Estimating the parameters of a photovoltaic array and solving equations of maximum power point using a numerical method and fuzzy controller

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1 Etimating the arameter of a hotovoltaic array and olving equation of maximum ower oint uing a numerical method and fuzzy controller Amin Taheri 1, Majid Dehghani 2 Deartment of Electrical Engineering, Najafabad Branch, Ilamic Azad Univerity, Najafabad, Iran Mail2amin2005@iaun.ac.ir 1, Dehghani@el.iaun.ac.ir 2 ABSTRACT Thi aer rooe the method of modeling and imulation of hotovoltaic (PV) array and achieve maximum ower oint (MPP) of them. Simulating a hotovoltaic array baed on the ingle-diode model ha been done in thi tudy. The firt objective i to find the arameter of nonlinear I-V equation by adjuting the curve in oen circuit, maximum ower and hort circuit oint bae on the ingle-diode model and the effect of the erie and arallel reitance. The variation of the arameter with change in irradiance and temerature i alo tudied. Then, equation of the maximum ower oint of hotovoltaic array have been introduced and olved uing ucceive under relaxation (SUR) numerical method. Afterward, diadvantage of SUR method have been identified and fuzzy controller ha been ued to get the maximum ower of the hotovoltaic array. Keyword: Photovoltaic array, Maximum ower oint tracker, Fuzzy controller 1. INTRODUCTION Due to a ignificant reduction in foil energy reource in recent year, a well a the ollution caued by wideread ue of them, the imortance of uing clean and renewable energy ha been revealed more and more. Solar ower i one of the clean, renewable and acceible energie. Develoed countrie have ulied a ignificant ortion of their energy need through green energie. Develoing countrie alo have taken aroriate te in the way of clean and renewable energie with regard to the future of foil fuel. So that the annual growth rate of exloitation of hotovoltaic (PV) ower on a global cale i more than 40%. Hence the renewable energie, eecially PV ower, will be a coniderable ource of energy in the near future. A hotovoltaic (PV) array directly convert unlight into the electricity. In recent year more efficient PV array ha been roduced and releaed on the market along with advancement in technology [1]. PV array reent nonlinear I-V characteritic with everal arameter. Although ome imortant arameter are alway announced by manufacturer, having acce to all arameter of PV array eem neceary for modeling and accurate deign of PV ytem. Due to the relatively low efficiency of PV array, getting the maximum ower of them i one of the major iue [2, 3, 4]. Although it i oible to connect a PV array directly to the mall load, like lighting ytem but the ower received from array will be coniderably lower than the cae in which array i connected to the load through a Maximum Power Point Tracking (MPPT) ytem [2]. Thi i the imortance of MPPT in high ower PV ytem. In thi tudy firtly equivalent circuit of PV cell have been introduced, imulating of a PV array baed on a ingle diode model i invetigated, equation of the PV array have been acquired and all the arameter related to PV array have been obtained uing Newton-Rahon method. Then maximum ower oint (MPP) equation defined and olved uing a ucceive under relaxation (SUR) method to achieve maximum ower oint. The MPPT ytem baed on the boot converter ha been deigned uing a fuzzy controller. Finally the MPPT method comare to each other. 1.1 Modeling the PV cell and the PV array The elementary of PV array i PV cell. In order to large outut voltage, PV array i formation from erie cell and For increaing outut current we need to connecting cell in arallel. For modeling a PV array firtly we need to model a PV cell. In recent year different equivalent circuit have been uggeted for modeling the behavior of PV cell. Such a ingle-diode, two-diode, and diode-caacitor model [3, 4, 5, 6]. Although two-diode model ha higher accuracy than ingle-diode, ingle-diode model ha demontrated good erformance in variou article [3, 8, 9]. In Error! Reference ource not found. ingle-diode model of PV cell ha been hown. 1

