Maximum of the absolute instantaneous error of all techniques after convergence of the algorithms.
1. Introduction
The increasing use of power electronic devices in power systems has been producing significant harmonic distortions, what can cause problems to computers and microprocessor based devices, thermal stresses to electric equipments, harmonic resonances, as well as aging and derating to electrical machines and power transformers [1–3]. The most important problems that have been reported in the literature concerns to the difficulty of the frequency control within the microgrids and the increase of the total harmonic distortion. These two factors may negatively impact on the protection system, power quality analysis and intelligent electronic devices (IEDs), in which digital algorithms assume that the fundamental frequency is constant. Based on this fact, there has been an increasing interest in signal processing techniques for detecting and estimating harmonic components of timevarying frequencies. Their correct estimation has become an important issue in measurement equipment and compensating devices. Although many methods have been proposed in the literature, it still remains difficult to detect and estimate harmonics of timevarying frequencies [4, 5]. The harmonic components (voltage or current) can change its frequencies due to continuous changes in the system configuration and load conditions, to the rapid proliferation of distributed resources, and to possibilities of new operational scenarios (e.g., islanded microgrids). Also the need for massive monitoring of networks is unquestionable within the concept of
The concept of timevarying harmonics came recently to the vocabulary of power systems engineers, because more and more nonlinear loads, with dynamic behavior, are being connected to power systems and the fundamental frequency is experiencing a large range of variation. These factors have putting in check the traditional stationary spectral analysis methods, and many techniques for improving harmonics measurement have been proposed in recent years. Parametric and nonparametric methods that commonly have been used by the community of signal processing have been applied to power system harmonic estimation. These methods have in common that they need to estimate the fundamental frequency to adjust some internal parameters, like filter coefficients. The challenge is producing a harmonic estimator with high convergence ratio, high accuracy, low computational burden and immunity to the presence of interharmonic: conditions that are not ease to simultaneously deal with. The most used technique for harmonics estimation is based on the discrete Fourier transform (DFT) [8, 10]. The DFT algorithm is attractive because of its low computational complexity and its simple structure. However, DFT does not perform well if power system frequency varies around the nominal value. Several other techniques have been proposed in the literature for harmonic estimation. However, the DFT still appear to be the preferred algorithm mainly due to its simplicity.
2. Power quality and smart grid
The term “
One of the promises of the
An adequate power quality should guarantees electromagnetic compatibility between all equipment connected to the grid. Then, an important issue for the successful and efficient operation of smart grids is the introduction of advanced, flexible, robust, and cooperative set of signal processing and computational intelligent techniques for power quality analysis. With this set of techniques, an effective and extensive smart monitoring system can be devised and deployed. Such a monitoring system have to allow for the monitoring of such as voltage, current, bidirectional energy consumption at distribution transformers, substations transformers, smart meters, distribution feeders, distribution switching devices, and strategically installed power quality monitors in the power systems.
3. Harmonic estimation techniques: Before and after smart grid
Consider the monitored power line signal, after processed by the analog antialiasing filter, be expressed by
where
Assuming
The goal of harmonic estimation techniques is to provide estimates of the parameters
Basically, we can say that before
4. Methods for estimating steadystate harmonics
Methods for estimating steadystate harmonics are more simple than timevarying ones, and its algorithms do not use information of the fundamental frequency of the signal under estimation. In what follows, we describe four algorithms.
4.1. Discrete fourier transform
The most common and the most used technique for steadystate harmonic estimation is the discrete Fourier transform (DFT) [810]. The DFT method is simple and easy to be implemented in monitoring systems, but its application for timevarying harmonics is not recommended.
Given the discrete signal
and
respectively, where,
and
in which
The DFT algorithm is very simple and its implementation is easy for realtime application. However, if the fundamental frequency is not nominal and constant, then the estimates can carry significant errors.
4.2. Demodulation
The demodulation technique presented in [15] can be used to estimate the parameters of harmonics as point out in [16]. In similar way to the DFT technique the demodulation technique can give erroneous results if its filter is fixed.
The kth harmonic parameters can be estimated by
and
in which
and
respectively, where
4.3. Goertzel
The Goertzel technique uses a secondorder infinite impulse response filter to estimate the parameters of the
The amplitude and phase of the kth harmonic is estimated, respectively, by
and
in which
and
where
4.4. Linear least squares
The linear least squares (LS) algorithm estimates several harmonics in one evaluation instead of a unique estimate [18]. Its advantage is the acquisition of several harmonics in only one evaluation. However the computational burden is high.
Basically, the LS algorithm has as a result the vector given by
Thus, the amplitude and phase of the kth harmonic (k ∈ [1, 2,...,
and
The vector
where
and
In this kind of technique, the number of harmonic has to be known a priori. Otherwise, the performance can be considered reduced, also, the computational complexity is higher due to the matrix operations.
5. Methods for estimating timevarying harmonics
Methods for estimating timevarying harmonics consider that not only the amplitudes and phases of harmonics change, but also the fundamental frequency, and consequently, the harmonics frequencies. Thus, the frequency estimation is generally required to improve the algorithms.
