Radial basis function (RBF) networks become very popular due to several important advantages over traditional Multi-Layer Perceptrons (MLP)[9]:

1. Locality of radial basis function feature extraction in hidden neurons, that allows usage of clustering algorithms and independent tuning of RBF network parameters.

2. Sufficiency of one layer of non-linear elements for establishing arbitrary input-output mapping

3. Solution of clustering problem can be performed independently from the weight in output layers

4. RBF network output in scarcely trained areas of input space is not random, but depends on the density of the pairs in training data set.

These proprieties lead to potentially quicker learning in comparison to multi-layer perceptrons trained by Back-Propagation (BP)[8]. In some extent, RBF networks allow us to actualize a classical idea about training layer by layer. Before beginning tracking operation using an adaptive neural network model (RBF-IIR), the unknown non-linear plan must be estimated according to the certain model. In this particular estimation process, the model consists of an adaptive neural network topology the Radial Basis function embedded in the hidden unites. In cascade with the network is a local IIR block structure as shown in figure 2. The IIR (Infinite Impulse Response Filter) synopsis network is used
to create double local network architecture that provides a computationally efficient method of training the system, and accordingly resulting in quick learning, and fast convergence. The approximated signal of the network v(«) can be modeled by:

yiky^a, 2{к-і)и(к)+^р, y(k-j)v(k) )

(=0 j= i

W’/ exp(

(2)

Where M оми a ше ню iiuiiiuei ui itstsu-iuiward delays and coefficient of the HR filter, respectively, N and bj are the number of feedback and delays and recursive filter coefficients, respectively. The u, v are the input and the co-input to the model at example k, respectively. Input v(k) is usually kept small for the feed back stability purpose, b is the bias value. The RBF network (Fig. 2) have a same structure as the MLP having only one hidden layer, the RBF is applied to the hidden layer [8] it is chosen as being Gaussian defined by its average ‘m’ and its a2 variance, the output layer can be linear or non-linear function. The determination of the network parameters has the same procedure as the MLP, it is also a universal approximator. That is to say a vector u having ‘i’ components Xj formed the input layer of the RBF and that is to say a hidden layer contained ‘h’ neurons and output layer, the output layer is given by:

Where my is the vector average of the hidden neurone ‘i’, and which the element mij is the weight between input ‘j’ and the hidden neurone ‘i’ . Ы2 is the variance from the hidden neurone ‘i’ and wy is the synaptic weights. Determination of the parameters my, ai, and wi is done by using the BP algorithm. The neural network parameters wi, mi, y, ai, ai and bi can be optimized in the LMS sense by minimizing the energy function, E over all example. Thus e(k)=y(k)-y(k). The energy function is defined by:

£.І2><Ц’ (3)

To minimize E we may use tne method of steepest descent which requires the gradients

,-£E-, 4е, 4е and 4е for updating the incremental changes to each particular

uwt omt, j uu i cat ub

parameter wi, y, a and b respectively. Gradient of E are:

(4) (5) (6) (7) (8)

f)

UWi M

-@£—=—2и*Х. ejz’Qptj—/№,/11 )ai(uj—mj)

от,,] m

■4Е=-шУ’е72,|гу, — m,,j If )(uj — m,, )(uj — m,, )T

от, ti 11 "

ІЕ-=-%(к)еШк-]) %—’ZgikMklHk-j)

The incremental changes of each parameter are simply the negative of their gradients,

z

z

z

z

Aw^F, Am=-$E, Дст =_Ж

ow cm ca

Аа=-Щ and АЬ=-Щ-Thus each oa ob

coefficient vector w, m, a, a and b of the network is update in accordance with the rule:

wij(k+1)=wt, j(k)+^.wAwt, j

m, j(k+1 )=mt, j(k)+2p. mAmt, j a t(k+1 )=at(k)+poAat, j at(k+1)=at(k)+paAat bt(k+1)=fo(k)+pbAfo

Where the subscripted p-values are fixed learning rate parameters.

■Kn)

v

Fig.2. Adaptive RBF network structure

Let y(n) and y(n) represent the current and predicted samples, respectively. We evaluate to estimate y(n) by a function g(y(n)

y(n-1), y(n-2),…………………. y(n-k)) so as to minimize the mean square error (MSE).

