C. A. Duque, P.M. Silveira, T. Baldwin, P. F. Ribeiro

Chapter 22

Novel Method for Tracking Time-Varying Power Harmonic Distortions without Frequency Spillover

C. A. Duque, P.M. Silveira, T. Baldwin, P. F. Ribeiro

22.1 Introduction

Although it is well known that Fourier analysis is in reality only accurately applicable to steady state waveforms, it is a widely used tool to study and monitor time-varying signals, which are commonplace in electrical power systems. The disadvantages of Fourier analysis, such as frequency spillover or problems due to sampling (data window) truncation can often be minimized by various windowing techniques, but they nevertheless exist. This chapter demonstrates that it is possible to track and visualize amplitude and time-varying power systems harmonics, without frequency spillover caused by time-frequency techniques. This new tool allows for a clear visualization of time-varying harmonics which can lead to better ways oftracking harmonic distortion and understanding time-dependent power quality parameters. It also has the potential to assist with control and protection applications.

While estimation technique is concerned with the process used to extract useful information from the signal, such as amplitude, phase and frequency; signal decomposition is concerned with the way that the original signal can be split in other components, such as harmonic, inter-harmonics, sub-harmonics, etc..

Harmonic decomposition of a power system signal (voltage or current) is an important issue in power quality analysis. There are at least two reasons to focus on harmonic decomposition instead of harmonic estimation: (a) if separation of the individual harmonic component from the input signal can be achieved, the estimation problem becomes easier; (b) the decomposition is carried out in the timedomain, such that the time-varying behaviour of each harmonic component can be observed.

Some existing techniques can be used to separate frequency components. For example, STFT (ShortTime Fourier Transform) and Wavelets Transforms [1], are two well known decomposition techniques. Both can be seen as a particular case of filter bank theory [2]. The STFT coefficients filters are complex numbers which generates a complex output signal whose magnitude corresponds to the amplitude of the harmonic component being estimated [2]. The main disadvantage of this method is to set up an efficient band-pass filter with lower frequency spillover. On the other hand, although wavelet transform utilizes filters with real coefficients, the common wavelet mothers do not have good magnitude response in order to prevent frequency spillover. Besides, the traditional binary tree structure is not able to divide the spectrum conveniently for harmonic decomposition.

Adaptive notch filter and PLL (Phase-Locked Loop) [3]-[4] have been used for extracting time-varying harmonics components. However, these methods work well only if a few harmonics components are present at the input signal. In the other case the energy of adjacent harmonics spills over each other and the decomposed signal becomes contaminated. In [5] the authors presented a technique based on multistage implementation of narrow low-pass digital filters to extract stationary harmonic components. Thus, no digital analytical technique has been proposed or used to track time-varying power systems harmonics without frequency spillover. Consequently, the method proposed in this chapter adequately utilizes filters banks to avoid the frequency mixture, particularly associated with time-varying signals.

Attempts to visualize time-varying harmonics using Wavelet Transform have been proposed in [6] and [7]. However, the structures were not able to decouple the frequencies completely.

The methodology proposed in this chapter is constructed to separate the odd and the even harmonic components, until the 15th. It uses selected digital filters and down-sampling to obtain the equivalent band-pass filters centered at each harmonic. After the signal is decomposed by the analysis bank, each harmonic is reconstructed using a non-conventional synthesis bank structure. This structure is composed of filters and upsampling that reconstructs each harmonic to its original sampling rate.

The immediate use for this method is the monitoring of time-varying individual power systems harmonics. Future use may include control and protection applications, as well as inter-harmonic measurements.

The chapter is divided into method description, odd/even harmonic extraction and simulation results.

22.2 Method description

Multirate systems employ a bank of filters with either a common input or summed output. The first structure is known as analysis filter bank [8] as it divides the input signal in different sub-bands in order to facilitate the analysis or the processing of the signal. The second structure is known as synthesis filter bank and is used if the signal needs to be reconstructed. Together with the filters the multirate systems must include the sampling rate alteration operator (up and down-sampling). Figure 22.1 shows two basic structures used in a multirate system, where the up and down-sampling factor, M and L, equal 2. Figure 22.1-a shows a decimator structure composed by a filter following by the down sampler and Fig. 22.1-b the interpolator structure composed by a up-sampler followed by a filter. The decimator structure is responsible to reduce the sampling rate while the interpolator structure to increase it. The filters H(z) and F(z) are typically band-pass filters.

Figure22.1- Basic structures used in multirate filter bank and its equivalent representation for L=M=2. (a) Decimator; (b) Interpolator

The direct way to build an analysis filter bank, in order to divide the input signal in its odd harmonic component, is represented in Fig. 22.2. In this structure the filter is a band-pass filter centered in the kth harmonic and must be projected to have 3dB bandwidth lower than 2f0, where f0 is the fundamental frequency. If only odd harmonics are supposed to be present in the input signal, the 3dB bandwidth can be relaxed to be lower than 5f0. Note that Fig. 22.2 is not a multirate system, because the structure does not include sampling rate alternation, which means that there is only one sampling rate in the whole system.

