# Amplitude Distribution of Partial Discharge Signals on Tunnel-Installed High-Voltage Cables
## 1 Introduction  
Partial discharge (PD) is an important characterization of the insulation defects in cross-linked poly-ethylene (XLPE) cables [1; 2; 3]. Through on-site or online detection of PD signals, the state of the XLPE cables can be evaluated without power supply interruptions, which can help find the early degradation of the cable as soon as possible, thus improving the reliability of the power grid [4; 5; 6; 7]. The high-frequency current method is commonly used in the PD detection of power cables. This method generally has a detection frequency band of 3-30 MHz [8]. For 110 kV and above HV XLPE cable systems, it is normal procedure to adopt a cross-bonding scheme, as shown in Figure 1. The high-frequency current transformers are installed on the cross-bonding connections of the cable metal sheaths through a non-contact method. These can detect the PD pulse current inside the cables without affecting the normal operation of the cable system [9; 10].  
As an increasing area of interest, the distribution of PD signals on the three-phase cross-bonding wires can be used as an important tool for PD location and eliminating the external disturbance [11; 12; 13]. However, previous studies are mainly based on fixed simulation parameters or a determined experimental environment, without considering the impact of cable laying parameters. In fact, the laying method and laying environment of HV cables have an inseparable relationship with the propagation of pulse current in the cable and will also affect the amplitude distribution of PD signals, which has not yet been seen in relevant research papers.  
In order to discuss the distribution characteristics of the PD signals on the cross-bonding wires under different laying conditions, it is necessary to establish a cable system model for theoretical analysis. Existing cable modeling tools such as ATP (alternative transients program) are mostly based on [PERSON]'s distributed parameter model of directly buried cables [14; 15]. This model uses [PERSON]'s formula [16] to calculate the earth return path impedance. However, [PERSON]'s formula is not suitable for the widely used tunnel-installed HV cable systems since it assumes that the conductors are completely surrounded by the earth. For underground tunnels, [PERSON] proposed a set of high-frequency formulas and low-frequency approximation formulas for calculating the earth return impedance [17]. [PERSON] et al. [18] clarified its physical meaning and application scope by theoretical analysis and case calculation. [PERSON]'s formulas have a good accuracy at high frequencies by taking the proximity effect and skin effect into consideration. In [19], the low-frequency approximation formulas are used to study the propagation of traveling waves in the cross-bonded cable system, but it does not involve the current amplitude distribution and the influence of the laying parameters.  
In this paper, a characteristic admittance model of three-phase cable systems is established based on [PERSON]'s formulas, and thus the formula of the amplitude distribution of PD pulse current on cross-bonding wires is obtained. On this basis, the influence of parameters such as tunnel radius, earth resistivity, cable center spacing, and the distance between the cable and the inner wall of the tunnel is studied. The model and conclusions is verified by experiments and on-site tests.