Trade-Off Judgement for Daylighting and Energy Consumption in the High and Large Space of the University Gymnasium in Beijing

Abstract

Taking the high and large space of the University of Science and Technology Beijing Gymnasium as this research object, this paper analyzes the influence of different window positions, window-to-wall ratio (WWR), solar heat gain coefficient (SHGC), heat transfer coefficient (K), and visible light transmittance (VT) on the total indoor energy consumption in winter and summer and obtains the relationship between the daylight factor and VT formed when the window is opened per unit area. Through energy consumption simulation, the variation law and calculation formula for indoor total energy consumption are obtained. The results show that the SHGC and K of the exterior window have a significant influence on the total energy consumption. By using the energy consumption simulation of different types of exterior windows, it is concluded that the SHGC of the south-facing window is negatively correlated with the variation of air conditioning energy consumption per unit area Δe_1,w, while the others are positively correlated. Moreover, the SHGC and K of the skylight have the most significant influence on the Δe_1,w. The total energy consumption decreases and then increases with the increase in the window area, and there is a lowest point, so the right side of the lowest point is less than or equal to 105% of the lowest total energy consumption as a reasonable window area zone. Finally, a progressive optimization method for weighing daylighting and energy consumption in university gymnasiums in Beijing is proposed.

1. Introduction

In recent years, with the rapid development of China’s higher education, gymnasiums have occupied an increasing proportion of the public buildings of universities [1], and many newly built campuses have built gymnasiums. As large public buildings, gymnasiums generally have large spans and large spaces [2], which greatly increase the energy consumption of office lighting, air conditioning systems, and other energy-using equipment. Zheng Yan et al. [3] conducted a survey on the University of Science and Technology Beijing (USTB) Gymnasium and concluded that the main energy resources of the gymnasium were office lighting and the air conditioning system, which together accounted for 84% of the total energy consumption.

At present, more and more transparent envelope structures, such as exterior windows, skylights, glass curtain walls, and other transparent materials, are widely used in the construction of university gymnasiums. Daylighting entering the room through transparent envelope structures can enhance the indoor visual effect, make people feel comfortable, and reduce lighting energy consumption [4,5,6,7]. Taking an office building as the simulation object, Peng Peng [8] concluded that the energy-saving rate when the lighting density decreased by 1 W/m² was equal to the energy-saving rate when the COP of the air conditioning system was increased by 0.3 or the indoor air conditioning temperature was increased by 1 °C. Hence, daylighting has great potential for energy savings [9]. However, the transparent envelope structures have poor shading capacity and a poor thermal insulation effect. If the window-to-wall ratio (WWR) of a building is too large, the energy consumption of air conditioning and heating will inevitably increase by a large margin [10,11,12].

Therefore, more and more scholars use various methods to find the optimal window design scheme based on the contradiction between daylighting and building energy consumption. Taking the lowest energy consumption of air conditioning, heating, and lighting in the classroom as the starting point, Futrell et al. [13] studied the impact of windows with different orientations on energy consumption and lighting. They adopted the Genopt clustering optimization algorithm to optimize them, concluding that the north window contributed the least to the indoor lighting effect and received the least solar thermal radiation. Konis et al. [14] developed a Passive Performance Optimization Framework (PPOF) and corresponding generative modeling workflow to inform early-stage design by optimizing building geometry, orientation, fenestration, and shading device geometry configurations in response to annual climate-based thermal comfort and daylighting performance outcomes. Motamedi et al. [15] raised an algorithm for the optimal skylight design of office buildings that combined lighting and energy consumption and compared different skylight ratios according to lighting energy consumption and air conditioning energy consumption to obtain the optimal skylight design. Yuan Fang et al. [16] put forward a building performance optimization process that uses parametric design, building simulation, and genetic algorithms to optimize design. They took a small office building as this research subject to verify the effectiveness of the optimization process. Lakhdari et al. [17] showed how the genetic algorithms used to search for high-performing design solutions could be applied to optimize the thermal, lighting, and energy performance of a middle school classroom in a hot and dry climate. Using a parametric approach and evolutionary multi-objective computation via the Octopus plug-in for Grasshopper, various windows-to-wall ratios (WWRs), wall materials, glass types, and shading devices were combined to derive potential solutions that achieve a good balance between daylight provision and thermal comfort while ensuring low energy consumption. Yan Li [18] used theoretical analysis and numerical simulation to obtain the variation trends of energy consumption, the calculation formulas of total building energy consumption under the different areas of transparent envelopes, and the evaluation method of the rationality of building windows.

Existing studies mainly focus on the relationship between lighting and energy consumption in small functional spaces, such as office spaces and classrooms, or in high and large spaces, such as railway stations. In contrast, few studies have been conducted on the balance between daylighting and energy consumption in high and large spaces, such as university gymnasiums. University gymnasiums and railway stations are different in space scale, flow density, equipment operation, interior space, and interior environment design requirements. Railway stations have a high occupant density, a long operating time, and a high frequency of equipment use. And their interior space structure is compact and open [19]. Compared with railway stations, university gymnasiums have a larger span, centralized activity space, and intermittent operation. In addition, the indoor environmental temperature requirements vary for different types of venues and competition events. Gymnasiums usually require a large amount of space to accommodate spectators, competition venues, and other facilities, making the indoor space relatively closed [20].

