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Some advanced parametric methods for assessing waveform distortion in a smart grid with renewable generation
EURASIP Journal on Advances in Signal Processing volumeÂ 2015, ArticleÂ number:Â 20 (2015)
Abstract
Power quality (PQ) disturbances are becoming an important issue in smart grids (SGs) due to the significant economic consequences that they can generate on sensible loads. However, SGs include several distributed energy resources (DERs) that can be interconnected to the grid with static converters, which lead to a reduction of the PQ levels. Among DERs, wind turbines and photovoltaic systems are expected to be used extensively due to the forecasted reduction in investment costs and other economic incentives. These systems can introduce significant timevarying voltage and current waveform distortions that require advanced spectral analysis methods to be used. This paper provides an application of advanced parametric methods for assessing waveform distortions in SGs with dispersed generation. In particular, the Standard International Electrotechnical Committee (IEC) method, some parametric methods (such as Prony and Estimation of Signal Parameters by Rotational Invariance Technique (ESPRIT)), and some hybrid methods are critically compared on the basis of their accuracy and the computational effort required.
1 Introduction
Currently, significant modifications are taking place in distribution systems as they move toward the future use of smart grids (SGs). The main objectives of SGs are the efficient use of energy, the reduction of losses in the system, improvement in the power quality (PQ), and to encourage the use of distributed generation, in particular renewable energy sources [14]. In this context, the increasing use of controllable and nonlinear loads and the new needs of liberalized markets impose new PQ requirements in order to avoid dangerous effects [57].
Among the disturbances mentioned above, the distortions of the voltage and current waveforms are of great interest, and they have been discussed extensively in the literature. Both types of distortions are due mainly to the extensive use of electronic power converters to supply loads and to connect dispersed generation (DG) units or electrical storage systems to the grid.
Photovoltaic systems (PVSs) and wind turbine systems (WTSs) are the fastest growing systems for meeting the requirements of dispersed generation. Their growth is due to the expected cost reduction and to active government policies in many countries that have encouraged their use in power grids in the last few decades. These DG units can be interfaced to the grid through either partially rated power converters or fullscale power electronic devices that can inject harmonic currents that cause voltage distortions [68]. Therefore, there is a pressing need to study the impacts of such DG systems on waveform distortions in distribution networks [9]. Thus, the main objective of this paper is to apply some of the advanced methods for the assessment of the waveform distortions that are caused by different configurations of PVSs and WTSs [1020].
PVSs and WTSs generate distorted, timevarying waveforms. As a result, great attention is required to evaluate the PQ indices to acquire information that must be deduced from the analyses of the spectral components of the waveforms and their locations as a function of time [5,21].
The International Electrotechnical Committee's (IEC) standard recommendations for signal processing use the Discrete Fourier Transform (DFT) over successive, rectangular time windows with the time duration set to 10 or 12Â cycles of the fundamental period for 50Hz systems or 60Hz systems, respectively, to evaluate the spectral components [22,23]. However, even though the standard method can provide a global quantification of the waveform distortion, it has limited ability to obtain more detailed information in the analysis of a single spectral component. This is due to some wellknown problems that characterize the DFT, i.e., the spectral leakage that arises when the duration of the time window is not correctly synchronized with the fundamental period of the power system [5].
Many solutions have been proposed in the relevant literature to overcome the spectral leakage problems by using DFTbased methods or parametric methods, such as Prony and Estimation of Signal Parameters by Rotational Invariance Technique (ESPRIT) [2430]. In particular, in [27,28], the SlidingWindow Prony method and the SlidingWindow ESPRIT method were used to provide an accurate estimation of both the harmonic and interharmonic components with highfrequency resolution, but the computational burden was excessive. Other authors have proposed hybrid methods that include DFT and parametric methods to analyze the different frequency bands of a signal separately to reduce the computational burden and provide acceptable accuracy [29,30].
Recently, some authors [31,32] proposed two new, modified slidingwindow parametric methods based on modifications of the Prony and ESPRIT signal models. In fact, these methods are based on the observation of the reduced time variability of the frequencies of the spectral components. Then, the frequency values of the spectral components were assumed to be constant over time, thereby reducing the number of unknown parameters of the signal model and the dimension of related equation systems to be solved.
The main aims of this paper can be summarized as follows:

To provide a depth application of some methods for the assessment of waveform distortion caused by PVS and WTS generators. In particular, the DFT method, parametric methods (Prony and ESPRIT), hybrid methods, and modified slidingwindow parametric methods are presented and critically compared on the basis of the accuracy of the results they produce and their computational burdens, taking into account both test and measured waveforms.

