Uludağ University Journal of The Faculty of Engineering, Vol. 25, No. 1, 2020 RESEARCH DOI: 10.17482/uumfd.592988 COMPARISON OF PRONY AND ADALINE METHODS IN INTER- HARMONIC ESTIMATION Nedim Aktan YALÇIN* Fahri VATANSEVER* Received: 17.07.2019; revised: 01.03.2020; accepted: 30.03.2020 Abstract: Especially in energy and power systems, harmonic estimation has crucial role. Many techniques developed in the subject of harmonic and inter-harmonic estimation. These techniques and methods include mathematical transformation (Fourier, Hartley, Hilbert-Huang, etc.) filters (adaptive, Kalman, etc.) and parametric methods (Prony, ADALINE, MUSIC, etc.) In realized study, performance of ADALINE and Prony methods are investigated in terms of harmonic and inter-harmonic prediction capability. Required data for simulations are produced from P&O MPPT algorithm for photovoltaic systems. Therefore, this model gives opportunity to try compared methods according to different harmonic intensity (closeness of harmonics to each other). At the result of different simulations, it is observed that Prony method is more preferable for low number of data and ADALINE produces more successful results than Prony method if there is high number of data and selection of high neuron size relatively. Keywords: Inter-harmonic, Prony method, ADALINE. Ara Harmonik Kestiriminde Prony ve Adaline Yöntemlerinin Karşılaştırılması Öz: Özellikle enerji ve güç sistemlerinde harmonik kestirimi, önemli rol oynamaktadır. Harmonik ve ara harmoniklerin kestirimi konusunda birçok yöntem ve teknik geliştirilmiştir. Bunlar arasında matematiksel dönüşümler (Fourier, Hartley, Hilbert-Huang vb.), filtrelemeler (adaptif, Kalman vb.) ve parametrik yöntemler (Prony, ADALINE, MUSIC vb.) yer almaktadırlar. Gerçekleştirilen çalışmada; Prony ve ADALINE yöntemlerinin ara harmonik kestirimindeki performansları incelenmiştir. Benzetimler için gerekli veriler, fotovoltaik sistemler için uygulanan P&O MPPT algoritmasından üretilmektedir. Böylece bu model; karşılaştırılan yöntemleri farklı harmonik içerik yoğunlukları (harmoniklerin birbirine yakınlıkları) açısından deneme olanağı vermektedir. Farklı benzetimlerle yapılan karşılaştırmalar sonucunda, veri sayısının düşük olduğu durumlarda Prony yönteminin daha tercih edilebilir olduğu; yüksek olduğu ve görece olarak çok sayıda nöronun kullanıldığı durumlarda da ADALINE yönteminin Prony yönteminden daha başarılı sonuçlar verdiği görülmüştür. Anahtar Kelimeler: Ara harmonik, Prony yöntemi, ADALINE. 1. INTRODUCTION In power systems, the harmonic problem has existed since the emergence of alternating current (AC) systems. Many studies have been proposed for definition and determination of this problem since beginning of 20th century (Bedell and Mayer, 1915; Bedell and Tuttle, 1906; Frank, 1910; Heartz and Saunders, 1954). Due to the fact that linear systems were generally used in the early stages of power systems, occurred harmonics were multiples of main harmonic * Bursa Uludağ University, Faculty of Engineering, Electrical-Electronics Eng. Dept., 16059 Bursa/Turkey Corresponding author: Nedim Aktan Yalçın (aktanyalcin@uludag.edu.tr) 405 Yalçın N. A., Vatansever F.: Comparison Of Prony And Adaline Method In Inter-Harmonic Estimation (fundamental harmonic) and revealing these types of harmonics could easily be performed with commonly known techniques such as Fourier Transform. For this reason, in the early period, harmonic analysis was not a conspicuous issue. But at the end of 20th