Aiming at the contradiction between daylighting and energy consumption in university gymnasiums in Beijing, this paper takes the high and large space of the University of Science and Technology Beijing (USTB) Gymnasium as this research object, simulates it by Ecotect software version 2010 to obtain the relationship between the daylight factor and visible light transmittance (VT) formed by skylights and side windows per unit area, and analyzes the variation law of total energy consumption with different transparent envelope structures by EnergyPlus software version 22.2. Ultimately, by analyzing the impact of daylighting, artificial lighting, and air conditioning energy consumption, a progressive optimization calculation process for weighing daylighting and energy consumption in university gymnasiums in Beijing is proposed. Use this calculation process to determine whether the specific window design scheme of university gymnasiums in Beijing is reasonable. If it is unreasonable, adjust the window design scheme and reevaluate until it is reasonable, which provides a theoretical basis for the window design scheme of university gymnasiums in Beijing.

2. Methodology

2.1. Model Establishment and Parameter Setting

Figure 1 shows the interior of the main stadium of the University of Science and Technology Beijing Gymnasium. The gymnasium model is shown in Figure 2. Furthermore, the thermal parameters of the envelope structure and internal disturbance parameters are set according to the code [21]. The illumination control and standard values of indoor daylighting for side and top lighting are set according to the standard [22]. The parameters for the base case are shown in

Table 1. Parameters for the base case.

Parameter		Description
Location		Beijing
Gymnasium model		Simplified as 75.1 m × 54.6 m × 23.75 m cuboid
Main competition zone		60 m × 40 m rectangle
Lighting zone		45 m × 30 m rectangle
All around		Trapezoidal stands
Stand height		10 m
Roof		Horizontal roof
Reflectance of an opaque envelope structure	Wall	0.750
	Roof	0.200
	Floor	0.580
K of opaque envelope structure	Wall	0.25 $\mathrm{W}/({\mathrm{m}}^2\cdot \mathrm{K})$
	Roof	0.25 $\mathrm{W}/({\mathrm{m}}^2\cdot \mathrm{K})$
Exterior window	Type	Double-glazed aluminum framed glass
	K	1.50 $\mathrm{W}/({\mathrm{m}}^2\cdot \mathrm{K})$
	SHGC	0.81
	VT	0.639
Critical illuminance of outdoor daylighting		5000 lux
Standard value of indoor daylighting	Side	300 lx
	Top	150 lx
Summer indoor air conditioning temperature		24 °C
Winter indoor air conditioning temperature		18 °C
Cooling operation period		6.1–9.30
Heating operation period		11.15–3.15
Artificial lighting power		20 W/m2
Occupant density		0.32 person/m2
Equipment power		10 W/m2
Operating time		08:00–22:00
Air change rate		0.5 per hour

.

2.2. Composition of Total Energy Consumption

Due to the previous research, it was found that the main energy sources of the University of Science and Technology Beijing (USTB) Gymnasium were office lighting and air conditioning systems. Therefore, this paper believes that the total annual energy consumption of the gymnasium is composed of lighting energy consumption and air conditioning energy consumption in summer and winter. This paper only discusses the impact of daylighting on the three types of energy consumption mentioned above, which can be divided into three parts: (1) the impact of window heating on indoor energy consumption; (2) changes in lighting energy consumption caused by daylighting; and (3) changes in energy consumption of air conditioning and heating caused by changes in lighting energy consumption. Figure 3 demonstrates the interaction between daylighting and indoor energy consumption.

As the window area increases, the solar heat gain into the room through the transparent envelope structure increases, and the heat transfer caused by the temperature difference between inside and outside through the glass also increases correspondingly. At the same time, the amount of daylight entering the interior also increases accordingly. Furthermore, the indoor daylight factor increases and the lighting energy consumption decreases accordingly, which leads to the reduction of heat gain from lighting, thus reducing the air conditioning energy consumption in summer but increasing air conditioning energy consumption (heating energy consumption) in winter.

The total energy consumption is analyzed as follows:

When the windows are closing, the total energy consumption $E_0$ is the sum of the air conditioning energy consumption $E_1$ and lighting energy consumption $E_2$, that is

$$E_0=E_1+E_2$$

(1)

When the windows are opening, the total energy consumption $E^{\prime }$ is the sum of the total energy consumption $E_0$ when the window is closed and the total energy consumption variation $\Delta E$ caused by the window opening.

$$E^{\prime }=E_0+\Delta E$$

(2)

The variation in total energy consumption $\Delta E$ caused by window opening is the sum of the variation in air conditioning energy consumption $\Delta E_1$ and the variation in lighting energy consumption $\Delta E_2$, that is

$$\Delta E=\Delta E_1+\Delta E_2$$

(3)

The variation in air conditioning energy consumption $\Delta E_1$ includes two parts: (1) the variation in air conditioning energy consumption $\Delta E_{1,W}$ caused by the change in heat transfer performance of the envelope structure (indoor and outdoor temperature difference heat transfer and solar radiation heat gain) after opening the window; and (2) the variation in air conditioning energy consumption $\Delta E_{1,L}$ caused by the reduction of lighting energy consumption after opening the window:

$$\Delta E_1=\Delta E_{1,W}+\Delta E_{1,L}$$

(4)

