Selçuk J. Appl. Math. Selçuk Journal of Vol. 5. No.1. pp. 39-48, 2004 Applied Mathematics
Estimation of the Solar Pond Temperatures Using Arti…cial Neural Network
Nalan Çiçek Bezir1, Arzu ¸Sencan2, Nuri Özek1, A.Kemal Yakut2 1Department of Physics, Science-Literature Faculty, Süleyman Demirel University,
32260 Isparta, Turkey;
e-mail: cicek@fef.sdu.edu.tr; e-mail: nozek@fef.sdu.edu.tr
2Department of Mechanical Education, Technical Education Faculty, Süleyman Demirel
University, 32260 Isparta, Turkey;
e-mail: sencan@tef.sdu.edu.tr; e-mail: akyakut@tef.sdu.edu.tr Received: April 21, 2004
Summary. In this paper, the temperatures at di¤erent depths of the solar pond in Yalvaç were predicted using Arti…cial Neural Network (ANN). Although the temperatures of the di¤erent depths of the solar pond with dimensions of 3.5x3.5x2 m, which was built in order to provide hot water for leather workshop, were measured, temperatures of the all points couldn’t be obtained. The tem-peratures of unmeasured points were predicted with the 90ANN method using existing temperature data. The predicted values of the ANN were compared with real values. As a result, the temperatures at unmeasured points of the solar pond were successfully predicted by the ANN method.
Key words: Solar energy, solar pond, arti…cial neural network.
1. Introduction
Solar ponds, which are developed being inspired by hot water (70 800C) in shallow lakes, are wide surfaced collectors whose depths are 1-2 meters that have water layers, fresh at the top and salty towards down. As seen in Figure 1, the physical construction of solar ponds resembles ‡at- plate collectors a lot. In plate collectors like black substance which sucks the radiation in ‡at-plate collectors, and glass covered chamber on it that provides heat insulation, solar ponds also have insulation zone between conventional zone which is formed
are with convection, and middle layers are without convection. Because of these similarities, the help of correlation that is developed for ‡at- plate collectors can do the e¢ ciency analysis of solar ponds.
Some theoretical studies on solar ponds were also done. Some of the most im-portant of these are the Weinberger [1], Rabl and Nielsen [2], Hull [3], Brinkworth and Hawladers [4]’s analytical studies on establishing temperature distribution in solar ponds.
The …rst scienti…c study on solar ponds in our country was done in 1978 by Kayal{ with a small, uninsulated solar ponds whose dimensions were 4.5 m x 4.5m x 1.5 m. In this study Kayal¬ did the physical analysis, modelling and economical analysis; it was noticed that the output for this uninsulated pond was about 16% , and the cost of solar ponds were 2.5 times cheaper than ‡at- plate collectors that produce hot water from solar energy. This study also proved that maintenance and preparation were quite easy [5].
In this study, temperatures in di¤erent depths in a salt gradient, insulated solar pond are measured experimentally and then the values, which couldn’t be measured, are estimated with the ANN method. The temperature measuring system in our study consists of Analogue-Digital Transformer Card with 16 input channels, a censor and a computer. In the system a temperature censor called LM35, which is an electronic element that changes with the tension emerged between temperature and the tips, is used .
Figure 1. A model solar pond
2. Solar Pond Model
An insulated, experimental solar pond whose side view is given in Figure 2 and the dimensions are 3.5x3.5 m2and 2 m depth was built in Isparta Yalvaç High School Campus area. As seen in Figure 2, the bottom and sidewalls of the pond were insulated very well.
Figure 2. Insulated solar pond model
In order to do temperature measurement, LM35 diodes are used as detector and the top of these diodes are covered with silicon in order not to be a¤ected from salty water. To be used in temperature measurement diodes calibrations are done. As seen in Figure 3, ten of the diodes, whose calibrations are done, are placed from below to above in the pond, two of them into the bottom concrete. The analogue data that comes from these detector diodes is registered in a computer, transforming digital temperature data by means of an AD/DA transformer and the diagrams that give the temperature distribution according to time are obtained by the Excel programme.
