are usually spread over complex surfaces
and in nonuniform fashions. The data that is generally available
includes scatter plots from which one can extract simple distributions that fit standard analytical functions such as elliptical
shapes. When correlation exists between the real and imaginary
BAGHZOUZ et al.: TIME-VARYING HARMONICS: PART II—HARMONIC SUMMATION AND PROPAGATION
283
Fig. 7. 14-Bus transmission system.
Fig. 6. Pdf of magnitude of sum of phasors with independent polar
components.
parts of an elliptical distribution, a multivariable expression of
the ellipse is needed as a model
(24)
Scatter plots of currents are also known to vary more erratically, depending of the type of loads connected to the local
supply. Several previous efforts analyzed a cluster of specific
loads, and evaluated the resulting harmonic current at the point
of common coupling: References [13] and [14] studied a set of
electric vehicle battery chargers where the initial state of charge
and recharge start time are considered random. Reference [15]
examined the diversity of a set computer loads due to difference
in phase angles. Finally, [16] studied a set of variable-speed air
conditioners with variable duty cycle. The main conclusion of
the above studies is that there is often significant harmonic cancellation and that the worst case (i.e., arithmetic sum) is often
too conservative, sometimes higher than the expected value by
an order of magnitude.
III. PROBABILISTIC HARMONIC VOLTAGES
When random harmonic currents are injected at multiple
nodes within a utility network, it is desired to determine the
characteristics of the resulting harmonic voltages at different
buses. In general, harmonic current injection depends on the
voltage supply. For most of nonlinear loads, however, this
effect of secondary order as long as the voltage distortion is
below 5% [17] which is often the case in practice. Therefore,
one can assume that the harmonic currents are independent of
the bus voltages, without much error.
In such a case, the resulting voltage expression at any bus
of a network with nodes is derived by simple linear circuit
theorems such as the superposition principle, i.e.
(25)
is the harmonic transfer impedance between buses
where
and . Note that the harmonic voltage above is a sum of weighted
independent harmonic currents: the pdf of each voltage component is equal to jpdf of the corresponding current but scaled
in magnitude and phase according to its transfer impedance.
Hence, the methods of the previous section can be utilized to
derive the pdf of .
In case where the harmonic current depends on the harmonic
voltage, an analytical relationship of the two signals is needed.
Unfortunately, such knowledge is limited as it is available only
for one type of nonlinear load, namely, the six-pulse rectifier.
Harmonic iterative power flow methods are available for deterministic studies [18] where the Jacobian matrix is many folds
larger than conventional power flow at fundamental frequency
and the convergence tends to be one order of magnitude longer
than those of the conventional power flow at fundamental frequency. Such a deterministic study has been extended to probabilistic ones [18].
For illustration purposes, one of the test systems selected
by the Task Force on Harmonics Modeling and Simulation
[19], namely the Test System no. 1 is selected for analysis.
The system is a 14-bus balanced transmission system whose
one-line diagram is shown in Fig. 7. The complete system
data and harmonic sources are listed in [19]. It is desired to
evaluate the fifth-harmonic voltage at bus 10 caused by the
fifth-harmonic currents at buses 3 and 8. The deterministic
pu at bus 3
values of these currents are
at bus 8. The current phase angles are
and
referred to a common reference (i.e., shifted according to the
load flow results). With the per-unit values of fifth-harmonic
and
both approximated to
transfer impedances
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 17, NO. 1, JANUARY 2002
IEEE Std. 519-1992 recognizes time variation of harmonics,
but the recommended voltage and current harmonic limits are
static and do not take into account actual randomness. Meanwhile, other standards include the variation by setting limits in
terms of percentile. For example, a CIGRE study [23] set limits
with 95 percentile (i.e., the limits are allowed to be exceeded 5%
of the time). Others included time duration of harmonic bursts
[24]. Universal limits will be accepted after harmonic effects on
equipment are quantified and well documented. Further work
on this subject is recommended.
V. CONCLUSION
Fig. 8.
Pdf fifth-harmonic voltage at bus 10.
, the resulting harmonic voltage magnitude at bus 10
pu.
Now let the above currents each contain an additional random
component with the following characteristics:
• Random component of current at bus 3: and vary inand
pu,
dependently and uniformly between
respectively.
• Random component of current at bus 8: and vary inand
,
dependently and uniformly between
respectively.
The surfaces covered by the above random phasors are similar
to those in Fig. 1(c) and (d). The resulting pdf of the magnitude of the harmonic voltage at bus 10 is derived by mean of
Monte Carlo simulation and it is shown in Fig. 8 (solid line).
The figure also displays two other curves illustrating the effect
of transfer impedance on the pdf of harmonic voltage. While the
curves remain nearly Gaussian, both the mean and standard deviations are sensitive to the relative magnitudes of the transfer
impedances.
is
IV. APPLICATIONS
Complete characterization of power system harmonic levels
will provide important information to electric utility engineers
and equipment designers. Some of the potential applications of
statistical indexes of harmonic voltages include a re-evaluation
of harmonic effects on equipment and a corresponding review of
existing recommended limits. The effect of harmonics on equipment is documented in [20] when static distortion is generally
assumed. The effect of harmonic randomness on temperature
rise and dielectric stress of electrical equipment is investigated
in [21], [22]. While each device has a different reaction to harmonics, it is generally acceptable to have bursts of short durations. Consequently, assuming worst case situations when designing equipment may be too conservative and more costly.
This paper has reviewed existing methods for harmonic summation and propagation in situations where harmonic sources
contain random components. Analytical explicit expressions of
resulting currents and voltages can be derived for simple cases
where the injected currents are independent and have uniform
or Gaussian distributions. The rest of the cases are considered
too complex to study analytically and one has to resort to either the central limit theorem when evaluating the sum of a
number of random variables, or the Monte Carlo simulation
which can handle practically any statistical problem. These statistical studies furnish more realistic data on the harmonic levels
in today’s distribution systems. Such data will be important
when evaluation harmonic effects on equipment and revising
current limits.
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A. Cavallini, photograph and biography not available at the time of publication.
A. E. Emanuel, photograph and biography not available at the time of publication.
M. Halpin, photograph and biography not available at the time of publication.
R. Langella, photograph and biography not available at the time of publication.
G. Montanari, photograph and biography not available at the time of publication.
K. J. Olejniczak, photograph and biography not available at the time of publication.
P. Ribeiro, photograph and biography not available at the time of publication.
S. Rios-Marcuello, photograph and biography not available at the time of publication.
F. Ruggiero, photograph and biography not available at the time of publication.
Y. Baghzouz, photograph and biography not available at the time of publication.
R. Thallam, photograph and biography not available at the time of publication.
R. F. Burch, photograph and biography not available at the time of publication.
A. Capasso, photograph and biography not available at the time of publication.
A. Testa, photograph and biography not available at the time of publication.
P. Verde, photograph and biography not available at the time of publication.