century, inter-harmonics emerged in power systems depending on the development of semiconductor technology and its technological applications which include non-linear loads such as converters, inverters, cycloconverters, rectifiers, etc. Inter-harmonics which were produced by non-linear loads couldn’t be determined with conventional harmonic analysis methods. This situation gave rise to many researches about inter-harmonic determination methods. Filtration and elimination of harmonics in power systems are very important in order to increase power quality factor. They are one of the reasons for creation of many methods in recent years (Bollen and Gu, 2006; Chang and Chen, 2010; Duhamel and Vetterli, 1990; Gonen, 1984, Jain and Singh, 2011; Kay and Marple, 1981; Robinson, 1982; Singh, 2009; Thomson, 1982). The other necessity of filtration and elimination of inter-harmonics is protecting circuits from resonance effect which is not foreseen with theoretical calculation according to conventional frequency analysis techniques. From classical perspective, necessary preventions are considered in design stage with taking into account just only integer multiples of main harmonic. Passive elements may destroy circuitry because of not consideration of resonance effect in non-integer multiples of main harmonic. These types of situations can be avoided with determination and filtration methods during system operation. Besides, except inter-harmonic determination algorithms, new models which try to explain how inter-harmonics occur are proposed to literature in recent years (Sangwongwanich et al., 2018; Testa et al., 2007). Inter-harmonic determination methods could be separated in three classes: parametric methods, non-parametric methods and hybrid methods (Figure 1) (Jain and Singh 2011). Parametric methods define signal parameters (amplitude, phase and frequency) with mathematical equation and solve this equation stochastically or deterministically (such as MUSIC, ADALINE, Kalman filter, ESPRIT, Prony method etc.) (Chang et al., 2009; Dash et al., 1999; Kalman, 1960; Prony, 1795; Roy and Kailath, 1989; Schmidt, 1986). On the other hand, non-parametric methods are such algorithms that could decompose signal’s attributes (amplitude, phase and frequency) without depending on mathematical equations (such as discrete Fourier transform, Hilbert-Huang transform, wavelet transform etc.) (Cooley and Tukey, 1965; Grossmann and Morlet, 1984; Huang et al., 1998; Mallat, 1989; Winograd, 1976). Lastly, hybrid methods are combination of parametric and non-parametric methods for determining signal components. The most important feature of hybrid methods are that their capability to blend advantages of parametric and non-parametric methods and creating more robust method. There are many hybrid methods which are created by combining parametric and non-parametric methods (Bettayeb and Qidwai, 2003; Hostetter, 1980; Mishra, 2005; Tarasiuk, 2004). Inter-harmonic estimation methods Parametric Non-parametric Hybrid Figure 1: Classification of inter-harmonic estimation methods In this study, Prony and ADALINE algorithms which are both parametric methods are compared in order to uncover their advantages to each other for different situations (such as different data size, closeness of harmonics in signal, etc.). Deterministic nature of Prony and stochastic approach of ADALINE to signal parameters are the most important difference between 406 Uludağ University Journal of The Faculty of Engineering, Vol. 25, No. 1, 2020 these methods. Second section of this paper summarizes basics of Prony and ADALINE algorithms. In third section, relevant simulations are realized and last section, the results are discussed. 