Estimating photovoltaic power generation: Performance analysis of artificial neural networks, Support Vector Machine and Kalman filter

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Estimating photovoltaic power generation: Performance analysis of artificial neural networks, Support Vector Machine and Kalman filter Raul V.A. Monteiro a,∗ , Geraldo C. Guimarães a , Fabricio A.M. Moura b , Madeleine R.M.C. Albertini b , Marcelo K. Albertini c a

Núcleo de Dinâmica de Sistemas Elétricos, Faculdade de Engenharia Elétrica, Uberlândia, MG, Brazil Faculdade de Computac¸ão, Universidade Federal de Uberlândia, Campus Santa Mônica, Uberlândia, MG, Brazil b Departamento de Engenharia Elétrica, Universidade Federal do Triângulo Mineiro, Uberaba, MG, Brazil c

a r t i c l e

i n f o

Article history: Received 23 May 2016 Received in revised form 13 October 2016 Accepted 21 October 2016 Available online xxx Keywords: Training algorithms Artificial neural network Support Vector Machine Kalman filter Photovoltaic Power generation

a b s t r a c t Current energy policies are encouraging the connection to the grid of power generation based on lowpolluting technologies, mainly those using renewable sources with distribution networks. Photovoltaic (PV) systems have experienced a wide and high increase in their adoption as an energy source over the last years. Hence, it has become increasingly important to understand technical challenges, facing high penetration of PV systems on the grid, especially considering the effects of uncertainty and intermittency of this source on power quality, reliability and stability of the electric distribution system. On the other hand, the connections for distributed generators, by PV panels, changes the voltage profile on low voltage power systems. This fact can affect the distribution networks onto which they are attached causing overvoltage, undervoltage, frequency oscillations and changes in protection design. In order to predict these disturbances, due to this PV penetration, this article analyzes seven training algorithms used in artificial neural networks, with NARX architecture, for the generated active power estimating, and thus the state of the distribution network onto which these micro generators are connected and then compare their best statistical results with the Support Vector Machine (SVM) and the Kalman filter (KF) techniques. The results show that the best training algorithm used for the ANN learning obtained a mean absolute percentage error (MAPE) of 0.02%, while the SVM and KF techniques obtained 0.33% and 3.41%, respectively. Taking in account the other statistical analysis, we concluded that artificial neural networks are more suitable for this type of problem than SVM and KF. In addition, performing the training process with cell temperature data improves the accuracy of the resulting estimations. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Distribution networks, traditionally seen as passive, or be it, designed for one way power flow from the substation to the end client, were not designed to support the insertion of generation units or distributed generation (DG). Hence, a number of studies have indicated that this integration can lead to technical and operational problems on the network. This fact points to the need for an understanding into how to provide this interconnection, paying attention to the best choice of bus. This along with a decision into the operation mode of the generators, in order to minimize

∗ Corresponding author. E-mail addresses: [email protected], [email protected] (R.V.A. Monteiro), [email protected] (G.C. Guimarães), [email protected] (F.A.M. Moura), rociodelpilar [email protected] (M.R.M.C. Albertini), [email protected] (M.K. Albertini).

possible impacts on the power quality and system stability. In addition, greater attention should be given to the intermittent nature of power generation by PV systems, as these depend on solar irradiance, which affects the reliability of the power supply. This condition requires that stand-alone applications, resort to power storage equipment, usually batteries. This accumulation process results in losses that go on to affect the system yield. The connection of this new power supply, PV, to the existing power system, is in fact a complex challenge for engineers. A generator based on solar energy does not respond to the variations in the conditions of the power system in the same manner as a traditional synchronous generator. Even in the absence of electromagnetic transients, the solar supply has exclusive features, such as high-speed response (low inertia) and high rates of power ramp up. The large-scale installation of PV supply, requires a more reliable means for its interconnection with the main power network.

http://dx.doi.org/10.1016/j.epsr.2016.10.050 0378-7796/© 2016 Elsevier B.V. All rights reserved.

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model [6], simulate [7], as well as plan the behavior and the operation of the distribution systems with distributed generation [8]. In this sense, the study of these factors becomes of the up most importance, in order that means are developed that can prevent and eventually avoid these types of effects. In this manner, for one to achieve greater accuracy over the interference of these DGs, artificial intelligence techniques can be used for the estimating of the PV power generation onto a grid. There are three of these techniques that have been widely used for this kind of subject, which are:

Fig. 1. Power demand versus PV generation.

Brazil is a country with a predominantly tropical climate with great potential for the production of electric power, by means of solar power. By evaluating only the West-Central and Southeast of the country, the average period of daily insolation is 7–8 h, with annual irradiation averages of 16–18 (MJ/m2 day) [1]. The generation of electric power through PV panels, in certain cases becomes a viable alternative, especially when analyzed from the point of view of communities far from energy distribution lines, and as such suffer limited access [2]. In 2012, ANEEL — (Agência Nacional de Energia Elétrica), approved the Regulatory Ordinance (Resoluc¸ão Normativa — REN 482/2012) that establishes the general conditions for access to micro and mini-generation of low voltage electric power distribution systems, supplied by renewable energy sources [3]. This scenario opens the expectation of a greater quantity of power generation through use of PV panels, as well as an increase in the search for such technology. In this scenario, the national electric power sector foresees a significant future increase in the generation of electric power through use of PV panels. However, the effects of this type of energy supply need to be better understood, as many of such effects are only noted when there is an increase this type of connection on the power network. At certain times of the year, as for example the Brazilian dry season, there is an increase in the demand for electric power. This is due to the intense use of air refrigerators. An alternative technique for establishing the balance between load-generation, for the energy utilities and power licensees, would be the use of the power supplied by the micro and mini-generation. However, the profile of the load for residential clients behaves in a very specific manner. During the day, the period in which the generation of electric power by PV is greater due to solar irradiation, is the period when consumers are not at their residences. Therefore, all the energy produced is injected onto the grid. This produces a considerable increase of active power available on the distribution network, thus producing a variation in the voltage level on the grid, in this case an increase in voltage [4]. Fig. 1 illustrates this case. The same phenomenon occurs in an inverse manner at night, or be it, with the reduction in production, and consequently of the active power injection onto the network, there may arise voltage sag at specific points of the feeder. Therefore, an overvoltage or an undervoltage caused by distributed generators inserted onto the electric energy distribution network, may end up damaging and compromising motor operation, electronic equipment, electro-electronic home appliances, as well as compromise the very structure of the network, in respect to distribution capacities of the electric conductors present. To this effect, studies have been undertaken in order to characterize [5],

