A prediction model based on neural networks for the energy consumption of a bioclimatic building

Energy and Buildings 82 (2014) 142–155

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Energy and Buildings journal homepage: www.elsevier.com/locate/enbuild

A prediction model based on neural networks for the energy consumption of a bioclimatic building R. Mena a,∗ , F. Rodríguez a , M. Castilla a , M.R. Arahal b a b

University of Almería, Agrifood Campus of International Excellence, ceiA3, CIESOL, Joint Center University of Almería – CIEMAT, Spain University of Sevilla, Dpto. Ingeniería de Sistemas y Automática, Escuela Técnica Superior de Ingeniería, Spain

a r t i c l e

i n f o

Article history: Received 10 April 2014 Received in revised form 30 May 2014 Accepted 18 June 2014 Available online 9 July 2014 Keywords: Electric demand prediction Prediction model Neural networks System identification

a b s t r a c t Energy in buildings is a topic that is being widely studied due to its high impact on global energy demand. This problem involves the performance of an adequate management of the energy demand, combining both convectional and renewable sources. To this end, the use of control strategies is an important tool. These control strategies can take advantage of knowledge of variables that act as disturbances in the closed loop scheme. Thus, it is of great importance the development of predictions of such variables. The main objective of this paper is to develop and assess a short-term predictive neural network model of the electricity demand for the CIESOL bioclimatic building, located in the southeast of Spain. The performed experiments show a quick prediction with acceptable final results for real data with a shortterm prediction horizon equal to 60 min and with a mean error of 11.48%. One-step ahead predictions and dynamic modeling simulations have also been evaluated. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Nowadays, the global energetic model is unsustainable in economic, social and environmental terms, where a sustainable model can be defined as that which “satisfies the actual needs without compromising the ability of future generations to satisfy their own needs” [1]. Consequently, due to the high impact of the production and consumption of electricity, it becomes increasingly necessary a proper electricity demand management to achieve energy efficiency. New saving mechanisms and control approaches, as demand side management [2], are able to optimize the electricity consumption maintaining, at the same time, the quality of the service and the satisfaction of the users’ needs. These mechanisms require an estimation of the load curve. Furthermore, the optimal exploitation of the renewable energy sources is another fundamental aspect with the aim to shift the load peaks whenever it is possible. In this line, over the past twenty years a special emphasis has been made in the construction of buildings with bioclimatic architecture, which can benefit from both solar energy and natural air flow for their use in natural heat and passive cooling thus reducing the intensive electricity consumption. In this paper, the CDdI-ARFRISOL-CIESOL (www.ciesol.es) building has been chosen

∗ Corresponding author. Tel.: +34 950 214083; fax: +34 950015129. E-mail addresses: [email protected] (R. Mena), [email protected] (F. Rodríguez), [email protected] (M. Castilla), [email protected] (M.R. Arahal). http://dx.doi.org/10.1016/j.enbuild.2014.06.052 0378-7788/© 2014 Elsevier B.V. All rights reserved.

for the study and development of a model based on neural networks for the short-term prediction of electric power demand. To achieve these objectives and to manage the electric energy demand in an efficient way, it is necessary to calculate an accurate and reliable prediction of the load curve. Such prediction will have to be made taking into account the most influential factors over the demand. There are numerous techniques that have been applied to the task of electricity demand prediction, such as engineering, statistical and artificial intelligence methods. A complete review of buildings energy prediction techniques may be viewed at [3,4]. Occasionally, it is common to find a combination of techniques [5]. Finally, some comparatives between different prediction techniques can be found in [6–10]. Engineering methods use physical principles to calculate the thermal dynamic and the energy behaviour on a building or in subcomponents level. They can be classified into detailed methods and simplified methods. Comprehensive methods use complex physical functions and thermal dynamics very elaborated to calculate, with accurate, the energy consumption of all the building’s components taking as input the information of the building and the environment like weather variables, construction details, building’s operation and equipment [11,12]. The International Organization for Standardization (ISO) has developed a standard for the calculation of the energy consumption for heating and cooling from a building and his components [13]. There is an updated list, maintained by the U.S. Department of Energy, with energy simulation tools [14]. Although these methods are accurate, they require a detailed

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Fig. 1. The CDdI-ARFRISOL-CIESOL building.

comprehensive parameters set from the building and the environment and these details are not always available. On the other hand, there are some simplified tools for the energy simulation [15,16]. Statistical methods try to correlate the energy consumption with the most influential variables. These empirical models are developed from the historical data which are used to train the model. The majority of the models developed are based on simplified variables, usually one or two weather variables [17–20]. Artificial intelligence methods are a research line that has been experiencing an increasing focus over the past years because of their good fit to this kind of problems. Artificial intelligence includes several techniques as artificial neural networks, fuzzy logic, support vector machine, genetic programming or a combination of them which is also known as hybrid system. Neural networks are widely used for the task of building energy consumption prediction because they are good at solving non-linear problems and fit well to this complex problem [21–23]. Fuzzy logic is a form of probabilistic logic that deals with reasoning that is approximate rather than fixed or exact, fuzzy logic’s success in these applications has been attributed to its ability to effectively model real world data [24,25]. Support vector machines are increasing their presence in research and industry over the last years, they are effective dealing with non-linear problems even with small historical data sets [26–28]. Finally, there are some cases where system information is only known partially, which it is called a gray system. In this way, a gray model can be used to analyze the energetic behaviour of the building when only incomplete or uncertain data are available [29–31]. To quantify energy demand inside buildings is, in general, a complex process. More specifically, in the case of the CIESOL building

there are several factors that influence on the energy consumption as weather variables, building’s construction, building’s occupants and their behaviour, lighting and/or HVAC (Heating, Ventilating and Air Conditioning) equipment utilization and their performance, etc. Therefore, in this paper a characterization of load in different conditions and categorization of such conditions has been developed. At the end, a prediction model based on artificial neural networks (ANN), which has been chosen by its distinctive features for this problem, has been obtained. The development process has been done following a concrete methodology for the proper architecture and structure selection for a NARX model and the selection of the embedding delay and the embedding dimension to reconstruct the state space. Finally, the obtained prediction model has been evaluated using a battery of tests spanning working and non-working days, different temperature conditions for winter and summer, different radiation conditions with cloudy and sunny days and also nightly consumption. This paper is organized as follows. In Section 2, CDdI-ARFRISOLCIESOL building is briefly described. Section 3 shows an analysis of the energy demand profile of the building used as case study. In Section 4, the methodology used to develop the model based on neural networks is widely described. Finally in Section 5, the results of different experiments are showed and discussed. 2. Scope of the research: the CDdI-ARFRISOL-CIESOL building The CIESOL building (Fig. 1) is a solar energy research center located in Almería, in the south-east of Spain (+36◦ 49 49.68 , −2◦ 24 25.92 , with an elevation of 27 m) where there is a

