Multi-Objective Optimization Design of Geometric Parameters of Atrium in nZEB Based on Energy Consumption, Carbon Emission and Cost

Abstract

Through the detailed design of the passive design of the geometric parameters of the atrium, it is beneficial to achieve the design goal of a nearly zero-energy building. In the architectural design stage, the geometric design parameters of the atrium are verified and evaluated with different objectives such as energy consumption, carbon emissions, and costs, and then the most appropriate solution according to different design requirements is selected, which can reduce energy consumption and save costs. This paper proposes a method to optimize the energy consumption of a building’s atrium. Taking Jinan City as an example, this paper conducted 1260 energy consumption simulations for buildings with different geometric parameters of the atrium, based on the investigation of the geometric scale and energy consumption of the multi-story office buildings with near-zero energy consumption in cold areas with atriums. The degree of influence of each parameter on building energy consumption was determined. Finally, the parameter selection combination with the best effect is proposed. The results show that the selected four parameters are significantly related to energy consumption, and a new atrium design parameter was found through the combined analysis of the parameters: the body shape coefficient of the atrium. It was found that the importance of atrium design parameters on building energy consumption is as follows: the body shape coefficient of the atrium, the height-span ratio of the atrium (DSR), the atrium building volume ratio (VR), the skylight area ratio (SR), the atrium width-to-depth ratio (FDR). Seven groups of optimal design parameters were obtained by analyzing the design decisions with energy consumption as the target. Taking carbon emission and cost as the targets, three groups of optimal design parameters were obtained according to the Pareto frontier solution set, such as DSR = 2, VR = 0.13, SR = 0.1, and FDR = 2.5. It provides some references and ideas for the optimization of the energy consumption of the atrium of multi-story nearly zero-energy office buildings in the cold regions of China.

1. Introduction

As the central space of public buildings, the atrium has the characteristics of publicity, artistry [1], and climate adaptability [2,3], which are of great relevance to people’s daily lives. Through the design of large-scale spaces such as atria and transparent enclosure structures, the problem of large and deep lighting is solved [4]; the entry of sunlight is also one of the natural factors that prevent diseases and increase red blood cells and haemoglobin, which is an indispensable factor for improving human health. Exposure to natural light plays a beneficial role in the growth and development of the body [5]. In cold areas, the climate is cold in winter and hot in summer, and ample indoor public activity space is more comfortable for users than outdoor public space. However, the climatic conditions constrain the structural parameters of buildings, which limits the expression of the building’s external form. Therefore, the artistic and spatial nature of the building is expressed through the interior design of the building space, and the sociality experienced by users in the public area is enhanced [6]. Compared with the temperature in other rooms in buildings, the atrium temperature can be too high in summer and too low in winter, so it is necessary to adopt active cooling and heating methods to ensure the comfort of the physical environment [7,8].

The research on nearly zero-energy buildings mainly focuses on selecting envelope structure parameters and active and passive architectural design methods. The European Union’s EPBD building energy standard aims to achieve zero energy consumption for all new buildings by 2020 [9]. Danish regulations stipulate that residential buildings’ annual heating and cooling capacity after 2020 should be less than 20 kW·h/(m²·a). The United States requires the technical and economic realization and matching of zero-energy buildings in 2020 [9]. In the design of building passive technology, strategies for building envelopes are often divided into systems such as opaque and transparent envelope insulation, roof insulation, coating, and other methods [10], and climate factors are considered in the design of thermal insulation, the balance of annual cooling and heating demand and costs [11]. Gupta et al. combined an optimization scheme for thermal insulation materials with excellent thermal performance in temperate climate regions based on thermal environmental comfort, energy consumption, and economic needs [12]. In some scenarios, passive design techniques are combined with natural ventilation design techniques [13,14]. Trombe walls are also a passive structure using solar energy technology. The design of glass surfaces and wall vents is conducive to cooling in summer and heat gain in winter [15]. Shading technology [16], solar photovoltaics [17], and other technologies are combined for nearly zero-energy building design.

The refined performance-oriented design of public building atria is conducive to reducing energy consumption and increasing comfort in the atrium. Due to the windows on the top of the atrium, the interior receives relatively more solar radiation than in other rooms, and many studies have investigated strategies for maintaining thermal comfort inside the atrium. Li Chuacheng et al. studied the patterns of temperature distribution in a tall space atrium through simulation and measurement [18]. Abd. Halid Abdullah et al. studied the improvement and optimization of the atrium thermal environment through two low-cost forms of shading and evaporative spray [19]. Tofigh Tabesh et al. studied the comprehensive strategies for energy-saving efficiency in the extensive use of the atrium and courtyard [20]. Leila Moosavi studied the efficiency of different cooling and passive strategies such as the stack effect, convective ventilation, and water walls [21]. Rojas et al. studied the impact of glazing materials, ventilation strategies, and forms of solar protection on the thermal demand of atria in a Mediterranean climate [22]. Steven et al. studied the thermal environment of the atrium under natural ventilation through experiments and simulations [23].

The design of the building atrium is often carried out in the program stage, and the atrium space design in the program stage significantly impacts building energy efficiency [24]. The design of the atrium form has different effects on the parameters of building cooling and heating [25]. Moosavi et al. studied the effects of atrium geometry, roof openings, and roof characteristics on building comfort and energy consumption [26]. Et studied the plan form of the atrium and pointed out that the building energy consumption for an atrium with a narrow and long shape is higher than that of an atrium with a square plan form [27]. Nasrollahi et al. studied the impact of the ratio of the atrium to the total building area on the energy-saving performance of buildings in a semiarid climate. Research shows that a plot ratio of 1:4 is the most effective choice for energy savings and light and thermal comfort in a semiarid environment [28]. Jaberansari et al. compared energy consumption for high-rise buildings with or without atria and concluded that semiarid areas facing west during office hours could reduce annual energy [29]. Wang Lan et al. studied the impact of skylight size on hotel building energy consumption under different atrium width-to-depth ratios in cold regions [30].