The practical problem concerning the structure shown in Fig. 22.2 is the difficulty to design each individual band-pass filter. This problem becomes more challenging when a high sampling rate must be used to handle the signal and the consequent abrupt transition band.

In this situation the best way to construct an equivalent filter bank is to utilize the multirate technique. Figure 22.3 shows how an equivalent structure to Fig.22.2 can be obtained using the multirate approach. The filters H0(z) and H1(z) are orthogonal filters [6], with the first one as a lowpass filter and the second one as a high-pass filter.

Figure22.2 – Analysis bank filter to decompose the input signal in its harmonics components.

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Figure 22.3- Multirate equivalent structure to the filter bank in Figure22.2

Figure 22.4 shows the amplitude response for the analysis bank. This figure was obtained using sampling rate equal to 226 samples by cycle and FIR filters of 69th order. These filters correspond to so-called orthogonal filter banks also known as power-symmetric filter banks [8].

Figure 22.4- Amplitude response

The main difference between the structure shown in Fig. 22.2, and the other one obtained using the multirate technique, is the decimator at the output of the bank, as shown in Fig. 22.5. In fact, Fig.22.5 is equivalent to Fig. 22.3. It was obtained using the multirate nobles identities [2] and moving the downsampler factor inside Fig.22.2 to the right side. The decimated signal at the output of each filter has a sampling rate 64 times lower then the input signal. To reconstruct each harmonic into its original sampling rate it is necessary to use the synthesis filter bank structure.

Figure22.5- Equivalent multirate analysis filter bank

22.3 The synthesis filter banks

The synthesis filter bank used here is a different implementation of the conventional one. As the interest is to reconstruct each harmonic instead of the original signal, the filter bank must be divided in order to obtain the corresponding harmonic. Figure 22.6 shows the filter bank utilized to reconstruct the harmonic components.

Figure22.6- Modified synthesis bank

It is important to remark that the frequency response of the synthesis filter bank is similar to Fig. 22.4.

The analysis of Fig. 22.4 reveals that the filter bank in this configuration is unable to filter even harmonics appropriately, whichmeans that these components will appear with the adjacent odd harmonics. To overcome this problem an additional filter stage is included. This filter is composed by a second order Infinite Impulse Response (IIR) notch filter.

22.4 Extracting even harmonics

To extract the even harmonics the same bank can be used together with a preprocessing of the input signal. Hilbert transform is then used to implement a technique known as Single Side Band (SSB) modulation [8]. The SSB modulation moves to the right all frequencies in the input signal by f0 Hz. In this way, even harmonics are changed to odd harmonics and vice-versa.

Figure 22.7 shows the whole system for extracting odd and even harmonics.

Figure 22.7 – Whole structure to harmonic extraction

22.5 Simulation results

This section presents two examples: the first is a synthetic signal, which has been generated in Matlab using a mathematical model, and the second is a signal obtained from the Electromagnetic Transient Program including DC systems (EMTDC) with its graphical interface Power Systems Computer Aided Design (PSCAD). This program can simulate any kind of power system with high fidelity and the resulting signals of interest are very close to physical reality.

Synthetic Signal

The synthetic signal utilized can be represented by:

(22.1)

Where h is the order (1 up to 15th) and A is the magnitude of the component, 0 is the fundamental frequency, and finally, f(t) and g(t) are exponential functions (crescent, de-crescent or alternated one) or simply a constant value . Besides, x(t) is partitioned in four different segments in such way that the generated signal is a distorted one with some harmonics in steady-state and others varying in time, including abrupt and modulated change of magnitude and phase, as well as a DC component. Figure 22.8 illustrates the synthetic signal.

The structure shown in Fig. 22.7 has been used to decompose the signal into sixteen different harmonic orders, including the fundamental (60 Hz) and the DC component.

Figure 22.9 shows the decomposed signal which its corresponding components from DC up to 11th harmonic. The left column represents the original components and the right column the corresponding components obtained through the filter bank.

For simplicity and space limitation the higher components are not shown. However, it is important to remark that all waveforms of the time-varying harmonics are extracted with efficiency along the time.

Naturally, intrinsic delays will be present during the transitions from previous to the new state. Figure 22.10 shows some components of both the original and decomposed signal in a shorttime scale interval.

Figure22.8 – Synthetic signal used

Simulated Signal

It is well known that during energization a transformer can draw a large current from the supply system, normally called inrush current, whose harmonic content is high.

Although today’s power transformers have lower harmonic content Table 22.1, shows the typical harmonic components present in the inrush currents [9]. These values are normally used as reference for protection reasons, but they do not take into account the time-varying nature of this phenomenon.

Table 22.1- typical harmonic content of the inrush current

Order / Content %
Dc / 55
2 / 63
3 / 26.8
4 / 5.1
5 / 4.1
6 / 3.7
7 / 2.4

In recent years, improvements in materials and transformer design have lead to inrush currents with lower distortion content [10]. The magnitude of the second harmonic, for example, has dropped to approximately 7% depending on the design [11]. But, independent of these new improvements, it is always important to emphasize the time-varying nature of the inrush currents.

