A Review of Aging Models for Electrical Insulation in Power Cables

Abstract

Electrical insulation is an integral part of power components. The aging of electrical insulation is an undeniable fact that limits the operational lifetime of power components. Apart from regular aging, abnormal stresses and the development of defects are real threats because of their contribution in accelerating the aging rate and thereby leading to a premature failure of the power components. Over the decades, various studies have been carried out to understand the aging behavior of electrical insulation mainly considering electrical and thermal stresses. Similarly, a number of mathematical (aging) models have been developed based on the theoretical and experimental investigations and evidences. However, a dependable formulation of the models that can provide more practical estimation of the insulation degradation profile has not been achieved yet. This paper presents a comprehensive review of the aging models considering single and multistress conditions. Further, the paper discusses possible challenges and barricades averting the conventional models to achieve a suitable accuracy. Finally, suggestions are provided that can be considered to improve the modeling approaches and their performance.

1. Introduction

While air provides the withstanding capability of 3 kV/mm between two conductors, the use of dielectric material increases this withstanding capability to the required voltage levels. In addition to increased electrical breakdown strength, the materials must be able to meet thermal and mechanical performance requirements [1]. Keeping performance in view, the reliability, cost, and environmental impacts are the major considerations during the insulation design of the components [2]. The operational lifetime of the power components depends on the reliability of the dielectrics. Insulation aging is a looming issue in all types of power components [3]. However, aging becomes more critical at medium or high-voltage operation as compared to low-voltage applications. During a survey of Stockholm, out of 1392 total failures, 263 were due to power transformers, 435 failures were caused by cables, and overhead lines were responsible for 20 failures where electrical insulation was considered as a significant cause of these failures [4]. Sudden failure of grid equipment can cause unplanned power outages, leading to huge economic and operational loss to customers and utilities. A better understanding of the aging behavior and its progression can help to foresee the incoming failures of critical network components, which in turn reduces the associated losses and interruption of the industrial, commercial, and social processes.

Considering the nature of the mechanism, aging can be divided into intrinsic and extrinsic processes. The former is a naturally occurring process that produces gradual irreversible changes in the insulation properties. This refers to an intuitive process which leads to unusual changes in physiochemical properties of the insulating materials that may affect a major part of the insulation. The chemical kinetic process causes the breaking of the chemical bonds. These bonds are continuously broken and rearranged under the influence of the electric field that forms moieties [5,6]. The moieties increase with time and collectively bring substantial changes to the microscopic and ultimately macroscopic properties of insulation, which may lead to accelerated degradation. Intrinsic aging normally takes longer to fail and is initiated due to cavities, defects, protrusions, and voids (CDPV), which are the result of poor manufacturing, transportation, installation, or operation and contribute to intrinsic aging in the future. These defects under the influence of thermal, electrical, ambient, and mechanical (TEAM) stresses pose a significant impact on insulation aging [7].

Diagnostic tests play an eminent role in evaluating the defects’ progression and impact of stresses. Research on the improvement of the proactive diagnostic capabilities has progressed well, especially considering the detection and location of insulation defects in the power components [8,9]. Similarly, prediction of lifetime has been a focus for the last 40 years [10,11,12]. It is well-understood that the progression of the insulation defects or insulation degradation is associated with the magnitude of stresses and condition of the insulation [13,14]. Various measurement methodologies are used to observe the insulation conditions, such as dielectric response in the time domain and frequency domain, infrared spectroscopy, degree of polymerization, moisture content, tensile strength and elongation, partial discharges (PD), and tan-delta [15]. The suitability of the adopted methodology is based on the type of equipment and the focus of diagnostics. The clarity in the interpretation of the measured quantities and their relationship with the aging mechanism or type of deterioration is vital for reliable diagnostics. Identification of the changing pattern as a reflection of progressing deterioration can provide a traceable behavior of insulation aging and prediction of its expected condition [16]. Due to stochastic characteristics of interest observed from the analyzed measurements, it is challenging to identify the degradation patterns. Therefore, reliable prediction of life expectancy of the component is a topic of increased interest for the power engineers [17].

Further in this article, Section 2 discusses aging from cause to consequence, typical cable defects, and the basic idea of the tests performed for the evaluation of insulation condition. Section 3 presents a holistic overview of various insulation aging models under thermal and electrical stress and discusses the threshold behavior of dielectric materials. Section 4 presents accelerated testing methods, limitations of available models in achieving general applicability, and prospective solutions for improved understanding of aging behavior.

2. Insulation Degradation in Solids and Aging Models

The degradation of insulation materials varies with their chemical composition, source (natural or synthetic), heat-resisting capability, and physical state, i.e., liquid, solid, and gas. Considering physical state, in liquids and gases degradation occurs due to contamination, solid impurities, and dissolved gasses. These impurities under the influence of continuous electric field lower the dielectric strength, which ultimately leads to the breakdown. In solids, electrical breakdown causes the final failure of insulation; however, electrical stress itself is not the dominant aging factor under normal operating conditions. The dominant degradation is caused by the combination of different stresses, among which thermal and mechanical stresses are more significant. Operational, physical, and environmental factors may lead to deteriorating effects and cause additional aging, such as the rise of temperature, bending or compression, contaminations or voids, moisture, etc. [18,19,20].

The effects mentioned above become the sources of different types of stresses that can be categorized as thermal, electrical, ambient, and mechanical stresses. Based on the magnitude of these stresses, the irreversible incipient degradation mechanisms are initiated in terms of oxidation, decomposition, electrical treeing, PDs, and space charges. As a result of these aging mechanisms, the material properties deteriorate, and dielectric strength is reduced at a localized level that is gradually extended across the insulation between conducting parts. Consequently, this leads to an electrical breakdown of the insulation, and the failure of the component occurs. An overview of the insulation degradation from initiation of aging into the possible failure is presented in Figure 1. The arrows indicate that the induced stresses are interrelated, as thermal stress is a result of mechanical stress. The bending and compression cause a temperature rise, which produces thermal stress. Cables operating in wet environments may cause corona discharges that increase electrical stress in cables. Mechanical stress relies on the operational and environmental conditions (harsh or mild). The damage to the cable sheath is due to mechanical stress, i.e., bending or compression, letting the moisture enter into the cable sheath, resulting in environment stress and vice versa. It also reduces the electrical performance of cable.

Considering underground cables as a critical component of the medium-voltage (MV) grid having dielectric a major part of its physical design, typical insulation defects are depicted in Figure 2. The most alarming defects lie within the main insulation between shielding and the conductor [21,22].

Water treeing is a serious concern in the extruded cables operating in wet environments where the potential aqueous infiltration can produce conditions that are favorable for the inception and development of water trees. These trees are effective areas inside cross-linked polyethylene (XLPE) insulation with reduced dielectric strength, and can therefore also contribute to aging, e.g., via consequent initiation of electrical treeing. Electrical trees are initiated when the insulation is subjected to high electrical stress for a sufficiently long time in the presence of (CDPVs). The electric trees are formed due to sustained PD activity, which causes gradual local disintegration of the insulation. Both water trees and electrical trees tend to expand in the direction of the electric field they are exposed to [23,24].

