This section uses a bottom-up approach to model the PLC channel to obtain the CFR of the system (Niu et al., 2021), where CFR is the voltage ratio between two nodes (transmitter and receiver). According to the transmission line theory, the characteristics of the transmission line are mainly expressed by the characteristic impedance and transmission constant, Z ; γ can be obtained from equations (1); (2), which are calculated according to the resistance, inductance, conductivity and capacitance (R, L, G and C) per unit length (Meng et al., 2004; Papaleonidopoulos et al., 2003; Paul et al., 1994). Z 0 = R + j 2 π f L G + j 2 π f C (1) γ = α + j β = R + j 2 π f L G + j 2 π f C (2)

The bottom-up channel modeling method needs to consider the network topology and simplify the network topology; As shown in Figure 2, the path between two nodes (Tx and Rx) in a typical network topology is called the trunk path (Tonello et al., 2009; Versolatto et al., 2010). All nodes on the trunk path are divided into n basic units according to the trunk path, and each basic unit has a unique trunk part and branch part; Further, the branch portion may have no branches, branches, or multiple branches. To simplify the calculation, the equivalent branch impedance of nodes with different branch levels is transmitted back to the trunk part (Paul et al., 2008). According to the bottom-up approach, each unit is divided into three parts, namely the first part, the second part and the branch part. Calculate the CFR of each basic unit. The detailed topology of each basic unit is shown in Figure 2.

As illustrated in Figure 2, the input impedance of the first part of the receiver can be obtained from the following equation: Z i n n = Z 1 n   Z p n + Z 1 n tanh γ 1 n l 1 n Z 1 n + Z p n tanh γ 1 n l 1 n (3) where Z 1 is the characteristic impedance, γ is the transmission constant, l is the line length, and Z p can be obtained from the following equation: Z P n = Z L n Z i n b n Z L n + Z i n b n (4)

Here, Z L is the impedance on the load side of the network, and Z i n b is the impedance on the branch side of the network. The CFR of the nth basic unit can be obtained by the voltage ratio of the receiving side to the transmitting side, expressed by the following equation: H n f = V l n V i n n = V l n V p n V p n V i n n (5) V l n V p n = 1 + Γ L n e γ 2 n l 2 n + Γ L n e − γ 2 n l 2 n (6) V p n V i n n = 1 + Γ p n e γ 1 n l 1 n + Γ p n e − γ 1 n l 1 n (7) Γ L n = Z L n − Z 2 n Z L n + Z 2 n (8) Γ P n = Z P n − Z 1 n Z p n + Z 1 n (9) Where Z 2 is the characteristic impedance of the line of the basic unit part 2,   γ 2 is the transmission constant of the line of the basic unit part 2,   l 2 is the line length of the line of the basic unit part 2, Γ P is the reflection coefficient of the basic unit part 1, Γ L is the reflection coefficient of the basic unit part 2.

Using these equation, we can obtain the CFR of the nth basic unit. Therefore, iterating from the back to front, the CFR of each basic unit can be determined. The total channel frequency response can be obtained using the following equation: H f = ∏ n = 1 N H n f   (10)

Similarly, the CFR between any two points can be determined by simply cascading the CFR of all the essential cells between the two points.

First, when a ground fault is generated, it will cause a transient change in the topology, and this change will cause a more drastic change in the CFR and impedance spectrum.

Since the ground fault is mainly manifested in one fault point, the fault is simulated by the resistance R f , the cable in case of fault is equated into a distributed parameter model, as displayed in Figure 3, to describe the transmission characteristics of the signal in the cable (Xie et al., 2017). The CFR in case of a cable fault is obtained through previously proposed channel modeling.

