## 3 Results and Discussion
### Numerical Tests on Linear Time-Invariant Individual System Models  
As the first test, we consider a stable linear ISM made of five stable linear subsystems connected as in Figure 2 where the subsystems (nodes) are numbered casually.  
previous paragraph about matrix \(K^{\prime}\), to the fact that the only-feed-forward system (i.e., with feedback arcs deleted) is unstable whenever we are involved with unstable subsystems. The feedback from the output \(y_{21}\) of subsystem B2 to input \(v_{12}\) of subsystem B1, and subsystem B1 itself stabilizes system B2: deleting this feedback it becomes unstable. Nevertheless, this is simply due to a modelling error: in fact, subsystems B1-B2 must be modelled as a one-subsystem as shown in Figure 3: system B12 represents a proper subsystem while subsystems B1 and B2 are only auxiliary ones used to simplify the graphical representation of B12 but not necessary in the whole ISMs.  
Inserting B12 in the ISM of Figure 2 instead of the couple B1-B2, the best ordering becomes (B3, B12, B4, B5). Matrix \(A\) becomes  
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{B3} & \multicolumn{3}{c}{B12} & \multicolumn{3}{c}{B4} & B5 \\  & \(-3\) & \(2\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) & \(0\) \\  & \(2\) & \(-3\) & \(0\) & \(0\)  
fluid (HPf). Both the CPf and the HPf are stored in insulated Cold and Hot Water Tanks (CWT, HWT). Moreover, warm water from RP and HP is stored in a Warm Water Tank (WWT) and recirculated: a control algorithm is supposed to ensure that the temperature \(T_{WT}\) of the WWT is always below the temperature of the refrigerant entering the expansion valve in the RP. As said the heat waste is used to pre-heat the warm fluid so an auxiliary Boiler is needed between the water tank and a final hot process to rise the water temperature to desired final temperature. In Figure 4, an outline of the chosen plant is shown with the primary input-output variables, pipes and pumps included in the model description below.  
External inputs:  
* \(T_{E}\): the external environment temperature; it is repeated for all subsystems,
* \(\dot{m}_{HPf}\) and \(\dot{T}_{HPf}\): flow rate and temperature of the HP fluid entering in the hot process,
* \(\dot{L}\): the power furnished to the refrigerant compressor in the refrigeration plant (remember that we are supposing a vapor compression cycle),
* \(\dot{m}_{CPf}\)\(T_{CPf}\): flow rate and temperature of the CP fluid entering in the cold process,
* \(\dot{m}_{CH_{4}}\): the methane flow rate feeding the Boiler (B). External outputs:
* \(\dot{T}_{HPfo}\): flow rate and temperature of the HP fluid leaving the hot process,
* \(T_{CPfo}\): flow rate and temperature of the CP fluid leaving the cold process.  
All the processes are interconnected through directed lines, representing mass and energy flows. Thus, the whole plant can be represented as a directed graph where the processes are the nodes and flows are the directed edges of the graph. Output pipes are included in the model equations.  
Moreover, all tanks (HWT, WWT, and CWT) are controlled such that the mass in each tank remains constant during the whole process. When the hypothesis of constant level in  
Figure 4: Outline of the simulated plant.  
the three tanks is removed, some of the process equations become non-linear Differential Algebraic Equations (DAEs).  
Now we are ready to assess equations and process variables for each subsystem independently on how they are connected.