Renewable Energy 36 (2011) 2508e2514 Content lit available at ScienceDirect Renewable Energy journal homepage: www.elevier.com/locate/renene Implementation of photovoltaic array MPPT through fixed tep predictive control technique Panagioti E. Kakoimo *, Antonio G. Klada Laboratory of Electrical Machine and Power Electronic, Faculty of Electrical and Computer Engineering, National Technical Univerity of Athen, 9 Iroon Polytechneiou Street, GR 15780 Athen, Greece article info abtract Article hitory: Received 22 November 2010 Accepted 28 February 2011 Available online 17 March 2011 Keyword: Incremental conductance algorithm Maximum power point tracking Photovoltaic array Predictive control technique Thi paper propoe the implementation of Photovoltaic (PV) array Maximum Power Point Tracker (MPPT) through Fixed Step-Model Predictive Controller (FS MPC). The propoed controller cheme i baed on the modified INcremental Conductance (INC) algorithm combined with the two-tep horizon FS MPC. The current baed INC algorithm i ubject to major modification in order to be capable of real time interaction between the MPPT and the controller obtaining ufficient information in one ampling time. The developed technique ha been incorporated into a model for the overall imulation of the performance of a PV array for olar energy exploitation and i compared to the conventional approach under olar radiation variation improving PV ytem utilization efficiency and enabling to optimize ytem performance. Thi tudy alo illutrate the effectivene of the propoed controller cheme under variou ky condition with a imulation model employing real olar radiation data. Ó 2011 Elevier Ltd. All right reerved. 1. Introduction Produced power from photovoltaic (PV) ytem can be delivered to the load by the implementation of a DC converter booting the level of the olar panel output voltage and attaining maximum energy extraction. Forcing the PV ytem to operate at the Maximum Power Point (MPP) located at the knee of the IeV characteritic contitute the main target of the controller operating the converter witch. A Maximum Power Point Tracker (MPPT) i alo required in order to track the MPP and upply the controller with the appropriate reference input. One of the mot widely ued MPPT i the INcremental Conductance (INC) algorithm impoing the reference output to the controller and achieving operation at the maximum power condition [1,2]. Conventional approach of uch an application demand the implementation of a proportional-integral (PI) controller characterized by two main drawback, the low tranient repone and the poible undeirable ocillation around the MPP. Specifically, PI controller require ufficient time for the ytem to reach teady tate operation increaing the time interval between two ucceive reference output from the MPPT and hence; deteriorating dynamic performance [3e5]. The interet in thi area i ignificantly growing in reearcher communitie focuing mainly on the MPPT efficiency improvement. * Correponding author. E-mail addre: panokak@gmail.com (P.E. Kakoimo). Fuzzy model-baed approach [6e8], genetic algorithm [9] and full gradient-baed technique have been enlited to improve MPPT performance obtaining ufficient information in one ampling time and thu peeding up MPPT operation [10]. However, the poibilitie of today microproceor facilitate alo the implementation of efficient control technique, achieving ignificant improvement almot independently of the adopted MPPT algorithm [11]. Such a control technique i the Model Predictive Controller (MPC) employed to olve a finite-horizon optimal control problem at each ampling intant and obtain control action for both the preent time and a future period [12,13]. MPC preent everal advantage over the conventional control technique uch a eay implementation and multivariable cae conideration [14,15], and i expected to improve PV ytem utilization efficiency under continuou change in olar radiation overcoming diturbance and uncertaintie [16]. The implementation of a PV array MPPT uing MPC combine two key of vital importance, peed and reliability, avoiding unacceptable ocillation depite the increaed peed. The mot obviou limitation in thee application are the required computational effort and the quality of the microproceor [12]. In thi paper the implementation of a PV array MPPT through Fixed Step (FS) MPC i preented for firt time. The two-tep horizon predictive control technique combined with the modified INC algorithm i initially analyzed. A particular methodology i then introduced propoing real time interaction between the MPPT 0960-1481/$ e ee front matter Ó 2011 Elevier Ltd. All right reerved. doi:10.1016/j.renene.2011.02.021