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Fig. 3. Comparison between energy consumption in working and non-working days. Fig. 2. Network of sensors in the CIESOL building.

semi-desertical Mediterranean climate. The building is located inside the campus of the University of Almería, and it is a mixed research center between CIEMAT (CIEMAT) and University of Almería, hosting research groups from both centers. This building has been designed and built within the framework of the research project PSE-ARFRISOL [32] whose main objective is the adaptation of bioclimatic architecture and solar energy in public buildings. Therefore, the CIESOL building has been built following a bioclimatic architecture criteria, for which it takes advantage of the benefits of solar energy and natural air flow for its use in natural heating and passive cooling. Furthermore, the building has a HVAC system based on solar cooling connected to a solar collector field, a hot water storage system, a boiler and an absorption chiller with its refrigeration tower [33]. On the other hand, the building has a photovoltaic power plant with a peak power of 9 kW to supply green power to the electricity grid of the building. Finally, it’s worth comment that some chemical research groups work at the building also, thus altering the energy behaviour of the system with their experiments. The building has an area of approximately 1100 m2 distributed in two floors. The lower floor consists of five offices, four laboratories, a kitchen, two bathrooms and a warehouse. On the upper floor it is located the director’s office, four laboratories and the conference room. There is also an environment, which occupies two floors, where is located all the machinery of the solar cooling installation. In addition, this building has all its enclosures monitored by a broad network of sensors (Fig. 2), whose data is stored in a database through different acquisition systems. However, this data may have any irregularities just as mismeasure, noise in the signals, discontinuities through the time series and different scales and range of values, so it was mandatory to make use of some preprocessing techniques as filtering, interpolation or normalization.

working and non-working days, using for that certain statistical parameters like arithmetic mean (¯x), standard deviation (), minimum and maximum of the electric power demand over different periods of time. First of all, a comparison between the energy consumption of a working and a non-working day had been performed, see Fig. 3. On the one hand, power demand in the working day starts to raise about at 8.00 am and it begins to decrease around 8.00 pm, so it can be established that energy consumption along working days is characterized by an office schedule. On the other hand, power demand in a non-working day is stationary. Due to inductive and research facilities, the power demand is around 20 kW. A comparison between the working and non-working days from a statistical parameters point of view can be observed in the table (see Fig. 3) where the arithmetic mean of the power demand for the working day is 25.42 kW while for the non-working day is 20.32 kW. Furthermore, the statistics for the working day show a higher variance and a higher peak value as would be expected. The other analysis performed inside the building was a comparison of the energy consumption through a week, from Monday to Sunday, along different seasons in a year. As can be observed in Fig. 5, all the seasons have a similar pattern, according to the energy demand profiles of working and non-working days commented previously. In addition, through the comparison of different seasons, as it is shown in Fig. 4, it can be observed that energy consumption for Spring and Summer is higher than for Winter and Autumn. This behaviour is derived from the typical semi-desertical

3. Analysis of the energy demand profile of the CDdI-ARFRISOL-CIESOL building In general, from an energy demand point of view, a building can be considered as a complex system composed of different kind of elements, such as HVAC systems, illumination, etc. More specifically, in this building one of most expensive systems, in energy consumption terms, is the solar cooling installation. Hence, it is necessary to perform a preliminary analysis of the data in order to obtain some conclusions. This analysis is focused on the study of demand profile patterns for different seasons and types of days,

Fig. 4. Comparison of the weekly energy consumption for each season.

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4. A neural network model for the prediction of the energy demand for the CDdI-ARFRISOL-CIESOL building The electricity demand prediction is a difficult task due to the complexity in the time series of demand which presents a high nonlinearity among the variables, as well as seasonality. In addition, it should be taken into account potential external variables which can influence the electric demand, specially weather variables. In this paper, a prediction tool based on artificial neural networks (ANN) has been developed. The choice of neural networks for this task has been made because of the complexity of the analyzed problem and the ANNs distinctive features: learning, self-adaptative, fault tolerance, flexibility and real time response [34–36]. The methodology used for the ANN forecasting model construction could be summarized in this way, although sometimes execution order differs: Fig. 5. Power demand of a day with the heat pump active in the CIESOL building.

Mediterranean climate of Almería which requires a greater use of the HVAC system for refrigeration than for heating, and thus the use of the absorption machine and refrigeration tower from the solar cooling installation. Furthermore, in Fig. 4 (inset table) it can be perceived that both the peak demand and the standard deviation, are similar for, on the one hand, Spring and Summer and, on the other hand, Winter and Autumn. Finally, the building energy consumption is dynamic and highly nonlinear, since it is focused to research. As an example, in Fig. 5, an atypical day with large oscillations and a high variance in the energy consumption can be observed and it can be checked at the table. In fact, there are days where power demand oscillates very quickly with a variance of approximately 40 kW in short periods of time. This special character will challenge the model construction. After an exhaustive analysis, it has been determined that such oscillating behaviour comes from a heat pump. After the performed analysis, a preliminary list of potential variables had been selected, see Table 1. After the performed preliminary analysis, two approaches have been proposed taking into account the knowledge of the system. Both of them are based on artificial neural networks (ANN) as can be seen in Fig. 6. The difference is that the first one uses all the information considered relevant to the process and another is a simplified version for comparison purposes which not takes into account the information from the solar cooling installation (dashed line). Table 1 Preliminary list of potential variables. Variable

Unit

Measurement range

Type of day (Working day/Non-working day) Hour of the day Outdoor temperature Outdoor humidity Outdoor solar radiation Outdoor wind speed Outdoor wind direction State of the pump B1.1 (Off/On) State of the pump B1.2 (Off/On) State of the pump B2.1 (Off/On) State of the pump B2.2 (Off/On) State of the pump B3.1 (Off/On) State of the pump B3.2 (Off/On) State of the pump B7 (Off/On) State of the boiler (Off/On) State of the absorption machine (Off/On) State of the cooling tower (Off/On) State of the heat pump (Off/On) Electric power demand Electric power injected by the PV plant

– – [◦ C] [%] [W/m2 ] [m/s]

{0, 1} [0, 23] [− 5, 50] [0, . . ., 100] [0, 1440] [0, 22] [0, 360] {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} [0, 85] [0, 9]

◦

– – – – – – – – – – – [kW] [kW]

1. 2. 3. 4. 5.

