Short-Circuit Fault Current Parameter Prediction Method Based on Ultra-Short-Time Data Window

Abstract

The prediction of short-circuit current parameters is essential for the adoption of short-circuit fault limiting techniques and the reliable cut-off of circuit breakers. In order to quickly and accurately predict the short-circuit current waveform parameters, a short-circuit fault current prediction method based on ultra-short-time data windows (UDWs) is proposed. First, a mathematical model for describing short-circuit faults is constructed and the characteristics of short-circuit currents are analyzed. Then, the principle of the UDW method for predicting short-circuit current waveform parameters is derived, the correctness of the principle is verified by setting-up an ideal signal through simulation, and the exponential and linear expressions fitted to the curve are analyzed and compared with the improved half-wave Fourier method for predicting current parameters. Finally, trend filtering technology is proposed to eliminate high-frequency interference and white noise interference. The results show that the ultra-short-time data window method can quickly and accurately predict the short-circuit current waveform parameters, where the exponential expression is a better fit to the waveform, and the trend filtering technique enables the elimination of high-frequency and white noise interference in the initial stages of prediction.

1. Introduction

Of all electrical system accidents, short-circuit faults are the most serious in terms of damage. The energy released by the short-circuit arc and the electric power generated by the short-circuit current will cause serious damage to electrical equipment, even explosion or fire [1,2]. In addition, the increasing short-circuit current will lead to the short-circuit at the outlet near the transformer, causing the winding deformation of the main transformer and other faults, leaving the transformer in a sub-health operation state, laying a great hidden danger for the safe operation of the power grid [3,4]. At present, the main way to deal with short-circuit faults is to extract effective information from the transient process to achieve early diagnosis of short-circuit faults, and then take certain measures to limit them [5,6,7]. If the pattern of short-circuit current changes can be accurately predicted within the shortest possible time after the occurrence of a short-circuit fault, the correct technical measures against short-circuit faults can be taken precisely on the basis of such predictions and the capacity of the different types of anti-short-circuit fault measures already available in the system to obtain the best anti-short-circuit fault effect [8,9]. When a short-circuit fault occurs in the line, the current may rise dozens or even hundreds of times in a few milliseconds, seriously endangering system safety [10,11]. Therefore, how to quickly and accurately predict the short-circuit current waveform parameters after the occurrence of a short-circuit fault, and take the best anti-short-circuit fault measures to minimize the damage caused by short-circuit faults and ensure the safe and stable operation of the power system is of great importance [12,13,14]. However, there are often high-frequency interference and white noise interference in the actual power system, which will have a greater impact on the accuracy of the prediction [15,16]. Therefore, it is necessary to fully consider these interfering factors, and strive for rapid identification while ensuring the accuracy of fault prediction.

The prediction of short-circuit current parameters is mainly focused on the prediction of the short-circuit current amplitude, while there is little literature on the identification of all parameters of the current waveform. In [17], a two-dimensional cloud model was used to predict the peak, but the two-dimensional cloud model is an uncertain prediction. Although the average relative error is low, the individual prediction error is greater than 5%, which is not conducive to practical application. In [18], the wavelet transform multi-resolution fast algorithm (MALLAT) was used, but this algorithm did not consider the attenuation of the DC component and the harmonic component of the current after the fault, and there will be a large deviation in the calculation of the linear relationship constant. In [19,20,21], neural network algorithms were used to predict peaks, but training required significant amounts of data, resulting in long training times and a tendency to fall into local optima. In [22,23], least squares was used to identify short-circuit current parameters, but this algorithm did not take into account the influence of harmonic factors, causing large errors in the waveform fitting process. In [24,25], the improved half-wave Fourier algorithm was used to fit the short-circuit current waveform, but the algorithm guarantees accuracy on the premise that the signal contains only odd harmonics, but when the content of even harmonics in the signal increases, the error in the algorithm’s calculation results also increases significantly.

In order to meet the requirements of both speed and accuracy, this paper proposes a short-circuit fault current parameter prediction method based on ultra-short-time data windows (UDWs). First, a mathematical model describing the short-circuit fault is established in Section 2. In Section 3, the principle of the UDW method used to predict short-circuit current waveform parameters is described in detail. In this section, the exponential and linear expressions in the UDW method are compared by simulation and analyzed in comparison with the improved half-wave Fourier method. In Section 4, trend filtering technology is proposed to eliminate high-frequency interference and white noise interference. A single-phase small-capacity prototype model is used to simulate a single-phase short-circuit fault for experimental verification, and the test results are compared and analyzed with the simulation results to further validate the feasibility and accuracy of the proposed UDW prediction of short-circuit current parameters in Section 5. Finally, conclusions are summarized in Section 6.

