Suggesting a stochastic fractal search paradigm in combination with artificial neural network for early prediction of cooling load in residential buildings

: Early prediction of thermal loads plays an essential role in analyzing energy-efficient buildings' energy performance. On the other hand, stochastic algorithms have recently shown high proficiency in dealing with this issue. These are the reasons that this work is dedicated to evaluating an innovative hybrid method for predicting the cooling load (CL) in buildings with residential usage. The proposed model is a combination of artificial neural networks and stochastic fractal search (SFS-ANN). Two benchmark algorithms, namely the grasshopper optimization algorithm (GOA) and firefly algorithm (FA), are also considered to be compared with the SFS. The non-linear effect of eight independent factors on the CL is analyzed using each model's optimal structure. Evaluation of the results outlined that all three metaheuristic algorithms (with more than 90 % correlation) can adequately optimize the ANN. In this regard, this tool's prediction error declined by nearly 23, 18, and 36 % by applying the GOA, FA, and SFS techniques. Also, all used accuracy criteria indicated the superiority of the SFS over the benchmark schemes. Therefore, it is inferred that utilizing the SFS along with ANN provides a reliable hybrid model for the early prediction of CL.


Introduction
Buildings, vehicles, and industry are the three primary energy consumption sectors globally [1][2][3]. Among those, buildings are consuming a considerable share, so that it is anticipated to reach over 30 % by 2040 [4]. On the other hand, people's high tendency to dwell in smart cities has resulted in the idea of developing energy-efficient structures [5,6]. Therefore, accurate analysis of buildings' energy performance (EPB) is a significant step toward this objective. In energy-efficient buildings, heating load (HL) and cooling load (CL) demand account for system energy consumption and are controlled by heating, ventilating, and air conditioning (HVAC) system [7] to provide convenient indoor air condition. It is well established that the cost and use of energy affect human lives every day. In this sense, many issues arise from the content of energy consumption such as acid rain, dependency on depleting supplies of fossil fuels, greenhouse gas emissions [8][9][10][11][12][13][14][15][16][17], climate change [18][19][20][21], as well as environmental concerns that come along with energy power supply [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. In recent years, various techniques have been used for the optimal design of the HVAC system [36][37][38][39][40][41][42][43][44]. Ghahramani, et al. [45], for instance, use a systematic model to optimize the performance of the HVAC system in office buildings in terms of temperature setpoints. Also, Ferreira, et al. [46] used a soft computing method for controlling the HVAC system. Implemented the proposed model led to about 50 % saving in energy consumption. The effect of façade parameters (e.g., solar reflectance and U-value) was investigated by Ihara, et al. [47], and it was deduced that solar heat gain coefficient (SHGC) could have the most significant impact on energy consumption reduction.
In this way, some drawbacks of forwarding modeling approaches (e.g., modeling in simulation packages), like large dimensions and inadequacy for occupied spaces [48], have driven engineers to employ artificial intelligence approaches for early estimation of energy [49][50][51]. Artificial intelligence is known as intelligence solutions demonstrated by machines, unlike the natural intelligence provided by animals and humans [52][53][54][55][56][57]. Artificial neural networks (ANN), as one of the most known AI-based solutions, have received increasing attraction recently [58][59][60][61][62]. More technically, deep learning-based [63][64][65][66], machine learning [67][68][69], decision making-based theories, feature selection-based solutions [70][71][72], extremer machine learning solutions [73][74][75][76], as well as hybrid searching algorithms that enhanced conventional multilayer perceptron like harris hawks optimization [77,78], whale optimizer [79,80], bacterial foraging optimization [81], chaos enhanced grey wolf optimization [82], moth-flame optimizer [74,83], many-objective sizing optimization [84][85][86][87][88][89], Driven Robust Optimization [90], ant colony optimization [91], and global numerical optimization [92]. These techniques are successfully employed in different aspects such as building design [93][94][95][96][97][98][99][100], image processing/classification [101][102][103][104][105][106][107][108], sustainability, and environmental concerns [24, [109][110][111]. Various studies are also performed on predicting the HL and CL of residential buildings [41,43,44,[112][113][114]. Zhou, et al. [44] used an ANN and the nonlinear autoregressive with exogenous inputs (NARX) concept for accurate analysis of thermal load in an academic building. In a similar effort, Koschwitz, et al. [115] implemented an approximation of long-term urban heating load using nonlinear autoregressive exogenous recurrent neural networks (NARX RNN). Roy, et al. [116] showed a better performance of neural networks (with a variance accounted for 99.76 %) compared to other conventional methods such as gradient boosted machines. Other predictors like adaptive neuro-fuzzy inference systems (ANFIS) have shown high robustness for dealing with EPB non-linear problems [117]. Pezeshki and Mazinani [118] concluded the superiority of ANFIS over typical fuzzy logic in buildings' thermal efficiency, as it enjoys the advantages of the ANN as well. Scholars like Naji, et al. [119] and Chou and Bui [120] have demonstrated the applicability of ANFIS and support vector-based methods. More recent studies have suggested utilizing conventional predictors with metaheuristic search schemes for various usages [43,121]. More significantly, for energy consumption modeling, such methods have gained high popularity. Satrio, et al. [122] found integrating ANN and multi-objective genetic algorithms a capable tool for optimizing the HVAC system. Moayedi, et al. [4] coupled an ANN with a firefly algorithm based on electromagnetism for predicting building energy consumption. The findings revealed their suggested model's superiority over conventional models like typical ANN, genetic programming, and extreme learning machine. Tien Bui, et al. [43] could reduce the HL prediction error of the ANN from 2.93 to 2.06 and 2.00 by applying the genetic algorithm and imperialist competition algorithm, respectively. This error also fell down from 3.28 to 2.09 and 2.10 for CL estimation. Particle swarm optimization (PSO) is another capable optimizer which has been widely employed for creating hybrid models [123,124]. Goudarzi, et al. [125] also used the PSO to develop a hybrid of autoregressive integrated moving average and SVR for energy consumption modeling. Moreover, in researches by Moayedi, et al. [126] and Moayedi, et al. [127], novel metaheuristic techniques (e.g., elephant herding optimization, Harris hawks optimization, and grasshopper optimization algorithm) were applied to enhance the prediction capability of the ANN. Despite the wide use of natureinspired metaheuristic techniques for optimizing the HVAC system, the wide variety of these techniques motivated the authors to investigate the applicability of a novel member of this family, namely stochastic fractal search in this paper. Based on our best knowledge, this algorithm has not been previously used in this field.