To explore and compare the theoretical waveform distortions introduced by PVS and WTS generators and the distortions detected by the considered methods in some current waveforms measured at the point of common coupling of PVS and WTS.
The paper is organized as follows. Section 1.1 describes waveform distortions in detail that result from the different configurations of the PVS and the WTS schemes. In Section 1.2, the SlidingWindow ESPRIT and Prony methods are exposed, and in Section 1.3, detailed descriptions of the slidingwindow hybrid methods and the slidingwindow modified parametric methods are presented. In Section 1.4, we describe several case studies that were based on synthetic and measured waveforms of PVSs and WTSs. This articleâ€™s conclusions are given in Section 2, and the DFT method is discussed in Appendix A.
1.1 Waveform distortion in PV systems and WT systems
It is well known that solar energy and wind energy currently are the most diffuse sources of renewable energy. The following subsections provide an overview of the waveform distortions caused by the most common PVS and WTS configurations. The overview deals with the primary spectral emissions that result from disturbances introduced by the specific nature of the system that is being considered, and it also deals with the secondary spectral emissions that are due to disturbances caused by other sources near the system, such as nonlinear loads and power communication signals [18]. Note that the distortions introduced by the PVSs and WTSs generally correspond to spectral components that are included in a wide range of frequencies.
For the sake of clarity, in the following, the spectral components up to 2Â kHz are referred as â€˜lowfrequency componentsâ€™, and the spectral components over 2Â kHz are referred as â€˜highfrequency componentsâ€™.
1.1.1 Photovoltaic systems
Photovoltaic panels are connected to the grid through an inverter that converts DC to AC and ensures that particular specifications for voltage and frequency are met and that other important tasks are performed, such as maximum power point tracking (MPPT) [10].
Threephase or singlephase inverters are used, with the former ensuring no zerosequence emission in the spectra of the waveforms. Multistring inverters also are available to obtain the combined benefit of the previous configurations [19].
FigureÂ 1 shows the most diffuse topology solutions for the inverters, i.e., (i) inverters with line frequency isolation transformers (FigureÂ 1a), (ii) inverters with highfrequency isolation transformers (FigureÂ 1b), and (iii) inverters without isolation transformers (FigureÂ 1c) [10,19].
Note that the different topologies and operating conditions of plants produce different waveform distortions. However, in general, it has been observed that every PVS has a low level of distorting spectral components that increases slightly as the amount of power being produced increases.
Practical experience has indicated that PVSs constituted by multiple inverters produce lower levels of emissions than PVSs that have a unique, largesize inverter [20]. The spectral emissions of PVSs at the point of common coupling (PCC) determine both the low and highfrequency spectral components.
The lowfrequency emissions can be due to both background distortion and overmodulation by the inverter. The amplitudes of these lowfrequency spectral components in a single photovoltaic plant are generally characterized by a current total harmonic distortion (THDi) smaller than 10%, but in resonance conditions, significant voltage distortions and problems for the electric network have been documented [19].
The highfrequency emissions basically are due to the PVS inverter and they are specifically due to the pulsewidth modulation (PWM) technique that is used. These emissions are always present during the power production of the system, and they are null when the inverter is turned off; moreover, for the current waveform of a single photovoltaic plant in ideal operating conditions, the amplitudes of these highfrequency spectral components are higher than those of the lowfrequency spectral components [18,19]. These spectral components depend on the type of inverter and on its switching frequency, and they are mostly harmonics that appear as sidebands that are centered around integer multiples of the switching frequency [19,33]. Since the switching frequency is generally in a range between 10 and 20Â kHz, actually, there still are no adequate and consolidated standards for these highfrequency emissions. Currently, however, the increasing number of spectral components above 2Â kHz has resulted in an increased intensity of research activities on this issue [19]. Note that, also at highfrequency, there are often spectral components due to the background voltage, which could become relevant for the series resonance effect.
1.1.2 Wind turbine systems
At the current time, the main large wind turbine schemes are (i) the fixedspeed wind turbine system, (ii) the doubly fed induction generator (DFIG), and (iii) the fullconverter wind turbine generator [17,34,35]. FigureÂ 2 shows the block schemes of the three configurations.
FigureÂ 2a shows a fixedspeed wind turbine system in which a squirrelcage induction generator (SCIG) works in a speedlimited range above the synchronous speed. An electronic soft starter generally is provided to avoid the increase of current during the startup phase. Also, a bank of capacitors is used to compensate for the reactive power.
The spectral emissions of this scheme primarily are caused by the action of the soft starter, which introduces odd harmonic components at low frequency. Triple components are usually due to voltage unbalances. Generally, for this configuration, the most significant components detected at the PCC are the 3rd, 5th, 7th, 9th, 11th, and 13th harmonic orders [17].
