application of advanced digital signal processing tools

International Journal of Electrical Engineering Education 46/3

Application of advanced digital signal processing tools for analysis of voltage events in power systemsEnrique Pérez and Julio BarrosDepartment of Electronics and Computing, University of Cantabria, Santander, SpainE-mail: [email protected]

Abstract This paper presents a digital signal processing laboratory developed for analysis of voltage events in power systems. The designed laboratory permits the analysis of simulated voltage events of different magnitude, duration and point-on-the-wave of beginning as well as the analysis of real voltage events from records taken by power quality monitoring equipment, using different signal processing tools such as r.m.s magnitude, Fourier analysis, Kalman fi ltering and wavelet analysis. This laboratory has been used as a benchmark in the development of new signal processing methods for the complete characterization of voltage events in power systems as well as a powerful teaching tool in electrical power quality courses.

Keywords digital signal processing; power quality; power systems; voltage events

Voltage dips, short interruptions and overvoltages are abnormal and sudden changes in the magnitude of supply voltage and at present they represent one of the most important power quality disturbances because of the effects they can produce on equipment connected to the power system. These voltage events are both un-predictible and unavoidable and their frequency of occurrence depends greatly on the type of power system and on the point of observation.

There is no uniform defi nition of these power quality disturbances. European Standard EN 50 160 uses the following defi nitions:1 A voltage dip is a sudden reduction of voltage supply to a value between 90% and 1% of the nominal voltage followed by a voltage recovery, with duration between 10 ms and 1 minute. A short interruption in voltage supply is defi ned as a condition in which the voltage supply is lower than 1% of the nominal voltage with a duration less than 3 min; otherwise the interruption is classifi ed as a long interruption, and fi nally, a temporary overvoltage is an overvoltage (>110% of the nominal voltage) of relatively long duration.

On the other hand, IEEE Standard 11592 defi nes a voltage sag (dip) as a short-duration voltage decrease at the power-system frequency to a value between 90% and 10% of nominal voltage. An interruption is defi ned when the supply voltage decreases to less than 10% of the nominal voltage for a period not exceeding 1 min, and fi nally a swell is defi ned as an increase in the r.m.s. voltage for durations from 0.5 cycles to 1 min, with typical magnitudes between 110% and 180%.

Voltage events are characterised by a pair of data, the magnitude and the duration. The magnitude of the voltage event is the lowest magnitude of supply voltage in the case of voltage dips and interruptions (the highest magnitude in the case of a voltage swell) measured during the event, and the duration of a voltage event is the time difference between its beginning and its end.

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212 E. Pérez and J. Barros


There are different methods for the estimation of the magnitude of supply voltage and its time evolution during a voltage event. The computation of r.m.s. voltage is the most simple processing tool and is the method proposed in power quality measurement standards3 but, as is shown in Ref. 4, it has limited performance espe-cially in the case of short duration voltage events. Other signal processing tools have been proposed for power quality applications in Refs 5 and 6. Among the most commonly used alternative methods for analysis of voltage events are Fourier analysis, Kalman fi ltering and wavelet analysis. A comparative study of the per-formance of these methods can be seen in Refs 7 and 8. The application of new digital signal processing techniques to exactly characterise the magnitude and dura-tion of voltage events is an important area of research for real-time monitoring and control of power systems. The purpose of this paper is to present the development and assessment of a laboratory designed to test different digital signal processing tools in the detection and analysis of voltage events in power systems, using the defi nitions of European Standard EN50160.

Digital signal processing tools for detection and analysis of voltage events

The digital signal processing tools considered in this paper for detection and analy-sis of voltage events are the r.m.s. method, Fourier analysis, Kalman fi ltering and wavelet analysis. The next subsections present a short review of the main charac-teristics of these processing methods.

R.m.s. methodThe r.m.s. magnitude of a voltage supply is computed in a digital system using the following equation:

VN

vrms ii

N

==∑1 2

1

where vi refers to the voltage samples and N is the number of samples taken in a window. The window size can be selected from a half-cycle to any multiples of half-cycles of the power system frequency. In the case of a voltage event it is neces-sary that the new value of the supply voltage after the event is entirely within the sampling window to obtain the correct r.m.s. value. Thus, depending on the window length and on the time interval for updating the values, the magnitude and the dura-tion of a voltage event can be very different as is reported in Refs 4, 9 and 10.

The r.m.s. method is simple and easy to implement, but shows a limited perfor-mance in the detection and in the estimation of the magnitude and duration of voltage events, mainly for short duration voltage events. Another limitation of this method is that it does not provide information about the phase angle or the point-on-wave where the event starts.

