*4.4. Computational Effort*

A drawback of the proposed WPD method could be the high number of operations that must be performed. At each level of decomposition, the number of filtering operation increased exponentially, although the decimation allowed reducing the number of samples that were convolved with the filter coefficients. In order to characterize the computational effort required, a comparison was performed by analyzing the same waveform, under the same conditions, with both methods. Although it did not represent an exhaustive study of the computational complexity, a comparison of the time required by the two methods on the same computer, for the same signal, and under the same operating conditions can be considered as an indication of the relationship between the computational effort of the two methods. The same analysis was repeated 1000 times for each of the 13 waveforms described in Section 4.1, and later, the 13,000 results were averaged. The results, obtained on a 3.00 GHz Intel Core i5-7400 CPU with 16 MB RAM computer, are reported in Table 3.

**Table 3.** Comparison of the average computational time for both methods on the same computer.


As can be seen, the time required by the proposed WPD algorithm resulted in being, on average, 148-times higher than FFT. However, it can be seen that, at least on the testing machine, the method had the capability of being performed online, since the time required for the calculation was less than the duration of the analyzed signal, i.e., 10 cycles of the power frequency (approximately 200 ms). A possible strategy to reduce the computation time is to implement the proposed method on an Field Programmable Gate Array (FPGA) since the structure, based on the iterative application of the same filter, is particularly suitable for hardware implementation and the time reduction could be significant, as demonstrated by recent works [23,30].

### **5. Analysis of Real Signals**

The newly proposed method, after being validated, was employed to analyze real waveforms. In order to obtain real waveforms, measurements were taken at the point of connection of an Active Front End (AFD) device. It as a three leg 50 kW converter, with a 20 kHz switching frequency. Current and voltage waveforms were acquired using a NI PXIe-6124 module (installed on an NI PXIe-1071 chassis), equipped with a Pico TA044 70 MHz 7000 V differential voltage probe and a Tektronix TCP2020 50 MHz 20 A AC/DC current probe. The assayed AFD was designed to obtain low power losses, high efficiency (>96%), and most importantly, to produce low harmonic distortion. At nominal power, the THD was less than 0.6%. Therefore, in order to obtain signals with higher harmonic content, the harmonic distortion was enhanced by operating the device at 5% of its nominal power. In this way, the fundamental current was greatly reduced, while the harmonic components were only slightly reduced, increasing the current THD up to 50%.

The top graphs of Figure 10a,b show the acquired voltage and current waveforms, respectively, with a total duration of 10 cycles of the fundamental (approximately 0.2 s), according to the IEC 61000-4-7 standard. It must be noted that in these cases, the error could not be calculated since the real energy content of each harmonic group was not known a priori. For this reason, the bottom plots of Figure 10a,b show the absolute difference between the RMS values obtained with the two methods, for each harmonic group, and not the error. Hence, it is not possible to know which method performed better, but only how different the results could be from each other. The voltage waveform in Figure 10a offers the possibility to analyze a mostly stationary signal. It can be observed that, in this case, the results obtained with the proposed WPD method were very similar to those obtained with the IEC method, and the differences between RMS values were close to zero for all harmonic orders. On the other hand, the current waveform in Figure 10b has a visible fluctuating character, offering the possibility to test the proposed WPD method with a real non-stationary signal. In this case, the differences between the two methods were higher, up to almost 1%. These results confirmed the conclusions of Section 4, i.e., that the WPD method was equally valid as the IEC Fourier strategy for analyzing stationary signals, but that differences arise under fluctuating conditions, where FFT is known to be inaccurate.

**Figure 10.** Analysis of real waveforms with a stationary (**a**) and non-stationary (**b**) character. The top graphs show the waveforms in the time domain, the middle graphs the harmonic content, and the bottom graphs the absolute difference between the proposed method and the IEC strategy.
