Chapter 23
Time-Varying Power Harmonic Decomposition using Sliding-Window DFT
P.M. Silveira, C. Duque, T. Baldwin, P. F. Ribeiro
23. 1 Introduction
While estimation techniques are concerned with the process used to extract useful information from a signal, such as amplitude, phase and frequency; signal decomposition is concerned with the way that the original signal can be split in individual components, such as harmonics, interharmonics, sub-harmonics, etc.
This chapter presents a time-varying harmonic decomposition using sliding-window Discrete Fourier Transform. Despite the fact that the frequency response of the method is similar to the Short Time Fourier Transform, with the same inherent limitation for asynchronous sampling rate and interharmonic presence, the proposed implementation is very efficient and helpful to track time-varying power harmonic. The new tool allows a clear visualization of time-varying harmonics, which can lead to better ways of tracking harmonic distortion and understanding time-dependent power quality parameters. It has also the potential to assist with control and protection applications.
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 individual harmonic components from the input signal can be achieved, the estimation task becomes easier; (b) the decomposition is carried out in the timedomain, such that the time-varying behavior of each harmonic component is observable.
Several techniques can be used to separate frequency components. For example, Short Time Fourier Transform (STFT) and Wavelets Transforms [1], are two well-known decomposition techniques. Both can be seen as a particular case of filter bank theory [2].
When the fundamental frequency is time varying and the sampling frequency is not synchronous other techniques must be used in order to obtain a good harmonic decomposition. For example, the adaptive notch filter and the Phase-Locked Loop (PLL) [3]-[4] have been used for extracting time-varying harmonics components. In [5] the authors presented a similar approach to [3] and [4], where an adaptive filter-bank, labeled as resonator-in-a-loop filter bank, was used to track and estimate voltage or current harmonic signal in the power system. In [6] the authors presented a technique based on multistage implementation of narrow low-pass digital filters to extract stationary harmonic components. The technique utilizes multirate approach for the filter implementation and needs to know the system frequency for the modulation stage.
Attempts to visualize time-varying harmonics using Wavelet Transform have been proposed in [7] and [8]. However, the structure was not able to decouple the frequencies completely. In [9] the authors presented a new methodology to separate the harmonic components using multirate and filter bank approach. The method is able to track time-varying power harmonic frequencies without frequency spillover.
However if the fundamental frequency is supposed to be constant and synchronous sampling frequency is assumed, and only harmonic component is present in the signal then the STFT is better fitted for harmonic separation.
The STFT uses filters with coefficients that are complex numbers, which generates a complex output signal, whose magnitude corresponds to the amplitude of the harmonic component into the band. If a rectangular window is used in the STFT a low computation burden recursive algorithm can be employed. This algorithm is well known as Recursive DFT or as Sliding Window DFT [10], [11].
This chapter presents a new structure for time varying harmonic decomposition using the Sliding window DFT and an efficient digital sinusoidal generator to reconstruct each harmonic. The advantage of this method compared to a previous one [9] is the low computational effort, no phase delay and a single cycle of transient time.
23. 2 Discrete STFT
Given a signal , the discrete STFT for harmonic h at time n is defined as [2],
(23.1)
where, is a suitably chosen window function (e.g., a rectangular window) of size L and
(23.2)
is the digital harmonic frequency in radian, and N is the total number of harmonics. The digital harmonic frequency is related with the real frequency (rad/sec) by the following expression,
(23.3)
where fs is the sampling frequency.
Equation (23.1) can be rewrite according to the following formula, for which a graphical representation can be seen in Fig.23.1,
(23.4)
Figure 23.1- Filtering interpretation of the STFT
According to the Fig. 23.1, the STFT can be interpreted as a convolution of the input signal with the impulse response hh(n) of a hth complex bandpass filter followed by a modulation, which is accomplished by an exponential signal. If the window is a real function, the impulse response of the filter will be a complex number. The modulating signal, after the filter stage, shifts the resultant spectrum to the left side by an amount of wk. Figure 23.2 illustrates how the STFT works when a signal is injected at the filter input. Figure 23.2.a shows the spectrum of x(n) and the filter . It is important to remark that the filter is not symmetric, since its coefficients are not real. Figure 23.2.b shows the filtered signal, which is a complex exponential. Then the modulation by shift left the spectrum in Fig.23.2b, translating it to zero frequency. The amplitude and phase can be computed from the modulated signal taking the module and the angle of it.
If a rectangular window is chosen to perform the STFT, the computational effort can be drastically reduced by using the recursive formulas to calculate the DFT [10] and [11].
In the theory of Fourier series, it is well known the equation (23.5), for real periodic signals. But another commonly encountered form is the rectangular one, given by (23.6).
(23.5)
(23.6)
Figure23.2- The complex exponential generated in the STFT.
In being so, the rectangular (quadrature) terms YCh(n) and YSh(n) can be obtained by using the structure shown in Fig. 23.3. The factor is used to obtain the term YCh(n) and to the term YSh(n).
Figure23.3- Recursive filter to compute the quadrature term YCh(n)
Based on this structure, the recursive equation can be easily written as:
(23.7.a)
(23.7.b)
23.3 The decomposition structure
Normally, the DFT recursive algorithm is used to extract and compute the amplitude and phase of the fundamental and other harmonics components [10], but not the waveform. Nevertheless, the main point of this work is, in fact, to obtain the fundamental waveform, as well as each individual harmonic. This task can be performed using the own equation (23.6). For real time implementation this can be obtained by two methods: (a) Table searching or (b) using a digital sine-cosine generator. The second approach is more effective and has been used to decompose and analyze some signals from power systems events.
