# Fault Sensing Using Fractal Dimension and Wavelet
## 3 The Improved Fractal Box Dimension (IFBD) Recognition Algorithm
### Definition of the Improved Fractal Box Dimension  
The traditional box dimension \(Dim\) (\(S_{r}\)) of the point set \(S_{r}\) in a linear space \(R^{n}\) is  
\[Dim\ (S_{r})=\lim_{n\rightarrow\infty}\frac{lnN_{n}\ (S_{r})}{ln2^{n}} \tag{3}\]  
where \(n\) is the length of a side of a square box which covers the point set \(S_{r}\). \(N_{n}(S)\) represents the minimum number of boxes containing \(S_{r}\).  
By using the above approximation method, the box dimensions of the fault components of phase \(A\), phase \(B\), and phase \(C\) are calculated in the condition of the phase \(A\) earth fault shown in Figure 1. \(Dim\ (S_{A})=1.56030\), \(Dim\ (S_{B})=1.57527\), \(Dim\ (S_{C})=1.54622\).  
For the curves in a plane, their box dimensions always range from one to two. To make the fault classification easier, we can enlarge the distances between the different classes of the 10 types of the short circuit faults. Based on experiments, we defined the improved box dimension \(F\) as follows.  
\[F=\tan\frac{Dim(S)}{E(Dim^{*})} \tag{4}\]  
where \"tan\" represents the tangent function, and \(E(Dim^{*})\) is the expectation of the traditional box dimension \(Dim\) of the phase current in the normal state, and it can be written as follows.  
\[E(Dim^{*})=\frac{1}{m}{\sum_{r=1}^{m}\ Dim\ (S_{r})} \tag{5}\]  
where \(S_{r}\) is the current signals of the power cable in normal state, \(r=1,2,\cdots,m\), and \(m\) is a positive integer greater than three.  
According to Equation (4), we have the improved box dimensions \(F_{A}\), \(F_{B}\), and \(F_{C}\) of the fault components of the three phase currents.  
\[F_{A}=\tan\frac{Dim(S_{A})}{E(Dim^{*})} \tag{6}\]  
\[F_{B}=\tan\frac{Dim(S_{B})}{E(Dim^{*})} \tag{7}\]  
Figure 2: (**a**) Three-phase currents of a single-phase \(A\) grounded fault; (**b**) Fault component of phase \(A\); (**c**) Fault component of phase \(B\); (**d**) Fault component of phase \(C\).  
\[F_{C}=\tan\frac{Dim(S_{C})}{E(Dim^{*})} \tag{8}\]  
where \(Dim(S_{A})\), \(Dim(S_{B})\), and \(Dim(S_{C})\) are the traditional box dimensions of the currents of phase \(A\), \(B\), and \(C\), respectively. \(E(Dim^{*})\) is the expectation of the traditional box dimension \(Dim\) of the phase current in the normal state. \"tan\" represents the tangent function.  
For the 10 detailed classes of the short circuit faults, the fault features using the traditional box dimension and the improved box dimension are shown in Figure 3a,b respectively. The horizontal axis represents the short circuit faults which include the 10 classes of short circuit faults. \(AG\), \(BG\), and \(CG\) are the single-phase short circuit faults. \(AB\), \(BC\), and \(AC\) are the phase-between short circuit faults. \(ABG\), \(BCG\), and \(ACG\) are the double-phase short circuit faults. \(ABC\) is the three-phase short circuit fault. The vertical axis is the fault feature values. The blue color represents phase \(A\), the green color represents phase \(B\), and the red color represents phase \(C\).  
From Figure 3, it is clear that the traditional box dimension features of the 10 classes are relatively crowded. Compared to the traditional box dimension features, the improved box dimension features of the 10 classes have larger distances among the single-phase fault and the double-phase fault, as well as the three-phase fault. Furthermore, the improved box dimension features have larger distances among the different single-phase faults of \(A\), \(B\), and \(C\). Obviously, the improved box dimension feature is better for short circuit fault recognition of the online power cable than the traditional box dimension feature.