# Energy-Saving Design of Building Envelope Based on Multiparameter Optimization

Authors: Weinan Gan; Yunzhong Cao; Wen Jiang; Liangqiang Li; Xiaolin Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The contradiction between the indoor environment and building energy consumption has been controversial. The design of building envelope involves many parameters such as window size and exterior wall material. These parameters have significant influence on building energy-saving design and indoor environment. In this paper, nondominant sorting genetic algorithm-II (NSGA-II) is utilized to calculate winter heat consumption, indoor total lighting energy consumption, and thermal comfort. The Pareto method is used to select the compromise solution and effective value of each building parameter. Different from other studies, we add more architectural design variables into the model calculation, which can bring architects more detailed energy-saving design content.


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## Body

## 1. Introduction

With global energy shortages, many countries have adopted corresponding energy policies, and global energy intensity declined by 1.8 percent in 2016, based on primary energy demand for gross domestic product (GDP). China's energy intensity has declined sharply, reflecting the continuing impact on energy efficiency policies [1]. But, the final energy consumption of buildings rose steadily from 119EJ in 2010 to 124EJ in 2016, at a rate that went beyond reduction in energy intensity. Along with the advancement of China's urbanization process, more than 97% of new buildings each year are high energy-consuming buildings, and the total energy consumption of buildings accounts for about 30% of the total energy consumption of Chinese society. China’s building energy consumption accounts for about 6.0% of global energy consumption, equal to the total energy consumption in the Middle East, twice that of Africa and twice that of Japan and South Korea combined [2]. In China, urban buildings are mainly divided into residential buildings and nonresidential buildings. Nonresidential buildings (public buildings) such as government buildings, commercial buildings, and school buildings account for a large proportion of building energy consumption, and the comprehensive energy consumption per square meter is more than twice that of residential buildings [3]. Therefore, energy conservation in public buildings is of great significance.The highest potential for green building energy-saving design lies in the renovation of building envelope structures to reduce the use of air conditioning system. Effective design scheme can not only respond to the needs of green energy-saving and long-term economic benefits brought by government investment but also meet the user’s comfort level. Through the local climatic conditions and site conditions, architects can maximize the control of architectural design and construction techniques [4]. Building envelope parameters can affect building energy performance and comfort to a considerable extent. Building envelope is usually composed of transparent and opaque components, and the solar energy and shading characteristics transmitted by materials’ heat conduction, windows, and the total projection rate of composite materials affect the cooling and heating load of buildings and energy consumption of lighting system [5, 6].The geometric configurations of buildings, such as the building volume, the aspect ratio of windows, and the window wall ratio [7, 8], are counted as additional factors influencing the energy load of buildings.In this article, indoor thermal performance, thermal comfort, and lighting energy consumption of NSGA-II are optimized by building geometry and physical boundary. Building energy consumption, indoor basic thermal comfort, and indoor use lighting are the three objectives of this study. The value range of each building parameter (design variable) is taken according to the national standard specification. The Pareto method selects the optimal solution set, and the different nondominated solutions correspond to discrete building parameters.


## 2. The Mathematical Model

We have built three objective functions. The first objective function (f1) includes the winter heat consumption of the enclosure and the summer cooling load. We use the heat consumption Q as the positive here. Therefore, to calculate their total energy consumption, we must subtract. The second (f2) is to calculate the total lighting energy consumption f2 for the whole year, and finally we use the PMV (predicted average voting) value f3 to analyze the thermal comfort of the room in summer. The design variables of these three objective functions have some coupling. It can be seen from the model of the objective function later. For example, the window design in f2 also affects f1. And f1 and f3 also have contradictory relations. More design variables help to provide a richer context for the architect, which is the novelty of this article. The model created in this article does not limit the maximum or minimum of the solution goal. This is to get more solutions. This solution set is more flexible and diverse. The final choice will be decided by the architect.(1)f1=−qsys+Q,f2=Eel,f3=PMV.


### 2.1. Lighting Energy Consumption

The assumption of diffuse reflection lighting conditions leads to another metric commonly referred to as the daylight factor (DF) [9], which is the ratio of indoor and outdoor illumination under standard CIE cloudy conditions. Due to the simplified sky model, given a certain daylight open configuration, regardless of direction, geographic location, and climate change, the DF at a particular point will remain approximately constant; for the purpose of design, the vector flux splitting method [10] is used. The DF consists of three separate components: sky (SC), external reflection (ERC), and internal reflection (IRC) components:(2)DF=SC+ERC+IRC.About complex design calculations, ERC can be determined by multiplying the external obstacle (SCobs, i.e., obstacles can be seen in the sky area) by the obstacle reflectivity (Robs, this paper takes a value of 0.2), and the calculation for IRC is also taken from [11]:(3)IRC=τjAjA1−RCRfw+5Rcw.In order to achieve a more practical design, additional calibration was performed in this paper, taking into account the maintenance factor (Mf), glass factor (G), and window frame factor (B), with values of 0.9, 1, and 0.85 [12], respectively. The final calculated DF is(4)DF=0.765SC+ERC+0.2SC.For the case where the calculation point is close enough to daylight, SC can be considered as the most important component of the three. SC is pure geometric calculation as shown in Figure1. The calculation equation of SC is as follows:.(5)SC=Ei,skyEO=Lz/3∫0θ′∫0β1+2sinθsinθcosθdθdψLz/3∫02π∫0π/21+2sinθsinθcosθdθdψ=314πβ−β′cosγ+27πarcsinsinβsinγ−17πsin2γsinβ′.Known:tanθ′=tanγcosψ,tanβ=LD,tanβ′=tanβcosγ=L/DH/D2+1.Figure 1
Diagram of the calculation of U from the vertical, effective daylight opening ABCD to D (from [13]).In the lighting design, there are roughly two possibilities for the position of the window relative to the calculation point; this paper refers to the position of the calculation point by Mangkuto [14]. The projection of the calculation point is located in the window area and internal or external (Figure 2); the height of the working surface is 0.75 m and is higher than the window sill position. Let the distance from the observation point to the left wall be x, which is calculated as follows:whenx≤D/2−Li/2,(6)SCa=SCLD=Ll−x+Li/2d,HD=Hh/2−0.75+Hj/2d−SCLD=Ll−x−Li/2d,HD=Hh/2−0.75+Hj/2d,whenD/2>x>D/2−Li/2,(7)SCb=SCLD=Ll−x+Li/2d,HD=Hh/2−0.75+Hj/2d−SCLD=Ll+Li/2d,HD=Hh/2−0.75+Hj/2d.Figure 2
Sectional view of the window wall, two possibilities for the position of the window relative to the calculated point (m).
(a)
(b)Annual lighting energy demand (EelKWh/m2/yr) refers to the total demand for electric lighting and energy lighting throughout the year. According to the national standard [15], the maximum value of the library building lighting power density limit target is 8 (W/m2), and each point of the illumination Ein is multiplied by the relevant DF multiplied by the outdoor diffuse, The selected value of the illuminance Eout is calculated. If the value of Ein is less than 300 lx (reading room illuminance standard value), then additional lighting is needed, and Ein at all points is averaged with a positive difference of 300 lx and multiplied by η = 37.5 lm/W (i.e. 300 lx per 8 W/m2) to obtain the desired illumination power density [14]. T is the total working time (8 hours/day × 5 days/week × 52 weeks/year), and we finally get the estimated Eel:(8)EelKWh/m2/yr=T1000mη∑i=1m300−Ein,i=0.55m∑i=1m300−Ein,i.


