Parametric Building Envelopes Rationalized in Terms of Their Solar Performance in a Temperate Climate

Abstract

The article presents a method of shaping unconventional building envelopes characterized by their effective solar performance in a temperate climate. An analysis related to the impact of geometric shape on the solar direct radiation falling on building envelopes was presented in terms of polyhedral forms. It was based on interdisciplinary issues located in the fields of solar radiation, unconventional forms of buildings, numerical simulations, and artificial neural networks. The elaborated method’s algorithm was employed to describe the relationships between the envelope systems and the amount of the radiation falling on these systems, identified during the performed simulations. Two novel parametric models were defined to execute the simulations. The first was an initial geometric model defined by a number of arbitrary independent variables. The second was defined by one dependent variable representing the quantity of the solar radiation falling on each envelope. The analysis carried out showed that the invented trapezoidal forms of envelopes allowed for better control of the incident solar radiation in relation to the forms generated with other methods. The invented trapezoidal forms of envelopes can increase the amount of their direct solar irradiation up to 63%, compared to prior pyramidal forms. Although the climatic loads used were related to Strzyżów characterized by the geographical coordinates 49.9 N and 21.9 E and located in Central Europe, it was possible to adapt the method to other meteorological boundary conditions by changing the values of the defined parameters. The resultant parametric solar model can be employed to search for many diversified discrete solar envelopes of buildings and rational arrangements of their external planes so that the direct solar radiation falling on these envelopes can be increased during cold periods and restricted during hot summer periods of the year.

1. Introduction

Relatively little direct solar radiation reaches building partitions located in Central Europe in the autumn–winter–spring period [1]. Therefore, the search for rational building forms is focused on creating special systems of façades and roofs, constituting external partitions, so that their arrangement guarantees an appropriate amount of their direct irradiation during the heating season. The part of the global solar radiation coming directly from the sun is called direct solar radiation, which can significantly affect the health and comfort of people staying inside buildings.

In Central Europe, the heating season lasts relatively long, about 8 months, and the direction and quantity of solar radiation change depending on the day and season. Therefore, more and more advanced structures of building envelopes taking into account different orientations, sizes, and slopes of individual external partitions are shaped to control the quantity and time of the direct radiation falling on their planes. The façade systems are supported by appropriate internal and external equipment, for example, static or moveable shades [2].

The trends appearing in the design process, aimed at optimizing the amount of solar irradiation, require creating parametric models of the building envelopes. Various procedures implementing these trends produce discrete models loaded during simulations of the solar operation of the envelopes. The simulation results are used to elaborate parametric solar models employed to search for effective unconventional solar solutions. The multitude of the input geometric and resultant solar parameters and the complexity of the parametric models need to use an artificial intelligence to describe the relationships identified during the simulations and to search for optimal geometric and solar solutions.

2. State of the Arts

2.1. Direct Solar Irradiation

The Earth changes its declination and rotates 360° about its axis a day. Therefore, the position of an observer located on the Earth changes in relation to the solar rays hitting the ground throughout the day or the year [3,4]. In the first case, the position of the Earth’s axis changes significantly during the year in relation to the plane on which the Earth moves around the sun, so an analysis of solar irradiation changes can be averaged on a monthly basis, and monthly average daily values should be taken into account. The monthly average daily radiation is obtained for a day in which the radiation conditions are similar or close to the average irradiation conditions appearing in the examined month [5].

The extraterrestrial solar radiation is calculated in relation to a horizontal plane [3]. The calculations are performed for a clear day [6], so they are insufficient. Therefore, a monthly average cloud cover was defined by Norris [7]. The relations between meteorological conditions and cloud cover were identified by Paltridge and Proctor [8]. For this purpose, the meteorological data were averaged over several years to obtain appropriate calculation accuracy.

In the second case, a daily analysis of the solar irradiation should be based on hourly or shorter time intervals. Each building envelope is usually considered a system of external partitions that do not change their positions, inclinations, and orientations, so the quantity and incident angle of the solar rays can change itself quickly during a day and even in an hour [3,4]. In most cases, the hourly analysis is used for passive and active systems to limit the energy supplied from the grid [9,10]. To obtain effective active and passive energy production systems, the orientation of all envelope planes toward the solar ray direction should be optimized [11,12,13]. Thus, the existing meteorological and environmental conditions must be taken into account [14,15].

The factors describing the rational use of building envelopes and their influence on the comfort in buildings were identified by Badarnah [16]. The potential of building envelopes to convert solar energy into comfortable internal conditions was presented by Desthieux et al. [17] and Soretta et al. [18].