The results show that the UDW method proposed in this paper can predict the short-circuit current waveform parameters quickly and accurately, and both the exponential and linear expressions fit well compared to the actual waveform, with the exponential expression fitting a curve closer to the actual curve. The proposed trend filtering technique can eliminate the high-frequency and white noise interference at the early stage of prediction.

2. Mathematical Model for Short-Circuit Fault

The most common short-circuit faults in transmission lines are single-phase ground short-circuit faults [26,27]. As shown in Figure 1, assuming that the supply voltage at t₀ (t₀ = 0) is used as the starting point for timing, the supply voltage can be set to u_s = U_msinωt. Assuming that the short-circuit impedance from the short-circuit point to the power supply is Rs + jωLs and the load impedance is R_L + jωL_L, and that a short-circuit fault occurs in the t_d system at some point after t₀:

According to the basic principles of the circuit, from moment t₀, the short-circuit current expression is

$$i_{\mathrm{s}}(t)=I_{bm}\sin (\omega t-{\phi }_1)+N_1{e}^{-\alpha (t-t_d)}=I_{bm}\sin (\omega t-{\phi }_1)+\left(i\left(t_d\right)-I_{bm}\sin (\omega t_d-{\phi }_1)\right)e^{-\alpha (t-t_d)}$$

(1)

In Equation (1), I_bm is the amplitude of the steady-state component of the short-circuit current, $I_m=U_m/\sqrt{R_s^2+{(\omega L_s)}^2}$. α is the short-circuit current transient component decay time constant, $\alpha =R_s/L_s$. φ₁ is the phase difference between the supply voltage and the steady-state component of the short-circuit current after a fault has occurred, ${\phi }_1=arctg\omega L_s/R_s$. N₁ is starting value of the transient component, t_d is the moment of short-circuit, and i(t_d) is the instantaneous value of the current at the moment of the short-circuit.

The short-circuit current can be divided into two parts: the steady-state component with sinusoidal changes and the attenuation transient component. The short-circuit current and bus voltage waveform are shown in Figure 2. It shows that after a short-circuit fault occurs, the current rises sharply.

3. Short-Circuit Current Waveform Parameters Prediction Based on UDW

The traditional least squares method for predicting short-circuit current parameters does not take into account the influence of harmonic factors and causes large errors, and the improved half-wave Fourier algorithm cannot accurately predict when the even harmonic content is too high. In response to these problems, this section presents a novel method for the prediction of short-circuit current waveform parameters, the Ultra-short-time Data Window (UDW) method. The principle of the UDW method for predicting short-circuit current waveform parameters is described in detail, and the predictive power of the two expressions present during the fitting of the short-circuit current curve is compared by simulation. To further validate the accuracy of the method, the UDW method is compared with the improved half-wave Fourier method, which leads to the final conclusions.

3.1. Prediction Principles

In practical power systems, the statistically obtained short-circuit current transient component decays out in about 8 to 10 cycles, i.e., in the time range of (160 to 200) ms. Therefore, even within a relatively short time frame after the short-circuit fault occurs, it is unscientific not to consider the changes in the transient components of the short-circuit current at all [28]. However, if the problem is considered within a very short time frame after the occurrence of a short-circuit fault, the change in the short-circuit current transient component within this time frame is negligible. Thus, for a very short time after the occurrence of a short-circuit, the short-circuit current can be approximated as

$$\begin{array}{cc}{i}_{\mathrm{s}}(t)& =I_{\mathit{bm}}\sin (\omega t-{\phi }_1)+N_1{e}^{-\alpha (t-t_d)}\hfill \\ & =I_{\mathit{bm}}\sin (\omega t-{\phi }_1)+\left(I_0-I_{\mathit{bm}}\sin (\omega t_d-{\phi }_1)\right)e^{-\alpha (t-t_d)}\hfill \\ & =I_{\mathit{bm}}\left(\sin \omega t\cos {\phi }_1-\cos \omega t\sin {\phi }_1\right)+\left(I_0-I_{\mathit{bm}}\left(\sin \omega t_d\cos {\phi }_1-\cos \omega t_d\sin {\phi }_1\right)\right)e^{-\alpha (t-t_d)}\hfill \\ & =X_1\sin \omega t-X_2\cos \omega t+\left(I_0-X_1\sin \omega t_d+X_2\cos \omega t_d\right)e^{-\alpha (t-t_d)}\hfill \end{array}$$

(2)

In Equation (2), X₁ = I_bmcosφ₁ and X₂ = I_bmsinφ₁.