6. 7. 8. 9.

Data collection Variable preselection Data preprocessing Training, validation and testing sets construction Neural networks paradigms (a) Architecture (b) Structure Variable selection Model order determination Neural network training Evaluation and implementation

Once that the data has been collected, and an initial preselection has been chosen (see Table 1), next step in the methodology is the preparation and division of the data sets for its use in the development of the ANN model. 4.1. Data-sets construction In order to obtain an appropriate neural network model, a set of historic data acquired at the building has been used. More specifically, the historic data set is composed by 700 000 points from 09/01/2010 to 02/29/2012, with a sample time equal to 1 min. In addition, the historic data has been split into three different data sets to train, validate and test the neural network model. The data division has been made manually due to discontinuities in time series and trying to get a balanced amount of data for each season and data set. The first data set is the Training Data Set (318 340 data points) and it is used for obtaining the ANN parameters using a Levenberg–Marquardt algorithm based on gradient descent [37]. The second one is the Validation Data Set (89 158 data points), which is necessary in the training and post-training processes in order to prevent the over-training in the ANN. The final data set is the Test Data Set (107 264 data points) whose objective is to provide an independent data set for benchmarking and testing purposes. The size of each data set can be checked in Fig. 7. 4.2. Neural network model Artificial neural networks (ANNs) have represented a new paradigm in the energy prediction. They work as a black-box model, thus, it is not necessary to have detailed information about the system because ANNs learn the relationship between input and output parameters by means of historical data. Another advantage is the ability to manage large complex systems with so many parameters interrelated between themselves, since they may ignore the excess of information with minor importance. More specifically, in ANN models, the model inputs are the number of neurons in the input layer, the model parameters are the number of neurons and the

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Fig. 6. Adopted approach based on ANNs.

values of interconnection weights, which do not have any physical meaning, in the hidden layers and, at last, the outputs are the number of neurons in the output layer [34]. In addition, the flexibility of ANN entails a high computational effort, since the number of nodes and the selection of weights must be decided by trial and error. In most applications, the iterative design procedure can be tackled using large amounts of data that are split in several sets, some of which are used for training/design and others to validate the solution [34]. The ANN consists of an input layer, one or more hidden layers and one output layer. In its simplest version, each neuron is connected to another neurons from a previous layer through adaptable synaptic weights, it is in this set of connection weights where the network knowledge is stored. The information processing through a neuronal unit is as follows. The neuron receives the weighted activation from another neurons as input connections, then these are added up and the sum is processed by the neuron activation function generating the neuron’s output. For each of these output

connections, the activation value is multiplied by the associated weight and it is passed to the next neuron [38,39]. Training is the process to calibrate the connection weights using an adequate learning method. The learning mode is supervised, since pairs of inputs and desired output are presented to the ANN in order to achieve that the network generates the desired output. When each training data record is presented, the ANN uses the input data to generate an output which is compared to the real one in the training data record. If there is any difference, the connection weights are modified in the direction in which the error is minimized. This happens in this way in the gradient descent algorithms, which is the type of algorithm that has been utilized in this work to train the ANNs, more specifically the Levenberg–Marquardt algorithm [37,40,41]. After that, the ANN checks all the input patterns, if the error is still greater than a maximum tolerable, the network runs another training cycle with all the training patterns until the error is within the permissible limits or until another stopping criterion will be satisfied. For example, in this work, an early stopping criterion has been used to stop the training process when the error on the validation set, which is measured in each training iteration, begins to raise even if training error is still decreasing, this is done with the goal of increasing the ANN generalization ability. Once training has been finished, the ANN stores the weights.

4.2.1. Architecture and structure selection In Lin et al. [42] it is noted that the learning of long temporal dependencies with gradient descent methods is more effective using a recurrent architecture named Nonlinear Autoregressive with eXogenous input (NARX) [43] than with the use of other models. This might be due to the fact that the NARX architecture is characterized by having as input vector two tapped delay lines, one for the input signal and the another for the output signal. The NARX scheme can be observed in Fig. 8, and it is mathematically expressed as it is shown in Eq. (1). y(n + 1) = Fig. 7. Manual data distribution over sets.

f (u(n), u(n − 1), . . ., u(n − du + 1); y(n), y(n − 1), . . ., y(n − dy + 1))

(1)

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trying to emulate recursively the dynamic behavior of the system which generates the nonlinear time series [46]. The multistep prediction and dynamic modeling tasks are much more complex than the one-step ahead prediction. It is in these complex tasks where the ANNs can have an important role, specifically the recursive neural architectures [47]. The multistep ahead prediction can be expressed as: yˆ (n + N|n) = f (ˆz (n + N − 1))

(5)

zˆ (n + N − 1) = (u(n), u(n − 1), . . ., u(n − du + 1),

(6)

yˆ (n + N − 1|n), yˆ (n + (N − 1) − 1|n), y(n), . . ., y(n − dy + 1))

Fig. 8. Non-linear autoregressive with exogenous inputs (NARX) architecture.

where u(n) ∈ Rm and y(n) ∈ Rp are the input and output signals at ¯ ¯ timestep n (u(n) and y(n) are in the form of vector), and du ≥ 1, dy ≥ 1 and du ≤ dy , correspond to the memory order for input and output signals, respectively. The non-linear mapping function f can be approximated, for example, by using a standard ANN as the multilayer Perceptron. The obtained connection architecture is called NARX network. Having in mind the prediction task, the state space of a dynamical system can be reconstructed according to Takens theorem [44,45] by: xˆ (n + 1) = [x(n), x(n − ), . . ., x(n − (dE − 1))]

(2)

where xˆ (n + 1) is the state space of the system reconstructed at the timestep (n + 1), dE is the embedding dimension and  is the embedding delay. Therefore, it is necessary to calculate the order of the model, this is the embedding dimension, du and dy for input and outputs signals, and the embedding delay . The Eq. (2) implements the embedding delay theorem [45]. According to this theorem, a series of past values in a vectorial space with dimension dE must give enough information to reconstruct the phase of an observable dynamic system. By doing this, the aim is to deploy the projection to a multivariable phase space whose topological properties are equivalent to those of the phase space generated by the real time series, while embedding dimension dE is big enough. In the one-step-ahead prediction task, the ANN must estimate the next time series state, without giving the feedback to the model input regressor. Namely, the input regressor contains only real data points from the time series. To achieve a larger prediction horizon, which is known as multistep-ahead prediction, the output of the model should be given as feedback to the input regressor in order to obtain a recursive prediction for a number of time steps in the future. In this case, the components of the input regressor, previously composed by actual time series values, are progressively replaced by predicted values forming a recurrent architecture. The one-step ahead prediction can be expressed as: yˆ (n + 1|n) = f (z(n))