The aim of aging studies and lifetime modeling is to relate the life with the associated stresses which can be achieved by experimental studies. Considering experimentation, accelerated-aging tests can be more efficient since the routine testing methods take a longer time. Typically, underground cables are designed for a service life of 30–40 years. Accelerated aging tests can be used to estimate an asset’s life in a short span with a suitable accuracy. IEEE 1407 defines well-established guidelines for accelerated tests for medium-voltage cables (5–35 kV), according to which accelerated water-treeing tests (AWTT) are performed where cables of various lengths are kept inside water-filled pipes from four months to one year. Later, the step AC breakdown voltage is applied, and insulation behavior is investigated. Another method where cables are aged in tanks instead of water pipes for a known period of time or until failure time is called the accelerated cable-life test (ACLT). In this method, the step AC voltage is applied under different temperatures and voltages. This method is very useful for the investigation of the overall aging process, i.e., from inception and growth to final failure [25]. IEEE 98 defines test procedures for thermal endurance (TE) characterization of insulating materials based on analytical methods, concluding that TE tests must be performed at three or more ranges of temperatures for a duration of more than 100 and 5000 h (about 7 months) for the shortest and longest test, respectively [26]. A shorter method for TE characterization purposed in [27] is based on the oxidation-stability measurements to estimate the activation energy (minimum energy required to initiate a chemical reaction) of degradation. Regarding measurement techniques, nondestructive tests such as insulation resistance, tan delta, and PD are comparatively popular.

IEC-60270 defines PD as an electrical discharge that partially bridges the insulation between conductors, which may or may not occur adjacent to a conductor. IEC also provides guidelines for PD measurement where the PD inception voltage (PDIV) is applied at nominal frequency i.e., 50 Hz, and PD pulses are measured at increasing voltages. These pulses can be measured by various sensors such as coupling capacitors [28]. Other devices such as high-frequency current transformer (HFCT), Rogowski coil, ultra-high-frequency (UHF) sensor, and electric-field sensors (D-dot) are also used for PD pulse measurement [14,29]. The results demonstrate the fault location, intensity, and type of PDs.

Modeling has made prominent progress from the Dakin physical model [5] to the Simoni multistress life models [30,31,32] and Dissado space-charge model [33]. These models predict insulation life under single or multiple stresses. The study of the insulation under multistress conditions can provide a more compressive aging response. Multiple stresses are not simply additive, but they are coactive and might have a direct or indirect impact; or the other way around, they might have sequential impact [34]. Although the accelerated-aging tests evaluate the insulation condition efficiently, the life estimation at a particular instance of service requires statistical tools and life models. Due to their significant degradation impact, thermal and electrical stress based aging models are discussed in the next section considering single-stress and multi-stress approaches, as shown in Figure 3. Multistress models are further classified into phenomenological, physical, and thermodynamic models.

3. Single-Stress Aging Models

3.1. Electrical-Aging Models

Electrical aging occurs due to the presence of free electrons and other charge carriers. The free electrons at low concentrations start colliding frequently with the lattice of dielectric materials and ionize molecules of insulating materials by transferring energy. At a high concentration, this collision liberates additional free electrons and collectively they produce an avalanche effect [35,36]. The excess of free electrons causes the current to flow within the insulation. At high stress, heating is produced due to the high current, which thermally deteriorates the material, thus contributing to aging. Introducing a relationship between electrical stress and time to breakdown, inverse power law (IPL) predicts the lifetime of the insulation as

(1)${\mathrm{t}}_{\mathrm{f} ={ \mathrm{aE}}^{-\mathrm{b}}}\hspace{1em}(\mathrm{a} > 0)$

where t_f is failure time, E is applied electrical stress, and a is the constant to be estimated by electrical life data. The parameter b is the voltage-endurance coefficient (VEC), which is the slope-of-life graph between applied electrical stress and life [37]. Life prediction through IPL is based on the accelerated-aging-test results, as it is the relation between electrical stress and time to failure [38]. It has been observed that electrical-endurance data or the relation between insulation life and applied stress can be best described using two-parameter Weibull distribution [39,40]. The cumulative density function for Weibull distribution is expressed as

(2)$\mathrm{f}\left(\mathrm{t}\right)=1-{\mathrm{e}}^{-[(\frac{\mathrm{t}}{\mathrm{p}}{)}^{\mathrm{q}}]}, \mathrm{t}\ge 0$

where f(t) is failure probability at applied stress or time, and p and q are the scale and shape parameters and are positive. The parameter p illustrates the voltage or (time) at which f(t) becomes 63.2%, whereas q is the measure of the dispersion of data such as variance in normal distribution and t is time to failure or breakdown value of electrical stress. Figure 4a is a graphical representation of IPL that depicts an inverse relation between electrical stress and insulation life; as electrical stress increases, the insulation life decreases [38]. It is observed that at lower value of electrical stress, the insulation life graph tends towards the electrical threshold E_O This behavior is best described by the exponential model, as shown in Figure 4b. The model represents the voltage endurance of the insulation based on a threshold value that permits faster and accurate results [41], expressed as

(3)${\mathrm{t}}_{\mathrm{f} ={ \mathrm{k} \mathrm{e}}^{\left(-\mathrm{nE}\right)}}$

where k and n are the constants ought to be found by electrical-endurance experimental data [38]. Both models have wide application for electrical endurance and are extensively used to study insulation life under single and multiple stresses.

In many cases, particularly when the diagnostic tests are extended for a long time, the electrical lines show their tendency towards the electrical threshold E_O. The life graph of polyethylene terephthalate (PET) film with surface discharge shows an S-shape curve in Figure 5a. The green curve (semi-log plot) follows IPL at first and later tends towards E_O at a low value of electrical stress, showing the practical existence of the electrical threshold [42]. The same material was tested in [43] at different temperatures, as shown in Figure 5b. It was observed that the threshold value decreased with increasing temperature; however, its presence was still there even at 130 °C. The tendency of the life graph towards the electrical threshold (E_O) can be due to the extrapolation of the experimental data or due to operating conditions of the insulation.

Materials that adopt this behavior can be expressed by adding a threshold value in both inverse-power-law exponential models. Thus (1) and (3) are extended as (4) and (5). Adding a threshold value mathematically in the life equation, e.g., E_O in (5), gives precise results, since it is impractical to include stress values below the threshold.

(4)${\mathrm{t}}_{\mathrm{f}}={\mathrm{t}}_{\mathrm{o}}{[\frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{O}}}]}^{-\mathrm{b}}$

(5)${\mathrm{t}}_{\mathrm{f}}={ \mathrm{t}}_{\mathrm{O}}\left[\frac{\mathrm{k}}{\mathrm{E}-{\mathrm{E}}_{\mathrm{O}}}\right]{\mathrm{e}}^{\left[-\mathrm{n} \left(\mathrm{E}-{\mathrm{E}}_{\mathrm{O}}\right)\right]}$

where E is applied voltage, t_f is insulation remaining life, and t_O is life at stress E_O [32]. Haideo also observed threshold behavior in dielectric materials such as polyethylene and polyethylene-terephthalate. His work illustrates that the life of electrical insulation follows a Z curve called flat Z characteristics, as shown in Figure 6. Three regions of representation are provided, while region I shows linear relation between life and applied stress following IPL; it is assumed that threshold E_T appears in the third region [44].