The variation in the transient component of the short-circuit current over an ultra-short period of time during the data window can be approximated by a linear expression. Therefore, in a short period of time after the short-circuit occurs, the short-circuit current can be approximated as

$$\begin{array}{cc}{i}_{\mathrm{s}}(t)& =X_1\sin \omega t-X_2\cos \omega t+\left(I_0-X_1\sin \omega t_d+X_2\cos \omega t_d\right)e^{-\alpha (t-t_d)}\hfill \\ & \approx X_1\sin \omega t-X_2\cos \omega t+\left(I_0-X_1\sin \omega t_d+X_2\cos \omega t_d\right)\left(1-\alpha \left(t-t_d\right)\right)\hfill \\ & =X_1{k}_s(t)-X_2{k}_c(t)+\left(I_0-X_1{k}_{sd}+X_2{k}_{cd}\right)\left(1-\alpha \Delta \left(t\right)\right)\hfill \\ & =X_1{k}_s(t)-X_2{k}_c(t)+\left(I_0-X_1{k}_{sd}+X_2{k}_{cd}\right)-\left(I_0-X_1{k}_{sd}+X_2{k}_{cd}\right)\alpha \Delta \left(t\right)\hfill \\ & =X_1{k}_s(t)-X_2{k}_c(t)+X_3-X_4\Delta \left(t\right)\hfill \end{array}$$

(3)

In Equation (3), k_s(t) = sinωt, k_c(t) = cosωt, k_sd = sinωtd, k_cd = cosωtd, ∆(t) = t − t_d, X₃ = (I₀ − X₁k_sd + X₂k_cd), X₄ = (I₀ − X₁k_sd + X₂k_cd), and α = X₃α. Only X₁, X₂, X₃, and X₄ are unknown quantities that can be found by sampling the values multiple times using the least squares method.

The relationship between X₁, X₂, X₃, and X₄ and the parameters to be sought in the short-circuit current waveform are

$$\left\{\begin{array}{l}{I}_{bm}=\sqrt{X_1^2+X_2^2}\\ {\phi }_1=arctg\left(X_1/X_2\right)\\ \alpha =X_4/X_3\end{array}\right.$$

(4)

From Equation (4), X₁, X₂, X₃, and X₄ are obtained, and so are the parameters to be obtained for the short-circuit current waveform.

Assuming that the sampling time interval is Δ, starting from a certain moment t₁, at n moments [t₁, t₂, t₃, t₄, …, t_N], and sampling to obtain n instantaneous values of short-circuit current [i_s1, i_s2, i_s3, i_s4, …, i_sN], then

$$t_j=t_1+\left(j-1\right)\Delta ;\hspace{0.33em}{t}_j+t_d=t_1+t_d+\left(j-1\right)\Delta ;\hspace{0.33em}{t}_j-t_d=t_1-t_d+\left(j-1\right)\Delta$$

(5)

In Equation (5), j is number of sampling points, j = 1, 2, 3, …, n. Constructing a function:

$$F={\displaystyle \sum _{j=1}^N{\left(i_{sj}-i_s(t_j)\right)}^2}$$

(6)

Then, the process of obtaining X₁, X₂, X₃, and X₄ is the process of obtaining the appropriate values of X₁, X₂, X₃, and X₄ to make the function F obtain the minimum value. The F derivative of X₁, X₂, X₃, and X₄, respectively, is such that

$$\left\{\begin{array}{l}\frac{\partial F}{\partial X_1}=-2{\displaystyle \sum _{j=1}^N\left(i_{\mathit{sj}}-i_s(t_j)\right)}{k}_s(t_j)\\ \frac{\partial F}{\partial X_2}=2{\displaystyle \sum _{j=1}^N\left(i_{\mathit{sj}}-i_s(t_j)\right)}{k}_c(t_j)\\ \frac{\partial F}{\partial X_3}=-2{\displaystyle \sum _{j=1}^N\left(i_{\mathit{sj}}-i_s(t_j)\right)}\\ \frac{\partial F}{\partial X_4}=-2{\displaystyle \sum _{j=1}^N\left(i_{\mathit{sj}}-i_s(t_j)\right)}\Delta \left(t_j\right)\end{array}\right.$$

(7)