All methods of calculating the solar irradiation of building envelopes must be based on (1) meteorological data delivered by ground stations located all over the world [19], including the stations grouped in the Baseline Surface Radiation Network [20], and (2) satellite images processed by various online tools employed in many fields of human activity [21,22].

The development of new technologies makes it possible to use internet tools as computer programs to calculate the amount of irradiation falling on every place in Europe [23,24,25]. The stationary meteorological stations provide high-quality solar radiation measurements but only on the specific small areas near these stations [26]. The data coming from satellite images [27,28] are less accurate; however, they allow for the consideration of almost all sites with any geographic coordinates other than those obtained from the ground meteorological stations [23,29]. Systems of maps can take the topographic conditions into account during various solar irradiation calculations [30,31].

The information on the methods of using the satellite data can be found on the internet [32,33]. Various methods of calculating the amounts of various types of solar irradiation have been elaborated by Muneer [5] and Perez [34]. The methods take the anisotropy of solar radiation into account [35]. Thus, their prediction is very accurate. Various error measures are also used for evaluating these predictions [3].

Hottel [36] elaborated a relatively accurate and well-known method for estimating direct solar radiation. Liu and Jordan [37] identified and described the main empirical relationships between direct and diffuse radiation by means of cumulative distribution curves defined for various latitudes and elevations. An accurate method of calculating the solar radiation falling on differently oriented envelope planes was elaborated by Muneer [38]. The method was based on direct solar radiation and meteorological data supplied by ground stations and satellites. It took into account cloud cover conditions. The second accurate method elaborated by Perez [34] was based on the diffuse solar radiation.

A method used for calculating solar radiation on the basis of satellite images was developed by Bocca et al. [38] and Müller et al. [39]. An estimation model implemented in PVGIS was developed by Muneer [5]. It contained various components, making it possible to distinguish between clear and overcast sky conditions and sunlit and shaded surfaces. The validation of the data was presented in the scientific article by Müller et al. [21,24] and on the website [32]. These accurate models were based on the anisotropy of solar radiation [40].

2.2. Solar Performance of Building Envelopes

Solar potential can be studied for building groups and individual buildings [41,42]. An analysis and description of the potential of urbanized city districts requires relatively complicated models and calculations [43]. However, in the case of individual models, the analysis and calculations are much more detailed.

A novel procedure was elaborated by Abramczyk [44] to analyze the influence of changes in complexity of building envelope forms on their direct solar irradiation. For that purpose, an initial cuboid envelope configuration was assumed as the basic C_b, whose external partitions were vertical and oriented in the west, south, east, and north directions, as shown in Figure 1a.

Several subsequent types of new derivative elevation systems C_di were derived from C_b so that the orientations (slopes and azimuths) of their partitions were changed, as shown in Figure 1b–d. For all derivative and basic configurations, the adopted cubature was constant to make them easy to compare. In addition, the surface areas of their elevation planes were changed with the help of a few discrete values. The numbers of the independent geometric variables (parameters) defining the subsequent derivative configurations C_di were regularly increased so that the quantity of the direct solar irradiation could be better controlled to obtain effective solutions.

The diagrams in Figure 2a–d present lines illustrating changes in direct solar irradiation caused by changes in the geometric parameters defining the considered elevation forms. Figure 2a shows the significant nonlinear influence of the azimuth θ on the solar irradiation I_dwu of the south façade wall of C_d1. Figure 2b presents the significant linear influence of the γ₅ angle (dependent on azimuth) of the side façade walls on the solar irradiation I_dT of the whole envelope C_d1. Figure 2c shows the significant nonlinear influence of the γ₄ inclination angle of the C_d2 roof on its solar irradiation I_dru. The line from Figure 2d represents the significant nonlinear influence of the γ₄ roof inclination angle of the envelope C_d2 on its solar irradiation I_dT.

In the case of the last derivative configuration C_d3, a new artificial neural network was developed to describe the relations between the geometric parameters defining C_d3 and direct solar irradiation found during the performed solar simulations. The invented neural network can be employed to search for the effective solar performance of the examined pyramidal type of unconventional building envelopes.

2.3. Parametric Artificial Neural Networks

The need for developing various procedures comparing the amount of direct solar irradiation incident on diversified unconventional building forms means that complex parametric systems of variously inclined and oriented external partitions are developed and simulated using different periods of time. These constantly developed procedures need to employ more and more independent variables defining the geometric, physical, geographical, and meteorological conditions and dependent variables accurately describing the created solar building models and their solar performance [45]. Thus, a regenerative design and parametric shaping with diversified machine learning models are used to search for effective unconventional building envelopes [46].