2. METHODOLOGY 2.1. Data Model In this study, data model is obtained from this reference (Sangwongwanich et al., 2018). This study reveals sources of inter-harmonics which are produced from Perturb and Observe Maximum Power Point Tracking (P&O MPPT) method which is implemented in photovoltaic (PV) systems. Investigated study derives mathematical expression for inter-harmonics based on realized experiments in the work. In the inspected study, it is shown that, frequency spectrum of inter- harmonics in PV systems expands if MPPT frequency is increased. Quick approximation to maximum power point with P&O MPPT causes enlargement in harmonic spectrum. Besides it is shown that amplitude of harmonics depends on step voltage which is the other parameter of P&O MPPT. 2.2. Prony Method Prony analysis is extended form of Fourier analysis and can reveal amplitude, phase and frequency of signals. A signal can be expressed as Equation (Eq.) 1 approximately (Hauer et al. 1990). It can be observed that damping coefficient 𝜎 has important role for revealing frequency components more successfully than Fourier analysis (Xiong et al., 2010). ∞ ?̂?(𝑡) = ∑A 𝑒𝜎𝑖𝑡i cos(2𝜔𝑓𝑖𝑡 + 𝜑𝑖) (1) 𝑖=1 If 𝑦(𝑡) is equally sampled with ∆𝑡 (𝑦(𝑡𝑘) = 𝑦(𝑘), 𝑘 = 0,1,… ,𝑁 − 1), strategy for obtaining solution can be summarized as below (Hauer et al. 1990). • Step 1: Produce a linear prediction model (LPM) based on observed data. • Step 2: Find roots of characteristic polynomial which are derived from LPM. • Step 3: Determine complex amplitude, phase and frequency values with using roots which are determined from Step 2. Eq. 1 can be reformed as Eq. 2. ∞ ?̂?(𝑡) = ∑B 𝑒𝜆𝑖𝑡i (2) 𝑖=1 If sampling time expressed as 𝑡𝑘, the statement could be written as Eq. 3. ∞ ?̂?(𝑘) = ∑ Bi𝑧 𝑘 1 , 𝑧𝑖 = 𝑒 𝜆𝑖𝑡 (3) 𝑖=1 In order to ensure ?̂?(𝑘) = 𝑦(𝑘) for all 𝑘, below equation should be solved using Eq. 3 for each 𝑡𝑘 and Bi and 𝑧𝑖 values should be determined (Hauer et al., 1990). 407 Yalçın N. A., Vatansever F.: Comparison Of Prony And Adaline Method In Inter-Harmonic Estimation 𝑎 01𝑧1 + … +𝑎𝑛𝑧 0 0 0 0 𝑛 𝑧1 𝑧2 … 𝑧𝑛 𝐵1 𝑦(0) 𝑎 𝑧 1 1 1 + … +𝑎𝑛𝑧 1 1 𝑛 𝑧1 𝑧 1 … 𝑧1 𝐵2 𝑛 2 𝑦(1)= [ ] = [ ] (4a) ⋮ … ⋮ ⋮ ⋯ ⋮ ⋮ ⋮ [𝑎 𝑧𝑁−1 + … +𝑎 𝑧𝑁−1] [𝑧𝑁−1 𝑁−1 𝑁−1 𝐵1 1 𝑛 𝑛 1 𝑧2 … 𝑧𝑛 ] 𝑁−1 𝑦(𝑁 − 1) Eq. 4a can be rewritten as Eq. 4b with matrix notation. 𝐙𝐁 = 𝐘 (4b) After finding 𝑧𝑖, 𝜆𝑖 values can be calculated with Eq. 3. All 𝑧𝑖 are roots of a polynomial with 𝑛. degree and their coefficients can be called 𝑎𝑖. This can be expressed mathematically as Eq. 5. 𝑧𝑛 − (𝑎1𝑧 𝑛−1 + 𝑎 𝑛−22𝑧 + ⋯+ 𝑎𝑛𝑧 0) = 0 (5) 1 × 𝑁 vector can be produced from Eq. 5 and expressed as Eq. 6. ?̅? = [−𝑎𝑛 − 𝑎𝑛−1 …−𝑎1 1 0…0] = [−𝐚 1 𝟎] (6) Eq. 7 can be written with implementing Eq. 6 to Eq. 4. ?̅?𝐘 = 𝑦(𝑛) − [𝑎1𝑦(𝑛 − 1) + ⋯+ 𝑎𝑛𝑦(0)] = ?̅?𝐙𝐁 (7) = 𝐵 𝑛1[𝑧1 − (𝑎1𝑧 𝑛−1 1 + 𝑎 𝑧 𝑛−2 2 1 + ⋯+ 𝑎𝑛𝑧 0 1 ) + ⋯ ] = 0 Last step in Eq. 7 is written because of providing Eq. 5 by each 𝑧𝑖. Comparing Eq. 5 and Eq. 7, it can be observed that Eq. 5 is general matrix equation and Eq. 7 is clearly rewritten form of Eq. 5 for each element (Hauer et al., 1990). Due to arbitrary choice of starting time, Eq. 7 can be reformed to Eq. 8. 