## 5 Results  
### Analysis of Optimal Solution's Objective Performance  
Based on the simulation results from EnergyPlus, the initial values for the optimized variables of the baseline building and the heating energy consumption per unit of building area are presented in Table 4. These values exceed the current standards for residential buildings, which require a 75% energy saving rate. Simulating the selected photovoltaic (PV) components reveals that from May to September, and as shown in Figure 6, the solar irradiance is abundant, gradually decreasing with the change in the sun's elevation angle, where the irradiance on an inclined surface is slightly higher than on a flat surface. The electricity generation from the PV components is lowest in November, with some energy loss through the inverter. From May to September, the output of electricity can reach up to 900 kWh. During the winter heating period, the PV components generate a total of 3603.23 kWh of electricity, which can be fed into the grid during the summer without the need for additional energy consumption. For setting optimization algorithm parameters, reference is made to related research. Under the premise of ensuring the accuracy of the Pareto solution set and reasonable computation time, the settings for the NSGA-II optimization algorithm are as follows: a population size of 50, a number of generations of 20, a crossover probability of 0.9, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Type**} & **Exterior Wall** & **Roof** & \multirow{2}{*}{**Floor Height**} & \multirow{2}{*}{**South-Facing**} & \multirow{2}{*}{**North-Facing**} & **Heating Load** \\  & **Additional** & **Additional** & & & **PWR** & **WWR** \\  & **Insulation** & **Layer** & & & & **(kWh/m2)** \\ \hline Baseline & \multirow{2}{*}{None} & \multirow{2}{*}{None} & \multirow{2}{*}{None} & \multirow{2}{*}{3.0} & \multirow{2}{*}{0.23} & \multirow{2}{*}{0.10} & \multirow{2}{*}{211.51} \\ Building & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Initial Performance of the Baseline Building. Figure 6: (a) Monthly output electricity; (b) monthly flat-plane irradiance. and a mutation probability of 0.1. The maximum number of iterations is used as the termination condition for optimization. After comprehensive optimization, the Pareto solution set under the control of objectives is obtained, as shown in Figure 7. Overall, it is evident that the various retrofit measure variables for the rural house all have significant optimization potential. After 20 generations of calculation, the multi-objective optimization tends towards convergence, yielding 64 Pareto optimal solutions for the rural house retrofit. As the cost of the rural house retrofit increases, the trend of reducing building energy consumption slows down from rapid to gradual. The solutions in the set all show heating energy consumption significantly better than the initial energy consumption values of the baseline building, with per unit area carbon emissions also lower than the initial carbon emission values of the baseline building. The trend in winter heating energy consumption and the global cost of rural house retrofit are inversely related, indicating that the optimization objectives are mutually constrained and cannot achieve optimum simultaneously. The diagram shows a series of retrofitting schemes obtained after 20 generations of optimization, which display varying degrees of trade-offs between cost reduction and energy consumption. These trade-offs reflect the mutual constraints of the optimization objectives. As a scatter plot representing different retrofitting schemes, each point on the plot indicates two key performance indicators of a retrofitting scheme: the vertical axis represents carbon emissions \(CE_{CO_{2}}\), and the horizontal axis represents the incremental global cost \(dC_{g}\). It is evident that there are schemes with \(dC_{g}<0\), indicating that these retrofitting schemes are economically viable.