Letting $\frac{dF}{d{I}_{bm}}=0$ and $\frac{dF}{d\alpha }=0$, a pair of joint cubic equations can be obtained. Joint cubic equations system:

$$\left\{\begin{array}{l}{\displaystyle \sum _{j=1}^N\left(i_{\mathit{sj}}-i_s(t_j)\right)}{k}_s(t_j)=0\\ {\displaystyle \sum _{j=1}^N\left(i_{\mathit{sj}}-i_s(t_j)\right)}{k}_c(t_j)=0\\ {\displaystyle \sum _{j=1}^N\left(i_{\mathit{sj}}-i_s(t_j)\right)}=0\\ {\displaystyle \sum _{j=1}^N\left(i_{\mathit{sj}}-i_s(t_j)\right)}\Delta \left(t_j\right)=0\end{array}\right.$$

(8)

Bringing in the expression for i_s(t_j) yields

$$\{\begin{array}{l}{X}_1{\displaystyle \sum _{j=1}^N{k}_s^2(t_j)}-X_2{\displaystyle \sum _{j=1}^N{k}_s(t_j)k_c(t_j)}+X_3{\displaystyle \sum _{j=1}^N{k}_s(t_j)}-X_4{\displaystyle \sum _{j=1}^N{k}_s(t_j)\Delta \left(t_j\right)}={\displaystyle \sum _{j=1}^N{k}_s(t_j)i_{\mathit{sj}}}\\ X_1{\displaystyle \sum _{j=1}^N{k}_s(t_j)k_c(t_j)}-X_2{\displaystyle \sum _{j=1}^N{k}_c^2(t_j)}+X_3{\displaystyle \sum _{j=1}^N{k}_c(t_j)}-X_4{\displaystyle \sum _{j=1}^N{k}_c(t_j)\Delta \left(t_j\right)}={\displaystyle \sum _{j=1}^N{k}_c(t_j)i_{\mathit{sj}}}\\ X_1{\displaystyle \sum _{j=1}^N{k}_s(t_j)}-X_2{\displaystyle \sum _{j=1}^N{k}_c(t_j)}+N{X}_3-X_4{\displaystyle \sum _{j=1}^N\Delta \left(t_j\right)}={\displaystyle \sum _{j=1}^N{i}_{\mathit{sj}}}\\ X_1{\displaystyle \sum _{j=1}^N{k}_s(t_j)\Delta \left(t_j\right)}-X_2{\displaystyle \sum _{j=1}^N{k}_c(t_j)\Delta \left(t_j\right)}+X_3{\displaystyle \sum _{j=1}^N\Delta \left(t_j\right)}-X_4{\displaystyle \sum _{j=1}^N{\Delta }^2\left(t_j\right)}={\displaystyle \sum _{j=1}^N{i}_{\mathit{sj}}\Delta \left(t_j\right)}\end{array}$$

(9)

Letting $a={\displaystyle \sum _{j=1}^N{k}_s^2(t_j)}$, $b={\displaystyle \sum _{j=1}^N{k}_s(t_j)k_c(t_j)}$, $c={\displaystyle \sum _{j=1}^N{k}_s(t_j)}$, $d={\displaystyle \sum _{j=1}^N{k}_s(t_j)\Delta \left(t_j\right)}$, $e={\displaystyle \sum _{j=1}^N{k}_s(t_j)i_{\mathit{sj}}}$, $f={\displaystyle \sum _{j=1}^N{k}_c^2(t_j)}$, $g={\displaystyle \sum _{j=1}^N{k}_c(t_j)}$, $h={\displaystyle \sum _{j=1}^N{k}_c(t_j)\Delta \left(t_j\right)}$, $i={\displaystyle \sum _{j=1}^N{k}_c(t_j)i_{\mathit{sj}}}$, $m={\displaystyle \sum _{j=1}^N\Delta \left(t_j\right)}$, $p={\displaystyle \sum _{j=1}^N{i}_{\mathit{sj}}}$, $q={\displaystyle \sum _{j=1}^N{\Delta }^2\left(t_j\right)}$, and $r={\displaystyle \sum _{j=1}^N{i}_{\mathit{sj}}\Delta \left(t_j\right)}$, the equations above can be rewritten as

$$\left\{\begin{array}{l}a{X}_1-b{X}_2+c{X}_3-d{X}_4=e\\ b{X}_1-f{X}_2+g{X}_3-h{X}_4=i\\ c{X}_1-g{X}_2+N{X}_3-m{X}_4=p\\ d{X}_1-h{X}_2+m{X}_3-q{X}_4=r\end{array}\right.$$

(10)

This can be written in matrix form as

$$AX=Y$$

(11)

In Equation (11), $A=\left[\begin{array}{cccc}a& -b& c& -d\\ b& -f& g& -h\\ c& -g& N& -m\\ d& -h& m& -q\end{array}\right]$; $X=\left[\begin{array}{c}{X}_1\\ X_2\\ X_3\\ X_4\end{array}\right]$; $Y=\left[\begin{array}{c}e\\ i\\ p\\ r\end{array}\right]$.