𝑦(𝑛 − 1) 𝑦(𝑛 − 2) … 𝑦(0) 𝑎1 𝑦(𝑛 + 0) 𝑦(𝑛 − 0) 𝑦(𝑛 − 1) … 𝑦(1) 𝑎2 𝑦(𝑛 + 1) [ ] [ ⋮ ] = [ ] (8) ⋮ ⋯ ⋮ ⋮ 𝑦(𝑁 − 2) 𝑦(𝑁 − 3) … 𝑦(𝑁 − 𝑛 − 1) 𝑎𝑁−1 𝑦(𝑁 − 1) Solution of Eq. 8 provides coefficients of polynomial at Eq. 5. After finding roots (𝑧𝑖) of the polynomial, 𝜆𝑖 eigenvalues could be detected. These operations complete Prony method’s first and second steps. In third step, complex amplitude and phase values (𝐵𝑖) are calculated (Hauer et al., 1990). 2.3. ADALINE ADALINE is an adaptive filter which is used for feature extraction, noise suppression and different application purposes (Widrow, 1960). In recent years, this method has widely used in power quality studies. ADALINE algorithm handles Eq. 8 in stochastic manner for finding polynomial coefficients and uses frequency information which is derived from the coefficients as input in another artificial neural network (ANN) design for detecting amplitudes and phases (Chang et al., 2009). In first ANN design, past signal values are fed to input of ANN and current signal value is fed to output. Therefore, ANN coefficients are trained for polynomial coefficients of Prony method. After finding polynomial coefficients, frequency components can be calculated. 408 Uludağ University Journal of The Faculty of Engineering, Vol. 25, No. 1, 2020 In second ANN design, found frequencies are inputs and signal values are outputs. With training second ANN, related parameters in Eq. 14 are detected. In this way, signal parameters are obtained. Frequency detection process is shown in Figure 2 and amplitude-phase finding process is shown in Figure 3. Eq. 9 shows instant value of predicted signal with ADALINE algorithm. 𝑘 𝑦𝑓(𝑖) = ∑𝑎𝑚(𝑘)𝑦(𝑖 − 𝑚) = −?̌? 𝑇(𝑘) ?̌?(𝑖 − 1) 𝑖=1 𝑎1(𝑘) 𝑦(𝑖 − 1) (9) 𝑎2(𝑘) 𝑦(𝑖 − 2)?̌?(𝑘) = [ ] , ?̌?(𝑖 − 1) = [ ] ⋮ ⋮ 𝑎𝑀(𝑘) 𝑦(𝑖 − 𝑀) Figure 2: Frequency estimation state of ADALINE (Chang and Chen, 2010) Figure 3: Amplitude and phase estimation state of ADALINE (Chang and Chen, 2010) 409 Yalçın N. A., Vatansever F.: Comparison Of Prony And Adaline Method In Inter-Harmonic Estimation Error which is occurred at system output is calculated with Eq. 10. 𝑒1(𝑖) = 𝑦(𝑖) − 𝑦𝑓(𝑖) (10) 𝑘 𝑂1(𝑘) = ∑𝜆 𝑘−𝑖𝑒21 (𝑖) (11) 𝑖=1 𝜆 statement in Eq. 11 is called forgetting factor and its values between 0 and 1. It is used for weighting instant error. Neuron coefficients of ANN are updated using Eq. 12. Changes in coefficients which depend on time are calculated from Eq. 13. In this study, update mechanism of coefficients is realized with Levenberg-Marquardt unlike Eq. 13 (Levenberg, 1944; Marquardt, 1963). ?̌?(𝑘 + 1) = ?̌?(𝑘) + Δ?̌?(𝑘) (12) 𝑘 𝜕𝑂1(𝑘) 𝑘−𝑖 Δ?̌?(𝑘) = = 2∑𝜆 𝑒1(𝑖) ?̌?(𝑖 − 1) 𝜕?̌?(𝑘) (13) 𝑖=1 Frequencies are obtained from 𝑎(𝑘) coefficients. Obtained frequencies are fed to ANN which is depicted in Figure 3. Output of second ANN is calculated with Eq. 14. Eq. 15 represents function to be minimized and Eq. 16 states error of ANN. 𝑀 𝑦𝑏(𝑘) = ∑(𝐴 ∗ 𝑚 cos𝜙 ∗ 𝑚 sin2𝜋𝑓 ∗ 𝑚𝑘Δ𝑡 + 𝐴 ∗ ∗ 𝑚 sin𝜙𝑚 cos2𝜋𝑓 ∗ 𝑚𝑘Δ𝑡) 𝑚=1 𝑀 = ∑(𝑤∗2𝑚−1 sin𝜃 ∗ ∗ ∗ ∗ 𝑚 + 𝑤2𝑚 cos𝜃𝑚) = w (k). x ∗(k) 𝑚=1 (14) w∗(k) = [𝑤∗ 𝑤∗ ∗ ∗1 2 ⋯ 𝑤2𝑀−1 𝑤2𝑀] 𝑤∗ ∗2𝑚−1 = 𝐴𝑚 cos𝜙 ∗ 𝑚 𝑤∗ ∗ ∗2𝑚 = 𝐴𝑚 sin𝜙𝑚 x∗(k) = [sin 𝜃∗ ∗1 cos 𝜃1 … sin 𝜃 ∗ cos𝜃∗𝑀 𝑀] 𝑘 𝑂2(𝑘) = ∑𝜆 𝑘−𝑖𝑒22 (𝑖) (15) 𝑖=1 𝑒2(𝑖) = 𝑦(𝑖) − 𝑦𝑏(𝑖) (16) Eq. 17 and Eq. 18 represent update mechanism of system and are used in analyzed paper. In this study, Levenberg-Marquardt algorithm is used instead of Eq. 15 (Levenberg, 1944; Marquardt, 1963). w∗(𝑘 + 1) = w∗(𝑘) − Δw(k) (17) 𝑘 𝜕𝑂2(𝑘) Δ𝑤(𝑘) = = −2∑𝜆𝑘−𝑖𝑒2 𝜕𝑤(𝑘) 2 (𝑖)𝑦(𝑖 − 1) (18) 𝑖=1 410 Uludağ University Journal of The Faculty of Engineering, Vol. 25, No. 1, 2020 Application steps which are used to predict polynomial coefficients by ADALINE algorithm are summarized below. • Produce ANN (Fig. 2) with random ?