Retrofitting Design of Residential Building Rooftops with Attached Solar Photovoltaic Panels and Thermal Collectors: Weighing Carbon Emissions Against Cost Benefits

Abstract

To reduce the carbon emissions of existing residential buildings while pursuing maximum cost benefits, a multi-optimization design method for the existing residential building rooftops, retrofitted by attaching the solar photovoltaic panels and thermal collectors, was proposed in the study. At first, the life cycle carbon emission and cost benefit of the retrofitted buildings were assigned as the optimization objectives, and the models of carbon emission and cost benefit were developed. Furthermore, a typical existing residential community located in the cold zone of China was selected to perform the multi-optimization based on the Grasshopper platform. Meanwhile, the laying area, laying angle, and allocation ratio of the solar photovoltaic panels and thermal collectors were selected as the design parameters. And then the best retrofitting solution suitable for the existing residential buildings was proposed. The results show that the weightings of the carbon emission of retrofitting life cycle are 42.68%, and that for the cost benefit is 57.32%. Significantly, there is a 31% reduction in carbon emissions compared to the building before retrofitting, and a 24.7% reduction in cost benefit.

1. Introduction

At this stage, the carbon emissions of the whole life cycle of buildings in China account for more than 40% of the total national carbon emissions, and the operational carbon emissions of buildings account for more than 50% of the total building carbon emissions [1]. Therefore, it is of great significance to reduce the carbon emissions of building operations. For photovoltaic panels, reduced cost, improved conversion efficiencies, and a high degree of integration with the building skin have laid the groundwork for a presence in the field of building retrofitting. PV systems in buildings are mainly divided into two types of applications. One type is the building-integrated photovoltaic (BIPV) system, which combines with building materials and directly forms part of the building, such as PV walls, PV rooftops, PV curtain wall, PV exterior window, and PV shading approaches [2]. The other is a building-attached photovoltaic system (BAPV), which mainly refers to solar PV panels additionally installed after the completion of the original building. Therefore, BAPV is more suitable for retrofitting projects because of its shorter construction period and lower installation cost [3]. In this paper, photovoltaic PV panels and solar thermal collectors, two emerging renewable energy utilization technologies, are poised for rapid development. Meanwhile, the integration of solar PV panels and thermal collectors on building rooftops can supplement electric energy and provide hot water for users.

In order to promote the development of photovoltaic and solar thermal collector technologies applied to building rooftops, many scholars have conducted different innovative research. Elaouzy and Fadar [4] demonstrated that solar PV panels and thermal collectors have the shortest discounted payback periods, the lowest levelized cost of energy and the highest saving-to-investment ratio compared to source heat pump systems, and green rooftops. Sun [5] analyzed the carbon reduction factors of PV rooftops and obtained the optimal combination of the design parameters under the case of the lowest carbon emission. Wang et al. [6] proposed a novel rural building PV/thermal wall and investigated the comprehensive performance of the demo building integrated with this wall. Furthermore, Li et al. [7] proposed an innovative adjustable PV green facade combining an adjustable PV blind system with a green facade. Kang et al. [8] utilized EnergyPlus to study the comprehensive thermoelectric performance of amorphous silicon PV windows with varying transmittances in typical cities across five climatic zones in China. Zhan et al. [9] developed a data prediction–MOO two-stage BIPV design model for PV rooftops and PV opaque facades. Balamurali et al. [10] present a solar concrete water heater as a structural rooftop component. Overall, the use of solar PV and heat collection technologies on the walls and rooftops of buildings is usually accepted to reduce the energy consumption of buildings.