### 2.2. Heat Consumption

The calculation of the basic heat consumption of the envelope structure refers to the national standard [16]. The heat consumption of the external protection structure of the universal layer is mainly composed of the outer window and the outer wall. The outer wall is composed of a concrete wall, interface mortar (or bonding layer), insulation layer (or insulation board), crack protection layer (or plaster layer), and veneer layer.The heat consumption of the enclosure consists of basic heat consumption (9) and additional heat consumption. The heat transfer coefficient is composed of the heat transfer coefficients of each structure, and their relationship is shown in formula (10). The additional heat consumption is determined as a percentage of the basic heat consumption. It mainly includes the orientation correction rate, the wind attachment rate, the external door attachment rate, the height addition rate of the building (except the stairwell), and the intermittent addition rate. The outdoor temperature air temperature is a harmonic time parameter with an angular frequency. Therefore, a simple prediction [17] can be performed using equation (10). The indoor air temperature is determined by the indoor setting temperature. And the interior design temperature is designed according to the level of thermal comfort, as can be seen from Table 1. The calculation method of the heat consumption of the envelope structure is shown in formula (11).(9)Q0=αFKtn−twn,(10)K=11/an+∑δ/aλ⋅λ+Rk+1/aw,(11)twn=twn˜+Δtwnsinωt,(12)Q=αFKtn−twn1+or+hh+od.Table 1
Indoor design temperature of air conditioning in the area where personnel stay for a long timetn.
Working condition of the category
Thermal comfort level
Interior design temperature (°C)
Heating conditions
I level
22∼24
II level
18∼22
Cooling conditions
I level
24∼26
II level
26∼28In formula (10), an represents the heat transfer coefficient of the inner surface of the envelope, aw represents the heat transfer coefficient of the outer surface of the enclosure, and aλ represents the correction coefficient of material thermal conductivity. Their values are shown in Tables 2 and 3, respectively [18].Table 2
Heat transfer coefficient of inner surface of envelopean.
Internal surface characteristics of envelope
a
n
W
/
m
2
⋅
K
A wall, floor or ceiling with a flat surface or ribbed projection,h/s≤0.3
8.7
A ribbed, well-shaped roof,0.2<h/s≤0.3
8.1
A ceiling with ribbed projections,h/s>0.3
7.6
A roof with a well-shaped projectionh/s>0.3
7.0Table 3
Heat transfer coefficient of outer surface of envelopeaw.
Outer surface characteristics of envelope
a
w
W
/
m
2
⋅
K
Exterior walls and roof
23
A floor above an unheated basement that communicates with outdoor air
17
A floor slab above a nonheated basement with a covered roof and windows on the exterior wall
12
A windowless, unheated floor above a basement
6


### 2.3. Cooling Load

The heat transfer process of the building envelope is a very complex process involving convection, conduction, and radiation. At the same time, the outdoor air temperature and heat radiation density also change greatly with the change of outdoor time, and the indoor air temperature also changes. In this paper, the heat transfer equation of external thermal mass and internal air is used to calculate the hourly cooling load of the enclosure. The following assumptions are made:(i)
The air distribution within the building, the temperature distribution of the internal thermal mass, and the internal surface temperature of the external thermal mass are balanced.(ii)
All thermal gain and heat generation in the building are assembled into a heat sourceE during the working period, while there is no room thermal gain during the nonworking period, and the radiant heat exchange between the heat source and other surfaces is ignored.(iii)
The direct thermal solar radiation gain and permeability thermal gain through the opening were ignored.(iv)
During the nonworking period, the air conditioning system is closed and night ventilation mode is adopted. The ventilation rate isQv.Based on the above assumptions, the heat balance equations of the inner surface temperature and the indoor air temperature of the outer envelope structure are as shown in equations (13) and (14) [17]:(13)MCm∂Tis∂t+h0A0Tis−T0+hiA0Tis−Ti=0,(14)qsys+hiA0Tis−Ti+E=0.The calculation of outdoor air temperature in summer is similar to formula (11):(15)T0=T0˜+ΔT0˜sinωt.Substitute equation (15) into equation (13):(16)MCm∂Tis∂t+h0+hiA0Tis=h0A0T0˜+hiA0Ti+h0A0sinωt.Letτ=MCm/ρCpQv, where τ is the time constant based on night ventilation rate Qv; λ0=h0A0/ρCpQv, where λ0 is the dimensionless external convective heat transfer; and λi=hIA0/ρCpQv, where λi is the dimensionless internal convective heat transfer.Substituteτ, λ0, and λi into formula (16):(17)ωτ∂Tis∂ωt+λ0+λiTis=λ0T0˜+λiTi+λ0ΔT0˜sinωt.λ
0 and λi, respectively, measure the external and internal convective heat transfer intensities of the surface of the outer envelope. In buildings with large external or internal convection heat transfer, the efficiency of internal or external convection heat transfer is higher than that of indoor air mixing.The general solution of equation (17) [17] is(18)Tisωt=λ0λ0+λiT0˜+λiλ0+λiTi+λiλ0λ0+λi2+ω2τ2ΔT0˜sinωt−β1+C1λie−λ0+λi/ωτωt,where C1 is a constant and β1=tan−1ωτ/λ0+λi is the relative delay parameter of cooling load with respect to outdoor air temperature; it is evaluated from 0 to π/2.Substitute equation (18) into equation (14) [17]:(19)qsys=−ρCpQvTE+λiλ0λ0+λiT0˜−Ti+λiλ0λ0+λi2+ω2τ2ΔT0˜sinωt−β1+C1λie−λ0+λi/ωτωt.In equation (18), q represents the cooling load that needs to meet the indoor hot air balance in summer and T represents the temperature rise caused by indoor heat source, TE=E/ρCpQv. The cooling load is mainly composed of three parts. The first part is the steady-state cooling load, which is jointly caused by the steady-state heat source and the heat gain generated by the temperature difference between indoor and outdoor through the convective heat transfer of hot substances. The second part is the periodic fluctuation cooling load Δqsys, whose amplitude depends on the fluctuation ΔT0˜ of outdoor air temperature. The third part (exponential part) is caused by its initial conditions, which will decay to 0 with the increase of time and external convective heat transfer or the decrease of time constant.