• Artificial neural networks (ANN): According to Ref. [9] a neural network is a massively parallel-distributed processor, consisting of simple processing units, which possess a natural tendency toward storing experimental knowledge and make it available for use. It resembles the brain in two aspects: 1—Knowledge is acquired by the network from its environment through a learning process; 2—Connection forces between neurons, known as synapses, are used to store the acquired knowledge. The learning process is performed through use of a learning algorithm, which has the objective of changing the synaptic weights, in a way that the artificial neural network adapts to and reaches the desired objective [9]. Due to its dissemination and performance in resolving problems with the aid of computers, artificial neural networks have been widely used as in Ref. [10], where the author uses a neural network to improve short-term load forecast. In Ref. [11], ANN were used to prediction of transformer oil breakdown, as well as for estimation of ground resistance [12] and short-time wind power forecasting [13]. • Support Vector Machine (SVM): Introduced by Vapnik and collaborators, the SVM is a type of computational learning, which is derived from statistics learning theory and the theory of dimension Vapnik–Chervonenkis. Based on the principle of the structural minimization of risk, the SVM shows a good performance in general and has been used on a large scale in a number of applications, such as for classification resolution, standard recognition, characterization of texts, regression, etc. [14–16]. In the electrical engineering studies, SVM has been used for fault detection [17] and classification [18], voltage stability monitoring [19], power quality classification [20], electricity market price forecast [21], short-term solar power prediction [22]. This paper proposes a least-square Support Vector Machine based model. The input of the model includes historical data of atmospheric transmissivity in a novel two-dimensional (2D) form the other meteorological variables, including sky cover, relative humidity, and wind speed. The output of the model is the predicted atmospheric transmissivity, which then is converted to solar power according to the latitude of the site and the time of the day. However, this model does not consider maximum ambient temperature, daily clear-sky, global radiation, average temperature. Recently a novel probabilistic forecasting approach was proposed to accurately quantify the variability and uncertainty of the power production from photovoltaic system [23], a linear programming based prediction interval construction model for PV power generation was constructed based on extreme learning machine (ELM) and quantile regression. Compared with conventional neural network methods, the miscellaneous hyper-parameters of which need to be adjusted, quite often leading to heavy computation, extreme learning machine has not only a relatively simple structure but also comparable accuracy. The main problem in the application of the ELM method is determining the optimum number of the hidden neurons to use. On the one hand, too many hidden neurons can significantly increase the computational complexity. On the other hand, too few would influence the accuracy. Moreover, considering the chaotic nature of weather systems may be that the linear model used does not attend the variable behavior then

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Fig. 2. Current–voltage curve for a photocell with different temperatures.

is better consider nonlinear model in order to get good results accuracy. • Kalman filter (KF): Theoretically the Kalman filter is an estimator of what is called the “linear quadratic problem”, which is the problem of estimating the instantaneous state of a Linear Dynamic system perturbed by a white noise, by using measurements linearly related to the state but corrupted by the white noise [24]. The applications of the Kalman filter involves various areas, but its utilization as a tool is almost exclusive to 2 proposes: estimation and estimators performance analysis [24,25]. For forecasting data, Kalman filter has been used to agricultural economics [26] and wind speed prediction [27,28], as examples.

The success of using SVMs for time series prediction is largely due to its good generalization ability and as a prediction technique, the artificial neural network has been widely applied in the solar power field, such as the global solar radiation prediction and the estimation of photovoltaic energy generation. However, the effectiveness of using SVMs, ANN and Kalmam filter technique for estimating the produced power of a photovoltaic module considering the cell temperature vary with variations in air temperature and irradiance has not been studied yet. In this paper, a performance analysis is proposed for the training algorithms used in artificial neural networks (ANN) based non linear models, which considers the evaluation of technical impacts. The diverse number of training algorithms from the neural network are subsequently tested and the performance of each is discussed, then the best performing algorithm is compared to the SVM and KF techniques. These included performance, processing and response time of these algorithms when used for temporal estimations using large quantities of data, also when using temperature data from the PV panels and solar irradiation variables over time along a particular horizon. The main contribution of this paper consists of finding a tool that takes the best performance from the network, minimizing or maximizing each technical aspect according to the interest of obtaining quicker responses to procedures in addition to possessing fewer errors, which moves toward a more realistic and diversified set of solutions for making decisions known as optimal solutions.

Given the specific combinatory nature of this problem, which requires an optimization tool capable of manipulating multiple objectives, the technical impacts will be evaluated simultaneously using a methodology based on the NARX concept (Nonlinear Autoregressive with External Input). Seven training algorithms were compared, which were available in the toolbox provided by MATLAB® : BFGS Quasi-Newton (BFG), Bayesian Regularization (BR), Conjugate Gradient with Powell/Beale Restarts (CGB), Polak–Ribiére Conjugate Gradient (CGP), Resilient Backpropagation (RP), Scaled Conjugate Gradient (SCG) and Levenberg–Marquardt (LM). SVM is a supervised technique based on Statistical Learning Theory and its training is based on empirical risk minimization [29]. This type of training aims to reduce error rate when employed to data not previously seen and to capture data behavior beyond its noise. This training goal is an improvement over ANN’s, which only aims to reduce error rate for the representation of training sample. In this case, a classical ANN training algorithm risks to err for future data if it has learned too much behavior of sample data and even included its noise. Thus, the SVM usually performs well in practice, while the ANN can present more errors, which causes overfitting of the training data. It is difficult to identify the number of ANN training times to avoid overfitting and this can be aggravated when the data set is small. Often, the SVM and well trained ANN are used together to build ensembles of predictors and evaluate the risk predictions. Moreover, this paper presented better results through the consideration of the cell temperature as an input variable. Compared to Ref. [30], the RMSE (%) obtained was the same, at 0.11, but the technique presented herein resulted in a regression coefficient r = 0.99, while that of the article resulted in 0.97 for estimations of predominantly sunny days. It is noted that the improvement of PV generation estimation techniques enable real-time applications for estimating the state of distribution networks involved in the concept of smart grids. As an example, one can mention the technique of Model Predictive Control (MPC), a technique which through the receding horizon philosophy can estimate the current state of a plant, in this case a distribution system, thereby facilitating decision-making on the part of system operators.