Many scholars have also explored the energy-saving potential of solar PV panels and thermal collectors for buildings using multi-objective optimization. Ascione et al. [11] optimized the primary energy consumption, initial investment and operating cost for hospitals. Qiao et al. [12] considered four variables, including building orientation, window size, window visible light transmittance, and type of PV to minimize both the annual net electricity cost and the extra investment cost of BIPV windows. Luo et al. [13] evaluated the office buildings in three scenarios of no PV, PV on the rooftops, and PV on both the rooftops and facade with the objectives of energy consumption, thermal comfort, embodied carbon and economy, and dynamic payback period. Chen et al. [14] selected building orientation, north–south window-to-wall ratio, north–south window type, and insulation thickness as optimized design variables and carried out the multi-objective optimization of minimizing carbon emissions of buildings and the active system. Jiang et al. [15] assessed the cost and payback period to determine the optimal combination of envelop and solar systems for rural residences in China. In summary, the combination of energy consumption and investment costs is closely followed in the multi-objective optimization for existing buildings.

The existing studies on building retrofitting have predominantly concentrated on the trade-off between energy consumption and investment costs. However, they remain insufficient in quantitatively elucidating the synergistic balance between carbon emissions and cost benefits. Furthermore, considering the rooftops are more exposed to the environment than outside facades and are sensitive to a significant amount of solar radiation at an exceptionally high angle. Therefore, in this study, a multi-optimization design method for existing residential building rooftops, retrofitted by attaching solar photovoltaic panels and thermal collectors, is presented. The innovations of this research are as follows:

(1)

Based on photovoltaic panels and collector synergistic rooftops, the simulation model of full life cycle carbon emission and cost benefit was developed to assess the carbon emission and cost benefit of existing building rooftops retrofitting.

(2)

NSGA-II multi-objective evolutionary algorithm with entropy-TOPSIS integrated workflow was proposed based on a parametric platform. It covers energy model development, multi-objective optimization, and multi-quasi-measurement decision-making, which provides technical support for retrofitting design of existing buildings.

(3)

The existing buildings in the cold zone of China were used for an objective, and the generalizability of the proposed method in case climate regions was explored through the performance verification of carbon emission and cost benefits.

2. Methodology

In building rooftop renovation projects, the application of solar photovoltaic panels and thermal collectors can significantly reduce the building’s dependence on traditional energy sources, even surplus electricity can be fed back to the power grid after meeting the needs of buildings. In addition, China provides policy support such as subsidies or feed-in tariff subsidies to encourage the use of renewable energy, all of which will help shorten the investment payback period. In this study, carbon emissions and cost benefits over the whole life cycle of retrofitted residential buildings were selected as the evaluation indices for multi-objective optimization to pursue the balance between environmental and economic benefits.

2.1. Models of Carbon Emission and Cost Benefit

The life cycle of retrofitted residential buildings includes the pre-retrofitting phase, retrofitting phase, post-retrofitting phase, demolition, and disposal phase [16]. This study focuses on the whole life cycle of solar PV panels and thermal collector retrofitted buildings. The life cycle carbon emissions of the retrofitted buildings for solar PV panels and thermal collectors are divided into four stages: production, transport and installation, operation, and decommissioning.

The environmental benefits are mainly increased by reducing carbon emission (CE) during the operation phase of the retrofitted residential buildings, which can be calculated as follows:

$$CE=C{E}_{\mathrm{S}}-C_{\mathrm{LCA}}$$

(1)

where CE_S is carbon emission from the operational phase in the remaining years of the retrofitted residential buildings, kgCO₂e; C_LCA is carbon emission from the retrofitting life cycle of the residential buildings, kgCO₂e.

$$C_{\mathrm{LCA}}=E_{\mathrm{PRE}}+E_{\mathrm{RE}}+E_{\mathrm{POST}}+E_{DEM}$$

(2)

where E_PRE, E_RE, E_POST, E_END, represent the carbon emissions of pre-retrofitting phase, retrofitting phase, post-retrofitting phase, demolition and disposal phase, kgCO₂e, respectively, which involves a specific calculation model as shown in

Table 1. Carbon emission model for each phase of retrofitting life cycle [17].