In a DFIG, the windings of the stator are connected directly to the grid, whereas, on the rotor circuit, there is a backtoback, partialscale static converter with a rated power that is roughly equal to 30% of the generator's power.
For this scheme, the spectral emissions in the grid are mainly due to the static converter. Three main types of spectral emissions can arise at the PCC, i.e., (i) inherent components, (ii) switching spectral components, and (iii) spectral components due to the unbalance conditions and/or background voltages [17,36].
The first category includes mostly lowfrequency components, and it is due to a nonsinusoidal air gap flux, which produces distorted voltages and currents at the frequencies f _{ k }â€‰=â€‰6â€‰k(1â€‰âˆ’â€‰s)â€‰Â±â€‰1f _{0,s }, with kâ€‰=â€‰1, 2, â€¦, and where s is the generator slip, and f _{0,s } is the fundamental frequency of the stator voltage. These spectral components are timevarying with the speed of the DFIG rotor.
The spectral components that belong to the second class are due to both the rotorside and gridside converters and are linked to the PWM technique that was used. They are mostly highfrequency spectral components, but in overmodulation conditions, lowfrequency components also can be detected. Under ideal conditions, the aforesaid harmonics and interharmonics are as follows: (i) for the grid side, in correspondence with the frequencies \( {f}_{k,m}^{\mathrm{PWM},g}=\left[k{f}_{sw,g}\pm m{f}_{0,s}\right] \), with k, mâ€‰=â€‰1, 2, â€¦ and where f _{0,s } is the nominal frequency of the system, and f _{sw,g } is the PWM switching frequency on the grid side; (ii) for the rotor side, in correspondence with the frequencies \( {f}_{k,m}^{\mathrm{PWM},r}=\left[k{f}_{sw,r}\pm m{f}_{0,r}\right] \), with k, mâ€‰=â€‰1, 2, â€¦ and where f _{0,r }â€‰=â€‰sf _{0,s } is the fundamental frequency of the waveform to generate for the rotor winding, and f _{sw,r } is the PWM switching frequency on the rotor side.
Note that the rotorside converter works at a frequency that is linked to the generator slip and that the aforesaid spectral components could be shifted because of the airgap coupling between the rotor current and the stator circuit [36].
The third type of spectral emissions is the spectral components that are introduced by notideal operating conditions and by the WTS's auxiliary, nonlinear loads, such as controllers and motors [12,17,36].
FigureÂ 2c shows the fullconverter windturbine generator scheme, which consists of a permanentmagnet synchronous generator connected to the grid by means of a fullscale, power electronic converter with a rated power that is 110% of the generator's power.
Also in this scheme, the emission of spectral components is caused mainly by the static converter, but unlike the previous case, there is no direct influence on the airgap flux, so the spectral content of the waveforms at the PCC is less. The most significant components are harmonics of 6â€‰kâ€‰Â±â€‰1 order (5th, 7th, 11th, 13th,â€¦) at low frequency, typical of a sixpulse, threephase bridge. However, for the high frequency, the typical spectral components of the PWM technique are detected. The Unbalanced condition and background voltage can introduce additional distortions [1217,34,35].
WTS spectral emissions appear to vary significantly when they are observed over a large period of time; this is because they depend on wind conditions and, as a result, on electrical production. In addition, the harmonic emissions increase as the amount of power generated increases.
1.2 Basic parametric methods for the assessment of timevarying waveform distortion
This section deals with two of the most popular parametric methods used to assess waveform distortion in power systems, i.e., the ESPRIT and the Prony methods, which are used as the basis of the other hybrid methods presented in Section 1.3.
1.2.1 The SW ESPRIT method
The ESPRIT method is one of the most wellknown subspace methods that model waveform samples by means a linear combination of M complex exponentials added to white noise r(n). In more detail, a given sequence of sampled data x(n) of size N is approximated by [37]:
where T _{ s } is the sampling time, and A _{ k }, Ïˆ _{ k }, f _{ k }, and Î± _{ k } are the amplitude, the initial phase, the frequency, and the damping factor of the kth complex exponential, respectively, which are the unknown parameters to be evaluated.
Model (1) can be written in a matrix form as:
where:
\( {h}_k={A}_k{e}^{j{\psi}_k} \) and with N _{ 1 }â€‰<â€‰N the selected order of the correlation matrix. The symbol Î¦ represents the rotation matrix, and its properties can be used to determine a solution for the system (2). In particular, problem (2) can be solved by introducing the matrix Åœ of the eigenvector associated with the first M eigenvalues of the correlation matrix R _{ x } and a matrix Î¨ that has the same eigenvalues of Î¦ and verifies the relationship:
where \( {\widehat{\boldsymbol{S}}}_1=\left[\begin{array}{cc}\hfill {\boldsymbol{I}}_{N1}\hfill & \hfill 0\hfill \end{array}\right]\widehat{\boldsymbol{S}},{\widehat{\boldsymbol{S}}}_2=\left[\begin{array}{cc}\hfill 0\hfill & \hfill {\boldsymbol{I}}_{N1}\hfill \end{array}\right]\widehat{\boldsymbol{S}} \) and I _{ Nâ€‰âˆ’â€‰1} is an identity matrix of order Nâ€‰âˆ’â€‰1. Then, by using the least squares approach, the matrix Î¨ can be estimated as:
Note that the eigenvalues \( {\widehat{\lambda}}_i \) of the matrix Î¨ are the elements of the main diagonal of Î¦, so \( {\widehat{\lambda}}_i={e}^{\left({\alpha}_i+j2\pi {f}_i\right){T}_s} \), and the unknown damping factors and frequencies can be computed as real and imaginary parts, respectively, of the natural logarithm of these eigenvalues. For the amplitudes, it is necessary to use another least squares approach in which the theoretical definition of the correlation matrix R _{ x } [28] is:
where A is the diagonal matrix of the squares of the unknown amplitudes, I is an identity matrix, and \( {\sigma}_w^2 \) is the variance of the white noise. In this way, by assuming that the noise r is negligible, the initial phases also can be evaluated by making appropriate substitutions in EquationÂ 2 and solving the modified equation.
If the analyzed waveform is timevarying, the SlidingWindow ESPRIT is used with a window that slides forward successively over time in order to obtain the timedependent estimates of the parameters of the ESPRIT model [5,28].
The reliability of the results and the computational burden of the ESPRIT method are dependent significantly on the number of exponentials M, the order N of the correlation matrix, and the sampling frequency [38].
1.2.2 The SW Prony method
The Prony method is another parametric method for spectral analysis. This method models the waveform samples by means of a linear combination of M complex exponentials. Specifically, a given sequence of sampled data x(n) of size N is approximated by [5]:
where \( {h}_k={A}_k{e}^{j{\psi}_k} \) and \( {z}_k={e}^{\left({\alpha}_k+j{\omega}_k\right){T}_s} \). To achieve this estimation of the unknown parameters in a traditional way, a severely nonlinear problem should be solved [5], but the Prony's intuition allows another approach to be used, i.e., the initial problem is divided into two systems of linear equations, which can be solved easily.
In the first step of the aforesaid approach, the following system of linear equations is solved to determine the damping factors and the frequencies:
System (7) is comprised of (Nâ€‰âˆ’â€‰M) linear equations, and the coefficients a(m) are the M unknowns to be computed. By imposing a(0)â€‰=â€‰1, system (7) can be written in matrix form as:
After the coefficients a(m) have been determined, the polynomial F(z) can be obtained:
The roots z _{ k } of F(z) are used to calculate the damping factors and the frequencies of each exponential by means of simple relationships.
The second step provides the amplitude and phase of each exponential by calculating the h _{ k }. This is possible by replacing the obtained z _{ k } in system (6) and, so, by solving another system of linear equations that in matrix form is [5]:
If the analyzed waveform is timevarying, the SlidingWindow Prony is used once again with sliding windows in order to obtain the timedependent estimates of the parameters of the Prony model [5,27]. In [27], an adaptive technique also was proposed in order to achieve an optimal and adaptive duration of the sliding window. The accuracy and computational burden of the Prony method depend significantly on the number of exponentials M, the number N of samples for analysis window, and the sampling frequency [38].
1.3 Advanced parametric methods for the assessment of timevarying waveform distortion
In this section, two types of advanced parametric methods for spectral analysis are analyzed: the Threestep SlidingWindow Hybrid method and the SlidingWindow Modified Parametric method. These methods are based on the parametric methods considered in Section 1.2 and are able to provide both accurate results and reduced computational burden.
1.3.1 The Threestep slidingwindow hybrid method
The Threestep SlidingWindow Hybrid method is a joint parametricDFT scheme that was developed in three stages in which either the slidingwindow (SW) ESPRIT, the SW Prony, or the SW DFT can be used alternatively to estimate (i) the fundamental and the interharmonic components in the 0 to 100Â Hz band, (ii) the harmonics, and (iii) the interharmonics at frequency fâ€‰>â€‰100â€‰Hz of power system waveforms [29,30], reducing the typical problems that characterize the SW DFT and the SW parametric method (i.e., spectral leakage problems and high computational efforts). FigureÂ 3 shows the block diagram of the threestep method.
In more detail, in the first step, the SW parametric method (Prony or ESPRIT) is applied to the output of a lowpass band filter of 0 to 200Â Hz, in order to estimate the power system's fundamental and the interharmonics in the frequency range from 0 to 100Â Hz. Then, the filtered waveform is approximated with a linear combination of exponentials according to either the (1) (SW ESPRIT) or (6) (SW Prony).
In the second step, the SW DFT is used to evaluate the harmonic components; in fact, there is also an estimation of the fundamental frequency among the outputs of the first step, so it is possible to set the duration of the DFT analysis window equal to an integer multiple of the fundamental period \( {\widehat{T}}_{\mathrm{fund}} \), greatly limiting the spectral leakage.
As shown in FigureÂ 3, the waveform analyzed by the synchronized DFT is obtained by subtracting from the original signal x(n) the reconstructed waveform \( {\widehat{\boldsymbol{x}}}_{\mathrm{l}.\mathrm{f}.}(n) \) that was obtained by the first step. The duration of the analysis window in this step can be set to 10Â cycles (12Â cycles) of the fundamental period \( {\widehat{T}}_{\mathrm{fund}} \), for 50Hz systems (60Hz systems), according to the IEC recommendations, but if it is required, it also is possible to select a different number of cycles [29]. We note that the aforesaid synchronization of the DFT time window guarantees a very accurate estimation of the harmonic components with a reduced computational effort, and this is attributable to the significant limitation of the spectral leakage due only to the possible presence of interharmonics at frequencies fâ€‰>â€‰100â€‰Hz in x _{ 1 }(n), which yields relatively low errors [5,29].
