Abstract—This paper presents a Chebyshev Functional Link Neural Network (CFLNN) based model for photovoltaic modules. There are two basic approaches to build a model – use an analyt-ical modeling technique or use an Artificial Neural Network (ANN) based method. However, both the analytical modeling technique and the traditional Multilayer Perceptron (MLP) mod-el have some disadvantages. For example, in the analytical model, the influence of irradiance and temperature on some parameters of the photovoltaic module, such as the parallel and series resis-tance and other uncertainty factors, are not taken into considera-tion. In the case of the multilayer neural network model, there is a large computational complexity in training the network and in its implementation. In order to overcome these advantages, we propose a CFLNN based model for solar modules. The proposed model not only reduces the complexity of the network due to the absence of hidden layers in the network configuration, but also shows better accuracy over the analytical modeling method. In the experimental section, the operating current predicted by CFLNN is compared with the outputs from other two modeling methods - MLP and the two-diode model. Finally, verification is performed using experimental datasets. The results show that the CFLNN modeling method provides better prediction of the out-put current compared to the analytical model and has a reduced computational complexity than the traditional MLP model.

Keywords- Chebyshev functional link neural network; photovoltaic arrays; multilayer neural network; Two-diode model

I. INTRODUCTION Due to the rapid decrease in the cost of photovoltaic (PV)

cells and advances in power electronic devices, solar energy, being an inexhaustible and clean renewable energy source, has gained in popularity. In our daily life, solar energy can be used in several forms - solar photovoltaic, solar heat, solar thermal electricity, and solar fuels. PV systems mainly consist of PV arrays / modules / cells, energy storage devices, converters, and the AC or DC electrical load. In order to increase the efficiency of the system, maximum power point tracking (MPPT) equip-ment is usually inserted between the converters and PV arrays to maximize the power output. Since the major challenge for PV systems is the highly nonlinear characteristics of current output versus voltage output, it is very important to build an accurate PV model when investigating the PV array characte-ristics, predicting the power output, and sizing the PV system.

Currently, the modeling methods for PV modules can be divided into two categories – the analytical method and the artificial intelligence method. The analytical methods are main-

ly dependant on the equivalent electrical circuit with five or seven-parameter models [1, 2]. If the solar module is described as a single-diode (1-D) model, it has five parameters which are the light-generated current, series resistance, parallel resistance, reverse saturation current of diode, and diode quality factor. If a two-diode model is used, an additional two parameters (intro-duced to compensate for the recombination loss in the deple-tion region) - reverse saturation current and diode quality fac-tor, are required to form a seven-parameter model. Usually, these five or seven parameters are calculated based on the rat-ings of the PV modules provided by the manufacturer under standard conditions, which rarely occur outdoors. A novel ap-proach to extract the five parameters under real environmental conditions based on the datasheet provided by the PV module manufacturers is given in [2]. Moreover, a three-diode based model is also proposed in [3], but it is rarely used because of its computational complexity. However, all of these models as-sume that the series resistance and the parallel resistance are constant. Actually, it is observed that all these parameters change with temperature and irradiance conditions [4]. For instance, when there is low irradiance or irradiance changes dramatically such as in cloudy days or late in the afternoon, the value of the parallel resistance ( ) is very low, which causes performance degradation due to the power reductions. There-fore, it is unrealistic to make this assumption in complex cir-cumstances. In addition, since there are many factors influen-cing the power output of the PV array systems such as solar cell/module mismatch losses, the failure of the MPPT algo-rithm under shading effects, ohmic losses, angular and spectral losses, and bad weather conditions, etc., it is difficult to model the characteristics of solar modules using the analytical method accurately. Moreover, they are sensitive to the initial estimation of these parameters and are computationally expensive.

Compared to analytical methods, artificial intelligence me-thods, such as the artificial neural network (ANN) [4, 5, 6], give better performance with no required knowledge of the internal system, and provide a compact solution for multivaria-ble problems. Usually, there are two methods for applying ANN technique in PV arrays. The first one is to estimate the classical five or seven circuit parameters initially and then ap-ply the predicted results into the traditional five or seven-parameter models. For example, Karatepe et al. [4] used the ANN based technique to predict the electrical circuit parame-ters and then built the one diode model for PV modules with these parameters. The second method is to estimate the current or voltage for the PV array directly from the irradiance and temperature conditions. For instance, Celik et al. [7, 8] used the generalized regression neural network to predict the operating current of mono-crystalline PV modules.