The multi-objective description and optimization of the solar performance of the unconventional multi-parameter building envelopes can be accomplished with the help of artificial intelligence, including neural networks and genetic algorithms, to control the energy use [47]. Renovating building envelope systems offers several diversified opportunities for reducing the life cycle costs and minimizing various negative environmental impacts. A number of artificial neural networks using the simulation-based multi-objective optimization are employed to predict the energy consumption models [48,49] and to evaluate the required thermo-modernization of habitats [50]. In addition, novel extreme learning machine methods are developed and modified [51]. Artificial intelligence tools are also used for the optimal implementation of wind energy to improve the off-grid energy supporting the development of sustainable cities [52].

Moayedi and Mosavi presented novel hybrid and intelligent methods employed to design environmentally friendly buildings [53]. To reduce and optimize the heating and cooling costs, several analytical optimization algorithms were mixed with artificial neural networks and genetic algorithms. To reduce the cooling energy needs of a public housing sector, a hybrid energy plus model was developed by Cheung et al. by means of artificial neural networks [54]. Ekici et al. proposed a multi-zone optimization methodology to support decision-making for high-rise buildings [55]. They proposed parametric modeling and simulations of high-rise buildings assisted by machine learning and optimization methods.

A solution to the complex problems concerning self-shaping building skins and multi-criteria optimization was proposed by Yi and Kim [56]. The models elaborated with artificial intelligence were more flexible and time efficient in relation to the conventional models employed during the energy simulations and the analysis related to the physical materials of transparent and opaque partitions, the sizes of elevation planes and windows, the indoor to outdoor temperature ratio, and the proportions between the elevation, roof, and cubature [57,58,59].

3. The Aim

The main aim is to present a new method based on an elaborated initial parametric geometric model of unconventional building envelopes used to: (a) simulate their thermal performance, (b) define a novel parametric solar model of these envelopes, and (c) search for novel effective discrete solar models of these envelopes. The method’s algorithm allows one to calculate the amount of direct irradiation falling on an envelope depending on its unconventional structure, constituting a system of flat facades and using the neural network elaborated on the basis of the relations found during the performed simulations.

The result of using this method is an analysis of the significance of the influence of the parametric initial model on the amount of the direct irradiation incident on the examined envelopes in the year. The new complex forms of envelopes predicted by the method will increase the number of the arbitrary independent variables defining the initial model. This analysis is related to the individual partitions and their mutual positions, which increase the number of the independent variables. Finally, this activity requires developing a new implementation of a parametric neural network to describe the relations identified during calculations, and appearing between the initial parametric geometric model and the resultant parametric solar model.

The presented algorithm will be easily extended by supplementing data coming from other ground stations and satellites and by analyzing more complex parametric building envelopes. The expected results of the use of the elaborated models and their modifications are related to the development of such unconventional rational building envelopes that the arrangement of their partitions will lead to: 1. rational quantities of the direct solar irradiation distributed uniformly over these partitions during the day and the year, and 2. technically justified arrangements of these partitions constituting reliable and attractive building structures.

4. Methodology

In the first step of the research algorithm, an initial, parametric, geometric model Cfp0 was defined by means of a set of eight linear parameters a_i (i from 1 to 8), constituting eight independent variables, as shown in Figure 3. The locations of all vertices and the shape of the form of an envelope can be determined on the basis of Cpf0.

In the second step, Cfp0 was transformed into an eight-parameter geometric model Cfp, employed to define the subsequent discrete models Cfdi of envelopes loaded during simulations. The Cfp model took into account selected proportions of the appropriate geometric elements of each envelope, which allowed for a simple comparison of the selected representatives of the envelopes in terms of their suitability for shaping effective solar forms.

In the third step, 180 simulations of the thermal performance of the discrete models Cfdi were performed. In the fourth step, the identification of the dependences existing between the independent and dependent variables was made, and then the significance of these relationships was analyzed on the basis of the obtained results. An analytical, quantitative description of the identified relations was carried out to define a resultant parametric solar model Cfsp.

The analysis of the results began with the search for the strength of the relations between the individual independent variables wi and the dependent variable Eso using the correlation coefficient rci, with a fairly high probability of over 95%. Therefore, the assumed significance level p was less than 0.05. The critical significance level pci was assumed to be less than 0.01. The closer the absolute value of the rci coefficient was to 1, the stronger the strength of the correlation between the independent variable wi and the dependent variable Eso,n. The letter n meant that the variable used normalized values. A negative value of rci indicated that the analyzed relationship was inversely proportional.