As described above, there are various studies on the effect of a single design parameter of the atrium and ventilation strategy on building energy consumption and comfort, whereas there is less research on the connection between the nearly zero-energy office building atrium geometry and energy consumption, cost, and carbon emissions. Existing international studies often focus on hot and mild regions, whereas research on cold regions is relatively scarce. Many studies focus on net-zero energy building practices and the optimization of the efficiency of renewable energy equipment and building envelope structures. There are few studies on the impact of architectural space design forms on energy consumption. This study focuses on the parameters of the atrium space design of near-zero energy buildings and conducts research from a new perspective of multiple objectives such as cost, carbon emissions, and design requirements. It can be considered in the decision-making at the initial stage of building design, which greatly affects whether it has renewable energy or the amount of renewable energy that needs to be used, thus reducing the loss of limited fossil fuel reserves or renewable energy. Therefore, it plays an important role in the sustainability of buildings and the environment.

This study aimed to screen and extract atrium design parameters from relevant quantitative studies by combining qualitative and quantitative methods and using correlation analysis to determine the influencing factors. Then, we explored the different degrees of influence of the changes in atrium design parameters on energy consumption for the optimized building and combined carbon emissions and costs to carry out multiobjective optimization. Finally, based on the simulation and optimization results, a parameter strategy for public building atrium design is proposed, which provides a reference for designing energy-saving atria in the architectural scheme stage (Figure 1).

As the location for this investigation, the city of Jinan (36.40° N, 117.00° E) in the cold area of China was chosen [31], and the basic nearly zero-energy office building model with atrium form was established as the research object. This study begins with the influence of parameters related to the structure of the atrium on energy consumption for cooling and heating near zero-energy office buildings in cold climates. The first section elaborates on the research object and methodology. The second section introduces the research methodology and establishes an energy consumption model. The third section consists of a three-step analysis of the simulation procedure and outcomes: setting public building simulation model settings and energy consumption simulation; statistical analysis of the link between atrium geometric parameters; and energy consumption multiple linear regression. The fourth section makes recommendations on design parameters for zero-energy office buildings in cold climates based on multiobjective optimization of the geometric parameters of the building atrium concerning cost and carbon emissions. This research takes the nearly zero-energy multistorey office building as the research model. The research model at this scale has a wide range of application scenarios for a considerable range of multistorey buildings, not limited to office buildings, such as industrial park office buildings, scientific research institutes, comprehensive university office buildings, universities for elderly individuals, and other public buildings.

2. Research Methodology and Model Building

2.1. Study Area

The typical characteristics of the climate of the cold areas in China are cold and dry in winter and hot in summer, with a large annual temperature range. In winter, insulation is needed to protect against the cold, while in summer insulation measures are needed to protect against the heat. For the atrium, solar radiation through skylights needs to be increased in winter, and sunshade measures are required in summer. This study takes Jinan, China, as the study area and investigates office buildings with atria in Jinan, taking the basic model of nearly zero-energy office buildings as the research object and studying the multiobjective optimization of its energy consumption, cost, and carbon emissions.

The architectural geometrical patterns of public buildings with atria differ from those of other types of buildings in the city and have a distinct impact on energy consumption. The compositional relationship between the atrium and the building room can be considered the basic unit form. The solar radiation entering the atrium from the skylight and the thermal connection between the room and the atrium are affected by the atrium’s shape. Therefore, when studying the energy consumption level of a building, the basic model is chosen as the simulation calculation research object. Combining the climatic characteristics of cold regions, striving for solar radiation and heat storage in winter, reducing solar radiation in summer, and increasing heat storage in winter and heat dissipation in summer have become the fundamental goals of the energy-saving design of atrium geometric parameters [32]. On this basis, the skylight area ratio, the height-span ratio of the atrium, the atrium building volume ratio, the atrium width-to-depth ratio, and the changing trend and energy consumption for the atrium were studied.

2.2. Research Methods

The methods used in building energy consumption research have matured and can be divided into simulation research through computer-aided tools, statistical research on data, and optimization decision research combining statistics and simulation. The research method logic of this study is shown in Figure 2. For the simulation study, this paper used Grasshopper 2021 for parametric modeling. Using the Hoybee + Ladybug 1.5.0 plug-in with EnergyPlus as the energy consumption calculation core, the relevant building energy consumption data were produced by linking the urban climate data as the default condition with the simulation program. This paper simulates the impact of different atrium geometric parameters on building energy consumption, employs statistical analysis to obtain the overall relative impact on energy consumption, and compares it with previous research data and standard specifications to validate the authenticity and accuracy of the simulated data and prepare for the next phase of correlation and regression analysis.

Commonly used multiobjective calculation techniques for optimization and decision-making software include annealing algorithms, genetic algorithms, and evolutionary algorithms such as Design Explorer, Octopus, Wallace Micro-GA, MATLAB, Galapagos, Multiopt, and GenOpt [33,34,35,36,37,38]. These applications can iteratively calculate and screen according to the algorithm, reduce the number of experiments, and finally form the Pareto frontier solution set. This research adopts the decision tool Design Explorer and the parametric multiobjective optimization tool Octopus. Using the simulation data and regression analysis obtained by Design Explorer according to the above steps, parameters were allocated and selected according to different atrium geometric design goals and energy consumption and carbon emissions optimization goals.