Phase Name	Carbon Emission Equation	Meaning
Pre-retrofitting phase	$E_{\mathrm{PRE}}=(Y_{\mathrm{buil}}/Y_{\mathrm{mate}}){\displaystyle \sum _{i=1}^{n}C{E}_{i,\mathrm{mate}}}$	Ybuil: Service life of building Ymate: Service life of solar PV panels and thermal collectors C: Areas of solar PV panels and thermal collectors used Ei,mate: Carbon emission factor for solar PV panels and thermal collectors, kgCO2e/unit
Retrofitting phase	$E_{\mathrm{RE}}=(Y_{\mathrm{buil}}/Y_{\mathrm{mate}}){\displaystyle \sum _{i=1}^n{T}_j\times }{\omega }_j$	M: Number of various types of construction machinery, unit Ci: Carbon emission factor for each machine, kgCO2e/unit
Post-retrofitting phase	$\begin{array}{l}{E}_{\mathrm{POST}}=\left[{\displaystyle \sum _{i=1}^n(E_{i}E{F}_i)-C_{\mathrm{p}}}\right]Y_{\mathrm{buil}}\\ E_i={\displaystyle \sum _{j=1}^n(E_{ij}-E{R}_{ij})}\end{array}$	Ei: Annual consumption of building energy type i, unit/a EFi: Carbon emission factors for energy type i Ei,j: Type i energy consumption for systems in type j, unit/a ERi,j: Amount of category i energy supplied by solar PV panels and thermal collectors, unit/a i: Type of end-use energy consumed in buildings j: Types of building energy systems CP: Annual carbon reductions from solar PV panels and thermal collectors, kgCO2e/a
Demolition and disposal phase	$E_{\mathrm{DEM}}=0.9(Y_{\mathrm{buil}}/Y_{\mathrm{mate}}-1)E_{\mathrm{pro}}$	Carbon emission from the demolition and disposal phase are 0.9 times that of the construction phase

.

The cost benefit was defined as the sum of the new cost benefit after deducting the incremental cost of renovation and the cost during the operational phase of the existing residential buildings over the whole life of the retrofitted building, demonstrated as follows [18]:

$$CB=P_{\mathrm{EXP}}+P_{\mathrm{EXC}}-IC+S_{\mathrm{ACT}}$$

(3)

where P_EXP is the revenue from surplus PV power generation sold to the grid throughout the life cycle of the retrofitted building, CNY, P_EXC is the revenue from surplus solar thermal collector heat generation throughout the life cycle of the retrofitted building, CNY, as shown in Equation (4), IC is incremental cost of building renovation, CNY, S_ACT is the cost during the operational phase of the existing residential buildings over the whole life of the retrofitted building, CNY.

$$P_{\mathrm{EXP}}={\displaystyle \sum _{i=1}^n(S_{\mathrm{p}}-P_{n,\mathrm{PV}})\times C_{\mathrm{EXP}}}$$

(4)

$$P_{\mathrm{EXC}}={\displaystyle \sum _{i=1}^n(S_{\mathrm{c}}-P_{n,\mathrm{PV}})\times C_{\mathrm{EXC}}}$$

(5)

where S_p is the power generation of PV panels in n-th year (kWh), S_c is heating generation of solar thermal collectors in n-th year (kWh), P_n,PV is substitution power of PV panels in n-th year, (kWh), C_EXP is grid-connected electricity prices for renewable energy generation (CNY/kWh), and C_EXC is measured price of heat consumption (CNY/kWh). To facilitate the calculation, the heat generated by the solar thermal collectors was converted into electricity for calculation.

$$S_{\mathrm{ACT}}={\displaystyle \sum _{i=1}^n(P_{n,\mathrm{PV}}+P_{n,\mathrm{PT}})}\times C_{\mathrm{E}}$$

(6)

where C_E is unit price of electricity consumption, CNY/kWh, P_n,PT is replacement from collectors in n-th year, kWh.

The total incremental cost of the retrofitting (IC) can be measured and can be expressed as:

$$IC=C_{\mathrm{A}}+C_{\mathrm{B}}+C_{\mathrm{C}}$$

(7)

where C_A is the cost of solar PV panels and thermal collectors, CNY; C_B is the construction cost of solar PV panels and thermal collectors, CNY; and C_C is the demolition cost of solar PV panels and thermal collectors, CNY.