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The paper is organized in the following manner: Section 2 presents a description of the PV cell, the curves for current and voltage are presented, along with the main constraints for cell performance and PV modules operational temperature, and Section 3 presents an information concerning data collection. Section 4 gives reasons for using this type of architecture for which a short introduction is given and provides citations for references to tested algorithm training theories. Section 5 provides a short explanation concerning the theory of SVM. Section 6 also provides a short explanation concerning the theory of KF. Section 7 introduces the indexes of the evaluated performance for the comparison of the algorithms analyzed. The results from the simulations are shown in Section 8, and finally, in Section 9 the conclusions for this paper are presented. 2. The effect of temperature on the photocell In semiconductors, the energy bandwidth decreases with the increase in temperature. In a photocell with a higher temperature, more of the photons possess sufficient energy to create p–n pairs. The open circuit voltage is strongly dependent on the temperature decreasing significantly and causing the short-circuit current to increase slightly. Fig. 2 demonstrates the effect of temperature on a photocell. The effect of the temperature on the power output of a PV photocell can be given by the influence upon the current I and the voltage V, as the maximum power is given by (1) [31]: Pm = Vm Im = (FF) Voc Isc

(1)

Subscript m refer to the maximum power point in the module’s I–V curve, while subscripts oc and sc denote the open circuit and short circuit values. In (1) FF is the fill factor. The efficiency for a PV panel can be represented by the traditional linear expression (2):





c = Tref 1 − ˇref Tc − Tref



(2)

in which Tref is the module’s electrical efficiency at reference temperature, Tref , and at solar radiation flux 1000 W/m2 . The coefficient ˇref is a temperature coefficient and with Tref are given by the manufacturer. The module operation temperature is represented by Tc (K). Moreover, according to Ref. [30], the cell temperature (◦ C) can be estimated for variations in air temperature and irradiance with a linear approximation, by (3): Tcell = T +

 NOCT − 20  0.8

G

(3)

where: T is the air temperature; G is the global irradiance; NOCT is the Nominal Operating Cell Temperature.

Table 1 Electrical characteristics of the solar panel. Electrical performance under standard irradiation conditions of 1000 W/m2 , AM = 1.5 and a cell temperature of 25 ◦ C Maximum power: Pmax (W) Maximum power voltage: Vmpp (V) Maximum power current: Impp (A) Open circuit voltage: VOC (V) Short circuit current: ICC (A) Maximum voltage (V) Temperature coefficient for VOC Temperature coefficient for ICC

235 W (0/+5%) 30,5 V 7.71 A 37 V 8.4 A 1000 V −0.4049%/◦ C 0.0825%/◦ C

Electric performance at 800 W/m2 , NOCT 20 ◦ C, AM = 1.5, wind speed 1 m/s Maximum power: Pmax (W) Maximum power voltage: Vmpp (V) Maximum power current: Impp (A) Open circuit voltage: VOC (V) Open circuit current: ICC (A)

172 W 27.7 V 6.2 A 33.9 V 6.8 A

accuracy of the analysis. The input data were collected from the Brazilian National Space Research Institute — INPE. Uberlândia is situated in the State of Minas Gerais in the mineiro triangle, close to the State of Goiás, characterized as a municipality with a tropical climate. It has an annual temperature of 21.5 ◦ C and an average irradiation of 6 kWh/m2 /day. The vegetation characteristic of the region is that of the cerrado and its variables [32]. Fig. 3(a) illustrates the average annual daily solar radiation in the city of Uberlândia [33]. Fig. 3(b) shows the power generated by a PV panel in accordance with the data collected over a period of 30 days, previously cited. The measured active power (W) data, in accordance with the input data previously cited, serves as objective data for the preparation of the artificial neural network. The relationship between the PV panel temperature and irradiation with its generated power can be seen in Ref. [34]. Seven solar panels were used, these were connected in series, and were of the JT235PC silicon polycrystalline type, fabricated by Jetion Solar [35], for which its characteristics are shown in Table 1. 4. Artificial neural network structure For the preparation of the artificial neural network, MATLAB software was used. This software provides artificial neural network models from its toolbox, of which the following can be cited: Fitting tools, pattern-recognition tool, clustering tool, time series tool. The use of the MATLAB environment and its successful applications can be proved by the studies made in Refs. [36,37]. For this research study, the tool used for temporal data estimation was the time series tool. The NARX neural network can be expressed by (4): yt+1 = f (xt , xt−1 , . . ., xt−n , yt , yt−1 , . . .yt−n, )

(4)

3. Solar data from the region of data collection For this study, climatic data was collected for irradiation (W/m2 ), air temperature (◦ C), temperature of the solar panel (◦ C), and the hours per day (h) for the month of October 2015, in the city of Uberlândia — MG (Brazil). These data served as input data for preparing and training the network. This month was chosen, due the fact that is the sunniest month of the data collected region. Thus, the data was modeled on an hourly basis with 3 exogenous numerical weather data sets. After the ANN training process, the same exogenous data were used as input for the ANN estimation, but now the data were collected for the month of October 2015. The same method was applied for the KF and SVM techniques. In this manner, the estimation studies were made for the month of November, where generated power data was also collected for the

where the next value of the output signal, yt+1 , is regressed using the previously measured values yt , yt−1 (active power measured), and input signals ut , ut−1 , (this is, environment temperature, cell temperature and time). The function f represents the neural network, where the training algorithms adapt the weights for each network connection. The NARX neural network is illustrated in Fig. 4. 4.1. Training algorithms The preparation of the network needs a training algorithm for the learning process of the artificial neural network. The learning consists of weight and thresholds adjustments of the neural network, until a specific criterion is satisfied. In the following, sum-

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Fig. 3. (a) Atlas solar-metric for Minas Gerais—average daily annual radiation. (b) Irradiance, cell temperature and generated power.

Fig. 4. Structure for the NARX network.

marized descriptions are given for some of the seven algorithms performed in this work. The Quasi-Newton method is based on the Taylor series of the second order. The iterative procedure is obtained through (5) [38]:

The Bayesian Regularization method changes the error performance function by attaching a standard deviation of the weights and the thresholds [40] and can be expressed by (6): F = ˇED + ˛Ew

wi = −A−1 gi i

(5)

where Ai is the Hessian matrix for the performance function of the iteration i and gi is a gradient vector. The method of Broyden–Fletcher–Goldfarb–Shannon (BFG), updates the Quasi-Newton method as a function of successive gradients of the performance function. The breakdown of this method can be seen in Ref. [39].

(6)

where ˛ and ˇ are the regularization parameters. Using (6) to minimize the performance error, enables the network to possess less weights and thresholds. This is equivalent to reducing the size of the network in such a way that it can respond smoothly, thus reducing over fitting. The successful application of Bayesian approach can be seen in Ref. [41]. The method of Conjugate Gradient with Powell/Beale Restarts (CGB) is used to check the orthogonality between the current gradient vectors and the prior to each epoch of the training algorithm

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Fig. 6. Support vectors and separating hyperplane.

Fig. 5. Training algorithm for Scaled Conjugate Gradient.