The correlation coefficients were used in the regression analysis of a search for an analytical description of the relationship between wi and Eso,n. Variables wi characterized by higher values of rci were first employed during the development of analytical formulas because they could have a greater impact on the accuracy of these formulas. The values of the calculated rci coefficients, the assumed significance levels p, and the values of the calculated explanatory variances R² allowed for the analysis of the influence of the variable wi on the variable Eso,n. Moreover, it was assumed that the accuracy of the results should not exceed 3.5%. Alternatively, the accuracy could be assumed at 5% if it was not possible to obtain that accuracy.

The following order of searching for analytical formulas was assumed. First, simple regression procedures were used. In the case of obtaining an unsatisfactory accuracy of these formulas, it was planned to use two-stage regression procedures, nonlinear estimation, and artificial neural networks.

The precision of the discrete thermal models determined with the regression procedures was too small. Therefore, one of the generally known artificial neural networks had to be implemented into the developed Cfsp model. The resultant solar model Cfsp was used to search for discrete solar Cfspi models of buildings operating efficiently in the respective solar environment.

The proposed parametric models, created with the help of the AutoCAD (version 2020, Autodesk, San Francisco, USA 2020, computer program [60], made it possible to change the mutual positions of all subsequent plane external partitions of envelopes by changing the discrete values of the respective independent variables. This action caused significant changes in the slope and orientation of the examined envelopes and led to changes in the amount of direct solar irradiation falling on the envelopes.

To analyze the relationships occurring between the adopted geometric variables and the calculated quantity of direct solar irradiation falling on the examined building envelopes, the following characteristics of the examined forms were defined, as shown in Figure 4a,b. The subsequent independent parameters constituting these characteristics were as follows. The variable a₁ was half of P₁P₄; a₂ was equal to B₁B₂; a₃ was half of P₂P₃; a₄ was equal to B₀B₁; a₅ was equal to B₀P₅; a₆ was the distance of P₅ from the plane P₁P₂P₃; and a₇ was the distance of B₄ from the plane P₁P₂P₃.

The seven-parameter pyramidal building forms were considered in the previous authors’ studies. In the present analysis, eight-parameter trapezoidal building forms were considered, where the parameters a_i (i = 1 to 8) are shown in Figure 5a–c. The parameters a₁ to a₄ were defined identically, as in the previous case. The remaining parameters a₅ to a₈ were differently defined due to four additional vertices, P₅, P₆, P₇ and P₈, appearing in all trapezoidal forms of Cfdi. The additional point B_p1 allowed us to define the following parameters: a₅ was equal to B₉B_p1, a₇ was equal to B₁B_p4, a₈ was equal to B₂B_p3, and a₆ was equal to B₅P₁₀.

The increase in the number of a_i parameters from 7 to 8 allowed us to easily control the width of the modeled envelopes and the proportions between the surface areas of the eastern wall P_c1 and the southern wall P_c2. Therefore, the influence of the set of the a_i parameters on the shape of each envelope and thus on the amount of its irradiation during the year was primarily studied and analyzed in this article. For each independent variable, a few discrete values were considered in the adopted range of their variability.

The preliminary eight-parameter linear model Cfp0(a_i) was transformed into an eight-parameter initial geometric, proportional model Cfp1(wi), where: the variable w8 was equal to a₁, w1 was the ratio of the P_c5’s surface area to the Cfdi’s cubature, w2 was the ratio of the P_c2’s surface area to the P_c5’s surface area, w3 and w4 were the inclination angles γ₁ and γ₃ of P_c1 and P_c3 to a horizontal plane, w5 was the inclination angle γ₄ of P_c4 to a horizontal plane, w6 was the angle γ₅ between the x-axis and the P₂P₃ line, and w7 was the ratio of the P_c3’s surface area to the P_c2’s surface area. These parameters constituted the surface and volume characteristics of envelopes. They allowed for the identification and quantitative description of the relations appearing between the form and irradiation of each considered envelope.

The principal axes x and y of the plane (x, z) of the global coordinate system [x, y, z] (Figure 5c) were determined by B₁B₂ and P₁P₂ edges of Cfp. In addition, P_c2 was defined as P₃P₄P₁₁P₁₂, P_c1 as P₂P₃P₁₁P₁₀, P_c4 as P₁P₂P₉P₁₀, and P_c5 as P₉P₁₀P₁₁P₁₂. If we assume the constant arbitrary values of the angles γ₁, γ₂, γ₃, γ₄, γ₅, cubature V_ref, surface area of the southern wall P_{ref_w2}, and surface area of the roof P_{ref_r}, then the values of the parameters a_i (i = 1 to 8) can be calculated using the system consisting of the following eight equations:

F₁(a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈) =V_ref

(1)

F₂(a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈) =P_{ref_w2}

(2)

F₃(a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈) =P_{ref_r}

(3)

F₄(a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈) =γ₄

(4)

F₅(a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈) =γ₂

(5)

F₆(a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈) =γ₃

(6)

F₇(a₁, a₂, a₃, a₄, a₅, a₆, a₇, a₈) =γ₅

(7)

a₁ = a_const

(8)

5. Results

Based on the created 180 different combinations of the selected values of eight independent variables, 180 simulations of solar operation of the examined envelopes under the direct solar radiation load were performed. The total energy Eso of the radiation incident on these envelopes during the experiment was calculated under the load characteristic of Strzyżów, localized to Central Europe. The geographical coordinates 49.879 N and 21.861 E corresponded to this location. The simulations were carried out using the Rhini/Grasshopper program [61]. The geometric and solar characteristics of several selected configurations of Cfsdi are presented in

Table 1. The normalized values of the geometric and solar characteristics of the selected Cfsdi configurations.

Config./Var.	w1	w2	w3	w4	w5	w6	w7	w8	Eso,n
Cfsd114	0.000	0.000	0.500	0.500	0.000	0.000	0.292	0.697	1.000
Cfsd115	0.000	0.000	0.500	0.500	1.000	0.000	0.830	0.090	0.097
Cfsd116	0.000	0.000	0.500	0.500	1.000	0.000	0.271	0.697	0.760
Cfsd117	0.000	0.000	0.500	0.500	0.000	1.000	0.907	0.242	0.204
Cfsd118	0.000	0.000	0.500	0.500	0.000	1.000	0.511	0.697	0.346
Cfsd119	0.000	0.000	0.000	0.000	0.000	0.000	0.780	0.191	0.100
Cfsd120	0.000	0.000	0.000	0.000	0.000	0.000	0.325	0.697	0.995
Cfsd121	0.000	0.000	0.000	0.000	1.000	0.000	0.303	0.697	0.757
Cfsd122	0.000	1.000	0.500	0.500	0.000	0.000	0.170	0.697	0.776
Cfsd66	0.000	1.000	0.000	0.000	0.000	0.000	0.399	0.320	0.776
Cfsd71	0.000	1.000	0.000	0.000	0.000	0.000	0.195	0.697	0.143
Cfsd72	0.000	0.000	0.500	0.500	0.000	0.000	0.292	0.697	0.758

. The first column provides the name of the respective Cfsdi configuration. The subsequent eight columns provide the normalized values of eight independent variables defining these configurations. The last column contains the normalized values of the dependent variable Eso. An analysis and discussion of the results obtained for Cfsdi were carried out in the following sections. The selected discrete models of Cfsdi are presented in Figure 6.

6. Analysis

6.1. Correlations

To examine the influence of each of the considered independent variables on the dependent variable Eso and the elaborated parametric solar model, a number of statistical procedures offered by the Statistica computer program [62] were used. The correlations between the dependent and independent variables calculated for the significance level p < 0.05 were considered to analyze this impact. The calculated values of the examined correlation coefficient are shown in

Table 2. Correlation coefficients between dependent and independent variables.

	rc1	rc2	rc3	rc4	rc5	rc6	rc7	rc8
Correlation Coefficient: rci	0.401	0.051	−0.422	−0.422	0.145	0.171	−0.563	0.789
Critical Significance Level: pci	0.000	0.499	0.000	0.000	0.052	0.000	0.000	0.000

. The parameters represented by rci (i from 1 to 8) were defined in the following way. The parameter rc1 was the coefficient defining the correlation between Eso and the ratio of the surface area of the wall P_c5 to the cubature of the respective configuration Cfsdi. In addition, (a) rc2 defined the correlation coefficient between Eso and the ratio of the surface area of P_c2 to P_c5, (b) rc3 described the correlation coefficient between Eso and γ1, (c) rc4 was related to the correlation coefficient between Eso and γ3, (d) rc5 defined the correlation coefficient between Eso and γ4, (e) rc6 indicated the correlation coefficient between Eso and γ5, and (f) rc7 defined the correlation coefficient between Eso and the ratio of the surface area of P_c3 to P_c2.

