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Sustainability 2022, 14, 10125 7 of 39 The partial load performance of cogeneration devices is represented by the set of linear relations presented in Equations (3) and (4). Hchp(m,d,h,j,u) = Khchp(m,d,h,1)·Echp(m,d,h,j,u) + Khchp(m,d,h,2)·Ochp(m,d,h,j,u) (3) Fchp(m,d,h,j,u) = Kfchp(m,d,h,1)·Echp(m,d,h,j,u) + Kfchp(m,d,h,2)·Ochp(m,d,h,j,u) (4) The coefficients Khchp and Kfchp have been obtained with a linear regression of the devices characteristic load curves, and the electric energy produced is limited above and below the device’s performance limits. The subscript “chp” represents the ICE and MGT devices for a given user. For that reason, in Equations (1)–(4), such subscript can be changed to “ice” or “mgt” to obtain the equations related to both devices. The ICE at the central unit is described by different equations because both load and size are decision variables, and it is then mandatory to introduce further constraints to maintain the problem a linear one. The ICE size and the relation between the operation and the device existence are expressed by Equation (5). Sice,lim,c(1)·Xice,c ≤ Sice,c ≤ Sice,lim,c(2)·Xice,c (5) Oice,c(m,d,h) ≤ Xice,c (6) The relations among the fuel consumed by the central ICE (Fice,c), the main (Eice,c), and the secondary (Hice,c) products are as follows: Fice,c(m,d,h) = Kfice,c(m,d,h,1)·Eice,c(m,d,h) + Kfice,c(m,d,h,2)·Oice,c(m,d,h) + Kfice,c(m,d,h,3)·ξice,c(m,d,h) (7) Hice,c(m,d,h) = Khice,c(m,d,h,1)·Eice,c(m,d,h) + Khice,c(m,d,h,2)·Oice,c(m,d,h) + Khice,c(m,d,h,3)·ξice,c(m,d,h) (8) where the variable ξice,c is introduced to set a linear equation with two independent vari- ables. Therefore, in order to avoid inconsistencies in the results when the engine is turned off, it is necessary to constrain ξice,c through the following equations: Sice,c + Sice,lim,c(2)·(Oice,c(m,d,h) − 1) ≤ ξice,c(m,d,h) ≤ Sice,c (9) Sice,lim,c(1)·Oice,c(m,d,h) ≤ ξice,c(m,d,h) ≤ Sice,lim,c(2)·Oice,c(m,d,h) (10) Eice,c(m,d,h) ≤ Sice,c (11) The central BOI is modelled in an analogous way to the central ICE, with the introduc- tion of the auxiliary variable ψboi,c (with a minimum load limit of Hboi,lim,c = 0.1). The fuel consumption, described by Equation (12), is affected by the BOI’s efficiency. Analogously, the CC, at the user level, is modelled through its COP. Both BOI and CC have no load limits. Fboi,c(m,d,h) = Hboi,c(m,d,h)/ηboi,c(m,d,h) (12) Hboi,lim,c·ψboi,c(m,d,h) ≤ Hboi,c(m,d,h) ≤ ψboi,c(m,d,h) (13) Sboi,c + Sboi,lim,c(2)·(Oboi,c(m,d,h) − 1) ≤ ψboi,c(m,d,h) ≤ Sboi,c (14) Sboi,lim,c(1)·Xboi,c ≤ Sboi,c ≤ Sboi,lim,c(2)·Xboi,c (15) The ABS devices are allowed to exist only at the user locations where there is the presence of ICE and MGT. Another important constraint is the cooling produced by the ABS, since it must not be greater than the heat supplied by both ICE and MGT (Equation (17)). Xabs(j,u) ≤ Xice(j,u) + Xmgt(j,u) (16) Cabs(m,d,h,j,u) ≤ Hice(m,d,h,j,u) + Hmgt(m,d,h,j,u) (17)PDF Image | Optimal Sharing Electricity and Thermal Energy
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