Furthermore, decision-making suggestions during design were made accordingly. Using Octopus, the atrium geometric design parameters after regression analysis were used as independent variables, and the establishment and life-cycle cost, carbon emission, and the area of the atrium and room were optimized as multiple objectives. Finally, the optimal solution set was calculated to achieve as little carbon emissions as possible at a lower cost when the atrium and room area were larger, and design decision-making suggestions were made according to the solution set.

2.3. Typical Model Building

2.3.1. Typical Model and Measured Model Object

In this study, several buildings were chosen from the office buildings in Jinan to investigate the plan of the atrium to obtain the range of energy-efficiency design parameters related to nearly zero-energy office buildings and provide data for establishing basic models and support for energy consumption simulation analysis. According to literature analysis, field visits, and mapping of the Jinan office buildings, the form of an atrium in the middle of the building was chosen. The choice of the closed inner atrium form as the research object was based on climatic considerations in cold regions; this form connects the rooms in all four directions, in contrast to other forms of atrium space. The solar radiation entering the atrium only enters through the skylight and not through the side windows of the facade, allowing a more in-depth analysis of the impact of the geometric design of the atrium on the building’s energy consumption.

The case selected for verification in this study was the prefabricated passive design of Shandong Jianzhu University, which is the first steel-structured prefabricated ultra-low energy office building in China. The building is six stories high and contains an atrium. According to the energy consumption monitoring of the background monitoring platform, we could get the cooling and heating energy consumption data month by year (

Table 1. Energy consumption measurement results (kWh/m²·a).

Month	Cooling and Heating Energy Consumption	Month	Cooling and Heating Energy Consumption
1	3.55	7	4.41
2	0.79	8	3.02
3	0.13	9	1.55
4	0.08	10	0.12
5	1.30	11	1.90
6	4.12	12	2.13

).

2.3.2. Verification of Energy Consumption Model

The verification of the energy consumption model uses the measured energy consumption data to verify the energy consumption simulation results of typical models. Taking a typical meteorological year as an example, the Ladybug and Honeybee plug-ins are used to simulate the energy consumption of cooling and heating hourly for typical model pairs. By analyzing the correlation between the energy consumption simulation data and the measured data, it is verified whether there is a significant correlation so that the reliability of the typical model can be verified.

Due to the climate data of a typical meteorological year used by the simulation software, the simulated operating conditions are different from the current operating conditions. Secondly, the measured building energy consumption is also related to the activities of building users. Therefore, based on the above factors, there is a certain error between the building energy consumption and the software simulation value. The Pearson correlation coefficient between the two was calculated by Origin software to be 0.867, and Sig. = 0.01 was less than 0.05 (Figure 3). It showed that the typical energy consumption model in this study is significantly correlated with the actual building operation energy consumption of ultra-low energy office buildings in cold regions in my country, and subsequent energy consumption simulations could be carried out.

3. Energy Consumption Simulation and Results

3.1. Simulation Parameter Setting and Simulation Process

3.1.1. Setting of Model Parameters

To ease variable control in the energy consumption simulation process and limit the influence of other variables on the results, the simulation object was selected as a standard and simple office building model. The establishment of the model was based on the size of the column grid and the design scale of a conventional multistorey near-zero energy office building. The model selection of the resume was six floors, and the floor height was set to 3.6 m. According to the actual investigation, the depth of the rooms on the north and south sides of the atrium was mostly a column spacing, so the column spacing was set to 7.5 m in the model, and the parameter connection of L₀ = L + 2 ∗ 7.5 was set. The width and depth of the other buildings and the atrium could be calculated and linked according to the selected parameters, such as the atrium width-to-depth ratio, the atrium building volume ratio, the atrium height-span ratio, and the skylight atrium area ratio.

Conventional energy consumption simulation modeling needed to be completed first. The second energy consumption simulation of the change in body shape parameters required the model to be modified. The relevant research under each parameter change needed to ensure that the relevant parameters did not change. The modeling work in this model used the Grasshopper battery pack for parametric modeling, which ensured that only one parameter (such as the height-span ratio) was changed when the relevant parameters (such as the atrium building volume ratio) remained unchanged (Table 1).

The meteorological data used to simulate energy use were obtained from the Energy Plus website. Principal setup parameters, including building parameters, HVAC parameters, and personnel activities parameters, were taken from the “Technical Standards for Nearly Zero Energy Buildings” (GB/T 51350-2019), “General Specification for Building Energy Conservation and Renewable Energy Utilization” (GB 55015-2021), “Design Standard for Energy Efficiency of Public Buildings” (GB 50189-2015), and “Henan Province Public Building Energy Efficiency Design Standard” (DBJ41/T 076-2016) (

Table 2. Model parameter settings.

Atrium Parameters	Value	Atrium Parameters	Value
Length	78 m	Visible light transmission ratio of exterior window	0.71
Width	22.6 m	Heat transfer coefficient of external windows	0.8
Height	3.6 m (6 floors)	Heat transfer coefficient of external walls	0.14
Length of the atrium	31.2 m	Heat transfer coefficient of external roofs	0.16
Width of the atrium	7.2 m	Window to wall ratio	0.2

). Norms and standards were set accordingly, including heating and cooling schedules in winter and summer (

Table 3. Building energy simulation parameter settings in this study.