The PV was selected as monocrystalline silicon cells with high power generation efficiency and long service life. The average annual power generation of PV panels can be calculated as follows [19]:

$$S_{\mathrm{p}}=H_A\times A_{\mathrm{p}}\times K_1\times (1-K_2)$$

(8)

where A_P is the laying area of PV panels, m²; H_A is the total solar irradiation on the light-gathering surface of PV panels, kWh/m²; K₁ is power generation efficiency of PV panels, %; and K₂ is system loss efficiency, %.

Considering the heat collection efficiency and ease of maintenance, it is appropriate to prioritize flat-plate type collectors. The average annual heat gain from the solar thermal collectors is calculated using Equation (9) [19]:

$$S_{\mathrm{C}}={\displaystyle \frac{A_{\mathrm{C}}\times J_{\mathrm{T}}\times (1-{\eta }_{\mathrm{c}})\times {\eta }_{\mathrm{cd}}}{3.6}}$$

(9)

where A_C is laying area of solar thermal collector, m²; J_T is total solar irradiation on the light-gathering surface of solar thermal collectors, MJ/m²; η_cd is annual average thermal efficiency of solar thermal collectors; and η_c is heat loss rate of the device such as pipes, pumps, tanks and other system, which is 15%, 10%, and 25% [20,21], respectively.

2.2. Multi-Objective Optimization

As shown in Figure 1, the multi-objective optimization study of the rooftop retrofitting design for the existing residential building in the community using solar photovoltaic panels and thermal collectors can be divided into three stages: development of the energy model, definition of objectives and design parameters, and optimization and decision-making.

In this study, a three-step process was performed to retrofit the rooftops of residential buildings using solar photovoltaic panels and thermal collectors.

Step 1: A site survey was conducted to collect building geometry information, envelope information, energy equipment data, and schedules of occupancy rate, appliance usage rate, and lighting activation rate, which were used to develop a typical model of an existing residential building in the community and create its energy model.

Step 2: Carbon emission was selected as the evaluation index for retrofitted residential buildings for multi-objective optimization. Meanwhile, EnergyPlus 24.2.0 was used to simulate the energy consumption of the retrofitted residential buildings in the community.

Step 3: A multi-objective optimization was performed using the Wallacei X 2.55module of Grasshopper 1.0.0007 embedded with the NSGA-II genetic algorithm, which is an evolutionary multi-objective optimization method with the ability to optimize non-linear and discrete decision variables and objective functions [22]. And then the Pareto frontier formed by non-dominated solutions was obtained. Furthermore, the optimal solution set was selected by making decisions on the Pareto frontier solution set using the entropy weight-TOPSIS method [23].

At first, the non-dominated set solved by NSGA-II multi-objective algorithm was normalized, as follows:

$$C{E}_q^{*}={\displaystyle \frac{\max (C{E}_q)-C{E}_q}{\max (C{E}_q)-\min (C{E}_q)}}$$

(10)

$$C{B}_q^{*}={\displaystyle \frac{C{B}_q-\min (C{B}_q)}{\max (C{B}_q)-\min (C{B}_q)}}$$

(11)

where $C{B}_q^{*}$ and $C{E}_q^{*}$ are q-th normalized values of the carbon emission and cost benefit in the Pareto frontier solution, respectively.

The weight of the two indices, carbon emission and cost benefit, was obtained as follows:

$$W_{\mathrm{CE}}={\displaystyle \frac{1-E_{\mathrm{CE}}}{(1-E_{\mathrm{CE}})+(1-E_{\mathrm{CB}})}}$$

(12)

$$W_{\mathrm{CB}}={\displaystyle \frac{1-E_{\mathrm{CB}}}{(1-E_{\mathrm{CE}})+(1-E_{\mathrm{CB}})}}$$

(13)

where the information entropy of each index was calculated according to Equations (14) and (15):

$$E_{\mathrm{CE}}=-k{\displaystyle \sum _{q=1}^{30}\frac{x_{\mathrm{CE},q}}{{\displaystyle \sum _{q=1}^{30}{x}_{\mathrm{CE},q}}}}\ln ({\displaystyle \frac{x_{\mathrm{CE},q}}{{\displaystyle \sum _{q=1}^{30}{x}_{\mathrm{CE},q}}}})$$