Substitute Equation (4) into Equation (3) to obtain

$$\Delta E=\Delta E_{1,W}+\Delta E_{1,L}+\Delta E_2$$

(5)

Since $\Delta E_{1,L}$ and $\Delta E_2$ are only related to the indoor light environment, which $\Delta E_{1,L}+\Delta E_2$ is denoted as $\Delta E_L$, then Equation (5) is simplified as

$$\Delta E=\Delta E_{1,W}+\Delta E_L$$

(6)

The total energy consumption $E^{\prime }$ after opening the window is:

$$E^{\prime }=E_1+E_2+\Delta E_{1,W}+\Delta E_L$$

(7)

$\Delta E_{1,W}$ can be obtained by changing the window area without changing the indoor lighting conditions (no lighting control) and calculating the variation in total energy consumption under different window design conditions; $E_1+E_2+\Delta E_L$ can be obtained by setting the amount of lighting that is required to be added under different window design conditions in the model without windows and calculating the total energy consumption. Both of these variables can be calculated by EnergyPlus.

$E_1+E_2+\Delta E_L$ is denoted as $E_{0,L}$, then the total energy consumption $E^{\prime }$ after opening the window is

$$E^{\prime }=\Delta E_{1,W}+E_{0,L}$$

(8)

Through the calculation of EnergyPlus, the variation law of $\Delta E_{1,W}$ and $E_{0,L}$ with the window area is obtained, and the total energy consumption $E^{\prime }$ under different window design conditions can be obtained by superposing $\Delta E_{1,W}$ and $E_{0,L}$, as shown in Figure 4.

3. Results and Discussion

3.1. Daylighting Simulation

The relationship between the daylight factor and VT was analyzed by Ecotect software version 2010. The WWR was set at 1% in the simulation condition, and the simulation results of daylight factors formed by the reference plane of the indoor lighting area under different VTs were obtained when the skylight and side window were opened per unit area, respectively, as shown in Figure 5.

The straight lines formed by the skylight and side window are fitted, respectively, and the fitting relation between the daylight factor (c) and VT formed when the window is opened per unit area is obtained.

Skylight: c = 1.92667(VT) + 0.09556

(9)

Side window: c = 0.365(VT) + 0.14417

(10)

3.2. Total Energy Consumption Simulation

This paper uses EnergyPlus to analyze the influence of different skylight ratios, solar heat gain coefficients (SHGC), K, and VT on total energy consumption (including lighting energy consumption and air conditioning energy consumption in summer and winter) under lighting control, as shown in Figure 6, Figure 7, Figure 8 and Figure 9. It can be seen from the figure that the SHGC and K of the exterior window have a significant influence on the total energy consumption. All material parameters are selected according to the code [21] and software material libraries.

3.3. Summer Air Conditioning Energy Consumption Simulation

This section analyzes the influence of different skylight ratios (SHGC, K, and VT) on air conditioning energy consumption in summer under lighting control, as shown in Figure 10, Figure 11, Figure 12 and Figure 13.

From Figure 10, it can be concluded that the air conditioning energy consumption in summer decreases and then increases with the increase in the skylight ratio, reaching its lowest point at 4%.

The reasons why air conditioning energy consumption in summer decreases first and then increases are:

(1) When the skylight ratio is small, the heat gain from lighting is the dominant factor. As the skylight ratio increases, the energy consumption and heat gain from lighting both decrease rapidly. The decrease in heat gain from lighting is beneficial for air conditioning energy consumption in the summer. In order to maintain indoor temperatures in the summer, air conditioning energy consumption decreases.

(2) As the skylight ratio continues to increase, the solar heat gain and heat transfer caused by the temperature difference are the dominant factors. At this time, although the reduction in lighting energy consumption gradually slows down, the solar heat gain and heat transfer caused by the temperature difference increase, which are not conducive to air conditioning energy consumption in the summer. To maintain indoor temperatures in the summer, air conditioning energy consumption increases rapidly.

(1) When the skylight ratio is the same, the air conditioning energy consumption in summer increases with a larger SHGC.

(2) The air conditioning energy consumption in summer under three different SHGCs decreases first and then increases with an increase in the skylight ratio.

The reason why air conditioning energy consumption in summer increases with a larger SHGC when the skylight ratio is the same as:

Under the same skylight ratio, the reduction in lighting energy consumption leads to the same reduction in air conditioning energy consumption in the summer. However, the solar heat gain entering the room increases with a larger SHGC, which is not conducive to air conditioning energy consumption in the summer.

(1) When the skylight ratio is the same, the air conditioning energy consumption in summer decreases with a larger K.

(2) The air conditioning energy consumption in summer under three different Ks decreases first and then increases with the increase in skylight ratio.

In summer, the daytime temperature is higher, and the air conditioning energy consumption increases. However, at night, the temperature of the outdoor air is lower than that of the indoor air. The larger the K is, the better the heat dissipation effect will be. When the decrease in daytime air conditioning energy consumption is smaller than the increase in nighttime air conditioning energy consumption, the air conditioning energy consumption in summer decreases with a larger K.