P.E. Kakoimo, A.G. Klada / Renewable Energy 36 (2011) 2508e2514 2509 and the controller improving ytem tranient repone under rapid change in olar radiation and i compared with the conventional approach. Moreover, thi tudy illutrate the effectivene of the propoed controller employing real olar radiation data for variou ky condition. Reult have hown that the overall ytem can attain high power converion efficiency. 2. Overall ytem configuration The overall ytem, a hown in Fig. 1, conit of the main following component: the PV array (A.), which generate power directly from olar radiation, the boot converter (B.), whoe witch i operated by the control cheme of the MPPT (F.) and the MPC (G.). Due to the fact that the firt priority of the boot converter control i MPP tracking, variation may appear in ytem output voltage (V C ) and therefore, an inverter AC to DC (D.) i then applied to provide energy to the network with table voltage. Control trategy in thi tudy i baed on the DC tep-up converter booting the level of the PV ytem output voltage, a well a determining the factor of maximum power exploitation. PV ytem output voltage (v PV ) and current (i PV ) meaurement are formed a input for the MPPT and the predictive controller. The MPPT reference output current (i*) and the converter output voltage (v C ) are alo deignated a input for the MPC in order to obtain ufficient information in one ampling time and operate the boot converter witch with the binary output of. The witch condition i determined by the value of the binary variable of, which i conidered a cloed when i equal to zero; while on the other cae i conidered a open. 3. Propoed control ytem analyi MPPT technique can be divided into three main categorie: lookup table method, perturbation and obervation and, computational method. For the purpoe of comparion and owing to it proven good performance, the INC algorithm claified in perturbation and obervation method, i modified and combined with the propoed predictive control technique and hence; i briefly introduced. 3.1. Modified INC algorithm Becaue of being eaily implemented the INC algorithm i the traditionally ued MPPT technique. The main drawback againt contemporary method are that at teady tate operation, the reference output varie between neighboring value and that under tranient phenomena, i not capable of tracking rapidly the MPP. Fig. 2 how the block cheme of the modified INC algorithm, where Fig. 2. Block cheme of the modified INC algorithm impoing the reference current i* to the controller. time k 1 correpond at the previou ampling time t 1, while k indicate the real time meaured value. The INC algorithm i ubject to two major modification. Firtly, the algorithm i modified to impoe the reference current to the controller (current baed) and econdly, the reference output i defined a the increment of the PV ytem current meaurement (i PV(k) ), and not a the increment of the previou ampling time reference current (i* (k 1) ). The latter modification make the ytem capable of deciding rapidly the right direction in PeV curve and following the MPP with larger tep epecially during variation. Tracking the MPP i baed on the derivative of the PV ytem output power (p PV ) with repect to the current (i PV ). The lope at the MPP i equal to zero determining the deirable operation point: vp PV ðv PV ; i PV Þ vi PV ¼ 0/v PV þ i PV, dv PV di PV ¼ 0 (1) 3.2. Predictive controller implementation The main concept of the FS MPC technique i the prediction of the future behavior of the controlled variable. The criterion of the control method i expreed a a cot function to be minimized. Fig. 3 how the DCeDC boot converter equivalent circuit for the two condition of the ideal witch. When the witch i conidered a open, the boot converter operation can be decribed by the well-known ytem of equation a follow: Fig. 1. Simplified chematic of the overall grid connected PV ytem configuration implementing MPPT through MPC technique. di PV ¼ 1 L,i PV þ 1 L,v PV (2)
