- Email: [email protected]
On the energy consumption in residential buildings
Energy and Buildings 34 (2002) 727–736
On the energy consumption in residential buildings G. Mihalakakou*, M. Santamouris, A. Tsangrassoulis Laboratory of Meteorology, Division of Applied Physics, Department of Physics, University of Athens, University Campus, Building Physics V, 15784 Athens, Greece Received 10 September 2001; accepted 1 November 2001
Abstract A neural network approach is used in the present study for modelling and estimating the energy consumption time series for a residential building in Athens, using as inputs several climatic parameters. The hourly values of the energy consumption, for heating and cooling the building, are estimated for several years using feed forward backpropagation neural networks. Various neural network architectures are designed and trained for the output estimation, which is the building’s energy consumption. The results are tested with extensive sets of non-training measurements and it is found that they correspond well with the actual values. Furthermore, ‘‘multi-lag’’ output predictions of ambient air temperature and total solar radiation are used as inputs to the neural network models for modelling and predicting the future values of energy consumption with sufficient accuracy. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Energy consumption; Solar radiation; Neural networks
1. Introduction The intelligent techniques such as neural networks or fuzzy logic methods can be designed and used for predicting and estimating a time series [1–4]. Although chaos prevents a long-term predictability, a short-time forecasting is possible and very promising results have been obtained by using an intelligent technique such as neural networks or fuzzy logic methods for non-linear modelling of multivariate time series [5]. Artificial neural networks are computational models which can be regarded as an attempt to simulate in a simpler way the structure and functions of the human brain. Neural networks belong to the class of ‘‘data-driven’’ approaches, instead of ‘‘model-driven’’ approaches. In the data-driven models the analysis depends only on the available data, with little rationalisation about possible interactions [6]. Relationships between variables, models, laws and predictions are constructed after building a machine, which simulates the considered data. The process of constructing such a machine based on available data is addressed by certain algorithms like ‘‘perceptron’’ [7] or ‘‘backpropagation’’ [8]. Various researchers proposed learning algorithms for time series prediction and they applied them to feed forward multilayered or recurrent neural networks [9–14]. * Corresponding author. Tel.: þ30-1-727-6841; fax: þ30-1-729-5282. E-mail address: [email protected] (G. Mihalakakou).
The objectives of the present paper are primarily to examine the ability of neural network systems to estimate the hourly values of energy consumption of a residential building using as inputs several meteorological parameters. The second objective is to examine the feasibility of the neural network system in predicting future values of energy consumption using as inputs ‘‘multi-lag’’ predicted values of ambient air temperature and total solar radiation time series. The paper is organised as follows: the first section contains the introductory part, the building’s description, the description of the database, and the neural network architecture are presented in the second section, while the results are discussed in Section 3, and the conclusions are given in Section 4.
2. Methodology 2.1. Description of the building The building is located at the north eastern area of the city of Athens. It stands by itself and all its facades are exposed. The area is not very densely populated and the traffic is rather low. It is a one-storey building and has a rectangular shape. The long axis is along the south-west–north-east while the smaller sides are facing north-east and south-west. The building has a surface of 200 m2 and its height is 3 m.
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The main building structure is made out of reinforced concrete. The roof and floor are insulated while the external walls are not insulated. Windows area covers the 15% of the building’s surface. All windows are single glazed, tinted glass with metal frames. The building is heated, cooled and ventilated by mechanical and electrical equipment while for artificial lighting, fluorescent lamps are used. 2.2. Database The time series generated in the present paper were the hourly values of the building’s energy consumption. The energy consumption time series include hourly values of the heating and cooling loads of the building for 6 years, (1993–1998). For the energy consumption estimation, the measurements of two climatic parameters were used: air temperature and total solar radiation. National Observatory of Athens, (NOA), provided the air temperature and total solar radiation hourly values. NOA is located on a hill at the centre of Athens (latitude ¼ 37:588N; longitude ¼ 23:438E; and altitude ¼ 107 m). Continuous observations of standard climatic parameters have been performed at this location since 1900. 2.3. Neural network architecture Artificial neural networks are computing systems, which attempt to simulate the structure and function of biological neurons. Neural networks generally consist of a number of interconnected processing elements or neurons. How the inter-neuron connections are arranged and the nature of the
connections determine the structure of a network. Neural networks can be classified according to their structures described below into the following two types [4]. 2.3.1. Feed forward networks In a feed forward network, the neurons are generally grouped into layers. Signals flow always from the input layer through to the output layer via unidirectional connections, the neurons being connected from one layer to the next, but not within the same layer. 2.3.2. Recurrent networks In a recurrent network, the outputs of some neurons are fedback to the same neurons or to neurons in preceding layers. Therefore, signals can flow in both forward and backward directions. A multi-layer feed forward neural network is shown in Fig. 1. The network consists of three layers: an input layer, an output layer and an intermediate or hidden layer. The dashed lines in Fig. 1 mean that there are more neurons in each layer than the represented in this figure. Fig. 2 shows the basic artificial neuron of the hidden layer. The time series estimation or prediction problem using a neural network approach can be separated into three successive steps or subproblems: model building or neural network architecture; the learning or training process; the testing or diagnostic checking. In the present study a multiple network based on backpropagation learning procedure is designed for estimating the building’s energy consumption. The selected neural
Fig. 1. Architecture of a neural network system.
