Chilled-water building cooling systems12 operate as indicated in the following sketch.

Water flows at a rate of F1 gallons per minute around the lower loop, entering the chiller at temperature T1,1 and being cooled to temperature T1,2 before passing through the building. An upper loop flowing at rate F2 gallons per minute absorbs heat within the chiller in a separate water stream, and passes it through an outdoor cooling tower to reduce its temperature from T2,1 to T2,2.
Tower output temperature T2,2 is a nonlinear function f1 of T2,1, F2, the tower fan speed S, and the ambient air temperature T0. The lower loop must absorb a load H of heat within the building (measured in water temperature difference times flow). Similarly, the heat exchanged between the two loops should balance within the chiller after adjusting for extra heat generated by the unit and its pumps, which is a constant multiple k of the electrical energy they consume. That energy is, in turn, a nonlinear function f2 of all four temperatures and both flow rates. Energy consumed in the cooling tower is another nonlinear function f3 of fan speed. On any given day, temperature T0, heat load H, constant k, and all 3 functions will be known, but other system variables can be controlled within ranges [Ti,j, Ti,j], [Fi, Fi], and [S, S ], respectively. Formulate an NLP model to decide how to operate such a system at minimum total electrical consumption.

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tower pump chiller condenser evaporator pump building