The Urms(1/2) magnitude is used in IEC standard 61000-4-30 for detection and evaluation of voltage dips, overvoltages and interruptions. The Urms(1/2) value is defi ned as the r.m.s. voltage measured over 1 cycle, commencing at a fundamental



DSP tools for analysis of voltage events 213


zero crossing, and refreshed each half-cycle. A voltage dip or an interruption begins when the Urms(1/2) magnitude is below the dip threshold and ends when the Urms(1/2) voltage is equal to or above the dip threshold plus the hysteresis voltage. On the other hand, a voltage swell begins when the Urms(1/2) voltage is above the swell threshold and ends when this magnitude is equal to or above the swell threshold minus the hysteresis voltage.

Fourier analysisThe application of the Discrete Fourier Transform (DFT) is the traditional method to obtain the fundamental and the harmonic components of a digital signal.

The DFT X(k) of a sampled signal is defi ned as:

X k x n j knNn

N( ) = ( ) −( )

=

−

∑ exp 20

1 π

where x(n) are the samples of a continuous time signal x(t) taken every Ts seconds.

The DFT produces a sequence of complex values X(k) whose magnitudes corre-spond to the discrete frequency components in x(n). The application of the DFT requires the input signal x(n) to be periodic, with a period of N samples. To avoid errors due to aliasing, the signal x(n) must be sampled with a sampling rate at least twice the highest frequency component within the signal.

The results of the DFT analysis are correct in the case of periodic and stationary signals, but they are incorrect in the case of non-stationary signals, as is the case for voltage events. A way to overcome the limitations of DFT for the analysis of non-stationary signals is the application of the Short Time Fourier Transform (STFT). The STFT partitions the signal into time segments, where it is considered stationary, applying the DFT within each segment.

The STFT of a discrete input is defi ned as

X k m x n w n m j knNn

, exp( ) = ( ) −( ) −( )∑ 2π

where w(n-m) in its simple form is a rectangular window function

w n mn m Notherwise

−( ) = ≤ −( ) ≤ −( ){1 0 10

for each window w(n-m), the STFT produces a sequence of complex values

X k m k N, , , , , . . . . ,( ) = −0 1 2 1

whose magnitudes correspond to the discrete frequency components of the input signal x(n). Once the window size is selected, the frequency resolution is fi xed for the whole spectrum. To provide different frequency resolution it is necessary to repeat the STFT for different window sizes.

Different authors have studied in Refs 11 and 12 the application of STFT for the detection and characterization of voltage events in power systems. The results





obtained show, as in the case of the r.m.s. method, its dependency on the time window width selected. An advantage of the STFT method over the r.m.s. method is that it gives information about the magnitude and phase-angle of the fundamental and harmonic components of voltage supply during the event.

Kalman fi lteringKalman fi lters have been used for the optimal estimation of electrical magnitudes in power systems from their sampled values. They provide the best estimation of these magnitudes with the smallest number of samples and in the shortest time. Voltage and current phasors, harmonic distortion, frequency deviations, voltage fl icker and other parameters can be computed using Kalman fi ltering.

The change in magnitude of the fundamental component of the voltage supply is used to detect and to analyse voltage events. An important advantage of Kalman fi ltering over the r.m.s. method is that the phase-angle of the fundamental component of voltage supply and the point-on-wave where the event begins and ends can also be determined. The effi ciency in the detection and analysis of voltage events using Kalman fi ltering depends on the model of the system used. References 13 and 14 compare the performance of linear Kalman fi lters of different order in the detection and analysis of voltage events.

Linear Kalman fi ltersKalman fi ltering uses a mathematical model of the signal in state variable form. Different models can be used in order to provide a more accurate representation of the system to obtain a better estimation of the magnitude and phase-angle of voltage supply. The fundamental component and different harmonic components in the voltage supply can be used as the state vector xk. Each frequency component requires two state variables, the components in phase and in quadrature with respect to its respective rotating reference, 2n being the total number of state variables to represent n harmonic components. The magnitude and phase angle are obtained from the two state variables for the fundamental and each harmonic component in the voltage supply included in the model.

One of the most critical issues related to the use of Kalman fi lters is the adequate selection of the noise covariance matrices (Q and R). These error covariance matri-ces act as tuning parameters to balance the dynamic response of the fi lter against the sensitivity to noise. Their theoretical values can be computed mathematically, but in many cases and especially for non-linear systems, these theoretical values do not produce the most accurate results. It has been demonstrated in Ref. 15 that the fi lter response depends more on the Q/R ratio than on their values.