E{[y(n)-g(y(n-1), y(n-2), y(n-3,……… y(n-k))]2}, where E is the expectation operator and k is

the order of the predictor the optimal non-linear prediction of the sample y(n) given the k previous sample y(n-1), y(n-2),….,y(n-k) is given by the conditional expectation [2].

y(n) = [y(n) 1 y(n -1). y(n — 2),……………. , y(n — k)] (9)

The conditional probabilities needed in Eq. (9) are not a priori. Sub-optimal solutions could be found using a suitable model for the source signal (data), and then computing the joint conditional probabilities. However, the lack of precise modeling precludes any useful solution by the direct use of Eq. (9). A neural network predictor does not minimize the conditional expectation directly, however, it does approximate this minimization to any desired precision [2]. In modeling the number of input and output neurons is determined by each application. However, the number of hidden layers and the number neurons within each layer must be adjusted during the learning phase so that the network can be trained efficiently. While, in theory, one hidden layer suffices for approximating a general input — output mapping, in practice however, the number of hidden layers and neurons is specific to the problem at hand. A constructed neural network must be trained to learn or model relationship between given input-out put data or measurement. Usually, a portion of the available data is used for training set. To be effective, the resulting model must be able to generalize, i. e must produce accurate results a new samples outside of the available training set. The performance of the network in processing the test set is used a measure of the network’s generalization capability and can thus be used on line to stop the training process. Figure 3 shows the diagram block of developed model. In this study from 365 patterns, set of 200 patterns was used for training of the network and 165 were used as for testing and validation for our model for each signal.

Fig.3. Diagram block of developed model

Results

26

10

0 20 40 60 80 100 120 140 160 18 Day

-5 0 10 .s5 0 20 J0 0 400 J00 0

0 20 40 60 80 100 120 140 160 180

lu (A) Useful current 1 -:Measured data.^Predicted data 0 20 40 60 80 100 120 140 160 180

-Measured data Vc (V):Generator PV voltage ..Predicted data 0 20 40 60 80 100 120 140 160 180

1 — Measured data H (Wh/m2):Radiation data ..:Predicted data

0 20 40 60 80 100 120 140 160 180 Day Fig.4. Comparaison between measured and predicted signals

Once a satisfactory degree of input-output mapping has been reached, the RBF-IIR network training is frozen and the set of completely is an unknown test data that was applied for validation. After simulation of many different structures, we found that the best performance is obtained with a one hidden layer with 4 neurons. The best model is validated by comparison between predicted results and actual values for different signals that are shown in figure 4. We observe that there is almost a complete agreement between the two series for different signals.

Table 1. Comparison between measured and predicted statistics Data Mean Variance Correlation Coefficient Relative Error Ta Ta 23.16 23.37 95.514 93.122 0.9941 0.0091 Tc Tc 22.98 23.18 97.652 92.121 0.9951 0.009 lu Iu 3.66 3.74 2.6525 1.6541 0.8336 0.020 ig ig 4.25 4.32 3.384 2.732 0.8225 0.021 Vb Vb 25.01 24.98 0.3214 0.2325 0.8432 0.017 Vc Vc 13.89 14.01 8.74 9.29 0.8424 0.018 H H 239 232 593 587 0.8525 0.015

Table 2. Training results from each network structures Neural Network Structures MSE # of Iterations Relative Error MLP 1x4x4x1 0.0954 954 0.087 2x6x1 0.074 841 0.060 RBF 1x2x1 0.0512 452 0.017 2x8x1 0.0433 490 0.013 RN 1x4x1 0.0354 525 0.015 MN 1x4x1 0.0515 748 0.032 RBF-IIR model 1x4x1 0.0087 250 0.0099 2x8x1 0.0072 320 0.0090

Tablel, displays the statistical features (mean, variance, correlation coefficient and RE) between the calculated data and those predicted by our model, it is found that there is no significant difference between the predicted and the measured parameter from the statistical features point of view. The correlation coefficient obtained for the validation data set is varied between 82 to 99%. In this respect, the closer to unity these values are the better the prediction accuracy. In order to illustrate the advantage of this model we made a comparison between different neural network structures like Multi-layer perceptron (MLP), Radial Basis Function (RBF), Recurrent Neural Network (RNN), Modular Network (MN). Therefore we have trying to finding the temperature model of the battery the results obtained is listed in table 2. According to this table one notices that this model have less low time calculation compared to the other neural network structures and also gives good results, which are approaches with the real results.

Conclusion

The main of this work is to train the RBF-IIR model to learn the prediction and modeling of the signals from stand-alone PV system. Once trained, the RBF-IIR estimates these signals faster. The validation of the model was performed with unknown signals data, which the network has not seen before. The ability of the network to make acceptable predictions even in an unusual day is an advantage of the present method. The estimation with correlation coefficient varied between 82 to 99 % was obtained. This accuracy is well within the acceptable level used by design engineers. The advantage of this model is to predict of different signal coming from the stand-alone prediction signals allow to analyzing and studying the performance of the PV systems and the sizing of PV system. Also this model have been compared between different neural networks structures, and given good results.

References

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[3] U. Amato, A. Andretta, B. Bartolli, B. coluzzi, V. Cuomo, F. Fontana, and C. Serio, "Markov process and Fourier analysis as a tool to describe and simulate solar irradiation”, Solar Energy, 37, pp197-2011986.

[4] M. Benghanem et A. Maafi, "Ajustement des donnees d’insolation d’Alger par un

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d’automatique, pp 53-56 , 1997.

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