### 2.4. Evaluation of Indoor Thermal Environment Model

PMV is the most comprehensive evaluation considering many factors of human thermal comfort. Table4 shows the thermal and cold ruler of ASHRAE corresponding to PMV. ISO defines the thermal comfort range as −0.5≤PMV≤0.5. In order to reduce energy consumption of building air conditioning, China reduces the comfort range to −1≤PMV≤1.Table 4
Thermal ruler of ASHRAE corresponding to PMV value.
Thermal sensation
Cold
Cool
Coolness
Moderation
Microwarm
Warm
Hot
PMV
−3
−2
−1
0
+1
+2
+2There are coupling effects among various factors affecting thermal sensation, for example, the increase of temperature can be compensated by the decrease of humidity or the increase of wind speed. ASHRAE defines thermal comfort as a psychologically satisfying thermal environment, which mainly includes six major factors, namely, air temperature, average radiant temperature, relative humidity, air velocity, human metabolism, and clothing [17]. Due to the complexity of PMV calculation, it is not conducive to engineering practice. Therefore, this paper chooses a simple calculation method for thermal environment assessment proposed [19].(20)PMV=2.43−3.67HB,(21)HB=Tsk−TopM1Ic1+Ia/fc1=Tsk−Top/Ic1+Ia/fc1M1.The molecule of formula (21) represents heat loss through sensible heat of skin, and the denominator represents total heat production of human body. Since invisible latent heat is inevitable, the value of the fraction must be less than 1, and its value represents the ratio of sensible heat dissipation to total heat production [19]. Tsk represents the average skin temperature. Since the skin temperature is relatively constant, the change range is very small in an environment close to thermal comfort. Here, the average skin temperature in the thermal comfort state is calculated at 33.5°C.According to ISO-7730 [20], when the wind speed is less than 0.2 m/s or the temperature difference between the average radiation temperature Tmrtand the air is less than 4°C, the calculation of the acting temperature Top is as follows:(22)Top=ATi+1−ATmrt,A=0.5,v<0.2m/s,0.6,0.2m/s≤v<0.6m/s,0.7,0.6m/s≤v<1m/s.I
cl represents thermal resistance of clothing. fcl represents the clothing area coefficient, and its calculation method is shown in (23). Ia represents the thermal resistance of air outside the garment, which is a function of air temperature and wind speed and is calculated according to formula (24) [19].(23)fcl=1.00+1.29Icl,Icl≤0.078,1.05+0.645Icl,Icl>0.078,(24)Ia=0.1550.61Ti+273/2983+1.9v298/Ti+273.T
mrt represents the average surface temperature of the area of j surfaces in a room, and its simplified calculation method is as follows:(25)Tmrt=∑j=05AjTis,j∑j=05Aj.


## 2.1. Lighting Energy Consumption

The assumption of diffuse reflection lighting conditions leads to another metric commonly referred to as the daylight factor (DF) [9], which is the ratio of indoor and outdoor illumination under standard CIE cloudy conditions. Due to the simplified sky model, given a certain daylight open configuration, regardless of direction, geographic location, and climate change, the DF at a particular point will remain approximately constant; for the purpose of design, the vector flux splitting method [10] is used. The DF consists of three separate components: sky (SC), external reflection (ERC), and internal reflection (IRC) components:(2)DF=SC+ERC+IRC.About complex design calculations, ERC can be determined by multiplying the external obstacle (SCobs, i.e., obstacles can be seen in the sky area) by the obstacle reflectivity (Robs, this paper takes a value of 0.2), and the calculation for IRC is also taken from [11]:(3)IRC=τjAjA1−RCRfw+5Rcw.In order to achieve a more practical design, additional calibration was performed in this paper, taking into account the maintenance factor (Mf), glass factor (G), and window frame factor (B), with values of 0.9, 1, and 0.85 [12], respectively. The final calculated DF is(4)DF=0.765SC+ERC+0.2SC.For the case where the calculation point is close enough to daylight, SC can be considered as the most important component of the three. SC is pure geometric calculation as shown in Figure1. The calculation equation of SC is as follows:.(5)SC=Ei,skyEO=Lz/3∫0θ′∫0β1+2sinθsinθcosθdθdψLz/3∫02π∫0π/21+2sinθsinθcosθdθdψ=314πβ−β′cosγ+27πarcsinsinβsinγ−17πsin2γsinβ′.Known:tanθ′=tanγcosψ,tanβ=LD,tanβ′=tanβcosγ=L/DH/D2+1.Figure 1
Diagram of the calculation of U from the vertical, effective daylight opening ABCD to D (from [13]).In the lighting design, there are roughly two possibilities for the position of the window relative to the calculation point; this paper refers to the position of the calculation point by Mangkuto [14]. The projection of the calculation point is located in the window area and internal or external (Figure 2); the height of the working surface is 0.75 m and is higher than the window sill position. Let the distance from the observation point to the left wall be x, which is calculated as follows:whenx≤D/2−Li/2,(6)SCa=SCLD=Ll−x+Li/2d,HD=Hh/2−0.75+Hj/2d−SCLD=Ll−x−Li/2d,HD=Hh/2−0.75+Hj/2d,whenD/2>x>D/2−Li/2,(7)SCb=SCLD=Ll−x+Li/2d,HD=Hh/2−0.75+Hj/2d−SCLD=Ll+Li/2d,HD=Hh/2−0.75+Hj/2d.Figure 2
Sectional view of the window wall, two possibilities for the position of the window relative to the calculated point (m).
(a)
(b)Annual lighting energy demand (EelKWh/m2/yr) refers to the total demand for electric lighting and energy lighting throughout the year. According to the national standard [15], the maximum value of the library building lighting power density limit target is 8 (W/m2), and each point of the illumination Ein is multiplied by the relevant DF multiplied by the outdoor diffuse, The selected value of the illuminance Eout is calculated. If the value of Ein is less than 300 lx (reading room illuminance standard value), then additional lighting is needed, and Ein at all points is averaged with a positive difference of 300 lx and multiplied by η = 37.5 lm/W (i.e. 300 lx per 8 W/m2) to obtain the desired illumination power density [14]. T is the total working time (8 hours/day × 5 days/week × 52 weeks/year), and we finally get the estimated Eel:(8)EelKWh/m2/yr=T1000mη∑i=1m300−Ein,i=0.55m∑i=1m300−Ein,i.