(3)

where z(n) = (u(n), u(n − 1), . . ., u(n − du + 1), y(n), . . ., y(n − dy + 1)) (4) If the prediction horizon tends to infinite, the input regressor will be only composed of estimated values at some time instant. In this case, the multistep prediction task is transformed into a dynamic modeling task, in which the ANN is an autonomous system

(7)

The structure of a ANN is defined by the number of hidden layers, the number of neurons in each one and in the output layer, and the activation function used in each hidden and output neuron. As ANN with only one hidden layer with tangent hyperbolic activation function and a output neuron with lineal activation function are universal approximators [48], more complex structures have not been taken into account. Therefore, it only remains to determine the number of neurons in the hidden layer, but taking into account that the number of neurons must be neither too low, as it can impede learning of dynamics, nor too big as it can lead to learning and noise overparameterization. Finally, based upon the model tests performed, the number of neurons in the hidden layer has been established in h = 10. 4.2.2. ANN inputs and size The selection of input variables has a critical role in the time series modeling. More specifically, in the case analyzed in this work, where generalization is influenced with such selection [49]. In time series prediction, past values of such time series and past or predicted values of another variables are used. When there is a group of independent variables, the selection of inputs has to be done in order to avoid the use of variables with few relevance in the prediction and, at the same time, including the most relevant ones. The methods for feature selection can be classified into two categories: these ones that are based on model testing and those that directly apply on the data without the need to build and test models. Among the methods based on model testing, the principal ones are: • Stepwise development. Adding up or removing input variable step by step, and testing the model performance gradually. • Ad hoc. Developing models with different input variables sets, and comparing the performance. • Global methods. Genetic algorithms are an example. These methods have the drawback of the time needed to be performed. In addition, it may occur that the effect produced by different input sets in model tests could be masked by the dependency of other parameters as the model structure (number of hidden nodes) or the parameters used in the training process which make difficult to isolate the model performance. On the other hand, there are free model methods which directly apply on the data. It can be distinguished: • Ad hoc. The inputs are manually set according to domain knowledge. • Analytical methods. A statistical method measures the strength in the relationship between the input and output variables. The most used method is the correlation which only takes into account the linear dependence between variables. Another

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Table 2 Percentage of lowest In in data blocks for each continuous variable. jth variable

% of lowest In in data blocks

Outdoor temperature Outdoor humidity Solar radiation Wind velocity Wind direction

35.29% 3.92% 45.09% 13.72% 1.96%

statistical methods, like the mutual information, are necessary when nonlinear dependence is suspicious to be present. In this work, it has been used analytical methods to measure the linear and nonlinear dependence by means of the correlation and mutual information, respectively. In addition, scatter-plots and model tests have been used in a complementary way. In addition, to select the most influential variables of an input set, it should be also taken into account the suppression of those interrelated variables to reduce the redundancy. However, given the relatively small initial inputs set (see Table 1), it has not been used a specific method for this task. 4.2.2.1. Measurement of linear dependence for input variable selection. In selection index method. The linear dependence between the continuous variables of the potential model inputs set and its output has been evaluated using an algorithm which provides a selection index In . This index is obtained directly from the data without the need to build any model [50]. In addition, it is not exclusively for linear dependencies, although it has been used with that purpose in this work. Given a data set composed of pairs of input–output values (xi , ti ), the input vector is composed of n input variables and xi,j is the jth component of the input vector for the data pair i. For each potential input j of the model, it can be obtained an indicator of its usefulness through the index In (x·,j ). This index measures the changes in the associated target value ti in reference to the changes in the variable xi,j . For this purpose, all the data pairs (xi,j , ti ) are sorted in such a way that new data pairs (xp,j , tp ) verify that xp,j ≤ xq,j for any p < q. The difference-based In estimator of residual variance can be estimated according to Eq. (8)

 n−1

In (x · ,j ) =

(t(i+1) − ti )2

(8)

i=1

Due to discontinuities in the time series data, the index In has been calculated independently for each contiguous data block. By this way, the variable with the lowest index will be selected for each data block, since it means that small changes in the input variable correspond to small changes in the target variable. As it can be seen in Table 2, both outdoor temperature and solar radiation are the variables with greater percentages of lowest index In , specially the solar radiation, which means that both variables are good predictors of the target variable. Scatter plots. They are a useful tool to visualize the dependence between two variables of a data set. In this way, it can be observed at a glance whether there is a correlation and if it is linear or nonlinear. The data points used for the scatter plots come from the training data set. As it can be observed in Fig. 9, both, outdoor temperature and humidity, show a nonlinear dependence, while direct solar radiation has a weak linear dependence. On the other hand, from the scatterplots of wind velocity and wind direction cannot be extracted a clear dependence. 4.2.2.2. Measurement of nonlinear dependence for input variable selection. The problem of previous statistical measures is that they

Table 3 Percentage of maximum mutual information in data blocks for each continuous variable. jth variable

% of max mutual information in data blocks

Outdoor temperature Outdoor humidity Solar radiation Wind velocity Wind direction

41.17% 27.45% 15.68% 13.72% 1.96%

do not reflect the nonlinear relationship that may exist between the inputs and the target variable. For this purpose, it is convenient to use another statistical measure as the mutual information. The mutual information is a measure of the dependence between two variables [51]. If the two variables are independent, the mutual information between them is zero. If the two variables are strongly related, that is, one is function of the other, the mutual information between them is large. There are another interpretations for the mutual information, for example the information stored in one variable about another one; also the degree of predictability of the second variable knowing the first one. All these interpretations are related with the same notion of dependence and correlation [52]. The mutual information between two variables xi and yj is defined as: I(xi ; yj ) = log

P(xi |yj ) P(xi )

(9)