The specific calculation formulas of X₁, X₂, X₃, and X₄ are derived in advance, which will minimize the solution calculation time. According to Clem’s rule [29,30] for solving systems of general linear equations, the solution of Equation (11) is

$$\begin{array}{l}{X}_1=\frac{{\Delta }_1}{{\Delta }^{\prime }}=\frac{\left|\begin{array}{cccc}e& -b& c& -d\\ i& -f& g& -h\\ p& -g& N& -m\\ r& -h& m& -q\end{array}\right|}{\left|\begin{array}{cccc}a& -b& c& -d\\ b& -f& g& -h\\ c& -g& N& -m\\ d& -h& m& -q\end{array}\right|};\hspace{1em}{X}_2=\frac{{\Delta }_2}{{\Delta }^{\prime }}=\frac{\left|\begin{array}{cccc}a& e& c& -d\\ b& i& g& -h\\ c& p& N& -m\\ d& r& m& -q\end{array}\right|}{\left|\begin{array}{cccc}a& -b& c& -d\\ b& -f& g& -h\\ c& -g& N& -m\\ d& -h& m& -q\end{array}\right|};\hspace{1em}\\ X_3=\frac{{\Delta }_3}{{\Delta }^{\prime }}=\frac{\left|\begin{array}{cccc}a& -b& e& -d\\ b& -f& i& -h\\ c& -g& p& -m\\ d& -h& r& -q\end{array}\right|}{\left|\begin{array}{cccc}a& -b& c& -d\\ b& -f& g& -h\\ c& -g& N& -m\\ d& -h& m& -q\end{array}\right|};\hspace{1em}{X}_4=\frac{{\Delta }_4}{{\Delta }^{\prime }}=\frac{\left|\begin{array}{cccc}a& -b& c& e\\ b& -f& g& i\\ c& -g& N& p\\ d& -h& m& r\end{array}\right|}{\left|\begin{array}{cccc}a& -b& c& -d\\ b& -f& g& -h\\ c& -g& N& -m\\ d& -h& m& -q\end{array}\right|}\end{array}$$

(12)

It can be seen from Equation (12) that simply by finding the values of a, b, c, d, e, f, g, h, i, m, p, q, and r, the values of X₁, X₂, X₃, and X₄ can be found with minimum computation time.The speed of calculation of the coefficients a, b, c, d, e, f, g, h, i, m, p, q, and r has a direct impact on the speed of obtaining the characteristic parameters of the short-circuit current waveform. In the following, the calculation of these coefficients is discussed in the hope that the fastest method of calculation can be obtained.

According to the previous analysis, coefficient a is calculated as:

$$\begin{array}{cc}a& ={\displaystyle \sum _{j=1}^N{k}_s^2(t_j)}\hfill \\ & ={\displaystyle \sum _{j=1}^N{\sin }^2(\omega t_j)}\hfill \\ & ={\displaystyle \sum _{j=1}^N{\sin }^2(\omega t_1+\omega (j-1)\Delta )}\hfill \\ & ={\sin }^2(\omega t_1){\displaystyle \sum _{j=1}^N{\cos }^2(\omega (j-1)\Delta )}+{\cos }^2(\omega t_1){\displaystyle \sum _{j=1}^N{\sin }^2(\omega (j-1)\Delta )}+0.5\sin (2\omega t_1){\displaystyle \sum _{j=1}^N\sin (2\omega (j-1)\Delta )}\hfill \end{array}$$

(13)

In Equation (13), ${\displaystyle \sum _{j=1}^N{\cos }^2(\omega (j-1)\Delta )}$, ${\displaystyle \sum _{j=1}^N{\sin }^2(\omega (j-1)\Delta )}$, and ${\displaystyle \sum _{j=1}^N\sin (2\omega (j-1)\Delta )}$ are only related to ∆ and N, and they were calculated when the short-circuit current detection system was developed. Letting ${\displaystyle \sum _{j=1}^N{\cos }^2(\omega (j-1)\Delta )}=K_1$, ${\displaystyle \sum _{j=1}^N{\sin }^2(\omega (j-1)\Delta )}=K_2$, ${\displaystyle \sum _{j=1}^N\sin (2\omega (j-1)\Delta )}=K_3$, a is simplified as

$$a=K_1{\sin }^2(\omega t_1)+K_2{\cos }^2(\omega t_1)+0.5K_3\sin (2\omega t_1)$$

(14)

As can be seen, a is quickly calculated once the detection start time t₁ has been determined.