Conjugate Gradient (CG). The orthogonality is checked in accordance with (7) [42]: T |gi−1 gi−1 | ≥ 0.2gi 2

(7)

where gi is the gradient vector and T the transpose matrix for this gradient vector. The (CG) algorithm is derived from the Steepest Descent (SD) algorithm, which is based on the Taylor series of the first order. A complete description of these two algorithms can be found [38]. The Polak–Ribiére Conjugate Gradient (CGP) algorithm updates a constant, also from the conjugate gradient algorithm, ˇi , as a scale product of the previous change on the gradient vector with the actual gradient. This is divided by the square of the previous gradient, expressed in (8) [38]: ˇi =

T g gi−1 i T gi−1 gi−1

(8)

where gi is the gradient vector and T the transpose matrix of this gradient vector. For the training algorithm resilient backpropagation, a complete description is given [42]. The training algorithm Scaled Conjugate Gradient is presented in Fig. 5 [43]: For the method of Levenberg–Marquardt, the changes to the weights (wij ) can be obtained by resolving (14) [43]: n 

˛ij wij = −

j=1

1 ∂E 2 ∂i

(14)

where “n” is the number of adaptable weights on the network, “E” the mean square error of the network, and ␣, a matrix whose elements are given by (15) [44]:



˛ij = 1 + ıij

N  ∂y (xk ) ∂y (xk ) k=1

∂wi

∂wj

(15)

The training algorithm LM, as the others presented here, is used to adjust the weights in the order that the neural network produces the required output for the input data [45]. 5. Support Vector Machine Consider the problem of the separation of  a series  of training vectors, which belong to two different classes xi , xj , . . ., (xl , yl ), where xi ∈ Rn is a vector resource and yi ∈ {+1,−1} is a class label. According to the SVM theory for non-linear classification, the original data are projected onto a particular space of high functional

dimensionality H by a non-linear map  : Rn → H. The SVM classifies data by finding the best hyperplane that separates all data points of one class form from those of the other class. The best hyperplane for an SVM means the one with the largest margin between the two classes. Margin means the maximal width of the slab parallel to the hyperplane that has no interior data points. Fig. 6 illustrates this idea. The problem of non-linear classification is transformed into a linear classification in the space  H[9–11]. By introducing the Kernel function K(xi , xj ) =  (xi ) ,  xj , it is not necessary to know in the most explicit sense the expression of  ( · ). The corresponding problem of non-linear optimizing classification is given by (16) and (17):

 1 Min wT w + C ␰i 2 l

(16)

1

 



yi wT ·  (x) + b ≥ 1 − ␰i

Subject to

␰i ≥ 0, i = 1, 2, . . .l

(17)

where C is the regularization parameter that controls the change between the maximization of the margin and the minimization of the training term error and ␰i is a slack factor for relaxing the restrictions of the rigid edges and the regularization constant C > 0 [46]. Due to the possible high dimensionality of the vector variable w, usually we solve the dual problem, Eqs. (18) and (19): 1 Min ˛T Q˛ − eT ˛ 2



Subjectto

(18) yt ˛ = 0

0 ≤ ˛i ≤ C, i = 1, 2, . . ., l

(19)

For this work, the Kernel radial function was used (20) [47]. exp(−||x − xi ||2 / 2 )

(20)

6. Time-varying Kalman filter The time-varying Kalman filter is a generalization of the steadystate filter for time-varying systems or LTI systems with non stationary noise covariance. The basics of Kalman filter can be seen in Ref. [24]. Given the plant state and measurement Eqs. (21) and (22): x [n + 1] = Ax [n] + Bu [n] + Gω [n]

(21)

yv [n] = Cx [n] + v [n]

(22)

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The time-varying Kalman filter is given by the recursions: Measurement update (23)–(25); and Time update (26)–(27) [24]:



xˆ [n|n] = xˆ [n|n − 1] + M [n] yn [n] − C xˆ [n|n − 1] M [n] = P [n|n − 1] C

 T

R [n] + CP [n|n − 1] C



 T −1

(23) (24)

P [n|n] = (I − M [n] C) P[n|n − 1]

(25)

xˆ [n + 1|n] = Aˆx [n|n] + Bu [n]

(26)

P [n + 1|n] = AP [n|n] AT + GQ [n] GT

(27)

In these equations: xˆ is used to indicate an estimated value; xˆ [n|n] is the updated estimate based on the last measurement yn [n] ; xˆ [n|n − 1] is the estimate of x [n] given past measurements up to yn [n − 1]; M[n] is the innovation gain; P[n] is the error covariance; Q [n] is the noise covariance; R [n] is the measurement noise covariance; Indices T refers to a transpose matrix. For this study the initial conditions are given by (28) and (29): x [1|0] = 0

(28)

P [1|0] = BQBT

(29)

7. Network preparation and performance indexes After processing the input and output data of the chosen neural network, it was trained through use of the seven algorithms previously cited. The capacity of the network used for training was 70%. For validation and test, 15% each. In all, from the input data and objective, 2820 data entries were used for the preparation and training of the network. The tests were performed on a PC with a 3.40 GHz Intel Core i7A Pentium® and 8 GB of RAM. In regards to the training of the neural network, one of the alternatives for resolving the problem of when to stop training is using the cross-validation technique. Therefore, instead of defining the exact number of epochs for adjusting the weights during training, one randomly divides the set into 3 sub-sets: training, validation and test. Through the use of this technique, the network is trained at each epoch, and with the weights already adjusted, it is tested with the validation subset and the estimation error is calculated at the end of the epoch. The aim here is to adjust the weights through the training subset data, then calculate the error using the data from the validation subset, thus supplying, different data to the network. Therefore, the cross validation error starts high, decreases to a certain point and then increases. While the validation error is decreasing, the network is generalizing, when the error starts to increase, at the same time the training error continues to decrease, the network starts to memorize inputs, thus losing its generalizing capacity. At this moment, the network should stop training. It is important to note that this technique, performed by MATLAB, stops only a few epochs later the moment it finds the best validation performance. The following transference functions for the hidden layers were tested: Linear, Log-Sigmoid and Tang-Sigmoid. That which obtained the best performance for the problem presented herein was Tangent-Sigmoid. The transfer function used for the output layer was the linear function. The number of neurons used in the hidden layer is 10 neurons. One can increase the number of neurons to over 20, since large quantities of hidden neurons offer more flexibility to the neural network, as the network will possess more parameters to optimize.

7

However, this results in a greater execution time from the network simulation. Linear regression was used as a data validation tool. This is a statistical process, which helps to deduct the relationship between a fixed number of dependent variables and independent variables. This analysis is useful in functional dependence studies between input and output factors, which implies that each input variable (x1 , x2 , x3 ,. . .) partially determines the level of the output variable y. Each value of the independent variable x is associated with the value of the independent variable y. A more detailed explanation for this tool can be found in Refs. [48] and [49]. 7.1. Performance indexes evaluation Another statistical tool used for obtaining the performance from the neural network under study is MAPE (Mean Absolute Percentage Error). MAPE is a precision statistical measurement in temporal series. This precision is represented in the form of a percentage [50–52] and can be defined as (30):



1y − x n x n

MAPE (%) =

× 100

(30)

i=1

where x is the expected output data and y the data estimated by the neural network, SVM and KF. Infinite values should not be considered. Finally, also as a statistical tool to be used for the performance our analysis, RMSE (Root Mean Square Error) is used. RMSE provides performance information in a short time for the correlation coefficient “r”, comparing the deviation extension from the estimated value from the real measured value [53–55]. The correlation coefficient “r” and RMSE are defined by (31) and (32), respectively:

n

r=



i=1

(yi − y¯ i ).(xi − x¯ i )

n (y i=1 i

n

− y¯ i ).