2 1.1.1 Modeling the PV array Fig. 1. Single-diode PV cell equivalent circuit including erie and arallel reitance Practical PV array are comoed everal PV cell which connected to each other in erie and arallel. For achieving the MPP of PV array firtly we need to find how I-V curve obtained. Regarding to Error! Reference ource not found. the current equation of PV array are a follow [5, 8, 9]: / I v I vn K i T G G n (1) In (1), I v i the roduced current of array regardle of the amount of erie and arallel reitance. I vn i roduced current by PV array in STC 1 condition in term of amere. ΔT=T T n i difference in temerature between STC condition and the environmental condition in term of Kelvin. G n i irradiance intenity in STC condition and equal to 1000W/m 2, and G i irradiance intenity in environmental condition in term of W/m 2. / I vn R R R I cn (2) In (2), R and R P are erie and arallel reitance in term of Ohm and I cn i the nominal hort circuit current of PV array. v ocn ocn t I0 I V / R / ex V / V / a / N 1 (3) In (3), I 0 i diode aturation current in term of amere, V t i thermal voltage of diode in term of volt and α i ideal diode contant. If array ha N arallel cell, equation (1) and (3) hould be corrected a: I PV = I v, cell N and I 0 = I 0, cell N [8]. V t i thermal voltage of diode which i deendent on junction temerature. Equation (4) ecifie thi ratio. Vt k T / q (4) In (4) q i electron charge ( C), k i Boltzmann contant ( J/K), and T i temerature in term of Kelvin. Outut current of PV array i calculated via (5) uing Kirchhoff' current law [8]. V R S I V R I I I v I 0 ex 1 Vt a R (5) In (5), V i outut voltage of PV array. A can be een from (5), outut current of PV array i alway deendent on nonlinear arameter. Furthermore, erie and arallel reitance are not contant and follow nonlinear equation. According to (6) and (7) uer bound of erie reitance and lower bound of arallel reitance can be calculated [9]. R _ max Vocn Vm / I m (6) R _ m in V m / I cn I m V ocn V m / I m (7) The amount of R and R are alo calculated via (8) and (9) [8, 10]: R V V I R m m m / V I V I ex V I R / V / N / a V I P m v m 0 m m t m 0 tc (8) R R r (9) 1 - Irradiance 1000 W/m 2, AM 1.5 ectrum, module temerature 25 C 2

3 In (9), P tc i the maximum outut ower of PV array under STC condition and r = Maximum outut ower of PV array under STC condition i calculated via (10) according to V m and I m from dataheet. Ptc Vm I m (10) Equation (11) how outut ower of one-diode model PV array under any irradiance and temerature circumtance [4, 8]: P mod el / I v I0 ex V I R / Vt / N / a 1 V V I R R (11) By uing the (1-11), the P-V and I-V curve of PV array under deire irradiance and temerature condition and amount of I v, I 0, R _max, R _min, R and R are acquired. The equation hould be olved via R = 0 and R = R _min firtly and the rimary amount of P model hould be obtained. By conidering P model = P tc the (1-11) are olved in term of each other with Newton-Rahon method. In order to tet the rooed method, information of a high quality olar array which i available in Iran market (Kyocera -KC200GT) ha been acquired according to the manufacturer dataheet. The amount of I v, I vn, I 0, R _max, R, R _min, R and P model ha been hown in Table 1. according to the rooed method [1] and the P-V and I-V curve ha been hown in Fig 2, 3. Thee reult how an aroriate erformance of the rooed method. Table 1. Calculated PV array arameter uing the rooed method Etimated item Secified arameter by data heet under STC condition I v P max 200W 10% 5% I vn V m 26.3V I I m 7.61A R _ max V ocn 32.9V R I cn 8.21A R _ min K v V / C R K i A / C P model n The behavior of PV array