It is often used logarithm base two, therefore, the most common unit of measure is the bit. Again, due to discontinuity in the data, the mutual information between the pairs of potential inputs and target variable has been calculated independently for each data block. Finally, in Table 3 can be seen for each variable, the percentages which obtain the maximum mutual information for each contiguous data block. In Table 3, it can be observed that outdoor temperature is the input with highest nonlinear dependence with the target variable. The outdoor humidity perhaps worth be taken into account, while the rest of variables have little nonlinear dependence. 4.2.3. ANN model input selection After several data analysis techniques and model tests the initial list of potential variables, see Table 1, has been reduced to the following ones: type of day, hour of the day (sine and cosine), outdoor temperature, outdoor solar radiation and the electric power consumed by the building, that is, the electric power demand added up with the electric power supplied by the photovoltaic plant. In addition, the variables of the actuators from the solar cooling installation has been also included into the model as they help to determine the dynamics of the energy consumption of the building. So the final list of inputs for the model is shown in Table 4. 4.2.4. ANN model order The next step is to select the order of the model inputs. For this purpose, and according to Eq. (2), it is necessary to obtain the embedding delay  and the embedding dimension d in order to reconstruct the state space [53]. The embedding delay  is unrelated with the traditional system pure delay L that is absent in the current dynamical system response. To find the optimum  embedding delay, it can be used the auto-correlation function or the average mutual information. The embedding delay  should be large enough in order to get a vector with independent information, but not so large to be totally independent. On the other hand, it should not be so small because it does not include new information. A common technique to obtain an optimum embedding delay  is to use the first local minimum of

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Fig. 9. Dependence of electric power demand over several variables.

the average mutual information function [54]. The average mutual information determines how much information has in common the measure x(t) at a given time with another instant x(t + t). This method has been executed with the three continuous variables included in the final inputs set (outdoor temperature, solar radiation and electric power demand) and the obtained results are shown in Fig. 10. As it can be observed, the theoretical optimum embedding delay for both outdoor temperature and solar radiation is around  = 400 samples and for the electric power demand is equal to  = 375 samples approximately. Despite the theoretical results of the method, empirical results based on model tests prove that an embedding delay  = 1 sample gives a better performance for this application. Finally, embedding delay for the variables type of day and hour of the day has been set to  = 1 sample due to their own conditions, and the embedding delay for the discrete variables of the actuators has been set to  = 1 sample due to system variability. The embedding dimension d is the minimum number of delay coordinates needed in order to assure that the trajectories of x(t) does not intersect in the d dimensions. In dimensions smaller than d, the trajectories can intersect because they are projected in few dimensions. If the dimension d is too large, noise can affect the prediction calculations. The method used to find a proper embedding dimension d is the False Neighbors Method [55]. This method can be described

using geometrical considerations: as d increases, the vectors that are near in the dimension d, get distanced themselves in the dimension d + 1. They are false neighbors in the dimension d. The false neighbors algorithm measures the percentage of false neighbors as embedding dimension d increases. The nearing points in d gets marked and those that get distanced widely in d + 1 are calculated. The function output is the percentage of false neighbors against the growing embedding dimension, and has a monotone decreasing curve. The optimal embedding dimension is when the curve drops below a determined threshold, usually around 0%. From Fig. 10, it can be deduced that the optimal embedding dimension is around d = 4 samples as maximum for both the temperature and the solar radiation. The inputs type of day and hour of the day have an embedding dimension d = 1 samples as maximum. Embedding dimension for the feedback variable has been established in dy = 3 samples as maximum according to model tests. The order for the discrete variables has been set using model tests and it is equal to d = 5 samples. This is summarized in Table 5.

4.3. Particular ANN with less input information In addition to the previous ANN model, another ANN has been developed for particular cases where information from the

Table 4 List of variables. Variable name

Unit

Measurement range

xi

Type of day (Working day/Non-working day)

– –

{0,  1}

x1

Hour of the day (Sine) Hour of the day (Cosine)

–

cos

x3

Outdoor temperature Outdoor solar radiation State of the pump B1.1 (Off/On) State of the pump B1.2 (Off/On) State of the pump B2.1 (Off/On) State of the pump B2.2 (Off/On) State of the pump B3.1 (Off/On) State of the pump B3.2 (Off/On) State of the pump B7 (Off/On) State of the boiler (Off/On) State of the absorption machine (Off/On) State of the cooling tower (Off/On) State of the heat pump (Off/On) Electric power demand added up with the electric power supplied by the PV plant

[◦ C] [W/m2 ] – – – – – – – – – – – [kW]

[− 5, 50] [0, 1440] {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} {0, 1} [0, 100]

sin

 1 241 2h , h = [0, 23]

2h , h = [0, 23] 24

x2 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17

150

R. Mena et al. / Energy and Buildings 82 (2014) 142–155

Embedding delay

1

Outdoor temperature Solar radiation Electric power demand

0.8 0.6 0.4 0.2 0 0

100

200

300

400 500 Embedding delay (τ)

600

700

800

False neighbors [%]

Minimum embedding dimension (τ = 1) 100

Solar radiation Outdoor temperature

50

0 0

1

2

3 Embedding dimension (d)

4

5

6

Fig. 10. Average mutual information with the three continuous variables for the estimation of the embedding delay and minimum embedding dimension for outdoor temperature and outdoor solar radiation.

actuators of the solar cooling installation is not available or is not convenient to be set. The philosophy is almost the same, but the input variables associated with the solar cooling installation have been removed from model inputs set. The rest of inputs are the same with equal order, except for the feedback variable (electric power demand) whose embedding dimension has been increased to dy = 20, the list of variables and their order used for this model can be checked in Tables 6 and 7, respectively. Finally, the structure has changed because the number of neurons in the hidden layer has been increased to h = 15. All these changes have been made according Table 5 List of variables with their order.

to different model tests performed in order to capture the system dynamics. A comparison of the structures used by the two proposed approaches can be observed in Table 8. According to Eq. 3 the prediction for one step can mathematically be developed as follows: the complete ANN is expressed in Eqs. (10) and (11) corresponds to the particular ANN. As it can be observed in these equations, the inputs and their optimum model order are used. More specifically, the first one (Eq. 10) uses the date time information (x1 ,x2 , x3 ), the outdoor environmental conditions (x4 , x5 ), the solar cooling installation states (x6 − x16 ) and the feedback of the output variable (x17 ). In addition, the appropriate order is used for each one of them. On the other hand, for the particular ANN (Eq. (11)), the information from the solar cooling installation is removed from the model and the order of the output variable (x6 ) is increased to 20

xi

x1

x2

x3

x4

x5

x6

x7

x8

x9

 d

1 1

1 1

1 1

1 4

1 4

1 5

1 5

1 5

1 5

xi

x10

x11

x12

x13

x14

x15

x16

x17

x7 (n), . . ., x7 (n − 4), x8 (n), . . ., x8 (n − 4),

 d

1 5

1 5

1 5

1 5

1 5

1 5

1 5

1 3

x9 (n), . . ., x9 (n − 4), x10 (n), . . ., x10 (n − 4),

yˆ (n + 1|n) =

f (x1 (n), x2 (n), x3 (n), x4 (n), . . ., x4 (n − 3), x5 (n), . . ., x5 (n − 3), x6 (n), . . ., x6 (n − 4),

x11 (n), . . ., x11 (n − 4), x12 (n), . . ., x12 (n − 4), Table 6 List of variables for the particular ANN.