The waveform \( {\widehat{\boldsymbol{x}}}_{\mathrm{harm}}(n) \) in FigureÂ 3 was obtained summing the estimated harmonics and using the results to compute the residual x _{ 2 }(n) by subtracting \( {\widehat{\boldsymbol{x}}}_{\mathrm{harm}}(n) \) from x _{ 1 }(n). The waveform x _{ 2 }(n) is analyzed in the third step by the same parametric method used in the beginning of the scheme.
Note that, in the first and third steps, the number of exponentials required in the model was considerably lower than the number needed in the application of SW ESPRIT or SW Prony to the full band signal. This positive outcome resulted from the explanation of the decomposition of the waveform in different frequency ranges separately analyzed in the first and third steps. As a result, the Threestep SW method has a significantly lower computational burden than SW ESPRIT or SW Prony while still providing highly accurate results.
In order to improve the global performances of the Threestep SW method, each step could have a different duration of the analysis window, and then, the results could be processed and associated with a common time interval.
1.3.2 The slidingwindow modified parametric method
The SlidingWindow Modified Parametric methods are based on a reduction of the unknown parameters in the signal model in the original parametric methods with obvious improvements in terms of computational efforts. In fact, starting from the consideration that the damping factors and the frequencies of spectral components in power system applications are slightly variable versus time [25,3943], the estimation of frequencies is conducted only a few times, and the damping factors are assumed to be constant along the entire signal to be analyzed [31,32].
Specifically, in [32], the frequencies of the spectral components were computed initially by the analysis of the first sliding window, the basis window, and then, the values that were obtained were assumed to be constant with time, and they were imposed as known quantities in the successive sliding windows (nobasis windows). The damping factors were assumed to be null in the SW MProny and fixed to previously calculated values during the analysis of the â€˜basis windowâ€™ in the SW MESPRIT [32].
We note that, to elude masking effects due to significant variation of the frequencies, it is expected that the estimation of the frequencies would be repeated periodically. If the deviation between the values obtained was greater than a fixed threshold, the frequencies are updated and the analysis restarts with these new values. We also note that a different choice is required for the damping factors in the Prony and ESPRIT methods. The damping factors, in fact, are able to take into account temporal variations of the amplitudes of the spectral components and to link two consecutive sliding windows smoothly.
When the duration of the windows is very short (one to two times the fundamental period), the damping factors have very low values, and their contribution is negligible. This is the case for the SW Prony, so for this method, the damping factors are set equal to zero. However, for the SW ESPRIT, the duration of the analysis window is significantly larger (four to five times the fundamental period), so in this case, the damping factors are assumed to be constant for the nobasis windows and equal to the damping factors estimated for the basis window [32].
FigureÂ 4 shows the block diagram of the SlidingWindow Modified Parametric methods.
In the first window (basis window for kâ€‰=â€‰0), a traditional parametric algorithm (TPA) is used to estimate all of the unknown parameters \( \left({A}_k^{\mathrm{BW}},{f}_k^{\mathrm{BW}},\ {\psi}_k^{\mathrm{BW}},\ {\alpha}_k^{\mathrm{BW}}\right) \), which, then, are stored; in this window, the research for the optimal values for M and N _{ 1 } or N (respectively for the ESPRITbased and Pronybased method) is effected, updating them if the reconstruction error overcomes a prefixed threshold. The same algorithm is also used after k _{ f } windows, where another basis window is generated in order to avoid the masking effect. The choice of the value k _{ f } is related to the particular nature of the analyzed waveform. In the nobasis windows, a modified parametric algorithm (MPA) is used only for the computation of the unknown amplitudes A _{ k } and initial phases Ïˆ _{ k } of the spectral components, since the frequencies are equal to \( {f}_k^{\mathrm{BW}} \), and the damping factors are forced to be null or equal to \( {\alpha}_k^{\mathrm{BW}} \) for the Prony and ESPRIT methods, respectively. Specifically, the equations solved in the MPA when ESPRIT is used are as follows:
i)
When Prony is used, the equation system solved in MPA is as follows:
ii)
The assumption of constant frequencies and damping factors inside the nobasis windows decreases the number of unknown parameters to be estimated, and the computational effort is reduced significantly compared to the classical Prony or ESPRIT approach [32].
1.4 Numerical applications
The basic and advanced parametric methods shown in Sections 1.2 and 1.3, respectively, were used to analyze the waveform distortion due to PVSs and WTSs, and the accuracy of their results and the computational burden of the methods were compared. Several numerical experiments were conducted, analyzing both synthetic and measured waveforms in many operating conditions of DG units, but for sake of brevity, only four case studies are reported in this section, and all of them are referred to the analysis of the current waveforms.
Specifically, the synthetic and measured currents of PVSs and WTSs were analyzed by using the following:

the IEC standard method (IECM);

the SlidingWindow ESPRIT method (SWEM);

the SlidingWindow Prony method (SWPM);

the Threestep SlidingWindow ESPRITDFTESPRIT method (SWEDEM);

the Threestep SlidingWindow PronyDFTProny method (SWPDPM);

the SlidingWindow Modified ESPRIT method (SWMEM);

the SlidingWindow Modified Prony method (SWMPM).
The results of each case study were referred to the same duration of signal, and the IECM computational burden was considered as a reference for the other methods. It is important to emphasize that, for the IECM, since its frequency resolution was fixed at 5Â Hz, the fundamental component was always detected in correspondence with 50Â Hz, so the harmonic components are estimated also as multiples of 50Â Hz.
All of the programs were conducted in MATLAB, and they were not optimized for computational speed because we were interested only in obtaining a rough and relative quantification of the efficiency of the different methods. The MATLAB programs were developed and tested on a Windows PC with an Intel i73770 3.4Â GHz and 16Â GB of RAM.
1.4.1 Case study 1: test signal of a photovoltaic system
The synthetic 3s waveform emulated a current at the PCC of a PVS that had a fullbridge, unipolar inverter. It was a sort of an â€˜acid test,â€™ since our aim was to observe the behavior of each of the methods in the analysis of a waveform with a very wide spectrum, i.e., exceeding a frequency of 20Â kHz. Specifically, the waveform was assembled assuming a frequency modulation index m _{ f } equal to 200 for the inverter PWM technique and the presence of all odd harmonics up to the 27th order for the lowfrequency components. The fundamental component was fixed at 50.02Â Hz, and it had an amplitude of 9 A. Also, a white noise with a standard deviation of 0.001 was added to the aforesaid components.
The sampling rate was 100Â kHz in order to provide the most appropriate operating conditions for the parametric methods that were used so that they could provide estimates of the spectral components around the order 2m _{ f }, which are the most significant introduced by the aforesaid type of PWM and whose amplitudes were fixed up to 12% of the fundamental^{a}. For the parametric methods, the error threshold was fixed equal to 10^{âˆ’5}, and for all of the methods, the window of analysis slides of 0.04Â s was used. In the second steps of both SWPDPM and SWEDEM, the analysis window was fixed equal to 5Â cycles of the fundamental period.
TableÂ 1 shows the average percentage errors in the estimates of the frequencies and amplitudes of five spectral components, particularly interesting in the comparison of the methods that were used. Specifically, the components that were examined were the fundamental and the harmonics of order 3rd, 11th, 401st, and 405th, with amplitudes equal to 1.6%, 6%, 12%, and 3% of the fundamental amplitude, respectively.
It was evident that SWEM, SWPM, SWMEM, and SWMPM always provided negligible percentage errors, with values almost of the same order of magnitude. As the first part of TableÂ 1 shows, the percentage errors obtained using the SWEDEM and the SWPDEPM also were very low; however, the second part of TableÂ 1 shows SWEDEM and SWPDEPM with higher errors for the amplitude than with the other basic and advanced parametric methods.
This behavior of the threestep methods in estimating the amplitudes of the fundamental and third harmonic was due to the presence of the lowpass band filter of 0 to 200Â Hz, which introduces a little attenuation in the amplitude of the components in the frequency band of 0 to 100Â Hz and slight distortions for the components near the filter cutoff frequency. However, it is important to emphasize that, for the other harmonic components (far from the filter cutoff frequency) both at low and high frequency, the errors in the amplitude were practically negligible. TableÂ 1 shows that the IECM errors were globally higher, both in terms of frequency and amplitude, and that, for the highfrequency components, the average percentage error in the estimation of amplitudes increased, reaching values greater than 90%.
This was due to the desynchronization between the duration of the time window and the fundamental period, which produces spectral leakage problems that increase as the harmonic order increases. TableÂ 2 shows the computational times obtained by using all of the methods to analyze the 3s waveform per unit of computational time required by IECM. It is evident that the basic parametric methods require greater computational time in order to provide accurate results; however, the advanced parametric methods reduced the computational effort significantly while simultaneously providing highly accurate results. Note that the SWEM (SWMEM) required less computational effort than the SWPM (SWPEM); this was due to undersampling the signal before analysis [44].
TablesÂ 3 and 4 are provided to make it clearer what occurred in terms of the accuracy and computational efforts when the sampling rate in SWEM and SWMEM remained at 100Â kHz. TableÂ 3 shows that there was an irrelevant gain in the accuracy of the results compared to the values shown in TableÂ 1.
However, the computational burdens of SWEM and SWMEM (TableÂ 3) were considerably greater than those observed for the same methods in TableÂ 2. This proves that the optimal sampling rate for each parametric method guarantees the best performance with respect to accuracy and computational time. It also proves that the accuracy of the results was not affected by exceeding the optimal sampling rate, but the computational effort increased significantly, especially at the very highsampling rates.
TableÂ 2 shows that the computational time of SWEDEM was greater than that of SWPDPM. This was because the presence of the filter in the first step resulted in some negligible spectral components at low frequency that made it impossible to reduce the signal sampling rate in the first step of SWEDEM and maintain an acceptable accuracy.
It was interesting to observe that, in this case study, the advanced parametric methods required computational times that were, in the worst case (i.e., the SWEDEM and the SWMPM), an order of magnitude greater than that needed by the IECM. SWPDPM and SWMEM had computational times that were the same order of magnitude as that of the IECM, although these methods provided better results than the method recommended by the IEC standards.
1.4.2 Case study 2: measurement of the current of the photovoltaic system
A 1s current waveform measured at the PCC of a 10kW, threephase inverter without an isolation transformer, which, together with another twininverter, is included in a PVS. The sampling rate was 10Â kHz, but in order to provide better operating conditions for the detection of spectral components by the parametric methods that were used, a resampling to 20Â kHz was used [44].
The error threshold for the parametric methods was fixed at 10^{âˆ’3}, and the window of analysis slides of 0.04Â s was used for all of the methods. In the second steps of the SWPDPM and the SWEDEM, the analysis window was fixed equal to 5Â cycles of the fundamental period. FigureÂ 5 shows the time trend of the measured current.
All of the methods that were used detected mainly lowfrequency spectral components. In fact, in the frequency band from 5 to 10Â kHz, the advanced methods that used the ESPRIT method and the DFT method detected the same components, but the values of amplitude were lower than the 0.05% of the fundamental, so they are not discussed here.
TableÂ 5 shows the average values of frequency and amplitude of some spectral components detected by all of the methods that were used. Among the most significant spectral components, TableÂ 5 reports the results related to the fundamental, a harmonic at about 1,000Â Hz, and an interharmonic at about 2,915Â Hz. They are good representatives that provide a perception of the behaviors of the methods that were used in the evaluation of the various types of spectral components.
First, as expected, it was evident that both harmonic and interharmonic components had low amplitudes and that each method was able to estimate them adequately, since the values that were obtained for both frequency and amplitude were in a narrow range.