(1) When the skylight ratio is less than 10%, the air conditioning energy consumption in summer decreases with a larger VT under the same skylight ratio; when the skylight ratio is greater than 10%, there is not much difference in air conditioning energy consumption in summer among the three VTs.

(2) The air conditioning energy consumption in summer under three different VTs decreases first and then increases with the increase in skylight ratio.

The reason why the air conditioning energy consumption in summer decreases with a larger VT when the skylight ratio is the same as:

The reduction in lighting energy consumption caused by high-transmittance glass is greater than that caused by low-transmittance glass. The reduction in lighting heat caused by the reduction in lighting energy consumption will be much reduced, which is beneficial for air conditioning energy consumption in the summer.

3.4. Winter Air Conditioning Energy Consumption Simulation

This section analyzes the influence of different skylight ratios (SHGC, K, and VT) on air conditioning energy consumption in winter under lighting control, as shown in Figure 14, Figure 15, Figure 16 and Figure 17.

From Figure 14, it can be concluded that the air conditioning energy consumption in winter increases first and then decreases with the increase in the skylight ratio, which is opposite to the reason why the air conditioning energy consumption in summer decreases first and then increases.

(1) When the skylight ratio is the same, the air conditioning energy consumption in winter decreases with a larger SHGC, which is opposite to summer.

(2) The air conditioning energy consumption in winter under three different SHGCs increases first and then decreases with the increase in the skylight ratio.

(1) When the ratio of skylights is the same, the air conditioning energy consumption in winter increases with a larger K.

(2) The air conditioning energy consumption in winter under three different Ks increases first and then decreases with the increase in skylight ratio.

(1) When the skylight ratio is the same, the air conditioning energy consumption in winter increases with a larger VT, which is opposite to summer.

(2) The air conditioning energy consumption in winter under three different Ks increases first and then decreases with the increase in skylight ratio.

3.5. Validation

Liu Yao [23] analyzed the influence of the skylight ratio, shading coefficient, heat transfer coefficient, and light transmission rate on the energy consumption under lighting control. The results showed that after changing different parameters, the total energy consumption and air conditioning energy consumption in summer first decreased with the increase in the skylight ratio and then increased with the increase in the skylight ratio after reaching the lowest point. In addition, the air conditioning energy consumption in winter increased first and then decreased with the increase in the skylight ratio. The energy consumption variation law simulated in this paper is the same as the above law. The objects of both studies are the large spaces of public buildings, and the influence of transparent envelope parameters on energy consumption is studied under lighting control, so this reference has certain comparative verification significance.

3.6. Determination of $\Delta E_{1,W}$

$\Delta E_{1,W}$ can be regarded as the variation in total energy consumption with the change of window area under no lighting control in the room. Therefore, it is necessary to separately simulate and analyze the variation laws of the total energy consumption of side windows and skylights in different orientations with the window-wall ratio under no lighting control, as shown in Figure 18 and Figure 19.

Due to the different areas for each orientation, the window area represented by the same WWR for each orientation is also different, and the impact of opening the window on the variation of air conditioning energy consumption per unit area $\Delta e_{1,W}$ cannot be directly compared according to the WWR for each orientation. Therefore, 1% of the roof area is taken as the unit area, and the $\Delta E_{1,W}$ for each orientation is converted into the unit window area for comparison. By increasing the window area (1% of the roof area) by one unit, the variation in air conditioning energy consumption per unit area $\Delta e_{1,W}$ for each orientation indicates that the total energy consumption of skylights increases by 1.83 kWh/m², the east-facing energy consumption increases by 0.26 kWh/m², the west-facing energy consumption increases by 0.39 kWh/m², the south-facing energy consumption increases by 0.15 kWh/m², and the north-facing energy consumption increases by 0.29 kWh/m².

The significant influencing factors of the $\Delta e_{1,W}$ are SHGC and K of the glass. Therefore, by selecting different types of exterior windows from software material libraries, EnergyPlus was used for energy consumption simulation to gain the variation of $\Delta e_{1,W}$ for each orientation under different SHGCs and Ks of the exterior. The results are shown in

Table 2. Variation in air conditioning energy consumption per unit area $\Delta e_{1,W}$.