It can also be noticed that during region I and region III, the rate, i.e., (dv/dt_f), is lower than in region III, which implies that the degradation rate is high in the transition phase, which is region II.

3.2. Thermal-Aging Models

Thermal stress is one of the prominent causes of insulation aging, as it produces irreversible and enduring reactions that eventually affect the dielectric properties of insulation, thereby increasing the dissipation factor or conductance. Dakin et al. [27] identified three processes that lead to thermal aging of insulation, as follows:

• increase in dissipation factor and conductance due to oxidation.

• brittle hardening of the insulation due to reduction in the plasticizer effect.

• depolarization of plastic insulation at elevated temperatures.

Montsinger also described aging as a function of time and temperature. According to him, an increase in temperature (8–10 °C), reduces equipment life by half [45]. The relationship between temperature and reaction rate of a chemical reaction was first aimed at by Arrhenius in Equation (6), expressing it as the rate of a chemical reaction directly proportional to the temperature [46].

(6)$\mathrm{K}={\mathrm{ke}}^{-\frac{E_a}{\mathrm{RT}}}$

where K is reaction rate, k is rate constant, T is absolute temperature, and R is general gas constant [47]. Here E_a is the activation energy that determines the rate of the chemical reaction. It is proportional to the slope of the line between the rate of reaction and the inverse of temperature. High activation energy represents slow chemical reaction. A modified form of (6) is proposed in (7), as (6) deals with the kinematics (reaction rates) only in gaseous form. [48]. However, expression (7) is able to support heterogenous reactions as well, i.e., reactions in which two or more forms of matter (liquids, gases, and solids) take part as reactants, and is presented as

(7)$\mathrm{K}=\frac{{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{\mathrm{h}}{ \mathrm{e}}^{-\frac{\Delta \mathrm{H}}{\mathrm{RT}}}{\mathrm{e}}^{\frac{\Delta \mathrm{S}}{\mathrm{R}} }$

where K_B and h are the Boltzmann and Planck’s constant, ∆H is activation enthalpy, ∆S is activation entropy. If life L_T is insulation life under thermal stress (assumed to be inversely proportional to the reaction rate), then (6) becomes

(8)${\mathrm{L}}_{\mathrm{T}}={\mathsf{\alpha }\mathrm{e}}^{\left(\frac{\mathsf{\beta }}{\mathrm{T}}\right)}$

where β = E_a/R is constant and function of activation energy of reaction [49]. The value of α is mainly dependent on the insulation characteristics and failure mode. The thermal graph shown in Figure 7 is obtained by extrapolation of data points and temperature indices, including temperature index TI [50,51].

The linearity of the thermal life graph is dependent on the constancy of rate of reaction at all ranges of temperature and on the assumption that the material only undergoes thermal stress. However, the tests performed under combined (electrical and thermal) stress showed that thermal life graph lines followed curved patterns [52,53]. Figure 8 presents the thermal lines of XLPE cable under different values of electrical stress.

It was observed that thermal lines adopt curvilinear behavior with increasing magnitude of electrical stress. The plots demonstrate that at the start, the thermal lines exhibit linear behavior, but at higher values of electrical stress, the lines tend towards thermal threshold Tt, which reflects the practical existence of the thermal threshold. Considering the threshold behavior, a modified form of (8) derived from the Eyring relationship proposed in [52] is expressed as

(9)$\mathrm{L}={\mathrm{K}}_1{\mathrm{T}}^{\mathsf{\gamma }}{\mathrm{e}}^{\left(-\frac{\mathsf{\beta }}{\mathrm{T}}\right)}{ \mathrm{e}}^{\left\{\left[{\mathrm{K}}_2+{\mathrm{K}}_3/{\mathrm{T}}_{\mathrm{T}}\left(\mathrm{S}\right)\right]\right\}}$

Here, S is stress other than thermal, which can be electrical or mechanical. It can be seen from (9) that for s = ≤0, there will be thermal aging only. K₁, K₂, K₃, and ϒ are parameters of the Eyring model, extracted by thermal-endurance data. In addition, Figure 8 shows that the insulation life reduces drastically at higher values of electrical stress even at low temperature, which indicates that electrical stress also influences thermal aging.

The models based on single (individual) stresses provide a clear understanding of the aging behavior and prediction of lifetime with good accuracy and flexibility. However, in a more practical scenario, insulating materials operate under multiple stresses, which are discussed in the next section.

4. Multistress Aging Models

During regular operation, equipment is exposed to multiple stresses simultaneously. Observing their combined impact provides a more comprehensive understanding of the aging behavior under operational conditions [46]. If any abnormality of thermal stress has primarily accelerated the aging rate due to the oxidation process in air, its interaction with electrical stress further increases the temperature and can initiate the treeing phenomenon. Conversely, the impact of electrical as the primary stress and thermal as the secondary stress substantially changes the PDIV and intensity of PD [30]. Thus, multistress models consider the synergistic effect of applied stresses that are multiplicative in nature, not additive. A generalized life model (10) for insulating materials (threshold and non-threshold) under multiple stresses, say, n, is expressed as in [49]:

(10)$\frac{\mathrm{l}}{{\mathrm{l}}_0}=\frac{{\mathrm{l}}_1}{{\mathrm{l}}_0}\frac{{\mathrm{l}}_2}{{\mathrm{l}}_0}\dots \dots \frac{{\mathrm{l}}_{\mathrm{n}}}{{\mathrm{l}}_0} \mathrm{C}({\mathrm{S}}_1{\mathrm{S}}_2,\dots \dots {\mathrm{S}}_{\mathrm{n}})$

where l is life under multiple stresses. Further, l₁, l₂ …, l_n are individual lives for stresses S₁, S₂ …, S_n and l₀ is life at a certain stage called reference life. The constant C is the correction factor, which is necessary to include since at higher values of stresses, the multiplicative model yields a solid reduction in the insulation lifetime [54]. Multistress models are based on Arrhenius and Eyring’s equations for thermal and mechanical endurance, inverse, and exponential models for electrical endurance. These models can be divided into three categories:

• Phenomenological Models.

• Thermodynamic Models.

• Physical Models.

4.1. Phenomenological Models

The empirical models are based on the aging phenomenon in accordance with the principal theories; however, their parameters hold no physical meaning. They are designed by analyzing the failure data of experiments or accelerated-aging-test results that cannot be found directly as material properties. Phenomenological life models are relatively successful in predicting the life of insulation at any stage. The major characteristics of these models can be stated as follows [55]:

• Depicts the relationship between stress (es) and life of the equipment based on the experimental data;

• Characterizes insulating materials based on experimental evidence rather than physical properties;

• Predicts life at any stage through extrapolation of the test data;

• Statistical and statistical or probabilistic approaches can be used to find model parameters.