In being so, a transformer energization case was simulated using EMTDC/PSCAD and the result is shown in Fig. 22.11.

Using the methodology proposed to visualize the harmonic content of the inrush current, Fig. 22.12 reveals the rarely seen time-varying behaviour of the waveform of each harmonic component, where the left column shows the DC and even components and the right column the odd ones. This could be used to understand other physical aspects not observed previously.

Figure22.9 – First column: original components, second column: decomposed signals.

Figure22.10 – Comparing original and decomposed component

Figure22.11 – Inrush current in phase A.

Table 22.1- typical harmonic content of the inrush current

Order / Content %
Dc / 55
2 / 63
3 / 26.8
4 / 5.1
5 / 4.1
6 / 3.7
7 / 2.4

22.6 Final consideration and future work

The methodology proposed has some intrinsic limitations associated with the analysis of inter-harmonics as well as with realtime applications. The Inter-harmonics (if they exist) are not filtered by the bank, so it would corrupt the nearby harmonic components. The limitation regarding with realtime applications is related to the transient time response produced by the filter bank. For example, in the presence of abrupt change in the input signal, such as in inrush currents, the transient response can last more than 5 cycles, which is not appropriate for applications whose time delay must be as short as possible. The computational effort for realtime implementation is another challenge that the authors are investigating. In fact the high order filter (69th) used in the bank structure demand high computational effort. By using multirate techniques to implement the structure it is possible to show that the number of multiplication by second to implement one branch of the analysis filter bank is about one million. Some low price DSPr (Digital Signal Processors) available in the market are able to execute 300 Million Float Point operations per second. This shows that, despite the higher computational effort of the structure, it is feasible to be implemented in hardware. However, new opportunities exist for overcoming the limitations and the development of improved and alternative algorithms. For example, the authors are investigating the possibility of extracting inter-harmonic components and developing a similar methodology based on Discrete Fourier Transform (DFT). The first results showsimilar visualization capabilities with the promise to provide a reduced transient time response and lower computational burden. This process has the potential of addressing specific protective relaying needs such as detecting a high impedance fault during transformer energization and detection of ferro-resonance.

Figure 22.12 – Decomposition of the simulated transformer inrush current.

22.7 Conclusions

This chapter presents a new method for time-varying harmonic decomposition based on multirate filter banks theory. The technique is able to extract each harmonic in the time domain. The composed structure was developed to work with 226 samples per cycle and to track up to the 15th harmonic. The methodology can be adapted through convenient pre-processing for different sampling rates and higher harmonic orders.

Acknowledgments

The authors would like to thanks Dr. Roger Bergeron and Dr. Mathieu van den Berh for their useful and constructive suggestions.

22.8 References

[1] Yuhua Gu, M. H. J. Bollen, “Time-Frequency and Time-Scale Domain Analysis,” IEEE Transaction on Power Delivery, Vol. 15, No. 4, October 2000, pp. 1279-1284.

[2] P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993.

[3] M. Karimi-Ghartemani, M. Mojiri and A. R. Bakhsahai, “A Technique for Extracting Time-Varying Harmonic based on an Adaptive Notch Filter,” Proc. of IEEE Conference on Control Applications, Toronto, Canada, August 2005.

[4] J. R. Carvalho, P. H. Gomes, C. A. Duque, M. V. Ribeiro, A. S. Cerqueira, and J. Szczupak, “PLL based harmonic estimation,” IEEE PES conference, Tampa, Florida-USA, 2007

[5] C.-L. Lu, “Application of DFT filter bank to power frequency harmonic measurement,” IEE Proc. of Generation Transmission and Distribution, Vol 152, No.1, January 2005, pp. 132-136.

[6] P. M. Silveira, M. Steurer, .P F. Ribeiro, “Using Wavelet decomposition for Visualization and Understanding of Time-Varying Waveform Distortion in Power System,” VII CBQEE, August 2007, Brazil.

[7] V.L. Pham and K. P. Wong, “Antidistortion method for wavelet transform filter banks and nonstationay power system waveform harmonic analysis,” IEE Proc. of Generation, Transmission and Distribution, Vol 148, No.2, March 2001, pp. 117-122.

[8] Sanjit K. Mitra, Digital Signal Processing – A computer-based approach, Mc-Graw Hill 2006, 3ª Edition.

[9] C.R. Mason, “The Art and Science of Protective Relaying,” John Wiley&Sons, Inc. New York, 1956.

[10] B. Gradstone, “Magnetic Solutions, Solving Inrush at the Source”, Power Electronics Technology, April 2004, pp 14-26.

[11] F. Mekic, R. Girgis, Z. Gajic, E. teNyenhuis, “Power Transformer Characteristics and Their Effect on Protective Relays”, 33rd Western Protective Relay Conference, Oct 2006.