̌?(𝑘) coefficients which are also ANN coefficients. • Find instantaneous output values according to Eq. 9. • Update ANN coefficients with Levenberg-Marquardt algorithm until desired stop condition is satisfied (error tolerance, maximum iteration value, etc.). • Find frequency values using Prony polynomial coefficients which are also ANN coefficients. • Assign the frequencies to input of second ANN (Fig. 3). • Find prediction signal with Eq. 14. • Calculate error value with Eq. 16. • Update second ANN using Levenberg-Marquardt until algorithm stop condition is satisfied. • After completion of updating stage, calculate amplitude and phase values from second ANN model. Shortly, it is seen that ADALINE comprises of two stages. In first stage, first four items are realized for finding frequencies and in second stage, last five items which are also called phase tuning steps are carried out for obtaining more approximate results to real amplitude and phase values. In this study, all steps are realized for finding frequency, amplitude and phase values and obtained result are compared with Prony method’s. 3. SIMULATIONS In this section, simulations are realized with MATLAB (MathWorks, 2019). Implemented data model is described in Methodology section. Obtained results are presented in figures and tables. In these simulations, base/fundamental/main frequency is selected as 50 𝐻𝑧, sampling frequency is 0.5 𝑚𝑠, simulation duration is 5 𝑠 ((10000) data samples). Dataset is obtained from harmonic model of PV (Photovoltaic) system which is derived from (Sangwongwanich et al., 2018) and based on Eq. 21. In this model, step voltage and MPPT frequency have major role in creation of inter-harmonics. Step voltage causes change in amplitude of inter-harmonics while MPPT frequency affects frequency spectrum of inter-harmonics. If step voltage increases, amplitude of inter-harmonics affected positively. If MPPT frequency increases, bandwidth of inter-harmonics expands. Based on these facts, there is a trade-off between, MPPT frequency and bandwidth of inter-harmonics. First one (step voltage) is important for quick response and second one (MPPT frequency) is crucial for spectrum of inter-harmonics. On the other hand, large step voltage guarantees of quick convergence of maximum power transfer point but causes high amplitude of inter-harmonics. Therefore, there is another trade-off between step voltage and response time. In conclusion, mathematical model of inter-harmonics which are created from PV systems is given in Eq. 21. In this equation, 𝑖𝑔 is produced current by MPPT system, 𝑓𝑔 is fundamental (main) frequency, 𝑓𝑛 is harmonic frequency which is created by MPPT frequency, 𝐴′𝑛 and ∅ ′ 𝑛 amplitude and phase value which are affected by step voltage. ∞ 𝐴𝑛 𝑖𝑔(𝑡) = ∑ [cos(2𝜋𝑡(𝑓𝑔 − 𝑓𝑛) + ∅𝑛) − cos(2𝜋𝑡(𝑓2 𝑔 + 𝑓𝑛) + ∅𝑛)] 𝑛=1 2𝑉𝑠𝑡𝑒𝑝 𝜋𝑛 2𝑉𝑠𝑡𝑒𝑝 (21) 𝑎𝑛 = sin ( ) , 𝑏𝑛 = cos(𝜋𝑛 − 1) 𝜋𝑛 2 𝜋𝑛 𝐴 = √𝑎2 𝑏 + 𝑏2, ∅ = tan−1 ( 𝑛 (2𝑛−1)𝑓 ), 𝑓 = 𝑀𝑃𝑃𝑇𝑛 𝑛 𝑛 𝑛 𝑛 𝑎𝑛 4 411 Yalçın N. A., Vatansever F.: Comparison Of Prony And Adaline Method In Inter-Harmonic Estimation Three simulations are realized using Eq. 21: 1st. Step voltage is 12 𝑉 and MPPT frequency is 5 𝐻𝑧 (Table-1 and Table-2). 2nd. Step voltage is 12 𝑉 and MPPT frequency is 20 𝐻𝑧 (Table-3 and Table-4). 