## 4 Objective Function Setting  
### Heating Carbon Emissions per Unit Building Area  
Surveys indicate that in the Wusu area, rural residents are busier in the fields during the summer, leading to lower occupancy rates during the day and infrequent use of cooling equipment. In contrast, heating is required throughout the day in winter. Therefore, this study targets the carbon emissions from heating per unit of building area during the heating season as the objective for retrofitting the baseline rural house. Meteorological parameters use the typical meteorological year data for Wusu City downloaded from the EnergyPlus official website, with other operational parameters as previously described.  
The carbon emissions of a building include those from construction, operation, demolition, and the production and transportation of building materials. After retrofitting, carbon emissions can be defined as the sum of the following components: (1) implicit and direct carbon emissions during the production and transportation of retrofit materials, and (2) direct and indirect carbon emissions from heating and cooling during the operational phase. A life cycle carbon emission analysis is applicable to new construction and may not fully apply to the retrofitting of existing buildings. The electricity consumption by solar photovoltaic systems, which is provided by the systems themselves without consuming external grid electricity, is therefore not considered in the carbon emission calculations, as solar systems contribute to reducing electricity consumption for heating.  
Currently, there are four main methods for calculating carbon emissions internationally: the measurement method, the input-output method, the material balance method, and the emission factor method [88]. The emission factor method calculates carbon emissions based on the statistical average amount of gas emitted per unit of product produced under normal economic and management conditions [89]. This paper focuses on the impact of different retrofit measures on carbon emissions during the heating period, paying particular attention to heating carbon emissions in the operational phase after retrofitting, using the emission factor method to calculate \(CE_{CO_{2}}\) values. Lower values indicate better environmental performance of the building. According to the basic formula revised by IPCC in 2006, \"Greenhouse gas emissions = Activity data \(\times\) Emission factor\", the accounting scope considers only CO\({}_{2}\) greenhouse gases, with the calculation formula as follows:  
\[CE_{CO_{2}}=E\times EF_{CO_{2}} \tag{14}\]  
In the formula, \(CE_{CO_{2}}\) represents the carbon emissions per unit area of heating for the rural house, measured in kgCO\({}_{2}\)/m\({}^{2}\); \(E\) is the heating energy consumption per unit building area, measured in kw\(\cdot\)h/m\({}^{2}\); \(CE_{CO_{2}}\) is the CO\({}_{2}\) emission factor for a specific type of energy, measured in kgCO\({}_{2}\)/KW\(\cdot\)h [90]. For electricity, \(EF_{CO_{2}}\) adopts the value from the \"2019 Annual Emission Reduction Project Regional Power Grid Baseline Emission Factor for China\" for the northwest regional power grid, which is converted to 0.8922 kgCO\({}_{2}\)/KW\(\cdot\)h.