Orientation	Exterior Window Type	SHGC	K [W/(m2·K)]	$\mathbf{\Delta }{\mathit{e}}_{1,\mathit{W}}$ (kWh/m2)
Crown	White single-layer glass window	0.72	5.39	2.120
	White hollow glass window	0.64	3.60	1.660
	White single-layer glass window	0.62	4.74	1.815
	White double-layer glass window	0.56	2.67	1.360
	White hollow glass window	0.55	2.68	1.365
	Low-E-coated hollow glass window	0.52	1.77	1.180
	Low-E-coated hollow glass window	0.42	1.60	1.015
East	White single-layer glass window	0.72	5.39	0.404
	White hollow glass window	0.64	3.60	0.322
	White single-layer glass window	0.62	4.74	0.362
	White double-layer glass window	0.56	2.67	0.250
	White hollow glass window	0.55	2.68	0.248
	Low-E-coated hollow glass window	0.52	1.77	0.198
	Low-E-coated hollow glass window	0.42	1.60	0.174
West	White single-layer glass window	0.72	5.39	0.520
	White hollow glass window	0.64	3.60	0.420
	White single-layer glass window	0.62	4.74	0.472
	White double-layer glass window	0.56	2.67	0.342
	White hollow glass window	0.55	2.68	0.336
	Low-E-coated hollow glass window	0.52	1.77	0.284
	Low-E-coated hollow glass window	0.42	1.60	0.240
South	White single-layer glass window	0.72	5.39	0.274
	White hollow glass window	0.64	3.60	0.188
	White single-layer glass window	0.62	4.74	0.252
	White double-layer glass window	0.56	2.67	0.146
	White hollow glass window	0.55	2.68	0.148
	Low-E-coated hollow glass window	0.52	1.77	0.096
	Low-E-coated hollow glass window	0.42	1.60	0.096
North	White single-layer glass window	0.72	5.39	0.514
	White hollow glass window	0.64	3.60	0.410
	White single-layer glass window	0.62	4.74	0.466
	White double-layer glass window	0.56	2.67	0.332
	White hollow glass window	0.55	2.68	0.330
	Low-E-coated hollow glass window	0.52	1.77	0.252
	Low-E-coated hollow glass window	0.42	1.60	0.218

.

Establish quadratic linear regression models of the SHGC and K to the $\Delta e_{1,W}$ for different orientations, as shown in Equations (11)–(15).

$$\mathrm{Crown}: \Delta e_{1,W}=1.4264SHGC+0.17624K+0.11571$$

(11)

$$\mathrm{East}: \Delta e_{1,W}=0.21212SHGC+0.04588K+0.01045$$

(12)

$$\mathrm{West}: \Delta e_{1,W}=0.31572SHGC+0.05037K+0.03011$$

(13)

$$\mathrm{South}: \Delta e_{1,W}=-0.08436SHGC+0.05397K+0.04692$$

(14)

$$\mathrm{North}: \Delta e_{1,W}=0.30082SHGC+0.05638K+0.00628$$

(15)

Standardizing the window area for each orientation to be the same as the skylight area, the equivalent skylight and the original window design scheme have the same $\Delta E_{1,W}$. The total energy consumption variation $\Delta e_{1,W}{}^{\prime }$ caused by the equivalent window per unit area represents the heat transfer characteristics of the glass. Since the window area and $\Delta E_{1,W}$ are linearly related, $\Delta e_{1,W}{}^{\prime }$ can represent the weighted average of $\Delta e_{1,W}$ for each orientation, that is

$$\Delta e_{1,W}{}^{\prime }=\frac{A_0}{A^{\prime }}(\frac{A_e}{A_0}\Delta e_{1,W}(e)+\frac{A_w}{A_0}\Delta e_{1,W}(w)+\frac{A_s}{A_0}\Delta e_{1,W}(s)+\frac{A_n}{A_0}\Delta e_{1,W}(n)+\frac{A_t}{A_0}\Delta e_{1,W}(t))$$

(16)

In the formula, e, w, s, n, and t respectively represent east, west, south, north, and crown; $A^{\prime }$ is the total area of the equivalent skylight, $A^{\prime }=A_e+A_w+A_s+A_n+A_t$; $A_0$ is the unit area (1% of the roof area).

3.7. Determination of $E_{0,L}$

The variation in $E_{0,L}$ is $\Delta E_L$, and $\Delta E_L$ is only related to the indoor light environment. Therefore, $E_{0,L}$ is expressed as the function of the daylight factor (C), $E_{0,L}=f(C)$, and the value of the daylight factor (C) represents the lighting characteristics of glass. With the increase in the window area, the daylight factor (C) increases linearly. The daylight factor $c^{\prime }$ formed by the unit equivalent skylight can also be written as the weighted average of the daylight factor c formed by the window per unit area for each orientation, that is

$$c^{\prime }=\frac{A_0}{A^{\prime }}(\frac{A_e}{A_0}{c}_e+\frac{A_w}{A_0}{c}_w+\frac{A_s}{A_0}{c}_s+\frac{A_n}{A_0}{c}_n+\frac{A_t}{A_0}{c}_t)$$

(17)

The curve of $E_{0,L}$ can be considered as to be solely related to the local meteorological conditions. The daylight factor is expressed in units of 2% to adjust the indoor lighting energy consumption according to 2%, 4%, 6%, …, and 30%. The variation curve of $E_{0,L}$ is shown in Figure 20.

As can be seen from Figure 20, $E_{0,L}$ decreases rapidly as the daylight factor increases, between approximately 2–20%. After that, its rate of decrease gradually slows down, eventually reaching a constant value. For different daylight factor intervals, the variation of $E_{0,L}$ is denoted as $\Delta E_{0,L}$, and its values are shown in

Table 3. Of each daylight factor interval.

Daylight Factor Interval	$\mathbf{\Delta }{\mathit{E}}_{0,\mathit{L}}$ (kWh/m2)
2~4%	−18.99
4~6%	−5.95
6~8%	−2.96
8~10%	−1.91
10~12%	−1.11
12~14%	−0.76
14~16%	−0.73
16~18%	−0.66
18~20%	−0.40
20~30%	>−0.40

. If the curves in each interval are approximated as straight lines, then $\Delta E_{0,L}$ is the average slope of the $E_{0,L}$ curves in different daylight factor intervals.