Extended Kalman fi ltersAn important aspect to be considered when using linear Kalman fi lters for the detec-tion and analysis of voltage events is how the fi lter responds to abrupt changes in voltage supply, as is the case in voltage events, which can make the fi lter unable to





track these changes. One solution proposed to compensate for this problem is the use of an Extended Kalman fi lter (EKF) to better estimate the non-linear process associated with a voltage event.

EKF is a modifi ed version of a linear Kalman fi lter that is applied in systems with non-linear process and measurement equations. In each step of the recursive algo-rithm, the non-linear equations are linearised at the latest estimate, using a fi rst-order Taylor series, to form a linear process, and then the linear Kalman fi lter model is applied.

Reference 8 shows the performance of different EKF models in the detection and analysis of voltage events. A 13-state EKF with the fundamental and odd har-monic components up to the 11th order of voltage supply and the power system frequency have been selected as the state vector. The magnitude and phase angle of fundamental and odd harmonic components are extracted from the instantaneous rectangular components of each harmonic frequency considered in the model of the system.

Wavelet analysisWavelet analysis is a powerful signal processing tool especially useful for the analysis of non-stationary signals. Wavelets are short duration oscillating waveforms with zero mean and fast decay to zero amplitude that are dilated and shifted to vary their time-frequency resolution. Reference 16 is a good introduction to the use of wavelet analysis in power systems.The continuous wavelet transform (CWT) of a signal x(t) with respect to a mother wavelet g(t) is defi ned as:

CWT a ba

x t gt b

adt,( ) = ( ) −( )−∞

∞

∫1

where a is the scale factor and b is the translation factor and both are continuous variables.

The coeffi cients of the CWT(a,b) at a specifi c scale and translation represent how well the original signal x(t) and the specifi c mother wavelet match. The set of all wavelet coeffi cients of a particular signal is the wavelet representation of this signal with respect to the mother wavelet selected.

Considering the mother wavelet as a window function, the scale factor corre-sponds to the window size. Narrow-band frequency components of the signal can be analysed by selecting small scale factors, whereas wide-band frequency compo-nents can be analysed using large scale factors. Thus, and in contrast to the STFT, the CWT permits the analysis of a signal with different frequency resolution.

The discrete wavelet transform (DWT) is the digital representation of the con-tinuous wavelet transform, and is defi ned as:

DWT m ka

x n gk nb a

am

om

mn

,( ) = ( ) −⎛⎝⎜

⎞⎠⎟∑1

0

0

0





where the scaling and translation parameters a and b are functions of an integer parameter m (a = a0

m and b = nb0a0m, giving rise to a family of daughter wavelets),

and k is an integer that refers to a specifi c sample in the input signal. The scaling parameter gives rise to a geometric scaling 1, 1/a0, 1/a0

2, . . . , that produces a logarithmic, in contrast to the uniform, frequency coverage of the STFT.

Selecting a0 = 2 (that is a0m = 1, ½, ¼, . . .) and b0 = 1, the DWT can be imple-

mented using a multistage fi lter bank with the wavelet function as the low pass fi lter and its dual as the high pass fi lter as is shown in Fig. 1. The outputs of these fi lters, after downsampling by two, represent the detail and the approximation version of the signal. The low frequency component is further split to obtain the other details of the input signal. By using this technique any wavelet can be implemented.

The key factor in the use of the DWT in power quality applications is the selection of the most adequate wavelet function for the type of disturbance to be analysed.17–23 Daubechie wavelets with 4 and 6 coeffi cients are reported in Refs 7, 22 and 23 to be the most suitable wavelet functions for detection and analysis of voltage events. According to Ref. 22 the output coeffi cients of the high pass fi lter of the fi rst level of the wavelet decomposition tree are insensitive to steady-state variation of the voltage supply but they present a high variation in magnitude associated with the transitions of the beginning and the end of a voltage dip.

Once a voltage event is detected, both the detail and approximation coeffi cients of the DWT are used to compute the r.m.s. magnitude of voltage supply during the event using the equation proposed in Ref. 21:

VN

xN

cN

d V Vrms ii

N

j kk

j kkj j

j jj j

= = + = += ≥ ≥∑ ∑ ∑∑ ∑1 1 12

1

2 2 2 20

00

0

, ,

where xi is the input signal (voltage samples during the event), cj,k and dj,k are respec-tively, the approximation and detail coeffi cients, j is the decomposition level and k the sampling point. Vj0 is the r.m.s. value of the lowest-frequency sub-band and Vj is the set of r.m.s. values of each frequency sub-band higher than or equal to the scaling level j0. Using this method a single r.m.s. magnitude of voltage supply is obtained during the event.