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Fig. 2. Presentation of a basic artificial neuron.
network architecture consists of one hidden layer of 15–22 log-sigmoid neurons followed by an output layer of one linear neuron. Linear neurons are those which have a linear transfer function while the sigmoid neurons use a sigmoid transfer function. Backpropagation networks use the log-sigmoid (logsig), or the tan-sigmoid (tansig), transfer function. Several learning techniques exist for optimisation of neural networks [15]. In the present neural network
approach learning is achieved using the backpropagation algorithm [8]. Mathematically, backpropagation is gradient descent of the mean square error as a function of the weights [16]. If the mean square error exceeds some small predetermined value, a new ‘‘epoch’’ (cycle of presentations of all training inputs) is started after termination of the current one. One of the main parameters of the backpropagation algorithm is the learning rate. The learning rate specifies the size of changes that are made in the weights and biases at
Fig. 3. Comparison of the measured with the neural network estimated energy consumption values for 2 years from the training set of data, (1993–1994), and for the months of July and January.
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each epoch. A learning rate of 0.5 was selected while the number of epochs varied between 3000 and 5000 in all cases.
3. Results and discussion 3.1. Estimation of the building’s energy consumption Hourly values of air temperature, total solar radiation as well as hourly values of the above described building energy consumption for 6 years (1993–1998) and for various months of the year were used for training and testing the network. Analytically 5 years, (1993–1997), were used for training the neural network and 1 year, (1998), was used for testing the training data. The network was trained over a certain part of the climatic data, and once training was completed, the network was tested over the remaining data. The input parameters of the neural network model were the following: air temperature hourly measurements in 8C, integrated hourly values of total solar radiation in MJ/m2.
The above two parameters have been selected as inputs to the neural models as they are the most significant climatic factors influencing the building’s energy consumption. The output was the energy consumption values for heating or cooling in W. Training is performed using hourly values of the input climatic parameters for the estimation of integrated hourly global solar radiation values for 5 years, (1993–1997) and for various months of the year. As learning occurs the mean square error decreases. Calculations have been performed for various months of the year and the following two time periods were selected for the presentation of results: The cold period of the year which consists of the months of November–March. During this period the estimated energy consumption consisted of the heating load of the building. The month of January was regarded as representative of the cold period for the result’s presentation. The warm period of the year which consists of the months of June–September. In this period the energy consumption valuesinclude thecoolingloadofthebuilding.Accordingly, the month of July was considered to be the representative month of the warm period for the presentation of results.
Fig. 4. Temporal variation of the estimated with the neural network and of the measured energy consumption values for one randomly selected from the training set day of July 1994 (4th), and for one randomly selected from the training set day of January 1995 (16th).
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Fig. 3 shows the comparison of the measured energy consumption values with the neural network estimated ones for 2 years from the training set of data (1993 and 1994) and for the months of July and January, respectively. As it can be seen from these figures there is a good agreement between measured and estimated values. For the month of July and for most cases, the energy consumption differences are less than 200 W while for the month of January these differences are mainly less than 5000 W. The correlation coefficients between the estimated and measured energy consumption values were found equal to 0.98 for the month of July and 0.96 for the month of January. The temporal variation of the estimated and measured energy consumption values for one randomly selected from the training set day of the warm period (4 July 1994), and for one randomly selected from the training set day of the cold period (16 January 1995), is shown in Fig. 4. In this figure, the continual line indicates the measured energy consumption values while the point symbols indicate the model’s estimations. As shown, there is a good agreement between the estimated and the measured data. Quite similar performance was observed for the whole training set of data.