Over the past three decades, a number of multistress models based on experimental results have been proposed. Among the first phenomenological multistress models, Fallou’s model presented in [56] is expressed as

(11)$\mathrm{L}={\mathrm{B}}_1[{\mathrm{e}}^{\left({\mathrm{B}}_2\mathrm{E}\right)}{\mathrm{e}}^{\frac{{\mathrm{C}}_1}{\mathrm{T}}}{ \mathrm{e}}^{\frac{{\mathrm{C}}_2\mathrm{E}}{\mathrm{T}}}]$

where B₁, B₂, C₁, and C₂ are the electrical constants, determined by experimental results. The interesting thing about this model is that each of the three terms in (11) represents the aging process, i.e., first term—electrical aging, the second term—thermal aging, and third term—aging under combined stress (14). The model is only valid for E > 0; thus, it does not consider the impact of threshold value or lower value of electrical stress near to zero. The exponential term in the model holds curvilinear properties at stresses near to zero, as shown in Figure 4b. Therefore, at a lower value of applied electrical stress, the extrapolation of the data in the model does not give accurate results. The major contribution in the field of multistress aging was made by Simoni [31]. The proposed models bear high credibility as they are developed based on broad research and experimental evidence. Simoni’s first multistress model is expressed as

(12)$\mathrm{L}={ \mathrm{K}}_{{\mathrm{n} \mathrm{L}}_0}\frac{e^{\left(-\mathrm{hE}-\mathrm{BT}+\mathrm{bET} \right)}}{\frac{{\mathrm{E}}_{\mathrm{G}}}{{\mathrm{E}}_{\mathrm{G}}{}_{\mathrm{to}}}+\frac{\Delta \mathrm{T}}{\Delta {\mathrm{T}}_{\mathrm{to}}}-1}$

where k_n is constant depending on insulation short-time characteristics, h and B are electrical (coefficient of inverse-power model) and thermal parameters, b is parameter for combined stress, E_G is electrical gradient, ∆T is change in temperature, and E_Gto and ∆T_to are electrical gradient and thermal threshold at room temperature, respectively. One of the major characteristics of Simoni models is the life surface, as shown in Figure 9: a three-dimensional graph that is a geometrical representation of life under combined stress. The surface intersection with axes gives electrical lifelines at constant temperature and thermal aging at constant electrical stress. Additionally, at constant life (L), the isochronal (hyperbolic) lines of thermal and electrical stress in correlation with selected life can also be acquired from the surface model.

Simoni’s observations during experiments proved to be a breakthrough in defining the threshold materials i.e., it is not material, which is threshold or non-threshold rather behaviour, depends on the value of applied stress. If at least one of the applied stresses is less than the threshold, e.g., at stress E < E_o the lifelines approach thermal threshold and vice versa. Thus, the material behaves as threshold material. If all applied stresses are higher than the threshold value, the material behaves as non-threshold material [33]. This led Simoni to propose a unique model which is valid for material behavior (threshold and non-threshold) expressed as

(13)$\mathrm{L}={\mathrm{L}}_0\frac{{\mathrm{e}}^{\left(-\mathrm{hE}-\mathrm{BT}+\mathrm{bET} \right)}}{{(\frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{to}}}+\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{to}}}-\left({\mathrm{bTt}}_{\mathrm{o} }\mathrm{ET}\right)/\left({\mathrm{h} \mathrm{E}}_{\mathrm{to}}{ \mathrm{T}}_{\mathrm{to}}\right)-1)}^{\mathsf{\beta }}}$

where

(14)$\mathsf{\beta }={\mathsf{\beta }}_{\mathrm{O}}\left[{\left(\frac{\mathrm{E}-{\mathrm{E}}_{\mathrm{to}}}{{\mathrm{E}}_{\mathrm{to}}}\right)}^2+{\left(\frac{\mathrm{T}-{\mathrm{T}}_{\mathrm{to}}}{{\mathrm{T}}_{\mathrm{to}}}\right)}^2\right]$

Equation (13) has seven parameters: L₀ is life at some reference point or at room temperature, E is applied electrical stress (E−E_to), and E_to and T_to are electrical and thermal thresholds at room temperature, respectively. The value of β_O, ultimately β, defines material behavior. It can be observed that by putting β = 0 in (13) the model represents the non-threshold behavior of insulating material, which means all stresses must be greater than or equal to corresponding threshold values, i.e., E ≥ E_to and T ≥ T_to. Yet if β > 0, the model represents threshold behavior, inferring that either of the applied stresses must be less than the respective threshold value, i.e., (E < E_Gto or T < T_to). Moreover, the values of β model validity rely on the following boundary conditions:

• In absence of electrical stress in other words (E ≤ E_Gto), the model must tend to the thermal model; however, at T ≤ T_to, the life function must tend to infinity.

• Likewise, in the absence of thermal stress the model must tend to the electrical model and at lower value of electrical stress (E ≤ E_to), the life must follow upward curvature (tend to infinity).

In the absence of both stresses, the model must converge to the linear model. Considering the versatility of this model, it can differentiate the behavior of materials based on the values of applied stresses in contrast with the threshold. It can also predict the life or aging in multiple scenarios given that the suitable test properties and criteria for failure are used for thermal and electrical-endurance data processing. However, there are higher chances of uncertainties calculating parameters through estimating methods such as maximum likelihood, even with the high volume of experimental data [57]. The phenomenological models have the capability to be transformed into probabilistic models based on the experimental data results [38,47,58]. A probabilistic model proposed in [59] can describe insulation aging of material subjected to thermal and electrical stress. The IPL-based model is expressed as

(15)$\mathrm{F} \left(\mathrm{t},\mathrm{E};\mathrm{T}\right)=1-e^{{\left[-\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{s}}}{\left(\frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{s}}}\right)}^{\mathrm{n}}\right]}^{\mathrm{q}\left(\mathrm{E},\mathrm{T}\right)}}$

where $n=\frac{{\mathrm{n}}_{\mathrm{i}}}{{\left(1-\frac{{\mathrm{E}}_{\mathrm{s}}-\mathrm{E}}{{\mathrm{E}}_{\mathrm{s}}-{\mathrm{E}}_{\mathrm{T}}}\right)}^{\Delta }}$. Here, n_i is the initial electrical-endurance coefficient, t is time to failure at stress E and ts is time to failure at electrical stress E_s, T represents thermal stress or temperature, and v U is shape parameter (v = v (T)). It can be observed that the value of the power term n is dependent on electrical stress E, temperature T, and v, which is different from previous models, dependent on E and T only. In addition, this model has no separate term for thermal stress like we have seen in previous models. Although it is a novel approach, implemented on XLPE-insulated cables, the general validity of the model for materials with threshold behavior still requires experimental evidence gathered on varied materials [45,46].