2510 P.E. Kakoimo, A.G. Klada / Renewable Energy 36 (2011) 2508e2514 Fig. 3. Boot converter equivalent circuit for the two tate of the ideal witch. (a) Open witch, ¼ 1. (b) Cloed witch, ¼ 0. dv C ¼ 1 C,i PV þ 1 R,C,v c (3) In cae of the cloed witch, the firt order term vanih and the previou equation ytem i of the following form: di PV dv C ¼ 1 L,v PV (4) ¼ 1 R,C,v C (5) The dicrete time ytem of equation can derive from (1)e(4) conidering the ampling frequency T, when the witch i open (5) and (6), or cloed (7) and (8). i PVðkþ1Þ ¼ i PVðkÞ T L,v CðkÞ þ T L,v PVðkÞ (6) v Cðkþ1Þ ¼ T C,i PVðkÞ þ 1 T,v R,C CðkÞ (7) i PVðkþ1Þ ¼ i PVðkÞ þ T L,v PVðkÞ (8) v Cðkþ1Þ ¼ 1 T,v R,C CðkÞ (9) The aforementioned dicrete time equation ytem can be expreed in matrix form a: 2 3 2 4 i PVðkþ1Þ v Cðkþ1Þ 3 5 ¼ 6 4 1, T L, T C 1 T R,C 7 5, 2 3 2 4 i PVðkÞ 5 6 þ 4 v CðkÞ T L 0 3 7 5,v PVðkÞ (10) Behavior of the controlled variable i PV and v C can now be predicted for the next ampling intant in order to obtain control action for both the preent time and a future period. One-tep horizon predictive controller input meaured value of i PV, v PV, and v C etimating future behavior of the controlled variable baed on the evaluation of a cot function. Evaluating the choen cot function two time, for each witch condition, the value of the binary variable can be computed in order of the predictive controller to decide which one direction in PeV curve mut be followed o a to atify the applied criteria a hown in Fig. 4. The determination of the cot function play a key role in MPC behavior contraining the deviation from the deirable value (i* and v*) andcanbeexpreeda: J¼n n¼0;1 ¼ w A, v C;¼nðkþ1Þ v* þ w B, i PV;¼nðkþ1Þ i* (11) where parameter w A and w B are in [1/V] and [1/A] unit, repectively. Fig. 4. Block cheme of the MPC technique operating witch tate. Furthermore, MPC technique provide the capacity of predicting ytem behavior for a future period of n-ampling intant obtaining neceary control action at preent time. Conidering n-tep horizon MPC i expected to extend ytem capability of avoiding undeirable ocillation at time t þ n becaue of a variation happened at time t, providing robutne to ytem behavior. Dicrete time ytem of equation for the n-tep horizon MPC i the following for the two witch condition, repectively a (6)e(9): i PVðkþnþ1Þ ¼ i PVðkþnÞ, T L,v CðkþnÞ þ T L,v PVðkþnÞ (12) v Cðkþnþ1Þ ¼, T C,i PVðkþnÞ þ 1 T,v R,C CðkþnÞ (13) In the cae of the two-tep horizon MPC the cot function i required to be evaluated four time, for each one combination for the binary variable at the repective ampling time t þ 1 and t þ 2 and ha the following form: Fig. 5. Time equence of the interaction between the controller (MPPT, predictive controller) and the controlled ytem. Time k correpond to the meaured value at the pecific ampling intant.
P.E. Kakoimo, A.G. Klada / Renewable Energy 36 (2011) 2508e2514 2511 Fig. 8. Equivalent circuit of the olar cell depending on temperature (T) and irradiance value (W/m 2 ). Fig. 6. Schematic diagram of the MPC proce for the two-tep horizon prediction. The dotted line correpond to MPPT output, while the black line correpond to the finally performed action. ¼m ¼ w C, v C;¼mðkþ2Þ v* þw D, i PV;¼mðkþ2Þ i* J n¼0;1&m¼0;1 þj ¼n (14) In order to calculate i PV at time k þ 2 from (12) and (13), there i need to etimate v PV at time k þ 1 for the reultant i PV at the ame ampling intant. The output voltage of the PV ytem can be etimated by uing a implified equivalent equation decribing PV ytem behavior. The PV output voltage (v PV ) doe not affect ignificantly controller deciion becaue of the involvement in the equation ytem for the two witch operation. Avoiding etimation error in the output PV ytem voltage the cot function at time k þ 1 can play a key role. In (14) the cot function J ¼ n contitute a regulator factor depending on the etimation error of the prediction for the output PV voltage at time k þ 1. Fig. 5 depict the proce of the propoed control cheme for the two-tep horizon MPC. MPPT at time k compare the tored value for time k 1 with the meaured one aving the recent value, and concurrently, yield and impoe the deirable reference current to the MPC. The latter input the meaured value and the reference current forcing the PV ytem to operate with the deirable current at time k þ 2. Fig. 6 depict the chematic diagram of the MPC proce for the two-tep horizon prediction conidering only one controlled variable. The dotted line correpond to the MPPToutput, which contitute the reference current for the controller to follow. At the firt tep the onetep horizon MPC had to decide between 0 and 1, whoe difference may not be ignificant. The two-tep horizon MPC decide among 00, 01, 10 and 11 evaluating the four correponding cot function and conidering the cot function of the previou tep at time k þ 1. The black line in Fig. 6 correpond to the finally performed action indicating the witch tate for each tep of the prediction. Proper configuration of the parameter in the cot function for the two cot function lead to a robut and tiff ytem, independent from the etimation error of the non-controlled variable. Fig. 7 how the evaluation of the cot function for the two witch condition and for the two tep of the MPC. Faded color repreent the combination whoe cot function value i higher for the econd tep than that of the combination with the black color. Conidering only the evaluation of the cot function for the econd tep then the choen combination may not be the mot appropriate depending on the etimation error of the non-controlled variable. Therefore, a combined cot function a in (14) involving the two tep can provide better ytem repone. Dahed line in Fig. 7 correpond to the cae where the evaluation of the cot function for the econd tep i taken into conideration with le ignificance than that of the firt tep. Depending on the difference between the two evaluated cot function for the firt tep the reulting witch condition may differ. 