The resistance R, inductance L, capacitance C, and conductance G at each unit length of the cable can be approximated by the following equation (van der Wielen et al., 2008): R ≈ 1 2 π μ 0 ω 2 1 r c ρ c + 1 r s ρ s (11) L ≈ μ 0 2 π ln r s r c + 1 4 π 2 μ 0 ω 1 r c ρ c + 1 r s ρ s (12) C = 2 π ε ln ⁡ ⁡ r s / r c (13) G = 2 π δ ln ⁡ ⁡ r s / r c (14) where the angular frequency ω = 2πf, r c , and r s are the cable core version diameter and shield inner radius, respectively, ρ c and ρ s are the cable core resistivity and shield resistivity, respectively, μ 0 is the vacuum permeability, ε is the dielectric constant of the dielectric, and δ is the dielectric conductivity. Knowing the distribution parameters of the cable, the channel can be modeled by the bottom-up presented approach above for the normal and fault cases. Its transmission characteristics change when a fault occurs, as displayed in Figure 3. When the value of R f is greater than 10 Z 0 , it is a high impedance fault, for R f less than 10 Z 0 , it is a low impedance fault, and Z 0 is the characteristic impedance of the cable (Lu et al., 2004). The high and low resistance fault sizes in this paper’s simulation section are set to 10 Z 0 and Z 0 , or 500 and 50 Ω, respectively.

Eq. 3 indicates that when a ground fault occurs, the branch and backbone structure of the network changes. Changes in structure will affect the topology and reflection coefficient of the network; changes in structure will also affect the impedance of each part. Under the influence of both, the input impedance will also change (Passerini et al., 2017). Therefore, whether a fault occurs from the perspective of input impedance can be determined, and the following equation is defined from the perspective of fault detection. Δ I N = Z N F − Z f a u l t Z N F × 100 % (15)

Here, Z N F is the input impedance under normal conditions, and Z f a u l t is under fault conditions.

However, no direct correlation was observed between CFR and input impedance in the aforementioned derivation, even though a certain mathematical relationship exists. In (Papaleonidopoulos et al., 2003), the input impedance was postulated to depend on the ratio between two functions (current transfer function and voltage transfer function) (Piante et al., 2019) as follows: H V = Z l Z i n H I = Z l Y i n H I (16)

Therefore, a relationship exists between the voltage transfer function and the input impedance, and the correlation is apparent when the current transfer function is constant as follows (Piante et al., 2019): H N F − H f a u l t H N F = Z l Y i n H I − Z l Y inf H I f Z l Y i n H I = Y i n − Y inf H I f H I Y i n (17) where H N F is the voltage transfer function under normal conditions, H f a u l t is the voltage transfer function under fault conditions. Furthermore, H I is the current transfer function under normal conditions, H I f is the current transfer function under fault conditions, and the change of input impedance before and after a fault is apparent,Y_in is the admittance value of the first part of the receiver. The fault can be monitored using the input impedance; as expressed in Eq. 17, the voltage transfer function can be used to observe the fault, and the frequency domain of the voltage transfer function is the CFR. Therefore, the fault can be detected from the perspective of the CFR, which can be measured by the PLC equipment.

Therefore, the input impedance and CFR allow the detection of faults, as shown below: Δ I N = | Z N F | − | Z f a u l t | | Z N F | × 100 % (18) Δ C F R = | H N F f | − | H f a u l t f | H N F f × 100 % (19)

After fault detection, fault location should be determined. When the network topology changes due to faults, enough fault information can be obtained from Eqs 18, 19, which can be used to determine the fault location. Since CFR is greatly affected, the fault can be determined from the perspective of input impedance. Moreover, the input impedance is obtained by using the impedance estimation technique, which does not require additional measurement equipment and is presented in Chapter three. After obtaining the input impedance, only the input impedance is subjected to the inverse fast Fourier transform (IFFT): Δ I N → I F F T Δ t (20)

To achieve fault positioning, the electromagnetic wave propagation speed in the cable should be calculated at high frequency. The faster the wave propagation speed is, the highest is close to the speed of light. At high frequencies, the propagation speed of electromagnetic waves is independent of frequency and is mainly determined by the primary transmission constants L and C of the cable line, and the electromagnetic wave speed can be expressed as follows: v = 1 L C (21)

Combining the wave velocity (21) with the input impedance Eq. 20 in time domain yields a function of distance. d v t = Δ t × v (22)

Therefore, The obtained equation is a function of distance and variation, from which information about the fault location can be obtained. The input impedance is obtained through iteration easily by using the modeling process. When a fault occurs, the input impedance at the fault point and the input impedance of the node after the fault point change. Therefore, the fault location can be determined according to this characteristic.

