# Design Optimization of a District Heating Network Expansion, a Case Study for the Town of Kiruna
## 3 Methodology
### The Optimization Problem  
The two objective functions to be minimized in the multi-objective optimization problem presented in this paper were the investment cost for the expansion of the DH network in Kiruna (\(C_{inv}\) in Equation (1)) and the operating cost to run the network after the expansion had been completed (\(C_{op}\) in Equation (2)).  
\[C_{inv}=\sum_{pipe}c_{pipe}\left(D_{pipe}\right)L_{pipe}+C_{bailer} \tag{1}\]  
\[C_{op}=\sum_{site\,res}c_{res}Q_{res,site}+\sum_{site}c_{cl}P_{pump,site} \tag{2}\]The investment cost is made of two terms: (i) the cost related to the pipes; and (ii) the cost related to the new bio-oil boiler. The investment cost for the pipes is expressed as the sum for all the pipes existing in the considered design solution of the cost per unit length of the pipe (\(c_{pipe}\), which depends on pipe diameter \(D_{pipe}\) and includes purchase, digging, welding and installation) multiplied by pipe length (\(L_{pipe}\)). The investment cost for the new bio-oil boiler (\(C_{boiler}\)) depends on its capacity.  
As in [12] the operating cost consists of two terms: (i) the cost for generating the requested heat, which is equal to the overall demand of end users plus the thermal losses along the network; and (ii) the cost for making the water circulate in the piping network, which is the sum of the costs for the pumping power required at the heat production sites to introduce the supply water mass flow rates into the network. In turn the cost for the generation of heat is equal to the sum at all heat production sites of the unit costs of the resources consumed in the boilers of the site (\(c_{res}\) in SEK/kJ, where _res_ could be municipal waste, bio-oil, biomass, electricity and oil) multiplied by the resource input in thermal power terms (\(Q_{res,site}\) in kW, which is greater than the heat output according to the efficiency of the boilers). It is worth noting that this term is calculated on the resource input required to generate the amount of heat supplied by the sites, so it implicitly takes into account the heat losses along the network and boiler efficiencies. The cost term related to the pumping power is expressed as the sum all heat production sites of the unit cost of electricity (\(c_{el}\) in SEK/kJ) multiplied by the pumping power consumed at the site (\(P_{pump,site}\) in kW).  
The decision variables of the optimization problem are:  
* The diameters for each pipe segment in the superstructure shown in Figure 4. These are 18 discrete variables that can assume the values of 0 (the pipe does not exist), 100, 200, 300 and 400 mm;
* The capacity of the new bio-oil boiler. This is another discrete variable that can assume the values between 7 and 18 MW at 1 MW intervals.  
Some constraints were also imposed to the values of the local pressures and temperatures in order to ensure the correct operation of the DH network. The maximum pressure difference within the supply network is not supposed to exceed 12 bars, considering the height of the nodes, while the temperature of the water supplied to a group of consumers is not allowed to be lower than 65 \({}^{\circ}\)C (for further details see [12]).