Table 1 provides the values of the critical significance levels, pci, calculated for the individual correlation coefficients, rci. Zero or very small values of these levels together with large rci values indicated a strong positive or negative correlation of the variables compared. Values of pci close to zero were considered small. On the other hand, values close to 1 or minus 1 were considered very large. Small rci values and pci values greater than 0.05 indicated a weak (small) relationship between the considered independent variable and Eso,n. In the conducted studies, the verification of the significance of the assumed hypotheses was assumed at a relatively small level of p less than 0.05, which ensured, together with 180 simulations, the appropriate quality of the obtained statistical results. Therefore, pci values significantly greater than 0.05 indicated a high uncertainty of the values of the calculated correlation coefficients. However, making a decision to take into account the impact of the considered independent variable on the resulting solar model of an envelope was dependent upon the expected accuracy of the simulated solar performance of this envelope, which was assumed at a strictly defined level defined in a further section.

From the analysis of the calculated correlation results, there was a strong relationship between the variable a₁ defining the envelope’s width and the quantity of direct solar radiation falling on the examined envelope. Therefore, the amount of solar irradiation hitting an envelope can be effectively controlled by changing the envelope’s width.

The absolute value of the correlation coefficient rc7 identified between Eso and w7 was still big but significantly smaller than in the previous case, which indicates that the relation between the ratio of the P_c3’s surface area to the P_c2’s surface area was strong. This dependence was inversely proportional, which reinforced the above-mentioned trend toward the strong dependence between the envelope width and the amount of its irradiation. The smaller the ratio of the surface area of the side wall P_c3 to the surface area of the southern wall P_c2, the greater the direct solar radiation hitting the building envelope. Therefore, we can make the assumption that an increase in the solar irradiation of each envelope can be effectively controlled by increasing the surface area of P_c2 accomplished by the envelope’s height or width.

The relatively high values of the correlation coefficients between the dependent variable Eso,n and the independent variables w1, w3, and w4 indicated the possibility of a strong relationship between the amount of irradiation and (1) the angles of inclination of the side and southern walls of the plane of the building’s base and (2) the ratio of the roof area to the building’s cubature. Therefore, the above relations should be taken into account first when the relationships between wi and Eso,n are going to be analyzed using regression procedures. In the case of these angles, the relationship was inversely proportional. However, in the case of the cubature, the relationship was directly proportional. We can also make the assumption that the quantity of the direct solar irradiation hitting the envelope may be effectively controlled by changing the inclination and surface areas of the walls and the roof in relation to the cubature of this envelope.

The following two dependencies turned out to be significantly weaker during searches using regression and estimation methods than the ones presented earlier; however, they are still relatively strong. The first was the correlation between Eso and the angle of inclination of the side walls to the southern wall. The second was the dependence between Eso and the angle of inclination of the roof to a horizontal plane. The relatively small values of rci and the relatively great values of pci indicated limited possibilities of controlling the amount of direct solar radiation using the above-mentioned angles.

At the end of the correlation analysis, we can state that the dependence between Eso and the ratio of the southern wall’s surface area to the roof’s surface area was not strong. Therefore, changes in the surface areas of the southern wall and the roof simultaneously and in the same proportions may not change the direct solar irradiation hitting the envelopes.

6.2. Regression

In the conducted analysis, the simple linear regression methods proved ineffective due to the obtained unsatisfactory modeling accuracy. Therefore, two-segment linear regression was performed. Its characteristics are given in

Table 3. The characteristics defining the two-segment linear regression, obtained at the significance level p < 0.05.

Var./Regr. Coeff.	B0	w1	w2	w3	w4	w5	w6	w7	w8
Fist Segment	0.072	0.157	0.030	−0.040	−0.040	0.059	0.075	0.011	0.198
Second Segment	0.440
Transition Point	0.359	−0.167	−0.044	−0.001	−0.001	0.009	−0.062	−0.124	0.397

. The explained variance factor R² was obtained at the level 0.85, and R was equal to 0.92.

These coefficients were relatively high but did not provide the satisfactory accuracy because the errors calculated for a number of the approximated values exceeded 40%, according to

Table 4. The geometric and solar characteristics of the selected Cfsdi configurations obtained using the two-segment linear regression method, obtained at the significance level p < 0.05.

Config./Var.	Eso,n	Eso,n Predictions	Residuals
Cfsd114	1.000	0.679	0.321
Cfsd115	0.097	0.118	−0.021
Cfsd116	0.760	0.691	0.069
Cfsd117	0.204	0.165	0.039
Cfsd118	0.346	0.251	0.095
Cfsd119	0.100	0.118	−0.018
Cfsd120	0.995	0.676	0.319
Cfsd121	0.239	0.160	0.079
Cfsd122	0.757	0.688	0.069
Cfsd66	0.776	0.650	0.126
Cfsd71	0.143	0.170	−0.026
Cfsd72	0.758	0.648	0.110

. R² showed what part of the variability of the dependent variable was explained by the variability of the considered independent variable in the range from 0 to 1. The normality of distribution of the residuals obtained as differences between the values of Eso obtained from simulations and the values predicted using the developed formula are presented in Figure 7a,b.