3.8. Trade-Off Judgement for Daylighting and Energy Consumption

The trade-off judgment for daylighting and energy consumption proposed in this paper is to find the window design condition with the lowest total energy consumption under daylighting conditions and judge the rationality of the window design scheme by the degree of actual total energy consumption deviating from the lowest one. Therefore, it is not necessary to calculate the entire curve, but only to calculate the actual total energy consumption $E^{\prime }(A)$ and the lowest total energy consumption $E^{\prime }(a)$, as shown in Figure 21.

(1) Actual total energy consumption

The actual total energy consumption is calculated as $E^{\prime }(A)=\Delta E_{1,W}(A)+E_{0,L}(A)$. After determining the window design scheme, the equivalent skylight characteristics and the actual total energy consumption at point A can be calculated, and then the actual total energy consumption $E^{\prime }(A)$ is calculated by substituting the value of point A into the trade-off judgment formula.

(2) Lowest total energy consumption

The formula for calculating the lowest total energy consumption is $E^{\prime }(a)=\Delta E_{1,W}(a)+E_{0,L}(a)$, find the position of point a with the lowest total energy consumption, and calculate the lowest total energy consumption $E^{\prime }(a)$ by substituting the value of point a into the trade-off judgment formula.

$E_{0,L}$ is a curve changing with the daylight factor (C). To unify the independent variables of $\Delta E_{1,W}$ and $E_{0,L}$, $\Delta E_{1,W}$ which are recorded as a curve changing with the daylight factor (C), the slope of $E^{\prime }$ is

$$\frac{d{E}^{\prime }}{dC}=\frac{d(\Delta E_{1,W})}{dC}+\frac{d{E}_{0,L}}{dC}$$

(18)

Since it is difficult to accurately fit the relation of $E_{0,L}$, the variation within a unit daylighting factor interval represents the average slope within the interval. To make the calculation results more accurate, 2% is taken as a unit daylighting factor:

$$\frac{d{E}^{\prime }}{dC}\approx \frac{\Delta E^{\prime }}{\Delta C}=0.02\times \frac{\Delta e_{1,W}{}^{\prime }}{c^{\prime }}+\Delta E_{0,L}$$

(19)

The derivative of the lowest point of the curve is 0. So let $\frac{d{E}^{\prime }}{dC}=0$, then the daylighting factor corresponding to the point a of the lowest total energy consumption can be calculated according to Equation (20).

$$0.02\times \frac{\Delta e_{1,W}{}^{\prime }}{c^{\prime }}+\Delta E_{0,L}=0$$

(20)

(3) Judge the rationality of window area

In this paper, the right side of the lowest total energy consumption is less than or equal to 105% of the lowest total energy consumption as a reasonable window area zone, and the rationality judgment of window area is shown in Figure 22.

3.9. The Progressive Optimization Calculation Process for Trade-Off Judgement

Since the optimal window area and form cannot be directly given in the architectural design, the specific lighting design scheme can be weighed based on the above trade-off judgment of daylighting and energy consumption. If the window design scheme is unreasonable, then adjust the window design scheme and re-judge until it is reasonable. In this paper, this method is called the progressive optimization method for weighing daylighting and energy consumption. Its calculation process is shown in Figure 23.

(1) Determine the window design scheme.

Determine the window area in each orientation and the VT, SHGC, and K of the exterior window glass, and substitute VT into Equations (9) and (10) to calculate the daylighting factor (c) formed by the skylight and side window per unit area in the indoor reference plane. Then, the SHGC and K are substituted into Equations (11)–(15) to calculate the $\Delta e_{1,W}$ in each orientation.

(2) Calculate equivalent skylight characteristics.

Substitute $\Delta e_{1,W}$ and c of each orientation into Equations (16) and (17), respectively, to calculate the $\Delta e_{1,W}{}^{\prime }$ and the daylighting factor $c^{\prime }$ formed by the per-unit equivalent skylight area.

(3) Calculate the actual total energy consumption

$E^{\prime }(A)=\Delta E_{1,W}(A)+E_{0,L}(A)$ is the formula for the actual total energy consumption. And the formula for $\Delta E_{1,W}(A)$ is $\Delta E_{1,W}(A)=\frac{A^{\prime }}{A_0}\Delta e_{1,W}{}^{\prime }$. According to the daylighting factor C(A) of the actual total energy consumption, $E_{0,L}(A)$ can be obtained by referring to Figure 20, where the daylighting factor $C(A)=\frac{A^{\prime }}{A_0}{c}^{\prime }$.

(4) Calculate the lowest total energy consumption.

The lowest total energy consumption is calculated as $E^{\prime }(a)=\Delta E_{1,W}(a)+E_{0,L}(a)$. Substitute $\Delta e_{1,W}{}^{\prime }$ and $c^{\prime }$ into Equation (20) to calculate $\Delta E_{0,L}$. Accordingly $\Delta E_{0,L}$, the daylighting factor C(a) corresponding to point a with the lowest total energy consumption can be obtained from Table 3. The variation in air conditioning energy consumption $\Delta E_{1,W}(a)=\frac{\Delta e_{1,W}{}^{\prime }}{c^{\prime }}\times C(a)$. And according to the daylighting factor C(a) of the lowest total energy consumption, $E_{0,L}(a)$ can be obtained by referring to Figure 20.