4.2. Physical Models

The major drawback of phenomenological models is the model validity, which solely depends on the experimental results since the parameters can only be found after the aging test. This process often takes longer and is difficult to handle a large amount of data using statistical means. On the other hand, physical models are based on the aging mechanisms, laws, and theories. These models support aging in insulation parameters of later models that have physical meaning and can be found directly from the measurement of physical quantities [55]. The physical model based on the PD phenomenon was initially proposed by Bahder et al. [60] as

(16)${\mathrm{L}}_{\mathrm{E}}=\frac{1}{{\mathrm{fb}}_1[{\mathrm{e}}^{{\mathrm{b}}_2\left(\mathrm{E}-{\mathrm{E}}_{\mathrm{tac}}\right)}-1]{ \mathrm{e}}^{{\mathrm{b}}_3\left({\mathrm{E}}_{\mathrm{b}}\right)}+{\mathrm{b}}_4}$

The b₁, b₂, b₃, and b₄ are constant terms and their values depend on frequency f and shape of applied stress i.e., impulse, DC, or AC. Here E_b is the voltage at which electrical breakdown occurs and E_tac is the threshold value of applied stress (AC). The value of E_tac is influenced by the shape and type of void or defect and insulation temperature. A similar model based on the tree growth and PD test is proposed in [61], expressed as

(17)$\mathrm{F}\left(\mathrm{t},{\mathrm{Q}}_{\mathrm{i}},\mathrm{E}\right)=1-e^{{\left[-\left(\frac{{\mathrm{tk}}_5(\mathrm{E}-{\mathrm{E}}_{\mathrm{T})}{}^{\mathrm{n}}}{\ln {\left[\left(\frac{{\mathrm{Q}}_{\mathrm{i}}}{{\mathrm{k}}_2}\right)+1\right]}^{\mathrm{d}}}\right)\right]}^{\mathsf{\beta }}}$

This expression describes the probability of insulation failure, where β and n are the parameters of Weibull distribution; n is also the electrical-endurance coefficient; d is tree depth and can be found from fractal characters of the tree; and k₂, k₅ are the constants, whose value relies on the material type and tree-growth phenomenon. The model is valuable in finding the residual life of the insulation at a chosen value of charge Q_i at time t.

Although the above models are based on the novel PD testing, the life estimation through this model depends on electrical tree growth. Knowing the fact that electrical treeing is one of the hazardous haps that severely affects insulation, it is therefore necessary to predict the insulation condition prior to the tree growth. In addition, this model needs sufficient PD data and processing tools for accurate estimation of the model parameters and constants.

4.3. Thermodynamic Models

Typically, thermodynamic models illustrate that the insulation degradation or aging is due to the reactions, activated by heat or temperature. These reactions form moieties that cause degradation. Both electric field and temperature are responsible for these reactions, as temperature provides free energy to electrons to overcome the barrier, and electrical stress reduces the height of that barrier [62]. Crine et al. [63] proposed a multi-stress model based on the thermodynamic approach, expressed as

(18)${\mathrm{t}}_{\mathrm{b}} \cong \frac{\mathrm{h}}{{\mathrm{K}}_B\mathrm{T}}{e}^{\left(\frac{\Delta \mathrm{G}-\mathrm{E}ƛ\mathrm{F}}{{\mathrm{K}}_B\mathrm{T}}\right)}$

∆G is Gibbs free energy, ƛ is scattering distance, and F is frequency. As seen from the equation as E→0, the life t_b will tend towards infinity. In this case, ƛ will be contingent on thermal quantities, i.e., temperature, ad must relate to physical properties (structure and size of micro voids and small cavities that grow in the material due to structural defects accelerated by electrons). The model presumes that electrons have sufficient energy to accelerate reactions that break the weakest bonds; this indicates either that there is a considerable number of defects or micro voids already present at the start of the aging process, or the presence of high-voltage stress or a simultaneous effect [63,64]. Another model proposed in [65] is based on the chemical characteristics of insulation. The breaking of chemical bonds under multiple stress(es), i.e., thermal, and electrical, form micro voids. These micro voids can defuse with each other and form cracks that contribute to deterioration that can be observed from the expression as

(19)$\mathrm{t}={{\displaystyle \int }}_{\mathrm{N}}^{{\mathrm{n}}_{\mathrm{c}}}\{\frac{{\mathrm{k}}_B\mathrm{T}}{\mathrm{h}}{[(e^{\left(\frac{-{\mathrm{U}}_{\mathrm{r}}\left(\mathrm{E}\right)}{{\mathrm{k}}_B\mathrm{T}}\right).\left(N-n\right)}-e^{\left(\frac{-{\mathrm{U}}_{\mathrm{b}\left(\mathrm{E}\right)}}{\mathrm{kT}}\right).\left(n\right)}]\}}^{-1}\mathrm{dn}$

Here, t is time to failure or time at which crack growth due to voids is initiated, n is number of bonds that are broken, n_c is their critical quantity, N is the quantity that can be breakable, and U_r(E) and U_b(E) are activation energies required for forming and breaking of bonds. Taking a similar concept Dissado described the role of space charges in insulation aging. Space charges are evolved in polymeric insulation as a result of charge injection by electrodes, where some charges may be trapped within the insulation. At high electrical stress, these trapped charges attain enough energy to break bonds thus changing chemical properties of a material at microscopic level and initialize degradation. The acceleration of degradation is a thermodynamic process [66]. This study focuses on the microscopic characteristics of insulating materials, which is different from the previous models, considering macroscopic characteristics as major cause of failure. Dissado el at. [67] proposed a model called DMM (Dissado–Mazzanti–Montanari), based on the space-charge degradation phenomenon under combined stress (thermal, and electrical) as

(20)$\mathrm{L} \left(\mathrm{E},\mathrm{T}\right)=\frac{\mathrm{h}}{2K_B\mathrm{T}}{ \mathrm{e}}^{\left[\frac{\frac{\Delta \mathrm{H}}{K_B}-\frac{{ {\mathrm{C}}^{\prime } \mathrm{E}}^{2\mathrm{b}}}{2}}{T}-\frac{\Delta \mathrm{S}}{k}\right]\ln \left[\frac{{\mathrm{A}}_{\mathrm{e}}\left(\mathrm{E}\right)}{{\mathrm{A}}_{\mathrm{e}}\left(\mathrm{E}\right)-{\mathrm{A}}^{*}}\right]{[\mathrm{Cos}\mathrm{h} (\frac{\frac{\Delta }{k}-{ {\mathrm{C}}^{\prime } \mathrm{E}}^{2\mathrm{b}}}{2T})]}^{-1}}$

where A represents degraded moieties, A* is the critical value of A (value at which large cavities are formed, sufficient for PDs and therefore the electrical tree), Ae (E) is A when equilibrium occurs between forward and backward reactions, and Δ is the difference of free energy between reactants and products. The constant b depends on charge-trapping and insulation-degradation quantities, and constant C’ depends on material properties such as element strain, permittivity, etc. Interesting results have been found with the implementation of (20) with certain assumptions. Firstly, it is assumed that the insulation holds a homogenous structure. As in the past, insulating materials were manufactured with imperfections (in small size and quantity), and thus can be considered homogeneous on macroscale level. In addition, it assumes that small voids are present during manufacturing. However, advanced manufacturing techniques produce extra clean materials, infers that there are low chances of defects at the manufacturing. On the other hand, charge carriers trapped in such materials could have a remarkable impact on aging. This concludes that there is an efficient need to investigate insulation degradation based on the space-charge-trapping mechanism in presently available materials. Secondly, the model predicts the inception of cracks and micro voids that result in PD or breakdown later, but not the failure time or breakdown itself. This means that the model is extremely useful in cable design and manufacturing, as in-service insulation is more concerned with failure time and lifetime prediction.