x13 (n), . . ., x13 (n − 4), x14 (n), . . ., x14 (n − 4),

Variable name

Unit

Measurement range

xi

x15 (n), . . ., x15 (n − 4), x16 (n), . . ., x16 (n − 4),

Type of day (Working day/Non-working day)

–

{0, 1}

x1

x17 (n), . . ., x17 (n − 2))

Hour of the day (Sine)

–

Hour of the day (Cosine)

–

Outdoor temperature Outdoor solar radiation Electric power demand added up with the electric power supplied by the PV plant

(10)

1  sin 24  1 2h , h = [0, 23]

x2

[− 5, 50] [0, 1440] [0, 100]

x4 x5 x6

cos

◦

[ C] [W/m2 ] [kW]

2h , h = [0, 23] 24

x3

Table 7 List of variables with their order for the particular ANN. xi

x1

x2

x3

x4

x5

x6

 d

1 1

1 1

1 1

1 4

1 4

1 20

Table 8 Comparison of the structures used in the two proposed approaches. Approach Characteristics

Complete ANN

Particular ANN

Number of inputs Number of hidden layers Number of neurons in the hidden layers Activation function in the hidden layers

17 1 10 Tangent hyperbolic 1 Linear Yes 3

6 1 15 Tangent hyperbolic 1 Linear Yes 20

Number of output neurons Activation function in the output neurons Output feedback to the input? Output feedback dimension

R. Mena et al. / Energy and Buildings 82 (2014) 142–155 Table 9 Battery of tests performed. Test

Day

Temperature

Radiation

Datetime

A B C D

Working day Working day Non-working day Working day

Winter Summer Summer Summer

Cloudy Cloudy Sunny Sunny

02/15/2011 06/30/2011 08/06/2011 09/14/2011

yˆ (n + 1|n) =

f (x1 (n), x2 (n), x3 (n), x4 (n), . . ., x4 (n − 3), x5 (n), . . ., x5 (n − 3), x6 (n), . . ., x6 (n − 19))

(11)

yˆ (n + N|n) =

f (x1 (n + N − 1|n), x2 (n + N − 1|n), x3 (n + N − 1|n), x4 (n + N − 1|n), . . ., x4 (n + (N − 3) − 1|n), x5 (n + N − 1|n), . . ., x5 (n + (N − 3) − 1|n), x6 (n + N − 1|n), . . ., x6 (n + (N − 4) − 1|n), x7 (n + N − 1|n), . . .x7 (n + (N − 4) − 1|n), x8 (n + N − 1|n), . . ., x8 (n + (N − 4) − 1|n),

100% n

n  E  . i=1  Yi 

• MAE (kW): Mean absolute error. It is defined by MAE = • ␴ (kW): Standard deviation of the error. • EMax (kW) : Maximum error.

1 n

n i=1

|E|.

For the prediction process, at timestep t = k, the ANN generates a prediction for the time interval from t = k + 1 to t = k + N, where N is the prediction horizon. Therefore, in the timestep t = k, the ANN receives as inputs for the prediction interval: the type and hour of the day, weather variables (outdoor temperature and solar radiation) and the state of the actuators from the solar cooling installation. After that, the ANN generates a recursive prediction feeding back the output in a closed loop. In the next subsections, results (see Table 10) for different prediction horizons (N = [1, 60, ∞]) will be commented, firstly for the complete ANN approach (Fig. 11) and following for the particular ANN approach (Fig. 12). 5.1. Prediction for one step ahead

x9 (n + N − 1|n), . . ., x9 (n + (N − 4) − 1|n), x10 (n + N − 1|n), . . ., x10 (n + (N − 4) − 1|n),

• Prediction error, or E: Firstly, the prediction error is defined as the difference between the prediction output or model response and the real known values or target values. Thus, being Yˆ the the vector with n predictions and Y the vector which contains n ˆ i − Yi . real values, the prediction error is defined by E = Y i=1 • CV-RMSE (%): Coefficient of variation of the root-mean-square deviation. It is usually used to compare the performance for different prediction models. It is defined by CV − RMSE = RMSE , Y¯ √ being RMSE the root mean square error RMSE = MSE which is n calculated by MSE = 1n i=1 (E)2 . • MAPE (%): Mean absolute percentage error. It is defined by  MAPE =

Finally, according to Eq. (5) the multistep prediction can mathematically be developed as follows. The full ANN approach is expressed in Eqs. (12) and (13) correspond to the particular ANN approach. The structure of the inputs and their order is the same than for the previous prediction task, but in this case the information is retrieved not from the current timestep n, but from the related timesteps in the future as this task is intended for multistep ahead prediction.

151

(12)

x11 (n + N − 1|n), . . ., x11 (n + (N − 4) − 1|n), x12 (n + N − 1|n), . . ., x12 (n + (N − 4) − 1|n), x13 (n + N − 1|n), . . ., x13 (n + (N − 4) − 1|n), x14 (n + N − 1|n), . . ., x14 (n + (N − 4) − 1|n), x15 (n + N − 1|n), . . ., x15 (n + (N − 4) − 1|n), x16 (n + N − 1|n), . . ., x16 (n + (N − 4) − 1|n),

Firstly, a prediction horizon N = 1 has been used. In this case, the output from the ANN is not feedback to its input given the short prediction horizon (one-step). The results are presented in Fig. 11 and in Table 10, where a comparison among different tests (Table 9) based on the statistical parameters presented previously has been performed. For all the tests performed, the results are excellent as can be observed in Fig. 11. The CV-RMSE is around [1–1.5 %], except for the day with the heat pump active in the building, where a CV-RMSE around 3% and a maximum error of 4.5 kW have been obtained as can be seen in Table 10. Anyway, results are excellent because of the short prediction horizon which has been used.