3rd. Step voltage is 24 𝑉 and MPPT frequency is 5 𝐻𝑧 (Table-5 and Table-6). In second simulation, it is expected that bandwidth of inter-harmonics is wider than the first, but amplitudes of inter-harmonics are same. In third simulation, it is expected that bandwidth of inter-harmonics is same as the first, but amplitudes of inter-harmonics are higher than the first. In all simulations, Prony coefficient size is selected 80 and ADALINE neuron size is firstly selected 80 and secondly selected 800. Selection of Prony coefficient size should be equal or greater than number of harmonics in signal. Selection criteria are entirely discussed in (Rabehi et al., 2019). In all tables it is obvious that Prony produces more successful results than ADALINE, if neuron size and coefficient size are selected same. But, when neuron size is increased, ADALINE gives better results than Prony. In first and third simulation, signal has closer inter-harmonics than second’s. Performance of Prony is weaker when inter-harmonics become closer as seen in Table-1, Table-5 and Table-3. ADALINE produces poor results when neuron size is selected 80. After neuron size is increased to 800, ADALINE gives better results (relative error converges to nearly zero) than itself and Prony as seen in Table-2, Table-4, Table-6. In order to uncover performance of ADALINE and Prony methods under low number of data size, it would be useful to realize two more simulations. In these simulations, simulation duration is 0.1 𝑠, (200 data samples), base frequency is 50 𝐻𝑧, sampling frequency is 0.5 𝑚𝑠, 𝑉𝑠𝑡𝑒𝑝 is 12 𝑉. In first, 𝑓𝑀𝑃𝑃𝑇 is 5 𝐻𝑧 (Table-7 and Table-8). In second, 𝑓𝑀𝑃𝑃𝑇 is 20 𝐻𝑧 (Table-9 and Table-10). Table 1. Obtained results with Prony for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟏𝟐 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟓 𝑯𝒛) Harmonic Estimated Relative error Harmonic Estimated Relative error frequency (Hz) frequency (Hz) (%) amplitude (V) amplitude (V) (%) 38.75 38.7544 0.0114 0.8541 0.7835 8.2660 41.25 41.5980 0.8436 0.4775 0.2442 48.8586 43.75 44.0236 0.6254 1.4235 1.5974 12.2164 46.25 ~ ~ 0.9549 ~ ~ 48.75 48.7451 0.0101 4.2706 4.2637 0.1616 50.00 50.0203 0.0406 22 24.4907 11.3214 51.25 ~ ~ 4.2706 ~ ~ 53.75 52.0224 3.2141 0.9549 2.2470 135.3126 56.25 56.1521 0.1740 1.4235 1.7843 25.3460 58.75 58.6921 0.0986 0.4775 0.5463 14.4084 61.25 61.2505 0.0008 0.8541 0.8473 0.7962 412 Uludağ University Journal of The Faculty of Engineering, Vol. 25, No. 1, 2020 Table 2. Obtained results with ADALINE for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟏𝟐 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟓 𝑯𝒛) Estimated frequency Estimated amplitude Harmonic Relative error (%) Harmonic Relative error (%) (Hz) (V) frequency amplitude 800 80 800 80 800 80 800 80 (Hz) (V) neurons neurons neurons neurons neurons neurons neurons neurons 38.75 38.7500 39.1353 ~0 0.9943 0.8541 0.8541 0.0142 ~0 98.3374 41.25 41.2500 ~ ~0 ~ 0.4775 0.4775 ~ ~0 ~ 43.75 43.7500 ~ ~0 ~ 1.4235 1.4235 ~ ~0 ~ 46.25 46.2500 45.5495 ~0 1.5146 0.9549 0.9549 0.0816 ~0 91.4546 48.75 48.7500 ~ ~0 ~ 4.2706 4.2706 ~ ~0 ~ 50.00 50.0000 50.0166 ~0 0.0332 22 22.0000 21.8280 ~0 0.7818 51.25 51.2500 ~ ~0 ~ 4.2706 4.2706 ~ ~0 ~ 53.75 53.7500 ~ ~0 ~ 0.9549 0.9549 ~ ~0 ~ 56.25 56.2500 55.1083 ~0 2.0297 1.4235 1.4235 0.0978 ~0 93.1296 58.75 58.7500 ~ ~0 ~ 0.4775 0.4775 ~ ~0 ~ 61.25 61.2500 61.0488 ~0 0.3285 0.8541 0.8541 0.0164 ~0 98.0799 Table 3. Obtained results with Prony for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟏𝟐 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟐𝟎 𝑯𝒛) Harmonic Estimated Relative error Harmonic Estimated Relative error (%) frequency (Hz) frequency (Hz) (%) amplitude (V) amplitude (V) 5 5.0053 0.1060 0.8541 0.8539 0.0234 15 14.9994 0.0040 0.4775 0.4779 0.0838 25 24.9979 0.0084 1.4235 1.4241 0.0421 35 35.0033 0.0094 0.9549 0.9555 0.0628 45 44.9975 0.0056 4.2706 4.2730 0.0562 50 50.0019 0.0038 22 21.9910 0.0409 55 54.9993 0.0013 4.2706 4.2897 0.4472 65 65.0001 0.0002 0.9549 0.9544 0.0524 75 75.0000 0 1.4235 1.4309 0.5198 85 85.0000 0 0.4775 0.4768 0.1466 95 95.0000 0 0.8541 0.8558 0.1990 Table 4. Obtained results with ADALINE for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟏𝟐 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟐𝟎 𝑯𝒛) Estimated frequency Relative error Estimated amplitude Relative error Harmonic Harmonic (Hz) (%) (V) (%) frequency amplitude 800 80 800 80 800 80 800 80 (Hz) (V) neurons neurons neurons neurons neurons neurons neurons neurons 5 5.0000 ~ ~0 ~ 0.8541 0.8541 ~ ~0 ~ 15 15.0000 ~ ~0 ~ 0.4775 0.4775 ~ ~0 ~ 25 25.0000 24.8274 ~0 0.6904 1.4235 1.4235 0.2220 ~0 84.4046 35 35.0000 ~ ~0 ~ 0.9549 0.9549 ~ ~0 ~ 45 45.0000 44.6511 ~0 0.7753 4.2706 4.2706 0.5184 ~0 87.8612 50 50.0000 50.0079 ~0 0.0158 22.0000 22.0000 21.9359 ~0 0.2914 55 55.0000 57.0697 ~0 3.7631 4.2706 4.2706 0.1084 ~0 97.4617 65 65.0000 ~ ~0 ~ 0.9549 0.9549 ~ ~0 ~ 75 75.0000 74.5606 ~0 0.5859 1.4235 1.4235 0.1153 ~0 91.9002 85 85.0000 84.6536 ~0 0.4075 0.4775 0.4775 0.0623 ~0 86.9529 95 95.0000 94.9985 ~0 0.0016 0.8541 0.8541 0.8592 ~0 0.5971 413 Yalçın N. A., Vatansever F.: Comparison Of Prony And Adaline Method In Inter-Harmonic Estimation Table 5. Obtained results with Prony for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟐𝟒 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟓 𝑯𝒛) Harmonic Estimated Relative error Harmonic Estimated Relative error frequency (Hz) frequency (Hz) (%) amplitude (V) amplitude (V) (%) 38.75 38.7537 0.0095 1.7082 1.7166 0.4917 41.25 41.7063 1.1062 0.9549 0.3420 64.1847 43.75 44.0675 0.7257 2.8471 3.1597 10.9796 46.25 ~ ~ 1.9099 ~ ~ 48.75 48.8739 0.2542 8.5412 10.9924 28.6985 50.00 50.2098 0.4196 22 27.3092 24.1327 51.25 52.1878 1.8299 8.5412 2.9788 65.1243 53.75 ~ ~ 1.9099 ~ ~ 56.25 56.1781 0.1278 2.8471 3.4041 19.5638 58.75 58.7065 0.0740 0.9549 1.1690 22.4212 61.25 61.2503 0.0005 1.7082 1.6085 5.8366 Table 6. Obtained results with ADALINE for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟐𝟒 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟓 𝑯𝒛) Estimated Relative error Estimated amplitude Relative error Harmonic frequency (Hz) (%) Harmonic (V) (%) frequency 800 80 800 80 amplitud 800 80 800 80 (Hz) neuron neuron neuron neuron e (V) neurons neurons neurons neurons s s s s 38.75 38.7500 38.8750 ~0 0.3226 1.7082 1.7082 0.8202 ~0 51.9845 41.25 41.2500 ~ ~0 ~ 0.9549 0.9549 ~ ~0 ~ 43.75 43.7500 44.0438 ~0 0.6715 2.8471 2.8471 0.5938 ~0 79.1437 46.25 46.2500 ~ ~0 ~ 1.9099 1.9099 ~ ~0 ~ 48.75 48.7500 ~ ~0 ~ 8.5412 8.5412 ~ ~0 ~ 50.00 50.0000 49.8815 ~0 0.2370 22 22.0000 11.6056 ~0 47.2473 51.25 51.2500 50.8578 ~0 0.7653 8.5412 8.5412 0.8474 ~0 90.0787 53.75 53.7500 ~ ~0 ~ 1.9099 1.9099 ~ ~0 ~ 56.25 56.2500 56.4622 ~0 0.3772 2.8471 2.8471 0.1165 ~0 95.9081 58.75 58.7500 ~ ~0 ~ 0.9549 0.9549 ~ ~0 ~ 61.25 61.2500 61.2116 ~0 0.0627 1.7082 1.7082 1.5284 ~0 10.5257 Table 7. Obtained results with Prony for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟏𝟐 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟓 𝑯𝒛) Harmonic Estimated Relative error Harmonic Estimated Relative error frequency (Hz) frequency (Hz) (%) amplitude (V) amplitude (V) (%) 38.75 38.7982 0.1244 0.8541 