(14)

$$E_{\mathrm{CB}}=-k{\displaystyle \sum _{q=1}^{30}\frac{x_{\mathrm{CB},q}}{{\displaystyle \sum _{q=1}^{30}{x}_{\mathrm{CB},q}}}}\ln ({\displaystyle \frac{x_{\mathrm{CB},q}}{{\displaystyle \sum _{q=1}^{30}{x}_{\mathrm{CB},q}}}})$$

(15)

where E_CE and E_CB are information entropies of carbon emission and cost benefit, respectively, $x_{\mathrm{CE},q}$ and $x_{\mathrm{CB},q}$ are carbon emissions, and cost benefit will be the probability of taking the q-th value. Furthermore, the weighting factors are calculated as follows:

$$V_{\mathrm{CE},q}=W_{\mathrm{CE}}\times C{E}_q^{*}$$

(16)

$$V_{\mathrm{CB},q}=W_{\mathrm{CB}}\times C{B}_q^{*}$$

(17)

The positive and negative ideal solutions were determined as (1, 1) and (0, 0), which are the maximum and minimum values of each normalized index, respectively.

$$d^{+}=\sqrt{{(1-V_{\mathrm{CE},q})}^2+{(1-V_{\mathrm{CB},q})}^2}$$

(18)

$$d^{-}=\sqrt{{(0-V_{\mathrm{CE},q})}^2+{(0-V_{\mathrm{CB},q})}^2}$$

(19)

The relative closeness coefficient of each non-dominated solution to the ideal solution was calculated, as follows:

$$C_q={\displaystyle \frac{d^{-}}{d^{+}+d^{-}}}$$

(20)

The larger the value of C_q, the better the evaluated solution is. And then, the optimal retrofitting design parameters for existing residential buildings using solar photovoltaic panels and thermal collectors can be obtained to weigh carbon emission against cost benefit.

3. Case Study

3.1. Information of the Case Buildings

The cold zone of China is mainly located in solar resource-rich areas [24]. Therefore, installing solar PV panels and thermal collectors on the rooftops of existing residential buildings in the community is promising. Therefore, the buildings of a residential area located in Tianjin of China, cold climate zone, were selected as a case to research, as shown in Figure 2. Tianjin belongs to the more abundant solar resource area, and the solar energy guarantee rate was taken as 50% [25]. The residential area was constructed in 2014, covering an area of 78,300 m², with a building area of 171,000 m² and a total number of 21 buildings and 940 households. Meanwhile, the residential buildings are divided into two parts, including thirteen 8-story slab buildings and eight 16-story tower buildings, where all residential buildings have a floor height of 3 m. In addition, the buildings use municipal electricity and central heating systems.

The internal spatial layout and interior wall performance have little influence on the overall carbon emissions of buildings. Therefore, in this study, the simplified model was developed in which the internal space of the buildings was divided into indoor spaces and stairwells [26]. When conducting building energy consumption calculations, the heating requirements of both stairwells and indoor spaces are usually taken into account. This is because, although stairwells do not directly serve as living spaces, they are part of the overall building’s energy consumption. The cooling requirements for stairwells may not be necessary. In energy consumption calculations, air conditioning is mainly used for cooling in summer, and stairwells may not require such equipment to maintain a comfortable indoor environment. The heat transfer coefficients of the building envelope are shown in

Table 2. Heat transfer coefficient of the building envelope.

Building Envelope	External Wall	Interior Wall	Roofing	External Window	Floor
Heat transfer coefficient, W/(m2·K)	1.47	2.05	3.32	5.6	4.0

. During the transitional seasons, when indoor conditions do not meet thermal comfort requirements, natural ventilation is achieved by opening windows. Indoor thermal disturbances are mainly affected by factors such as occupants, equipment, and lighting, the parameters of which for the case buildings during the operation are shown in

Table 3. Parameters of the case buildings during the operation.

