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Sustainability 2022, 14, 10125 8 of 39 The heat pump modelling is a bit more complicated since, besides its existence (Equation (18)), its operation should be managed properly, as the heating and cooling production cannot happen at the same time (Equations (19)–(21)). The linear equations regarding the heating (Equation (22)) and cooling (Equation (23)) production as a function of the electricity input are also presented. Equations (24)–(26) represent the operation limits, as well as the limitation about the electricity input for both operation modes. Xhp(j,u) ≤ Xhp(j − 1,u) (18) Ohp,h(m,d,h,j,u) ≤ Xhp(j,u) (19) Ohp,c(m,d,h,j,u) ≤ Xhp(j,u) (20) Ohp,h(m,d,h,j,u) + Ohp,c(m,d,h,j,u) ≤ 1 (21) Hhp(m,d,h,j,u) = Khp(m,d,h,u,1)·Ehp,h(m,d,h,j,u) + Khp(m,d,h,u,2)·Ohp,h(m,d,h,j,u) (22) Chp(m,d,h,j,u) = Khp(m,d,h,u,3)·Ehp,c(m,d,h,j,u) + Khp(m,d,h,u,4)·Ohp,c(m,d,h,j,u) (23) Shp,lim(m,d,h,u,1)·Ohp,h(m,d,h,j,u) ≤ Ehp,h(m,d,h,j,u) ≤ Shp,lim(m,d,h,u,2)·Ohp,h(m,d,h,j,u) (24) Shp,lim(m,d,h,u,1)·Ohp,c(m,d,h,j,u)≤Ehp,c(m,d,h,j,u)≤Shp,lim(m,d,h,u,2)·Ohp,c(m,d,h,j,u) (25) Ehp(m,d,h,j,u) = Ehp,h(m,d,h,j,u) + Ehp,c(m,d,h,j,u) (26) PVp and STp plants are proportional to the user and central unit available surface sizes. They are estimated a priori through the hourly insolation, inclination, and orientation angle of installed panels. 2.2. District Heating and Cooling Network In order to have an effective working DHCN, the user location has to be geographically near the DHCN pipelines (within the distance of about 1.5 km) in order to avoid large amounts of thermal losses through the pipelines themselves. The definition of the DHCN layout and the maximum capacity of pipelines (considering the whole system operation) are two of the aims of the DG energy system optimization, since the network strongly affects the optimal solution. Equation (27) describes the heat flowing into each DHCN pipeline: . Qp = Ap ·vp ·ρp ·cp ·∆t (27) in which the velocity vp is supposed to be constant (ranging from 1.5 to 2.5 m/s), so that . the heat flowing through the pipeline (Qp) is a function of the pipeline cross section area (Ap) and the temperature difference ∆t between the inlet/outlet of the pipeline itself. This temperature difference is assumed to be fixed (ranging into the 15–25 ◦C interval), as is the network temperature, while the pipeline length and maximum flow ratio introduced by the model are constant. The network layout and size are decision variables for which the pipeline flow rate limits, and the superstructure are the constraints. The thermal losses are expressed by Equation (28) and depend on the pipeline length lp(u,v) and a coefficient of proportionality (δt). pt(u,v) = δt·lp(u,v) (28) Another important constraint is the one represented by Equation (29). It does not allow the model to connect two users (e.g., user u and user v) with two pipelines sending thermal energy. Xtp(u,v) + Xtp(u,v) ≤ 1 (29) The maximum heat flow rate is constrained by the pipelines size, while the energy flow into each pipeline is bounded between a lower and an upper limit: Hnet,lim(1)·Xtp(u,v) ≤ SH,net(u,v) ≤ Hnet,lim(2)·Xtp(u,v) (30)PDF Image | Optimal Sharing Electricity and Thermal Energy
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