A Fuzzy Logic Concept for Predicting the Seasonal Thermal Performance of Building Envelopes Based on Structural and Geographical Parameters

Abstract

The current practice of building thermal retrofitting is based on the outcomes of energy audits that make use of standardized tabulated information mapped on a structure or structural elements under inherently very specific conditions. Therefore, it provides very limited outcomes for further analysis, especially when decision making procedures are required to particularize the retrofitting strategy. This paper introduces a novel fuzzy logic approach for predicting the thermal performance of building walls that can be used in practice to partially substitute time-consuming and costly energy audits or complex computational analyses. The objective of this concept is to forecast the annual heating energy demands of buildings and to identify the potential energy savings that could be achieved by applying thermal retrofitting measures based on limited resources and information obtained from maps, blueprints, and/or a simple site inspection. For this purpose, a sample knowledge base was created using a validated computational model of the heat and moisture transport in the multilayered wall assemblies. Then, the fuzzy logic model was introduced to predict the thermal performance of selected walls. Our comparison of the fuzzy model outputs with simulated data proved the potential of using the proposed concept as an efficient and straightforward tool for predicting the seasonal thermal performance of building envelopes and to partially replace the current practice, which requires the utilization of building energy audits.

1. Introduction

The energy performance of buildings is affected by a variety of factors, including climatic conditions, structural parameters, the energy efficiency of heating or cooling systems, user behavior, etc. The structural parameters of buildings include material selection and composition (i.e., material layer sequences, the thermal properties of the used materials, the positioning of thermal insulation and its thickness, etc.) as well as design parameters, typically represented by wall to window ratio, solar shielding and overhangs, façade and roof type, wall orientation, etc. The thermal performance of building envelopes plays a role of the utmost importance in forming the total energy consumption of buildings since space heating or cooling represent a large percentage (35–70%) of total energy demand [1]. Typically, the thermal performance of the wall is described via thermal transmittance (U-value) [2,3,4]. The quantification of the U-value is based on a steady-state heat flux calculation. According to standardized methods, the steady-state conditions are achieved using design value quantities to approximate severe conditions typical for a given location. Such an approach, on the one hand, aims to provide a sufficient level of safety to the calculations, thus preventing the building’s designers from underestimating the local environmental loads, which can lead, in the worst scenarios, to the insufficient thermal protection of the occupants. On the other hand, the thermal performance of buildings varies over time, as the boundary conditions evince seasonal changes during the year. Although this phenomenon is typical for any building located anywhere on the planet, it is impossible to take it into consideration using the U-value. Calculating thermal transmittance according to the standardized methods is based on averaging the thermal performance over a certain period (typically one year) and provides global mean quantities, which are straightforward and practical on one hand but might be too oversimplified when performance in a finer time resolution is needed [5]. Therefore, more complex approaches should be sought to quantify the seasonal aspects of thermal performance of buildings during the year [6,7,8,9,10] to provide more complex inputs for the building design.

The inherent uncertainty contained in the results of energy audits for real-life structures has been expressed successfully with help of fuzzy set theory in many studies focusing on assessing the energy performance of buildings [11,12,13]. Fuzzy logic has also been successfully employed for the energy management of buildings, as seen in [14,15,16,17,18]. It should be noted that the use of fuzzy set theory is quite common in solving engineering problems which are more theoretical than the basic fuzzy logic control applications [19,20,21]. The big advantage of fuzzy set theory is that it can be applied to predict outputs of both linear and non-linear systems. Additionally, it can be used to predict the responses of studied systems in the conditions with limited access to experimental and/or in situ data. The application of fuzzy set theory can be easily adapted to various conditions or dynamically changing knowledge bases. Therefore, fuzzy logic seems to be an ideal tool for the estimation of wall thermal performance under various conditions represented either by location or different sets of structural parameters.