The IECM provided an amplitude value of about 1 order of magnitude lower than the other methods only for the interharmonic detection, although the component was almost a multiple of 5Â Hz (the IECM frequency resolution). This phenomenon was compatible with the observation in the previous case study; in fact, it confirmed that spectral leakage increases at high frequencies.
TableÂ 6 shows the computational time, per unit of the computational time required by IECM, obtained by analyzing the current waveform by all of the methods. The advanced parametric methods always required less computational time than the basic parametric methods. In particular, all of the advanced parametric methods required a computational time that was 2 orders of magnitude less than those of SWEM and SWPM.
Also, in this case, the SWMEM was the method that came the closest to matching the performance of IECM with respect to computational time. Note that, for the analysis of the measured current, all of the advanced parametric methods required computational times that were only 1 order of magnitude greater than that required by IECM.
1.4.3 Case study 3: test signal of wind turbine system
The synthetic 1s waveform emulated a current at the PCC of a WTS that had a doubly fed induction generator. Also, this case study was a sort of â€˜acid test,â€™ since our aim was to observe the behavior of each of the methods in the analysis of a waveform with a spectrum that exceeded a frequency of 2Â kHz and that had embedded successive harmonic and interharmonic components.
The waveform was assembled assuming a frequency modulation index m _{ f } of 40, and the fundamental frequency and amplitude were fixed at 50.02Â Hz and 5.4 A, respectively. FigureÂ 6 shows the lowfrequency and the highfrequency spectrum of the test signal. White noise with a standard deviation of 0.001 was added to the signal in order to make the waveform more realistic and to stress the performance of the spectral analysis methods that were used.
The sampling rate was 20Â kHz. For the parametric methods, the error threshold was fixed at 5â€‰Ã—â€‰10^{âˆ’8}, and for all of the methods, the window of analysis slides was 0.04Â s. In the second steps of both the SWPDPM and the SWEDEM, the analysis window was equal to 10Â cycles of the fundamental period.
TableÂ 7 shows the average percentage errors in the estimates of the frequency and amplitude of the five spectral components, which are particularly interesting for comparing the methods that were used. Specifically, the components that were examined were the fundamental, the 5th, and the 38th order of harmonic and two interharmonics at 74.79 and 382.35Â Hz.
In this case study, it was observed that the SWPM, in order to provide reasonably accurate results, requires a very high number of exponentials M, and as a result, it also requires a high value of N for the optimal window of analysis. This is due to the contemporaneous presence of noise interference and many small spectral components in the signal. In fact, since the Prony signal model does not include noise, Pronybased methods require an increasing number of M in the presence of so many small spectral components and noise. Basically, the SWPM adds many spectral components to the real spectrum, since it also must individuate the noise spectrum to approach the analyzed waveform adequately.
However, it is easier for the SWEM to detect the spectrum, since noise is accounted for in its model.
However, as shown in TableÂ 7, both the SWPM and the SWEM provide better results in terms of accuracy, with average percentage errors of the same order of magnitude.
Obviously, noise interference also was a problem for the SWMPM, which produced less accurate results than the SWMEM; however, it performed well in detecting the spectral components, with errors of the same order of magnitude as the basic parametric methods (error of less than 0.021% for the estimation of frequency and less than 0.9% for the estimation of the amplitude). The second part of TableÂ 7 shows that the SWPDPM and the SWEDEM had some difficulties in estimating the amplitudes of the components up to the fifth harmonic; as underlined in case study 1, this was due to the effect of the lowpass band filter. However, the first part of TableÂ 7 shows that the average percentage errors for these methods were comparable to the best results obtained by the basic methods.
However, in the estimations of both the frequency and the amplitude, the IECM had the largest errors, especially for the interharmonic components, which were detected with average percentage errors in the frequency of 0.2% and 0.6% for the components near 74.79 and 382.35Â Hz, respectively. For the same components, the average errors of amplitude were even worse, at 3% and 31%, respectively.
TableÂ 8 shows the computational time required to analyze the test signal by all of the methods. Also in this case, the SWEM and the SWPM appear to be the slowest methods, whereas the advanced parametric methods had computational times that were significantly less than those of SWEM and SWPM. In fact, in the best case, they were only 1 order of magnitude greater than the time required by IECM. In particular, once again, SWMEM is the method that required a computational time that was the closest to that of IECM.
1.4.4 Case study 4: measured signal of wind turbine system
A 6s current waveform measured during the soft starting of a fixedspeed wind turbine was analyzed. The original sampling rate was 2,048Â Hz, so a resampling at 10Â kHz was conducted.
For the parametric methods, the error threshold was fixed equal to 10^{âˆ’7}, and the window of analysis slides was 0.02Â s for all of the methods. In the second steps of both the SWPDPM and the SWEDEM, the analysis window was fixed equal to 3Â cycles of the fundamental period. FigureÂ 7 shows the highly timevarying trend of this waveform, which allowed us to test and compare methods also in the analysis of a nonstationary waveform.
As was predictable based on the theoretical considerations reported in Section 1.1, all of the methods identified the 3rd, 5th, 7th, and 11th harmonics as significant components, in addition to the fundamental. The amplitudes of the aforesaid components were characterized initially by an increasing trend up to a maximum value, after which the descending phase began and continued until each component achieved its steadystate value.
TableÂ 9 reports the maximum peak values of amplitude for the fundamental, the fifth, and the seventh harmonics that were detected by the methods that were used.
The results presented in TableÂ 9 provide evidence that the IECM had significant spectral leakage since the estimated values always were considerably lower than those obtained by the parametric methods. These lasts, in fact, estimate, for each component, values that differ slightly from each other.
Note that, as expected, the behaviors of the time trends of the frequencies were different; in fact, all of the parametric methods had spectralcomponent frequencies with negligible time variations.
TableÂ 10 shows the computational time required by all of the methods to analyze the current's waveform. The results were similar to those in the previous case study. It is important to observe that the SWMEM required a computational time that was only double that of IECM.
2 Conclusions
In this paper, we analyzed some methods used to estimate waveform distortions caused by photovoltaic and wind turbine systems. An overview of the waveform distortions caused by the most common configurations of PVSs and WTS schemes was also provided.
The theoretical aspects of the basic and advanced parametric methods proposed in literature were presented, focusing on their high accuracy while, at the same time, emphasizing how the advanced parametric methods have the advantage of having computational times that are significantly less than those required by the basis parametric methods.
The basic parametric methods that were considered were the SlidingWindow ESPRIT method and the SlidingWindow Prony method. The advanced parametric methods were the following:

the Threestep SlidingWindow ESPRITDFTESPRIT method;

the Threestep SlidingWindow PronyDFTProny method;

the SlidingWindow Modified ESPRIT method;

the SlidingWindow Modified Prony method.
These methods were used in the spectral analysis of synthetic and measured currents of PVSs and WTSs in order to compare them in terms of the accuracy of the results they produced and computational efforts for both stationary and nonstationary waveforms. The waveforms were analyzed also by the IEC standard method, which we used as a reference for comparisons of computational times.
The case studies that were considered highlighted the importance of an adequate sampling rate for the parametric methods and the importance of the effect of noise on the detection of spectra by the Pronybased method when the signal components have very small amplitudes. In fact, in these cases, the ESPRITbased methods can estimate the spectral components more easily, requiring a lower computational burden than the methods that use the Prony model.
All of the parametric methods were shown to have high levels of accuracy, exceeding that provided by the IECM, especially when the waveform spectrum was very wide. The IECM, in fact, failed in the detection of both interharmonic components and highfrequency harmonics.
Conversely, the advanced parametric methods took less computational effort than the basic parametric methods, with computational times of the same order of magnitude or no more than 1 order or magnitude greater than that of the IECM. The aforesaid numerical experiments also demonstrated that the SWMEM generally provided the best compromise between highaccuracy results and low computational effort.
It is important to note that, for very wide spectra, the threestep solution could inspire a new method for spectral analysis in which separating the waveform into low and highfrequency components could result in the rapid and accurate detection of the spectral components of both the aforesaid bands by adapting the duration of the analysis window to the lowest frequency of each band.
3 Appendix A  IEC method
The standards IEC 6100047 [23] and IEC 61000430 [22] propose to use, for the spectral analysis of the nonstationary waveforms, the SlidingWindow Discrete Fourier Transform, which consists of applying the DFT to consecutive windows that slide forward successively over time.
If x(n) is a waveform sampled data, it is possible to obtain the corresponding Npoint DFT, X(k), as:
The SW DFT, X _{ m }(k), by definition, follows from the (13), introducing, for the sampled waveform x(n) with length L, a window function w(n), generally rectangular, with size Nâ€‰<â€‰L:
where m is the starting time instant.
It is worth to observe that the aforesaid window function is very important both in terms of analysis resolution and of spectral leakage. Specifically, it is important to choose adequately the time duration T _{ w } of the analysis window; in fact, for a sampling interval T _{ s }, it is T _{ w }â€‰=â€‰NT _{ s }, and the frequency resolution of the spectrum is fâ€‰=â€‰1/T _{ w }, so the selected value N has to be a compromise to obtain a good resolution both in time and in frequency. According to the IEC standard, for a rectangular window, the compromise is a time duration T _{ w } equal to 10 and 12Â cycles of fundamental period, respectively, for 50Hz systems and 60Hz systems [22,23].
Moreover, selecting a synchronized window with a T _{ w } equal to an integer multiple of the waveform fundamental period, it is also possible to avoid the spectral leakage phenomenon, which is the most significant problem of the DFT method, because it impacts negatively on the analysis results accuracy. In this regard, the time duration recommended by the IEC standards becomes generally inadequate, especially in the presence of interharmonic components.
3.1 Endnote
^{a}Note that, as shown in [44], if fs is the chosen sampling rate, the Prony's method is able to detect a maximum component at a frequency of fs/4, whereas the ESPRIT method can estimate components up to fs/2. Then, the signal was undersampled at 50Â kHz before the application of SWEM and SWMEM.
Abbreviations
 DER:

distributed energy resources
 DFIG:

doubly fed induction generator
 DFT:

Discrete Fourier Transform
 DG:

dispersed generation
 IEC:

International Electrotechnical Committee
 IECM:

IEC standard method
 MPPT:

maximum power point tracking
 PCC:

point of common coupling
 PQ:

power quality
 PVS:

photovoltaic system
 PWM:

pulsewidth modulation
 SCIG:

squirrelcage induction generator
 SG:

smart grid
 SW:

slidingwindow
 ESPRIT:

estimation of signal parameters by rotational invariance technique
 SWEDEM:

Threestep SlidingWindow ESPRITDFTESPRIT method
 SWEM:

SlidingWindow ESPRIT method
 SWMEM:

SlidingWindow Modified ESPRIT method
 SWMPM:

SlidingWindow Modified Prony method
 SWPDPM:

Threestep SlidingWindow PronyDFTProny method
 SWPM:

SlidingWindow Prony method
 THDi:

current total harmonic distortion
 WTS:

wind turbine system
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Alfieri, L. Some advanced parametric methods for assessing waveform distortion in a smart grid with renewable generation. EURASIP J. Adv. Signal Process. 2015, 20 (2015). https://doi.org/10.1186/s1363401501950
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DOI: https://doi.org/10.1186/s1363401501950