Techniques to Accelerate Power Quality Analysis Based on DWT

Ileana-Diana Nicolae Andreea Chiva

Department of Computers and Information Technology University of Craiova

Craiova, Romania nicolae_ileana@software.ucv.ro, chivaandreea@gmail.com

Petre-Marian Nicolae Marian-Ştefan Nicolae

Dept. of Electrical, Energetic and Aero-Spatial Engineering University of Craiova

Craiova, Romania pnicolae@elth.ucv.ro, snicolae@elth.ucv.ro

Abstract— The paper deals with techniques to accelerate the

DWT power quality analysis. Firstly a non-spline interpolation schematic is proposed for the evaluation of the first components near the right border of the signal. Comparisons are made between power quality indices, shape of details waveforms and runtimes, considering 3 cases. The first 2 cases correspond to data acquired on stand and correspond to a driving system using chopper and DC motor. The 3rd case deals with sets of currents and voltages acquired from the secondary winding of the excitation system used to supply a generator. 4 methods were used for calculation. The 1st one - FFT, is used to generate “reference data”. The others rely on 3 distinct original functions used to implement DWT using a Daubechy filter of length 8. The 1st function does not use interpolation, the 2nd one uses spline interpolation and the 3rd one, uses interpolation relying on mean slopes. Similar results were generated by all methods, but the newly introduced function, especially useful when the analyzed signal’s periodicity is questionable, exhibited halved runtimes. Finally an original multithreading technique is presented. Savings of 45% of runtimes were revealed by tests.

Keywords— Discrete wavelet transforms, Power quality, Signal processing algorithms, Parallel computing.

I. INTRODUCTION DWT analyzes the signal at different frequency bands with

different resolutions by decomposing the signal into an approximation containing coarse and detailed information. DWT employs two sets of functions which are associated with low pass and high pass filters. The decomposition of the signal into different frequency bands is obtained by successive high pass and low pass filtering of the time domain signal. The original signal s[n] is first passed through a half-band high pass filter g[n] and a half-band low pass filter h[n]. h[n] removes all frequencies that are above half of the highest frequency, while g[n] removes all frequencies that are below that frequency [1]. Afterward successive decompositions of the approximations are made, with no further decomposition of the details. Fig.1 depicts a 3 levels decomposition [2]. cAi denotes the approximation vector whilst cDi denotes the detail vector on the i-th level. The most significant frequencies from the original signal appear with high magnitudes in that specific region of the DWT signal including them, with the preservation of their time localization.

Fig. 1. A decomposition on 4 levels using DWT.

Power quality is a permanent concern nowadays, continuous efforts being done to find more efficient methods for its analysis [3]-[6].

For the analysis of quasi-stationary regimes, corresponding to highly distorted waveforms with small variations of magnitude and harmonic contents in three-phase systems, DWT is a valuable tool. The Fast Fourier Transform (FFT) can be used for stationary regimes in order to provide “reference data” for the values obtained for power quality indices when DWT-based theories are used [7], [8].

II. ANALYZED DATA AND ANALYSIS METHODS

A. Analyzed data The first 2 sets of analyzed data, acquired on stand using a

data acquisition system with a sampling frequency of 19200 Hz, correspond to a driving system using chopper and DC motor. The first set corresponds to unfiltered currents. A partial representation is depicted by Fig. 2. The second set (partially reproduced by Fig. 3) contains currents with a reduced harmonic content and the corresponding voltages. Data were acquired. The third set of data is defined by sets of currents and voltages acquired from the secondary winding of the excitation system used to supply the main generator from a power plant. The first 200 samples of the analyzed three-phase systems of voltages and currents are depicted by Figs. 4 and 5.

978-1-4799-4161-2/14/$31.00 ©2014 IEEE

Fig. 2. First set of analyzed data (partial representation)

Fig. 3. Second set of analyzed data (partial representation).

Fig. 4. 3rd set of data: voltages.

Fig. 5. 3rd set of data: currents.

B. Analysis methods Firstly an original program, based on Fast Fourier

Transform, was used to analyze all sets of data.

Our previous studies [9] were concerned with 10 levels DWT-based decompositions, using Daubechy filters of length 8, implemented as original functions called “dwm” and “dwi” respectively. dwm assumes that the analyzed signal is periodic, whilst dwi performs interpolations relying on spline techniques provided by the MATLAB library , being also applicable to non-periodic signals.

The filters’ coefficients are calculated as described in [10], [11] and details on their implementation are given in [12].