In practice, when retrofitting measures are designed, they usually follow a specific energy audit, which, in essence, represents a method to identify the energy consumption of a specific structure. Such audits should provide valuable data to assess the impact of the intended intervention. For those considering retrofitting to achieve passive energy standard dwellings, typically by adding new insulating layers to existing structural components, valid energy audits are essential. Unfortunately, most of these audits prioritize financial feasibility to assure their competitiveness on the market. Such feasibility is usually achieved by exploiting the standardized tabulated information and mapping them (often inappropriately) on a structure or a structural element under inherently very specific conditions (e.g., construction material parameters, elevation, orientation, boundary conditions, solar gains, etc.). Therefore, they provide very limited outcomes for further analyses, especially when decision making procedures are required to particularize the retrofitting strategy.

This situation has become even more aggravated amidst today’s global economic instability and unpredictability, even when a valid energy audit result is of major importance. The factors which make thermal retrofitting rather difficult for deciding how to proceed are the volatility of prices and stocks of building and insulating materials. If the prices are too high or the material is not available (which was the real situation on the market 1–2 years ago), then one can expect that the decision making process will become an integrated part to set priorities in the retrofitting strategy rather than insulate the building as a whole. Amidst the limited and unpredictable market conditions, such priorities will be primarily focused on minimizing the material inputs and costs while achieving the highest efficiency of applied measures. Typically, the most susceptible and resistant parts of buildings, cost-efficient thicknesses of insulation layers for each building element, suitable types of insulation material, etc., will be identified, each under locally unique environmental and geographical conditions.

The above-described issues within the field bring new questions that should be addressed and, at the same time, define a gap in the current practice that should be filled. In general, the common efforts that are being made for the overall improvement of energy audits (and their validity, complexity, cost- and time-effectiveness, etc.) are limited and often immediately devalued by specific key factors and circumstances that are implicitly associated with this agenda. For example, only authorized professionals or engineers are allowed to conduct energy audits, which requires skills and professional resources (tools, equipment, software). Additionally, energy audits typically have several attributes, such as relatively high prices, due to the necessity of adapting to specific conditions in each case.

The issues described above define the key points that should be addressed whenever a new solution is being sought or developed. New solutions should turn the key issues, (i.e., limitations) into the key features (i.e., benefits) that will grant benefits to the potential applications in the future.

In this paper, a novel solution to the problems described above is proposed. This solution is based on the concept of an easy-to-use numerical tool with an implicit (“black box”) approximation of the complex numerical model that is used for the analysis of thermal performance rather than simple calculations or steady-state models as defined in the national standards. The approximation should provide an environment for end users that is friendly, straightforward, and intuitive, requiring neither a priori knowledge nor professional skills so that it can be instantly used by any non-professionals or laymen. For this reason, in this paper, a novel concept that uses a fuzzy logic model as a key element of the entire solution is introduced. In principle, fuzzy logic models are built on a robust knowledge base that is generated using a validated computational model or from field data. In the case of this study, a validated computational model was used to build a knowledge base from simulations with two different load-bearing materials, with each having two thicknesses, four insulation thicknesses, two site elevations, and eight different orientations, meaning that we had 256 different simulations that needed to carried out. Then, the prediction of the thermal performance of building walls with various parameters can be carried out simply via fuzzy interference, all without the need to use sophisticated computational tools. Since this fuzzy logic-based solution provides numerous perspectives for further investigations (such as wall composition and material solution, material layer sequence—i.e., interior, exterior insulation layer, sandwich panels, etc.—window-to-wall ratio, air infiltration rate, solar heat gain coefficient of the glazing, ground albedo, etc.), it is far beyond the scope of this manuscript to cover them all. Therefore, this paper aims to describe the main fuzzy logic concept and demonstrate its utility within the field of energy auditing.