Figure 3. Temperature Distribution Measurement System in a Model Pond
3. Arti…cial Neural Network
ANN is a system that is modelled according to the human brain. It tries to solve the problems, which can’t be solved with the classical methods, like the operating system of the human brain.
Figure 4a shows a simple model of neuron proposed by McCulloch and Pitts (1943), usually called an M-P neuron [6]. This arti…cial neuron computes a weighted sum of its n input signals, xj, for j=1,2,. . . n, to generate an output of 1 if this sum is above a certain threshold otherwise an output of 0 results.
(1) y = ' 0 @ n X j=1 wijxj 1 A
where ' is a unit step function at 0, and wij is weight associated with the jth input. Equivalently, the threshold could be considered as another weight w0=
attached to the neuron with a constant input x0= -1, thus
(2) y = ' 0 @ n X j=1 wijxj 1 A
As seen in Figure 4b, a typical neural network consists of three layers: input layer, hidden layer and output layer. The ANN elements are connected to each other with a sequence of layers. In the Figure 4b, xo; x1; x2; : : : ; xn are input layer nodes; h1; h2; : : : ; hnare hidden layer nodes; o1; o2; : : : ; onare output layer nodes. In a typical neural network each node is fully connected to the next layer nodes. Input layer feeds data to the network, therefore it is not a computing layer since it has no weight and activation function. Output layer represents the output response to a given input. Generally logistic function is used in the hidden layer. For the simplicity activation function of the hidden layer to output layer may be chosen linear [6,7].
The back-propagation algorithm is a supervised iterative training method for multilayer feedforward nets with a di¤erentiable non-linear function. A typical non-linear function used is the sigmoid function de…ned by
(3) '(x) = 1
1 + exp( x)
The back-propagation algorithm minimizes the mean square di¤erence be-tween the network output and the desired output. The error function that the back-propagation algorithm minimizes can be expressed as follows:
(4) E = 1 2P X p X k (dpk opk) 2
where p is the pattern index, k is the index of elements in the output vector, ddk is the kth element of the pth desired pattern vector, opkis the kth element
of the output vector when pattern p is presented as input to the network, and P is total number of training patterns. Minimizing the cost function represented in Eq.(4) results in an updating rule to adjust the weights of the connections between neurons. The weight adjustment of the connection between neuron i in a layer m and neuron j in layer m+1 can be expressed as follows:
(5) wji= joi
where i is the index of units in a layer m, is a small positive constant called the learning rate, oi is the output of unit I in the mth layer, and is the delta error term back-propagated from the jth unit in layer m + 1 de…ned by
j = [dj oj]oj[1 oj]; if neuron j is in the output layer
j = yj[1 yj] X
k
kwkj;
if neuron j is in a hidden layer where k is the index of neurons in the layer (m + 2), ahead of the layer that contains neuron j.
Choosing a small learning rate leads to slow rate of convergence, and too large leads to oscillation. A simple method for increasing the rate of learning without oscillation is to include a momentum term 4 wji(n) that determines the e¤ect of past weight changes on the current direction of movement in the weight space, where n is the iteration number, and as a small positive constant. Thus the weights update rule is
(6) wji(n + 1) = joi+ wji(n)
The iterative process of presenting an input-input pair and updating the weights continues until the error function reaches a pre-speci…ed value or the weights no longer change. In that case the training phase is done and the network is ready for testing and operation.
Figure 4. (a) A typical neuron and (b) neural network arhtitecture
Theoretically the numbers of hidden neurons are increased so that the ANN can approximate more complicated non-linear function if it is piecewise contin-uos. However, it is not the case for all problems. Systems having fewer neurons in the hidden layer converge quicker. More than one hidden layer can also be used in order to model a complicated system [7,8].