Methodology 2.1 ANN
As suggested by McCulloch and Pitts [128], the basic idea of ANNs roots in the biological neural system of humans. The structure of this model comprises several processors named neurons connected by so-called synapses "weight." As a potential advantage of this model, the ANNs tries to analyze the non-linear association of a set of input-output data by establishing a stage of mathematical relationships [129]. Figure 1 shows the structure of one of the most popular notions of ANNs, namely multi-layer perceptron (MLP). As is seen, this network is composed of one hidden layer. However, based on the problem's complexity, it can contain two or more hidden layers [130,131]. Input nodes receive the data. Each node assigns (i.e., multiplies) a specific weight factor (W), and the obtained value is added to a bias factor (b). Lastly, an activation function (f) is applied to the whole term to generate the proposed neuron's response. It is mathematically expressed by Equation 1: (1)

Stochastic fractal search
Proposed by Salimi [132], stochastic fractal search (SFS) is a recently-developed metaheuristic algorithm. This method is based on the fractal mathematical concept and the diffusion feature. Mosbah and El-Hawary [133] used this approach for training an ANN. The SFS draws on two major processes, namely diffusing and appraising, for handling the optimization task. The appraising process is a random manner leading to exploiting features. The diffusing stage is for exploiting features. It increases the chance of a search agent catching the global minimum and also keeps the agent from getting trapped in local solutions. It is carried out by a static diffusion procedure, which indicates that only the elite agent is taken into consideration. Therefore, other solutions (belonging to the rest of the agents) are discarded [134].
Assuming pb and pi, respectively, as the elite and typical agent, the Gaussian walk in the diffusion process is expressed as follows: in which G (mn , δ) indicates the Gaussian distribution function with mean mn and standard deviations δ. z shows the iteration, and two terms a1 and a2 are random values varying from 0 to 1. Also, mnp and mnBp equal to and , respectively. More information about the SFS algorithms is detailed in similar studies like [135,136].

