Chapter 2 – Relationship Between
Probability Distribution and Spectral
Analysis of Non-Stationary Random
Processes and the Concept of
Evolutionary Spectrum
Paulo F. Ribeiro and Carlos Duque
3.1 Introduction
The concept of harmonic decomposition can only be applied to steady state, periodic
waveform functions. In real life however, voltage and currents are constantly changing with
time. The pragmatic way of dealing with time-varying functions is to extend the time window
and capture the variations along the time. However, this method can have some significant
limitations and loss of physical meaning depending on the time-varying nature of the waveform
analyzed. This section suggests the application of some spectral analysis and probability
distribution concepts, in an integrated way, for a better understanding of the nature of timevarying harmonics and possibly as a more precise way to treat time-varying harmonics and
validate harmonic summation studies.
Harmonic decomposition is a very convenient, but artificial way to manipulate nonsinusoidal waveforms. Therefore, they only “exist” so to speak, and have physical meaning
when associated to a periodic waveform or an instant in time, and they are characterized by a
certain magnitude and phase angle.
Thus, for example, changes in the waveform can be
attributed to the phase angles only. As a consequence analyzing the magnitudes only of the
harmonic components for time-varying distortion may suffer from severe limitations and lead
to misinterpretation of the phenomena.
This is particularly relevant when dealing with
waveform sensitive harmonic distortion problems rather than heating effects for which the
magnitudes alone are sufficient.
Traditionally we have overlooked the problem and applied statistical and probabilistic
methods indiscriminately to time-varying harmonic distortion.
But how can time-varying harmonic distortion be understood and analyzed more
precisely? This chapter draws from some previous mathematical derivations and analysis (1)
and attempts to apply to power quality issues associated to harmonic distortion under timevarying conditions (2).
First the similarity between spectral analysis and probability distribution functions are
observed. Secondly, the concept of generalized frequency is presented and finally the concept
of evolutionary spectra is discussed. A simple application is made to illustrate the usefulness of
the concept/approach.
3.2 - Concept 1 – Similarities Between Spectral Analysis and Probability
Distribution Functions
A curious engineer may have already noticed the similarity in shape between the graphs
used to describe spectral analysis and the graphs of probability distribution functions as
illustrated below in Figures 1 and 2.
10
ft j
10
5
0 0
10
10
100
j
100
Figure 1 – Continuous Spectrum of a Non-Periodic Waveform
10
10
lower
upper
5
f
0
0
30
20
 29.883
10
0
int
10
20
30
29.883
Figure 2 – Probability Distribution Function of a Non-Stationary Process
This similarity is no coincidence. Indeed it can be demonstrated (1) that the normalized
integrated spectrum has the same properties of a probability distribution function.
This concept gives us the confidence necessary to work with spectral analysis of nonperiodic time-varying functions.
However, on the other hand, depending on the shape of continuous spectra (probability
distribution function) one may conclude that probabilistic methods applied to harmonics may
or may not be mathematically possible.
This evolutionary concept, therefore, can be used to
screen and validate harmonic analysis of time-varying waveforms.
3.3 - Concept 2 – The Generalized Concept of Frequency
In order to illustrate the generalized concept of frequency as presented by (1) suppose that X(t)
is a deterministic function which has the form of a damped sine wave below and illustrated in
Figure 1.
t
X( t)
A e

2
2


 cos 0 t  
40
vi
 40
0
i
128
2
Figure 3 – Damped Sine Wave Function
If one caries out a Fourier analysis of X(t) one sees that it contains all frequencies as illustrated
in Figure 2.
6
5
ft j
ft j
0 0
100
50
0
50
 128
j
100
150
128
Figure 4 – Fourier Analysis of a Damped Sinusoidal (including negative frequencies)
In fact, the Fourier transform of X(t), as seen from Figure 2, consists of two Gaussian functions, one
centered on w0 and other on (-w0), the width of these functions being inversely proportional to the
tion which has the
a damped
sine
wave X(t) as a sum of sine and cosine functions with constant amplitudes, we
Inform
otherofwords,
if we
represent
need to include components at all frequencies. However, we can equally well describe X(t) by saying
it consists
of two "frequency"
components,
each having a time varying amplitude of
er analysis of X(t) that
we see
that it contains
all frequencies
-
consists of two Gaussian functions, one centered on 
se functions being inversely proportional to the parameter .
as a sum of sine and cosine functions with constant amplitudes,
all frequencies. However, we can equally well describe X(t) by
ency" components, each having a time varying amplitude of
ocal behavior of X(t) in the neighberhood of the time t0, this is
e. if the interval of the observation was small compared with  , X(t)
on with a frequency . and amplitude .
ngful the function X(t) must possess what we can loosely describe
characterize this property by saying that the Fourier transform of
t
Ae