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The neural network’s predictions were checked by comparing its results with the actual values of a testing set of data which consists of the hourly energy consumption measured values for the year 1998. The temporal variation of the estimated from the network energy consumption values and of the testing set measured values for one randomly selected day of July 1998 (1st) and of January 1998 (14th) are presented in Fig. 5. Again, the continual line indicates the measured energy consumption values while the point symbols indicate the model’s estimations. As shown the neural network estimated values perform well on the testing set of measurements. Quite similar performance has been observed for the whole set of the testing data. Fig. 6 shows the comparison between the estimated hourly values and the measured ones of energy consumption for the testing set of data. Again, the months of July and January of 1998 were selected for the presentation of the results. The correlation coefficients are 0.96 for July and 0.94 for January. Similar performance is observed for the whole set of the testing data. The present results are very encouraging and the neural network approach is found able to simulate and estimate the energy consumption time series with sufficient accuracy.
Fig. 5. Temporal variation of the estimated with the neural network and of the measured energy consumption values for one randomly selected from the testing set day of July 1998 (1st), and for one randomly selected from the training set day of January 1998 (14th).
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Fig. 6. Comparison of the measured with the neural network estimated energy consumption values for the year of the testing set of data, (1998), and for the months of July and January.
3.2. Prediction of the building’s energy consumption using ‘‘multi-lag’’ predicted input data 3.2.1. ‘‘Multi-lag’’ predictions of air temperature Air temperature values were predicted for the month of July 1998 using 20 h measurements as inputs for the first output prediction [17]. Analytically, the measured past 20 values of the ambient air temperature time series were used for the prediction of the next value of time series by the neural network. This predicted value is used as one of the lagged inputs for the next prediction of two time steps into the future. Similarly, the predicted values at this second time step as well as the previous time step are used as lagged inputs for the next prediction of three time steps into the future. This is the ‘‘multi-lag’’ prediction where the predicted values are appended to the network input database and are used to predict future values. The term ‘‘multi-lag’’ prediction is used to distinguish it from the ‘‘one-lag’’ prediction where the prediction of future values was based only on past measured values. For instance, if the network is used to predict the eighth value T(8) from the measured temperature values T(1)–T(7), then the next neural network prediction T(9) is made using as inputs T(2)–T(7) and T(8), and the subsequent network prediction T(10) is made using the temperature values T(3), T(4), T(5), T(6), T(7), T(8), and
T(9). In the beginning, seven measured values were used as inputs for the first prediction. Furthermore, it was observed that the more the measured data were added as inputs for the first output, the longer the predictions were made. About 7–25 past temperature data were used in the present study for the prediction of the first output. The prediction was performed for the month of July 1998, and it was found that by using the ambient air temperature measured values for 20 h, it is possible to predict hourly temperatures for the next 12 days with sufficient accuracy. Results are shown in Fig. 7, where it can be seen the temporal variation of the measured and predicted ambient air temperature values for almost 300 h, in July 1998, using 20 h measurements as inputs for the first output prediction. The selected days were 15–18 of July. In this figure the solid line indicates the measured temperature values, while the plus symbols indicate the model’s predictions. As shown, the predicted values perform well with the measured ones for more than 250 h. For this time period the root mean square error was found equal to 0.52 8C. 3.2.2. ‘‘Multi-lag’’ predictions of total solar radiation Furthermore, ‘‘multi-lag’’ predictive neural networks were used for total solar radiation prediction [18]. In the beginning, five measured total solar radiation values were
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Fig. 7. Temporal variation of the measured and predicted air temperatures for 14 days of July 1998 (5–18), using 20 h measurements as inputs for the first output prediction.
used as inputs for the first prediction, and the ‘‘multi-lag’’ prediction results were not satisfactory. Furthermore, the number of measured values used as inputs for the prediction of the first output is increased in order to improve the performance of the method and the five input values became six, seven, etc. About 6–13 past solar radiation values are used as inputs for the prediction of the first output. This work has been done for nearly every month of the whole set of testing data (1998) and it was found that using the radiation measured values for 14 h, it is possible to predict with sufficient accuracy, the total solar radiation values 6–15 days in advance for all the summer months. Fig. 8 shows the temporal variation of the measured and predicted total solar radiation values for the same case of July 1998 (5–18), used in the air temperature prediction, with 13 h radiation measurements as inputs for the first output prediction. As shown from this figure, for the first 11 days time period the predicted values are in close agreement with the measured ones. For this time period the root mean square error was found equal to 0.22 MJ/m2. Using the present neural network systems for ‘‘multi-lag’’ predictions of ambient air temperature and total solar radiation, it is not always possible to predict future values for in
any case and under any climatic conditions. However, various climatic parameters such as the total solar radiation or the air temperature can be predicted for small periods with very promising results using a non-linear method such as neural network approaches. In the present research the predicted period is mainly concentrated in the summer months. Summer months are characterised as warm and very dry in the Mediterranean area, and they consist of clear and sunny days, usually without weather phenomena. So, air temperature and total solar radiation time series can be modelled and quite longer time predictions can be achieved for this period of the year. 3.2.3. Predictions of energy consumption Finally, the building’s energy consumption has been estimated with the neural models for the above considered time period of July 1998, (5–16). The input parameters of the neural network models are the following: the predicted, in paragraph 3.2.1, values of ambient air temperature for July 1998, the predicted, in paragraph 3.2.2, values of total solar radiation for July 1998.
