The XLPE cable commonly works in the temperature range of 60–80°C, which is higher than the temperature of environment. It leads to an increasing temperature gradient between the outer sheath and core of the cable. Thus, the XLPE insulating layer between the outer sheath and the core is non-uniformly aged.

So as to analyze conveniently, this study regards some parts with little temperature difference as an isothermal area, and sequentially numbered, which conforms to T ₁ > T ₂ > T ₃ > … > T _min .

The extended Debye model uses a resistor and a capacitor in series to equivalent the relaxation polarization process of the dielectric. However, the extended Debye model with a series of resistor and capacitor circuits is not sufficient to reflect the complex polarization process. Therefore, a set of exponential functions containing multiple relaxation elements is established, which corresponds to the parallel connection of multiple resistance and capacitor branches in series in the circuit model. Thus, the complex polarization process can be represented by the sum of the functions. The extended Debye model is shown in Figure 1, which is regarded as the most equivalent model for aged cables, where R ₀ and C ₀ are the insulation resistance and vacuum capacitance of XLPE respectively. i is branch number, R _i and C _i (1 < i < n) is the lossy relaxation polarization at different relaxation times, the aforementioned parameters can be calculated by polarization and depolarization current (Saha et al., 2005; Kumar and Mahajan, 2010).

The current obtained by the galvanometer is polarization current i _p (t) when the DC voltage U ₀ is applied at both ends of the sample. If the DC power is disconnected after t _c time and the two ends of the sample are shorted, the measured current will be depolarization current i _d (t).

Apparently, the insulation layer of the cable determines its insulation performance. Thus, in order to simplify the process of the experiment, XLPE discs are used to prepare non-uniform aging samples which can simulate non-uniform aging of cables conveniently.

The disk samples are made of 10 kV XLPE material from Nordic Chemical Company, which has a thickness of 1.5(±0.05) mm and a radius of 16(±0.2) cm. The disk samples are cleaned with anhydrous ethanol to remove impurities on the surface and dried in a vacuum dryer oven for 48 h at a temperature of 80°C and a pressure of 50 pA to eliminate the effects of water vapor, cross-linking by-products, and mechanical stress in order to achieve the desired cable insulation state.

Crystallization of the XLPE material can be destroyed by thermal aging under long-term effect (White et al., 2000). In this study, accelerating thermal aging is used to produce the XLPE samples with various aging degrees at the temperatures of 130, 140, and 150°C, last for 7, 15, 25, and 40 days at each temperature, all samples are numbered as shown in Table 1. Furthermore, each non-uniform aging sample consists of three discs samples aged with different temperatures and the same aging time. For example, a non-uniform aging sample at 140°C, 15 days is composed of N₂, N₆, and N₁₀ samples. On the contrary, three N₆ samples constitute a uniformly aged sample at 140°C for 15 days. All sample numbers and production methods are shown in Table 2, where X and Y represent the non-uniform sample and uniform sample, respectively.

Especially, preliminary experiments found that there are always three isothermal areas in the modified Debye model if the non-uniform sample consists of more than three layers. Therefore, this study decided to make the non-uniform thermal aging samples with three layers.

To improve the stability of the PDC data, samples in Table 2 are placed in three-electrode and measured at 60°C. The i _p (t) and i _d (t) can be measured under a DC voltage of 200 V (polarization voltage) for 1,500 s (polarization depolarization time).

Furthermore, to avoid current interference, the measurements should be performed in an electromagnetic-free environment. Thus, the three-electrode is grounded and housed in a metal shielding box in this study.

Figure 3 shows the depolarization current curve of the non-uniform samples. Since switching may lead to impulse current, in this study, the curve is recorded after 10 s. It is easy to discover from the Figure 3 that the sample current amplitude is at the order of pA. The current is roughly divided into two parts according to the inflection point. The front section decreases rapidly is regarded as the high-frequency part, while the middle and rear section stabilizes gradually, called the low-frequency part. In addition, with the increase of aging time, the critical point of the two parts gradually moves to the left, and the current value becomes larger when it is relatively stable. This result could be explained by the fact that thermal aging increases the number of defects in the XLPE and destroys the cross-linked structure, and produces more aging products, charged particles, and molecular groups. Thus, the discharge process is accelerated and the conductivity is increased. For non-uniform aging samples, these characteristics are similar to the existing uniform aging research results (Baral and Chakravorti, 2013).

Although the internal changes of the XLPE caused by uniform and non-uniform thermal aging are essentially the same, the different aging processes could change the distribution of materials that influence the charge movement, such as dipole pairs and charged particles, which leads to difference between the uniform and non-uniform aging results. It can be seen from Figure 4 that the depolarization current of X is larger than that of Y in the high-frequency part at any aging time. Furthermore, in the low-frequency part, the two current values are gradually close to each other or even coincide. In addition, the coincidence degree of samples with different aging modes decreases with an increase in aging. It can be inferred that the non-uniform aging causes the density imbalance of the materials involved in the charge movement in the insulation. This phenomenon leads to the larger current in the initial discharge than that in the uniform aging sample. When the current decays in the later stage, the effect of different density becomes weaker.