4. The Developed Model With Arti…cial Neural Network
The ANN modelling consists of two steps: First step is to train the network; second step is to test the network with data, which were not used for training. The processing of adaptation of the weights is called learning [8]. During the training stage the network uses the inductive-learning principle to learn from a set of examples called the training set. Learning methods can be classi…ed as supervised and unsupervised learning. In supervised learning, for each input neuron there is always an output neuron. The weights are updated for every set of input/output data. These are most commonly used the ANNs. However, for the unsupervised learning it is enough only to have input neurons. The network learns to adapt using experience gained from previous inputs [9].
In this study; temperatures of solar pond, which is established experimen-tally, are estimated depending on time, pond depth, solar radiation and air tem-perature with the ANN. The network selected in this study has a feedforward structure and back-propagation algorithm. Time, pond depth, solar radiation and air temperature were chosen as input. For output layer, pond temperature was chosen.
In order to an optimum architecture, di¤erent hidden neurons were assigned. Each architecture was trained with same training set. After each training, mean square errors (MSE) were determined. The highest MSE belongs to the best architecture has 4 input neurons, 1 hidden layer with 3 neurons, and 1 output neuron (Figure 5). Learnig rate of 0.001 was chosen and epoch was limited with 10000 after training.
Figure 5. ANN model used in application
All measurements of the solar pond, which was established experimentally in Yalvaç, were done for the year 2001 and went on the whole year. In this study the data used for training the ANN was obtained from Yalvaç Meteorology station for September between 700and 900, and the measurement results for the other months and by the ANN method, desired temperature values can be estimated. In the experiment, the temperature in 10,30,50,70,90,110,130,150 and 190 cm depth were measured with censors. The temperature values that couldn’t be measured in di¤erent depths were estimated using the ANN method. In September the measurements were done and we have 127 data. 92 of this data set were used for training, and other 25 used for testing. In order to train the network, back-propagation algorithm and sigmoid activation function were selected. Regression between experimental and simulating data was determined. As shown in Figure 6, regression value is approximately 0.98.
Figure 6. Regression curve of data set
5. Conclusion
The ANN is alternative solution method that has been used recently to solve the complex problems, which are not linear.
In this study, the ANN method was used for estimating the temperature values of the salt gradient solar pond in Yalvaç For this purpose the data received from Yalvaç Meteorology station was used. In the study the regression value between real values and the values calculated with the ANN method was found as 0.9892. The results obtained from this study show that the ANN method could be used successfully on estimating the temperature in di¤erent depths in a solar pond.
References
1. Weinberger H. (1964): The physics of the solar pond, Solar Energy, Vol. 8, No: 3. 2. Rabl, A., Nielsen E. (1974): Solar ponds for space heating, Solar Energy, Vol. 17, pp. 1-12, Great Britain.
3. Hull, J.R. (1980): Computer simulation of solar pond thermal behavior, Solar Energy, Vol. 25, pp. 33-40, Britain.
4. Hawlader, M. N. A., Brinkworth, B. J. (1981): An analysis of the non convecting solar pond, Solar Energy, Vol. 27, No:3, pp. 195-294, Great Britain.
5. Kayal¬, R. (1980): Experimental solar pond in Çukurova area and observation of various parameters, Ç. U. Science Institute, Master thesis, 63p.
7. Tsoukalas, L. H., Uhrig, R. E. (1997): Fuzzy and Neural Approaches In Engineering, John Wiley& Sons Inc.
8. Fu, L. M. (1994): Neural Networks in Computer Intelligence, McGraw-Hill Inter-national Editions.
9. Dorvlo, A. S., Jervase, J. A., Al-Lawati, A. (2002): Solar radiation estimation using arti…cial neural network, Applied Energy, 71, 307-319.
10. Kalogirou, S. A. (2001): Arti…cial neural networks in renewable energy systems applications (a review), Renewable and Sustainable Energy Reviews, 5, 373–401.
