Home Thermal Modeling : Cooling Energy Consumption and Costs in Saudi Arabia

Objectives: The consumption of electricity and its costs are expected to be increased in Saudi Arabia due to its rapid growth in population. As the Kingdom is characterized by extreme hot climate, a massive amount of electricity consumed by the residential sector goes to power air conditioners. To control this huge amount of energyconsumedin homes, thermal models have been generated with two or more parameters. Methodology: The households’ surveys have been conducted in order to collect the data. The Non-linear regression analysis has been carried out to obtain the outcomes of study. Moreover, household surveys have been conducted for data collection. The grid algorithm and the non-linear regression have been used to learn the parameters in the model to simulate the weather in Saudi Arabia. The temperature loggers have been placed in the houses to observe the behavior of residents of using cooling system. The web forecast has been used to analyze the temperature of cities on hourly basis. Results: Simple thermal model has been built using two parameters by applying the grid and non-linear regression methods for data fitting. Then the thermal model with envelope has also been created using four parameters by applying non-linear regression method for data fitting. Conclusion: It has been evaluated through outcomes that thermal model with envelope is better as compared to simple thermal model. Moreover, the data fitting by non-linear regression method has also been observed to perform better than data fitting by grid method.


Introduction
Energy consumption has been enhancing throughout the world, as a result of which, majority of the countries are facing unprecedented expansion in electricity infrastructure.A stable agriculture and irrigation system also tends towards a great need of energy consumption.It has been evaluated that the modern agricultural techniques are widely helpful for retrieving outcomes related to the severe energy related complications.A past study indicated that the implementation of modern irrigation system can easily improve the energy consumption aspects in an effective way (Valipour, 2015a).Residential sector is the third-largest major consumer of energy in the world, which represents almost 27% of total consumption (Lausten, 2008).During the time span of 2010, the electricity expansion has been made in residential sector with 426 Mtoe (17.84 EJ).This expansion of electricity has experienced almost 29% of dramatic increase in the last decade.Thus, the enhancement in energy efficiency of electrical equipments is not only influential for the conservation of energy in residential areas, but it also helps in reducing the load on electrical generators (Nejat, Jomehzadeh, Taheri et al., 2015).There are many challenges regarding engineering structures that can influence the future needs of energy consumption.For example, many factors within a region can affect the overall energy consumption of that region such as weather, population etc, also energy efficiency and incorporation of energy resources with transmission of electricity (Valipour, 2015b).Taking into consideration the context of Saudi Arabia, the consumption of electricity and its costs are expected to enhance due to its growing population year by year (World Population Statistics, 2013).Saudi Arabia lies in the tropics between 16 ○ N and 32 ○ N latitudes and 37 to 52 ○ E longitudes, due to which it is one of the hottest countries with low humidity in the world.The climate of Saudi Arabia comprised of extreme heat and aridity (Krishna, 2014).Due to extreme heat and hot climate of Saudi Arabia, the residential sector has been observed to consume a massive amount of electricity to power air conditioning (Saudi Electricity Company, 2016).It is important to find solutions to minimize the consumption of air conditioners or cooling systemsin the Kingdom of Saudi Arabiain order to save the energy consumption.Moreover, these solutions involve saving the overstated costs, which are expected to be spent by Saudi Arabia.
In order to minimize and control the consumption of cooling and costs in residential homes, the first step is to study their environment thermal parameters that affect the work of cooling systems.The thermal models for residential homes have been established to further perform the study.Thermal modeling helps to understand the generation or loss of cooling and heating in buildings.The flows of cooling or heating throughout a building can be related to temperature variations inside the building.Therefore, in order to minimize cooling or heating energy consumption in home, the thermal model of that home must be built (Ryder-Cook, 2009).
Home thermal model is a mathematical formula that describes internal temperature variations according to the effect of home thermal parameters.There are several parameters that affect the cooling or heating in home.Some of these parameters are thermal production power, thermal leakage rate, home envelope, i.e. the thickness of home walls and solar heat.The power of thermal production depends on the cooling or heating system brand, capacity, and engines.The thermal leakage rate depends on wall envelope thickness and ventilations.Leakage through ventilations can occur in different ways such as leakage through windows, wall gaps, and deliberate ventilations (Rogers, Maleki, Ghosh et al., 2011).The effect of solar heat depends on the orientation of outside facade and the position of sun (Nassiopoulos, Kuate, & Bourquin, 2014).Besides this, some other parameters, such as thermal capacity of room air and heat power generated by internal sources can be also considered in thermal models.Internal sources include heating provided by cooking and other energy use, as well as the heat generated by people (Guo, Li, Poulton et al., 2008;Nassiopoulos, Kuate, & Bourquin, 2014).Here, the thermal models are categorized on the basis of parameters involved in modeling formula.First, there is simple thermal model, which has been created according to thermal production power and thermal leakage rate parameters.Different thermal models can be created by adding the effect of any or many thermal parameters to simple thermal model formula.The effect of home envelope parameters is added to produce thermal model with envelope.According to the situation, adding applicable thermal parameters would give more precise thermal models (Rogers, Maleki, Ghosh et al., 2011).
Over the last few years, thermal models of buildings have been studied by different scientists and institutions to provide appropriate solutions to save the energy.An intelligent agent has been introduced by Rogers, Maleki, Ghosh et al. (2011) at the University of South Hampton in UK, in which the primary objective is to control a home heating system within a smart grid.The agent has been established with the task of learning thermal properties of home.It uses Gaussian processes to predict environmental parameters over the next 24 hours.The agent then provides real-time feedback to householders concerning the cost and carbon emissions of their heating preferences.Hagras, Packham, Vanderstockt et al., (2008) proposed a novel agent-based system for the management of energy in commercial buildings.This system has been entitled as Intelligent Control of Energy (ICE).In order to learn inside/outside conditions of thermal as well as to control the cooling/heating system, different techniques have been used by the researchers.Some of these techniques are fuzzy systems, neural networks, and genetic algorithms, which are beneficial in minimizing the costs associated with energy and also to maintain customer comfort.Moreover, a Gaussian Adaptive Resonance Theory Map (gARTMAP) has been presented by Mokhtar, Liu, & Howe (2014) for the management of heat system in buildings.The proposed model is an artificial neural network, which helps to predict and categorize the required control output by using non-linear regression.Therefore, the step towards saving energy in all previous works has been observed to occur by learning thermal characteristics of building, creating the appropriate thermal models, and predicting user consumption behavior, and then adjusting the heating/cooling system dependently.
Previous literature is mainly focused on diversified algorithms and traditional irrigation method, which could be helpful for fulfilling the consumption related aspects.However, the energy consumption rate is continuously increasing at a constant rate across the globe.Therefore, it is necessary to know about the impact of modern agricultural and irrigation systems for resolving energy demands.The significance of home thermal modeling should be identified for reducing the complications related with energy consumptions.