Building Parameters		Activity and Equipment Parameters
Heat transfer coefficient of roofs	0.176 W/(m2·k)	Number of occupants per unit area	Office	0.1 persons/m2
			Atrium	0.01 persons/m2
Heat transfer coefficient of external walls	0.546 W/(m2·k)	Calculated number of air changes	1.098 H−1
Heat transfer coefficient of external windows	2.14 W/(m2·k)	Electricity consumption	6.88 W/(m2·k)
Window to wall ratio	0.4	HVAC parameters	Winter heating temperature	18 °C
Building structure	Reinforced concrete construction		Summer cooling temperature	26 °C

). To eliminate the impact of heating and cooling techniques on simulation results, the study assumed that the HVAC cooling and heating coefficient COP in the building was 4.5, and centralized cooling and heating methods were adopted.

3.1.2. Simulation Process

After confirming the input parameters for the energy consumption simulation, Ladybug + Honeybee (Mostapha, PA, USA) was used to set the parameters of the energy consumption model established in the previous step. The four design parameters identified in the grouping were used for simulation to obtain different energy consumption metrics. The procedure was repeated until all individual variables had been evaluated, after which the correlating data were collected (Figure 4).

The main components of the L+H battery set built inside Grasshopper are the office atrium study model and the atrium design variables battery set related to energy consumption. The red frame shows a portion of the set atrium design parameters, which can be changed at any time by using the slider for the study. The Boolean toggle is then set to True to start the energy simulation for the building simulation with the current parameter settings. After a period of calculation, the simulation engine obtains the energy consumption data of heating and cooling under the parameter setting, records the data, and automatically starts the simulation after modifying the design parameters until all the set design parameters are studied.

3.2. Simulation of Atrium Design Parameters

3.2.1. The Simulation Results of the Skylight Area Ratio

The area of the skylight (SR) determines the amount of solar radiation entering the atrium through the window and the energy consumption for air conditioning in summer and winter [39]. The size of the window affects the light environment of the room. Although light infiltration is desirable, it is more important to reduce energy consumption in cold regions [40]. In summer, the skylight in the atrium transmits more solar radiation; in winter, the skylight allows too much solar radiation during the day. However, heat loss may occur through the skylight due to its high heat transfer coefficient [41]. Therefore, the impact of various skylight area ratios on heating and cooling energy consumption was studied.

To ensure the influence of different window area ratios on the heating and cooling energy consumption, when establishing the energy consumption model, we had to first ensure that the other three factors of each group remained unchanged. The simulation was the same in the following three considerations. According to the familiar skylight area ratio of the office building atrium, the proportional connection of 7 scales from 0.2 to 0.8 was established.

Table 4. The simulation results of skylight ratios (kWh/m²·a).

Type	DSR = 1.7				DSR = 1.8
SR	Heating	Cooling	Total	CO2	Heating	Cooling	Total	CO2
0.2	6.650	12.940	19.590	11.681	6.887	13.127	20.013	11.934
0.3	6.678	13.012	19.690	11.741	6.916	13.198	20.114	11.994
0.4	6.707	13.082	19.789	11.800	6.945	13.268	20.214	12.053
0.5	6.738	13.151	19.889	11.860	6.975	13.338	20.313	12.113
0.6	6.768	13.218	19.986	11.918	7.005	13.405	20.410	12.170
0.7	6.798	13.283	20.081	11.974	7.035	13.470	20.505	12.227
0.8	6.829	13.346	20.175	12.030	7.066	13.533	20.599	12.283

demonstrates that the heating and cooling energy consumption per unit building area increases continuously with the skylight area ratio. When the skylight area ratio is changed from 20% to 80%, the cooling energy consumption increases by 0.4, and the heating energy consumption increases by 0.2. It can be seen that, in this case, the skylight area ratio has a greater influence on the cooling energy consumption than the heating energy consumption. The energy consumption per unit building area increases when the top-opening ratio increases, mainly because more solar radiation and heat enter the atrium through the more oversized skylight. Under suitable conditions, the skylight area ratio at the top of the building atrium should be reduced when the building is constructed.

3.2.2. Simulation Results of the Atrium Height-Span Ratios

The height-to-span ratio (DSR) of the atrium affects the degree of solar radiation entering the atrium [42]. In addition, the DSR of the atrium affects the area of the interior that can be irradiated by solar radiation, which is related to the local solar elevation angle. On the other hand, the DSR controls the plane form and space form of the atrium in the longitudinal section. People in a space enclosed by buildings will have different psychological feelings due to their different positions, distances, and angles from the building space [43]. If the spatial scale is too small, it will make people feel depressed; if the scale is too large, it will make people feel empty and unfriendly. According to the column network structure of the office building, the height-span ratio of the atrium is set to 1.5, 1.6, 1.7, 1.8, and 1.9.

Based on the simulation data for energy consumption shown in

Table 5. The simulation results of height-span ratios (kWh/m²·a).

Type	DSR = 1.7				DSR = 1.8
DSR	Heating	Cooling	Total	CO2	Heating	Cooling	Total	CO2
1.5	6.232	12.700	18.932	11.289	6.508	6.262	12.770	19.031
1.6	6.470	12.892	19.362	11.546	6.462	6.500	12.962	19.462
1.7	6.707	13.082	19.789	11.800	6.413	6.738	13.151	19.889
1.8	6.945	13.268	20.214	12.053	6.363	6.975	13.338	20.313
1.9	7.182	13.457	20.639	12.307	6.313	7.213	13.526	20.739
2	7.421	13.646	21.067	12.562	6.264	7.451	13.715	21.166

above, the heating and cooling energy consumption per unit of building area presents an upward trend with increasing building distance. When the height-span ratio of the building atrium is increased from 1.5 to 2, the cooling energy consumption increases by 0.946, and the heating energy consumption increases by 1.189. It can be seen that, in this case, the effect of the skylight area ratio on the cooling energy consumption is not much different from the heating effect, which has a more significant impact on heating. Under suitable conditions, the height-span ratio of the building atrium should be reduced when the building is created.