Fig. 8. Temporal variation of the measured and predicted total solar radiation for 14 days of July 1998 (5–18), using 14 h measurements as inputs for the first output prediction.
Fig. 9. Temporal variation of the measured and predicted energy consumption of the considered building for 12 days of July 1998 (5–16) using as inputs the predicted values of air temperature and total solar radiation.
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Fig. 10. Temporal variation of the relative error (%) between measured and predicted energy consumption for the 12 days of July 1998 (5–16).
The neural models are designed and trained for the estimation of the building’s energy consumption, which obviously for the month of July consists of the building’s cooling load. Fig. 9 shows the temporal variation of the measured and predicted energy consumption values for the 12 days of July (5–16). As shown, for the first 11 days, (5–15), the predicted values are in close agreement with the measured ones. This can be explained by the fact that for the first 11 days the input data performed well with the corresponding measurements. Fig. 10 shows the temporal variation of the relative error (%RE) between measured and predicted energy consumption values for the considered 12 days of July 1998 shown in Fig. 9. %RE ¼
Emeas Epred 100 Emeas
where Emeas and Epred are the measured and predicted from the neural network model energy consumption values respectively. For the first 11 days, 90% of the %RE values fall between 8 and 15%, (Fig. 10).
4. Concluding remarks Neural network models were trained in the present study to learn the hourly energy consumption values of a typical
residential building located in Athens. Remarkable success has been achieved in making accurate predictions of future values. Analytically: 1. The building’s energy consumption was estimated for several years, using as inputs to the neural models climatic parameters such as the ambient air temperature and the total solar radiation. From this investigation it was found that the neural network approach is able to estimate with remarkable success the building’s energy consumption values for both the warm and the cold period of the year. 2. ‘‘Multi-lag’’ predictions were performed for ambient air temperature and total solar radiation using the predicted values to the input database in order to model future air temperature and solar radiation values. From the calculations, it was observed that it is possible to predict with sufficient accuracy, the ambient air temperature and total solar radiation values for several days in advance for the warm period of the year. 3. The energy consumption of the building was predicted with sufficient accuracy for several days of the warm period of the year using as inputs the ‘‘multi-lag’’ predicted values of the ambient air temperature and total solar radiation.
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References [1] J.D. Farmer, J.J. Sidorowich, Predicting chaotic time series, Physics Reviews Letters 59 (1987) 845–849. [2] D. Teodorescu, Time series-information and prediction, Biological Cybernetics 63 (1990) 477–485. [3] A. Cichocki, R. Unbehauen, Neural Networks for Optimisation and Signal Processing, Wiley, Stuttgart, 1993. [4] D.T. Pham, X. Liu, Neural Networks for Identification, Prediction and Control, Springer, Berlin, 1995. [5] M. Li, K. Mehrotra, C.K. Mohan, S. Ranka, Sunspot numbers forecasting using neural networks, in: Proceedings of the IEEE Symposium on Intelligent Control, 1990, pp. 524–529. [6] K. Chakraborty, K. Mehrotra, C.K. Mohan, S. Ranka, Forecasting the behaviour of multivariate time series using neural networks, Neural Networks 5 (1992) 961–970. [7] F. Rosenblatt, Principles of Neurodynamics, Spartan Press, Washington, DC, 1961. [8] D.E. Rumelhart, G.E. Hinton, R.L. Williams, Learning internal representations by error propagation, in: D.E. Rumelhart, J.L. McClelland (Eds.), Parallel Distributed Processing, MIT Press, Cambridge, 1986, pp. 318–362. [9] F.S. Wong, Time series forecasting using backpropagation neural networks, Neurocomputing 2 (1991) 147–159. [10] J.T. Connor, D.R. Martin, L.E. Atlas, Recurrent neural networks and robust time series prediction, Transaction on neural networks 5 (1994) 240–253.