Building the Simple Thermal Models
In order to build thermal models, the mathematical formulations are required to be generated.In this case, the standard thermal model has been considered, in which the day is divided into a set of discrete time slots t ∈ Time.
As this work is about minimizing the energy consumption of cooling systems, it has been assumed that air conditioner cools the home.The thermal production power of air conditioners measured in Kilo Watts is given by CP.The variable COOLon∈ {1, 0} is defined for every t ∈ Time, such that COOLon = 1, if the air conditioner is actively producing cool, and otherwise 0. The value of COOLon in any time slot t is given by: On the basis of aforementioned equation, the amount of cooling energy COOLrate has been measured in C ̊ that delivered by the cooling system in any time slot t.
Where, α is the value of thermal leakage rate, is external temperature in the time slot t, and is internal temperature in the time slot t.Thus, the internal temperature in any time slot t after delivering this cooling energy is given by: Assuming that the air conditioner is controlled by a timer and thermostat, the variable that represents the air conditioner timer value is defined, TIMERon t ∈ {1, 0} for every t ∈ Time and given by: For all t ∈ Time and TIMERon t = 1, the thermostat acts to keep internal temperature T int of homes at the set point of thermostat T set by: Where, ∆ induces hysteresis, due to which thermostat cannot cycle on continuously at the set point (Rogers, Maleki, Ghosh et al., 2011;Lin, Middelkoop, & Barooah, 2012;Ma, Borrelli, Hencey et al., 2012).