 RMSE =

i=1

(31)

(xi − x¯ i )

n (y − x)2 I=1

n

(32)

where x and y are the expected and estimated data, respectively, x¯ i and y¯ i are the measured values of x and y, and where n is the total number of values. The lower the MAPE and the RMSE values, the better will be the neural network performance [53]. 8. Simulation results and discussions For the training algorithms, after the input data is inserted (solar irradiation, environment temperature, and temperature on the solar panels) along with the objective data (measured active power), the training was performed, and the first result analyzed was the number of necessary epochs, in order that the network be trained and validated in accordance with the training algorithm that was being used. A higher number of epochs for validation, training and tests, results in more time needed to prepare the network. The network training and preparation time is directly linked to the training algorithms, since it is the error reduction stimulated by these that go on to be the criterion for the cross-validation. By analyzing Fig. 7, it is possible to note that the performance of some algorithms can be affected by the degree of precision required for the desired approximation. This figure illustrates the number of epochs versus the mean square error. Noted also is that the LM algorithm error falls much more with the epochs than other analyzed algorithms. The BR algorithm, even after 168 epochs was not validated, and for this reason, it was not added in Fig. 7. This occurred due to the

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Fig. 7. Performance of training algorithms.

Fig. 8. Linear regression for the training algorithms.

training algorithm not reaching the condition where the stop criteria is satisfied. However, it cannot be said that the training failed. In order of best performance for the cross-validation of the algorithms, the algorithm that obtained the fastest training time was the CGP, which took only 6 epochs for training, validation and test. The next best algorithm, with 21 epochs, is the CGB, followed by the SCG algorithm in third place with 36 epochs. The RP algorithm needed 75 epochs for the training process, and the LM algorithm needed 134 epochs. Following on from these is the BR algorithm, which needed 168 epochs, and finally, with the slowest time, the BFG algorithm with 354 epochs needed to perform the process. The second index to be analyzed is linear regression, resulting from the estimated data and the measurements, in accordance with each training algorithm. To achieve this, the power estimation was performed, which was generated in a 72 h interval, for future data

values that correspond to input data used for the preparation and training of the network. Fig. 8 illustrates the regressions obtained. These data values were compared with real measured active power data. The BR algorithm presented the best linear regression from among all those evaluated with a value of r = 0.9999. This was followed by the LM algorithm, which presented a regression coefficient of r = 0.9998. The SCG algorithm resulted in a regression coefficient of r = 0.9992. Next in line comes the CGB algorithm with a regression coefficient of r = 0.9977. The fifth best regression went to the BFG algorithm with r = 0.9967. This followed by the RP algorithm with regression coefficient of r = 0.9958. Finally, showing the worst linear regression from among all the evaluated training algorithms comes the CGP algorithm with r = 0.9919.

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Fig. 9. Measured power (W) and estimated power (W) during an interval of 72 h for the BFG (a) and BR (b) training algorithms.

Fig. 10. Measured power (W) and estimated power (W) during an interval of 72 h for the CGB (a) and CGP (b) training algorithms.

Fig. 11. Measured power (W) and estimated power (W) during an interval of 72 h for the RP (a) and SCG (b) training algorithm.

Fig. 12. Measured power (W) and estimated power (W) during an interval of 72 h for the LM training algorithm.

In corroboration with that previously shown, Figs. 9–12 illustrate the comparison between the measured data and the estimated data for the BFG, BR, CGB, CGP, RP, SCG and LM, respectively. However, to address the possibility of a generalization of the ANN training by BR and LM algorithms, that is, the ANN copies the inputs to the outputs, other tests were performed using more diverse data. For these tests, data was used from 3 days with intermittent irradiance, or be it, partial cloudy days, were chosen. The results are shown in Fig. 13(a) and (b). A more detailed study concerning variable data for irradiation forecasting can be seen in Ref. [56]. One notes that by means of Fig. 13(a) and (b), the measured and the estimated active power were in fact very accurate. Therefore, even with diverse data, the response was the same as that shown previously in Figs. 12 and 9(b), respectively. The linear regression,

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Fig. 13. Measured power (W) and estimated power (W) for 3 days with intermittent irradiance. (a) For the LM training algorithm. (b) For BR training algorithm.

Table 2 Synthesis of the data analyzed in accordance with the training algorithms. Training algorithms

Epochs for the validation of the ANN

Regression coefficient

MAPE (%)

RMSE (%)

BFG BR CGB CGP LM RP SCG

354 168 21 6 134 75 36

0.9967 0.9999 0.9977 0.9919 0.9999 0.9958 0.9992

10.37 0.02 5.28 6.27 0.31 7.85 6.33

9.06 0.11 6.13 10.41 0.74 8.76 5.15

Table 3 ANN structure with BR training algorithm performance results. Technique

Number of layers

Hidden neurons

Tappeddelaytime

Activation function/ hidden layer

Activation function/ output layer

Epochs

Regression coefficient

MAPE (%)

RMSE (%)

Training role (%)

Validation role (%)

Testing role (%)

BR

2

10

2

Tangent-Sigmoid

Linear

168

0.9999

0.02

0.11

70

15

15

MAPE and RMSE were also calculated and the results were the same as those already cited along with those cited below. As in the latest performance indexes analyzed, one has the MAPE and the RMSE. The BR training algorithm obtained the lowest analyzed indexes with a MAPE of 0.02% and a RMSE of 0.11%. The LM algorithm presented a MAPE of 0.31% and a RMSE of 0.74%. With a MAPE of 5.28% and a RMSE 6.13% comes the CGB. Next comes a MAPE of 6.27% and a RMSE of 10.41% associated with the CGP algorithm. In the case of the SCG algorithm, the MAPE was 6.33% and the RMSE 5.15%. The training RP algorithm resulted in a MAPE of 7.85% and a RMSE of 8.76%. An analysis of the BFG algorithm showed a MAPE of 10.37% and a RMSE of 9.06%. It is natural that the RMSE of some algorithms are higher than their MAPE, once that on the RMSE, one has a squared sum, which does not subtract the negative indexes of the sum, contrary to that which occurs on the MAPE. Fig. 14 illustrates the results for the obtained MAPEs and RMSEs. Table 2 summarizes the results obtained in our experiments for ANN training algorithms. The algorithm that on average obtained across all indexes, the best performance for the problem put forward was the BR algorithm. In second place was the LM algorithm, and third place was the CGB algorithm. Leaving the fourth spot to the SCG algorithm, fifth to the CGP algorithm, sixth place to the RP algorithm, and seventh place to the BFG algorithm. Table 3 summarizes the ANN structure used for this study with the results for the best training algorithm analyzed. The number of hidden neurons, tested in a range, was found to be negligible in the final results, and for reaching these results, the default MATLAB value was used, which is 10 neurons. In addition,