The impedance estimation technique is categorized into the following steps, as displayed in Figure 4.

First, the CFR is obtained by modeling the channel in a bottom-up approach. A close relationship exists between the input impedance and CFR; therefore, machine learning techniques are used to predict the input impedance, using the CFR as the input and the input impedance as the output. However, some error exists when using the CFR directly as the input. Therefore, feature extraction is performed for the CFR using VMD. In this study, the impedance estimation technique was introduced using a set of CFR data as an example, and a set of channel frequency response data is displayed in Figure 5.

The CFR was then subjected to VMD decomposition and performed on the CFR data in Figure 5A to obtain six sets of eigenmode functions (IMF components), as displayed in Figure 5B. High-frequency components exist in the IMF of Figure 5B, and the high-frequency components are used as the input produces errors in the prediction results. Therefore, the IMF should be analyzed. Experimental results revealed that the first three components of the VMD decomposition were selected for superior prediction.

In the third step, machine learning is used to predict impedance. Experimental results have revealed the first three quantities in the IMF after decomposition using VMD are closely related to the impedance; therefore, after removing the curve’s high frequency noise, the first three quantities are selected as the input to the decision tree, and the amplitude and phase of the impedance are used as the output. The data from the preliminary modeling are compared to verify the accuracy of the impedance estimation technique. The results are displayed in Figure 6, which revealed that the impedance estimation technique exhibits excellent impedance prediction results; The results show that the predicted value from the simulation procedure is also in line with the set real value.

Impedance estimation techniques can predict impedance accurately, and the expected input impedance can be used for fault detection. When the CFR is obtained, the CFR can be used for fault detection. The impedance estimation technique can be used predict the impedance, and the input impedance can be used for fault detection and location. By contrast, the CFR can be obtained directly by using power line communication equipment.

In this chapter, The simulation model is built in matlab software, and the transmission line model is modeled and processed by combining the bottom-up modeling method and the theory of transmission line. Combined with the change of network topology under fault conditions, the CFR of the receiving end is changed. According to the relationship between CFR and input impedance, the VMD and machine learning methods are used to predict the impedance. Then, according to the change of impedance spectrum under normal and fault conditions, the fault is monitored and the fault is located. First, a random topology using measured load data and random trunk and branch lengths is constructed, and then any set of data from it is taken for detailed analysis. Second, the fault detection is performed by CFR data and impedance data, and the fault detection effects of the two different methods are compared and analyzed, and finally, the fault location effect using the predicted input impedance is demonstrated.

The subsequent simulation will use a single-core power cable as the object of study, and simulations have been performed based on the actual parameters of the cable, and Figure 7 shows the simplified structure of the single-core cable (Wagenaars et al., 2010).

The parameters of 240-mm² single-core cross-linked polyethylene cable (XLPE cable) are presented in Table 1, and the properties of XLPE cable are listed in Table 2.

Given the above parameters, the cable can be modeled according to the cable modeling method described previously.

First, 500 sets of CFR data under normal conditions are generated. The trunk length of each basic unit is randomly generated in the range of 1000m–2000m, the branch length is randomly generated in the range of 500m–1000m, and the load impedance is set to the measured load impedance data Xiaoxian et al. (2007). The 500 group CFR data is shown in Figure 8, each set of CFR data showed a trend of oscillation attenuation with the increase of frequency.

To reflect the randomness of the extraction model, taking out any group from 500 groups of random data, and its topology is shown in Figure 9. The total length of the network backbone was 2591m, with two branches; branch lengths were 524 and 841 m respectively. The red color represents the backbone network, and it is divided into 579 m AB segment, 1172 m BC segment, and 783 m CD segment.

First, the network displayed in Figure 9 is simulated to obtain the CFR under normal conditions, as displayed in Figure 10A.

Based on the impedance estimation technique verified above, the system input impedance under normal conditions is predicted, and the prediction effect is shown in Figure 10B.

