# Controlled Fault Current Interruption Scheme for Improved Fault Prediction Accuracy
## 2 Short-Circuit Current Prediction Algorithm
### Half-Cycle Elimination Method  
In the short-circuit current expression, the period component is a sinusoidal function. According to the properties of sinusoidal functions, two sinusoidal functions differing in phase by half a period have opposite function values. \[\sin(\omega t+X_{2}+T_{1}/2)=-\sin(\omega t+X_{2}) \tag{4}\]  
Similarly, when the third and fifth harmonics differ in phase by half a fundamental cycle, their corresponding sinusoidal function values change to the opposite number. \[\left\{ \begin{array}{ll}\sin(3\omega t+X_{6}+T_{1}/2)=-\sin(3\omega t+X_{6} )\\ \sin(5\omega t+X_{8}+T_{1}/2)=-\sin(5\omega t+X_{8})\end{array} \right. \tag{5}\]  
The sinusoidal term can be eliminated by adding the short-circuit current to the data with a difference of half a cycle. If the superposition of the short-circuit current data with a difference of half a cycle is expressed in \(I(t)\), the non-periodic part of the superposition after addition can be deduced as follows:  
\[ \begin{array}{ll}I(t)&=i_{\mathrm{s}}(t)+i_{\mathrm{s}}(t+\frac{T_{1}}{2}) \\ &=X_{3}\mathrm{e}^{-\frac{t}{X_{4}}}+X_{3}\mathrm{e}^{-\frac{t+T_{1}/2}{X_{4 }}}\\ &=X_{3}\mathrm{e}^{-\frac{t}{X_{4}}}+X_{3}\left(\mathrm{e}^{-\frac{T_{1}}{2X_{ 4}}}\times\mathrm{e}^{-\frac{t}{X_{4}}}\right)\\ &=X_{3}\left(1+\mathrm{e}^{-\frac{T_{1}}{2X_{4}}}\right)\mathrm{e}^{-\frac{t} {X_{4}}}\end{array}  \tag{6}\]  
For ease of calculation, both sides of the equal sign of Equation (6) can be obtained by taking the logarithm after taking the absolute value at the same time, as follows:  
\[\ln(I(t))=\ln(|X_{3}|(1+\mathrm{e}^{-\frac{T_{1}}{2X_{4}}}))-\frac{t}{X_{4}} \tag{7}\]  
After transforming Equation (6) into the form of an exponential function to Equation (7) in the form of a primary function, RLS is applied to find the values of its parameters; this has a small amount of computation and occupied storage space and has real-time processing capability [21]. The formula for RLS is as follows:  
\[\left\{\begin{array}{ll}\hat{\theta}(k+1)=\hat{\theta}(k)+K(k+1)\left[y(k+ 1)-\varphi^{T}(k+1)\hat{\theta}(k)\right]\\ K(k+1)=\frac{P(k)\varphi(k+1)}{1+\varphi^{T}(k+1)P(k)\varphi(k+1)}\\ P(k+1)=\big{[}I-K(k+1)\varphi^{T}(k+1)\big{]}P(k)\end  
Then, the values of \(a\) and \(b\) can be found using RLS, and the values of the coefficients to be determined, \(X_{3}\) and \(X_{4}\), can be found after the derivation and substitution of \(T_{1}=20\)ms into Equation (7). \[\left\{ \begin{array}{l}X_{3}=\text{sign}(I(t))\cdot\frac{\theta}{1+\text{e}^{ b\times 40\Omega}}\\ X_{4}=-\frac{1}{b}\end{array} \right. \tag{10}\]  
where the \(\text{sign}(I(t))\) function is a symbolic function, representing the positive and negative of \(I(t)\), so as to complete the calculation of the parameters of the short-circuit current non-periodic component. The steps for calculating the periodic component are as follows: First, compute the DC component and subtract it from the sampled data to obtain the periodic component. Next, extract half a cycle of data. Then, take the opposite values of the extracted data to generate the latter half-cycle, forming a complete cycle. Finally, apply the fast Fourier transform (FFT) algorithm to calculate the amplitude and phase of the periodic component. After calculating the parameters of the short-circuit current, since the short-circuit current data are discrete points, it is not possible to directly obtain the time coordinates of the current over-zero point, so the time coordinates of the over-zero point are obtained by linear interpolation. As shown in Figure 2, when the half-cycle elimination method is used to calculate the short-circuit current, prediction failure occurs under special fault phase angles. In this case, the periodic component is basically correct, but the time constant in the non-periodic component is calculated incorrectly, leading to a rising value of the predicted short-circuit current. The reason for this fault is that it is too much affected by noise at special phase angles.