ˆ − 1|n),. . ., xˆ 17 (n + (N − 2) − 1|n)) xˆ 17 (n + N yˆ (n + N|n) =

f (x1 (n + N − 1|n), x2 (n + N − 1|n), x3 (n + N − 1|n), x4 (n + N − 1|n), . . ., x4 (n + (N − 3) − 1|n), x5 (n + N − 1|n), . . ., x5 (n + (N − 3) − 1|n),

(13)

xˆ 6 (n + N − 1|n), . . ., xˆ 6 (n + (N − 19) − 1|n)) 5. Results and discussion All the experiments have been performed using the Test Data Set, whose data has not been used for training nor validating the model. The performed experiments are referenced in Table 9. This battery of tests has been designed with the aim to check the ANN response with a representative set of conditions including working and non-working days, different temperature conditions for winter and summer, different radiation conditions with cloudy and sunny days and also a test for a day with the heat pump active in the building. For reference, the nomenclature used for the performance indexes and statistical parameters are:

5.2. Prediction for 60 steps ahead In this case, the same battery of tests has been executed using a different prediction horizon. More specifically, a prediction horizon equal to 60 (N = 60) has been chosen taking into account the time when the energy prices changes, approximately each hour, and the temperature dynamics of the most characteristic rooms of the building evaluated by LTI models which provided a time constant close to 60 min. As can be seen in Fig. 11, results are good except for the day with heat pump activated. This day, the ANN recognizes the electric power demand dynamics but it is wrongly dimensioned; with a CVRMSE of 33.29% and a maximum error of 33.74 kW. For the rest of days, the CV-RMSE is around [9–12 %] and the maximum error is not higher than 8 kW. All these results may be checked in Table 10. 5.3. Prediction for infinite steps ahead (simulation) Finally, the last benchmarking has been done using a prediction horizon N =∞, which is also known as dynamic modeling or simulation. The goal is to know the ANN model capacity to react to

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Table 10 Results obtained for different prediction horizons with both approaches: complete and particular ANN. Test

CV-RMSE (%)

MAPE (%)

Variation (kW)

MAE (kW)

 (kW)

EMax (kW)

Results for complete ANN 1-step ahead prediction 1.00 A B 1.25 1.33 C D 3.16

0.80 1.04 1.06 2.08

[19.12–40.88] [18.47–42.49] [13.42–23.05] [18.80–55.93]

0.21 0.28 0.19 0.73

0.17 0.22 0.14 0.76

1.18 1.49 0.99 4.47

60-step ahead prediction 12.72 A 8.99 B C 9.60 D 33.29

10.67 8.22 6.57 20.46

[19.12–40.88] [18.47–42.49] [13.42–23.05] [18.80–55.93]

3.11 2.21 1.17 8.10

1.71 1.39 1.29 7.69

8.15 7.68 6.98 33.74

∞-step ahead prediction (simulation) A 16.37 B 9.23 C 11.76 D 30.37

13.60 8.47 8.33 18.89

[19.12–40.88] [18.47–42.49] [13.42–23.05] [18.80–55.93]

4.02 2.27 1.50 7.37

2.15 1.42 1.52 7.03

9.77 7.85 7.03 34.50

Results for particular ANN 1-step ahead prediction A 1.00 1.06 B 1.17 C 2.41 D

0.81 0.87 0.91 1.73

[19.12–40.88] [18.47–42.49] [13.42–23.05] [18.80–55.93]

0.22 0.24 0.16 0.60

0.17 0.19 0.13 0.55

1.19 1.25 0.90 2.77

60-step ahead prediction 9.37 A B 12.35 5.49 C D 26.10

6.26 7.96 4.32 14.83

[19.12–40.88] [18.47–42.49] [13.42–23.05] [18.80–55.93]

1.86 2.55 0.78 5.98

1.84 2.52 0.62 6.39

8.15 11.48 2.95 32.29

∞-step ahead prediction (simulation) A 26.65 B 20.70 C 8.02 36.92 D

19.27 15.17 5.99 26.58

[19.12–40.88] [18.47–42.49] [13.42–23.05] [18.80–55.93]

6.01 4.80 1.10 10.12

4.36 3.60 0.95 7.15

14.67 15.50 4.16 32.92

different environment situations with specific conditions and without the knowledge of the real status, this is done by feeding back the ANN’s output back to its input recursively. As can be seen in Fig. 11, the obtained results are not bad. The ANN responds in an appropriate way and it is able to capture the dynamics of the system for the different type of tests (Table 9). For the test (A), a cloudy winter working day, the output from the ANN follows the dynamics of the real time series, although the prediction is a bit underestimated and may be due to the specific conditions

of that day (cloudy and colder than the average day of Almeria), yet the CV-RMSE is 16.37% and the maximum error is 9.77 kW. For test (B), a cloudy summer working day, the ANN responds quite good to the building dynamics and produces a very accurate prediction with a CV-RMSE of 9.23% and a maximum error of 7.85 kW. Test (C) is a sunny summer non-working day and thus the electric power demand is more stationary, the ANN response is very good with a CV-RMSE of 11.76% and a maximum error of 7.03 kW. Finally, in test (D), the simulation results are worse because the ANN captures 50 Power demand [kW]

Power demand [kW]

50 40 30 20 10 00:00

08:00 16:00 Time [HH:MM]

20

Real power demand Prediction (1−step) Prediction (60−step) Prediction (Simulation)

08:00 16:00 Time [HH:MM]

24:00

08:00 16:00 Time [HH:MM]

24:00

80 Power demand [kW]

Power demand [kW]

30

10 00:00

24:00

30 25 20 15 10 00:00

40

08:00 16:00 Time [HH:MM]

24:00

60 40 20 0 00:00

Fig. 11. Prediction results for different tests using the complete neural network.

R. Mena et al. / Energy and Buildings 82 (2014) 142–155

50 Power demand [kW]

Power demand [kW]

50 40 30 20 10 00:00

08:00 16:00 Time [HH:MM]

Power demand [kW]

Power demand [kW]

30 20

08:00 16:00 Time [HH:MM]

24:00

08:00 16:00 Time [HH:MM]

24:00

60

20

10 00:00

40

10 00:00

24:00

25

15

153

Real power demand Prediction (1−step) Prediction (60−step) Prediction (Simulation)

08:00 16:00 Time [HH:MM]

24:00

50 40 30 20 10 00:00

Fig. 12. Prediction results for different tests using the particular neural network.

the energy consumption dynamics but underestimating it, the CVRMSE is 30.37% and the maximum error is 34.50 kW. In general, the simulation results (prediction when the prediction horizon tends to infinity N =∞) obtained by the ANN model are good. The most challenging scenario is when the heat pump is active on the building producing an unstable energy load curve along the day. All these results may be checked in Table 10.

