# Air Conditioning Load Forecasting and Optimal Operation of Water Systems
## 2 Methods
### Operation Optimization of Air Conditioning Water System
#### 2.3.3 Objective Function Constraints  
1. Water chiller load rate is given by Equation (8). \[0.1\leq PLR\leq 0.9\] (8)
2. Water chiller temperature is given by Equations (9)-(12). \[7\ ^{\circ}\mathrm{C}\leq T_{w0}\leq 12\ ^{\circ}\mathrm{C}\] (9) \[15\ ^{\circ}\mathrm{C}\leq T_{ci}\leq 35\ ^{\circ}\mathrm{C}\] (10) \[T_{w0}\leq T_{wi}\] (11) \[T_{ci}\leq T_{co}\] (12) where \(T_{w0}\) is the outlet temperature of chilled water, \({}^{\circ}\mathrm{C}\); \(T_{ci}\) is the inlet temperature of cooling water, \({}^{\circ}\mathrm{C}\); \(T_{ci}\) is the outlet temperature of cooling water, \({}^{\circ}\mathrm{C}\); and \(T_{wi}\) is chilled water inlet temperature of water chiller, \({}^{\circ}\mathrm{C}\).
3. Chilled and cooling water flow constraints are given by Equations (13) and (14). \[0.4F_{cp,0}\leq F_{cp,all}\leq 1.2F_{cp,0}\] (13) \[0.4F_{wp,0}\leq F_{wp,all}\leq 1.2F_{wp,0}\] (14) where \(F_{cp,0}\) is the rated flow of cooling water system at 600 m\({}^{3}\)/h, and \(F_{wp,0}\) is the rated flow of chilled water system at 400 m\({}^{3}\)/h.
4. Heat-exchange constraints between equipment are given by Equations (15)-(18). \[Q_{t}=Q_{ch}+P_{ch}=cm_{F_{cp,all}}(T_{ci}-T_{co})\] (15) \[F_{cp,all}=\frac{Q_{ch}+P_{ch}}{c\rho\ (T_{ci}-T_{co})}\] (16) \[Q_{ch}=cm_{F_{wp,all}}(T_{wi}-T_{iw})\] (17) \[F_{wp,all}=\frac{Q_{ch}}{c\rho\ (T_{wi}-T_{w0})}\] (18) where \(Q_{t}\) is the heat dissipation of cooling tower, kW-h; \(Q_{ch}\) is the refrigerating capacity of the water chiller, kW-h; \(P_{ch}\) is the power of water chiller, kW; \(c\) is the specific heat of water, ignoring the change of specific heat caused by the change of water temperature, at 4.18 kJ/(kg \({}^{\circ}\)C); and \(\rho\) is the density of water, at 1000 kg/m\({}^{3}\), ignoring the change of water density caused by the change of water temperature.
5. Optimal cooling amplitude of cooling tower is given by Equation (19). \[t_{co}-t_{wb}=3.5\ ^{\circ}\mathrm{C}\] (19)where \(t_{co}\) is the outlet temperature of cooling water, \({}^{\circ}\)C, and \(t_{wb}\) is the outdoor air wet-bulb temperature, \({}^{\circ}\)C.  
After determining the objective function and constraints of water system energy consumption optimization, the operation parameter optimization problem of air conditioning water system can be transformed into a constrained nonlinear optimization problem with the lowest operation energy consumption of air conditioning water system under the conditions of disturbance variables and constraints. The mathematical model of the optimization problem is given by Equation (20):  
\[\left\{\begin{array}{c}\begin{array}{c}minf(x)\\ s.t.\ c(x)\leq 0\\ c_{eq}(x)=0\\ AX\leq b\\ A_{eq}(x)=b_{eq}\\ lb\leq x\leq ub\end{array}\right. \tag{20}\]  
where \(f(x)\) is the objective function; \(x\) is the optimization vector; \(c(x)\) and \(c_{eq}(x)\) are column vectors composed of nonlinear functions; \(A\) is the constraint coefficient matrix of linear inequality; \(b\) is the right end vector of linear inequality constraint; \(A_{eq}\) is a linear equality constraint coefficient matrix; \(b_{eq}\) is the right end vector of linear equality constraint; and \(lb\) and \(ub\) are the lower and upper bounds of the optimization variable \(x\), respectively.