## 3 Optimization Procedure by PDG  
### Network Economic Cost  
The average annual cost of the pipe network (\(D\)) can be expressed as Equation (3) [16; 19; 30]:  
\[D=C_{0}^{p}\frac{r\left(1+r\right)^{L}}{\left(1+r\right)^{L}-1}+C_{w}^{p}+C_{m }^{p}+C_{f}^{p}, \tag{3}\]  
where \(C_{0}^{p}\) represents the initial investment in pipeline construction; \(C_{w}^{p}\) represents the heat loss cost of the pipe; \(C_{m}^{p}\) represents the depreciation and maintenance cost of the pipe;  
Figure 8: Connected graph G of the study area using the Energy Station-to-Load Points model. Figure 9: Network topology plan guided by ESLP. and \(C_{f}^{P}\) represents the operating cost of the pump. \(r\) is the annual interest rate and \(L\) is the life of the project. Additionally, Equation (4):  
\[C_{o}^{P}=\sum_{i=1}^{n}C(d_{i})l_{i}, \tag{4}\]  
where \(C(d_{i})\) represents the construction cost of pipe section \(i\) per meter of pipe length, including supply and return water pipeline; \(d_{i}\) represents the pipe diameter of section \(i\); \(l_{i}\) represents the pipe length of segment \(i\). The heat loss costs are calculated by Equation (5):  
\[C_{w}^{P}=\sum_{i=1}^{n}\frac{4\pi\lambda t_{c}l_{i}T_{p}C_{e}}{ln\frac{d_{i} +2\beta_{i}+\beta_{i}}{d_{i}+2\beta_{i}}\times COP\times 10^{7}}, \tag{5}\]  
where \(\lambda\) is the thermal conductivity coefficient of thermal insulation material coated on the pipeline; \(t_{c}\) is the average temperature difference between inside and outside the water conveyance pipeline; \(T_{p}\) is the annual maximum working time of circulating pump; \(C_{e}\) is the electricity price; \(\delta_{i}\) is the thickness of the pipe; \(\delta_{it}\) is the thickness of the insulation layer; COP is the energy efficient ratio of the system. The depreciation and maintenance costs are calculated by Equation (6):  
\[C_{m}^{P}=\Big{(}\mu_{1}^{P}+\mu_{2}^{P}\Big{)}C_{o}^{P}, \tag{6}\]  
where \(\mu_{1}^{P}\) represents the equipment depreciation rate, and \(\mu_{2}^{P}\) represents the maintenance cost proportional coefficient. The pumping costs are calculated by Equation (7):  
\[C_{f}^{P}=\sum_{i=1}^{n}2.78\times 10^{-7}\frac{Q_{i}\cdot H_{Pi}}{\rho\cdot \eta_{P}}\cdot T_{p}C_{e}, \tag{7}\]  
where \(C_{f}^{P}\) represents the annual operating cost of the circulating pump; \(Q_{i}\) represents the calculated flow rate of pipe segment \(i\); \(H_{Pi}\) represents the head of circulating pump; \(\rho\) represents the density of the heat vector and water is used as the heat vector in this text. \(\eta_{P}\) represents the motor efficiency of the circulating pump; \(T_{p}\) represents the number of working hours of circulating water pump calculated in the next year; \(C_{e}\) is the price of electricity. If the pump power set in each branch pipe segment is fully used to overcome the flow resistance, then the head value of circulating pump in the pipe segment can be expressed by the resistance loss in the pipe segment. The resistance loss of pipe section consists of two parts: the resistance loss due to friction and the local resistance loss of the fluid, as expressed in Equation (8):  
\[H_{Pi}=\Delta P_{i}=\Delta P_{mi}+\Delta P_{ji}, \tag{8}\]  
where \(\Delta P_{mi}\) represents the resistance loss along the pipe segment i; \(\Delta P_{ji}\) represents the local resistance loss of section \(i\). The resistance loss along the pipe is caused by the friction between the fluid and the wall surface of the pipe in the straight pipe. Its calculation formula is Equation (9), as follows:  
\[\Delta P_{mi}=\gamma\cdot\frac{l_{i}}{d_{i}}\cdot\frac{\overline{v}^{2}\rho}{ 2}, \tag{9}\]  
where \(\overline{v}\) is the velocity of thermal media, \(\gamma\) is the resistance coefficient along the inner wall of the pipe, which can be determined by schifflinson formula, Equation (10):  
\[\gamma=0.11\bigg{(}\frac{K}{d_{i}}\bigg{)}^{0.25}, \tag{10}\]where \(K\) represents equivalent roughness of pipe inner wall, which can be determined directly according to the design specification of urban heating pipe network.