# Improved Solar Operation Control for a Solar Cooling System of an IT Center
## 2 Materials and Methods
### CE Method and CE Controller  
The control strategy described in Section 3.2 to start the solar heat operation (SHO) of an SAC system as early as possible and to operate it in SHO mode as long as possible under the constraint of a maximum electricity demand \(w_{\rm ACCA}<w_{\rm CCCA}\) necessitates a calculation procedure for the part load behavior of an absorption chiller. For this purpose, the characteristic equation method was applied. In contrast to the characteristic equation method used by [PERSON] et al. (2010), the method in this contribution is general for \(\rm H_{2}O/LiBr\) single-effect chillers (1998). By means of a revised heat transfer calculation in the absorber and desorber as compared to the established method (cf. [17]), the variation of thermodynamic losses under part load conditions can be accounted for in an explicit calculation procedure. According to the improved method, the cooling capacity \(\rm Q_{E}\) of an absorption chiller is a linear function of a characteristic temperature difference \(\rm\Delta\Delta t_{i}^{*}\):  
\[\rm\Delta\Delta t_{i}^{*}=t_{Di^{*}}(1-K_{1})-t_{Ai^{*}}(1-K_{2})+t_{Ei^{*}}(1 -K_{3}), \tag{2}\]  
\[\rm Q_{E}=K_{4}\cdot\Delta\Delta t_{i}^{*}, \tag{3}\]  
\[\rm Q_{D}=K_{5}\cdot\Delta\Delta t_{i}^{*}+K_{6}\cdot\Delta\Delta t_{min,i}^{*}, \tag{4}\]  
\[\rm\Delta\Delta t_{min,i}^{*}=K_{1}\cdot t_{Di}-K_{2}\cdot t_{Ai}+t_{Ei^{*}}( K_{3}-1). \tag{5}\]  
The characteristic temperature difference \(\rm\Delta\Delta t_{i}^{*}\) combines the inlet temperatures \(\rm t_{Xi}\) (where \(\rm X~{}=~{}D,E,C,A\) holds for the main heat exchangers, desorber, evaporator, condenser, and absorber). The coefficients \(\rm K_{1}\) to \(\rm K_{3}\) in Equation (2) account for the phase equilibrium data of the respective working pair (i.e., \(\rm H_{2}O/LiBr\)) in combination with the external and internal heat capacity flow rates \(\rm W_{X}\) and heat transfer capabilities \(\rm Y_{X}=U_{X}\cdot A_{X}\). Hence, the characteristic temperature difference \(\rm\Delta\Delta t_{i}^{*}\) also includes all the information of the load-dependent and load-independent losses. The capacity of the chiller (i.e., its 'thermal size') is described by slope parameters \(\rm K_{4}\) and \(\rm K_{5}\), respectively. Finally, the coefficient \(\rm K_{6}\) scales the minimum driving heat \(\rm Q_{D,min}=K_{6}\cdot\Delta\Delta t_{min,i}^{*}\), which results mainly from the limited internal heat recovery in the solution heat exchanger. The minimum driving heat \(\rm Q_{D,min}\) is also not constant. It can be calculated with a second characteristic temperature difference \(\rm\Delta\Delta t_{min,i}^{*}\), which is also a function of the independent external inlet temperatures \(\rm t_{Xi}\) and coefficients \(\rm K_{1}\) to \(\rm K_{3}\)[12]. The determining equations for all coefficients \(\rm K_{N}\) with \(\rm N=1\ldots 6\) can be found in [13] or [18]. For the absorption chiller at UBA, the values are depicted in Table 3 at the design conditions specified in Table 1.  
Moreover, for control purposes, the external evaporator outlet temperature \(\rm t_{Eo}\) is of interest rather than \(\rm t_{Ei}\). Instead of \(\rm\Delta\Delta t_{i}^{*}\), a modified characteristic temperature difference \(\rm\Delta\Delta t^{*}\) can be derived analytically, which combines the hot and cooling water inlet temperature and chilled water outlet temperature:  
\[\rm\Delta\Delta t^{*}=t_{Di^{*}}(1-K_{1})-t_{Ai^{*}}(1-K_{2})+t_{Eo}.(1-K_{3}). \tag{6}\]  
Since the same coefficients \(\rm K_{1}\) to \(\rm K_{3}\) (with their respective physical meaning) are applied, a well-defined conversion factor \(\rm K_{Eo}=\left(1-(K_{1}-K_{2})\cdot K_{4}/W_{E}\right)^{-1}\) is available to transform Equations (3) and (4) into Equations (7) and (8) for use with \(\rm\Delta\Delta t^{*}\) instead of \(\rm\Delta\Delta t^{*}_{i}\).