Chebyshev Functional Link Neural Network-Based Modeling and Experimental Verification for Photovoltaic Arrays

Lian Lian Jiang1, Douglas L. Maskell1 and Jagdish C. Patra2

1School of Computer Engineering, Nanyang Technological University,Singapore. 2Faculty of Engineering & Industrial Sciences, Swinburne Univer-sity of Technology, Hawthorn, Australia. Email: ljiang2@e.ntu.edu.sg, asdouglas@ntu.edu.sg, JPatra@swin.edu.au.

U.S. Government work not protected by U.S. copyright

WCCI 2012 IEEE World Congress on Computational Intelligence June, 10-15, 2012 - Brisbane, Australia IJCNN

In this paper, the latter method is used to estimate the cur-rent output from the PV array. In comparison with the analyti-cal and classical modeling methods, the ANN based method provides better prediction performance because it takes second order effects into account [9]. Most of the ANN methods use a Multilayer Perceptron (MLP) with Back-Propogation (BP) algorithm to predict the current or power output for the PV module or array. However, MLP has some disadvantages such as a higher computational complexity and a longer training time. Since the Functional Link ANN (FLANN) features fast convergence speed, simplicity, and good accuracy, it is very suitable for applications of nonlinear system modeling [10, 11]. The Chebyshev Functional Link Neural Network (CFLNN) used in this paper is a class of FLANN, which is computationally more efficient than traditional MLP networks [10, 11]. In our previous work [12, 13, 14], we have success-fully used the FLANN in modeling electrical characteristics of dual junction solar cells, the estimation of external quantum efficiency, and the MPPT problem. In this paper, we use CFLNN to predict the current output of PV modules by input-ting irradiance, temperature of the modules and measured vol-tage under real environmental conditions. In order to verify the proposed CFLNN based model, the current output of PV mod-ules is predicted with the experimental data measured from the PV panel setup installed on the building rooftop. The results indicate that the CFLNN based model shows advantages over MLP in both training time and computational complexity, and has better accuracy than the two-diode analytical modeling method.

The remaining portion of the article is organized as follows: In Section II, the traditional two-diode model of solar cells is introduced. The principle of CFLNN and its structure for appli-cation in modeling of PV modules are explained in Section III. The experimental setup for the electrical data acquisition and the process of network training are presented in Section IV. In Section V proposed CFLNN-based model for PV modules is verified using experimental data, and the results of current pre-diction and comparative analysis are presented. Finally, con-clusions and our future work directions are given in Section VI.

II. TRADITIONAL TWO-DIODE MODEL FOR SOLAR MODULES

In this section, the two-diode model for solar module and the commercial specifications of the PV module that we used in this paper are presented. A PV cell consists of a silicon P-N junction, which converts solar energy directly into electricity by photovoltaic effect. Multiple cells/modules are connected together in series or in parallel to get higher voltage or higher current. Assemblies of solar cells make solar modules, and multiple modules in series or parallel make PV arrays. In order to compare the performance of current prediction by the pro-posed CFLNN based model, the two-diode model is described below.

We consider that the PV array contains Nss modules in series, Npp modules in parallel, and Ns solar cells in series in each PV module. The equivalent electrical circuit of two-diode solar cell model is shown in Fig.1 [1]. From the equivalent circuit, the output current of the PV is given as

pddpv IIIII −−−= 21 , (1)

where I is the current output of PV array, Ipv is the light gener-ated current, Id1 and Id2 is the current going through two diodes and Ip is the leakage current through parallel resistant Rp. By substituting each element of (1) with corresponding electrical expressions, (1) can be re-written as

p

s

sst

spp

sst

spppppv

RIRV

NVaIRVNI

NVaIRVNINII

λλλ

λ

+−−+−

−+−=

]1) [exp(

]1)[exp(

202

101

, (2)

where V is the output voltage of PV array, λ = Nss / Npp, Rs and Rp are the series and parallel resistances, respectively, Vt is the thermal voltage of the two diodes (Vt = Ns k T / q), k is the Boltzman constant, q is the electron charge, and a1, a2 are the diode ideal constants. The light generated current is given by