(5) Determine whether the window design scheme is in a reasonable zone.

The rationality judgment of the window design scheme is divided into the following three situations:

When C(A) < C(a), it is in the insufficient window area zone.

The actual total energy consumption point A is to the left of the lowest total energy consumption point a, and the actual total energy consumption is in the declining stage of the total energy consumption curve. At this time, if the window area is increasing, the actual total energy consumption will decrease, so it is in the insufficient window area zone.

When C(A) ≥ C(a) and $E^{\prime }(A)\le 1.05E^{\prime }(a)$, it is in the reasonable window area zone.

The actual total energy consumption point A is on the right of the lowest total energy consumption point a, the actual total energy consumption is in the rising stage of the total energy consumption curve, and the actual total energy consumption is less than 105% of the lowest total energy consumption, so it is in the reasonable window area zone.

When C(A) ≥ C(a) and $E^{\prime }(A)$ > $1.05E^{\prime }(a)$, it is in the excessive window area zone.

The actual total energy consumption point A is on the right of the lowest total energy consumption point a, and the actual total energy consumption is in the rising stage of the total energy consumption curve. At this time, if the window area is increasing, the actual total energy consumption will increase, and the actual total energy consumption is more than 105% of the lowest total energy consumption, so it is in the excessive window area zone.

3.10. Example of Trade-Off Judgement

This section takes a gymnasium in Beijing as an example to illustrate how the progressive optimization method for weighing daylighting and energy consumption can be applied in practical situations.

The main lighting forms, size, side window to wall ratio, skylight ratio, VT, SHGC, and K of the gymnasium are shown in

Table 4. Parameter for the gymnasium.

Name	Parameter
The main lighting forms	East and west side windows and skylights
Size	75 m × 55 m × 25 m
Side window-to-wall ratio	East and west side: 70%
Skylight ratio	5%
Glass material	VT = 0.639
	SHGC = 0.81
	K = 1.50 $\mathrm{W}/({\mathrm{m}}^2\cdot \mathrm{K})$

.

(1) Determine the window design scheme.

By substituting the VT of the glass into Equations (9) and (10), the daylighting factor (c) formed by the skylight and side windows per unit area indoors can be calculated.

Skylight:

$$c_t=1.92667\times 0.639+0.09556=1.33\%;$$

East and west side windows:

$$c_e=c_w=0.365\times 0.639+0.14417=0.37\%.$$

By substituting SHGC and K of the glass into Equations (11)–(13), the variation of air conditioning energy consumption per unit area $\Delta e_{1,W}$ for orientation can be calculated.

Skylight:

$$\Delta e_{1,W}(t)=1.4264\times 0.81+0.17624\times 1.5+0.11571=1.54 \mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2;$$

East side window:

$$\Delta e_{1,W}(e)=0.21212\times 0.81+0.04588\times 1.5+0.01045=0.25 \mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2;$$

West side window:

$$\Delta e_{1,W}(w)=0.31572\times 0.81+0.05037\times 1.5+0.03011=0.36 \mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2.$$

(2) Calculate equivalent skylight characteristics.

By substituting the daylighting factor (c) and $\Delta e_{1,W}$ into Equations (16) and (17), the total energy consumption variation $\Delta e_{1,W}{}^{\prime }$ and the daylighting factor $c^{\prime }$ caused by the equivalent skylight per unit area can be calculated.

$$\Delta e_{1,W}{}^{\prime }=\frac{0.05\times 75\times 55\times 1.54+0.7\times 55\times 25\times (0.25+0.36)}{0.05\times 75\times 55+2\times 0.7\times 55\times 25}=0.42 \mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2;$$

$$c^{\prime }=\frac{0.05\times 75\times 55\times 1.33+2\times 0.7\times 55\times 25\times 0.37}{0.05\times 75\times 55+2\times 0.7\times 55\times 25}=0.36\%;$$

$$\frac{A^{\prime }}{A_0}=\frac{0.05\times 75\times 55+2\times 0.7\times 55\times 25}{0.01\times 75\times 55}=52.$$

(3) Calculate the actual total energy consumption

The daylighting factor corresponding to the actual total energy consumption at point A: $C(A)=\frac{A^{\prime }}{A_0}{c}^{\prime }=52\times 0.46=23.92\%$;

$$\Delta E_{1,W}(A)=\frac{A^{\prime }}{A_0}\Delta e_{1,W}{}^{\prime }=52\times 0.42=21.84 \mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2;$$

According to the daylighting factor C(A) of the actual total energy consumption, $E_{0,L}(A)$ can be obtained by referring to Figure 20: $E_{0,L}(A)=211.75$ $\mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2$;

The actual total energy consumption at point A:

$$E^{\prime }(A)=\Delta E_{1,W}(A)+E_{0,L}(A)=21.84+211.75=233.59 \mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2.$$

(4) Calculate the lowest total energy consumption.