[11] T. Hondou, Y. Sawada, Analysis of learning processes of chaotic time series by neural networks, Progress in Theoritical Physics 91 (1994) 397–402. [12] P.K. Dash, G. Ramakrishna, A.C. Liew, S. Rahman, Fuzzy neural networks for time series forecasting of electric load, IEE Proceedings in General Transmission and Distribution 142 (1995) 535– 544. [13] E. Eisenstein, I. Kanter, D.A. Kessler, W. Kinzel, Generation and prediction of time series by a neural network, Physics Reviews Letters 74 (1995) 6–9. [14] S. Kalogirou, C. Neocleous, S. Michaelides, C. Schizas, A time series reconstruction of precipitation records using artificial neural networks. In: Proceedings of EUFIT’97, Aachen, Germany, 1997 pp. 2409–2413. [15] D.E. Rumelhart, J.L. McClelland, Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol. I and II, MIT Press. Cambridge, MA, 1986. [16] A.S. Weigend, B.A. Huberman, D.E. Rumelhart, Predicting the future: a connectionist approach, International Journal of Neural Systems 1 (1990) 193–209. [17] G. Mihalakakou, M. Santamouris, D. Asimakopoulos, Modelling the ambient air temperature time series using neural networks, Journal of Geophysics Research 103 (1998) 509–517. [18] G. Mihalakakou, M. Santamouris, D. Asimakopoulos, The total solar radiation time series simulation in Athens, using neural networks,, Theoretical and Applied Climatology 66 (2000) 185–197.
On the energy consumption in residential buildings G. Mihalakakou*, M. Santamouris, A. Tsangrassoulis Laboratory of Meteorology, Division of Applied Physics, Department of Physics, University of Athens, University Campus, Building Physics V, 15784 Athens, Greece Received 10 September 2001; accepted 1 November 2001
Abstract A neural network approach is used in the present study for modelling and estimating the energy consumption time series for a residential building in Athens, using as inputs several climatic parameters. The hourly values of the energy consumption, for heating and cooling the building, are estimated for several years using feed forward backpropagation neural networks. Various neural network architectures are designed and trained for the output estimation, which is the building’s energy consumption. The results are tested with extensive sets of non-training measurements and it is found that they correspond well with the actual values. Furthermore, ‘‘multi-lag’’ output predictions of ambient air temperature and total solar radiation are used as inputs to the neural network models for modelling and predicting the future values of energy consumption with sufficient accuracy. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Energy consumption; Solar radiation; Neural networks
1. Introduction The intelligent techniques such as neural networks or fuzzy logic methods can be designed and used for predicting and estimating a time series [1–4]. Although chaos prevents a long-term predictability, a short-time forecasting is possible and very promising results have been obtained by using an intelligent technique such as neural networks or fuzzy logic methods for non-linear modelling of multivariate time series [5]. Artificial neural networks are computational models which can be regarded as an attempt to simulate in a simpler way the structure and functions of the human brain. Neural networks belong to the class of ‘‘data-driven’’ approaches, instead of ‘‘model-driven’’ approaches. In the data-driven models the analysis depends only on the available data, with little rationalisation about possible interactions [6]. Relationships between variables, models, laws and predictions are constructed after building a machine, which simulates the considered data. The process of constructing such a machine based on available data is addressed by certain algorithms like ‘‘perceptron’’ [7] or ‘‘backpropagation’’ [8]. Various researchers proposed learning algorithms for time series prediction and they applied them to feed forward multilayered or recurrent neural networks [9–14]. * Corresponding author. Tel.: þ30-1-727-6841; fax: þ30-1-729-5282. E-mail address: [email protected] (G. Mihalakakou).
The objectives of the present paper are primarily to examine the ability of neural network systems to estimate the hourly values of energy consumption of a residential building using as inputs several meteorological parameters. The second objective is to examine the feasibility of the neural network system in predicting future values of energy consumption using as inputs ‘‘multi-lag’’ predicted values of ambient air temperature and total solar radiation time series. The paper is organised as follows: the first section contains the introductory part, the building’s description, the description of the database, and the neural network architecture are presented in the second section, while the results are discussed in Section 3, and the conclusions are given in Section 4.
2. Methodology 2.1. Description of the building The building is located at the north eastern area of the city of Athens. It stands by itself and all its facades are exposed. The area is not very densely populated and the traffic is rather low. It is a one-storey building and has a rectangular shape. The long axis is along the south-west–north-east while the smaller sides are facing north-east and south-west. The building has a surface of 200 m2 and its height is 3 m.
