# Dynamical Networks Modelling Applied to Low Voltage Lines with Nonlinear Filters
## 4 Results, Discussion and Perspectives
### Discussions, Perspectives and Conclusions  
Nonlinear filter behaviour in the real-world is difficult to predict and we can only assume a range of possible responses. However, a simulation model can have better results if we compute the signals dynamically and therefore we have a narrow range of results to work with. Looking to the varistor plots in Appendix A, Appendix B, and Appendix C, clamping zones vary according to the physical structure of each component, and with the processes running on top of these structures. The complete communication system, i.e., line, sender, receiver, filter, and interferences, create a complex adaptive system. Comparing the three different varistors used in our simulations, one can realize the slight result differences when each filter is operating, even when they have the same nominal voltage clamping. Additional features, as the inherent capacitance, can affect the outcome (in these cases, positively if we consider they can smooth the surges). Another interesting result happens when lightning interference strikes the system (not unusual at all). In this case, the curve after the varistor filter slowly varies, decaying as the strike decays, until the point where the nonlinear filter suddenly becomes linear again (see the plots at the right side at Figures 11-13). This happens to agree with the transition from the varistor nonlinear region to the linear region, as shown in Figure 10, region B. The case study presented is a one-dimensional model and does not show signals in the surroundings of the line. If the line is an antenna, for example, how can we predict the near- and far- electric and magnetic fields? The one-dimensional model can be extended to two-dimensional or three-dimensional models, applying the same process of reflected and incident signals travelling through the connections. Figure 14 illustrates these models considering Cartesian coordinates, which are useful for electric wave propagation--several models were developed to solve these problems and presented in [8; 9; 12]. If the problem is not Cartesian coordinate dependent, the dimension of the resultant matrix will consider each node degree. Each reflection will be the result of a function, dependent on the node characteristics. This process was used when we modelled the varistor, considering its internal nonlinear characteristics to give a response for each incident signal. The nonlinear filter was applied to the line to guarantee communication sustainability (the communication robustness, in other words). The varistor adds to a specific node a transitory structural characteristic: if a threshold of instability is achieved, i.e., there is an unacceptable variation in the process, the internal parameters of the node changes to return to a \"normal\" condition, correcting the instability. The acceptable condition, for this case study, means a range of typical signals recognizable by both sender and receiver. It is interesting to notice that the nonlinear component transitorily changes the internal structure of a node, changing its features to provide sustainability for the process. Here we  
Figure 14: One, two and three-dimensional grids, where an excited node (input) sends signals to its neighbours, starting an iterative time-step incident/reflection process. These examples show static network structures, with dynamical processes running on top of them. see the two factors mentioned before: (1) the physical structure and (2) the processes running on top of it. Figure 15 illustrates these two components, structure and processes. The processes corresponding to the dynamic component, and the structure is the static counterview. Strictly, the structure is not static but is considered like this in the model when we freeze one moment, \(t\), and run the processes on top of it to connect \(t\) to \(t+1\). At \(t+1\) we could, or not, have a new structure, and again the process re-starts. The idea in this work is that system robustness is simultaneously dependent on the structure robustness and the process robustness.