## 2 Materials and Methods
### The New Approach to Individual Subsystems Models: General Aspects  
As briefly inferred in the previous section, when dealing with ISMs, the structure of the problem can be directly defined by its model graph where the i-th subsystem, \(i=1\ldots N\), can be described by a subset of DAEs as in (7), here repeated for readability:  
\[\Phi_{i}\big{(}t,x_{i}(t),\dot{x}_{i}(t),y_{i}(t),v_{i}(t),u_{i}(t),p_{i}\big{)} =0. \tag{7}\]  
All the subsystems are interconnected to each other by connecting their outputs to corresponding inputs of other subsystems. A graph \(G(V,E)\) represents this structure where \(V\) is the vertex set of the \(N\) nodes, and \(E\sqsubseteq V\times V\) is the edge set (directed edges). Each node \(N_{i}\left(N_{i}\in V,\ i=1\ldots N\right)\) corresponds to a single subsystem. A directed edge \(\left(N_{i},N_{j}\right)\in E\) exists if a scalar output of the subsystem \(N_{i}\) is connected to a scalar input of subsystem \(N_{j}\). As said above, in this study, without loss of generality, we do not consider self-loops; that is, every subsystem \(N_{i}\) is assumed not to be connected to itself, i.e., \(\left(N_{i},N_{i}\right)\
otin E\). Each subsystem \(N_{i}\) is assumed to admit a DAEs representation as in Equation (7). Thus, the graph incidence matrix can be considered the incidence matrix of the system of DAEs. The subset of DAEs (7) of each subsystem can be solved using the previously described methods applied directly to the subsystem's definition. This method allows joining different methods according to the context (e.g., causal or acausal methods, general numerical methods, etc.) to which each subsystem belongs. This way, we describe each subsystem as representing independently of any other subsystem in the ISMs. The great advantage is that we can substitute it whenever needed with a different representation (i.e., different DAEs for that subsystem and different solution methods), making maintenance and evolution of the whole ISM easy to implement. Another advantage is that we do not need to know the content of each subsystem. It is sufficient to have a black box with known inputs and outputs: this becomes particularly helpful when we ask an external firm to model a particular component without necessarily asking for open access to its complete content. This representation belongs to the ISM framework more generally than previously conducted, as each subsystem is modelled and solved individually and independently on other models. The graph structure coordinates all these solvers to determine the whole ISM's dynamical evolution efficiently. Therefore, one of the main novelties of our approach is that each subsystem (or a subgroup of them as shown below) could be solved separately in a proper order based on the structure of the graph, making it possible for the following procedure that simplifies ISM simulation. Precisely, each output of a subsystem (or group of them: [PERSON]. [PERSON] et al. [55]) becomes the input of the next subsystem in the chosen ordering. Matrix \(M\) can be structured as a block matrix to better describe what just said, each block representing a node (subsystem) and corresponding connections of the subsystem to another one, then represented as the sum of a strictly block lower triangular matrix \(M^{\prime}\), and a strictly block upper triangular matrix \(M^{\prime\prime}\). This way, if no self-loops are considered then it is the diagonal blocks that are null and not the individual diagonal elements of \(M\). The same obviously occurs for both \(M^{\prime}\) and \(M^{\prime\prime}\): \(M^{\prime}\) represents all the feed-forward block connections in the considered node ordering, while \(M^{\prime\prime}\) represents all feedback block connections.