4. Reult and dicuion Conidering a typical PV ytem configuration the propoed control technique ha been teted under abrupt change in olar irradiance. In order to illutrate the effectivene of the introduced control cheme real olar radiation data have been employed for a day with the poradic preence of cloud. 4.1. PV ytem configuration Serie and parallel combination of the ideal olar cell model compoe the PV array, whoe baic mathematical equation are briefly introduced. The equivalent circuit conit of one current ource, two exponential diode and two reitor, one i parallel and the other one i in erie with the generated current a hown in Fig. 8. The output current of the olar cell can be computed by [17e19]: vþi,r vþi,r i ¼ i ph i o1, e a 1,v t 1 i o2, e a 2,v t 1 vþi,r (15) R p Fig. 7. The four combination of the witch condition for the two-tep horizon MPC and the evaluation of the repective cot function. Fig. 9. IeV characteritic of the PV ytem for two different irradiance level with marked repective maximum power point A and B for 1000 and 1200 W/m 2, repectively.
2512 P.E. Kakoimo, A.G. Klada / Renewable Energy 36 (2011) 2508e2514 Fig. 10. PeI characteritic of the PV ytem for two different irradiance level. where i ph i the photovoltaic current generated by olar irradiation, i o1, i o2 are the D1, D2 revere aturation current, a 1, a 2 are the diode ideal contant, v t i the thermal voltage and v i the terminal voltage. In cae of the parallel and erie combination of olar cell, i ph, i o1, i o2 can be multiplied by N P and v t by N S, where N P and N S are the parallel and erie connection of cell, repectively. From the evaluation of (15), IeV characteritic of the examined PV ytem configuration for two different value of irradiance level, 1000 and 1200 W/m 2, can be obtained a hown in Fig. 9. With letter A i marked the MPP of the IeV characteritic for olar radiation equal to 1000 W/m 2. Under an abrupt change in olar radiation from 1000 to 1200 W/m 2 the ytem i expected to operate, after the MPPT contribution, at point C, which i the MPP for 1200 W/m 2 olar irradiance. PV ytem current (i PV ) at point A and C i 18.6 and 22.3 A, repectively. Fig. 10 how the PeI characteritic of the PV ytem for the two aforementioned olar radiation level, where the repective MPP can be eaily oberved. Fig. 11. MPPT output reference current (i*) under irradiance variation from 1000 to 1200 W/m 2 for each different approach. (a) Conventional approach. (b) Simplified MPC. (c) Propoed MPC. 4.2. Invetigation among the preented approache In the propoed control methodology, MPPT reference output contitute the real time input in the predictive controller. The controller provided with the computed reference output (i*)by the MPPT operate uitably the boot converter witch conidering two-tep horizon prediction in one ampling period. Contrarily, the PI controller demand ufficient time for the ytem to reach teady tate operation, increaing time interval between two ucceive reference output from the MPPT and thu deteriorating ytem dynamic performance under abrupt and continuou variation. In order to illutrate the benefit from the propoed control technique three different approache are examined under olar irradiance variation: conventional approach (PI controller interact with the traditional MPPT requiring ufficient time interval), the implified MPC (MPC i configured a the conventional approach) and the propoed MPC (real time interaction between the modified MPPT and the MPC). Table 1 ummarize the main characteritic of the compared methodologie. Table 1 Characteritic of the preented approache. Conventional approach Simplified MPC Propoed MPC Controller type PI controller MPC MPC Interaction with MPPT t interval t interval T Reference current i* (t þ 1) i* (t) þ i inc i* (t) þ i inc i PV(t) þ i inc MPPT increment i inc Sytem parameter Same Fig. 12. PV ytem output current (i PV ) under irradiance variation from 1000 to 1200 W/m 2 for each different approach. (a) Conventional approach. (b) Simplified MPC. (c) Propoed MPC.