Parameter	Value	Unit	Reference
Equipment power density	3.8	W/m2	[28]
Lighting power density	6	W/m2	[28]
Air changes per hour	0.5	h−1	[28]
Quota for hot water	50	L/person·d	[21]
Water heater inlet/outlet water temperature	10/60	°C	[21]
Personnel density	0.0027	person/m2	[27]

. The air-conditioning period was set from 15 June to 15 September each year, with an indoor design temperature of 26 °C. The heating period was selected from 15 November to 15 March of the following year, with an indoor design temperature of 18 °C, and the heating system was set to operate in a 24 h mode. Moreover, the occupancy rate of people, usage rate of appliances, and lighting activation rate are shown in Figure 3 [27].

The life cycle retrofitting costs of solar PV panels and thermal collectors are shown in Figure 4. The residential electricity price in Tianjin was 0.49 CNY/kWh, the PV feed-in tariff was 0.3655 CNY/kWh, and the PV subsidy policy was 0.03 CNY/kWh. The power generation efficiency of PV panels was taken as 0.15, and the annual average thermal efficiency of the thermal collectors was 0.9. The life cycle of the PV panels was 25 a, and that of the thermal collectors was 15 a [29]. The degradation rate of the power generation efficiency for the PV panels was calculated as 2.5% attenuation in the first year and 0.7% attenuation in the next year and every year thereafter [30], and the thermal efficiency of the thermal collectors was calculated as 4% attenuation in the first year and 2% attenuation in the next year and every year thereafter [31]. At the same time, the PV system used the self-generation and self-consumption, and the residual power in the grid mode.

In this study, the building energy consumption simulation was mainly based on the Grasshopper, a visual programming software based on the Rhino platform, which is commonly used by architects for parametric design and building performance simulation [32]. The physical environment of the site was constructed in Rhino and Grasshopper based on a site-survey. The Ladybug plugin 1.8.0 calls the CSWD database and imports the hourly meteorological data of Tianjin, China.

Table 4. Irradiance in Tianjin throughout the year.

Month	January	February	March	April	May	June
Irradiance (W/m2)	14.73	16.49	18.23	17.63	19.50	17.98
Month	July	August	September	October	November	December
Irradiance (W/m2)	15.50	15.89	17.38	16.41	13.81	12.61

shows the annual radiant illuminance in Tianjin. Ladybug and Honeybee were used to calculate the building energy consumption. There are three main phases for the energy consumption simulation: development of the site physical model, development of the energy model, and installation of the solar PV panels and thermal collectors on building rooftops. The simulation process of building energy consumption is shown in Figure 5.

3.2. Design Parameters and Constraints

The main parameters that affect the average annual power generation of PV panels are solar radiation intensity, laying angle, PV panel area and power generation efficiency [33]. Meanwhile, the average annual heat gain of the thermal collectors was also affected by the total solar radiation intensity, laying angle, thermal collector area, and annual average thermal efficiency. The allocation ratio of area covered by solar PV panels and thermal collectors depends on the electric energy and thermal energy consumption of buildings. In general, electric energy consumption peaks in the summer, and thermal energy consumption reaches its peak in the winter. Therefore, in the study, laying angle and area of the solar PV panels and thermal collectors were screened as the design parameters, and the optimal allocation ratio of the area covered by solar PV panels and thermal collectors was studied.

As shown in Figure 6, the layout of solar PV panels or thermal collectors was oriented along the direction of the rooftop surface of the buildings, with the length of the rooftop surface in the slope direction as the boundary, thereby controlling the area of PV panels installation by adjusting the length of the rooftop surface where the solar PV panels or thermal collectors were laid out. With the bottom edge of the solar PV panels or thermal collectors as the axis and the rooftop surface as the starting plane, the solar PV panels or thermal collectors were rotated along the axis according to its laying angle. The rooftops of the case buildings were divided into six types of slopes. The total area of solar PV panels or thermal collectors and thermal collectors is the sum of the areas for the six slope types. Therefore, the laying area of solar PV panels (A_p) and thermal collectors (A_c) can be expressed as Equation (21) and Equation (22), respectively.