Slightly preferable results were obtained by using a nonlinear estimation method. However, they did not offer a satisfactory result, even in the case of very complex formulas. The best accuracy was obtained using the following very complex analytical formula:

w9 = b0 + b1 · w1 + b2 · w2 + b3 · w3 + b4 · w4 + b5 · w5 + b6 · w6 + b7 · w7 + b8 · w1 · w1 + b9 · w7 · w7 + b10 · w2 · w2 + b11 · w1 · w7 + b12 · w1 · w7 · w2 + b13 · w7 · w7 · w7 + b14 · w1 · w2 + b15 · w2 · w7 + b16 · w1 · w5 + b17 · w1 · w6 + b18 · w7 · w5 + b19 · w7· w 6 + b20 · w1· w6 · w7 + b21 · w7 · w5 · w1 + b22 · w7 · w6 · w1 · w5 + b23 · w8 + b24 · w8 · w8 · b25 · w7 · w8 + b26 · w8 · w4 · w5 · w6

(9)

where bj (j = 1 to 26) represents the constants, and wi (i = 1 to 8) represents the independent variables. The dependent variable w9 represents Eso,n. In this case, the explained variance factor R² was 0.881. The normality of the obtained residuals is presented in Figure 8.

The inaccuracy obtained for Cfsdi (i from 1 to 180) exceeded 20%. In one case, it was about 50%, according to

Table 5. The geometric and solar characteristics of several selected Cfsdi configurations obtained with the help of nonlinear estimation, obtained at the significance level p < 0.05.

Config./Var.	Eso,n	Eso,n Predictions	Residuals
Cfsd114	0.407	0.23	0.177
Cfsd115	0.097	0.107	−0.010
Cfsd116	0.760	0.826	−0.066
Cfsd117	0.204	0.196	0.008
Cfsd118	0.346	0.397	0.051
Cfsd119	0.100	0.130	−0.030
Cfsd120	0.995	0.785	0.210
Cfsd121	0.239	0.240	−0.002
Cfsd122	0.757	0.795	−0.038
Cfsd66	0.776	0.624	0.152
Cfsd71	0.144	0.288	−0.144
Cfsd72	0.758	0.624	0.134

. Therefore, it became necessary to develop a more accurate artificial neural network, presented in the next subsection.

6.3. Parametric Neural Network

Based on the simulation results, an artificial neural network characterized by the satisfactory accuracy was developed. The characteristics of this network are given in

Table 6. The main characteristics of the developed neural network and several selected Cfsdi configurations obtained using this network.

MLP 8-11-1	BFGS 125	Learning Error	0.00015
Learning Quality	0.995	Testing Error	0.00083
Testing Quality	0.977	Validation Error	0.00047
Validation Quality	0.991
Hidden Activation Function	Than	Activation Function	Exponential
Configuration	Eso_n	Predicted Eso_n	Residuals
Cfsd114	1.000	0.978	0.022
Cfsd115	0.097	0.126	−0.029
Cfsd116	0.760	0.823	−0.063
Cfsd117	0.204	0.180	0.024
Cfsd118	0.346	0.386	−0.040
Cfsd119	0.100	0.119	−0.019
Cfsd120	0.995	0.971	0.024
Cfsd121	0.239	0.242	−0.003
Cfsd122	0.757	0.759	−0.002
Cfsd66	0.776	0.775	0.001
Cfsd71	0.144	0.134	0.010
Cfsd72	0.758	0.751	0.007

. The presented diagrams illustrate the correctness of the selection of this network.

The automatic option provided by the statistics computer package Statistica [62] was selected to implement an accurate parametric artificial neural network. The BFSG learning algorithm elaborated by Broyden, Fletcher, Goldfarb, and Shanno was selected to achieve a high quality of learning, testing, and validation. This algorithm is one of the family of quasi-Newtonian algorithms based on the finite differences of objective function gradient approximations [62].

The developed optimal neural network MLP 8-11-1 was based on 125 BFSG learning algorithms. It was composed of three hidden layers. The first layer consisted of eight neurons. The second layer consists of 11 hidden neurons. The third layer contained one neuron. This network was chosen from many standardized networks, taking into account the following characteristics: (1) the high values of its training, testing, and validation quality coefficients: 0.995, 0.977, and 0.991; (2) the great normality of the calculated residuals; and (3) the relatively symmetric histograms of the residuals, as shown in Figure 9a,b.