2. Materials and Methods

2.1. Implementation Strategy

The main objective of this work was to present a method for estimating the annual total energy savings that could be achieved by applying retrofitting measures in buildings with only a minimal amount of information obtained from maps, a brief communication with the house owner, or from a simple site inspection. To develop an instrument for such predictions, several structural and material solutions need to be selected for the analysis, which is based on the investigation of various structural aspects and/or the environmental effect of the solutions included. The outputs of such an analysis initially create a knowledge base, which the fuzzy model will subsequently be based on, as discussed elsewhere in this paper. The nature of this analysis itself determines the computational modeling to be a perfect instrument for the creation of this knowledge base rather than collecting experimental data from a large number of various sites. However, the utilization of a validated computational model is crucial to fill in the database with data that are as accurate and reliable as possible.

The material solutions for the family houses investigated for this work involved two different types of load-bearing materials. The traditional/historical solution was based on a mixed stone/brick masonry, while the more advanced one was based on the incorporation of aerated concrete blocks (AAC). Following these material solutions, each load-bearing structure was considered with additional insulation layers of specific thicknesses in the wall assembly to simulate potential thermal retrofitting measures. The insulation material was represented solely by expanded polystyrene (EPS). The structural aspects included the specific thicknesses of each material layer, wall orientation, and site elevation. Variations of material solutions and structural aspects created a sufficiently diverse range of building designs typically used for one- or two-story buildings across the Czech Republic. A summary of the material solutions and structural aspects considered in this research study is given in

Table 1. Summary of material solutions and structural aspects.

Load-Bearing Material	Wall Thickness	Bulk Density	U-Value Estimation	Additional Insulation
Stone/brick	750 mm	2244 kg·m−3	2.06 W·m−2·K−1	0 mm
				100 mm
				200 mm
Stone/brick	1000 mm	2244 kg·m−3	1.74 W·m−2·K−1	0 mm
				100 mm
				200 mm
AAC	300 mm	500 kg·m−3	0.37 W·m−2·K−1	0 mm
				100 mm
				200 mm
AAC	500 mm	500 kg·m−3	0.23 W·m−2·K−1	0 mm
				100 mm
				200 mm

. Here, the U-value roughly estimated from the thermal conductivity of load-bearing material in the dry state is simply divided by the wall thickness. Such estimations provide safe values for subsequent analyses, as neither surface resistance nor finishing layers are taken into consideration.

2.2. Fuzzy Logic Model

The proposed model is based on fuzzy logic, which was introduced in 1965 [22] and, over the years, has become a preferred tool for the definition of numerical models built to describe ill-defined or vaguely defined processes. The vagueness in the description is caused by either the lack of hard experimental or measured data, which are subsequently substituted with human estimates or via verbal definitions of trends and tendencies which are impossible to prove via measurements or difficult to define analytically.

The knowledge base in this research was constituted over a set of simulated data for selected locations in the Czech Republic that were reasonably robust to demonstrate the applicability of the fuzzy model. For this purpose, a validated computational model [23], which is described further in the following subsection, was implemented. A scheme of the fuzzy logic model is shown in Figure 1. The input values of the model are represented by four parameters, namely (i) the estimated thermal transmittance (U-value) of the uninsulated wall, (ii) the thickness of additional thermal insulation layers, (iii) site elevation, and (iv) wall orientation. The output value of the fuzzy logic model is the annual thermal performance per unit area of the wall, which can be further used for the calculation of potential energy savings or to evaluate the efficiency of applied retrofitting measures. The decision making process is based on the Takagi–Sugeno [24] scheme, which can describe a generally nonlinear system by operating with a set of IF–THEN rules to represent the relations between the inputs and outputs.