0.9145 7.0718 41.25 ~ ~ 0.4775 ~ ~ 43.75 42.6588 2.4942 1.4235 1.4606 2.6063 46.25 ~ ~ 0.9549 ~ ~ 48.75 48.0929 1.3479 4.2706 0.6170 85.5524 50.00 50.2069 0.4138 22 23.0777 4.8986 51.25 ~ ~ 4.2706 ~ ~ 53.75 54.7418 1.8452 0.9549 2.1413 124.2434 56.25 ~ ~ 1.4235 ~ ~ 58.75 58.5625 0.3191 0.4775 0.5367 12.3979 61.25 61.2827 0.0534 0.8541 0.8112 5.0228 414 Uludağ University Journal of The Faculty of Engineering, Vol. 25, No. 1, 2020 Table 8. Obtained results with ADALINE for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟏𝟐 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟓 𝑯𝒛) Harmonic Estimated Relative error Harmonic Estimated Relative error frequency (Hz) frequency (Hz) (%) amplitude (V) amplitude (V) (%) 38.75 39.7806 2.6596 0.8541 2.0346 ~ 41.25 ~ ~ 0.4775 ~ ~ 43.75 ~ ~ 1.4235 ~ ~ 46.25 ~ ~ 0.9549 ~ ~ 48.75 ~ ~ 4.2706 ~ ~ 50.00 50.5711 1.1422 22 26.8375 21.9886 51.25 52.0037 1.4706 4.2706 4.1420 3.0113 53.75 ~ ~ 0.9549 ~ ~ 56.25 ~ ~ 1.4235 ~ ~ 58.75 ~ ~ 0.4775 ~ ~ 61.25 61.9046 1.0687 0.8541 0.1627 80.9507 Table 9. Obtained results with Prony for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟏𝟐 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟐𝟎 𝑯𝒛) Harmonic Estimated Relative error Harmonic Estimated Relative error frequency (Hz) frequency (Hz) (%) amplitude (V) amplitude (V) (%) 5 ~ ~ 0.8541 ~ ~ 15 ~ ~ 0.4775 ~ ~ 25 22.6672 9.3312 1.4235 2.3468 64.8613 35 28.3178 19.0920 0.9549 0.2694 71.7876 45 44.5807 0.9318 4.2706 0.0852 98.0050 50 49.7010 0.5980 22 12.7131 42.2132 55 53.4735 2.7755 4.2706 9.8848 ~ 65 65.0177 0.0272 0.9549 0.9985 4.5659 75 75.0039 0.0052 1.4235 1.4204 0.2178 85 85.0001 0.0001 0.4775 0.4775 0 95 95.0000 ~0 0.8541 0.8541 0 Table 10. Obtained results with ADALINE for MPPT simulation (𝑽𝒔𝒕𝒆𝒑 = 𝟏𝟐 𝑽, 𝒇𝑴𝑷𝑷𝑻 = 𝟐𝟎 𝑯𝒛) Harmonic Estimated Relative error Harmonic Estimated Relative error frequency (Hz) frequency (Hz) (%) amplitude (V) amplitude (V) (%) 5 ~ ~ 0.8541 ~ ~ 15 ~ ~ 0.4775 ~ ~ 25 29.6377 18.5508 1.4235 8.8180 519.4591 35 31.3281 10.4911 0.9549 7.9631 733.9198 45 48.0850 6.8556 4.2706 27.8884 553.0324 50 49.0331 1.9338 22 52.9238 140.5627 55 ~ ~ 4.2706 ~ ~ 65 ~ ~ 0.9549 ~ ~ 75 76.9587 2.6116 1.4235 2.6924 89.1394 85 81.9129 3.6319 0.4775 1.4876 211.5393 95 95.0148 0.0156 0.8541 0.4253 50.2049 415 Yalçın N. A., Vatansever F.: Comparison Of Prony And Adaline Method In Inter-Harmonic Estimation In Table 7-10, it is observed that Prony method produces better results than ADALINE. Due to the fact that usage of low number of data, it could be said that ADALINE couldn’t have enough data to train itself. In order to get higher efficiency for training, neuron size of ADALINE is selected 100 (This neuron size gives best performance for last simulations). In short, these simulations show that Prony produces better results for low number of data size. This makes Prony more powerful for analysis of transient response. 4. RESULTS In this study, ADALINE and Prony methods which are both classified as parametric methods for harmonic estimation are compared. In realized simulations, data are produced from mathematical model of P&O MPPT algorithm for PV systems. This model gives opportunity to compare inspected methods under different harmonic density (closeness to each other) situations. It is difficult to be detected harmonics by Prony method as they get closer to each other. But detection performance of ADALINE method remains more stable under same conditions with Prony method. 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