STC

STCscSTCpvpv GGTTKII ))(( _ −+= , (3)

where Ipv_STC represents the light generated current under stan-dard test conditions (STC) with temperature TSTC = 25 , and irradiance GSTC = 1000 (w/m2), and the constant Ksc is the short circuit current coefficient. The reverse saturation current of the diode is given by

1)/)exp(( _

_0201 −Δ+

Δ+==

tocSTCoc

scSTCsc

VTKVTKI

II , (4)

where the constant Koc is the open circuit voltage coefficient, Isc_STC is the short circuit current under STC, and Voc_STC is the open circuit voltage under STC. The equations (1) - (4) are adopted from [1, 15]. In fact, (1) can be represented as an im-plicit and nonlinear function, I = f (I, V) with seven unknown parameters (a1, a2, Ipv, I01, I02, Rs, and Rp). Thus, after obtaining the values of these seven parameters, this nonlinear equation can be solved using standard Newton-Raphson method for a given voltage value.

Therefore, the process of generating model is implemented as follows: with the PV module parameters provided by manu-factures, Ipv, I01, I02, Rs, and Rp are calculated using the method with fast convergence given in [1]. Here, we choose the value of diode ideal constants a1 and a2 to be 1 and 1.2 [1], respec-tively. Therefore, the calculated values of Rs and Rp are 1.12 Ω and 1100.43 Ω, respectively. Since seven parameters, a1, a2, Ipv, I01, I02, Rs, and Rp, are obtained, the current output of PV mod-ule is computed with the measured voltage values.

Fig. 1. Equivalent electrical circuit of two-diode solar cell model with seven parameters

III. CHEBYSHEV NEURAL NETWORK BASED MODELING FOR SOLAR MODULES

In this section, the detailed principle of CFLNN and the proposed model for predicting the current output for PV arrays are addressed.

Before explaining the CFLNN, here the theory of FLANN [10] is introduced first. The FLANN consists of a Functional Expansion Block (FEB) and a single-layer perceptron as shown in Fig. 2. The FEB is used to transfer the input va-riables from a lower dimension to a higher dimension. Many method can be used as this expansion function, such as trigo-nometric polynomials, Chebyshev polynomials, and other or-thogonal polynomials [10, 11]. Since Chebyshev polynomials have advantages of easy to compute, better prediction capa-bility in many applications, efficient and flexible hardware implementation, they are chosen in this paper. After being transferred into a higher dimension by FEB, the expanded inputs are directly connected to the output layer, which is acti-vated by the hyperbolic tangent transfer function. During the training process, the weights between the expanded inputs and the output layer are adjusted until the cost function reaches a particular limit. Here, the Levenberg - Marquardt (LM) algo-rithm is used to train the network. Since there is no hidden layer in the network, the computational complexity is largely reduced compared to MLP network. In order to illustrate the structure of CFLNN and the process of training network, let the input vector, X, be given by

,],,[ 321 ′= xxxX (5)

where x1, x2, x3 are the attributes of input samples, and [.]' de-notes transposition. After going through the FEB, the inputs are expanded by the following Chebyshev polynomials [10]

,188)(,34)(

,12)(,)(

,1)(

244

33

22

1

0

+−==

−===

ppp

ppp

pp

pp

p

xxxLx-xxL

xxLxxL

xL

(6)

where xp is one of the attributes for the network inputs to the FEB, -1< xp <1, p = {1, 2, 3}. The remaining higher dimension Chebyshev polynomials can be generated using recursive for-mula given by ),()(2)( 21 pnpnppn xLxLxxL −− −= (7) where n is the dimension of the expanded space for each input attribute. If the input vector has p attributes, then the dimension of the expanded space, N, is n × p + 1. The constant, 1, means the bias in the expanded space.