$$A_{\mathrm{p}}=p(A_1+A_2+A_3+A_4+A_5+A_6)$$

(21)

$$A_{\mathrm{c}}=(1-p)(A_1+A_2+A_3+A_4+A_5+A_6)$$

(22)

where p is the allocation ratio of the area covered by solar PV panels and thermal collectors, namely, the proportion of PV panels area to total rooftop area, and A₁, A₂, A₃, A₄, A₅, A₆ are the areas of solar PV panels or thermal collectors on a slope of α₁, α₂, α₃, α₄, α₅, α₆, respectively, as shown in Equations (23)–(28).

$$A_1=\left\{\begin{array}{l}246.08-50d_1\left(0\le d_1\le 0.82\right)\\ -8.26{d_1}^2-24.54d_1+204.65\left(0.82<d_1\le 4.92\right)\end{array}\right.$$

(23)

$$A_2=-0.49{d_2}^2-2.99d_2 + 15.59\left(0\le d_2\le 3.39\right)$$

(24)

$$A_3=-0.44{d_3}^2-2.15d_3+14.03\left(0\le d_3\le 3.72\right)$$

(25)

$$A_4=\left\{\begin{array}{l}536.64-80d_4\left(0\le d_4\le 1.12\right)\\ -2.84{d_4}^2-48.14d_4+447.59\left(1.12<d_4\le 3.39\right)\end{array}\right.$$

(26)

$$A_5=-0.72{d_5}^2-3.43d_5+21.04\left(0\le d_5\le 3.53\right)$$

(27)

$$A_6=-0.23{d_6}^2-3.98d_6+72.72\left(0\le d_6\le 11.19\right)$$

(28)

The retrofitting design of residential buildings aims to add the solar PV panels and thermal collectors. As shown in Figure 6, the design parameters for the case buildings in the community involve the low-level PV panel laying angle and small-slope laying area, intermediate-slope laying area, large slope laying area, high-level PV panel laying angle and small-slope laying area, intermediate-slope laying area, large-slope laying area. In addition, the proportion of PV panels area to total laying rooftop area was defined. The optimization boundary conditions of the design parameters are shown in

Table 5. Range of design parameters.

Parameters		Unit	Range of Values	Step
X1	Low-level PV panel laying angle	°	0.00–20.00	0.01
X2	A1	m2	0.00–28.00	0.01
X3	A2	m2	0.00–31.00	0.01
X4	A3	m2	0.00–67.00	0.01
X5	Angle of laying of high-rise PV panels	°	0.00–20.00	0.01
X6	A4	m2	0.00–42.00	0.01
X7	A5	m2	0.00–146.00	0.01
X8	A6	m2	0.00–209.00	0.01
X9	PV solar thermal laying area ratio	—	0.0–1.0	0.1

, where all the design parameters are all continuous variables.

4. Results and Discussion

4.1. Analysis of Solution Sets

The multi-objective optimization was used for the existing residential building rooftop, attached solar photovoltaic panels, and thermal collectors, in the case that community was carried out to weigh carbon emission against cost benefit, where the number of populations was set to 30 and the number of iterations was set to 50 generations. Considering that the optimization mechanism of engine always seeks to minimize the objective function value, the negative value of CB was selected as the objective function of optimization. Due to the vast search space of the optimal solutions, the range of objective function values is large. Therefore, in the optimization process, the optimal solutions are randomly constructed by a large population of individuals. After crossover and mutation, the objective values for each generation of individuals decrease, and the evolutionary process continuously converges until the allowed number of optimization iterations is reached.

Figure 7 provides the variation of carbon emissions and carbon benefits at different generations. Clearly, in the preliminary iterations, the crossover probability is too large, resulting in the solution population not having converged yet, which makes the fluctuation range of the average values of CE and CB larger. As the number of iterations increases, the average values of both the CE and CB gradually become stable. Meanwhile, the cost benefits are highest at the 50th iteration.