## 2.2. Heat Consumption

The calculation of the basic heat consumption of the envelope structure refers to the national standard [16]. The heat consumption of the external protection structure of the universal layer is mainly composed of the outer window and the outer wall. The outer wall is composed of a concrete wall, interface mortar (or bonding layer), insulation layer (or insulation board), crack protection layer (or plaster layer), and veneer layer.The heat consumption of the enclosure consists of basic heat consumption (9) and additional heat consumption. The heat transfer coefficient is composed of the heat transfer coefficients of each structure, and their relationship is shown in formula (10). The additional heat consumption is determined as a percentage of the basic heat consumption. It mainly includes the orientation correction rate, the wind attachment rate, the external door attachment rate, the height addition rate of the building (except the stairwell), and the intermittent addition rate. The outdoor temperature air temperature is a harmonic time parameter with an angular frequency. Therefore, a simple prediction [17] can be performed using equation (10). The indoor air temperature is determined by the indoor setting temperature. And the interior design temperature is designed according to the level of thermal comfort, as can be seen from Table 1. The calculation method of the heat consumption of the envelope structure is shown in formula (11).(9)Q0=αFKtn−twn,(10)K=11/an+∑δ/aλ⋅λ+Rk+1/aw,(11)twn=twn˜+Δtwnsinωt,(12)Q=αFKtn−twn1+or+hh+od.Table 1
Indoor design temperature of air conditioning in the area where personnel stay for a long timetn.
Working condition of the category
Thermal comfort level
Interior design temperature (°C)
Heating conditions
I level
22∼24
II level
18∼22
Cooling conditions
I level
24∼26
II level
26∼28In formula (10), an represents the heat transfer coefficient of the inner surface of the envelope, aw represents the heat transfer coefficient of the outer surface of the enclosure, and aλ represents the correction coefficient of material thermal conductivity. Their values are shown in Tables 2 and 3, respectively [18].Table 2
Heat transfer coefficient of inner surface of envelopean.
Internal surface characteristics of envelope
a
n
W
/
m
2
⋅
K
A wall, floor or ceiling with a flat surface or ribbed projection,h/s≤0.3
8.7
A ribbed, well-shaped roof,0.2<h/s≤0.3
8.1
A ceiling with ribbed projections,h/s>0.3
7.6
A roof with a well-shaped projectionh/s>0.3
7.0Table 3
Heat transfer coefficient of outer surface of envelopeaw.
Outer surface characteristics of envelope
a
w
W
/
m
2
⋅
K
Exterior walls and roof
23
A floor above an unheated basement that communicates with outdoor air
17
A floor slab above a nonheated basement with a covered roof and windows on the exterior wall
12
A windowless, unheated floor above a basement
6