P.E. Kakoimo, A.G. Klada / Renewable Energy 36 (2011) 2508e2514 2513 4.2.1. Sytem behavior under abrupt olar irradiance variation Preented approache have been teted under an abrupt increae of olar radiation by 20% examining tranient behavior and maximum energy exploitation. Fig. 11 how MPPT output reference (i*) for the three different approache. The conventional technique and the implified MPC are not capable of following rapidly the change in olar radiation becaue of the required time interval for the ytem to remain table. Furthermore, ocillation around the MPP can be oberved for thee two approache, but not with the ame intenity for the implified MPC. Conventional approach for the ame ytem etting preent ocillation due to ytem delay to reach teady tate operation. PI gain adopted have been obtained from different imulation carried out a the mot appropriate for thi ytem configuration. In contrat, the propoed MPC due to the real time interaction with the MPPT preent advantage over the conventional technique tracking the MPP with ignificantly increaed peed and thu aving power energy. Fig. 12 how that the output PV ytem current (i PV ) follow accurately the reference current for the method involved MPC. The mathematical character of the MPC enable ytem to behave the ame way under any tranient phenomenon tracking accurately the MPP, and doe not neceitate the reevaluation of control ytem parameter. Contrarily, through PI controller it i poible to preent teady tate error, due to the fact that the PI gain cannot guarantee ame tranient repone under different ytem condition keeping concurrently increaed peed. Fig. 13 depict the produced power from the PV array and the total ytem output power. Total power diipation comparing the three control cheme i higher for the conventional and implified Fig. 14. Meaured time erie of the olar irradiance level at NTUA campu (Augut 2010). approach conidering the time required for the ytem to reach MPP. However, method involving MPC, the implified and the propoed method, a already mentioned, are not affected ignificantly from tranient phenomena; though converge preciely and almot independently at the MPP. Conidering the examined variation ytem achieve high level of MPPT efficiency of about 99.86%, while the conventional approach efficiency i of 99.36%, without conidering power diipation during tranient repone. The total amount of energy i ignificant auming continuou operation all over the year. 4.2.2. Solar irradiance variation under variou ky condition Previou analyi ha hown that the propoed MPC i featured by it capability of achieving both better tranient repone and higher PV ytem utilization. In order to illutrate the effectivene of the propoed control cheme real olar radiation data meaured at the National Technical Univerity Campu in Athen (NTUA), have been employed. Meaurement of olar irradiance level under three different weather condition have been carried out. Fig. 14 depict the meaured time erie of the olar radiation for a unny day without the preence of cloud and, therefore, the differentiation of the three preented control cheme i marginal. Fig. 15 and 16 how the meaured time erie of the olar irradiance level under the poradic preence of cloud. Under cloudy ky condition olar radiation fluctuate with abrupt change neceitating the MPPT control to be reliable and accurate overcoming uch difficultie and diturbance and increaing ytem converion efficiency. Fig. 16 refer to a winter day where the olar irradiance level are ignificantly low. The reult derived from the olar radiation data reveal that the propoed MPC achieve converion ytem efficiency of about Fig. 13. Overall ytem generated power under irradiance variation from 1000 to 1200 W/m 2. (a) Conventional approach. (b) Simplified MPC. (c) Propoed MPC. Fig. 15. Meaured time erie of the olar irradiance level at NTUA campu (April 2010).
