## 4 Discussion
### Mathematical Tools--A Further Research Avenue
However, all voltage sag matrixes, which are constructed according to current concept (including this paper), will be hard to understand through a topological prism, especially in the case of large power systems (in particular, ring topology) even with the best possible numeration. Therefore, let us introduce the improvement of the method on a typical BRELL transmission grid by designing schemes given in classical literature [59], and begin with the simplest three-phase fault case. The chosen schemes and their voltage sag vector in the case of a three-phase short-circuit occurring at node No. 4 are given in Figure 80. But the overall situation becomes much more understandable with the construction (proposal) of these vectors' topological extensions (which are also shown in the same figure): for example, in the case of Figure 80a, it is clear that fault location (node No. 4) is fed by nodes No. 1, No. 3, No. 5, and No. 7, and in the case of Figure 80b--by No. 1, No. 3, and No. 5. The centers of loops are depicted (represented) with inner gray cells. Let us continue the investigation of the proposed topological extension, and introduce a new analysis concept based on field theory. As has been concluded before, since voltage sags propagation strongly correlates with power-flow direction, the situation can be modeled as an elementary vector field, where the flux can be pointed either to a node or from a node. In normal operation mode, the divergence of each ordinary node will be equal to zero (according to [PERSON]'s current law), the divergence of generation nodes will be greater than zero (source of the field), and the divergence of end-user nodes will be lower than zero (sink of the field). In the case of a short-circuit, the divergence of the fault node will become lower than zero due to the flux leakage to the ground. According to the general definition, the divergence of a vector field at a point \(x_{0}\) is defined as a limit of the  
Figure 78: Detail coefficients of multilevel decomposition. Figure 79: Signal details displayed in superimposed mode, where darker colors correspond to the higher levels. closed surface integral when the volume of this surface shrinks to zero. Considering this, the definition for a cell (grid node) \(x_{0}\) of a voltage sag matrix is derived as follows:  
\[\text{div}\mathbf{F}|_{x_{0}}\triangleq\lim_{V\to 0}\frac{1}{|V|}\underset{J_{S}}{ \overset{\oint}{\oint}}(\mathbf{F},d\mathbf{S})=\frac{\partial F_{x}}{\partial x }+\frac{\partial F_{y}}{\partial y}+\frac{\partial F_{z}}{\partial z}\Rightarrow \text{div}\mathbf{S}|_{x_{0}}\triangleq\sum_{i}\mathbf{S}_{i}+\sum_{j}\mathbf{ S}_{j}\,, \tag{39}\]  
where \(\mathbf{F}\)--the continuously differentiable vector field composed of \(F_{x}\), \(F_{y}\), and \(F_{z}\); \(V\)--volume with boundary \(S\); \((\mathbf{F},d\mathbf{S})\)--the scalar product of \(\mathbf{F}\) and the infinitesimal element \(d\mathbf{S}\) of vector area \(\mathbf{S}\) (defined as the unit normal \(\mathbf{\hat{n}}\) scaled by the area, i.e., \(d\mathbf{S}=\mathbf{\hat{n}}d\mathbf{S}\)); \(\mathbf{S}\)--the vector field of the apparent power-flow composed of \(S_{x}\); \(\mathbf{S}_{i}\)--the apparent power-flow flowing to \(x_{0}\); \(\mathbf{S}\)--the apparent power-flow flowing from \(x_{0}\). Another field operator--curl (also known as a rotor)--could also be useful as a supplementary parameter in the gray cells of the topological extension (see Figure 80). It can show whether the electric current flow forms a closed loop (circulation), which can also be important for the prediction of voltage sag behavior. It is obvious that the existence of circulation in all rings is limited by [PERSON]'s current law, hence it is possible only in some--probably, in the most important (critical) rings. Thus, in normal operation mode, the current can either form a closed loop (either clockwise or counterclockwise) or not--which means a total of three different conditions.