Similarly, the other parameters can be quickly calculated as follows:

$$\left\{\begin{array}{l}b=0.5K_4\sin (2\omega t_1)+0.5K_3\cos (2\omega t_1)\\ K_4={\displaystyle \sum _{j=1}^N\cos (2\omega (j-1)\Delta )}\end{array}\right.$$

(15)

$$\left\{\begin{array}{l}c={\displaystyle \sum _{j=1}^N{k}_s(t_j)}=K_5\sin (\omega t_1)+K_6\cos (\omega t_1)\\ K_5={\displaystyle \sum _{j=1}^N\cos (\omega (j-1)\Delta )}\\ K_6={\displaystyle \sum _{j=1}^N\sin (\omega (j-1)\Delta )}\end{array}\right.$$

(16)

$$\left\{\begin{array}{l}d=K_5(t_1-t_d)\sin (\omega t_1)+K_6(t_1-t_d)\cos (\omega t_1)+K_7\sin (\omega t_1)+K_8\cos (\omega t_1)\\ K_7={\displaystyle \sum _{j=1}^N(j-1)\Delta \cos (\omega (j-1)\Delta )}\\ K_8={\displaystyle \sum _{j=1}^N(j-1)\Delta \sin (\omega (j-1)\Delta )}\end{array}\right.$$

(17)

$$f=K_1{\cos }^2(\omega t_1)+\hspace{0.17em}{K}_2{\sin }^2(\omega t_1)-0.5K_3\sin (2\omega t_1)$$

(18)

$$g=K_5\cos (\omega t_1)-K_6\sin (\omega t_1)$$

(19)

$$h=K_5(t_1-t_d)\cos (\omega t_1)-K_6(t_1-t_d)\sin (\omega t_1)+K_7\cos (\omega t_1)-K_8\sin (\omega t_1)$$

(20)

$$m=N(t_1-t_d)+0.5N(N-1)\Delta$$

(21)

$$q=N{(t_1-t_d)}^2+N(N-1)(t_1-t_d)\Delta +(2N^3-3N^2+N){\Delta }^2/6$$

(22)

The other parameters, $e={\displaystyle \sum _{j=1}^N{k}_s(t_j)i_{\mathit{sj}}}$, $i={\displaystyle \sum _{j=1}^N{k}_c(t_j)i_{\mathit{sj}}}$, $p={\displaystyle \sum _{j=1}^N{i}_{\mathit{sj}}}$, and $r={\displaystyle \sum _{j=1}^N{i}_{\mathit{sj}}\Delta \left(t_j\right)}$, need to be calculated on-site after the instantaneous value of the short-circuit current has been obtained.

In summary, the values of a, b, c, d, e, f, g, h, i, m, p, q, and r are obtained based on Δ, t_d, i(t_d), t₁, i(t₁), [t₁, t₂, t₃, t₄, …, t_N], and [i_s1, i_s2, i_s3, i_s4, …, i_sN]; X₁, X₂, X₃, and X₄ are obtained; short-circuit current waveform parameters I_bm, φ₁, and α are obtained; the fitting of the short-circuit current curve is completed.

3.2. Algorithm Validation

The setting of the ideal short-circuit fault signal in Matlab simulation software is

$$i=40\sin (100\pi t-\frac{\pi }{2})-35\sin (-\frac{\pi }{2})e^{-22t}$$

(23)

The sampling frequency is set to 10 kHz and a short-circuit fault occurs at t = 0 s. Accordingly, the current values at the moment of fault occurrence and the instantaneous values of the short-circuit current [i_s1, i_s2, i_s3, i_s4, …, i_sN] corresponding to the n moments [t₁, t₂, t₃, t₄, …, t_N] after the fault occurrence are obtained.

From Equation (3), based on the UDW waveform parameter prediction method, there are two expressions in the process of fitting the short-circuit current curve, namely the exponential expression and linear expression. Assuming n = 10, the predicted curve (exponential), the predicted curve (linear), and the actual curve are shown in Figure 3. It can be seen from the figure that both the waveform curve predicted by the exponential expression and the linear expression fit the actual waveform curve very well, but the curve fitted using the exponential expression is closer to the actual curve. The reason for this result is that the linearization of the expression leads to a bias in the identification of the time constants.

The UDW method is carried out within a very short period of time after the occurrence of a short-circuit, so the value of n cannot be very large. However, if the value of n is too small, the error of curve fitting will be comparatively large. Therefore, the total number of samples n is set to 8–12. As shown in Figure 4 and

Table 1. Error in waveform parameters when n is taken as different values.