in maximum ower oint The outut ower of a PV array i alway deendent on intenity of olar irradiance, angle of irradiance and temerature of PV array. Outut ower of the array i directly related to irradiance intenity but it ha a revere relation with temerature of PV array. Hence, high irradiance intenity and low temerature are the bet condition for generating energy from PV array. By defining a vector form of definition of voltage and current of PV array, behavior diagram of PV array i achieved regarding variou temerature and irradiance condition uing (1-11). P-V and I-V diagram of PV array (model: Kyocera KC200GT) in different irradiance condition at 25 C ha been hown in Fig.2,

4 Fig. 2. I-V diagram in different irradiance condition at 25 C Fig. 3. P-V diagram in different irradiance condition at 25 C If the intenity of olar irradiance be contant and the temerature of PV array be variable, P-V and I-V diagram of PV array will be different. Fig.4 and 5 how the behavior of above PV array in irradiance condition of 1000W/m 2 and different temerature condition. Fig. 4. P-V diagram in 1000W/m 2 and variable temerature Fig. 5. I-V diagram in 1000W/m 2 and variable temerature 3. Equation of decribing the maximum ower oint A can be een from Fig.2 and 4, the maximum ower oint (MPP) i a oint which the maximum voltage Vm and current Im i received from PV array with regard to the temerature and irradiance condition. The equation of maximum ower oint according to the irradiance intenity and temerature condition ha been hown in (12, 16) [11]:,,, Vm G T Voc G T I m G T R N Vt U (12) Ic( G,)( T,)( Im,) G T R R Vm G T U Ln Ic( G,)( T,) R R Voc G T (13) In (12), V m (G,T) i the voltage of maximum ower oint regarding temerature and irradiance condition, V OC (G,T) i the oen circuit voltage of the PV array with regard to temerature and irradiance condition, I m (G,T) i the current of maximum ower oint according to temerature and irradiance condition, R i erie reitance, N and V t are the number of erie cell in PV array and thermal voltage of diode reectively. In (13), R i the amount of arallel reitance, I c (G,T) i the hort circuit current of the PV array regarding irradiance and temerature condition. With reect to the reult from [6]: 4

5 G Ic ( G,) T Ic, n G, n Tn KiT G n (14) Voc, n ( Gn,)( Tn,) Voc G T K v T (15) Uing the (14) and (15), value of I c (G,T) and V oc (G,T) are calculated in term of intended irradiance and temerature condition. In thee equation, I c, n (G n,t n ) and V oc, n (G n,t n ) are hort circuit current and oen circuit voltage of PV array reectively in STC condition. ΔT=T T n i the difference of temerature between STC condition and intended condition in term of Kelvin, G n i irradiance intenity in STC condition which i equal to 1000W/m 2, and G i the amount of irradiance intenity under deired condition in term of w/m 2.I m i current value of PV array calculated via (17) with regard to the irradiance and temerature condition of (16) [11]: m I G, T Vm G,( T,)/( Q G,)/ T N Vt R V m G T R 1( Q,) G T/( R/) N V R R R t (16) Q ( G,)( T,)()( I c,) G T R R Voc G T Vm ( G,)( T,)( I m,) G T R Voc G T ex N Vt (17) Equation (12-17) are decribing the maximum ower oint. Thee equation are nonlinear and require reetitive olving method or intelligent algorithm to be olved. Among reetitive method, Newton-Rahon and Gau-Seidel method eem aroriate with regard to their imle imlementation. But thee method are raidly diverging, becaue equation are everal arametric. So we need a way to be reitant againt divergent to olve the equation of the maximum ower oint, and becaue equation are everal arametric it i aroriate to have a imle algorithm rooed method. The rooed numerical method to olve the equation i SUR [12, 13, 14]. In fact, thi i generalized Gau-Seidel method. The difference i that in Gau-Seidel X (k+1) = F(X k ), k = 1, 2, 3,,i laced at any te of reetition, but