Fig. 1 Three-level wavelet decomposition tree.





Hybrid method wavelet analysis – Kalman fi lteringThe use of the detail coeffi cients of the fi rst level of the wavelet decomposition tree for the analysis of real voltage events can lead to a number of false detections due to their high sensitivity to high-frequency noise and other power quality disturbances in voltage supply. In Refs 7 and 14 several examples are reported where the wavelet analysis incorrectly identifi es transitions that do not correspond to real voltage events.

For this reason a new method has been proposed using DWT and EKF simultane-ously to improve the detection properties of wavelet analysis. In this new method DWT is used for detection and estimation of the time-related parameters of a voltage event and EKF is used for confi rmation of the event and for estimation of the mag-nitude and phase angle of voltage supply during the event. A voltage event is con-fi rmed if the fundamental component of voltage supply computed using the EKF is outside of the accepted voltage range variation (± 10% of nominal voltage). If the EKF does not confi rm the voltage event, then it is discarded. The details and the performance of this hybrid method can be seen in Ref. 8.

Only the detail coeffi cients of the fi rst level of the wavelet decomposition tree, d1(n) in Fig. 1, are computed. Each new value of d1(n) is compared with the voltage detection threshold selected in order to detect a possible event in the input signal. Simultaneously with DWT an EKF is applied to the voltage samples to confi rm the beginning and the end of the event and to compute the magnitude and phase-angle of voltage supply during the event. The 13-state EKF previously reported has been selected for this purpose.

Digital signal processing laboratory

The laboratory has been implemented in Matlab and, as was previously stated, it can be used in two different modes: simulation of voltage events and analysis of real voltage events. In both cases the user can select among different signal process-ing tools for detection and for computation of the magnitude and duration of a voltage event.

Detection and analysis of simulated voltage eventsIn the case of simulation of voltage events, the software developed enables the selection of the sampling frequency (samples/cycle), magnitude (in percentage of nominal voltage), duration (in samples) and point-on-the-wave of the beginning of the event (in samples from zero-crossing) as can be seen in Fig. 2. Once the para-meters of the voltage event are selected, the system generates an event record made up of two cycles of voltage waveform pre-event of 1 p.u. magnitude, the voltage event itself and three cycles of voltage supply post-event. Then the user must select one of the different signal processing tools for the analysis of the event.

R.m.s. methodThere are many combinations of window sizes and updating intervals that can be selected in the computation of the r.m.s. magnitude of voltage supply. In the





(a)

(b)

Fig. 2 User interface for analysis of simulated voltage events using (a) r.m.s. standard method, (b) STFT method, (c) 12-state Kalman fi ltering and (d) wavelet analysis.





(c)

(d)

Fig. 2 Continued





case of the r.m.s. method the user of the laboratory can select between the IEC standard method and the continuous r.m.s. method (‘RMS std’ and ‘RMS cont’ in Fig. 2). In the former the Urms(1/2) magnitude is computed, whereas in the con-tinuous r.m.s. method the same time window as in the standard method is used but refreshing the values with each new sample taken. The continuous r.m.s. method presents the highest computational burden but also the highest accuracy of the r.m.s. methods.

Fourier analysisIn the case of Fourier analysis only one selection is possible using this laboratory, the STFT is applied using 1 cycle sampling window and updating the results with each new sample taken.

Kalman fi lteringIn the case of selecting Kalman fi ltering as the signal processing tool, the user can select between different Kalman fi lter models:

• A two-state Kalman fi lter, modelling only the fundamental component of voltage supply (‘Kalman 2’ in Fig. 2).

• A low pass fi lter followed by a two-state Kalman fi lter (‘Kalman 2 + LPF’ in Fig. 2).

• A 12-state Kalman fi lter model including the fundamental component of voltage supply and the odd harmonic components up to 11th order (‘Kalman 12’ in Fig. 2).

The user can also select the ratio Q/R to study its effect in the convergence of the fi lter selected.