Fig. 14. MAPEs and RMSEs obtained by training algorithms.

for the tapped-delay-time (TDLs) this was also true, and the TDLs used were 2. These analyzes were also performed in Ref. [57] and the results obtained for the ANN structure configurations were the same as those presented herein. With the performed algorithms analyzes in hand, the next objective is the presentation of the temporal estimation by the SVM. One notes, in Fig. 15(a), the linear regression between the estimated and the measured active power and Fig. 15(b) illustrates the power measured in (W) and the power estimated (W) by the SVM. The data from the SVM used, along with the result of its regression coefficient, MAPE and RMSE are given in Table 4. For the SVM used, a linear regression of 0.9987 was obtained, or be it, a regression slightly inferior to those found for the training

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Fig. 15. (a) Linear regression for the SVM. (b) Results for SVM: training with first 72 h and testing with the next 72 h. Table 4 Data from the SVM used for the estimation of data. Type SVM

C

i

Regression coefficient

MAPE (%)

RMSE (%)

Kernel radial

1

0.001

0.9987

0.33

3.57

Table 5 Data from the KF used for the temporal estimation of data. Type KF Time-varying

Regression coefficient

ECB

0.9943

1.030

ECA 0.1587

Table 6 Resume table for the best resulting estimation tools. MAPE (%) 3.41

RMSE (%)

Technique

Regression coefficient

MAPE (%)

RMSE (%)

3.19

BR SVM KF

0.9999 0.9987 0.9943

0.02 0.33 3.41

0.11 3.57 1.19

algorithms BR and LM. However, the MAPE for the SVM resulted in 0.33% and the RMSE in 3.57%. One notes from Fig. 15 that comparatively speaking of Figs. 9 and 12, by using the ANN algorithms mentioned above obtained a better performance than the SVM. By means of Fig. 16(a), one notes the error between estimated data (in red) and measured data (in black) for a 700 data sample, which is the same sample number used to train the ANN. Fig. 16(b) illustrates the residual errors for this estimation. Finally, one shows the results for the Kalman filter estimation. Fig. 17(a) illustrates the power measured in (W) and the power estimated (W) by the KF. One notes that for the KF the values closer to zero are better suited than those encountered in the SVM response, Fig. 15. However, the linear regression, Fig. 17(b), for the KF resulted in 0.9943, which shows a worse result than those from the SVM, LM and BR training algorithms. The following results show the analysis for the implemented KF performance. Fig. 18(a) compares the measurement error (blue line) with the estimation error (green line). One notes that the noise level has been significantly reduced. This can be confirmed by the error covariance before (ECB), 1.0300, and the error covariance after (ECA) filtering, 0.1587. By means of Fig. 18(b), one observes that the output did indeed reach a steady state in approximately 3 samples. The best covariance error is the one that reduces to zero, the error between the measured and the estimated values. Table 5 shows the statistical results for the KF technique obtained. 8.1. Comparison of three technique used to estimate generated active power Table 6 summarizes the indices statistical results for a better comparison between the most efficient ANN training algorithm, SVM and KF.

9. Conclusion The aim of this paper was to compare the most widespread and used ANN training algorithms for the temporal estimation of data. Then following on with a comparison using two data prediction methods, which are becoming more widely used, the SVM and KF. For a relatively large quantity of data as that used in the neural network analyzed in this paper, the NARX network architecture, the training algorithm that obtained the best performance from among all those analyzed was the Bayesian Regularization. The Levenberg–Marquardt algorithm, although obtaining the second best performance, needed less training and preparation time for the neural network. If time is a limiting factor then is recommended the use of LM training algorithm. In terms of prediction accuracy, by comparing the performance of the two best training algorithms from the ANN with the performance of the SVM, it was noted that for the estimation of diversified data, the ANN model significantly outperformed the SVM-based model. This is due to the fact that the Kernel function has an intrinsic limitation when data samples with heavy noise or which are non-linear are included in the data set and this can therefore lead to a bad performance of the SVM. Moreover ANN has better ability of capturing nonlinear and time-varying nature of the data than the SVM model does However, the SVM technique showed to respond very well for the problem analyzed, as one can see by means of its statistics results. In addition, the SVM model achieved a better performance than the Kalmam Filter model, which is largely due to the good generalization ability of SVM. The KF showed the worst results among all the verified techniques, but in general responded with very good accuracy too. However, by means of the results of the regressions coefficients from the 3 techniques, one sees that all of them are pretty suitable

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Fig. 16. (a) Estimated versus measured power data. (b) Residual errors for the SVM estimation. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.)

Fig. 17. (a) Linear regression for the KF. (b) Results for KF: training with first 72 h and testing with the next 72 h.

Fig. 18. (a) Measurement errors from the KF. (b) Covariance error for the KF.

for the type of analysis performed in this study but, for a better accuracy, the use of the training algorithms denominated as Bayesian Regularization or the Levenberg–Marquardt is recommended. Therefore, this study also shows that for a better accuracy and reduction of uncertainties from the results, for the training process and the use of other estimation techniques, the PV cell temperature should be considered. This reduced the errors for the analyzed tech-

niques even when the estimations were made for partially cloudy days. Acknowledgments The authors would like to thank CAPES for their financial support as well as the support given by the Universidade Federal de Uberlândia.