The parameters of the modified Debye model and extended Debye model of the X₁, X₂, X₃, and X₄ samples are shown in Table 3. R _j (k) and τ _j (k) of the same branch decline gradually with the decrease of isothermal area numbers. Because a smaller number represents a higher aging temperature of XLPE samples, it results in higher polarization strength, faster polarization response speed, and worse insulation conditions. In addition, the biggest change occurs in the parameters of the 3rd branch, as given in Table 3. It is generally believed that the largest time constant branch characterizes the characteristics of solid insulation, and these parameters are calculated first during identification. Therefore, the parameters are most sensitive to the degree of non-uniform aging.

Compared with the modified Debye model, the time constant of each branch in the extended Debye model is between the maximum and minimum isothermal areas of the extended Debye model, and the polarization resistance is approximately equal to the sum of the polarization resistances of isothermal areas, as shown in Eq. 4: { τ j ( 1 ) < τ j < τ j ( k ) R j ≈ ∑ k = 1 m j R j ( k ) , (4)

where τ _j and R _j are time constant and polarization resistance of the jth branch in the extended Debye model, respectively. If all isothermal areas of any branch in Figure 2 are combined, the polarization resistance is equivalent to that of the branch corresponding to the extended Debye model. The results strongly prove that the modified Debye model is correct in analyzing the aging condition of XLPE.

However, the characteristic variable of XLPE thermal aging insulation condition evaluation proposed in the existing literature is mostly based on the parameters of the extended Debye model (Saha et al., 2005; Kumar and Mahajan, 2010). In order to determine the accuracy of the extended Debye model in this respect, the deviation rate of each isothermal area is given in Table 4 and expressed as Eq. 5: Δ X = τ j − τ j ( k ) τ j ( k ) × 100 % . (5)

It can be seen from Table 3 that each isothermal area in the No. 1 branch of X₃ has the smallest deviation rate, which can be basically ignored. The |ΔX| of No. 2 and No. 3 branches increases with aging time, and it is negative in the No. 3 isothermal area. This confirms that the insulation condition of lower heating temperature parts is often underestimated when using the extended Debye model. Meanwhile, the parts with higher heating temperature are overestimated. This situation leads to errors in the evaluation results. Therefore, the non-uniform thermal aging effect of cable should be considered when adopting the PDC method, and the modified Debye model should be used to analyze it.

It is unavoidable that a large number of chemical impurities and structural defects can be produced during XLPE production and manufacture. These defects will form different trap levels which capture charge carriers, in the forbidden band of polymers. According to the J.G. Simmons theory (Simmons and Tam, 1973), under a high electric field, the carriers escaping from traps will move out of the medium directionally and rapidly, which become hot electrons. The most important thing is that the generation and energy of hot electrons are determined by the density and depth of traps, which are deeply affected by aging. Therefore, the aging condition must be inferred from the change of traps in XLPE. It provides a theoretical basis for quantitatively expressing the XLPE aging condition.

The PDC method needs to analyze the region occupied by electrons above the Fermi level. At constant temperature, the relationship between the depth of the electron trap and time can be expressed as Eq. 6 (Watson, 1988; Watson, 1995): | E c − E tn | = kt ⁡ ln ( vt ) , (6) where |E _c - E _tn | is the electron trap energy level, k is the Boltzmann constant, and v is the electron vibration frequency. In addition, variation of the depolarization current I _d with time is shown in Eq. 7: I d ( t ) = qdKT 2 t f 0 ( w ) ⋅ N ( E ) , (7) where f ₀ (w) is the initial density of the electron trap, N (E) is the density of the trap energy level, q is the charge of the electron, and d is the thickness of insulation. Obviously, I _d (t) is directly proportional to N (E), and I _d (t) can be used to study the density of electron trap energy level in the insulating materials. In this study, the third-order exponential decay function is used to fit I _d (t) (Ye et al., 2016), as shown in Eq. 8: I d ( t ) = ∑ j = 1 3 α j e − t τ j , (8) where i represents three polarization types, i = 1 represents the subject polarization of the insulation material, i = 2 represents the interfacial polarization between crystalline and amorphous states, and i = 3 represents the interfacial polarization between various ions and groups after aging, α _i is coefficient, and τ _i is the time constant. Consequently, the charge can effectively characterize XLPE aging degree by aging factor A (Birkner, 2004), as shown in Eq. 9: A = Q ( τ 3 ) Q ( τ 2 ) . (9)

