*3.1. System Description*

The schematic diagram is shown in Figure 5, which is a typical primary-constant and secondary-variable chilled water system. The specifications of its main components are presented in Table 1. Four critical temperatures are controlled by PI controllers, namely the supply cooling water temperature (*Tscw*), the supply chilled water temperature at the primary (*Tschw*,*prm*) and secondary loops (*Tschw*,*sec*), and the supply air temperature (*Tsa*). *Tscw* is controlled through varying the cooling tower fan frequency; *Tschw*,*prm* is controlled by modulating the refrigerant flow rate; *Tschw*,*sec* is controlled by modulating the water flow rate; *Tsa* is controlled by AHUs. The set-points of the above four critical temperatures are optimized and reset in real-time to achieve the best energy efficiency.

**Figure 5.** Schematic of the air-conditioning (AC) system.

**Table 1.** Specifications of the main components.


Note: *Ma* is the air flow rate; *Mw* is the water flow rate; *ev* and *cd* stand for evaporator and condenser respectively.

#### *3.2. Optimal Control Problem Formulation*

For all-electric AC systems without thermal storage, the minimization of the energy use can be simplified as the minimization of the system total power at each time instance [16]. In this case study, the decision variables that significantly affect the energy efficiency of the system operation are *Tscw*, *Tschw*,*prm*, *Tschw*,*sec* and *Tsa*. Thus, the real-time optimal control problem is formulated as

$$\left(T^\*\_{\text{scrw}}, T^\*\_{\text{schw}, \text{prmu} \prime}, T^\*\_{\text{schw}, \text{scc} \prime}, T^\*\_{\text{sa}}\right) = \arg\min \, P\_{\text{sys}, \text{tot}} \Big(T\_{\text{scrw} \prime}, T\_{\text{scmu}, \text{prmu} \prime}, T\_{\text{scbnu}, \text{scc} \prime}, T\_{\text{sa}}\Big) \tag{5}$$

where *Psys*,*tot* = *Pch*,*tot* + *Pct*,*tot* + *Ppump*,*tot* + *PAHU*,*tot*.

The system total power can be written as a function of these four decision variables

$$P\_{sys,tot} = f\left(T\_{scw\nu} \, T\_{schw,prmu\nu} \, T\_{schw,sc\nu} \, T\_{sa}\right) \tag{6}$$

where *f*(·) is always a nonlinear function. The decision variables are limited in their feasible ranges, which are treated as operational constraints (shown in Equations (7)–(10)).

Table 2 shows the values of the operational constraints.

$$T\_{scw,lower} \le T\_{scw} \le T\_{scw,upper} \tag{7}$$

$$T\_{schw,prm} \le T\_{schw,prm} \le T\_{schw,prm,upper} \tag{8}$$

$$T\_{\text{schw}, \text{scc}} \le T\_{\text{schw}, \text{scc}} \le T\_{\text{schw}, \text{scc}, \text{upper}} \tag{9}$$

$$T\_{sa,lower} \le T\_{sa} \le T\_{sa,upper} \tag{10}$$

**Table 2.** Operational constraints of the case AC system [11].

