## 3 Results and Discussion
### Numerical Tests on Linear Time-Invariant Individual System Models
#### 3.3.1 Mathematical Models of Each Machine in the Chosen Plant System  
Boiler (B) and Corresponding Output Pipe  
The water coming into the boiler from the WWT is heated by burning methane. The governing equation is  
\[BoilerB:\ T_{Bo}=(1-\eta_{PB})\bigg{(}T_{Bi}+\frac{\eta_{B}h_{CH_{4}}}{\dot{m}_ {Bi}c_{w}}\dot{m}_{CH_{4}}\bigg{)}+\eta_{PB}T_{E}\]  
where \(\dot{m}_{B},T_{Bi}\) and \(T_{Bo}\) are, respectively, the mass flow rates and the temperatures of the water entering the boiler and leaving the output pipe, \(\eta_{PB}\) is the pipe heat exchange efficiency (the smallest possible if the pipe is well insulated; this is true for all pipes and the same efficiency is used here), \(\eta_{B}\) is the boiler efficiency, \(h_{fCH_{4}}\) is the Lower Heating Value (LHV) of methane, \(c_{w}\) is the specific heat of water, and \(T_{E}\) is the external temperature. In terms of internal input \(v_{1}=T_{Bi}\), external inputs \(u_{1}=T_{E}\) and \(u_{2}=\dot{m}_{CH_{4}}\), and output \(y_{1}=T_{Bo}\) (in this case there are no ODEs, so no state variable is needed; anyway is useful to add a dummy constant state \(x_{1}=x_{1}^{(0)}\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{\dot{ \dot{\dot{\dot{\dot  
\[H\text{P}:\ y_{21}=\eta_{H\text{P}f}x_{2}+\Big{(}1-\eta_{H\text{P}f}\Big{)}u_{3}.\]  
\[H\text{WWT output pipe}:\ y_{22}=(1-\eta_{PHP})x_{2}+\eta_{PH}u_{1}.\]  
Warm Water Tank (WWT) and Corresponding Output Pipes  
In the WWT (of total mass \(M_{WT}\) and temperature \(T_{WT}\)), there are two streams of water, respectively, on HP and RP sides; the water coming from the HP side is cooler than water in the WT, so it is a cooling fluid for the WWT, while the water coming from the RP is warmer than the water in the WWT, and it is a heating fluid for the WT. Differently from the HWT and the CWT, the WWT is considered not perfectly insulated. The governing equations are  
\[\text{WWT}:\frac{d(T_{WT})}{dt}=\frac{\dot{m}_{H\text{WWT}}}{M_{WT}}T_{HPWTi}+ \frac{\dot{m}_{B\text{WWT}}}{M_{WT}}T_{RPWTi}-\frac{\dot{m}_{W\text{T}}}{M_{WT }}T_{WT}-\frac{\dot  
In terms of internal input \(v_{4}= \begin{bmatrix}T_{RPCi}\\ T_{RPEi}\end{bmatrix} \), external inputs \(u_{1}=T_{E}\) and \(u_{4}=\dot{L}\) and outputs \(y_{4}= \begin{bmatrix}T_{RPCo}\\ T_{RPEo}\end{bmatrix} \) (in this case there are no ODEs, so no state variable is needed; anyway, it is useful to add a dummy constant state \(x_{4}=x_{4}^{(0)}\)) from the above equations become  
\[\textit{Dummy state equation}:\ \dot{x}_{4}=0.\]  
\[\textit{Condenser side}:\ y_{41}=(1-\eta_{PC})v_{41}+\frac{(1+COP)}{\dot{m}_{ RPC}c_{w}}u_{4}+\eta_{PC}u_{1}.\]  
\[\textit{Evaporator side}:\ y_{42}=(1-\eta_{PE})v_{42}-\frac{COP}{\dot{m}_{ RPE}c_{w}}u_{4}+\eta_{PE}u_{1}.\]  
Cold Process (CP) and Output Pipe  
In the CP, the cold water refrigerates the CPf (transferred heat: \(\dot{Q}_{CP}\)). This water is, therefore, heated (it resides in a tank of the total mass \(M_{CP}\) and temperature \(T_{CP}\)) and it leaves the CP going to the output pipe. The governing equations are  
\[\textit{CWT}:\ \frac{dT_{CP}}{dt}=-\frac{\dot{m}_{CP}}{M_{CP}}T_{CP}+\frac{ \dot{m}_{CP}}{M_{CP}}T_{CPi}+\frac{\dot{Q}_{CP}}{M_{CP}c_{w}}.\]  
\[\textit{CP}:\ \eta_{CPf}\dot{m}_{CPf}c_{PCPf}\Big{(}T_{CP}-T_{CPfi}\Big{)}=- \dot{Q}_{CP}=\dot{m}_{CPf}c_{PCPf}\Big{(}T_{CPfo}-T_{CPfi}\Big{)}.\]  
\[\textit{CWT output pipe}:\ T_{CPo}=(1-\eta_{PCP})T_{CP}+\eta_{PCP}T_{E}.\]  
where \(\dot{m}_{CP}\), \(T_{CPi}\) and \(T_{CPo}\) are, respectively, the mass flow rate and the temperatures of the water entering the CP and leaving the output pipe (RP side), \(\dot{m}_{CPf}\), \(T_{CPfi}\) and \(T_{CPfo}\) are respectively the mass flow rate and the temperatures of the CPf entering and leaving the CP (CPf side), \(\eta_{PCP}\) the output pipe heat exchange efficiency (RP side) and \(\eta_{CPf}\) CP heat exchanger efficiency (CPf side). In terms of state \(x_{5}=T_{CP}\), internal input \(v_{5}=T_{CPi}\), external inputs \(u_{1}=T_{E}\)_and_\(u_{5}=T_{CPfi}\) and outputs \(y_{5}=