The traditional MLP network is shown in Fig. 3. The out-puts of the jth neuron from the hidden layer are calculated as

∑=

×=N

iijijj xWGx

1

)(φ , (8)

where Gj is the activation function for the jth node in the hid-den layer, Wji is the weight connecting the jth hidden node with

the ith input xi. In the output layer, the kth network output is calculated as

))((1∑

=

×=H

jjkjkk xWGY φ , (9)

where Gk is the activation function for the kth node in the out-put layer, Wkj is the weight connecting kth output node with the jth node in hidden layer. Therefore, with these space transfor-mations, the input space {x1, x2, x3., xN} is transferred into another space, {φ1(x), φ2(x), φ3(x), ..., φH(x)}, with a higher di-mension. Thus, the linear discrimination becomes easier in this transformed space with a higher dimension than the original input space. Instead of learning to find the learnable arbitrary functions in the hidden layer, the Chebyshev polynomials are used to expand the original inputs.

∑

Fig. 2. Basic configuration of the FLANN

x1

x2

W1

Y

+1(bias)

...

+1(bias)

Hiddenlayers

Outputlayer

x3 ...

+1(bias)

W3

...

Inputs

Fig. 3. Structure of the MLP

Based on various trials on different functional expansion of Chebyshev polynomials, we found that adding some cross products of expanded inputs gives better performance. For ex-ample, we added some products of separate component, x1x2, x1x3, x2x3, x1x2x3. The detailed 2-stage functional expansion block is shown in Fig. 11 in the Appendix. The total number of the expanded dimension was chosen to be 29 (29 = 7×4+1). We suppose that the expanded input L = [L0, L1, L2, ..., L28]'. By using LM training algorithm, the weights (W), and the biases between inputs and outputs are updated until the cost function is minimized within a limit. The cost function for the tth sam-ple data is given by

2)(21

ttt OTE −= , (10)

where Tt is the target and Ot is the network output

)tanh()( ttt UUfO == , (11)

where the input of the hyperbolic tangent transfer function is given by

ttt LWU ′= , (12)

where Wt=[wt,0, wt,1, wt,2, ..., wt,28]' is the weight vector for the tth sample.

IV. EXPERIMENTAL VERIFICATION OF CHARACTERISTICS FOR SOLAR MODULES

In this section, the experimental architecture of data acqui-

sition for the PV system and the network training process for the two types of ANNs (i.e., MLP and CFLNN) used in this study are presented.

A. CFLNN based model for solar array The proposed CFLNN model for solar array is shown in Fig.

4. The inputs are the irradiance, module temperature, and vol-tage measured from the terminal of PV modules. The output is the current output from the PV arrays. Before using the CFLNN to predict the current output, the network is trained and tested with the measured data. In order to get the realistic dataset samples to train the network and verify the performance accuracy for the CFLNN modeling approach, the outdoor test equipments were set up on the rooftop of the department build-ing in Singapore (1.34οN, 103.68οE).

B. Experimental setup The experimental PV system for data acquisition, as shown

in Fig. 5, consists of PV modules, storage batteries, electrical load, and the MPPT connected between the PV modules and the load. The data logger is used to record the measured values of the current, voltage, irradiance, and temperature. The elec-tricity source contains two 80W mono-crystalline PV modules in parallel and each PV module contains 72 solar cells con-nected in series. The parameters of the mono-crystalline PV modules we used in the experiment are listed in Table I. Two 65Ah/12V of sealed type lead-acid batteries in series are used to storage the excess energy generated by the PV modules dur-ing daytime. The batteries can provide the load enough energy when there is no enough sunlight on the PV modules. The commercial MPPT controller (SS-MPPT-15 L) which is used in this system to maximize the power output from the PV mod-ules keeps batteries from overcharge problem. A 120W/24V light bulb was used as the system load. Fig. 6 shows the PV modules situated on rooftop of the building and titled at an angle of 20° facing south. The PV output voltage and current are measured at an interval of one second, and recorded by a data logger for further processing. Thermocouples are mounted at the back of the modules to sense the temperature of the PV modules, which makes temperature measurement more accu-rate than the values measured from the environment. The PV modules provide the electricity for the load and charge the bat-

teries during daytime. The load, the light bulb, consumes power from the battery at night. Therefore, the advantages of this PV testing system are that there is no requirement of discharging the batteries manually and that the data is recorded with the good accuracy of one second compared with some other works [7, 8, 16].