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The main building structure is made out of reinforced concrete. The roof and floor are insulated while the external walls are not insulated. Windows area covers the 15% of the building’s surface. All windows are single glazed, tinted glass with metal frames. The building is heated, cooled and ventilated by mechanical and electrical equipment while for artificial lighting, fluorescent lamps are used. 2.2. Database The time series generated in the present paper were the hourly values of the building’s energy consumption. The energy consumption time series include hourly values of the heating and cooling loads of the building for 6 years, (1993–1998). For the energy consumption estimation, the measurements of two climatic parameters were used: air temperature and total solar radiation. National Observatory of Athens, (NOA), provided the air temperature and total solar radiation hourly values. NOA is located on a hill at the centre of Athens (latitude ¼ 37:588N; longitude ¼ 23:438E; and altitude ¼ 107 m). Continuous observations of standard climatic parameters have been performed at this location since 1900. 2.3. Neural network architecture Artificial neural networks are computing systems, which attempt to simulate the structure and function of biological neurons. Neural networks generally consist of a number of interconnected processing elements or neurons. How the inter-neuron connections are arranged and the nature of the
connections determine the structure of a network. Neural networks can be classified according to their structures described below into the following two types [4]. 2.3.1. Feed forward networks In a feed forward network, the neurons are generally grouped into layers. Signals flow always from the input layer through to the output layer via unidirectional connections, the neurons being connected from one layer to the next, but not within the same layer. 2.3.2. Recurrent networks In a recurrent network, the outputs of some neurons are fedback to the same neurons or to neurons in preceding layers. Therefore, signals can flow in both forward and backward directions. A multi-layer feed forward neural network is shown in Fig. 1. The network consists of three layers: an input layer, an output layer and an intermediate or hidden layer. The dashed lines in Fig. 1 mean that there are more neurons in each layer than the represented in this figure. Fig. 2 shows the basic artificial neuron of the hidden layer. The time series estimation or prediction problem using a neural network approach can be separated into three successive steps or subproblems: model building or neural network architecture; the learning or training process; the testing or diagnostic checking. In the present study a multiple network based on backpropagation learning procedure is designed for estimating the building’s energy consumption. The selected neural
Fig. 1. Architecture of a neural network system.
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Fig. 2. Presentation of a basic artificial neuron.
network architecture consists of one hidden layer of 15–22 log-sigmoid neurons followed by an output layer of one linear neuron. Linear neurons are those which have a linear transfer function while the sigmoid neurons use a sigmoid transfer function. Backpropagation networks use the log-sigmoid (logsig), or the tan-sigmoid (tansig), transfer function. Several learning techniques exist for optimisation of neural networks [15]. In the present neural network
approach learning is achieved using the backpropagation algorithm [8]. Mathematically, backpropagation is gradient descent of the mean square error as a function of the weights [16]. If the mean square error exceeds some small predetermined value, a new ‘‘epoch’’ (cycle of presentations of all training inputs) is started after termination of the current one. One of the main parameters of the backpropagation algorithm is the learning rate. The learning rate specifies the size of changes that are made in the weights and biases at
Fig. 3. Comparison of the measured with the neural network estimated energy consumption values for 2 years from the training set of data, (1993–1994), and for the months of July and January.
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each epoch. A learning rate of 0.5 was selected while the number of epochs varied between 3000 and 5000 in all cases.
3. Results and discussion 3.1. Estimation of the building’s energy consumption Hourly values of air temperature, total solar radiation as well as hourly values of the above described building energy consumption for 6 years (1993–1998) and for various months of the year were used for training and testing the network. Analytically 5 years, (1993–1997), were used for training the neural network and 1 year, (1998), was used for testing the training data. The network was trained over a certain part of the climatic data, and once training was completed, the network was tested over the remaining data. The input parameters of the neural network model were the following: air temperature hourly measurements in 8C, integrated hourly values of total solar radiation in MJ/m2.
The above two parameters have been selected as inputs to the neural models as they are the most significant climatic factors influencing the building’s energy consumption. The output was the energy consumption values for heating or cooling in W. Training is performed using hourly values of the input climatic parameters for the estimation of integrated hourly global solar radiation values for 5 years, (1993–1997) and for various months of the year. As learning occurs the mean square error decreases. Calculations have been performed for various months of the year and the following two time periods were selected for the presentation of results: The cold period of the year which consists of the months of November–March. During this period the estimated energy consumption consisted of the heating load of the building. The month of January was regarded as representative of the cold period for the result’s presentation. The warm period of the year which consists of the months of June–September. In this period the energy consumption valuesinclude thecoolingloadofthebuilding.Accordingly, the month of July was considered to be the representative month of the warm period for the presentation of results.