At continuously monitored signals, even if they have a non-periodic or quasi-periodic nature, no interpolations are required for the currently analyzed signal (denoted by CS) when the wavelet decomposition is performed for the first level, because usually the required values beyond the right edge of CS can be provided by the data acquisition system after few time steps (6 in the case of a filter of length 8). Unfortunately for all remaining decomposition levels (with ranks>1), interpolations must be performed. This is because at non-periodic or quasi-periodic signals no data can be made available by the data acquisition system relative to the future values of approximation vectors, which are calculated, not acquired signals. Considering the decimation phenomena, an interpolation-free decomposition for all 9 remaining levels of a signal with uncertain periodicity should require the knowing of 6 x 29 values beyond the right edge of CS, which in real-time analysis should introduce an unacceptable time delay.

The spline interpolations involve significant runtimes. The mean runtimes corresponding to full dwi-based decompositions of the signals from Figs 2 and 3 were 2.59 times higher than those corresponding to dwm-based decompositions. For the signals from Figs. 4 and 5 [12] the situation was even more dramatic, a ratio of 4.25 being revealed by full decompositions (fewer calculation points were used and the impact of interpolation became more visible).

This made us consider a different, non-spline, interpolation schematic in order to accelerate the decomposition process. This schematic [13] is described by Fig. 6. In this case, the estimation of sn+1 when it is separated by “step” seconds from sn requires the following calculations:

tg1=(sn-2-sn-3)/step;

tg2=(sn-sn-1)/step;

mean_tg=(tg1+tg2)/2;

sn+1= sn+mean_tg x step.

Fig. 6. Non-spline interpolation schematic.

III. ACCELERATING DWT DECOMPOSITIONS WITH SPLINE-FREE INTERPOLATION SCHEMATICS

A. Qualitative results Tables I...III present the values of node-zero, non-zero and

RMS for all phase voltages and currents calculated for all sets of data using fft, dwm, dwi [9] and dwns (the newly introduced function, performing non-spline interpolations). The differences between the values calculated with dwm and respectively those calculated with fft were calculated with:

100_/)__(_ ⋅−= fftvalfftvaldwmvaldwmdiff (1)

Similar formulas were used for the values generated by dwi and dwns respectively.

To see if the new interpolation technique generates values with acceptable deviations from those generated by the values yielded by dwm and dwi respectively, we calculated the synthetic data gathered by Table IV. Each cell contains values mediated over all phases from the same set.

The differences dwns versus dwm and respectively dwi versus dwm were found to be very close, revealing no systematic pattern: positive, negative or even zero mean values of differences were obtained with no correlation to the phase number or type of power quality index.

B. Analysis of waveforms shape The quantitative evaluations are in good agreement with the

conclusions drawn from the shape-related analysis of the

TABLE I. COMPARATIVE RESULTS FOR POWER QUALITY INDICES AT THREE-PHASE SYSTEMS - 1ST SET OF DATA

Method Phase 1 Phase 2 Phase 3

Value [A] % diff Value [A] %

diff Value [A] % diff

Ief

dwm 218.75 -0.12 218.63 -0.07 218.17 0.06 dwi 218.69 -0.15 218.72 -0.03 218.11 0.04

dwns 218.88 -0.06 218.93 0.07 218.08 0.02 fft 219.01 218.78 218.03

Ij0

dwm 3.44 -2.27 3.81 0.26 4.85 20.65 dwi 3.46 -1.7 3.76 -1.05 4.85 20.65

dwns 3.48 -1.14 4.17 9.74 5.01 24.63 fft 3.52 3.8 4.02

In

dwm 218.78 -0.11 218.67 -0.06 218.23 0.08 dwi 218.72 -0.14 218.76 -0.02 218.17 0.05