5.1. Discrete fourier transform with sampling frequency control
The main weakness of the DFT is to estimate the harmonics when the sampling frequency is not synchronous with the fundamental frequency. In order to guarantee this synchronism we can control the sampling frequency as shown in Fig. 1. This method reduces and can also eliminate the errors caused by the mismatch between the fundamental frequency and the sampling frequency, however, it requires a robust and controllable ADC converter and a frequency estimation algorithm. As a result, its use is not recommended.
5.2. Discrete fourier transform with signal resample
An alternative to the problem of synchronization of the sampling frequency when the sampling frequency is constant and not controllable is resampling the original signal before the harmonic estimation with the DFT. The drawback of this approach is the high computation complexity required by the resample process. Also, a frequency estimation technique is required to control the resampling process. Fig. 2 shows the block diagram of this strategy.
5.3. Discrete fourier with window
An interesting way of improving the DFT algorithm is using a window in each block of data before the evaluation of the DFT. The windowing of the data can deal with the spectral leakage of the DFT caused by the frequency deviation by adding some computational burden to the algorithm. Some windows generally used are the triangular and Hanning. The coefficients of a Hanning window are computed from the following equation:
The triangular window has its coefficient given by
for
for
5.4. Demodulation
An interesting method based on demodulation technique for estimating timevarying harmonics is presented in [16]. This technique can provide very accurate estimates with a reasonable computational complexity.
The block diagram of the demodulation technique is depicted in Fig. 3. The LP blocks implement identical lowpass filters and the blocks COS and SIN implement the demodulation signals expressed by
and
respectively. The term
where
The blocks AMP and PHAS implement, respectively, the expressions
and
respectively, where
In this technique, it is applied a approach to control the demodulation signals (blocks COS and SIN) and the frequency response of the lowpass filters (blocks LP) by the power frequency estimate, which is implemented by the block FREQ. The low pass filters are finite impulse response (FIR) filters which are controlled by a frequency estimator.
5.5. Non linear least squares
The nonlinear least squares (NLS) uses the same expressions of the linear least squares for evaluate the harmonics [18]. The advantage of this approach is the improving of the estimates related to the linear version, however, the additional searching of the optimal frequency introduces additional delay in the technique and computational burden.
The NLS algorithm test several values of
Thus, with the optimal
6. Performance analysis
In order to analyze the performance of the described techniques the following signal is considered:
where
Fig. 4 shows time estimations of the amplitude of the 3rd harmonic considering a 50% drop in the amplitude of signal when the fundamental frequency is equal to 60 Hz. Estimation delays of 2 cycles of the fundamental component are noted because the twocycles version of each technique was considered. However, when the fundamental frequency is set to 60.5 Hz such techniques exhibit significant errors in the estimates (time variations), as can be seen in Fig. 5. Otherwise, the timevarying techniques significantly improve the estimates. These results are depicted in Fig. 6. Table 1 shows the maximum of the absolute instantaneous error of all techniques after convergence of the algorithms (after the 50% drop in the amplitude of signal).
The best results are achieved with the demodulation and NLS techniques. Considering only these last two techniques, the errors were evaluated when the fundamental frequency varies between 59.5 Hz to 60.5 Hz for the 25th harmonic (See Fig. 7). Also, it is important to note that the improvement achieved by the DFT with hanning and triangular windows is significant compared with the standard DFT method as can be seen by Figs. 5 and 6. In order to better show this improvement, the errors, considering the DFT, DFT with triangular window and DFT with hanning window, were evaluated when the fundamental frequency varies between 59.5 Hz and 60.5 Hz for the 25th harmonic (Fig. 8).


DFT  4.7655 
Goertzel  4.7655 
LS  4.7655 
Demodulation (steadystate)  4.7655 
DFT with hanning window  0.7479 
DFT with triangular window  0.2296 
NLS  0.0842 
Demodulation (timevarying)  0.0660 
7. What is next and needed?
Most existing enduser equipment (computer, television, lamps, etc) emit almost exclusively at the lower odd integer harmonics, but there are indications that modern devices including certain types of distributed generators emit a broadband spectrum [11–14]. The measurement of these low levels of harmonics at higher frequencies will be more difficult than for the existing situation with higher levels and lower frequencies. This might require the development of new measurement techniques including a closer look at the frequency response of existing instrument transformers. Consequently, harmonic estimation of higher order harmonics will be very important and needed. In this case the sampling frequency should be increased to satisfy the Nyquist criterion and faster
Power electronic based photovoltaic solar and wind energy equipment may emit disturbances causing voltage fluctuations and unbalance. These types of electric sources will have large presence in the future grids very large. In order to deal with this new scenario, the harmonic estimation algorithms must be immune to higher voltage fluctuations and interharmonics.
An important issue associated with smart grid in regarding to harmonic estimation is the real time estimation of several harmonics instantaneously, including higherorder harmonics. Higherorder harmonics will be more and more important to estimate due its influence in sensitive electronics devices. Also, the dynamic and diversity of smart grid will demand different set of techniques to analyze the behavior of the timevarying harmonics. For deal with these issues, the use of reconfigurable hardware that allow the exchange of features between existing monitoring devices is of ultimate importance.
8. Concluding remarks
Several methods and techniques were developed so far for estimating steadystate and timevarying harmonics. Although several techniques can deal with timevarying harmonics, the implementation of them is incipient. However, the needs and demands related to
Although
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