Benchmark optimizers
As mentioned, the grasshopper optimization algorithm (GOA) and the firefly algorithm (FA) is used to create benchmark hybrids of ANN. Like any other algorithms, these techniques implement their certain search method to attain a globally optimal solution for a given problem [137]. As the name implies, the GOA is inspired by the herding behavior of grasshoppers in nature. It was designed by Saremi, et al. [138] in two significant steps, including exploration and exploitation. The herd members (i.e., the grasshoppers) fly to find food ( Figure 2), where their swimming movement is affected by three parameters of social relationship, gravity force, and wind advection. More details about the GOA can be found in previous studies [139][140][141].
The FA is also a capable nature-inspired search scheme proposed by Yang [142]. The FA represents the social flashing behavior of fireflies. The intensity variation of light and attractiveness formulation are two crucial ingredients of this algorithm [143]. Three basic rules of the FA are: (i) the individuals are unisex and are attracted to each other regardless of gender, (ii) the more brightness, the higher attractiveness, and (iii) the objective function of the problem determines the brightness of a firefly. More mathematical information about the fireflies' interaction is well-explained in references like [144][145][146].

Data and statistical analysis
As is known, using reliable data plays an essential role in the proper development of intelligent models. In the case of this study, the information of a residential building is used. The database is produced by a computer simulation in Ecotect [147] environment. Tsanas and Xifara [148] generated it, and it is now available on http://archive.ics.uci.edu/ml/datasets/Energy+efficiency. By analyzing 12 different buildings (with a total volume of 771.75 m 3 ), the HL and CL of 768 cases are collected when relative compactness (RC), surface area (SA), wall area (WA), roof area (RA), overall height (OH), orientation, glazing area (GA), and glazing area distribution (GAD) are taken as influential parameters (i.e., independent factors). The statistical description of these parameters (in terms of average value, sample variance, standard error, minimum, and maximum) are presented in Table 1. Figure 3 also illustrates the relationship between the CL and the mentioned parameters. In order to develop the proposed models, the dataset should be divided into two parts, namely training and testing groups. In this work, it is done with respect to the broadly taken proportion of 80:20. In other words, out of all 768 data, 80 % (i.e., 614 samples) are used in the training phase for analyzing the relationship between the CL and influential parameters, and the remaining 20 % (i.e., 154 samples) are considered as testing data. This group is given to the trained networks to assess their generalization capability.

Results and discussion
This study aims to evaluate the applicability of three notions of metaheuristic algorithms, namely stochastic fractal search, grasshopper optimization algorithm, and firefly algorithm in the early estimation of cooling load in energy-efficient buildings. In this research, the algorithms are contributed to the problem through optimizing the parameters of an ANN used for estimating the CL by analyzing environmental parameters. The prediction results of the MLPNN tool, as well as the GOA-ANN, FA-ANN, and SFS-ANN ensembles, are presented and discussed in this part.