Larger A indicated a deeper level of traps and more serious aging conditions. However, it can be inferred from the preceding paragraph that there is a difference between non-uniform and uniform aging, whether from the PDC curve or Debye model parameters. In order to make assessment results more realistic and accurate, it is necessary to express I _d (t) by using Eq. 10: I d ( t ) = ∑ j = 1 , k = 1 3 α j ( k ) e − t τ j ( k ) , (10) where is the pre-exponential factor of the kth isothermal area in the jth branch, I _d (t) is equal to the sum of currents in the three branches of the modified Debye model, and the current of each branch is superimposed by the current of each isothermal zone. Therefore, the modified aging factor is expressed as Eq. 11: A mod ≈ ∑ k = 1 3 α 1 ( k ) τ 1 ( k ) + ∑ k = 1 3 α 2 ( k ) τ 2 ( k ) ( 1 − 1 e ) + ∑ k = 1 3 α 3 ( k ) τ 3 ( k ) ( 1 − e − τ 2 ( 1 ) + τ 2 ( 2 ) + τ 2 ( 3 ) τ 3 ( 1 ) + τ 3 ( 2 ) + τ 3 ( 3 ) ) ∑ k = 1 3 α 1 ( k ) τ 1 ( k ) + ∑ k = 1 3 α 2 ( k ) τ 2 ( k ) ( 1 − e − τ 3 ( 1 ) + τ 3 ( 2 ) + τ 3 ( 3 ) τ 2 ( 1 ) + τ 2 ( 2 ) + τ 2 ( 3 ) ) + ∑ k = 1 3 α 3 ( k ) τ 3 ( k ) ( 1 − 1 e ) . (11)

Table 5 shows A and A _mod of the samples. First of all, it can be noted that A _mod and A both increase with the degree of aging, and the degree of change becomes small gradually. It shows that the insulation performance of cable changes rapidly in early stages and is slow in the end. Second, A _mod is larger than A in each stage, and the gap between the two is widening, which means with the influence of temperature gradient on insulation is greater with the aggravation of aging. Finally, it is interesting that compared with the criteria in Table 6 (Gao et al., 2015), A indicates a good insulation stage of the cable, while A _mod indicates that the cable has been moderately aged for 15 days when the aging time is 7 days or 15 days. Consequently, in the qualitative assessment, there are usually some qualitative assessment errors for cables in the middle aging stage. In summary, A _mod characterizes the non-uniform aging condition of cable insulation more accurately, and a PDC test should be carried out for the cable in the middle of the operation, so as to ensure the stability of the power system to a greater extent.

The aging will directly affect the mechanical properties of the cable. It is well known that the elongation at break reflects the mechanical aging degree of XLPE, and the elongation at break retention EAB is a recognized standard to judge the insulation failure of XLPE. The insulation performance of the cable has been lower than the standard of safe operation when EAB = 50%. In order to further improve the accuracy of A _mod, this study analyzes the relationship between the insulation and mechanical properties. The EAB is calculated by using Eq. 12, and the corresponding relationship between A _mod, A, and EAB is established as shown in Figure 5. EAB = k k 0 , (12) where k ₀ is the elongation at the break of the unaged sample. The fitting curve of A and EAB shown in Figure 5 is derived by using Eq. 13: A = 3.643 × exp ( − EAB 0.315 ) + 1.408 . (13)

The fitting curve of A _mod and EAB is calculated by using in Eq. 14: A mod = 3.826 × exp ( − EAB 0.339 ) + 1.383 . (14)

If the value of the state boundary point of X₄ is brought into Eq. 13, the corresponding elongation at break can be calculated. Meanwhile, in a perfect state, 0.833 and 0.667 divide EAB into three sections. It is not difficult to find from Figure 6 that the error between the perfect value and the measured value is small, which shows that there is a corresponding relationship between mechanical characteristic quantity and electrical characteristic quantity. Based on this point, A _mod can be divided into four stages according to EAB, as shown in Table 7.

As mentioned before, A _mod is supposed to be more effective as a dielectric parameter for analyzing the non-uniform thermal aging state of XLPE cable. In order to verify the accuracy of the proposed evaluation reference, four XLPE cables in operation are considered as the research object of this part. The depolarization current and A _mod are shown in Figure 7.

In Figure 7, the PDC curve characteristics of cables with different service lives basically conform to the results in 5.1. In the early stage of life, although the service times of S₂ and S₃ are the same, S₃ has entered the early stage of aging. In addition, S₄ is a retired cable. Referring to Table 5, decommission will occur with A _mod = 2.21. The aforementioned results are consistent with reality. Meanwhile, considering that the A _mod evaluation criteria are based on mechanical characteristics, Figure 8 shows the measured values and calculated values of EAB for cables. The small error range further proves that A _mod is accurate and reliable to evaluate the aging degree of XLPE cables.