dwns 218.91 -0.05 218.97 0.07 218.14 0.04 fft 219.03 218.81 218.06

Uef

dwm 93.15 2.69 94.51 2.29 92.38 2.1 dwi 94.28 3.94 95.12 2.95 92.44 2.17

dwns 93.89 3.51 94.86 2.67 92.42 2.14 fft 90.71 92.39 90.48

Uj0

dwm 38.63 -6.62 40.45 -3.87 38.84 -3.39 dwi 37.23 -10.01 39.7 -5.66 38.17 -5.06

dwns 37.26 -9.93 39.64 -5.80 38.24 -4.88 fft 41.37 42.08 40.2

Un

dwm 100.84 1.13 102.8 1.26 100.22 1.22 dwi 101.37 1.66 103.07 1.53 100.01 1.01

dwns 101.01 1.30

102.81 1.27

100.02 1.02

fft 99.71 101.52 99.01

TABLE II. COMPARATIVE RESULTS FOR POWER QUALITY INDICES AT THREE-PHASE SYSTEMS - 2ND SET OF DATA

Method Phase 1 Phase 2 Phase 3

Value [A] % diff Value [A] %

diff Value [A] % diff

Ief

dwm 217.67 -0.05 217.44 0 216.9 0.01 dwi 217.63 -0.03 217.44 0 216.71 -0.07

dwns 217.9 0.07 217.63 0.09 216.72 -0.07 fft 217.74 217.43 216.87

Ij0

dwm 4.37 -2.46 4.04 4.94 4.43 7.34 dwi 4.35 -2.9 3.99 3.64 4.29 3.95

dwns 4.76 6.25 4.08 5.97 4.5 8.96 fft 4.48 3.85 4.13

In

dwm 217.71 -0.03 217.47 0 216.84 -0.03 dwi 217.68 -0.05 217.48 0.01 216.75 -0.07

dwns 217.95 0.08 217.67 0.10 216.76 -0.07 fft 217.78 217.46 216.91

Uj0

dwm 84.73 0.75 87.08 0.57 84.88 0.64 dwi 84.71 0.73 87.11 0.6 84.75 0.49

dwns 84.87 0.92 87.16 0.66 84.88 0.64 fft 84.1 86.59 84.34

Un

dwm 2.77 -33.73 2.84 -29.7 3.12 -38.34 dwi 2.74 -34.45 2.84 -29.7 3.06 -39.53

dwns 2.88 -31.10 2.87 -28.96 2.87 -43.28 fft 4.18 4.04 5.06

Uj0

dwm 84.77 0.78 87.13 0.52 84.92 0.51 dwi 84.75 0.76 87.16 0.55 84.8 0.37

dwns 84.92 0.96

87.21 0.61

84.93 0.52

fft 84.11 86.68 84.49

TABLE III. COMPARATIVE RESULTS FOR POWER QUALITY INDICES AT THREE-PHASE SYSTEMS - 3RD SET OF DATA

Method Phase 1 Phase 2 Phase 3

Value [A] % diff Value [A] %

diff Value [A] % diff

Ief

dwm 1015.45 -0.43 1069.16 -0.88 1012.01 -0.10dwi 1016.87 -0.57 1066.67 -0.64 1010.53 0.05

dwns 1016.92 -0.58 1066.84 -0.66 1011.80 -0.08fft 1011.06 0 1059.85 0 1011.03 0

Ij0

dwm 974.96 -0.71 1027.39 -1.25 971.36 -0.32dwi 977.63 -0.99 1025.88 -1.10 970.77 -0.26

dwns 977.68 -0.99 1025.95 -1.11 971.98 -0.38 fft 968.08 0 1014.7 0 968.26 0

In

dwm 283.89 2.61 295.94 3.20 283.95 2.40dwi 279.77 4.02 292.13 4.44 280.67 3.53

dwns 279.77 4.02 292.53 4.31 281.07 3.39 fft 291.49 0 305.71 0 290.94 0

Uef

dwm 501.85 -0.62 517.26 -0.1 524.48 -0.19dwi 501.77 -0.56 517.37 -1.02 524.78 -0.13

dwns 501.84 -0.61 517.39 -1.03 524.91 0.10 fft 498.8 - 512.14 - 525.46 -

Uj0

dwm 490.31 -0.33 506.65 -0.79 513.61 0.21dwi 490.28 -0.32 506.91 -0.84 514.09 0.12

dwns 490.35 -0.33 506.94 -0.84 514.22 0.09 fft 488.72 502.7 514.7

Un

dwm 106.99 -7.28 104.23 -6.43 106.23 -0.44dwi 106.76 -7.05 103.49 -5.68 105.42 0.32

dwns 106.78 -7.07 103.49 -5.68 105.41 0.33 fft 99.73 0 97.93 0 105.76 0

decomposition vectors generated with different functions, as depicted by Figs. 7...10. In these figures cyan is used for waveforms generated by dwi, red is used for waveforms generated by dwns and dark blue is used for waveforms generated by dwm.

TABLE IV. SYNTHETIC DATA GENERATED AS AVERAGE OVER ALL PHASES FOR PERCENT RELATIVE DIFFERENCES BETWEEN VALUES

GENERATED WITH DIFFERENT DWT BASED FUNCTIONS

Set index

Power Quality

index

1st set 2nd set 3rd set

dwns vs dwm [%]

dwi vs dwm [%]

dwns vs dwm [%]

dwi vs dwm [%]

dwns vs dwm [%]

dwi vs dwm [%]

Ief 0.05 0.06 0.04 0.07 -0.03 0.05 Ij0 4.64 4.93 3.83 5.53 0.07 0.05 In 0.05 0.06 0.06 0.07 -1.21 0.09

Uef 0.40 -0.24 0.09 0.13 0.04 0.01 Uj0 -2.36 0.04 -1.00 -0.01 0.06 0.02 Un -0.01 -0.20 0.09 0.14 -0.56 0.00

Fig. 7. 1st set of data. Almost identical details were generated by all

functions, so almost zero differences are generated.