Parameters	Actual Parameters	Error %
		n = 8	n = 9	n = 10	n = 11	n = 12
Ibm	40 A	5.63	4.40	0.15	5.12	6.07
φ1	90°	3.09	1.73	0.07	3.44	5.31
α	22	3.45	0.04	0.95	3.72	5.41

, when n = 10, the predicted fitted waveform is closest to the predefined waveform, with the lowest errors in I_bm, φ₁, and α. Because the sampling frequency is 10 kHz, the prediction is only 1 ms of the sampling data, which can achieve the purpose of rapid and accurate prediction.

In order to further verify the accuracy of this method, the UDW method is compared with the improved half-wave Fourier method [25]. From the comparison of the two waveform prediction methods in

Table 2. The comparison of the two waveform prediction methods.

Parameters	Error %
	Improved Half-Wave Fourier	UDW
Ibm/A	0.61	0.40
φ1/°	2.60	0.07
α	1.98	0.95

and Figure 5, it can be seen that the UDW method is the most accurate for the prediction of short-circuit current waveform parameters I_bm, φ₁, and α. Two methods can satisfy the purpose of fast prediction of current parameters, but the improved half-wave Fourier method calculates the steady-state component and transient component separately, which takes longer to calculate than the other.

4. Removal of Interference in Short-Circuit Current Signal

In actual work, in the initial stage of the short-circuit current, it is often accompanied by high-frequency interference with high frequency but fast attenuation speed, and there is also noise interference in the actual sampling process [15,16]. Due to the existence of these disturbances, the parameter estimation of the short-circuit current will be larger, biased, or even impossible. For this reason, this paper proposes an interference cancellation method based on trend filtering technology. First, the principles of the trend filtering technique are described. It is then applied to the removal of high-frequency interference and white noise interference, and finally the final conclusions are drawn through simulation.

4.1. Principle of Trend Filtering Technique

It is assumed that for any moment t, the true value of the short-circuit current at that moment is related not only to the ‘collected short-circuit current value’ at that moment, but also to the ‘collected short-circuit current value’ at a certain time before and after that moment, and that the true value of the short-circuit current at that moment can be obtained by a weighted average of the above related ‘collected short-circuit current value’. The equation is

$$i_{{}_k}^{\left(1\right)}=\frac{{\displaystyle \sum _{j=k-m}^{k+m}{k}_j{i}_j^{(0)}}}{{\displaystyle \sum _{j=k-m}^{k+m}{k}_j}}$$

(24)

In Equation (24), i_k⁽¹⁾ is the short-circuit current value at the moment of t_k after filtering, i_j⁽⁰⁾ is the short-circuit current value at the moment of t_j before filtering, and k_j is the weight. The above equation indicates that the short-circuit current value i_k⁽¹⁾ at the moment of t_k can be calculated by averaging the weights of the sampling value of short-circuit current i_k⁽¹⁾ at the moment of t_k, sampling values of m short-circuit currents i_k−m⁽⁰⁾, i_k−m−1⁽⁰⁾, …, i_k−1⁽⁰⁾ before the moment of t_k, and sampling values of m short-circuit currents i_k+1⁽⁰⁾, i_k+2⁽⁰⁾, …, i_k+m⁽⁰⁾ after the moment of t_k.

The distribution of the weights k_j follows the principle of symmetry, i.e., with k_k as the center of symmetry, i.e., k_k−1 = k_k+1, k_k−2 = k_k+2, k_k−m = k_k+m. The value of k_j can be considered in accordance with the binary principle. Concretizing it:

For the case of m = 1:

$$i_{{}_k}^{\left(1\right)}=\frac{{\displaystyle \sum _{j=k-1}^{k+1}{k}_j{i}_j^{(0)}}}{{\displaystyle \sum _{j=k-1}^{k+1}{k}_j}}=\frac{i_{k-1}^{(0)}+2i_k^{(0)}+i_{k+1}^{(0)}}{4}$$

(25)

For the case of m = 2:

$$i_{{}_k}^{\left(1\right)}=\frac{{\displaystyle \sum _{j=k-2}^{k+2}{k}_j{i}_j^{(0)}}}{{\displaystyle \sum _{j=k-2}^{k+2}{k}_j}}=\frac{i_{k-2}^{(0)}+2i_{k-1}^{(0)}+4i_k^{(0)}+2i_{k+1}^{(0)}+i_{k+2}^{(0)}}{10}$$

(26)

For the two endpoints of the waveform, special treatment is required. The starting point is i₁⁽¹⁾ = i₁⁽⁰⁾ and the finishing point is i_N⁽¹⁾ = i_N⁽⁰⁾.