C. Data acquisition After a long-term measurement, a large number of datasets

were obtained. Each dataset includes the global horizontal irra-diance, the temperature of PV modules, the PV terminal vol-tage, and the current output. It is important to observe that se-lection of datasets for training neural network has a significant influence on the performance of the neural network. Moreover, the accurate prediction of current or voltage of a photovoltaic module is very critical for estimating the energy production from the photovoltaic system. Therefore, in order to train and verify the proposed CFLNN model for PV module, we select four datasets. Their characteristics are shown in Table II where

TABLE I SPECIFICATION OF THE PV MODULE

Panel name SL80CE-36M

Short circuit current ( Isc )

2.44A

Maximum power (Pmax)

80W Number of cells in

each module ( in series )

72

Optimum voltage (Vmp)

35.1V Temp. coefficient of Isc A /

0.976 10

Optimum current (Imp)

2.28A Temp. coefficient of Voc V /

-0.16416

Open circuit voltage (Voc)

43.2V Module efficiency 15%

Fig. 4. CFLNN model for the solar array

Fig. 5. Configuration of the PV system for data acquisition

each dataset on different day is denoted with numbers - 1, 2, 3, and 4, respectively. The minimum irradiance for each day is zero. Before using these datasets, the average values of meas-ured datasets in every one minute from 8:20 AM to 5:20 PM were calculated to filter the signal noise. Finally, 540 groups of data in each dataset are generated for each dataset. In this expe-riment, since the dataset on July 19, 2011 contains enough in-formation for network training, it is used to train the network. Other three datasets were applied to test the model. It is impor-tant to point out that large number of datasets is required to train the network when using the network to predict the current output for a long-term running. Otherwise, the network needs to be trained frequently after a certain running period.

V. RESULTS AND DISCUSSION In this section, in order to verify that CFLNN has (i) lower

computational complexity than traditional MLP, and (ii) good accuracy as MLP but better than analytical two-diode model, two experiments are conducted using MATLAB. In addition, the computational complexities of proposed CFLNN model and the MLP model are compared, and the performances of current prediction by CFLNN, MLP, and the two-diode analytical model are also analyzed.

A. Experiment 1 This experiment compares the computational complexity of

CFLNN and MLP during training process. The way of evaluat-ing computational complexity is to calculate the execution time for each epoch during training period. For each dataset, twenty runs were conducted for each algorithm - MLP and CFLNN. The common training parameters for these two algorithms are set to be the same. For example, the training goal of the MSE between the normalized current predicted by network and the

normalized target current output was set to 0.001, the maxi-mum training iterations was 200, the learning rate was 0.2, and other parameters were set with the default values. Since theo-retically any nonlinear function can be approximated by a MLP with a hidden layer by adding enough hidden neurons, the number of hidden nodes in MLP was set to 10. The style of training was batch training, in which the weights and biases only adjust after the complete set of training samples have been applied to the network. Because the initial values of weights between each layer are randomly generated at the beginning of training process, the results of weights and biases are different after each training. In order to evaluate the general performance, we run each algorithm for 20 times for each dataset.

Thus, after training the neural network by dataset of July 19, 2011, the maximum, minimum and average number of itera-tions to converge within 20 runs are calculated as shown in Table III. We observe that the average number of iterations by CFLNN is about two times less than that by MLP, and that the average execution time of one epoch by CFLNN is about 3.8 times lower than that by MLP. The execution time of function-al expansion in CFLNN, 7.27×10-2 ms, can be ignored. There-fore, the execution time of each epoch by CFLNN is far less than that by MLP. Even though nowadays the time of offline training is not so important, it still gives an advantage of FLANN to take a shorter time to train the network and when the algorithm is implemented in hardware, CFLNN is more computationally efficient than MLP. In addition, the number of adders and multipliers in one iteration can also be calculated to compare the complexity of MLP and CFLNN by using equa-tions provided in [10].