A digital sine-cosine generator is presented in [11]. It can be implemented using the following state equations:
(23.8)
where s1(n) is a sine function and s2(n) is a cosine function, and the initial states must be correctly set.
The advantage of this approach is that the sine and cosine signal can be used at the same time in both, decomposition and reconstruction tasks.
Figure 23.4 presents the block diagram of the core structure that extracts the hth harmonic. For extracting N harmonics it is necessary to employ N cores as shown in Fig.23.4. The advantage of this structure can be summarized as following:
i)low computational effort, suitable for real time decomposition implementation;
ii)no phase delay;
iii)transient time of only one cycle.
The disadvantage of this structure comes from the limitation of the DFT:
i)a synchronous sampling is needed;
ii)interharmonics produce estimation error of the quadrature components.
Figure23.4- The core structure for extracting the hth harmonic.
23. 4 Simulation results
In order to present some results of tracking time-varying harmonics using the structure shown in Fig. 23.4, two examples are considered: (a) a synthetic signal, which has been generated in Matlab using a mathematical model, and (b) a signal obtained from the “Electromagnetic Transient Program including DC systems” (EMTDC) with its graphical interface known as “Power Systems Computer Aided Design” (PSCAD). This program can simulate power systems with high fidelity and the resulting signals of interest are very close to physical reality.
23.4.1 Synthetic Signal
The synthetic signal utilized can be represented by:
(23.8)
Where h is the order (1st 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 portioned 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 23.5 illustrates the synthetic signal.
Figure. 23.5 – Synthetic signal used.
The structure composes of 16 cores like Fig. 23.4 has been used to decompose the signal into sixteen different harmonic orders, including the fundamental (60 Hz) and the DC component.
Figure 23.6 shows the decomposed signal from 4th up to 7th harmonic components. 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 other 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 axis.
Naturally, intrinsic transients will be present during the transitions from previous to the new state. Figure 23.7 shows both the original and the estimated components (DC to 3rd harmonic) in a short time scale interval. The transient of one cycle is shown in each output.
23.4. 2 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 23.1 shows the typical harmonic components present in the inrush currents [12]. 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 23.1- 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
Figure23.6 – First column: original components, second column: decomposed signals.
Figure23.7 – Comparing original and decomposed component.
In recent years, improvements in materials and transformer design have lead to inrush currents with lower distortion content [13]. The magnitude of the second harmonic, for example, has dropped to approximately 7% depending on the design [14]. 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.23.8.
Figure23.8 – Inrush current in phase A.
Using the methodology proposed to visualize the inrush current, the Fig.23.9 reveals the rarely seen time-varying behavior of the waveform of each harmonic component. This could be used to understand other physical aspects not observed previously. In Fig. 23.9, the left column shows the DC and even components and the right column the odd components.
Figure23.9 – Decomposition of the simulated transformer inrush current.
23.5 Conclusions
This chapter presents a method for time-varying harmonic decomposition based on sliding window DFT. The technique is able to extract each harmonic in the time domain. The methodology has the advantages of low computation burden, no phase delay and a short transient time and can be used as a useful tool for real time applications. The disadvantages are inherent to all DTF based algorithms, i.e., they need synchronous sampling and areinfluenced by the presence of interharmonics. However some strategies can be used to guarantee synchronization, such as the use of the Phase Looked Loop (PLL) to estimate the fundamental frequency. The influence of the interharmonics can be minimized through the choice of other windows, with smaller Main Lobe width and higher Side-Lobe Attenuation. The cost to be paid is the increasing of the computational effort.
23. 6 References
[1] Yuhua Gu, M. H. J. Bollen, “Time-Frequency and Time-Scale Domain Analysis,” IEEE Trans. on Power Delivery, Vol. 15, No. 4, Oct. 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, Aug. 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] H. Sun, G. H. Allen, and G. D. Cain, “A new filter-bank configuration for harmonic measurement,” IEEE Trans. on Instrumentation and Measurement, Vol. 45, No. 3, June 1996, pp. 739-744.
[6] C.-L. Lu, “Application of DFT filter bank to power frequency harmonic measurement,” IEE Proc. Gener. , Transm,. Distrib., Vol 152, No. 1, Jan. 2005, pp. 132-136.
[7] 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, Aug. 2007, Brazil.
[8] V.L. Pham and K. P. Wong, “Antidistortion method for wavelet transform filter banks and nonstationay power system waveform harmonic analysis,” IEE Proc. Gener., Transm., Distrib., Vol 148, No. 2, March 2001, pp. 117-122.
[9] C. A. Duque, P. M. Silveira, T. Baldwin, and P. F. Ribeiro, “Novel method for tracking time-varying power power harmonic distortion without frequency spillover,” IEEE 2008 PES, July 2008, Pittsburgh, PA, USA.
[10] Sanjit K. Mitra, Digital Signal Processing – A computer-based approach, Mc-Graw Hill 2006, 3ª Edition.
[11] R. Hartley, K. Welles, “Recursive Computation of the Fourier Transform", IEEE Int. Symposium on Circuits and Systems, Vol.3, 1990. pp. 1792 -1795.
[12] C.R. Mason, The Art and Science of Protective Relaying, John Wiley & Sons, Inc. New York, 1956.
[13] B. Gradstone, “Magnetic Solutions, Solving Inrush at the Source”, Power Electronics Technology, April 2004, pp 14-26.
[14] F. Mekic, R. Girgis, Z. Gajic, E. teNyenhuis, “Power Transformer Characteristics and Their Effect on Protective Relays”, 33rd Western Protective Relay Conference, Oct. 2006.