5. Discussion

Improved understanding of the insulation aging and life prediction is essential for the reliable operation of power components. Development of defects as a result of manufacturing or certain stress reaching above the specified range causes initiation of the degradation mechanisms, leading to final breakdown. The complexity of the mechanisms makes it challenging to develop well-defined degradation models to predict the lifetime or failure time of the affected component with a reliable degree of accuracy.

This paper provides a comprehensive review of aging models, highlighting the competencies and barricades of the models under single and/or multiple stresses. The single-stress approach is relatively simple in terms of experimental analysis, data interpretation, and relating insulation life to the corresponding stress. However, more practically, the power components operate in multistress environments; thus, predicting lifetime under single stress can be indecisive in many cases. Multistress models offer improved estimation of insulation life, keeping in view variation of insulation characteristics. Phenomenon-based models provide good understanding of insulation-aging behavior, as model terms rely on the experimental results. Although model parameters have no physical meaning, these models can be very helpful in the comparison of different insulating materials. On the other hand, physical models consider the aging mainly caused by electrical stress. It can be noticed in (16) and (17) that the models represent electrical and fractal quantities only. Thermodynamic models—particularly the space-charge approach—bear high significance in predicting the development of defects (voids and cavities) that result from space charge, but the insulation operating in-field is much concerned about the time-to-failure approach instead.

Therefore, the available multistress models are quite useful in predicting the insulation life. However, there are certain observations that limit their general applicability:

• It has been observed that most of the models are material-oriented, as their validity is examined on certain materials. It can be seen in Table 1 that most of the models are tested on XLPE and ethylene-propylene rubber (EPR) cables. However, insulation characteristics or aging behavior varies with material type, manufacturing techniques, operating environments, and level of stress applied. The life of cables with the same insulation installed in different networks is different. This depends on the environment and operational conditions. As the operational conditions keep on changing, the degradation rate changes hence the time to failure will change. This challenges the general validity of models.

• Experimental surveys [43,68] show a wide variation in the life graphs of various insulation materials when they come under single or multiple stress. Due to randomness in experimental data points, the graph does not follow a clear pattern. Thus, it is quite unwieldy to fit data accurately in one model. Although the model in [33] is valid for threshold and non-threshold behavior of dielectric materials, the life equation relies on several parameters. Almost all models contain parameters, indicated in Table 1, that can be found by different statistical methods such as maximum likelihood, linear regression, etc. The parameter resembles variables in an equation. It is cumbersome to handle multiple variables when it comes to experimental analysis of combined thermal and electrical stress.

• Considering the operation of the new components being proliferated in the modern grid, such as power electronics converters (PEC), it is found that available models do not incorporate their effect. A model proposed in [69] considers the impact of PEC and describes overall aging or degradation rate as the sum of electrical stress (peak-to-peak value of voltages) at nominal frequency (50 Hz) and the impulse frequency expressed as

(21)${\mathrm{L}}_{{\mathrm{f}}_1}=[{\left(\raisebox{1ex}{${ {\mathrm{U}}^{\prime }}_{\mathrm{pk}}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{pk}$}\right.\right)}^{\mathrm{n}} \left(\raisebox{1ex}{${\mathrm{f}}_1$}\!\left/ \!\raisebox{-1ex}{$50$}\right.\right)]$

where U_pk/pk is peak-to-peak voltage, n is VEC determined from experimental results, and f₁ is the frequency at which insulation life is to be found. Unlike conventional models, it assumes no synergism between stresses, i.e., between voltage and impulse frequency. In addition to that the model does not consider the impact of thermal stress, the model terms represent the impact of electrical stress and frequency only.

The operational lifetime of the equipment can be followed by the insulation’s remaining life in terms of time to failure or likelihood of failure. A comprehensive aging model can be developed based on the most representative combination of both time to failure and activity to failure and considering pros and cons of reviewed aging models mentioned in Table 2. Any hard event or abnormal stress conditions can shorten life and unexpected failure may occur. To predict the breakdown moments, it should be more useful to observe the outcome of degradation instead of the mechanism of degradation. Observing the outcome of the degradation can provide a direct indication of the insulation conditions. PD measurements provide a direct illustration of the insulation conditions. Initial studies conducted by the authors demonstrate the characteristics of PD signals such as PD magnitude and pulse-repetition rate analyzed from the PDIV up to the breakdown of the insulation due to the gradual increase in electrical stress [16]. The variation in these characteristics provides a cognizable pattern during various stages of degradation. Further characteristics of the PD signals in time and frequency domain should be investigated to understand the aging behavior in terms of insulation lifetime.

The measurement methodology can play a key role in improving the accuracy of life prediction capability. Nondestructive testing is considered safe and cost-effective without causing damage to the equipment or interruption in the plant’s operation. The role of accelerated aging tests is crucial in predicting insulation life in a short time span. However, the deviation of results from real operation tests or routine tests is a matter of concern, since accelerated-aging tests are performed at elevated levels of stress, which do not take into account all the dynamics associated with real operations. A term called accelerated factor A can be used to correlate accelerated-aging tests to normal routine tests [70], is expressed as

(22)$A=\frac{t_{normal}}{t_{accelerated }}$

where the times t_normal and t_accelerated are those at which the property (under test) achieves the same value after the normal and accelerated-aging test. A high value of A shows big variation in results of the normal and accelerated-aging tests.

In addition to nondestructive testing, destructive testing is needed to observe in-depth degradation behavior. Destructive testing allows for understanding the pattern of changing PD characteristics up to a certain stage of degradation. Based on the authors’ own experience during recent destructive testing, the proximity of failure provides distinct behavior that must be observed.

6. Conclusions

This paper reviews various aspects of insulation aging in solids, including theoretical and experimental studies describing the aging phenomenon, aging mechanisms, useful measurement techniques, and their role in the development of efficient aging models. Major contributions in the field of insulation life modeling were made during the late 1980s and 1990s. While continuous advancements are being made on proactive diagnostic solutions, no major breakthroughs have been observed during the last two decades in developing well-defined formulation to determine the aging indices. Further, the conventional aging models discussed in this paper highlight the limitations of available models in achieving accuracy and general viability. The major limitations of the models include their viability when tested on specific materials, and also that most of the model studies do not incorporate modern grid-operating conditions. The impulsive growth in hybrid girds, DC links, impact of power electronic converters, and EV charging has raised concerns over the reliablity of insulation. As there is not enough research on insulation behavior under these conditions, future work will be oriented towards investigating all these major aspects concerning modern grid operation. The insulation samples can be tested under IEEE 1407 guidelines where samples of cable are placed in tanks for four to six months, and later, a partial-discharge test will be conducted under different scenarios. PD activity offers greater visibility and continuous tracking of the insulation-degradation process. The statistical analysis of PD data from inception to breakdown will lead to the development of the aging model. Due to the close relationship between the characteristics of the PD activity and the mechanisms of insulation degradation, PD measurement can determine the state of the components more effectively. Low-cost nonintrusive measurement sensors such as Rogowski coils, available data-acquisition solutions, ease of installation, and simplicity of the analysis makes PD measurement an affordable methodology for aging studies. For today’s developing grid, it is imperative to develop models that enable the achievement of pragmatic, well-defined, and accurate formulation for predicting the incipient faults before the equipment failure causes an unexpected outage.