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References [1] Agência Nacional de Energia Elétrica—ANEEL, Atlas de energia elétrica do Brasil, 2a ed., Brasília—DF, 2005. [2] Ce Shang, D. Srinivasan, T. Reindl, An improved particle swarm optmization algorithm applied to battery sizing for stand-alone hybrid power systems, Int. J. Electr. Power Energy Syst. 74 (2016) 104–117, Elsevier http://dx.doi.org/10. 1016/j.ijepes.2015.07.009. [3] M.K. Junior, A.V. Soares, P.F. Barbosa, M.E.M. Udaeta, Distributed generation in Brazil: advances and gaps in regulation, IEEE Lat. Am. Trans. 13 (8) (2015) 2594–2601 http://dx.doi.org/10.1109/TLA.2015.7332137. [4] J. Descheemaeker, V.J. Ryckeghem, C.V. Steenberge, C. Debruyne, J. Desmet, Incentives and technical considerations related to increased voltage tolerance in low voltage distribution grids, ICHQP. Romênia. Mai (2014) http://dx.doi. org/10.1109/ICHQP.2014.6842813. [5] G. Salazar, D. Carrión, Characterization and modeling of the efficiency of photovoltaic systems, IEEE Lat. Am. Trans. 13 (8) (2015) 2580–2586 http://dx. doi.org/10.1109/TLA.2015.7332135. [6] E.N. Conceic¸ão, K.M. Silva, Modeling and simulation of the protection of distribution feeders in ATP, IEEE Lat. Am. Trans. 13 (5) (2015) 1392–1397 http://dx.doi.org/10.1109/TLA.2015.7111994. [7] A.V. Netto, Planning of network system for the distribution and transmission areas of electric energy, IEEE Lat. Am. Trans. 13 (1) (2015) 345–352 http://dx. doi.org/10.1109/TLA.2015.7040668. [8] A. Barin, L. Canha, A. Abaide, R. Machado, Methodology for placement of dispersed generation systems by analyzing its impacts in distribution networks, IEEE Lat. Am. Trans. 10 (2) (2012) 1544–1549 http://dx.doi.org/10. 1109/TLA.2012.6187598. [9] H. Simon, Neurals Networks: a Comprehensive Foundation, Prentice Hall, Inc., 1999. [10] A.S. Khawaja, M. Naeem, A. Anpalagan, A. Venetsanopoulos, B. Venkatesh, Improved shot-term load forecasting using bagged neural networks, Electr. Power Syst. Res. 125 (2015) 109–115 http://dx.doi.org/10.1016/j.epsr.2015. 03.027. [11] M.A.A. Wahab, Artificial neural network-based prediction technique for transformer oil breakdown voltage, Electr. Power Syst. Res. 71 (2004) 73–74 http://dx.doi.org/10.1016/j.epsr.2003.11.016. [12] F.E. Asimakopoulou, G.J. Tsekouras, I.F. Gonos, I.A. Stathopulos, Estimation of seasonal variation of ground resistance using artificial neural networks, Electr. Power Syst. Res. 94 (2013) 113–121 http://dx.doi.org/10.1016/j.epsr. 2012.07.018. [13] N. Amjadya, F. Keyniaa, H. Zareipourb, Short-term wind power forecasting using ridgelet neural network, Electr. Power Syst. Res. 81 (2011) 2099–2107 http://dx.doi.org/10.1016/j.epsr.2011.08.007. [14] Z. Qizhong, Gene selection and classification using non-linear kernel support vector machines based on gene expression data, IEEE Complex Medical Engineering, CME (2007) http://dx.doi.org/10.1109/ICCME.2007.4382018. [15] H. Han, J. Dang, E. Ren, Comparative study of two uncertain support vector machines, IEEE Advanced Computational Intelligence (ICACI) (2012) 388–390 http://dx.doi.org/10.1109/ICACI.2012.6463192. [16] E.C.C. Tsang, D.S. Yeung, P.P.K. Chan, Support vector machines for solving two-class problems, Machine Learning and Cybernetics International Conference (2003) 2–5. [17] A. Swetapadma, A. Yadav, Electrical power and energy systems directional relaying using support vector machine for double circuit transmission lines including cross-country and inter-circuit faults, Int. J. Electr. Power Energy Syst. 81 (2016) 254–264 http://dx.doi.org/10.1016/j.ijepes.2016.02.034. [18] J. Dalei, K.B. Mohanty, Electrical power and energy systems fault classification in SEIG system using Hilbert-Huang transform and least square support vector machine, Int. J. Electr. Power Energy Syst. 76 (2016) 11–22 http://dx. doi.org/10.1016/j.ijepes.2016.02.034. [19] K.S. Sajan, V. Kumar, B. Tyagi, Electrical power and energy systems genetic algorithm based support vector machine for on-line voltage stability monitoring, Int. J. Electr. Power Energy Syst. 73 (2015) 200–208 http://dx.doi. org/10.1016/j.ijepes.2015.05.002. [20] Ö. Yıldırım, B. Eris, Electrical power and energy systems optimal feature selection for classification of the power quality events using wavelet transform and least squares support vector machines, Int. J. Electr. Power Energy Syst. 49 (2012) 95–103 http://dx.doi.org/10.1016/j.ijepes.2012.12.018. [21] X. Yan, N.A. Chowdhury, Electrical power and energy systems mid-term electricity market clearing price forecasting utilizing hybrid support vector machine and auto-regressive moving average with external input, Int. J. Electr. Power Energy Syst. 63 (2014) 64–70 http://dx.doi.org/10.1016/j.ijepes. 2014.05.037. [22] J. Zeng, W. Qiao, Short-term solar power prediction using a support vector machine, Renew. Energy 52 (0) (2013) 118–127 http://dx.doi.org/10.1016/j. renene.2012.10.009. [23] C. Wan, J. Lin, Y. Song, Z. Xu, G. Yang, Probabilistic forecasting of photovoltaic generation: an efficient statistical approach, IEEE Trans. Power Syst. PP (99) (2016) 1 http://dx.doi.org/10.1109/TPWRS.2016.2608740. [24] M.S. Grewal, A.P. Andrews, Kalman Filtering: Theory and Practice Using MATLAB, vol. 5, California State University at Fullerton, 2001. [25] S.K. Singh, N. Sinha, A.K. Goswami, N. Sinha, Electrical power and energy systems several variants of Kalman filter algorithm for power system

[26]

[27]

[28]

[29] [30]

[31]

[32]

[33] [34]

[35] [36]

[37]

[38]

[39]

[40]

[41]

[42]

[43]

[44] [45]

[46]

[47]

[48]

[49]

[50]

