## 2 Materials and Methods
### The New Approach to Individual Subsystems Models: General Aspects
This approach leads to a scheme middle way between pure explicit and pure implicit ones: using only known values (at time step \(t_{n}\)) of the subsystems output variables, i.e., \(y_{n+1}\), for feed-forward edges (matrix \(M^{\prime}\)) and \(y_{n}\) for feedback edges (i.e., outputs not yet calculated at time step \(t_{n+1}\): matrix \(M^{\prime\prime}\)), the discretized version of the matrix Equation (3b) becomes:  
\[v_{n+1}=M^{\prime}y_{n+1}+M^{\prime\prime}y_{n} \tag{8}\]  
Therefore, it is desired to partition the subsystems (i.e., the nodes of the graph representing the ISM) into sequential groups [56], each group containing the minimum number of subsystems. Only feed forward of information exists from each group to the groups that follow (or as many as possible whenever a not empty minimum feedback arc set exists, see discussion below). With this partitioning, the equations of each group can be easily solved simultaneously. The result is used to solve the equations of the following groups in the selected ordering. A minimum number of subsystems is desired in each group because this minimizes the number of equations that must be solved simultaneously. For complex ISM, an algorithm [57; 58] is necessary to accomplish the partitioning. No symbolic or other type of manipulation is required. The partitioning process is only the first step toward a solution and remains to solve each partitioned group's coupled equations. If feedbacks are not altogether cancelled, the corresponding variables are such that it is possible to execute all the subsystems of the group in the selected order if these variables had known values (e.g., from previous time steps or from measured variables in the case of mixed simulation-measurement real-time system). If a group contains non-linear subsystem, then a definitive solution is not generally possible, and iteration procedures must be used separately for each group to reduce the computational effort. These groups can be considered the proper subsystems, while their sub-subsystems become auxiliary ones (see the second example in Section 3.2). Solving ISM without taking advantage of the structure of the equations of such interconnected systems would become highly computationally inefficient. On the other hand, the smallest possible set of variables should be chosen for each partitioned group to reduce the computational effort. The choice of a minimum number of such variables can be shown as equivalent to a common problem in the theory of directed graphs, determining a minimum feedback arc set for each group graph. The following definitions, taken from [PERSON] [59], will help to make this identification:  
* A feedback arc set is a set of arcs that leave the resultant graph circuit free if removed for a directed graph. * A feedback arc set is a minimum if no other feedback arc set for that graph consists of a smaller number of arcs. * A sequential ordering \(R\) of a graph of \(N\) nodes is a permutation of the sequence \(1\ldots N\)In general, a pure sequential evaluation of the above type (i.e., empty feedback arc set) is possible only if the graph is free of directed loops. The sorting procedure should act in two steps. The first step is to partition the system graph nodes into groups as outlined above whenever possible. A preliminary simplification of the system graph is possible, reducing the complexity of the partitioning process: the source nodes and the arcs which leave them (external inputs \(U\)) can never be iteration variables, as they represent known quantities. Therefore, the source nodes can be considered as the first group of the partition. Similarly, external outputs (e.g., \"measured\" outputs) can be considered the last group (sink nodes). If possible, the partitioning procedure is applied to the subgraph, called the Reduced ISM (RISM), obtained by deleting source and sink nodes from the original graph.