3.2.3. The Simulation Results of the Atrium Building Volume Ratios

The atrium building volume ratio (VR) affects the degree to which the solar radiation heat enters the atrium space and diffuses in the surrounding direction, as well as the thermal influence between the surrounding rooms and the atrium [44] (Figure 5). This index parameter represents the connection between the tall transition space and the spatial layout of the building [45]. The longwave radiation heat transfer between the inner surfaces of the atrium is an essential factor affecting the heat exchange process in the atrium space. It is affected by material composition, solar radiation angle, time, and other factors. The longwave radiation heat transfer also accounts for a relatively large proportion of the total heat [45]. According to the column network structure of the office building, values of 0.12, 0.13, 0.14, 0.15, and 0.16 were set for the atrium building volume ratios.

It can be seen from the simulation results that with the increase in the volume ratio of the building atrium, the total energy consumption per building unit and heating energy consumption both show an increasing trend (

Table 6. The simulation results of the atrium building volume ratios (kWh/m²·a).

Type	DSR = 1.7				DSR = 1.8
VR	Heating	Cooling	Total	CO2	Heating	Cooling	Total	CO2
0.12	6.205	12.649	18.854	11.243	6.476	6.231	12.707	18.938
0.13	6.454	12.863	19.317	11.519	6.445	6.482	12.927	19.409
0.14	6.707	13.082	19.789	11.800	6.413	6.738	13.151	19.889
0.15	6.966	13.304	20.270	12.087	6.38	6.999	13.379	20.378
0.16	7.229	13.531	20.760	12.379	6.346	7.265	13.611	20.876

). When the height-span ratio of the building atrium changes from 0.12 to 0.16, the cooling energy consumption increases by 0.882, and the heating energy consumption increases by 1.024. In this case, the effect of the skylight area ratio on the cooling energy consumption is different from the heating effect. The effect on system energy consumption is significant. This shows that when the conditions are suitable, the volume ratio of the building atrium should be reduced when the building is designed. Choosing a building scheme with a smaller atrium is more conducive to energy conservation and emission reduction.

3.2.4. The Simulation Results of the Atrium Width-to-Depth Ratios

The atrium width-to-depth ratio (FDR) of the atrium also affects energy consumption for the building [46]. From the perspective of architectural design, different width and depth ratios determine whether the plan of the building atrium is narrow or square [47] (Figure 6). The width-to-depth ratio of the atrium controls the plane form and space form of the atrium in the cross-section. According to the column network structure of the office building, values of 2.5, 2.6, 2.7, 2.8, 2.9, and 3 are set for the atrium width-to-depth ratio.

According to the above energy consumption simulation data (Figure 7) (

Table 7. The simulation results of atrium width-to-depth ratios. (a) Description of heating consumption data. (b) Description of cooling consumption data. (kWh/m²·a).

Type	SR = 0.4				SR = 0.5
FDR	Heating	Cooling	Total	CO2	Heating	Cooling	Total	CO2
2.5	6.808	13.181	19.989	11.919	6.412	6.838	13.250	20.088
2.6	6.756	13.129	19.885	11.858	6.413	6.786	13.199	19.985
2.7	6.707	13.082	19.789	11.800	6.413	6.738	13.151	19.889
2.8	6.663	13.037	19.700	11.747	6.414	6.693	13.107	19.800
2.9	6.621	12.996	19.617	11.698	6.415	6.651	13.066	19.717
3	6.583	12.957	19.540	11.652	6.415	6.612	13.027	19.639

), heating and cooling energy consumption per unit building area decreases as the atrium width-to-depth ratio increases. When the aspect ratio of the building atrium increases from 2.5 to 3, the cooling energy consumption increases by 0.225, and the heating energy consumption increases by 0.224. It can be seen that, in this case, the effect of the skylight area ratio on the cooling energy consumption is the same as the heating energy consumption. This finding indicates that, in the case of suitable conditions in cold regions, the atrium width-to-depth ratio of the building should be increased as much as possible when creating a building, and a narrower and longer atrium plane shape should be selected.

3.3. Sensitivity Analysis of Geometric Parameters of the Atrium

In this part, SPSS 26 [31] was used to conduct 5 ∗ 6 ∗ 6 ∗ 7 simulation results for four sets of layout parameters a total of 1620 times (

Table 8. Different parameter analogue values.

Atrium Design Parameter	Value
Face Depth Ratio	2.5; 2.6; 2.7; 2.8; 2.9; 3.0
Volume Ratio	0.12; 0.13; 0.14; 0.15; 0.16
Depth Span Ratio	1.5; 1.6; 1.7; 1.8; 1.9; 2.0
Skylight Ratio	0.2; 0.3; 0.4; 0.5; 0.6; 0.7; 0.8

), and a cubic multiple linear regression analysis for cooling and heating energy consumption was carried out. The building’s heating and cooling energy consumption is the study’s dependent variable, while the geometric characteristics of the atrium are the independent variables. Based on statistical criteria determined using the equation for multiple linear regression [48], it can be observed that the above four parameters have a linear connection.