Fig. 4. Temporal variation of the estimated with the neural network and of the measured energy consumption values for one randomly selected from the training set day of July 1994 (4th), and for one randomly selected from the training set day of January 1995 (16th).
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Fig. 3 shows the comparison of the measured energy consumption values with the neural network estimated ones for 2 years from the training set of data (1993 and 1994) and for the months of July and January, respectively. As it can be seen from these figures there is a good agreement between measured and estimated values. For the month of July and for most cases, the energy consumption differences are less than 200 W while for the month of January these differences are mainly less than 5000 W. The correlation coefficients between the estimated and measured energy consumption values were found equal to 0.98 for the month of July and 0.96 for the month of January. The temporal variation of the estimated and measured energy consumption values for one randomly selected from the training set day of the warm period (4 July 1994), and for one randomly selected from the training set day of the cold period (16 January 1995), is shown in Fig. 4. In this figure, the continual line indicates the measured energy consumption values while the point symbols indicate the model’s estimations. As shown, there is a good agreement between the estimated and the measured data. Quite similar performance was observed for the whole training set of data.
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The neural network’s predictions were checked by comparing its results with the actual values of a testing set of data which consists of the hourly energy consumption measured values for the year 1998. The temporal variation of the estimated from the network energy consumption values and of the testing set measured values for one randomly selected day of July 1998 (1st) and of January 1998 (14th) are presented in Fig. 5. Again, the continual line indicates the measured energy consumption values while the point symbols indicate the model’s estimations. As shown the neural network estimated values perform well on the testing set of measurements. Quite similar performance has been observed for the whole set of the testing data. Fig. 6 shows the comparison between the estimated hourly values and the measured ones of energy consumption for the testing set of data. Again, the months of July and January of 1998 were selected for the presentation of the results. The correlation coefficients are 0.96 for July and 0.94 for January. Similar performance is observed for the whole set of the testing data. The present results are very encouraging and the neural network approach is found able to simulate and estimate the energy consumption time series with sufficient accuracy.
Fig. 5. Temporal variation of the estimated with the neural network and of the measured energy consumption values for one randomly selected from the testing set day of July 1998 (1st), and for one randomly selected from the training set day of January 1998 (14th).
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Fig. 6. Comparison of the measured with the neural network estimated energy consumption values for the year of the testing set of data, (1998), and for the months of July and January.
3.2. Prediction of the building’s energy consumption using ‘‘multi-lag’’ predicted input data 3.2.1. ‘‘Multi-lag’’ predictions of air temperature Air temperature values were predicted for the month of July 1998 using 20 h measurements as inputs for the first output prediction [17]. Analytically, the measured past 20 values of the ambient air temperature time series were used for the prediction of the next value of time series by the neural network. This predicted value is used as one of the lagged inputs for the next prediction of two time steps into the future. Similarly, the predicted values at this second time step as well as the previous time step are used as lagged inputs for the next prediction of three time steps into the future. This is the ‘‘multi-lag’’ prediction where the predicted values are appended to the network input database and are used to predict future values. The term ‘‘multi-lag’’ prediction is used to distinguish it from the ‘‘one-lag’’ prediction where the prediction of future values was based only on past measured values. For instance, if the network is used to predict the eighth value T(8) from the measured temperature values T(1)–T(7), then the next neural network prediction T(9) is made using as inputs T(2)–T(7) and T(8), and the subsequent network prediction T(10) is made using the temperature values T(3), T(4), T(5), T(6), T(7), T(8), and
T(9). In the beginning, seven measured values were used as inputs for the first prediction. Furthermore, it was observed that the more the measured data were added as inputs for the first output, the longer the predictions were made. About 7–25 past temperature data were used in the present study for the prediction of the first output. The prediction was performed for the month of July 1998, and it was found that by using the ambient air temperature measured values for 20 h, it is possible to predict hourly temperatures for the next 12 days with sufficient accuracy. Results are shown in Fig. 7, where it can be seen the temporal variation of the measured and predicted ambient air temperature values for almost 300 h, in July 1998, using 20 h measurements as inputs for the first output prediction. The selected days were 15–18 of July. In this figure the solid line indicates the measured temperature values, while the plus symbols indicate the model’s predictions. As shown, the predicted values perform well with the measured ones for more than 250 h. For this time period the root mean square error was found equal to 0.52 8C. 3.2.2. ‘‘Multi-lag’’ predictions of total solar radiation Furthermore, ‘‘multi-lag’’ predictive neural networks were used for total solar radiation prediction [18]. In the beginning, five measured total solar radiation values were
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Fig. 7. Temporal variation of the measured and predicted air temperatures for 14 days of July 1998 (5–18), using 20 h measurements as inputs for the first output prediction.