Author Contributions

M.C.: literature review, conceptualization, writing—original draft, writing—review and editing, visualization; M.S.: conceptualization, writing—review and editing, supervision, resources, funding acquisition; I.K.: writing—review and editing; A.H.: writing—review and editing; P.T.: writing—review and editing; I.P.: writing—review and editing, supervision. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Table

Figure 1. Aging factors, causes, mechanisms, effects, and consequences for solid insulation.

Figure 2. Typical defects in extruded power cables.

Figure 3. Classification of insulation-aging models.

Figure 4. Relationship between applied stress and insulation life for varnished polyamide films using (a) inverse life graph; (b). exponential life graph adapted with permission from [40].

Figure 5. Electrical life curves for PET, approaching threshold EO, (a) at constant temperature; (b) at different temperatures adapted with permission from [43].

Figure 6. Representation of flat Z characteristics of insulation life in three regions adapted with permission from [44].

Figure 7. Thermal life graph between temperature and insulation life adapted with permission from [51].

Figure 8. Thermal lifelines of XLPE cable at different values of electrical stress adapted with permission from [53].

Figure 9. Life surface graph representing linear and isochronal lines adapted with permission from [31].

Table 1

Summary of reviewed multistress aging models.

Model No.	Model Equation	Model Type andNo. of Parameters	Validity Tested On	References
1.	$\mathrm{L}={\mathrm{B}}_1[{\mathrm{e}}^{\left({\mathrm{B}}_2\mathrm{E}\right)}{\mathrm{e}}^{\frac{{\mathrm{C}}_1}{\mathrm{T}}}{ \mathrm{e}}^{\frac{{\mathrm{C}}_2\mathrm{E}}{\mathrm{T}}}]$	Phenomenological4	Poly propylene-oil system	[56]
2.	$\mathrm{L}={ \mathrm{K}}_{{\mathrm{n} \mathrm{L}}_0}\frac{{\mathrm{e}}^{\left(-\mathrm{hE}-\mathrm{BT}+\mathrm{bET} \right)}}{\frac{{\mathrm{E}}_{\mathrm{G}}}{{\mathrm{E}}_{\mathrm{G}}{}_{\mathrm{to}}}+\frac{\Delta \mathrm{T}}{\Delta {\mathrm{T}}_{\mathrm{to}}}-1}$	Phenomenological6	Composite (Nomex-Mylar-Nomex), cycloaliphatic epoxy resin and polyurethane resin	[31]
3.	$\mathrm{L}={\mathrm{L}}_0\frac{{\mathrm{e}}^{\left(-\mathrm{hE}-\mathrm{BT}+\mathrm{bET} \right)}}{{(\frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{to}}}+\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{to}}}-\left({\mathrm{bTt}}_{\mathrm{o} }\mathrm{ET}\right)/\left({\mathrm{h} \mathrm{E}}_{\mathrm{to}}{ \mathrm{T}}_{\mathrm{to}})\right)-1)}^{\mathsf{\beta }}}$	Phenomenological7	XLPE Cables and (Composite N-M-N)	[33]
4.	$\mathrm{F} \left(\mathrm{t},\mathrm{E};\mathrm{T}\right)=1-{\mathrm{e}}^{{\left[-\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{s}}}{\left(\frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{s}}}\right)}^{\mathrm{n}}\right]}^{\mathrm{q}\left(\mathrm{E},\mathrm{T}\right)}}$	Phenomenological5	XLPE Cables	[59]
5.	${\mathrm{L}}_{\mathrm{E}}=\frac{1}{{\mathrm{fb}}_1[{\mathrm{e}}^{{\mathrm{b}}_2\left(\mathrm{E}-{\mathrm{E}}_{\mathrm{tac}}\right)}-1]{ \mathrm{e}}^{{\mathrm{b}}_3\left({\mathrm{E}}_{\mathrm{b}}\right)}+{\mathrm{b}}_4}$	Physical5	XLPE, PE (Polyethylene) and (EPR) Cables	[60]
6.	$\mathrm{F}\left(\mathrm{t},{\mathrm{Q}}_{\mathrm{i}},\mathrm{E}\right)=1-{\mathrm{e}}^{{\left[-\left(\frac{{\mathrm{tk}}_5(\mathrm{E}-{\mathrm{E}}_{\mathrm{T})}{}^{\mathrm{n}}}{\ln {\left[\left(\frac{{\mathrm{Q}}_{\mathrm{i}}}{{\mathrm{k}}_2}\right)+1\right]}^{\mathrm{d}}}\right)\right]}^{\mathsf{\beta }}}$	Physical4	XLPE Cables	[61]
7.	${\mathrm{t}}_{\mathrm{b}} \cong \frac{\mathrm{h}}{{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{\mathrm{e}}^{\left(\frac{\Delta \mathrm{G}-\mathrm{E}ƛ\mathrm{F}}{{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}\right)}$	Thermodynamic1	Polyethene (HMW-PE), Cross linked PE, XLPE and EPR	[63]
8.	$\mathrm{t}={{\displaystyle \int }}_{\mathrm{N}}^{{\mathrm{n}}_{\mathrm{c}}}\{\frac{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{h}}{[({\mathrm{e}}^{\left(\frac{-{\mathrm{U}}_{\mathrm{r}}\left(\mathrm{E}\right)}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right).\left(\mathrm{N}-\mathrm{n}\right)}-{\mathrm{e}}^{\left(\frac{-{\mathrm{U}}_{\mathrm{b}\left(\mathrm{E}\right)}}{\mathrm{kT}}\right).\left(\mathrm{n}\right)}]\}}^{-1}\mathrm{dn}$	Thermodynamic2	XLPE Cables	[65]
9.	$\mathrm{L} \left(\mathrm{E},\mathrm{T}\right)=\frac{\mathrm{h}}{2{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{ \mathrm{e}}^{\left[\frac{\frac{\Delta \mathrm{H}}{{\mathrm{K}}_{\mathrm{B}}}-\frac{{ {\mathrm{C}}^{\prime } \mathrm{E}}^{2\mathrm{b}}}{2}}{\mathrm{T}}-\frac{\Delta \mathrm{S}}{\mathrm{k}}\right]\ln \left[\frac{{\mathrm{A}}_{\mathrm{e}}\left(\mathrm{E}\right)}{{\mathrm{A}}_{\mathrm{e}}\left(\mathrm{E}\right)-{\mathrm{A}}^{*}}\right]{\left[\mathrm{Cos}\mathrm{h} \left(\frac{\frac{\Delta }{\mathrm{k}}-{ {\mathrm{C}}^{\prime } \mathrm{E}}^{2\mathrm{b}}}{2\mathrm{T}}\right)\right]}^{-1}}$	Thermodynamic8	PET and PP Specimen and XLPE cables	[6]
10.	${\mathrm{L}}_{{\mathrm{f}}_1}=\left[{\left(\raisebox{1ex}{${ {\mathrm{U}}^{\prime }}_{\mathrm{pk}}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{pk}$}\right.\right)}^{\mathrm{n}}\left(\raisebox{1ex}{${\mathrm{f}}_1$}\!\left/ \!\raisebox{-1ex}{$50$}\right.\right)\right]$	Phenomenological1	Enameled Wires	[69]