From Section 3.2 of this paper, the building heating and cooling energy consumption have a linear connection with each parameter, which is related to the skylight and is positively correlated with the atrium area ratio, atrium height-span ratio, and atrium building volume ratio, and negatively correlated with the atrium width-to-depth ratio. The four significant parameters above were chosen as independent variables for the multiple linear regression analysis of energy consumption Y according to the results of the representativeness and correlation analyses of independent variables: the ratio of skylight to atrium area, atrium height-span ratio, atrium building volume ratio, and atrium width-to-depth ratio. The model established for public buildings with atrium parameters and the cooling energy consumption index was obtained as follows:

Y = a₁X₁ + a₂X₂ + a₃X₃ + a₄X₄ + b

where X₁ is the atrium width-to-depth ratio; X₂ is the atrium building volume ratio; X₃ is the atrium height span ratio; X₄ is the skylight atrium area ratio; Y is the total energy consumption for cooling and heating; Y₁ is the heating energy consumption; Y₂ is the cooling energy consumption.

3.3.1. Multiple Regression Equation of Each Parameter on Heating Energy Consumption

Multiple linear regression computations provide the following linear regression equation for heating:

Y₁ = 0.128 − 0.47X₁ + 26.538X₂ + 2.367X₃ + 0.299X₄

R² = 0.994 (

Table 9. Multiple linear regression model fit statistics of Model 1.

Model	R	R2	Adjusted R2	Durbin-Watson	Sig.
Heating	0.997	0.994	0.994	0.421	0.00

), showing that these four factors can determine 99.4% of the building’s cooling and heating data. The significance of the four variables is 0.00, which demonstrates that all four variables can significantly affect total energy consumption. The VIF = 1, which is less than 5 and indicates no multicollinearity among the four factors (

Table 10. Multiple linear regression equation analysis of Model 1.

Model	Unstandardized Coefficients		Standardized Coefficients Beta	t	Sig.	Collinearity Statistics
						Tolerance	VIF
	B	Std. Error 4
(Constant)	0.128	0.026		4.963	0.000	1.000	1.000
FDR	−0.47	0.007	−0.143	−67.431	0.000	1.000	1.000
VR	26.538	0.084	0.667	315.192	0.000	1.000	1.000
DSR	2.367	0.007	0.719	339.564	0.000	1.000	1.000
SR	0.299	0.006	0.106	50.231	0.000	1.000	1.000

). The residuals of the model have a normal distribution, suggesting that the error of this equation falls within a reasonable range (Figure 8).

3.3.2. Multiple Regression Equation of Each Parameter on Cooling Energy Consumption

Multiple linear regression computations provide the following linear regression equation for cooling:

Y₂ = 7.651 − 0.465X₁ + 23.058X₂ + 1.877X₃ + 0.679X₄

R² = 0.994 (

Table 11. Multiple linear regression model fit statistics of Model 2.

Model	R	R2	Adjusted R2	Durbin-Watson	Sig.
Cooling	0.997	0.994	0.994	0.421	0.00

), showing that these four factors can determine 99.4% of the building cooling and heating data. The significance of the four variables is 0.00, which proves that all four variables can significantly affect total energy consumption. The VIF = 1, which is less than 5 and indicates no multicollinearity among the four factors (

Table 12. Multiple linear regression equation analysis of Model 2.

Model	Unstandardized Coefficients		Standardized Coefficients Beta	t	Sig.	Collinearity Statistics
						Tolerance	VIF
	B	Std. Error 4
(Constant)	7.651	0.022		347.179	0.000	1.000	1.000
FDR	−0.465	0.06	−0.164	−78.048	0.000	1.000	1.000
VR	23.058	0.072	0.673	320.626	0.000	1.000	1.000
DSR	1.877	0.006	0.661	315.174	0.000	1.000	1.000
SR	0.679	0.005	0.280	133.497	0.000	1.000	1.000

). The residuals of the model have a normal distribution, suggesting that the error of this equation falls within a reasonable range (Figure 9).

3.3.3. Multiple Regression Equation of Each Parameter on Cooling and Energy Consumption

Multiple linear regression computations provide the following linear regression equation for cooling:

Y₃ = 7.779 − 0.935X₁ + 4.24X₁ + 49.6X₂ + 0.98X₃

R² = 0.995 (

Table 13. Multiple linear regression model fit statistics of Model 3.

Model	R	R2	Adjusted R2	Durbin-Watson	Sig.
Total	0.997	0.994	0.994	0.448	0.00

), showing that these four factors can determine 99.5% of the building’s cooling and heating data. The significance of the four variables is 0.00, which proves that all four variables can significantly affect total energy consumption. The VIF = 1, which is less than 5 and indicates no multicollinearity among the four factors (

Table 14. Multiple linear regression equation analysis of Model 3.

Model	Unstandardized Coefficients		Standardized Coefficients Beta	t	Sig.	Collinearity Statistics
						Tolerance	VIF
	B	Std. Error 4
(Constant)	7.779	0.47		164.659	0.000	1.000	1.000
FDR	−0.935	0.013	−0.153	−73.233	0.000	1.000	1.000
VR	49.596	0.154	0.673	321.695	0.000	1.000	1.000
DSR	4.244	0.013	0.695	332.462	0.000	1.000	1.000
SR	0.978	0.011	0.188	89.705	0.000	1.000	1.000

). The residuals of the model have a normal distribution, suggesting that the error of this equation falls within a reasonable range (Figure 10).