Simple Thermal Model
The simple thermal model seems to be dependent on air temperature, which generally increases over the time slot due to cooling system thermal production power and decreases due to home thermal leakage: Where, denotes internal temperature in the time slot t that is predicted by thermal model (Rogers, Maleki, Ghosh et al., 2011).The simple thermal model presented above is primarily depending on two parameters: the thermal production power of air conditioner CP and thermal leakage rate of homeα.Before the creation of thermal model, it is necessary to learn CP and α since these two parameters are not measured, so they have to be estimated.The process of learning parameters is called data fitting.Generally, the data fitting is aimed at describing the data by a simpler mathematical principle that is as close as possible to the real data.Then, the fit yields the parameters in corresponding mathematical formula.Commonly, the root mean squared errors used as a measure for deviation between observed and predicted data.Thus, the data fitting means to find a minimum for root mean squared error (Betzler, 2003).The root mean squared error (RMSE) is given by: Where, X obsrv is the real observed value and X pre is the predicted value by a model at time unit i (Brennan, 2013).
In this study, different information should be known for data fitting.Observations over 24 hours of internal temperatures and external temperatures have been collected.The time periods when air conditioner timer is on and thermostat set points are needed to find the time periods when the air conditioner provides active cooling, a mentioned in Formula (5).These 24 hours observations are used by Formula (6) to predict the evolution of internal temperature over the same period initializing at = ; where is observed internal temperature in the time slot t = 1.Then, the error in this prediction E is given by: Thus, the values which minimize this error are considered as the best estimates of CP and α (Rogers, Maleki, Ghosh et al., 2011).
cis.ccsenet.For the computation of cooling cost of air conditioners, the pseudo code has been given below: Algorithm (4) input C  # air conditioner cooling capacity input B  # specific hours budget // Compute air conditioner consumption by Kilowatt for one hour // Each Ton of cooling capacity consume 1.6 Kilowatt/Hour consumption = 1.6 * C // Compute the cooling cost for one month // Only 60% of each consumption segment considered because that the air conditioner consumption represents 60% of Saudi homes bills cost 0 sum 0 for c = 1:30 // loop through month days for k = 1:B // loop through each day cooling hours sum = sum + consumption ifsum<= 60% of 1 st segment then cost = cost + (1 st segment cost * consumption) else ifsum<= 60% of 2 nd segment then cost = cost + (2 nd segment cost * consumption) elseif .else ifsum>60% of highest segment then cost = cost + (highest segment cost * consumption) end end monthly_cost = cost / 100; // convert from Halalah to Riyals The estimated air conditioner consumption cost has been calculated for House G on June 8 th from thermal model developed in Figure 4.The outcomes demonstrate a consumption cost of 160.96Riyals per month, considering that the air conditioner capacity is equal to 2 Tons.Many researches have used these regression practices which have been used in this paper, Valipour, (2015c) utilized the transfer-based models and evaluate the data by regression analysis in his study related to crop evapotranspiration.Khoshravesh et al., (2015) has propsed different regression models to analyze the monthly reference evapotranspiration.Similarly, Valipour, (2014a;2014b;2012) has also used these models to analyze the results.

Results
Statistical measurements are needed to evaluate the efficiency of thermal modeling and data fitting to be accurate and authentic.The residuals and norm of residual analysis is a measure, which is often used for the efficiency of data fit for the comparison of different data fits.The residuals are the differences between the observed data and corresponding predicted fit data.The residuals plot can give insight into the efficiency of a fit by examining it visually (MathWorks, 2014).If the model fit to the data is correct, the residuals will approximate only experimental errors that exhibit random arrangement of positive and negative residuals.This is helpful in generating a statistical relationship among the observed and predicted values.But if the model fit to data is inappropriate, the positive residuals may tend to cluster together at some parts of graph; whereas, negative residuals cluster together at other parts.Such clustering indicates that the observed values differ systematically from the predicted values (Motulsky & Ransnas, 1987).Therefore; if the residuals appear to behave randomly, it suggests that the model fits the data well.On the other hand, if non-random structure is evident in residuals, it is a clear sign that the model fits the data poorly.However, the more the residuals are randomly distributed; the best is the fit (Féménias, 2003).The norm of mathematical object is a quantity that describes the length, size or extent of this object.Mathematically, a more precise vector t has been denoted as: The norm of ties defined as: (13) Thus, the norm of residuals is square root of the sum over squared residuals.The norm of residuals varies from 0 to infinity with smaller numbers, which indicates better fit; whereas, zero indicates a perfect fit (Betzler, 2003).

