Relationship Between Probability Distribution and Spectral Analysis of Non-Stationary Random

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

2.1Introduction

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 being analyzed. This section suggests the application of spectral analysis and probability distribution concepts, in an integrated way, for a better understanding of the nature of time-varying 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 non-sinusoidal 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 time-varying 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.

2.2 - 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 2.1 and 2.2.

Figure 2.1 – Continuous Spectrum of a Non-Periodic Waveform

Figure 2.2 – 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 non-periodic 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.

2.3 - Concept 2 – The Generalized Concept of Frequency

In order to illustrate the generalized concept of frequency as presented by (2.1) suppose that X(t) is a deterministic function which has the form of a damped sine wave below and illustrated in Figure 2.1.
(2.1)

Figure 2.3 – 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 2.2.

Figure 2.4 – Fourier Analysis of a Damped Sinusoidal (including negative frequencies)

In fact, the Fourier transform of X(t), as seen from Figure 2.2, consists of two Gaussian functions, one centered on w0 and other on (-w0), the width of these functions being inversely proportional to the parameter .

In other words, if we represent X(t) as a sum of sine and cosine functions with constant amplitudes, we need to include components at all frequencies. However, we can equally well describe X(t) by saying that it consists of two "frequency" components, each having a time varying amplitude according to equation (2.2) below

(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.For example 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.

2.4 - 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 summationwill not result in accurate estimations.

2.5- 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 be 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 2.5 which may characterize the voltage at the PCC of and arc furnace.

Figure 2.5 – Waveform of Voltage With Time-Varying Harmonics

The FFT of the previous waveform is illustrated in Figure 2.6 and clearly shows the dominance of two frequencies: the 60 Hz and the 180 Hz.

Figure 2.6 – FFT of Voltage With Time-Varying Harmonics

From Figure 2.6 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 and linear summation methods could be applied. At other frequencies the level of confidence and meaning of the results are not to be trusted.

2.6- Conclusions

The concept of evolutionary spectrum applied to time-varying harmonic distortion seems like 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.

2.7- 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.