In summary, the significance values of the X₁ atrium width-to-depth ratio, X₂ atrium building volume ratio, X₃ atrium height-span ratio, and X₄ skylight atrium area ratios are all less than 0.05, which indicates that each regression coefficient of the parameter is statistically significant. Furthermore, the regression coefficient X₃ > X₂ > X₄ > X₁ shows that the atrium height-span ratio has the greatest impact on energy consumption for nearly zero-energy office buildings in cold areas, followed by the atrium building volume ratio, followed by the skylight atrium area, and finally the atrium width-to-depth ratio (Figure 11). VIF < 10 shows the absence of an apparent relationship between parameters. The residuals of the model follow a normal distribution, suggesting that this equation’s error is likewise within an acceptable range These findings demonstrate the validity of the notion that these four atrium geometry factors impact energy usage. The linear regression equation derived from this model is thus statistically significant [49,50].

3.3.4. Multiple Regression Equation and the Shape Coefficient of the Atrium and Heating and Cooling Energy Consumption

For noncubic complex building spaces (Figure 12), the regression analysis of energy consumption may not be carried out in the form of the atrium width-to-depth ratio, height-span ratio, atrium building volume ratio, and skylight area ratio. When simulating the energy consumption calculation, it may not be possible to only establish a cubic model for simple modeling energy consumption analysis. Therefore, based on the thermal process of the building and its surrounding environment and the thermal cycle of the atrium and surrounding rooms, a new design parameter of the atrium, that is, the body shape coefficient of the atrium (AVR), is proposed (Figure 13). The AVR is expressed as the ratio of the contact area between the atrium and the surrounding rooms to the volume of the atrium. The above assumptions are verified by regression with energy consumption. In this typical model, its expression is as follows:

AVR = 2(W + L) H/W·L·H

After adjusting the regression data by replacing the atrium height-span ratio and the atrium width-to-depth ratio with the body shape coefficient of the atrium, the following equations were developed:

Y = 5.156 + 1.566X₁ + 49.585X₂ + 0.978X₃

R² = 0.994, proving that these four factors can determine 99.4% of the building’s cooling and heating data. The significance of the four variables is 0.00, which demonstrates that all three variables can significantly affect the total energy consumption. The VIF = 1, which is less than 5 and indicates no multicollinearity among the four factors.

The significant index Sig. values of the X₁ body shape coefficient of the atrium, X₂ atrium building volume ratio, X₃ skylight area ratio, and constant are all less than 0.05, which indicates that the respective regression coefficients of the body shape coefficient of the atrium and atrium building volume ratio and the skylight area ratio are statistically significant (

Table 15. Multiple linear regression model fit statistics of Model 4.

Model	R	R2	Adjusted R2	Durbin-Watson	Sig.
Total	0.997	0.994	0.994	1.555	0.00

).

Furthermore, the regression coefficient X₁ > X₂ > X₃ (Table 15). This indicates that the impact of the body shape coefficient of the atrium on energy consumption for nearly zero-energy office buildings in cold regions is greater than those of the atrium building volume than the skylight area ratio. The influence of different body shape coefficient ratios on energy consumption is due to the change in the internal space of the atrium, which leads to a shift in the spatial functional attributes of the building, which in turn affects the transformation of the total energy consumption of the building. The residuals of the model follow a normal distribution, suggesting that this equation’s error is likewise within an acceptable range. This finding demonstrates the validity of the notion that these four atrium geometry factors impact energy usage (

Table 16. Multiple linear regression equation analysis of Model 4.

Model	Unstandardized Coefficients		Standardized Coefficients Beta	t	Sig.	Collinearity Statistics
						Tolerance	VIF
	B	Std. Error 4
(Constant)	5.156	0.034		153.639	0.000	1.000	1.000
AVR	1.566	0.005	0.711	317.961	0.000	1.000	1.000
VR	49.595	0.165	0.673	300.700	0.000	1.000	1.000
SR	0.978	0.012	0.188	83.851	0.000	1.000	1.000

, Figure 14). The linear regression equation derived from this model is thus statistically significant [51,52].

4. Proposed Plans for the Design of Public Building Atria in Cold Regions

4.1. Design Decisions Based on Energy Consumption and Carbon Emissions

The calculation results are classified, screened, and analyzed by Design Explorer software based on the correlation analysis and multiple linear regression of the four atrium parameters and building heating and cooling requirements. Architects can use this approach to discuss the selection of parameters in a specific situation according to the actual design project combined with the goals and make calculations and decisions according to different energy consumption goals. The following part is a subconditional discussion based on 1620 optimizations, which can be used as a reference for designing public buildings with atria in cold regions.

4.1.1. Situation 1

The skylight atrium area ratio, atrium height-span ratio, atrium face-to-depth ratio, and skylight ratio are not limited in the design project and are given priority in design. If the minimum energy consumption and carbon emissions are taken as the optimal target values, the maximal atrium aspect ratio and the smallest skylight atrium area ratio, atrium height-span ratio, and atrium width-to-depth ratio can be selected according to the situation, as shown below (Figure 15,

Table 17. Decision Scheme 1.

FDR	VR	DSR	SR	Heating	Cooling	Total	CO2
3	0.12	1.5	0.2	5.653	12.103	17.756	10.588

).

4.1.2. Situation 2

When the maximum skylight area ratio is considered in the scheme design when energy consumption and carbon emission are selected as the optimal target values, the skylight area ratio with the maximal atrium aspect ratio can be selected according to the situation, and the minimum atrium height-span ratio, the aspect ratio of the atrium to the depth, and the selection of the decision parameters can be selected according to the situation, which is shown in Figure 16 (

Table 18. Decision Scheme 2.

FDR	VR	DSR	SR	Heating	Cooling	Total	CO2
3	0.12	1.5	0.8	5.802	12.445	18.247	10.88068

).