The unorthodox manner of the model flow scheme shown in Figure 1 was chosen due to limitations in 3D visualization, as 4D fuzzy inferencing needs to be described. Each of the four input parameters and the output are expressed by fuzzy sets. The number of the fuzzy sets reflects the number of recognized typical (and thus representative) values of the input parameters and their respective outputs. Four fuzzy sets ranging from 0.23 to 2.06 W·m⁻²·K⁻¹ are used for the U-value to describe the thermal performance of the non-insulated perimeter walls, three fuzzy sets represent the insulation layer thicknesses in the range from 0 to the applicable maximum thickness of 200 mm, two fuzzy sets are used for elevation—representing the boundaries of the elevation of the 90% of the family houses in Czechia, denoted as lowland (200 m) and highland (500 m)—and nine fuzzy sets are used for the orientation of the houses and their walls, with the simulated increment of 45° in the range of 360°. The output value was represented by five fuzzy sets ranging from the minimum of 60 kWh to the maximum of 260 kWh. The proposed model is also capable of providing the resulting annual total energy and annual total energy savings in the form of fuzzy numbers so that the effect of the uncertainty contained within the input values can be assessed. The input fuzzy numbers are defined by their mean values, and their respective boundaries are given as a percentage of the mean value, while the membership functions are considered linear (i.e., triangular fuzzy numbers). The calculation is then performed using the so-called α-cuts when the fuzzy logic model is used as a “black box” function, producing crisp value outputs from crisp value inputs. The membership functions of the input values and the output values are then obtained using the maximum-of-minima principle when the computation runs using the “black box” function are performed for all boundary combinations.

2.3. Computational Model

The computational model used to evaluate the thermal performance of typical perimeter walls is based on a two-scale modeling approach that has been validated in the past [25]. Such an approach requires two stand-alone physical models, which are interconnected during executions to exchange simulation data. In general, one model provides data on transient heat transfer through the opaque elements of the building envelope, while the other one estimates heat gains and losses through the transparent parts of the envelope and accounts for them in regard to the steady-state thermal equilibrium of the interior space. The basic idea of the modeling approach is depicted in Figure 2, while a diagram of the computational procedure is shown in Figure 3.

The model also focuses only on the perimeter walls of the house, as thermal retrofitting is mainly performed by adding new insulation layers to the external faces of buildings. In this research project, the modified Künzel’s mathematical model [23] was implemented into the in-house computer code SIFEL [26,27] to simulate transient heat transfer using the balance equations listed below:

$$\frac{dH}{dT}\frac{\partial T}{\partial t}=\mathrm{div}\left(\lambda \mathrm{grad}T\right)+L_v\mathrm{div}({\delta }_p\mathrm{grad}{p}_v),$$

(1)

$$\left[{\rho }_w\frac{dw}{d{p}_v}+\left(\psi -w\right)\frac{M}{RT}\right]\frac{\partial p_v}{\partial t}=\mathrm{div}\left[D_g\mathrm{grad}{p}_v\right]$$

(2)

In the above equations, H—enthalpy density (J·m⁻³); L_v—latent heat of evaporation of water (J·kg⁻¹); M—molar mass of water vapor (kg·mol⁻¹); p_v—partial pressure of water vapor in the porous space (Pa); t—time (s); T—absolute temperature (K); δ_p—water vapor diffusion permeability (s); λ—thermal conductivity (W m⁻¹ K⁻¹); ρ_w—density of water (kg·m⁻³); and ψ—total open porosity (-). The indoor space model requires several input parameters to describe the thermal response, and these parameters are listed in

Table 2. Indoor space model parameters.

Parameter	Value
Room width (m)	5.0
Interior heat capacity (per 1 m2 of the wall) (J·m−2·K−1)	2.6·105
Solar radiation (W·m2)	data from TRY
Glazing ratio (-)	0.3
SHGC of the windows (-)	0.6
Air infiltration rate (h−1)	0.6
Heating setpoint (°C)	20.0

.