The input impedance changes when a grid ground fault occurs. By monitoring the input impedance at point B in Figure 9, a fault between BD can be observed because when a fault occurs in the BD segment, it causes the input impedance at point B to change. A 500-Ω high-impedance fault and a 50-Ω low-impedance fault are set in the BD segment, and the fault distance is set randomly in the BD segment; 300 sets are set in the BC segment, and 200 groups are set in the CD segment; therefore, 500 sets of fault data are generated, and 500 sets of input impedance and CFR data are shown in Figures 11A, B.

Figure 11A displays a low-impedance fault with a fault impedance of 50 Ω, and Figure 11B displays a high-impedance fault with a fault impedance of 500 Ω. The 500 sets of CFR data and 500 sets of input impedance amplitude data used in the study are shown. When the CFR data and the input impedance data are obtained, the data are calculated by using Eqs 18, 19 above to obtain the Δ value characterizing the system variation obtained from the CFR and input impedance, respectively. The Δ-values for the low-impedance fault and high-impedance fault cases are displayed in Figures 11C, D.

Figure 11C details a low impedance fault with a fault impedance of 50 Ω, and Figure 11D indicates a high-impedance fault with a fault impedance of 500 Ω. Five hundred sets of Δ(CFR) and Δ(IN), obtained from CFR and input impedance, respectively.

When the Δ value is obtained, Δ(CFR) and Δ(IN) are analyzed separately. If the peak value of Δ is more significant, the system exhibits more apparent changes and can be easily monitored. The peak values of 500 sets of Δ data are illustrated in Figure 12. Eventually, the Δ data can be analyzed to obtain fault information.

Figure 12A shows the data for Δ(CFR), Figure 12B shows the data for Δ(IN), and Figure 12C displays a comparison between Δ(CFR) and Δ(IN) for the low-impedance case. A comparison of the effectiveness of two of these techniques for monitoring faults is shown in Table 3 Figures 12A, B reveal that both different technologies have good monitoring results for high and low resistance faults, the peak value of low impedance faults is always higher than that of high impedance faults, so low impedance faults are easier to monitor. This method can be applied to monitor LIFs and HIFs. As displayed in Figure 12C, a branch exists at 1751 m in the topology. When using the input impedance for fault detection, a clear stratification is observed in Δ(IN), and also can roughly identify the fault before or after the branch; however, this effect is not clear when using CFR. And the Δ(CFR) is more significant than Δ(IN) when using CFR for fault detection. As illustrated in Figures 12C, D and Figure 12, 500 groups of random faults Δ Values have specific peaks; therefore, Δ Value can be used to detect faults.

When using the aforementioned location method for fault location, the input impedance changes when the network changes; which provides sufficient information about fault location. The topological network displayed in Figure 9 was used for fault location. Figure 13 shows the input impedance data (with actual as well as predicted values) for faults occurring at 1,000 and 1,500 m with 50 and 500 Ω. The input impedance predicted using impedance estimation techniques is then used for fault location.

When fault location is performed, the PLC device acquires the real-time CFR and estimates the current input impedance waveform from the real-time CFR. Then the IFFT of the impedance is obtained from Eq. 20. The wave speed can be obtained from Eq. 21. From the wave speed, a function (22) of distance can be obtained, by which the fault point can be located. Figure 14 shows the localization results for different fault impedances.

The figure shows that there is a clear wave peak at the location of the 1,000 and 1,500 m point, the peak contains the location information of the fault point, the position corresponding to the peak is the position of the fault point, so the fault can be located by peak finding. Among them, the results of fault impedance localization are shown in Table 4.

As presented in the table, the positioning, although a specific error exists, from the perspective of using the input impedance to locate the fault, the fault can be an approximate location of the calculation. This fault monitoring and location algorithm has a positive effect on high and low resistance fault monitoring and can be applied in fault monitoring.

If the fault occurs on the branch, the above method can also be used to detect and locate the fault. First, the receiver is connected to the branch end, the position of the transmitter remains unchanged, and the original network topology is changed. The original branch part can be regarded as the backbone part of the new topology. Then, the same method as above is used to achieve Fault detection and location.