4.1.3. Situation 3

When considering the maximal height-span ratio of the building in the design of the scheme, energy consumption and carbon emissions are selected as the optimal target values. According to the situation, we can choose the maximal atrium width-to-depth ratio, height-span ratio, and the smallest atrium building volume ratio and skylight area ratio. The choice of its decision parameters is shown in Figure 17 (

Table 19. Decision Scheme 3.

FDR	VR	DSR	SR	Heating	Cooling	Total	CO2
3	0.16	1.5	0.2	6.497	12.810	19.307	11.51301

).

4.1.4. Situation 4

Considering the large height-span ratio of the building in the scheme design, the suggestions for selecting energy consumption and carbon emissions as the optimal target values are as follows. According to the situation, we can choose the maximal atrium width-to-depth ratio and height-span ratio and the smallest atrium building volume ratio and skylight area ratio. The choice of decision parameters is shown in Figure 18 (

Table 20. Decision Scheme 4.

FDR	VR	DSR	SR	Heating	Cooling	Total	CO2
3	0.12	2	0.2	6.642	12.892	19.534	11.648

).

4.1.5. Situation 5

In this scheme’s design, there is no priority design consideration for the above four parameters, but the artistry and energy savings of the atrium are comprehensively considered. It is possible to select a moderate atrium width-to-depth ratio, atrium building volume ratio, atrium height-span ratio, and skylight area ratio according to the situation and make appropriate selections according to the building properties. Recommendations for the selection of decision parameters are shown in Figure 19 (

Table 21. Decision Scheme 5.

FDR	VR	DSR	SR	Heating	Cooling	Total	CO2
2.7	0.15	1.6	0.5	6.745	13.178	19.922	11.880
2.8	0.15	1.6	0.8	6.798	13.344	20.142	6.798
2.8	0.15	1.5	0.8	6.546	13.144	19.690	11.741

).

4.2. Multiobjective Optimization and Decision-Making Related to Cost, Carbon Emissions and Design Area

The above design analysis and decision-making only consider the influence of atrium parameters on energy consumption. In reality, economic conditions will be considered; therefore, the whole life-cycle cost is considered the optimization goal here. The whole life-cycle cost is the sum of the initial investment of the building and the current value of the operation cost during the whole life cycle. Since this study mainly studies the impact of the geometric parameters of the atrium on the target, only the skylight area ratio under its influence, the average cost per square meter of the nearly zero-energy building, and the cost of energy consumption in the Chinese market are considered in the cost calculation. The relevant calculation parameters are as follows (Table 21 and

Table 22. Parameters related to the calculation of the whole life cycle.

Parameters	Value
Building life	65 years
The service life of the skylight	20 years
Construction cost per square meter	2400 CNY
Converted electricity price	0.5 CNY/kW·h

):

The parametric multiobjective optimization tool Octopus was used in this study (

Table 23. Performance parameters and cost of skylight for different options (U = heat transfer coefficient of the skylight.

Model	U	SHGC	Cost
1	1	0.37	2160
2	1.3	0.31	1800
3	1.8	0.3	1440
4	2.5	0.41	900

). When the independent variables were set as the above four atrium design parameters and the dependent variables were set as cost and carbon emission and atrium area, the multiobjective optimization operation could be accurately performed in detail. The calculation process included the optimization calculation to achieve the minimum cost, the minimum carbon emission, and the maximum construction area of the atrium and the room in the whole life cycle.

Table 24. Octopus optimized parameter settings.

Optimization Parameters	Value
Elitism	0.5
Mut. Probability	0.2
Mutation Rate	0.9
Crossover Rate	0.8
Population Size 20	20
Max Generation	200

shows the setting parameters of the genetic algorithm in this study. After 152 iterations, Octopus ended the computation, and the Pareto boundary solution (

Table 25. Pareto frontier solution. FDR = atrium face depth ratio; VR = atrium volume ratio; DSR = atrium depth span ratio, SR = skylight ratio; U = heat transfer coefficient of the skylight; Cost = life-cycle costing; CO₂ = carbon dioxide emissions; S₁ = atrium area; S₂ = room area.

Group	FDR	VR	DSR	SR	U	Cost	CO2	S1	S2
1	2.5	0.13	2	0.1	2.5	2422.94	11.63	291.6	11,708
2	2.5	0.12	2	0.1	1	2430	11.30	292	12,830
3	2.5	0.16	2	0.1	2.5	2426.04	12.59	291.6	9185

, Figure 20) was obtained. By comparative analysis, four groups can achieve energy savings and emission reduction at a lower cost and meet the requirements of the building area ratio.

5. Research Implications and Conclusions

This study provides a parameter optimization concept based on energy-saving goals for designing nearly zero-energy office buildings in cold regions. Through correlation analysis and multiple linear regression, the correlation and different degrees of influence of the connections between building atrium design parameters and energy consumption were studied.

The parametric design method of energy consumption simulation was used for cold regions, and the design parameters of the atrium of public buildings were optimized with carbon emission, life-cycle cost, and atrium and building area as multiple objectives. Finally, the optimal design parameter combination and optimal solution set of atrium design factors in cold regions were obtained, which addresses the limitations of traditional scheme design based on experience and simulation and provides a more systematic, diverse, and scientific design decision-making method.

Through analysis and regression verification, the body shape coefficient of the atrium is proposed, which is related to the design of the atrium. With the increase in the body shape coefficient of the atrium, energy consumption for nearly zero-energy office buildings in cold regions shows an increasing trend. For a complex atrium space form, the energy consumption model cannot be simplified based on a simple atrium of the same volume, and the factor of the body shape coefficient of the atrium needs to be considered.