used as inputs for the first prediction, and the ‘‘multi-lag’’ prediction results were not satisfactory. Furthermore, the number of measured values used as inputs for the prediction of the first output is increased in order to improve the performance of the method and the five input values became six, seven, etc. About 6–13 past solar radiation values are used as inputs for the prediction of the first output. This work has been done for nearly every month of the whole set of testing data (1998) and it was found that using the radiation measured values for 14 h, it is possible to predict with sufficient accuracy, the total solar radiation values 6–15 days in advance for all the summer months. Fig. 8 shows the temporal variation of the measured and predicted total solar radiation values for the same case of July 1998 (5–18), used in the air temperature prediction, with 13 h radiation measurements as inputs for the first output prediction. As shown from this figure, for the first 11 days time period the predicted values are in close agreement with the measured ones. For this time period the root mean square error was found equal to 0.22 MJ/m2. Using the present neural network systems for ‘‘multi-lag’’ predictions of ambient air temperature and total solar radiation, it is not always possible to predict future values for in
any case and under any climatic conditions. However, various climatic parameters such as the total solar radiation or the air temperature can be predicted for small periods with very promising results using a non-linear method such as neural network approaches. In the present research the predicted period is mainly concentrated in the summer months. Summer months are characterised as warm and very dry in the Mediterranean area, and they consist of clear and sunny days, usually without weather phenomena. So, air temperature and total solar radiation time series can be modelled and quite longer time predictions can be achieved for this period of the year. 3.2.3. Predictions of energy consumption Finally, the building’s energy consumption has been estimated with the neural models for the above considered time period of July 1998, (5–16). The input parameters of the neural network models are the following: the predicted, in paragraph 3.2.1, values of ambient air temperature for July 1998, the predicted, in paragraph 3.2.2, values of total solar radiation for July 1998.
Fig. 8. Temporal variation of the measured and predicted total solar radiation for 14 days of July 1998 (5–18), using 14 h measurements as inputs for the first output prediction.
Fig. 9. Temporal variation of the measured and predicted energy consumption of the considered building for 12 days of July 1998 (5–16) using as inputs the predicted values of air temperature and total solar radiation.
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Fig. 10. Temporal variation of the relative error (%) between measured and predicted energy consumption for the 12 days of July 1998 (5–16).
The neural models are designed and trained for the estimation of the building’s energy consumption, which obviously for the month of July consists of the building’s cooling load. Fig. 9 shows the temporal variation of the measured and predicted energy consumption values for the 12 days of July (5–16). As shown, for the first 11 days, (5–15), the predicted values are in close agreement with the measured ones. This can be explained by the fact that for the first 11 days the input data performed well with the corresponding measurements. Fig. 10 shows the temporal variation of the relative error (%RE) between measured and predicted energy consumption values for the considered 12 days of July 1998 shown in Fig. 9. %RE ¼
Emeas Epred 100 Emeas
where Emeas and Epred are the measured and predicted from the neural network model energy consumption values respectively. For the first 11 days, 90% of the %RE values fall between 8 and 15%, (Fig. 10).
4. Concluding remarks Neural network models were trained in the present study to learn the hourly energy consumption values of a typical
residential building located in Athens. Remarkable success has been achieved in making accurate predictions of future values. Analytically: 1. The building’s energy consumption was estimated for several years, using as inputs to the neural models climatic parameters such as the ambient air temperature and the total solar radiation. From this investigation it was found that the neural network approach is able to estimate with remarkable success the building’s energy consumption values for both the warm and the cold period of the year. 2. ‘‘Multi-lag’’ predictions were performed for ambient air temperature and total solar radiation using the predicted values to the input database in order to model future air temperature and solar radiation values. From the calculations, it was observed that it is possible to predict with sufficient accuracy, the ambient air temperature and total solar radiation values for several days in advance for the warm period of the year. 3. The energy consumption of the building was predicted with sufficient accuracy for several days of the warm period of the year using as inputs the ‘‘multi-lag’’ predicted values of the ambient air temperature and total solar radiation.
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