in SUR X (k+1) = (1-W) X k + W F(X k ), k = 1, 2, 3,, and W < 1 are laced at any te of reetition. W i common ratio in thi equation. If there i W = 0 then SUR method will be changed to Gau-Seidel method. It hould be noted that convergence in thi method deend on the right choice of the baic value of arameter of the iue [14,15]. In the following equation (12-17) for the KC200GT olar array at 800W/m 2 irradiance and 25 C temerature are olved uing the rooed method. Reult are hown in Table 2. Table 2. Reult of olving the maximum ower oint equation uing SUR method Q I V Reetition m U m A can be een from Table 2, the rooed method could converge in the firt hae of reetition. According to Table 2, the numerical method han t had high reciion due to the election of the initial value. In the cae of changing irradiance and temerature condition, initial value need to be ecified again for uing the method. The mot aroriate initial value are ecified baed on trial and error method. Beide, the rooed method i diverged in ome cae of irradiance and temerature, which make the rooed method inaroriate to be ued in ractical ytem. That i why intelligent algorithm with low convergence time are ued for deigning MPPT ytem and uually achieve good reult. 4. Deigning a MPPT ytem baed on fuzzy controller Different algorithm have been uggeted for deigning MPPT ytem. Turbulence and obervation, INC, neural network, fuzzy controller are examle of thi cae. [2,4]. The uroe of deigning MPPT i a ytem with the ability of receiving the maximum ower from PV array with uitable velocity and maximum efficiency. Among the above method, turbulence and obervation ha imle algorithm, but the quality of outut ower i omehow reduced due to the ubtantial degree of turbulence in it. Alo thi method ha higher convergence time than other method. 5

6 Algorithm baed on neural network have ignificant imoible mode. Such that adjuting thee algorithm ha become a major iue [3]. Meanwhile, fuzzy controller ha rovided high quality outut ower in addition to aroriate velocity [4,10]. Fuzzy controller ha been ued to get the maximum ower from PV array (Kyocera KC200GT). Firt in thi controller error E[K] and change in error CE[K] have been meaured uing (18) and (19), then fuzzy controller ha been deigned uing thee two arameter. In fact, in equation (18) and (19) voltage of the rior te i comared to reent voltage of the ytem. E () K Pv ()( K P1) v K I v ()( K I1) v K (18) CE()()( K E1) K E K (19) Fig. 6. Memberhi function E[K] Fig. 7. Memberhi function CE[K] Table 3. Fuzzy rule E(K)/CE(K) Fig. 8. Outut memberhi function In (18) and (19), P PV (K) and I PV (K) are ower and current of PV array reectively. Diagram of fuzzy memberhi function ha been hown in Fig.6, 7 and (8). Fuzzy rule are alo ecified in Table 3. In Fig.6, 7 the memberhi function of E[K] and CE[K] have been conidered a a triangle, o controller can have a har reond toward inut. Outut memberhi function are conidered a a Gauian, o controller can have lower noie in it outut in addition to it teady outut. The range of function E[K] and CE[K] are conidered large enough in controller. A long a the amount of E[K] and CE[K] are large, controller will be able to demontrate an aroriate erformance. The controller i configured in uch a way that in the cae of noiy condition and radiation of 200 and 1000W/m 2 at temerature 25 C ha the highet efficiency. Function 5in(100πt)+3 ha been ued a a noie function alied to PV array [10]. A boot converter, along with fuzzy controller, ha to uly voltage of V in the ower of 200w. Parameter of the boot converter have been ecified in Table 4. Power 200W Table 4. Parameter of the boot converter Voltage Switching L C 2 freq V 20 KHZ 75 H 10 F C F 6