7. Discussion

The Pareto diagram, shown in Figure 10, presented the correlations between all individual independent variables and the dependent variable Eso,n from the highest to the lowest value. It described the possibility of a very strong dependence of the parameters w7 (the area ratio of the side wall to the southern wall) and w8 (the width of envelopes) on Eso,n with a relatively high probability, where the significance level was less than 0.05. On the other hand, the relation between the parameter w2 (the proportion of the southern wall’s area to the roof’s area) and Eso,n can be considered insignificant, with a high adopted probability.

The analysis of the diagram presented in Figure 11 showed that the dependence between the width of the trapezoidal envelopes and the amount of their direct solar irradiation was very strong. Therefore, the most effective way to control solar irradiation relies on changing those variable values that can force change in the width of the envelopes. The proposed novel trapezoidal forms of envelopes also allowed free control of the width to length or height ratio of the envelopes. The developed forms of elevations differed radically from the pyramidal forms because one vertex of a pyramidal form was replaced by four vertices, as shown in Figure 4a,b.

Such a complication of the building envelope forms required increasing the number of independent variables, which complicated the parametric description of the relations implemented into the developed method’s algorithm. The complication was particularly evident while looking for the relations between the initial parametric geometric model and the resultant parametric solar model. Therefore, it was necessary to use artificial intelligence to obtain the desired accuracy in the description of the identified simulation relationships. As a result, the parametric artificial neural network MLP 8-11-1 was developed, which additionally allowed us to search for the unconventional envelope’s forms operating effectively in the solar external environment.

Finally, the developed neural network was used to visualize the impact of the independent variable w₈ on the dependent variable Eso (and Eso,n), where w₈ defines the width of the envelope, Eso defines the solar irradiance of the envelope, and Eso,n represents the normalized values of this irradiance, as shown in Figure 12a,b. The normalized values of the remaining six variables all equated to 0.00. On the other hand, the independent variable w₇ was constantly equal to 0.325. The lines presented in these diagrams indicated a very significant, nonlinear effect of the envelope’s width w₈ on the amount of solar irradiance of the tested trapezoidal forms, which confirmed the results of the statistical analyses performed earlier. These results also showed consistency with the results of other studies obtained for pyramidal forms, as shown in Figure 2d. The difficulty in performing a precise direct comparison of these results followed the fact that in the case of the pyramidal envelopes, a significant part of each roof was covered with photovoltaic panels.

The analysis carried out showed that the invented trapezoidal forms of envelopes allowed for better control of the incident solar radiation in relation to the forms generated with other methods. The invented trapezoidal forms of envelopes can increase the amount of their direct solar irradiation up to 63%, compared to prior pyramidal forms.

8. Conclusions

The elaborated parametric method used to simulate and predict the solar performance of the unconventional trapezoidal forms of building envelopes was based on the invented parametric geometric model. The model led to creating diversified individual discrete models loaded during simulations to obtain discrete resultant solar models. The obtained discrete solar models were employed to develop the resultant parametric solar model used to search for rational solar envelopes operating effectively in a temperate environment.

The developed relationships between the elaborated parametric geometric and solar models were calculated based on the relationships found during the computer simulations to search for discrete solar models of building envelopes operating effectively in a temperate climate. The observed relationships were accurately reproduced using an implementation of the artificial neural network MLP-8-11-1, whose training quality was 0.995, testing quality was 0.977, and validation quality was equal to 0.991. The invented method made it possible to obtain great accuracy of the possible predictions. The identified training, testing, and validation accuracies were 0.0002, 0.0008, and 0.0005, respectively.

The developed parametric models and neural network proved that the introduction of one additional parameter to the parametric characteristics of the above-mentioned models allowed for the effective control of the width and rational direct solar irradiation of the trapezoidal envelopes, compared to the pyramidal ones. If we treat these envelopes as eight-parameter systems of differently oriented flat partitions, with different azimuths and surface areas, then we will be able to search effectively for envelopes characterized by optimal solar performance.

It is therefore possible to freely control the amount of the above-mentioned irradiation using a parametric artificial neural network defining envelopes as parametric systems of building structure partitions subjected to direct solar irradiation of the entire envelope in the building. Further research is planned to separate the amount of direct solar radiation falling on the considered envelopes during each month, so as to obtain information on the control of this irradiation using a parametric system of envelope partitions. More precise control of the irradiation can be obtained by using diversified façade systems composed of various plain or shell partitions. These activities are intended to increase the solar irradiation of parametric envelopes during cold periods and to reduce the irradiation during the summer period.