## 4 Results and Discussion  
### Results  
According to the calculation and analysis of multiple iterations, the diameter of the pipeline in the targeted area is distributed within the interval of DN100\(-\)DN200. The pipeline segments in the area can be divided into three levels, with three different nominal values (DN100, DN150 and DN200). The results of pipe diameter grading are shown in Figure 12.  
In this way, the weight of any pipe segment in the target area can be updated to \(D_{i}\) according to Equation (18), and the original problem is transformed into a minimum spanning tree problem with network annual cost as the weight. The Prim algorithm is used to solve for the network topology that gives the minimum annual cost. It should be noted that in the process of pipe diameter grading, the pipe diameter is regarded as a fuzzy variable, denoted as \"fuzzy pipe diameter\", not the pipe segment diameter in the final scheme. There are two reasons for this:  
First, the velocity of the fluid in the pipe is a variable which changes positively as a function of pipe diameter. According to the relationship between the flow rate and the pipe diameter, the selection of the appropriate pipe diameter will fluctuate within a certain range. In addition, since pipe diameters in the studied area do not vary significantly, it can be described by a fuzzy variable with permitted fluctuation.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & **Values** & **Units** & **Parameters** & **Values** & **Units** \\ \hline \(r\) & 6\% & & \(\delta_{DN100}\) & 2.5 & mm \\ \(C(d_{DN100})\) & 609.41 & RMB/m & \(\delta_{DN150}\) & 3.5 & mm \\ \(C(d_{DN150})\) & 864.46 & RMB/m & \(\delta_{DN200}\) & 4.5 & mm \\ \(C(d_{DN200})\) & 1206.26 & RMB/m & \(\delta_{tDN100}\) & 25 & mm \\ \(\lambda\) & 0.06 & W/(m-K) & \(\delta_{tDN150}\) & 30 & mm \\ \(L\) & 20 & year & \(\delta_{tDN200}\) & 40 & mm \\ \(t_{c}\) & 10 & “C & \(\mu_{1}^{\mathrm{b}}\) & 0.01 & \\ \(T_{p}\) & 3360 & h & \(\mu_{2}^{\mathrm{b}}\) & 0.015 & \\ \(C\epsilon\) & 0.6 & RMB/(kWh) & \(\rho\) & 1000 & kg/m\({}^{3}\) \\ COP & 4.2 & & \(\eta_{p}\) & 0.6 & \\ \(\overline{v}\) & 0.8\(-\)1.2 & m/s & \(\alpha\) & 0.2 & \\ \(K\) & \(5\times 10^{-4}\) & m & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Table of simulation references.  
Figure 12: The result of the Pipe Diameter Grading in the study area.  
After the topology scheme of the pipe network is calculated, the diameters of each pipe segment are determined according to Equations (13) and (14), described in Section 3.2. Afterwards, the \"fuzzy pipe diameter\" estimated by the pipe diameter grading is adjusted accordingly, to iterate for the optimal solution.  
It can be found that the method of pipe diameter grading is feasible, and the final pipe network construction plan with minimum annual cost as the optimization goal is shown in Figure 13. The optimization results with the shortest path and minimum cost are compared in Figure 14. Although the latter increases the total pipe length by 1.88%, the cost decreases by 2.38%. It can be concluded that the method introduced in this paper is efficient and adaptable for optimizing the design of district energy system pipe networks.