Evaluating Results of Simple Thermal Model
A model of simple thermal model has been created by the grid and non-linear regression methods of data fitting have been used.The norm of residuals has been calculated for each thermal model data fit.The outcomes obtained through these calculations are shown in Table 2 and Table 3 (Note 4).The norm of residual values has been observed to be resulted by using the grid method range of 92-412, and the overall average of calculated values is 203.01, as shown in Tables 2 and 3.However, the norm of residuals values resulted from using the non-linear regression range between 20-69 and the average of overall calculated values is 32.04.According to the resulted averages, the method of non-linear regression provides better fits than using the grid method for data fitting.

Evaluating Thermal Model with Envelope Results
It has been evaluated through outcomes that the method of non-linear regression is much better than the grid method for data fitting.Thermal model with envelope is built here by the non-linear regression method of data fitting using the collected data for this study.In Table 4, the norm of residuals is calculated for these thermal models data fit to be compared with the norm of residuals values calculated for simple thermal model data fit created by non-linear regression method, which are shown in Table 3.The norm of residuals values range between 20-52 and the average value is 29.48.Results of the thermal model with envelope are close to, but smaller in comparison with the results of simple model.The plot of residuals for previous models has been examined visually, and it exhibits random behavior in the cases of simple thermal and thermal with envelope models by non-linear regression method of data fitting.However, in the case of simple model by the grid method of data fitting, it is close to making a pattern.It has been observed from earlier sections that if residuals appear to behave randomly, then the model fits the data well.So, this examination proves that the method of non-linear regression is better as compared to the grid method for data fitting.For previo cases of s However, 6, and 7 cl Figure sho

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In first sta through no among the method as model and the norm envelope.compared The algorithms used in this study are helpful in analyzing the results.Grid technique has also been introduced for assessing the outcomes.Similarly, Mallick (2014) applied algorithm technique in analyzing the results of weather conditions in Saudi Arabia, which not only estimated the heterogeneous areas but also focused on homogenous areas.It can be said that energy consumption is directly proportional to the weather conditions, hot weather tend to consume more energy in the form of different appliances.This study can be compared by the study of Foucquier et al. (2013), who described the energy consumption majorly in buildings in European Union.

Conclusion
The concepts of thermal modeling and data fitting have been discussed in the present study.It has been argued that thermal models can be created with two or more parameters.Also, different methods for data fitting can be applied.First, simple thermal model has been developed with the help of two parameters by applying two methods for data fitting.These methods include non-linear regression and grid methods.Second, thermal model with envelope has been created using four parameters by applying non-linear regression method for data fitting.Moreover, different thermal models have been tested, and the evaluation and comparison of results have been made.From the comparison of results, it has been observed that thermal model with envelope functions more appropriately as compared to simple thermal model.Besides this, the outcomes also revealed that the method of non-linear regression of data fitting provides more appropriate results in contrast with data fitting by the grid method.Also, the method for computation of estimated cost of cooling from thermal model has been introduced and implemented.The results related to the other regions under similar or different climates can only be assessed through diverse collection of data.The results of this study cannot be extended for other regions because of territory differences.
The research has been limited to the energy consumption for a small number of households.However, there is still a space for more studies regarding commercial and residential buildings situated in Saudi Arabia.There are many options that are being utilized in other states like microCCHP, vacuum insulation, LHTES, etc. Therefore this tends to the research related to test the feasibility in the built environment in Saudi Arabia.Moreover, the study has been restricted to the thermal models of buildings in Saudi Arabia for only a province.The study could be conducted in some major areas of energy consumption within Saudi Arabia or other major energy-consumed regions in the world.The future studies have the space to follow the increased solar energy projects within Saudi Arabia and the energy management processes.Aside from solar projects, the future researches can follow the geothermal, wind and biomass alternatives proposed for buildings.There are many solutions existing can be useful as free of cost or less expensive, such as by designing the building by taking in account the incorporating passive design aspects which have a vital effect on the consumption of energy. Figu Figures 5, 6 and 7 clarify plots of residuals for different thermal models described above for House G on June 18 th .Each figure shows thermal model and the residuals of its plot.cis.ccsenet.
The consumption cost calculated each 30 days.

Table 2 .
Norm of residuals values for simple thermal models created by grid method of data fitting

Table 3 .
Norm of residuals values for simple thermal models created by non-linear regression method of data fitting

Table 4 .
Norm of residuals values for thermal models with envelope created by non-linear regression method of data fitting