## 2.3. Cooling Load

The heat transfer process of the building envelope is a very complex process involving convection, conduction, and radiation. At the same time, the outdoor air temperature and heat radiation density also change greatly with the change of outdoor time, and the indoor air temperature also changes. In this paper, the heat transfer equation of external thermal mass and internal air is used to calculate the hourly cooling load of the enclosure. The following assumptions are made:(i)
The air distribution within the building, the temperature distribution of the internal thermal mass, and the internal surface temperature of the external thermal mass are balanced.(ii)
All thermal gain and heat generation in the building are assembled into a heat sourceE during the working period, while there is no room thermal gain during the nonworking period, and the radiant heat exchange between the heat source and other surfaces is ignored.(iii)
The direct thermal solar radiation gain and permeability thermal gain through the opening were ignored.(iv)
During the nonworking period, the air conditioning system is closed and night ventilation mode is adopted. The ventilation rate isQv.Based on the above assumptions, the heat balance equations of the inner surface temperature and the indoor air temperature of the outer envelope structure are as shown in equations (13) and (14) [17]:(13)MCm∂Tis∂t+h0A0Tis−T0+hiA0Tis−Ti=0,(14)qsys+hiA0Tis−Ti+E=0.The calculation of outdoor air temperature in summer is similar to formula (11):(15)T0=T0˜+ΔT0˜sinωt.Substitute equation (15) into equation (13):(16)MCm∂Tis∂t+h0+hiA0Tis=h0A0T0˜+hiA0Ti+h0A0sinωt.Letτ=MCm/ρCpQv, where τ is the time constant based on night ventilation rate Qv; λ0=h0A0/ρCpQv, where λ0 is the dimensionless external convective heat transfer; and λi=hIA0/ρCpQv, where λi is the dimensionless internal convective heat transfer.Substituteτ, λ0, and λi into formula (16):(17)ωτ∂Tis∂ωt+λ0+λiTis=λ0T0˜+λiTi+λ0ΔT0˜sinωt.λ
0 and λi, respectively, measure the external and internal convective heat transfer intensities of the surface of the outer envelope. In buildings with large external or internal convection heat transfer, the efficiency of internal or external convection heat transfer is higher than that of indoor air mixing.The general solution of equation (17) [17] is(18)Tisωt=λ0λ0+λiT0˜+λiλ0+λiTi+λiλ0λ0+λi2+ω2τ2ΔT0˜sinωt−β1+C1λie−λ0+λi/ωτωt,where C1 is a constant and β1=tan−1ωτ/λ0+λi is the relative delay parameter of cooling load with respect to outdoor air temperature; it is evaluated from 0 to π/2.Substitute equation (18) into equation (14) [17]:(19)qsys=−ρCpQvTE+λiλ0λ0+λiT0˜−Ti+λiλ0λ0+λi2+ω2τ2ΔT0˜sinωt−β1+C1λie−λ0+λi/ωτωt.In equation (18), q represents the cooling load that needs to meet the indoor hot air balance in summer and T represents the temperature rise caused by indoor heat source, TE=E/ρCpQv. The cooling load is mainly composed of three parts. The first part is the steady-state cooling load, which is jointly caused by the steady-state heat source and the heat gain generated by the temperature difference between indoor and outdoor through the convective heat transfer of hot substances. The second part is the periodic fluctuation cooling load Δqsys, whose amplitude depends on the fluctuation ΔT0˜ of outdoor air temperature. The third part (exponential part) is caused by its initial conditions, which will decay to 0 with the increase of time and external convective heat transfer or the decrease of time constant.


## 2.4. Evaluation of Indoor Thermal Environment Model

PMV is the most comprehensive evaluation considering many factors of human thermal comfort. Table4 shows the thermal and cold ruler of ASHRAE corresponding to PMV. ISO defines the thermal comfort range as −0.5≤PMV≤0.5. In order to reduce energy consumption of building air conditioning, China reduces the comfort range to −1≤PMV≤1.Table 4
Thermal ruler of ASHRAE corresponding to PMV value.
Thermal sensation
Cold
Cool
Coolness
Moderation
Microwarm
Warm
Hot
PMV
−3
−2
−1
0
+1
+2
+2There are coupling effects among various factors affecting thermal sensation, for example, the increase of temperature can be compensated by the decrease of humidity or the increase of wind speed. ASHRAE defines thermal comfort as a psychologically satisfying thermal environment, which mainly includes six major factors, namely, air temperature, average radiant temperature, relative humidity, air velocity, human metabolism, and clothing [17]. Due to the complexity of PMV calculation, it is not conducive to engineering practice. Therefore, this paper chooses a simple calculation method for thermal environment assessment proposed [19].(20)PMV=2.43−3.67HB,(21)HB=Tsk−TopM1Ic1+Ia/fc1=Tsk−Top/Ic1+Ia/fc1M1.The molecule of formula (21) represents heat loss through sensible heat of skin, and the denominator represents total heat production of human body. Since invisible latent heat is inevitable, the value of the fraction must be less than 1, and its value represents the ratio of sensible heat dissipation to total heat production [19]. Tsk represents the average skin temperature. Since the skin temperature is relatively constant, the change range is very small in an environment close to thermal comfort. Here, the average skin temperature in the thermal comfort state is calculated at 33.5°C.According to ISO-7730 [20], when the wind speed is less than 0.2 m/s or the temperature difference between the average radiation temperature Tmrtand the air is less than 4°C, the calculation of the acting temperature Top is as follows:(22)Top=ATi+1−ATmrt,A=0.5,v<0.2m/s,0.6,0.2m/s≤v<0.6m/s,0.7,0.6m/s≤v<1m/s.I
cl represents thermal resistance of clothing. fcl represents the clothing area coefficient, and its calculation method is shown in (23). Ia represents the thermal resistance of air outside the garment, which is a function of air temperature and wind speed and is calculated according to formula (24) [19].(23)fcl=1.00+1.29Icl,Icl≤0.078,1.05+0.645Icl,Icl>0.078,(24)Ia=0.1550.61Ti+273/2983+1.9v298/Ti+273.T
mrt represents the average surface temperature of the area of j surfaces in a room, and its simplified calculation method is as follows:(25)Tmrt=∑j=05AjTis,j∑j=05Aj.


## 3. Case Study

### 3.1. Building Information

The proposed library is located in Chengdu, Sichuan province, with the coordinates of 30.67° north latitude and 104.02° east longitude. The building floor is the sixth floor, and the model verification object in this paper is the standard floor. Figure3 shows the library’s 3D model and standard layer. The standard floor consists of a reading room and a catalogue room. The entire area has a length of 21.27 m, a width of 11.9 m, and a height of 4.5 m. The total wall area is 221.4 m2, the floor area is 253.16 m2, and the ceiling area is 266.9 m2. The remaining initial input values are rendered in Table 5. What needs to be noted here is that in this calculation, we calculate Pareto values of 10 times of a day in winter and summer, respectively. Here, the average outdoor temperature in Chengdu is twn˜=2.7°C on a certain day in winter and T = 2 on a certain day in summer.Figure 3
3D view of the library and reading room plan.Table 5
The remaining initial input data.
Initial input data
E
out
,
s (lx)
12000
or
0.1
od
0.15
α
1
a
n
W
/
m
2
⋅
K
8.7
a
w
W
/
m
2
⋅
K
23
a
λ
1.2
h
i
W
/
m
2
⋅
K
2.2
T
0
˜ (°C)
31
E (W)
534
M
1
W
/
m
2
116
I
cl
m
2
C
°
/
W
0.0575
C
p
kJ
/
kg
⋅
K
0.98
hh
4
R
k
m
2
⋅
K
/
W
0.1
C
m
kJ
/
kg
⋅
K
1.05
t
wn
˜ (°C)
2.7
Δ
t
wn (°C)
3
h
0
W
/
m
2
⋅
K
5.5
Δ
T
0
˜ (°C)
3
ρ
kg
/
m
3
1.2
C
1
−2.39
v
m
/
s
0.2
T
sk (°C)
33.5
Q
v
m
3
/
h
400
C
39What needs to be explained here is that the value of constantC1 is obtained according to the case study in literature [17]. Of course, C1 can be different values for the general solution of ordinary differentiation. When the indoor temperature is 26°C, we calculated the value of C1for the verification of this model. In order to simplify the calculation, this paper assumes that the windows are integral and ignores the influence of different window positions on indoor lighting. We have set a total of 625 lighting calculation points in the room, as shown in Figure 4.Figure 4
Spatial profile of calculation points (m).