Fig. 8. 1st set of data. dwm generates higher values compared to dwi and

dwns, which are almost identical.

Fig. 9. 2nd set of data. dwm generates lower values compared to dwi and

dwns, which are almost identical.

As one can see, no shape-related pattern could be derived relative neither to the decomposition level, nor to the phase or type of waveform (voltage versus current).

C. Analysis of runtimes 300 consecutive decompositions over all 10 levels were

performed for data from Figs. 2 and 3. The analysis of runtimes obtained in the programming environment MATLAB revealed mean runtimes of a single phase full decomposition equal to 3.85 msec when using dwns, of 7.67 msec when using dwi and respectively of 2.96 msec when using dwm.

Fig. 10. 2nd set of data. No systematic rule can be deduced relative to

differences between waveforms generated with different functions.

The small runtimes and the results of shape and accuracy analysis allow us to recommend the use of the new function – dwns, as a faster alternative, providing a similar degree of accuracy both for non-stationary and quasi-stationary regimes respectively. The runtime required by dwns is reduced to a half as compared to the case when spline interpolations are used and is only approximately 30% higher than then that required by dwm, which is interpolation free.

IV. MULTITHREADING IMPLEMENTATION MEANT TO ACCELERATE DWT DECOMPOSITION AT THREE-PHASE SYSTEMS

Java is a proper and widely used programming environment when multithreading techniques are to be implemented. Therefore we conceived a complex program to study the possibility to accelerate the wavelet decompositions for three-phase systems.

Our starting point was to perform in parallel the decomposition of all currents with that of all voltages, over each decomposition level [13]. The waveforms depicted by Fig. 2 were used for the simulations, but the shape of waveform has no influence over the runtime and therefore for any data the runtimes will be the same, considering the same number of calculation points and machine architecture. For the runtimes obtained through our program based on a DWT decomposition with dwm for a filter of length 8, the number of calculation points was equal to 12288, the processor has the configuration from Fig. 11 and the available quantity of installed RAM memory was of 4 GB.

Fig. 12 depicts the runtimes corresponding to 10 sets of 100 consecutive decompositions for 3 periods of 6 waveforms (3 currents and 3 voltages).

The mean runtime per set when using the multithreading technique for the first two levels of decomposition and a mono-threading technique for the last 8 levels was of 2267.3 msec, whilst that corresponding to the mono-threading, classical implementation for all decomposition levels, was of 4084.3 msec. The choice of using multi-threading only for the first 2

Fig. 11. Configuration of employed machine

Fig. 12. Runtimes corresponding to 10 sets of 100 consecutive decompositions.

levels was imposed by the effect of the decomposed segment’s length over the multithreading efficiency, considering the decimation phenomenon (the decomposed vector’s length is reduced by a factor of 2 with each level).

The runtime was clearly reduced when using multi-threading with 45%.

V. CONCLUSIONS DWT can be used to perform multi-periods data segments’

analysis, providing a better global picture of the power quality indices and reliable results even for the cases of slightly variable parameters.

When the analyzed waveforms cannot be considered as “perfect stationary”, interpolations must be performed to estimate the values beyond the right edge of the currently decomposed approximation vector.

The spline interpolations involve significant runtimes. The newly derived function (dwns) relying on an interpolation schematic based on averaged slopes corresponding to points from the analyzed data segment’s right edge was tested from 3 points of view: accuracy when applied to the evaluation of power quality indices, shapes of decomposition vectors and runtimes. Savings of 50 % were obtained for the runtimes required by the non-spline interpolation based function (dwns) as compared to those when spline interpolations were used, under similar accuracy-related performances and shapes of waveforms corresponding to the decomposition vectors.

Moreover, the runtime required by dwns is only approximately 30% higher than then that required by dwm, which is interpolation free. This makes dwns a better option for analysis when the nature of the analyzed regime cannot be defined with certitude as perfect stationary.

Our original multi-threading implementation of the DWT decomposition for the first 2 levels when the interpolations-free function (dwm) is used exhibits runtime savings of 45% as compared to the classic mono-threading implementation.

ACKNOWLEDGMENT This paper got support from the grant no. 448 / CEOSE

under the frame of “Research projects in partnership between universities/RD institutes and companies”.

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