Calculate $\Delta E_{0,L}$ by substituting $\Delta e_{1,W}{}^{\prime }$ and $c^{\prime }$ into $0.02\times \frac{\Delta e_{1,W}{}^{\prime }}{c^{\prime }}+\Delta E_{0,L}=0$: $\Delta E_{0,L}=-1.83$ $\mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2$;

According to $\Delta E_{0,L}$, the daylighting factor C(a) corresponding to point a with the lowest total energy consumption can be obtained from Table 3: $C(a)=10\%$;

$$\Delta E_{1,W}(a)=\frac{\Delta e_{1,W}{}^{\prime }}{c^{\prime }}\times C(a)=\frac{0.42}{0.0046}\times 0.1=9.13 \mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2;$$

According to the daylighting factor C(a) of the lowest total energy consumption, $E_{0,L}(a)$ can be obtained by referring to Figure 20: $E_{0,L}(a)=216.44$ $\mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2$;

The lowest total energy consumption at point a:

$$E^{\prime }(a)=\Delta E_{1,W}(a)+E_{0,L}(a)=9.13+216.44=225.57 \mathrm{k}\mathrm{w}\mathrm{h}/{\mathrm{m}}^2.$$

(5) Determine whether the window design scheme is in a reasonable zone.

From the above results, the trade-off judgment result between daylighting and energy consumption in the gymnasium can be determined. As shown in Figure 24, C(A) > C(a) and $E^{\prime }(A)\le 1.05E^{\prime }(a)$. The actual total energy consumption is in the rising stage of the total energy consumption curve, and the actual total energy consumption is about 105% of the lowest total energy consumption. Therefore, the window design scheme of the gymnasium is in the reasonable window area zone.

4. Conclusions

This paper simulates the building energy consumption for the high and large space of the main stadium of the University of Science and Technology Beijing Gymnasium and makes a trade-off judgment on its daylighting and energy consumption. The main conclusions are as follows:

(1) The relationship between the daylight factor and VT formed by skylights and side window openings per unit area is obtained by simulation.

(2) The University of Science and Technology Beijing Gymnasium is a high and large space. If changing the SHGC, K, and VT under lighting control, the total energy consumption and air conditioning energy consumption in summer will first decrease and then increase with the rise of the skylight ratio. However, the variation in air conditioning energy consumption in winter is opposite to them. Air conditioning energy consumption in the summer is the dominant factor affecting the variation in total energy consumption.

(3) Under no lighting control in the room, by changing the areas of the side window and skylight, the total energy consumption of the gymnasium increases with the rise of the WWR, and the total energy consumption is in a linear relationship with the WWR, in which changing the area of the east and south side windows has little impact on the total energy consumption. If the daylighting effect is strengthened, the east and south side windows should be added.

(4) The SHGC and K of the exterior window have a significant impact on the total energy consumption. Through energy consumption simulation with different types of exterior windows, it is concluded that only the SHGC of the south-facing window is negatively correlated with the $\Delta e_{1,W}$, and its regression coefficient is −0.08436, while the other oriented windows are positively correlated. Compared with other orientations, the SHGC and K of the skylight have the most significant influence on the $\Delta e_{1,W}$.

(5) $E_{0,L}$ decreases rapidly as the daylight factor increases, between approximately 2–20%. After that, its rate of decrease gradually slows down, eventually reaching a constant value.

(6) The total energy consumption first decreases and then increases with the rise of the window area, and there is a lowest point, so the right side of the lowest total energy consumption is less than or equal to 105% of the lowest total energy consumption as a reasonable window area zone. When C(A) < C(a), it is in the insufficient window area zone; when C(A) ≥ C(a) and $E^{\prime }(A)\le 1.05E^{\prime }(a)$, it is in the reasonable window area zone; when C(A) ≥ C(a) and $E^{\prime }(A)$ > $1.05E^{\prime }(a)$, it is in the excessive window area zone. And a progressive optimization calculation process for weighing daylighting and energy consumption in university gymnasiums in Beijing is proposed. Use this calculation process to determine whether the specific window design scheme of university gymnasiums in Beijing is reasonable. If it is unreasonable, adjust the window design scheme and reevaluate until it is reasonable.

5. Prospect

The shortcomings of this paper and the suggestions for further research:

(1) Restricted by on-site requirements and objective measurement conditions, the lack of measurement data for verification is one of the shortcomings of this paper. In further research, a comparative analysis can be carried out based on the measurement data of other similar buildings. Although this method cannot completely replace the on-site measurement data, it can still provide a certain degree of support for the reliability of the simulation results.

(2) Although this paper explores the effect of different window design positions (WWR, SHGC, K, and VT) of the transparent envelope on the total energy consumption, the final foothold is the influence of the window area on the total energy consumption. In the next step of research, specific optimizations will be made for other thermal performance parameters.

(3) This paper only considers gymnasiums with enclosed windows, which use mechanical ventilation systems to control indoor airflow. In the following research, the use of windows for natural ventilation will be considered, and the ventilation aspects will be explained in detail.

(4) For modeling convenience, only one window is set up in this paper according to the WWR. In the following study, the impact of the number, shape, and distribution of windows on energy consumption will be considered.