13

harmonic estimation, Int. J. Electr. Power Energy Syst. 78 (2015) 793–800 http://dx.doi.org/10.1016/j.ijepes.2015.12.028. L. Xiaoyun, Z. Xiaoling, X. Xian, Application of Kalman filter in agricultural economic forecasting, in: IEEE International Conference on Systems, Man, and Cybernetics, pp. 2185–2189. http://dx.doi.org/10.1109/ICSMC.1996.565487. Y. Tian, Z. Hu, Wind speed forecasting based on Time series—Adaptive Kalman filtering algorithm, in: IEEE Far East Forum onNondestructive Evaluation/Testing (FENDT). http://dx.doi.org/10.1109/FENDT.2014.6928287. K. Chen, J. Yu, Short-term wind speed prediction using an unscented Kalman filter based state-space support vector regression approach, Appl. Energy 113 (2013) 690–705 http://dx.doi.org/10.1016/j.apenergy.2013.08.025. C. Cortes, V. Vapnik, Support-vector networks, Mach. Learn. 20 (1995) 273–297 http://dx.doi.org/10.1007/BF00994018. A. Mellit, S. Saglam, S.A. Kalogirou, Artificial neural network-based model for estimating the produced power of a photovoltaic module, Renew. Energy 60 (2013) 71–78 http://dx.doi.org/10.1016/j.renene.2013.04.011. E. Skoplaki, A.G. Boudouvis, J.A. Palyvos, A simple correlation for the operating temperature of photovoltaic modules of arbitrary mounting, Sol. Energy Mater. Sol. Cells 92 (11) (2008) 1393–1402 http://dx.doi.org/10.1016/j. solener.2008.10.008. UFU—Universidade Federal de Uberlândia. A cidade de Uberlândia. Available in http://www0. ufu.br/catalogo novo/idiomas/pt/cidade.htm (Accessed January 2016). CEMIG—Companhia Energética de Minas Gerais. Atlas Solarimétrico de Minas Gerais. Belo Horizonte: Cemig, 2012. 80 p. R. Dufo-lópez, J.M. Lujano-rojas, J.L. Bernal-Agustín, Comparison of different lead-acid battery lifetime prediction models for use in simulation of stand-alone photovoltaic systems, Appl. Energy 115 (2014) 242–253 http:// dx.doi.org/10.1016/j.apenergy.2013.11.021. JETION SOLAR. Available in http://www.jetionsolar.com/ (Accessed November 2015). M. Valipour, M.E. Banihabib, S. Mahmood, R. Behbahani, Comparison of the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir, J. Hydrol. 476 (2012) 433–441 http://dx.doi.org/10.1016/j.jhydrol.2012.11.017. M. Valipour, Optimization of neural networks for precipitation analysis in a humid region to detect drought and wet year alarms, Meteorol. Appl. 100 (November) (2015) 91–100 http://dx.doi.org/10.1002/met.1533. M. Forouzanfar, H.R. Dajani, V.Z. Groza, M. Bolic, S. Rajan, Comparison of feed-forward neural network training algorithms for oscillometric blood pressure estimation, 4th International Workshop on Soft Computing Applications (SOFA) (2010) 119–123 http://dx.doi.org/10.1109/SOFA.2010. 5565614. M.J. Er, F. Liu, Genetic Algorithms for MLP neural network parameters optimization, 2009 Chinese Control and Decision Conference (2009) 3653–3658 http://dx.doi.org/10.1109/CCDC.2009.5192353. X. Li, D. Wang, A sensor registration method using improved Bayesian regularization algorithm, International Joint Conference on Computational Sciences and Optimization (4) (2009) 195–199 http://dx.doi.org/10.1109/CSO. 2009.447. M. Khoshravesh, M. Ali, G. Sefidkouhi, M. Valipour, Estimation of reference evapotranspiration using multivariate fractional polynomial, Bayesian regression, and robust regression models in three arid environments, Appl. Water Sci. 5 (2015) 1–12 http://dx.doi.org/10.1007/s13201-015-0368-x. J.-S.R. Jang, C.-T. Sun, E. Mizutani, Neuro-Fuzzy and Soft Computing: A Computational Approach to Learning and Machine Intelligence, Prentice-Hall, NJ, 1997. M. Riedmiller, H. Braun, A direct adaptive method for faster backpropagation learning: the RPROP algorithm, IEEE International Conference on Neural Networks (1993) 586–591, vol. 1 http://dx.doi.org/10.1109/ICNN.1993. 298623. P.E. Andries, Computational Intelligence: An Introduction, second ed., Wiley, 2007. G. Lera, M. Pinzolas, A quasi-local levenberg-marquardt algorithm for neural network training, in: Neural Networks Proceedings, 1998. IEEE World Congr. Comput. Intell. 1998 IEEE Int. Jt. Conf., vol. 3, pp. 2242–2246, 1998. http://dx. doi.org/10.1109/IJCNN.1998.687209. L. Lei, Z. Gao, W. Dinl, Fuzzy multi-class support vector machine based on binary tree in network intrusion detection, Electrical and Control Engineering (ICECE) (2010) 1043–1046 http://dx.doi.org/10.1109/iCECE.2010.264. F. Kaytez, M.C. Taplamacioglu, E. Cam, F. Hardalac, Electrical power and energy systems forecasting electricity consumption: a comparison of regression analysis, neural networks and least squares support vector machines, Int. J. Electr. Power Energy Syst. 67 (2015) 431–438 http://dx.doi. org/10.1016/j.ijepes.2014.12.036. A. Mellit, A.M. Pavan, A 24-h forecast of solar irradiance using artificial neural network: application for performance prediction of a grid-connected PV plant at Trieste, Italy, Sol. Energy 84 (5) (2010) 807–821 http://dx.doi.org/10.1016/j. solener.2010.02.006. S. Pulipaka, F. Mani, R. Kumar, Modeling of soiled PV module with neural networks and regression using particle size composition, Sol. Energy 123 (2016) 116–126 http://dx.doi.org/10.1016/j.solener.2015.11.012. E. I˙ zgi, A. Öztopal, B. Yerli, M.K. Kaymak, A.D. S¸ahin, Short–mid-term solar power prediction by using artificial neural networks, Sol. Energy 86 (2) (2011) 725–733 http://dx.doi.org/10.1016/j.solener.2011.11.013.

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G Model EPSR-4926; No. of Pages 14 14

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[51] A.N. Celik, Artificial neural network modelling and experimental verification of the operating current of mono-crystalline photovoltaic modules, Sol. Energy 85 (10) (2011) 2507–2517 http://dx.doi.org/10.1016/j.solener.2011. 07.009. [52] L. Olatomiwa, S. Mekhilef, S. Shamshirband, K. Mohammadi, D. Petkovic, C. Sudheer, A support vector machine-firefly algorithm-based model for global solar radiation prediction, Sol. Energy 115 (2015) 632–644 http://dx.doi.org/ 10.1016/j.solener.2015.03.015. [53] S. Bhardwaj, V. Sharma, S. Srivastava, O.S. Sastry, B. Bandyopadhyay, S.S. Chandel, J.R.P. Gupta, Estimation of solar radiation using a combination of Hidden Markov Model and generalized Fuzzy model, Sol. Energy 93 (2013) 43–54 http://dx.doi.org/10.1016/j.solener.2013.03.020. [54] L. Mentaschi, G. Besio, F. Cassola, a. Mazzino, Problems in RMSE-based wave model validations, Ocean Model. 72 (2013) 53–58 http://dx.doi.org/10.1016/j. ocemod.2013.08.003.

[55] N.D. Kaushika, R.K. Tomar, S.C. Kaushik, Artificial neural network model based on interrelationship of direct, diffuse and global solar radiations, Sol. Energy 103 (2014) 327–342 http://dx.doi.org/10.1016/j.solener.2014.02.015; D. Bernecker, C. Riess, E. Angelopoulou, J. Hornegger, Continuous short-term irradiance forecasts using sky images, Sol. Energy 110 (2014) 303–315 http:// dx.doi.org/10.1016/j.solener.2014.09.005. [56] G.M. Tina, S. De Fiore, C. Ventura, Analysis of forecast errors for irradiance on the horizontal plane, Energy Convers. Manag. 64 (2012) 533–540 http://dx. doi.org/10.1016/j.enconman.2012.05.031. [57] A.G.R. Vaz, B. Elsinga, W.G.J.H.M. Van Sark, M.C. Brito, An artificial neural network to assess the impact of neighbouring photovoltaic systems in power forecasting in Utrecht, the Netherlands, Renew. Energy 85 (2016) 631–641 http://dx.doi.org/10.1016/j.renene.2015.06.061.

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