### 3.2. Data Input

We set a total of 20 design variables, which will provide architects with more abundant energy-saving building design parameters. The value range of these variables meets the national design standards [15, 16, 18], as shown in Table 6.Table 6
The remaining initial input data.
Variable
Symbol
Minimum
Maximum
Window height (m)
H
w
2
4
Window width (m)
L
w
10
18.25
Thickness of concrete (m)
δ
c
0.2
0.3
Thickness of mortar (m)
δ
m
0.02
0.06
The thickness of insulating material (m)
δ
i
0.01
0.06
The thickness of the window pane (m)
w
t
0.003
0.012
Heat transfer coefficient of concreteW/m⋅K
λ
c
0.1
1.74
Heat transfer coefficient of the whole window with closed space and window frame area accounting for 20%W/m⋅K
λ
w
1.07
5.8
Heat transfer coefficient of concrete
λ
m
0.76
0.93
Heat transfer coefficient of insulating materialW/m⋅K
λ
i
0.025
0.08
Set indoor temperature in summer (°C)
S
T
24
28
Floor reflection ratio
r
f
0.1
0.5
Wall reflection ratio
r
w
0.3
0.8
The reflection ratio of window glass
r
g
0.07
0.84
Ceiling reflection ratio
r
c
0.6
0.9
The total transmittance of the window
τ
j
0.4
0.84
Density of concretekg/m3
ρ
c
300
2500
Density of mortarkg/m3
ρ
m
300
1800
The density of insulating materialkg/m3
ρ
i
20
180
Set the indoor temperature in winter (°C)
w
T
18
24


### 3.3. Method

NSGA-II is based on the evolution of the “individual” group. When the algorithm performs nondominated sorting, each individual in theN-sized population is compared against M objective functions and N − 1 individuals in the population. NSGA-II needs to save two quantities during sorting:(i)
The number of dominantnp: this amount is the number of all individuals who can dictate the individual p in the feasible solution space(ii)
The controlled individuals gatheredSp: this quantity is a collection of all individuals in the feasible solution space that is dominated by individual pIn this paper, a total of three objective functions are set, which are in conflict with each other and cannot be compared. It is impossible to obtain the global maximum or minimum like single-objective problems. According to the objective function (fitness function) proposed above, NSGA-II is used to obtain a compromise solution set (nondominant solution) with initial input data and design variables, which does not favour any objective function. Its flow is shown in Figure5.Figure 5
Flowchart of NSGA-II.The optimizer was drawn up and run by Python 3.7 with a crossover probability of 0.8, a mutation probability of 1/20, an overall size of 100, and a maximum number of iterations of 200 generations.


### 3.4. Results

It should be reiterated here thatf1 is the total output result of the basic heat consumption of typical climate in Chengdu in winter and the cooling load of typical climate in summer. We calculated 11 moments in winter and summer, respectively. And f2 represents the annual lighting energy consumption per hour per square meter.In this process,f1 and f2 have a certain coupling, for example, the change of window size will affect the change of f1 and f2 output at the same time. Therefore, as f1 changes in calculation, f2 will also change correspondingly. In total, we obtained the Pareto solution sets at 11 different moments in a day and the values of corresponding design variables.According to the calculation result of the above case, for the convenience of the architect to more quickly get detailed building energy efficiency design data, we propose that when two different choices are adopted in the minimum value at 11 moments, the output value of each building parameter is as shown in Tables7 and 8.Table 7
The value of each design variable whenf1 takes the minimum value.
Parameters of the category
The values
The height of the window (m)
3.38
The width of the window (m)
12.48
Thickness of concrete (m)
0.3
Heat transfer coefficient of concrete
0.1
Thickness of mortar (m)
0.06
Heat transfer coefficient of mortar
0.89
The thickness of insulating material (m)
0.06
Heat transfer coefficient of insulating material
0.03
The thickness of the window pane (m)
—
Heat transfer coefficient of window
5.8
Set indoor temperature in summer (°C)
28
The reflection ratio of the floor
0.1
The reflection ratio of the wall
0.3
Window reflection ratio
0.84
Ceiling reflection ratio
0.9
The total transmittance of the window
0.74
Density of concrete (kg/m3)
1533
Density of mortar (kg/m3)
1799
Density of insulating material (kg/m3)
132
Set indoor temperature in winter (°C)
19Table 8
The value of each design variable whenf2 takes the minimum value.
Parameters of the category
The values
The height of the window (m)
4
The width of the window (m)
17.67
Thickness of concrete (m)
0.2
Heat transfer coefficient of concrete
0.11
Thickness of mortar (m)
0.02
Heat transfer coefficient of mortar
0.76
The thickness of insulating material (m)
0.06
Heat transfer coefficient of insulating material
0.08
The thickness of the window pane (m)
0.012
Heat transfer coefficient of window
5.7
Set indoor temperature in summer (°C)
24
The reflection ratio of the floor
0.5
The reflection ratio of the wall
0.8
Window reflection ratio
0.84
Ceiling reflection ratio
0.9
The total transmittance of the window
0.83
Density of concrete (kg/m3)
1947
Density of mortar (kg/m3)
1371
Density of insulating material (kg/m3)
162
Set indoor temperature in winter (°C)
18Finally, we give the PMV output results of different selection methods. As can be clearly seen from Figure6, when f2 at each moment is selected as the minimum value, the interval of PMV remains at −0.5∼0.5. In the other case, all values are higher than 0.5. However, in China, in order to make buildings more energy efficient, the range of PMV is extended to −1∼1. Accordingly, these two kinds of choices are to accord with indoor thermal and comfortable requirement. The main reason for this gap is that we only calculate thermal comfort in summer. But, the computational models used are the same in winter and summer. Here, we give more specific design variables of building parameters.Figure 6
PMV output results of different selections.To evaluate the effectiveness of NSGA-II in this article, we use the spacing (SP) [21] method to verify the consistency of the solution set. Firstly, given a solution set A=a1,a2,…,aN,(26)SPA=1N−1∑i=1Nd¯−d1ai,Aai2,where d¯ represents the average of d1a1,A/a1,d1a2,A/a2,…,d1aN,A/aN and d1ai,A/ai represents the L1 norm distance (Manhattan distance) of ai and established A/ai.(27)d1ai,A/ai=mina∈A/ai∑j=1maij−aj,where m represents the number of objectives and aij represents the solution ai of the jth objective. When the SP value is close to 0, an excellent consistent solution is obtained. Table 9 shows the SP values at different moments.Table 9
SP values of solutions at different moments.
Moment
SP
8:00
0.014
10:00
0.067
12:00
0.655
14:00
0.454
16:00
0.665
18:00
0.133
9:00
0.147
11:00
0.011
13:00
0.075
15:00
0.085
17:00
0.624From the values ofSP at each of the above moments, it can be seen that the solutions at each moment have certain stability. Therefore, we have put forward the most suitable options for architects in the solution set at these moments.