5.4. Prediction results using the particular neural network In this section, results based on different tests (Table 9) for different prediction horizons (N = [1, 60, ∞]) using the particular ANN with less input information will be presented. This ANN is useful under certain situations where information from solar cooling installation is not available or is not convenient to be set.

5.4.1. Prediction for one step ahead As in the previous approach, the first case is to use the particular ANN with a short prediction horizon, N = 1. In this case, the obtained results are excellent for all the tests executed as it can be seen in Fig. 12 and in Table 10. The CV-RMSE obtained is not higher than 2.41% and the maximum error obtained is 2.77 kW in the performed tests. In comparison with the previous approach using information from the solar cooling installation, the results are almost identical and with an excellent performance.

5.4.3. Prediction for infinite steps ahead (simulation) For the case of simulation, that is, using a prediction horizon N =∞, the results are shown in Fig. 12. As it can be observed, for test (A) the ANN output is poor because the dynamics are represented but they are very underestimated. This may be due to the lack of information from the solar cooling installation, the CV-RMSE is 26.65% and the maximum error is 14.67 kW. A similar situation occurs with test (B) because the dynamics is fairly represented but they are underestimated, the CV-RMSE is 20.70% and the maximum error is 15.50 kW. In test (C), the results are good because it is a non-working day and thus the electric load curve is almost stationary, the CV-RMSE is 8.02% and the maximum error is 4.16 kW. Finally, in test (D), the results are very bad because the dynamics are poorly represented due to the lack of information from the heat pump state and, in addition, the energy consumption is underestimated, the CV-RMSE is 36.92% and the maximum error is 32.92 kW. The rest of statistical parameters may be checked in Table 10. In general, the absence of information from the solar cooling installation is noticeable as the ANN model prediction output is worse and with less dynamic representation. In comparison with the previous approach using information from the solar cooling installation, the results for this simplified approach are worse than for the full ANN approach and in addition, this approach loses the capacity to represent the dynamics of the building energy consumption because of the lack of system information.

6. Conclusions 5.4.2. Prediction for 60 steps ahead A prediction horizon of N = 60 has been studied with this particular ANN for the same reasons which were commented previously. As it is shown in Fig. 12, the prediction results are very acceptable for tests (A), (B) and (C) where a maximum CV-RMSE of 12.35% and a maximum error of 11.48 kW are obtained. However, the results obtained for test (D) are a bit worse because the ANN underestimate the power demand although it captures the dynamics acceptably. The rest of statistical parameters can be checked in Table 10. In comparison with the previous approach using information from the solar cooling installation, the results for this approach are slightly better which could be due to the length of the prediction horizon and the greater importance of weather variables over the solar cooling installation, at least in these cases.

In this work, a model based on ANNs for the electric demand prediction of a solar energy research center, the CIESOL building, has been presented. After a careful design of the system identification which has lead to the determination of the most relevant factors on the electric demand and the detection of possible consumption patterns, the construction of the ANN model has been done following a methodology to take into account several important factors. This study has revealed that the more influential factors on the building energy consumption are the outdoor temperature and the solar radiation, and in addition, information from the solar cooling installation is very valuable, specially the state of the heat pump. In fact, two approaches have been proposed, one taking into account all the information related to the solar cooling installation and a simplified approach without it.

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In order to analyze the properly behaviour, different tests have been performed. These tests include different types of days and energy consumption patterns. In addition, different prediction horizons have been evaluated, more specifically one-step ahead prediction (N = 1 min), 60-step ahead prediction (N = 60 min) and multistep ahead prediction also known as dynamic modeling or simulation (N =∞ min). For the complete approach, results for N = 1 are excellent with a CV-RMSE of [1–3 %]. For N = 60, the results are good with a CV-RMSE of [9–30 %]. For N =∞, the CV-RMSE is [9–30 %]. In general, the obtained results are good and the ANN recognizes the building dynamics. Furthermore, a second ANN prediction model has been developed with the aim of using less input information regarding the solar cooling installation. Hence, this ANN is intended to be used in certain situations where information from solar cooling installation is not available or it is not convenient to be set. This ANN model has been evaluated with the same battery of tests than the previous ANN model and with the same prediction horizons. For N = 1, the CV-RMSE is around [1–2.40 %]. For N = 60, the CV-RMSE is around [5–26 %]. For N =∞, the CV-RMSE is around [8–36 %]. In general, results are good but the ANN prediction model loses capacity for the representation of the building energy consumption dynamics. Considering the complexity of the building system which is composed of different kind of elements, such as HVAC systems, illumination, solar cooling installation, chemistry laboratories, etc., the results are acceptable both for the complete ANN prediction model and the reduced version. In addition, the obtained ANN model allows to generate a quick prediction of the building electric demand with acceptable results as a function of the complexity of the system. Therefore, it is a suitable tool for the electric energy demand prediction of this building which could be integrated with different control techniques. Moreover, the development of this work has allowed the identification of the system and the recognition of different energy demand profiles which could facilitate future analysis and works. In addition, the methodology used in this paper can be extrapolated to other similar problems. Future works could study the disaggregation of the energy system into different subsystems which could facilitate the building energy consumption dynamics recognition and load prediction, specially that which comes from the solar cooling installation and that it is difficult to predict. In addition, another techniques, specially artificial intelligence like support vector machines or genetic programming, could be used to compare the performance. Acknowledgements The authors are very grateful to Andalusia Regional Government (Spain), for financing this work through the Programme “Formación de personal docente e investigador predoctoral en las Universidades Andaluzas, en áreas de conocimiento deficitarias por necesidades docentes (FPDU 2009)”. This is a programme cofinanced by the European Union through the European Regional Development Fund (ERDF). This work has been partially funded by the following projects: PSEARFRISOL PS-120000-2005-1 and DPI2010-21589-C05-04 (financed by theSpanish Ministry of Science and Innovation and EU-ERDF funds). PHB2009-0008 (financed by the Spanish Ministry of Education; CNPq-BRASIL; CAPESDGU 220/2010). The authors would like to thank all companies and institutions included in the PSE-ARFRISOL project. References [1] W.E.C. United Nations Department of Economic & Social Affairs, World Energy Assessment – Energy and the Challenge of Sustainability, United Nations Development Programme, 2000. [2] G. Strbac, Demand side management: benefits and challenges, Energy Policy 36 (2008) 4419–4426.

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