2
2
Indeed, if we were to examine the local behavior of X(t) in the neighborhood of the time t0, this is
precisely what we would observe, i.e. if the interval of the observation was small compared with a , X(t)
would appear simply a cosine function with a frequency w0 and amplitude as indicated before .
Caution: For the term frequency to be meaningful the function X(t) must possess what we can loosely
describe as an oscillatory form, and we can characterize this property by saying that the Fourier
transform of such a function will be concentrated around a particular point w0.
In conclusion, if we have a non-periodic function X(t) whose Fourier transform has an absolute
maximum at a point w0, we may define w0 as "the frequency" of this function, the argument being that
locally X(t) behaves like a sine wave with conventional frequency w0, modulated by a "smoothly
varying" amplitude frequency.
3.4 - Concept 3 – The Evolutionary Spectra
The evolutionary spectra have essentially the same physical interpretation as the
spectra of stationary processes.
The main distinction being that whereas the spectrum of a stationary process describes
the power-frequency distribution for a whole process (over all time), the evolutionary spectrum
is time dependent and describes the local power-frequency distribution at each instant of time.
The theory of evolutionary spectra is the only one which can preserve the physical
interpretation for non-stationary processes.
The evolutionary spectrum is a continuously
changing spectrum or in other words, a time-dependent spectrum.
It is not practical to estimate the spectrum at every instant of time. But if we assume
that the spectrum is changing smoothly over time then, by using estimates which involve only
local functions of the data, we may attempt to estimate some form of “average” spectrum of
the process in the neighborhood of any particular time instant.
Thus in order to guarantee the confidence of probabilistic methods applied to
harmonics the concept of evolutionary spectrum can be applied when the FFT shows separate
dominant frequencies. In this case the frequencies behave as sine waves at a particular point
in time.
When these conditions are not satisfied probabilistic methods applied to harmonic
summation, etc., will not result in accurate estimations.
3.5- Considerations and Applications
The concept of evolutionary spectra can then be applied to investigate and characterize
the behavior of time-varying harmonic loads such as arc furnaces and electronic drives as well
as aggregate harmonic loads. This approach, which needs to further investigated and applied
to a number of loads and systems conditions, may help to more precisely determine the
behavior of time-varying harmonic distortion.
As a simple example let us consider the following waveform in Figure 5 which may
characterize the voltage at the PCC of and arc furnace.
1.191
vi
 0.896
0
i
N
1
Figure 5 – Waveform of Voltage With Time-Varying Harmonics
The FFT of the previous waveform is illustrated in Figure 6 and clearly shows the
dominance of two frequencies: the 60 Hz and the 180 Hz.
2
ft j
2
1
0 0
50
10
100
150
200
j
250
300
256
Figure 6 – FFT of Voltage With Time-Varying Harmonics
From Figure 6 and using the concept of evolutionary spectrum one can confidently say
that the behavior around these two frequencies at a certain instant in time is sinusoidal can
could linear summation methods could be applied. At other frequencies the level of confidence
and meaning of the results are not be trusted.
3.6- Conclusions
The concept of evolutionary spectrum applied to time-varying harmonic distortion
seems a useful approach, and may help the power quality engineer to better understand the
nature of such variations and properly utilize analytical tools to predict their behavior. Further
investigations and applications using practical waveforms together with field measurements are
necessary to validate this approach.
3.7- References
(1) Spectral Analysis and Time Series, M.B. Priestly, Academic Press, 1981.
(2) Paulo F. Ribeiro, Evolutionary Spectral For Dealing with Time-Varying Harmonic
Distortion, Presentation for the Task Force on Probabilistic Aspects of Harmonics, IEEE PES
Summer Meeting, Chicago, July 23, 2002.