The indoor model calculates solar heat gains (SHG) and heat losses or gains via air infiltration (AI) according to Equations (3)–(5) and provides the temperature step change based on the user-defined thermal mass of the interior space (ITM) (i.e., the interior heat capacity):

$$SHG=AbsCoef\left(Q_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}+Q_{\mathrm{d}\mathrm{i}\mathrm{r}}\right)\cdot G\cdot SHGC,$$

(3)

$$Q_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}=I_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}\cdot {\left(\cos ({\iota }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}/2)\right)}^2+\alpha \left(I_{\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}}+I_{\mathrm{d}\mathrm{i}\mathrm{r}}\right)\cdot {\left(\sin ({\iota }_{\mathrm{w}\mathrm{a}\mathrm{l}\mathrm{l}}/2)\right)}^2,$$

(4)

$$Q_{\mathrm{d}\mathrm{i}\mathrm{r}}=r\cdot I_{\mathrm{d}\mathrm{i}\mathrm{r}}$$

(5)

In the above equations, AbsCoef—absorption coefficient (-); Q_diff, Q_dirr—partial heat gain or loss from diffuse and direct solar radiation, respectively (J·m⁻²); I_diff, I_dirr—direct and diffuse solar irradiation from TRY (W·m⁻²); G—glazing ratio (-); ι_wall—wall inclination (rad); α—ground albedo (-); r (s)—orientation factor calculated from the location coordinates and elevation for the particular time of the year.

If the indoor temperature variation during the year is known, the heating and/or cooling energy demands can be easily calculated by introducing their respective setpoints. The heating demands are evaluated as energy related to 1 square meter of the perimeter wall that needs to be supplied to heat up the interior to the desired temperature level. In this respect, each one of the studied walls under different boundary conditions were assessed to generate the data required for the analysis by the fuzzy logic model.

3. Results and Discussion

The general idea of the proposed work is to provide end users with a sophisticated and, at the same time, easy-to-use tool for the fast assessment of thermal retrofitting efficiency under particular conditions defined by structural parameters and geographical locations. The proposed work has been conducted for demonstrational purposes with a limited knowledge base only. However, this knowledge base can be extended very easily by adding data covering a series of different structural or material parameters, which would significantly extend the application areas.

A series of computational simulations was conducted to generate a simple yet sufficiently robust knowledge base. In total, two different load-bearing materials, each one with two thicknesses, four insulation thicknesses, two site elevations, and eight different orientations were investigated, amounting to 256 computational simulations that needed to be carried out. Each simulation covered a period of three consecutive years, with boundary conditions being represented by TRY (Test Reference Year) for given locations, while the energy assessment was carried out for the last year of the simulation. The first two years were simulated to obtain the appropriate initial conditions for the pf simulation of the previous year. In this way, the knowledge base was created and set for the subsequent application of the fuzzy logic model. In its simplest form, the model can be used to produce estimations of heating energy demand per square meter of the perimeter wall Q_h (kWh·m⁻²·year⁻¹) as a crisp value for a given set of crisp inputs representing both structural and geographical parameters.

$$Q_h=f\left(U,\gamma ,e,i\right),$$

(6)

where U (W·m⁻²·K⁻¹) is the U-value, γ (°) is the wall orientation, e (m a.s.l.) is the elevation above the sea level, and i (mm) is the insulation thickness. Sample crisp output data for several random sets of input parameters are shown in

Table 3. Selected outputs of the fuzzy model.

U (W·m−2·K−1)	γ (°)	e (m a.s.l.)	i (mm)	Value (kWh·m−2·Year−1)
0.5	90	235	0	80.26
0.6	120	405	80	93.75
1.3	45	390	150	81.34
0.8	275	390	50	93.29
1.9	145	250	190	81.87
1.8	50	290	100	77.27
1.2	80	330	40	88.41

.

If two of the four input variables in Equation (6) are set as constant values, then surface responses can be easily plotted, giving the end user basic information on thermal performance of the wall, as well as general dependencies related to given environmental conditions. Examples of surface plots are shown in the following figures.