Hybridizing the MLPNN using SFS, FA, and GOA
In order to optimize an ANN using optimizer algorithms, they should be mathematically synthesized [149,150]. But before doing this, the best structure of the ANN should be determined. To this end, the influential parameters, especially the number of hidden neurons, should be optimized. Based on previous studies, as well as the authors' experience, tangent-sigmoid (Tansig) is selected as the activation function of the ANN neurons. Then, ten different MLPNN structures (with the number of hidden neurons from 1 to 10) are tested, and it was shown that six neurons generate the most reliable hidden layer. Next, the predictive structure of the proposed MLPNN is given the GOA, FA, and SFS algorithms to construct the hybrid models. Given the training dataset, these algorithms try to find highly optimized weights and biases to build an efficient MLPNN. A total of 1000 repetitions is considered for each algorithm to have enough convergence. For each iteration, an evaluation is carried out by means of an objective function which is set root mean square error (RMSE) in this study. The RMSE is formulated as follows: where the number of involved data is shown by Q, CLi observed and CLi predicted stand for the CL values obtained by Tsanas and Xifara [148] and our intelligent models, respectively. The size of the working population is an influential factor for the performance of the implemented ensemble models. Thus, a trial and error process is taken to ensure that the best population size is used during the optimization. Each model is implemented with nine population sizes of 10, 25, 50, 75, 100, 200, 300, 400, and 500 and the RMSE of the latest iteration is recorded as the best response. The results are shown in Figure 4-(a). As is seen, the best optimization results for the GOA, FA, and SFS algorithms with 400, 100, and 400 population sizes, respectively. The optimization curves of the mentioned structures are also shown in Figure 4 -(b), (c), and (d). These models are then used to estimate the CL.

Accuracy criteria
Along with the RMSE, mean absolute error (MAE) is used to report the error of prediction in both training and testing phases. Besides, the accommodation of the results is evaluated by the coefficient of determination (R 2 ) which varies from 0 to 1. The larger the R 2 is, the higher the regression between the results becomes. These two indices (i.e., MAE and R 2 ) are expressed by the Equations 6 and 7, respectively.
where the average of CLi observed values are addressed by ̅̅̅̅ observed.

Accuracy evaluation
In the training phase, assessing the products showed a good understanding of the relationship between the CL and considered inputs for all used predictors. A comparison between the measured and predicted values of CL is also presented in Figure 5. It can be seen that the models can correctly forecast the overall CL pattern. In addition, the used error criteria indicate a reasonable error in the learning process.    (Table 2).  Moreover, the evaluation of the accuracy is done by measuring the error of testing data. In this sense, Figure 7 -(a) to (h) depicts the calculated error (= measured CL -predicted CL)   Table 3 gives a summary of the results. Based on the resulted accuracy criteria, three major points can be deduced: (i): All used artificial intelligence models are capable enough to learn properly, and subsequently, estimate the CL pattern for buildings with unseen conditions.

Presenting the neural predictive formula
As explained, all evidence asserted the superiority of the SFS technique in optimizing the ANN in this study. It means that the solution matrix (i.e., ANN weights and biases) proffered by this scheme develops a more flexible neural network. Therefore, in this section, the SFS-ANN ensemble is presented in the form of a neural-metaheuristic formulation to predict the HL by taking the effective parameters into the equation. It is expressed by Equation 8, which is made of six weights and one bias term belonging to the unique output neuron. As can be seen, this formula needs to receive some inputs shown by A, B, …, F. These parameters symbolize the response of the hidden neurons (see Figure 1), which are calculated by Equation 9. Considering the number of input parameters and hidden neurons, the main matrix comprises eight columns and six rows which are multiplied by the inputs matrix. Next, the bias matrix is added, and the Tansig activation function is applied.

Conclusions
In this work, for the first time, three robust optimizers of stochastic fractal search, grasshopper optimization algorithm, and firefly algorithm were applied to the problem of the cooling load early prediction. These algorithms were used to optimize the performance of a neural network by adjusting the computational parameters. The findings revealed, first, the applicability of neural computing for analyzing the non-linear relationship between the CL and eight environmental parameters. Secondly, all three metaheuristic algorithms could promisingly optimize the ANN for the betterment of accuracy. Last but not least, from a comparison among the results, it was concluded that the SFS-based hybrid model enjoys more accuracy than GOA and FA. The governing neural-metaheuristic relationship of the SFS-ANN was also presented in the form of a viable formula for mathematically predicting the CL.

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Chen, Y.; He, L.; Li, J.; Zhang, S., Multi-criteria design of shale-gas-water supply chains and production systems towards optimal life cycle economics and greenhouse gas emissions under uncertainty.