The hypervolume of the non-dominated solution set is one of the important indicators for evaluating the convergence and diversity of the non-dominated solutions in the multi-objective optimization process [34]. The larger the value of hypervolume, the better the optimization solutions. Figure 8 gives the change in hypervolume during the multi-objective optimization process. It can be found that the value of hypervolume starts to stabilize near the iteration of about 39 generations. Thus, the quality of the non-dominated solution exhibits pronounced superiority.

It is difficult to obtain optimal solutions for the objective functions at the same time because of the contradictions between the optimization objectives. Therefore, the solution set of multi-objective optimization is often a series of non-dominated solutions, usually called the Pareto frontier. The distribution of Pareto solutions for the multi-objective optimization is shown in Figure 9.

4.2. Selection of Optimal Solution

After obtaining the Pareto frontier, the entropy-TOPSIS was used to decide on the optimal solution.

Table 6. Weights of the two objectives.

Parameter	Information Entropy	Information Utility Value	Weighting Factor
Carbon emissions	0.9892	0.0108	42.68%
Cost benefits	0.9855	0.0145	57.32%

shows the information entropy and weights of the two optimization objectives by the entropy-TOPSIS. It can be seen that the weight of carbon emissions accounts for a proportion of 42.68%, whereas cost benefit accounts for 57.32%. Therefore, cost benefit was set as the primary decision-making objective, and solution with the highest potential for cost benefit was prioritized, while ensuring that carbon emissions meet sustainable development objectives. Furthermore, the positive and negative ideal solutions were determined, and the Euclidean distances of non-dominated solutions and the relative closeness of each solution were calculated. And then, the solution with the largest relative closeness coefficient was chosen as the optimal solution.

The TOPSIS comprehensive evaluation method was employed to select the optimal solution. The non-dominated solutions were then ranked according to the size of relative closeness coefficients as shown in Figure 10. Furthermore, the optimal objectives and design parameters are shown in

Table 7. Optimization solution.

Parameter	X1 (°)	X2 (m2)	X3 (m2)	X4 (m2)	X5 (°)	X6 (m2)	X7 (m2)	X8 (m2)	X9	CE (kgCO2e)	CB (CNY)
Value	20.00	120.21	13.51	12.50	2.08	482.24	20.72	1.92	0.2	1.9228 × 107	5.6425 × 106

.

For the case study, there have been significant improvements in carbon emissions and cost benefits after the optimal retrofitting design. Compared to the residential building before the retrofitting, a 31% reduction in carbon emission over the remaining life of the buildings, and a 24.7% reduction in cost during the operational phase of the existing residential buildings in the case community were revealed.

5. Conclusions

In the study, an optimization design method for the existing residential building rooftops retrofitted with the installation the solar photovoltaic panels and thermal collectors was proposed to weigh the carbon emission and cost benefit. Notably, the mathematical models of the carbon emission and cost benefit were developed. Furthermore, a case study for the typical residential community, located in the cold zone of China, was conducted. The main conclusions are as follows:

(1)

Carbon emission is the primary objective of the retrofitting design for the existing residential building rooftops by using solar photovoltaic panels and thermal collectors, of which the weight of the carbon emission based on the entropy value method is 42.68%, while that for the cost benefit is 57.32%. Cost benefit should be set as the primary decision-making objective, while ensuring that carbon emissions meet sustainable development objectives.

(2)

The optimal installation parameters for the solar photovoltaic panels and thermal collectors are as follows: the optimal allocation ratio of the area covered by solar PV panels and thermal collectors is 0.2. Moreover, the solar photovoltaic panels and thermal collectors on the lower building have a laying angle of 20°, and that on the higher building have a laying angle of 2.08°. It provides a reference for retrofitting and installation of solar photovoltaic panels and thermal collectors in existing buildings.

(3)

Compared to the residential building before the retrofitting, a 31% reduction in carbon emission over the remaining life of the buildings, and a 24.7% reduction in cost during the operational phase of the existing residential buildings in the case community were revealed. The reliability of the research method is verified, and an empirical evidence basis is laid for the promotion of solar photovoltaic panels and thermal collectors in the retrofitting of existing buildings.