7 In Fig.10 the rototye of Boot converter and fuzzy controller ha been hown. The voltage, current and the ower of boot converter are hown at Ohmic load at Fig.9. Fig. 9. a Fig. 9. b Fig. 9. c Fig. 9. a.voltage b. Current and c. ower of boot converter in 1000W/m 2 irradiance intenity and 25 C temerature Fig. 2. Prototye of Boot converter and fuzzy controller A can be een in Fig.9, the fuzzy controller ha had an aroriate erformance deite a noiy condition in 1000W/m 2 irradiance intenity and 25 C temerature. Thi ytem ha received the maximum ower from PV array and ha tranmitted it to the load with an aroximate accuracy of %97. The controller convergence time i about econd. The tranient of the controller ha been hown in Fig.9 too. A can be een the voltage, current and ower of PV array ha ick in rimary condition, it caue that the olar array can rovide high intant ower. In Table 4 the comarion between the rooe method and other algorithm ha been hown [2, 3, 10]. 7

8 Table 4. Comarion between the rooe method and other tyical algorithm Tye of controller Time of convergence Noie reitance Efficiency SUR large no ~ 80% Prooe fuzzy method ye ~ 97% P&O ~ no 80% - 90% INC ~ no 85% - 92% 5. Concluion Thi aer ha analyzed the develoment of a method for the mathematical modeling of PV array and finding the MPP of them. The firt objective i achieve all arameter of a PV array and P-V, I-V curve according to the roduct dataheet uing ingle-diode model equation. Then equation of the maximum ower oint have been defined and olved baed on the SUR numerical method. In term of change in intenity of olar irradiance and temerature, thi method loe it effectivene. Therefore, a fuzzy controller ha been deigned. Fuzzy controller had an accetable erformance comare to SUR method and it had convenient oeration in noiy condition. REFERENCES [1] Kyocera. (n.d.). KC200GT high efficiency multicrytal hotovoltaic module. htt:// [2] Ali Murtazaa, M. C. (January 2014). A maximum ower oint tracking technique baed on bya diode mechanim for PV array under artial hading. Energy and Building, ScienceDirect, [3] Mohamed A. Eltawil, Z. Z. (June 2013). MPPT technique fo r hotovoltaic alication. Renewable and Sutainable Energy Review, ScienceDirect, [4] Feng Wang, X. F. (2014). Analyi of Unified Outut MPPT Control in Subanel PV Converter Sytem. IEEE TRAACTIO ON POWER ELECTRONICS, VOL. 29, NO. 3. [5] Nichola D. Benavide, P. L. (July 2008). Modeling the Effect of Voltage Rile on the Power Outut of Photovoltaic Module. IEEE TRAACTIO ON INDUSTRIAL ELECTRONICS JULY 2008, VOL. 55, NO. 7. [6] B. Chitti Babu, a. S. (July 2014). A Novel Simlified Two -Diode Model of Photovoltaic (PV) Module. IEEE JOURNAL OF PHOTOVOLTAICS, VOL. 4, NO. 4. [7] Youef A. Mahmoud, W. X. (December 2013). A Parameterization Aroach for Enhancing PV Model Accuracy. IEEE TRAACTIO ON INDUSTRIAL ELECTRONICS, VOL.60, NO. 12. [8] Marcelo Gradella Villalva, J. R. (May 2009). Comrehenive Aroach to Modeling and Simulation of Photovoltaic Array. IEEE TRAACTIO ON POWER ELECTRONICS, VOL. 24, NO. 5. [9] Yun Tiam Tan, D. S. (December 2004). A Model of PV Generation Suitable for Stability An alyi. IEEE TRAACTIO ON ENERGY CONVERSION, VOL. 19, NO. 4. [10] R.Mahalakhmi, A. K. (2014). Deign of Fuzzy Logic Baed Maximum Power Point Tracking Controller for Solar Array for Cloudy Weather Condition. Conference of Power and Energy Sytem toward Sutainable Energy. [11] Abir Chatterjee, A. K. (Setember 2011). Identification of Photovoltaic Source Model. IEEE TRAACTIO ON ENERGY CONVERSION, VOL. 26, NO. 3. [12] Mohlenkam, T. Y. (December 2014). Introduction to Numerical Method and Matlab Programming fo r Engineer. Deartment of Mathematic, Ohio Univerity. [13] Won Young Yang, W. C.-S. (2005). APPLIED NUMERICAL METHODS USING MATLAB. A JOHN WILEY & SO, INC., PUBLICATION. DOI: [14] Fochi, P. (Avril 2004). Numerical method for etimating linear ec onometric model. PhD. Thei. Neuchatel, Switzerland. [15] F. Ghani, M. D. (July 2011). Numerical determination of araitic reitance of a olar cell uing the Lambert W- function. Solar Energy ciencedirect,