## 3.1. Building Information

The proposed library is located in Chengdu, Sichuan province, with the coordinates of 30.67° north latitude and 104.02° east longitude. The building floor is the sixth floor, and the model verification object in this paper is the standard floor. Figure3 shows the library’s 3D model and standard layer. The standard floor consists of a reading room and a catalogue room. The entire area has a length of 21.27 m, a width of 11.9 m, and a height of 4.5 m. The total wall area is 221.4 m2, the floor area is 253.16 m2, and the ceiling area is 266.9 m2. The remaining initial input values are rendered in Table 5. What needs to be noted here is that in this calculation, we calculate Pareto values of 10 times of a day in winter and summer, respectively. Here, the average outdoor temperature in Chengdu is twn˜=2.7°C on a certain day in winter and T = 2 on a certain day in summer.Figure 3
3D view of the library and reading room plan.Table 5
The remaining initial input data.
Initial input data
E
out
,
s (lx)
12000
or
0.1
od
0.15
α
1
a
n
W
/
m
2
⋅
K
8.7
a
w
W
/
m
2
⋅
K
23
a
λ
1.2
h
i
W
/
m
2
⋅
K
2.2
T
0
˜ (°C)
31
E (W)
534
M
1
W
/
m
2
116
I
cl
m
2
C
°
/
W
0.0575
C
p
kJ
/
kg
⋅
K
0.98
hh
4
R
k
m
2
⋅
K
/
W
0.1
C
m
kJ
/
kg
⋅
K
1.05
t
wn
˜ (°C)
2.7
Δ
t
wn (°C)
3
h
0
W
/
m
2
⋅
K
5.5
Δ
T
0
˜ (°C)
3
ρ
kg
/
m
3
1.2
C
1
−2.39
v
m
/
s
0.2
T
sk (°C)
33.5
Q
v
m
3
/
h
400
C
39What needs to be explained here is that the value of constantC1 is obtained according to the case study in literature [17]. Of course, C1 can be different values for the general solution of ordinary differentiation. When the indoor temperature is 26°C, we calculated the value of C1for the verification of this model. In order to simplify the calculation, this paper assumes that the windows are integral and ignores the influence of different window positions on indoor lighting. We have set a total of 625 lighting calculation points in the room, as shown in Figure 4.Figure 4
Spatial profile of calculation points (m).


## 3.2. Data Input

We set a total of 20 design variables, which will provide architects with more abundant energy-saving building design parameters. The value range of these variables meets the national design standards [15, 16, 18], as shown in Table 6.Table 6
The remaining initial input data.
Variable
Symbol
Minimum
Maximum
Window height (m)
H
w
2
4
Window width (m)
L
w
10
18.25
Thickness of concrete (m)
δ
c
0.2
0.3
Thickness of mortar (m)
δ
m
0.02
0.06
The thickness of insulating material (m)
δ
i
0.01
0.06
The thickness of the window pane (m)
w
t
0.003
0.012
Heat transfer coefficient of concreteW/m⋅K
λ
c
0.1
1.74
Heat transfer coefficient of the whole window with closed space and window frame area accounting for 20%W/m⋅K
λ
w
1.07
5.8
Heat transfer coefficient of concrete
λ
m
0.76
0.93
Heat transfer coefficient of insulating materialW/m⋅K
λ
i
0.025
0.08
Set indoor temperature in summer (°C)
S
T
24
28
Floor reflection ratio
r
f
0.1
0.5
Wall reflection ratio
r
w
0.3
0.8
The reflection ratio of window glass
r
g
0.07
0.84
Ceiling reflection ratio
r
c
0.6
0.9
The total transmittance of the window
τ
j
0.4
0.84
Density of concretekg/m3
ρ
c
300
2500
Density of mortarkg/m3
ρ
m
300
1800
The density of insulating materialkg/m3
ρ
i
20
180
Set the indoor temperature in winter (°C)
w
T
18
24


## 3.3. Method

NSGA-II is based on the evolution of the “individual” group. When the algorithm performs nondominated sorting, each individual in theN-sized population is compared against M objective functions and N − 1 individuals in the population. NSGA-II needs to save two quantities during sorting:(i)
The number of dominantnp: this amount is the number of all individuals who can dictate the individual p in the feasible solution space(ii)
The controlled individuals gatheredSp: this quantity is a collection of all individuals in the feasible solution space that is dominated by individual pIn this paper, a total of three objective functions are set, which are in conflict with each other and cannot be compared. It is impossible to obtain the global maximum or minimum like single-objective problems. According to the objective function (fitness function) proposed above, NSGA-II is used to obtain a compromise solution set (nondominant solution) with initial input data and design variables, which does not favour any objective function. Its flow is shown in Figure5.Figure 5
Flowchart of NSGA-II.The optimizer was drawn up and run by Python 3.7 with a crossover probability of 0.8, a mutation probability of 1/20, an overall size of 100, and a maximum number of iterations of 200 generations.