Wavelet analysisSelecting wavelet analysis as the signal processing tool, the user can choose between Daubechies with 4 or 6 coeffi cients (‘DWT db4’ and ‘DWT db6’ in Fig. 2), reported in the literature as the most suitable wavelet functions for detection and analysis of voltage events. In any case, the output coeffi cients of the high pass fi lter of the fi rst level of the wavelet decomposition tree, d1(n) in Fig. 1, are selected for the detection of the beginning and the end of the voltage event, whereas the detail and approxima-tion coeffi cients of the fi rst level of the wavelet decomposition tree, d1(n) and c1(n), are used for the computation of the r.m.s. magnitude of voltage supply during the event, applying the method described in the earlier section entitled ‘Wavelet analysis’.

Hybrid methodTwo possible hybrid methods can be selected using the laboratory. Both use the same wavelet analysis but different Kalman fi lter modelling (‘Wave-Ekalman1’ and ‘Wave-Ekalman2’ in Fig. 2).

As a result of the analysis of the simulated voltage event using one of the above signal processing tools, the software computes the magnitude, duration and





point-on-the-wave of the beginning of the voltage event and presents the waveform of the simulated voltage supply with the event and its ideal r.m.s. magnitude and, depending on the processing method selected, the r.m.s. voltage magnitude, the magnitude of the fundamental component of voltage supply during the event, or the time evolution of the coeffi cients of the wavelet analysis.

As an example, Fig. 2 shows the user interface for the case of a 35% magnitude and 35.15 ms duration (225 samples using 128 samples/cycle) and 33.75 degrees point-on-the-wave of beginning (12 samples from zero-crossing) simulated voltage dip computed using the r.m.s. standard method, STFT method, 12-state linear Kalman fi lter and wavelet analysis with Daubechies with 6 coeffi cients.

From the results reported in Fig. 2, the accuracy of each method can be assessed in the detection and analysis of this voltage event. The delay in the detection of the beginning of the voltage event and the error in the estimation of its duration using the r.m.s. standard method is noticeable. In both cases the standard method gives the worst performance of any of the methods under study. In contrast, the wavelet analysis shows the best performance of any of the methods in the detection and analysis of this simulated and ideal voltage event.

In the case of short duration voltage events, as is reported in Refs 4 and 7, the performance of the r.m.s. standard method and the STFT method worsen consider-ably, failing in the detection of a range of short duration and low magnitude voltage events, whereas Kalman fi ltering and wavelet analysis-based methods maintain their performance.

Analysis of real voltage events

In this working mode, the user can fi rst select from a list of power quality disturbance recordings and within each one then choose the voltage phase to be analysed. Again, as in the simulation mode, the standard r.m.s. analysis, Fourier analysis, Kalman fi ltering, wavelet analysis and hybrid analysis can be performed on the selected register. As a result the voltage waveform and the r.m.s. magnitude, the magnitude of fundamental component of voltage supply or the coeffi cients of the wavelet analysis are shown to the user.

Figure 3 shows the user interface for the analysis of a voltage event recorded in our low-voltage distribution network using the r.m.s. standard, STFT and DWT-EKF methods previously described. The event recorded is a two-step voltage dip of rela-tively constant magnitude and less than four cycles' duration.

As was the case in the simulation of voltage events, Fig. 3(a) shows the long delay in the detection of the voltage dip using the r.m.s. standard method (about 20 ms from the visual inspection of the event record). On the other hand, Fig. 3(c) shows how the detail coeffi cients of the fi rst level of the wavelet decomposition tree clearly point out the beginning and the end of the voltage dip, which is confi rmed using the Kalman fi lter. Other peak values of these coeffi cients over the detection threshold, due to a new transition in voltage supply, are rejected as a possible voltage event using the method proposed. The duration and the magnitude of this voltage event were 74.38 ms and 54.20% respectively.





(a)

(b)





Conclusions

The paper presents a laboratory for analysis of voltage events in power systems using advanced digital signal processing tools. The laboratory permits the simulation and the analysis of real voltage events, showing the performance of different methods in the detection and characterization of voltage events. This laboratory has been successfully used as a benchmark for the development of new and more effi cient signal processing methods for the complete characterization of voltage events in power systems.

Acknowledgements

The authors wish to thank the Spanish Ministry of Education and Science, National Plan for R+D+I (2004–2007), for its support of this research project under grant DPI2006-15083-C02, of which the present paper is a part.

References

1 European Standard EN50160: Voltage characteristics of electricity supplied by public distribution systems (Cenelec, 1999).

(c)

Fig. 3 User interface for analysis of real voltage event using (a) r.m.s. standard method, (b) STFT method and (c) DWT-EKF method.





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