## 3 Results and Discussion
### Performance Analysis  
Now we are ready to compare the other methods (Full implicit and Full explicit schemes) regarding stability conditions when unstable subsystems are present (but, of course, stabilized by suitable feedback), the possibility of a block substitution, and the possibility of choosing a particular node ordering in solving each subsystem.  
In particular, we have seen that even if Full Implicit schemes are unconditionally stable (see Table 2), they do not allow the substitution of a subsystem. We may wish to substitute the unstable subsystem B2 of the second example in the previous paragraph,  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Values** & **Parameters** & **Values** \\ \hline \(\hat{m}_{H}\) & 2 kg/s & \(c_{w}=c_{CPf}=c_{HPf}\) & 4.187 kJ/kgK \\ \(\hat{m}_{W}\) & 2 kg/s & \(h_{fd4}\) & 50,000 kJ/kg \\ \(\hat{m}_{C}\) & 2 kg/s & \(\eta_{B}=\eta_{CPf}=\eta_{HPf}\) & 0.98 \\ \(M_{HP}\) & 100 kg & \(COp\) & 3 \\ \(M_{WT}\) & 50 kg & \(T_{c}\) & 25\({}^{\circ}\)C \\ \(M_{CP}\) & 200 kg & \(\hat{m}_{CH4}\) & 0.0001 kg/s \\ \(\eta_{P}\) & 0.01 & \(\hat{L}\) & 43 kW \\ \(U_{WT}\) & 0 kW/m\({}^{2}\)K & \(A_{WT}\) & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: Constants used in the simulation.  
Figure 5: Trend of the temperatures in the Hot and Cold Processes: analytical and simulated results are compared.  
e.g., only modifying some internal parameters, with the stable system B2 of the first example. However, this needs to change the iterative algorithm as it must consider all the equations together that change at every substitution. A fully Explicit algorithm indeed allows both node ordering (but with no preference for optimality) and block substitution as subsystems are solved separately. However, it becomes unstable in the presence of unstable subsystems as feedback is entirely cancelled. On the other hand, the proposed new method allows both optimal (suboptimal) ordering, speeding up the solving algorithm if compared with Full Implicit methods, and block substitution as done in the second example where the two subsystems B1 and B2 are substituted with the block B12 without altering the global solution algorithm.  
Therefore, no symbolical manipulation is needed to obtain the state variables and convert a DAE model into an iterative numerical model, nor to solve the system of DAEs as a whole. As said the most common and effective approach consists in preliminary determining an ordering in the subsystem's execution. However, they are mainly based on the system's nature rather than on its structure. On the other hand, when dealing with ISMs, the structure of the problem can be directly defined by its model graph, and this is what is achieved with the proposed technique.  
Finally, the following steps allow to schematize the procedure:  
1. decompose the entire system into individual subsystems (all stable when studied individually);
2. identify the status, input, and output variables of each single subsystem;
3. model each individual subsystem, including an individual numerical solver;
4. identify the connections between the output variables of each subsystem with the input variables of other subsystems;
5. represent in a graph all the subsystems (nodes) and connections (arcs);
6. search for the optimal (or suboptimal) order of the nodes of the graph;  
solve each subsystem in the order identified, transferring the values of the output variables to the input variables to which they are connected.