## 3.4. Results

It should be reiterated here thatf1 is the total output result of the basic heat consumption of typical climate in Chengdu in winter and the cooling load of typical climate in summer. We calculated 11 moments in winter and summer, respectively. And f2 represents the annual lighting energy consumption per hour per square meter.In this process,f1 and f2 have a certain coupling, for example, the change of window size will affect the change of f1 and f2 output at the same time. Therefore, as f1 changes in calculation, f2 will also change correspondingly. In total, we obtained the Pareto solution sets at 11 different moments in a day and the values of corresponding design variables.According to the calculation result of the above case, for the convenience of the architect to more quickly get detailed building energy efficiency design data, we propose that when two different choices are adopted in the minimum value at 11 moments, the output value of each building parameter is as shown in Tables7 and 8.Table 7
The value of each design variable whenf1 takes the minimum value.
Parameters of the category
The values
The height of the window (m)
3.38
The width of the window (m)
12.48
Thickness of concrete (m)
0.3
Heat transfer coefficient of concrete
0.1
Thickness of mortar (m)
0.06
Heat transfer coefficient of mortar
0.89
The thickness of insulating material (m)
0.06
Heat transfer coefficient of insulating material
0.03
The thickness of the window pane (m)
—
Heat transfer coefficient of window
5.8
Set indoor temperature in summer (°C)
28
The reflection ratio of the floor
0.1
The reflection ratio of the wall
0.3
Window reflection ratio
0.84
Ceiling reflection ratio
0.9
The total transmittance of the window
0.74
Density of concrete (kg/m3)
1533
Density of mortar (kg/m3)
1799
Density of insulating material (kg/m3)
132
Set indoor temperature in winter (°C)
19Table 8
The value of each design variable whenf2 takes the minimum value.
Parameters of the category
The values
The height of the window (m)
4
The width of the window (m)
17.67
Thickness of concrete (m)
0.2
Heat transfer coefficient of concrete
0.11
Thickness of mortar (m)
0.02
Heat transfer coefficient of mortar
0.76
The thickness of insulating material (m)
0.06
Heat transfer coefficient of insulating material
0.08
The thickness of the window pane (m)
0.012
Heat transfer coefficient of window
5.7
Set indoor temperature in summer (°C)
24
The reflection ratio of the floor
0.5
The reflection ratio of the wall
0.8
Window reflection ratio
0.84
Ceiling reflection ratio
0.9
The total transmittance of the window
0.83
Density of concrete (kg/m3)
1947
Density of mortar (kg/m3)
1371
Density of insulating material (kg/m3)
162
Set indoor temperature in winter (°C)
18Finally, we give the PMV output results of different selection methods. As can be clearly seen from Figure6, when f2 at each moment is selected as the minimum value, the interval of PMV remains at −0.5∼0.5. In the other case, all values are higher than 0.5. However, in China, in order to make buildings more energy efficient, the range of PMV is extended to −1∼1. Accordingly, these two kinds of choices are to accord with indoor thermal and comfortable requirement. The main reason for this gap is that we only calculate thermal comfort in summer. But, the computational models used are the same in winter and summer. Here, we give more specific design variables of building parameters.Figure 6
PMV output results of different selections.To evaluate the effectiveness of NSGA-II in this article, we use the spacing (SP) [21] method to verify the consistency of the solution set. Firstly, given a solution set A=a1,a2,…,aN,(26)SPA=1N−1∑i=1Nd¯−d1ai,Aai2,where d¯ represents the average of d1a1,A/a1,d1a2,A/a2,…,d1aN,A/aN and d1ai,A/ai represents the L1 norm distance (Manhattan distance) of ai and established A/ai.(27)d1ai,A/ai=mina∈A/ai∑j=1maij−aj,where m represents the number of objectives and aij represents the solution ai of the jth objective. When the SP value is close to 0, an excellent consistent solution is obtained. Table 9 shows the SP values at different moments.Table 9
SP values of solutions at different moments.
Moment
SP
8:00
0.014
10:00
0.067
12:00
0.655
14:00
0.454
16:00
0.665
18:00
0.133
9:00
0.147
11:00
0.011
13:00
0.075
15:00
0.085
17:00
0.624From the values ofSP at each of the above moments, it can be seen that the solutions at each moment have certain stability. Therefore, we have put forward the most suitable options for architects in the solution set at these moments.


## 4. Conclusion and Recommendations for Future Work

This paper introduces a design method of building energy conservation. To implement the model, the nondominant sorting genetic algorithm-II (NSGA-II) is written in the Python environment. In the design questions raised, the design parameters include window size, glass material and specification, floor veneer material, wall veneer material, external wall building material and specification, etc. In addition, the three objective functions of building thermal load, indoor thermal comfort, and lighting energy consumption are considered. Finally, 1100 solution sets are obtained at 11 different moments. In these solution sets, we choose two representative solution sets, which are, respectively, based on the minimum value of building cooling and heating load and the minimum value of lighting energy consumption.NSGA-II has certain limitations. At present, in the research of energy-saving optimization, many scholars still use NSGA-II to solve three target problems and get excellent results [22–24]. It has developed for nearly 20 years, and the established objective functions are basically no more than three. Due to the proportion of nondominated individuals in the population will rise of power series with the increase of the target space dimension [25]. Therefore, we will adopt more efficient methods to address optimization problems in future studies.The construction plan set out in the present paper does not mean specific cost analysis because cost analysis involves the impact of labour costs, construction technology, and local material price fluctuations. Identifiable cost issues can be further analysed on a case-by-case basis. Finally, the contradiction between building energy consumption and indoor environment is closely related to related climate characteristics and renewable energy. We will continue to explore the application and benefits of renewable energy in energy-efficient buildings in the next study.



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*Source: 5261869-2019-12-10.xml*