In general, it is difficult to achieve a good filtering effect after one such trend filtering, so multiple iterations of filtering can be carried out: that is, the output quantity of the previous filtering is used as the input quantity for the next filtering, and another filtering is carried out, and so on, until the desired effect is achieved.

4.2. Elimination of High-Frequency Interference and White Noise Interference of Short-Circuit Current

Assuming that the sampling time interval is Δ, starting from a certain moment t₁, at N moments [t₁, t₂, t₃, t₄, …, t_N], N short-circuit current instantaneous values are sampled [i_s1, i_s2, i_s3, i_s4, …, i_sN]. The first part of these current values contains high-frequency interference, while the later ones are the real short-circuit current values. This is reflected in the waveform diagram in Figure 6b. From Figure 6c–f, after multiple filters by trend filtering technology, the high-frequency components are almost completely eliminated, and finally, a short-circuit current waveform without high-frequency components is obtained. Moreover, in the part of the original waveform that does not contain high-frequency components, the filtering hardly changes the original waveform. Therefore, it can be considered that the rapid elimination of high-frequency interference can be achieved by performing multiple trend filtering on only those parts of the original waveform that contain high-frequency components. When using the ultra-short-time data window method to predict the short-circuit current waveform parameters, as there are not much data to be collected, all the collected data can be trend-filtered.

There is also the problem of noise interference during the actual sampling. To this end, wgn (White Gaussian Noise) is used to add white noise interference to the original fault current waveform, and the intensity of the noise is set to 20 dB, as shown in Figure 7b. From Figure 7c–f, it can be seen that after the trend filtering technique has been filtered several times, the white noise interference is almost completely eliminated and the filtered waveform is obtained almost identical to the original waveform, so the trend filtering technique is very obvious for eliminating the short-circuit current.

5. Measured Short-Circuit Current Waveform

The tests in this paper use a single-phase, small-capacity prototype model to simulate a single-phase short-circuit fault. The experimental circuit diagram is shown in Figure 8. u_s stands for 220 V mains AC. T stands for regulator. K stands for divider controller. R stands for line resistance, and the resistance value is much smaller than the inductance value. L stands for the current limiting inductance value, L = 12 mH. The 220 V power supply is turned on, the 120 V voltage is output through the voltage regulator T, the A622 current probe (TEKTRONIX, Beaverton, OR, USA) is used to measure the short-circuit current, the sampling frequency is 10 kHz, and the number of samples required for prediction is set to N = 10. The DP05054 oscilloscope (TEKTRONIX, Beaverton, OR, USA) is used to measure and collect the line current waveform in real time, and Matlab software is imported to compare and analyze the test results with the simulation results to further verify the feasibility and accuracy of the UDW prediction short-circuit current parameters proposed in this paper.

The short-circuit current waveform predicted using the UDW is compared with the measured waveform, from Figure 9a, and the measured short-circuit current waveform is almost the same as the predicted short-circuit current curve. The measured short-circuit current is 54.1255 A. The predicted short-circuit current peak is 51.4164 A, the error is 5%, and the prediction time used is 1 ms. The reason for the large error is because the interference in the signal is too large. The trend filtering technique is used to eliminate disturbances in the short-circuit current, giving a peak-predicted short-circuit current of 53.2205 in Figure 9b, with an error of 1.67%. It can be seen that the short-circuit current predicted by the ultra-short-time data window can quickly and accurately predict the waveform and amplitude of the short-circuit current.

6. Conclusions

In this paper, a prediction method based on an ultra-short-time data window is proposed to study the prediction of short-circuit current parameters when a short-circuit failure occurs on a transmission line. Through the analysis of simulation and measured results, the following conclusions are obtained:

• The UDW method has very clear advantages in fitting short-circuit current waveforms to achieve fast and accurate prediction of the waveform parameter. The exponential expressions in the UDW method fit curves closer to the actual curve, with errors of 0.15% for I_bm, 0.07% for φ₁, and 0.95% for α.
• Comparing the UDW method with the modified half-wave Fourier method, it is verified that the UDW method has a shorter prediction time and higher accuracy. The improved half-wave Fourier method has a higher error when the even harmonics increase, whereas the UDW method does not have this problem, so the UDW method is more versatile and has a higher prediction accuracy. The Improved Half-Wave Fourier method calculates the steady-state and transient components separately, which results in a long calculation time, whereas the UDW method requires only 1 ms of sampled data, so the prediction time is shorter.
• Trend filtering technology can realize multiple trend filtering on the sampled data in the initial stage of prediction to achieve the purpose of quickly eliminating high-frequency interference and white noise interference, and improve the accuracy of prediction without affecting the rapidity of prediction.