Figure 4 shows the effect of adding an insulation layer to the existing uninsulated wall at a given elevation. The highest energy demands can be observed for the north- and north–east-oriented walls. Logically, the heating energy demand increases with decreasing insulation thickness. The wall orientation has a significant effect for uninsulated walls, where the difference between the heating energy demands for the north- and south-oriented walls can reach up to 20 kWh/m² of the wall annually for constructions with higher U-values. Figure 5 demonstrates the effect of elevation on the annual heating energy demands associated with one square meter of the 150 mm insulated wall. It is apparent from Figure 5 that increasing elevation shifts the energy demand proportionally. The effect of wall orientation is apparent from Figure 5 as well, in accordance with Figure 4. The difference in annual heating energy demands between the north- and south-oriented walls can reach up to 15%, depending on the elevation and material composition of the wall.

Estimations of potential energy savings (calculated per square meter of the perimeter wall) can be considered as the most practical outcomes that can be derived the proposed fuzzy model. In the practical application, the total energy savings should be related to the floor area rather than the wall surface, which gives a practical value that supports the final decision in the formulation of the retrofitting strategy. However, the normalization of fuzzy model outputs to the floor area would require another calculation step with prior knowledge obtained from blueprints or site surveys, which is beyond the scope of this work at the moment. Examples of annual energy savings for different elevations are shown in Figure 5. For this study, the annual energy savings were calculated as follows:

$$Q_{savings}=f\left(U,\gamma ,e,i\right)-f\left(U,\gamma ,e,0\right)$$

(7)

Assuming the retrofitting is based on the addition of a new insulation layer to the existing uninsulated wall, Equation (7) can be rewritten as follows:

$$Q_{savings}={f\left(U,\gamma ,e\right)}_{i+},$$

(8)

where i+ (mm) is the thickness of the newly added insulation layer. The effect of added insulation layer thickness on the annual energy savings per square meter of the perimeter wall is shown in Figure 6. Here, two additional insulation thicknesses are plotted and shown against each other, namely i+ = 50 mm and 150 mm. Both cases were studied at an elevation of e = 350 m a.s.l. The data in Figure 6 provide valuable and useful information that can help to efficiently manage and control the retrofitting strategy and help to select the most susceptible walls, the type of insulation material, design the appropriate thickness of insulation layer, etc.

4. Conclusions

The results presented in this paper highlight the potential of the implementing the fuzzy logic approach for predicting the thermal performance of building envelopes. The implementation of such an approach can contribute to the overall improvement of the energy audits, as it can be used by the wider public without the need to have professional skills or a priori experience. Additionally, the application of this fuzzy logic tool does not require any other professional resources, such as professional tools or equipment, which would otherwise be needed when a standard audit is being carried out.

The example applications in this paper were demonstrated on a knowledge base built upon several varied parameters, such as the type of load-bearing materials, wall thickness, thermal insulation thickness, wall orientation, and elevation. With such a setting, the method allows one to easily determine potential energy savings based on selected thermal insulation thicknesses or to identify the most susceptible walls with respect to their orientation.

The combination of fuzzy model outputs with economic data and indicators such as the market prices of building materials, energies or fuels, near-future trends, etc., can efficiently support final decisions on retrofitting strategies by providing new insights such as cost optimization and estimations of the average ROI (return on investment) period or the overall rentability of the proposed measures.

For further applications, the knowledge base can be extended by adding a series of new data, such as new site locations, new types of load-bearing materials (or even introducing the thermal conductivities of the load-bearing materials as the only additional parameter), different types of insulating materials, glazing percentage, glazing parameters, air infiltration rate, etc. Introducing these additional parameters will significantly extend the application range of the proposed method, which can be conducted very easily and could be used by investors and/or non-professionals as an initial step prior to the energy audits, which will still need to be carried out by official authorities.

Also, the proposed method opens a new field for commercialization, as fuzzy logic techniques may be easily implemented into end-user solutions within stand-alone PC or mobile device applications.