B. Experiment 2 This experiment measures the accuracy of the PV output

current by three methods - two-diode model, MLP and CFLNN. The mean absolute percentage error (MAPE) and the mean squared error (MSE) are used to assess the performance of each method. MAPE is defined by the formula [7, 8]

1001

1

×⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧ −

= ∑=

D

i meas

predmeas

III

DMAPE (13)

where Ipred is the predicted current by CFLNN, MLP or two-diode model, Imeas is the measured current output (target out-put), and D is the number of data points in one dataset. The MSE is defined by the formula

∑=

−=D

ipredmeas II

DMSE

1

2)(1 (14)

Fig. 6. PV modules installed on the building rooftop

TABLE IIIMPORTANT STATISTICS OF DATASETS MEASURED FROM EXPERIMENTAL SETUP

Dataset ( # )

Date Irradiance (W/m2) Temperature ( ) Voltage (V) Current (A) Max. Avg. Max. Avg. Max. Avg. Max. Avg.

1 May 8, 2011 983.59 743.87 52.94 46.94 32.16 30.71 4.22 3.14 2 May 11, 2011 1093.10 621.26 57.26 47.94 34.46 29.98 5.18 2.82 3 July 19, 2011 1064.00 548.62 51.71 42.49 35.51 30.68 4.99 2.48 4 July 24,2011 986.89 496.92 50.59 42.14 33.48 31.11 4.35 2.19

The training dataset are randomly divided into three parts, of which 80% is used for training, 10% for validation, and 10% for testing. After training the networks - MLP and CFLNN, the regression plot regarding to the targets (measured current) re-lative to the outputs predicted from the networks are shown in Fig. 7 and Fig. 8. The parameter R represents the correlation between the target output and the predicted output by networks. The ideal value of R equals one. We can see from Fig.7 and Fig.8 that the points are almost located on the line of best re-gression, which means the both networks predict the current for training dataset very well.

By applying daily dataset to each model, the values of MAPE and MSE for the current output predicted by three dif-ferent modeling methods - CFLNN, MLP and two-diode model, are calculated as shown in Table IV. From the table, we can get the following results:

• For dataset 1, the values of the minimum MAPE and MSE by CFLNN and MLP are about nine times lower than that by the Two-diode model. Whereas, the aver-age MAPEs by MLP and CFLNN are almost five times less, compared to the Two-diode model.

• For dataset 2, even though the average value of MSE produced by MLP and CFLNN is slightly larger than that by the Two-diode model, we still find that the minimum MSE by these two ANN methods are smaller than that by the Two-diode model.

• For dataset 3, obviously, we can see that the values of the minimum and average of MAPE by intelligent me-thods are only half of that by Two-diode model. At the same time, the minimum and mean of the MSE

produced by CFLNN and MLP are also much smaller than by Two-diode model.

• For dataset 4, all the values of MSE and MAPE by MLP and CFLNN are smaller than that by the Two-diode model. For example, the MAPE produced by Two-diode model is 5.73%, which is about 1.6 times as that by MLP and about 1.5 times as that by CFLNN. Similarly, the values of MSE by these two neural net-works methods are all smaller than that by analytical method - Two-diode model.

In order to have a clear look, the daily current output by the three methods from 8:20 AM to 5:20 PM is also presented in Fig. 9 and its enlarged part is shown in Fig. 10 to emphasize the detailed transition points. Here, we only show the perfor-mance of one dataset of July 24, 2011. For other datasets we illustrate the MAPE and MSE values in Table IV. From the curves in Fig. 10, it is clear that the current values by the two-diode model are overestimated. However, MLP and CFLNN give better prediction of the current. For the analytical method, the failure to build PV array model can be caused by many uncertainties such as panel aging, module/cell mismatch, inac-curacy of datasheet provided by manufactures, and so on.

The above results indicate that the models with artificial intelligence methods give better current prediction and smaller error compared to Two-diode modeling method. Compared to MLP model, CFLNN offers lower computational complexity.

VI. CONCLUSIONS AND FUTURE WORK

In this paper, we proposed using a Chebyshev neural net-work for the modeling PV modules. The accuracy and compu-

(a) (b) Fig. 7. Normalized target and estimated daily current for training dataset on July 19, 2011, (MLP method).