2514 P.E. Kakoimo, A.G. Klada / Renewable Energy 36 (2011) 2508e2514 Reference Fig. 16. Meaured time erie of the olar irradiance level at NTUA campu (December 2009). 95.7% compared to the maximum energy exploitation, while the implified approach attain lower efficiency of 94.1% conidering a cloudy day a the olar radiation data employed. 5. Concluion Thi tudy focue on the controller technique in order to attain maximum energy exploitation by applying modification to one of the mot widely ued MPPT, thu a PV array MPPT through predictive control technique ha been developed. The capacity of the MPC of being upplied with the reference current by the MPPT at one ampling time enable high tranient repone under abrupt change in olar irradiance, preented uually under cloudy ky condition. Solar irradiance varie continuouly and abruptly under uch ky condition neceitating method for harneing maximum energy conidering operation all over the year. Propoed control cheme efficiency ha been illutrated by employing real olar radiation data into the imulation model. Appendix. Table S1 ummarize main pecification for the examined PV ytem configuration. Table S1 Main ytem pecification. Inductance, L (mh) 20 Capacitance, C (mf) 50 MPPT increment (ma) 100 Time interval (MPPT) (m) 0.5 Time interval (MPC) (m) 50 Sampling time, T (m) 50 [1] Houamo I, Locment F, Sechilariu M. Maximum power tracking for photovoltaic power ytem: development and experimental comparion of two algorithm. Renewable Energy 2010;35:2381e7. [2] Maoum M, Dehbonei H, Fuch E. Theoretical and experimental analye of photovoltaic ytem with voltage and current-baed maximum power-point tracking. IEEE Tranaction On Energy Converion 2002;17: 514e22. [3] Ropp ME, Gonzalez S. Development of a MATLAB/Simulink model of a inglephae grid-connected photovoltaic ytem. IEEE Tranaction On Energy Converion 2009;24:195e202. [4] Pandey A, Dagupta N, Mukerjee AK. High-performance algorithm for drift avoidance and fat tracking in olar MPPT ytem. IEEE Tranaction On Energy Converion 2008;23:681e9. [5] Pan C, Juan Y. A novel enorle MPPT controller for a high-efficiency microcale wind power generation ytem. IEEE Tranaction On Energy Converion 2010;25:207e16. [6] Chiu C. TeS fuzzy maximum power point tracking control of olar power generation ytem. IEEE Tranaction On Energy Converion; 2010: 1e10. [7] Gounden N, Annpeter S, Nallandula H, Krithiga S. Fuzzy logic controller with MPPT uing line-commutated inverter for three-phae grid-connected photovoltaic ytem. Renewable Energy 2009;34:909e15. [8] Larbe C, Aït Cheikh S, Obeidi T, Zerguerra A. Genetic algorithm optimized fuzzy logic control for the maximum power point tracking in photovoltaic ytem. Renewable Energy 2009;34:2093e100. [9] Chen L, Tai C, Lin Y, Lai Y. A biological warm chaing algorithm for tracking the PV maximum power point. IEEE Tranaction On Energy Converion 2010;25:484e93. [10] Syafaruddin, Karatepeb E, Hiyamaa T. Polar coordinated fuzzy controller baed real-time maximum-power point control of photovoltaic ytem. Renewable Energy 2009;34:2597e606. [11] Sala V, Alono-Abellá M, Chenlo F, Olía E. Analyi of the maximum power point tracking in the photovoltaic grid inverter of 5 kw. Renewable Energy 2009;34:2366e72. [12] Corté P, Kazmierkowki MP, Kennel RM, Quevedo DE, Rodríguez J. Predictive control in power electronic and drive. IEEE Tranaction On Indutrial Electronic 2008;55:4312e24. [13] Kouro S, Corte P, Varga R, Ammann U, Rodriguez J. Model predictive controlda imple and powerful method to control power converter. IEEE Tranaction On Indutrial Electronic 2009;56:1826e38. [14] Khalid M, Savkin A. A model predictive control approach to the problem of wind power moothing with controlled battery torage. Renewable Energy 2010;35:1520e6. [15] Hua C, Wu C, Chuang C. A digital predictive current control with improved ampled inductor current for cacaded inverter. IEEE Tranaction On Indutrial Electronic 2009;56:1718e26. [16] Teng T-P, Nieh H-M, Chen J-J, Lu Y-C. Reearch and development of maximum power tranfer tracking ytem for olar cell unit by matching impedance. Renewable Energy 2010;35:845e51. [17] Villalva MG, Gazoli JR, Filho ER. Comprehenive approach to modeling and imulation of photovoltaic array. IEEE Tranaction On Power Electronic 2009;24:1198e208. [18] Andrade Da Cota B, Lemo J. An adaptive temperature control law for a olar furnace. Control Engineering Practice 2009;17:1157e73. [19] Armtrong S, Hurley WG. A new methodology to optimie olar energy extraction under cloudy condition. Renewable Energy 2010;35:780e7.