-1 -0.5 0 0.5 11-1

-0.5

0

0.5

1

Normalized target

Nor

mal

ized

cur

rent

out

put Training: R=0.99781

DataFitY = T

-0.5 0 0.5 1

-0.5

0

0.5

1

Normalized target

Nor

mal

ized

cur

rent

out

put Test: R=0.99698

DataFitY = T

(a) (b) Fig. 8. Normalized target and estimated daily current for training dataset on July 19, 2011, (CFLNN method).

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

Normalized target

Nor

mal

ized

cur

rent

out

put Training: R=0.99784

DataFitY = T

-1 -0.5 0 0.5-1

-0.5

0

0.5

Normalized target

Nor

mal

ized

cur

rent

out

put

Test: R=0.998

DataFitY = T

Fig. 9. Measured and predicted current by three methods - FLANN, MLP,and Two-diode model with environment condition on July 24, 2011.

8:20 11:20 14:20 17:200

1

2

3

4

5

time

Cur

rent

(A)

CFLNNMLPTwo-DiodeMeasured

Fig. 10. Blow up of Fig. 9 from 9:00 to 11:15 on July 24, 2011.

9:00 9:27 9:54 10:21 10:48

1

1.5

2

2.5

3

timeC

urre

nt (A

)

CFLNNMLPTwo-DiodeMeasured

tational complexity of the proposed model is demonstrated by comparing the test results with other two modeling methods - MLP and the two-diode analytical model. The analytical mod-eling method is relatively simple to implement. However, it ignores the influence of solar irradiance and cell temperature on parallel, series resistance and other uncertainty factors, such as panel aging, module/cell mismatch, manufacturer datasheet inaccuracy and the random choice of initial conditions, etc., Therefore, large errors can be associated with analytical model-ing. Despite the fact that MLP can efficiently model the solar modules as anticipated, it is computationally complex when implemented in hardware and requires a longer training time.

The CFLNN-based model proposed in this paper considers all the variations in the system parameters due to environmen-tal conditions, system uncertainties, and aging effects. The out-put current is predicted only by inputting the irradiance, tem-perature, and voltage values measured from the experimental setup. It gives better accuracy than analytical method and less computational complexity than MLP network. Finally, the ef-fectiveness of this new method was verified by testing several experimental datasets. Therefore, the proposed model is suita-ble to model solar arrays.

With these encouraging results from using CFLNN for modeling PV arrays, our future research work will include: (1) investigating the modeling of PV modules under shaded condi-tions, (2) optimizing the number of dimensions of the expanded input space by using some intelligent methods, (3) applying this CFLNN technique to the maximum power point tracking problem, and (4) comparing the effect of different polynomials such as Legendre, Gaussian, Sigmoid, power, trigonometric series, etc. for better approximation of the current output.

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TABLE IIICOMPARISON OF EXECUTION CHARACTERISTICS FOR MLP AND CFLNN DURING TRAINING PROCESS WITHIN 20 RUNS EACH

Network Min. iterations Max. iterations Avg. iterations Avg. execution time / epoch (ms)

Network structure

MLP 7 105 40 435.40 3-10-1

CFLNN 6 34 19 113.45 29-1

TABLE IVMAPE AND MSE OF PREDICTED CURRENT BY TWO-DIODE, MLP, AND PROPOSED CFLNN MODEL UNDER 20 RUNS

Parameters Dataset 1 Dataset 2 Dataset 3 Dataset 4

Two-diode model MLP CFLNN Two-diode

model MLP CFLNN Two-diode model MLP CFLNN Two-diode

model MLP CFLNN

Minimum MAPE (%) --- 1.00 1.26 --- 3.37 3.75 --- 3.87 4.19 --- 3.55 3.69

Mean MAPE (%) 9.47 1.78 1.40 6.24 4.58 3.91 9.23 4.34 4.43 5.73 3.95 3.86

Minimum MSE --- 0.011 0.012 --- 0.0057 0.008 --- 0.0063 0.0074 --- 0.0085 0.0088

Mean MSE 0.095 0.032 0.013 0.0079 0.014 0.009 0.014 0.0085 0.0075 0.015 0.011 0.011

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Appendix:

Fig. 11. The structure of Two-stage expansion block in CFLNN