Table 2

Pros and cons of reviewed multistress aging models.

Model No.	Pros	Cons
1.	Model parameters can be found from experimental results. Model terms represent individual and synergetic effect of electrical and thermal stress. Contain nominal number of parameters.	Only valid for non-threshold materials.
2.	Valid for both threshold and non-threshold materials. Model parameters can be found through experimental results. The model includes term for reference life i.e., l0.	Increased number of parameters can increase the complexity in evaluation of the results.
3.	The model includes additional term (β) that describe material behavior. Valid for both threshold and non-threshold materials. Model parameters can be found through experimental results. The model includes term for reference life i.e., l0.	Increased number of parameters can increase the complexity of model.
4.	The model terms represent impact of time besides electrical and thermal stress. Model terms only define synergetic effect of thermal and electrical stress that reduces the size of equation and parameters. Predicts life under thermal and electrical stress.	The model doesn’t represent the impact of individual stress. There is no separate term for thermal or electrical stress.
5.	Model terms consider impact of frequency. Nominal number of parameters. Parameter can be found by measuring physical quantities by performing PD test.	The model terms´b1, b2, b3 and b4 require prior knowledge of physical properties of materials. There is no term that directly represent thermal stress.
6.	Model terms have physical meaning i.e., depth of electrical tree d and charge q. Model can predict life at chosen value of charge. Nominal number of parameters	Model parameters rely on occurrence of electrical tree. Thus, life prediction prior to tree is not possible.
7.	Model terms includes the impact of microscopic properties of insulation, the quantity n defines number of broken bonds. Minimum number of parameters	Model terms rely on chemical properties of insulation. Thus, difficult to extract through convential methods. The assumptions made about micro-voids in the model can impact the accuracy of the model.
8.	The model is useful in analyzing insulation behavior at microscopic level as model terms also define the quantity of broken bonds i.e., n defines number of broken bonds.	The assumptions made about the already broken bonds in the model can impact the accuracy of the model. Increased number of parameters can increase the complexity of model.
9.	Well-suited for cable design and manufacturing process as model terms are extracted from microscopic properties of insulation. Predicts time for development of micro-voids.	The model can’t predict time to breakdown or aging at macroscopic level. Increased number of parameters can increase the complexity of model.
10.	Consider the impact of modern grid operating conditions i.e., impulse, and high frequency switching. The parameter n can be calculated through experiments. Minimum number of parameters.	Model terms does not consider impact of thermal stress. The model tested only on the enameled wires where insulation thickness is small.

References

1. Saleem, M.Z.; Akbar, M. Review of the Performance of High-Voltage Composite Insulators. Polymers; 2022; 14, 431. [DOI: https://dx.doi.org/10.3390/polym14030431]

2. Campo, E.A. 4—Electrical Properties of Polymeric Materials. Selection of Polymeric Materials; Plastics Design Library, Willian Andrew Inc.: Norwich, NY, USA, 2008; pp. 141-173. [DOI: https://dx.doi.org/10.1016/B978-081551551-7.50006-1]

3. Garros, B. Ageing and reliability testing and monitoring of power cables: Diagnosis for insulation systems: The ARTEMIS program. IEEE Electr. Insul. Mag.; 1999; 15, pp. 10-12. [DOI: https://dx.doi.org/10.1109/57.776939]

4. Bertling, L.; Eriksson, R.; Allan, R.N.; Gustafsson, L.Å.; Åhlén, M. Survey of causes of failures based on statistics and practice for improvements of preventive maintenance plans. Proceedings of the 14th Power Systems Computation Conference; Seville, Spain, 24–28 June 2002; pp. 1-7.

5. Dakin, T.W. Electrical Insulation Deterioration Treated as a Chemical Rate Phenomenon. Cond. Affect. Success Main Line Electrif.; 1948; 67, pp. 113-122. [DOI: https://dx.doi.org/10.1109/T-AIEE.1948.5059649]

6. Mazzanti, G.; Montanari, G.C.; Dissado, L.A. Electrical aging and life models: The role of space charge. IEEE Trans. Dielectr. Electr. Insul.; 2005; 12, pp. 876-890. [DOI: https://dx.doi.org/10.1109/TDEI.2005.1522183]

7. Densley, J. Ageing mechanisms and diagnostics for power cables—An overview. IEEE Electr. Insul. Mag.; 2001; 17, pp. 14-22. [DOI: https://dx.doi.org/10.1109/57.901613]

8. Van der Wielen, P.C.J.M. On-Line Detection and Location of Partial Discharges in Medium-Voltage Power Cables. Ph.D. Dissertation; Eindhoven University of Technology: Eindhoven, The Netherlands, 2005.

9. Yaacob, M.M.; Alsaedi, M.A.; Rashed, J.R.; Dakhil, A.M.; Atyah, S.F. Review on partial discharge detection techniques related to high voltage power equipment using different sensors. Photon Sens.; 2014; 4, pp. 325-337. [DOI: https://dx.doi.org/10.1007/s13320-014-0146-7]

10. Tshiloz, K.; Smith, A.C.; Mohammed, A.; Djurović, S.; Feehally, T. Real-time insulation lifetime monitoring for motor windings. Proceedings of the 2016 XXII International Conference on Electrical Machines (ICEM); Lausanne, Switzerland, 4–7 September 2016; pp. 2335-2340. [DOI: https://dx.doi.org/10.1109/ICELMACH.2016.7732847]

11. Zaeni, A.; Khayam, U.; Viviantoro, D. Methods for Remaining Life Prediction of Power Cable based on Partial Discharge with Regard to Loading Factor Calculation and Voltage Variation. Proceedings of the 2019 2nd International Conference on High Voltage Engineering and Power Systems (ICHVEPS); Denpasar, Indonesia, 1–4 October 2019; pp. 180-185.

12. Srinivasan, M.; Krishnan, A. Prediction of Transformer Insulation Life with an Effect of Environmental Variables. Int. J. Comput. Appl.; 2012; 55, pp. 43-48. [DOI: https://dx.doi.org/10.5120/8755-2658]

13. Bahadoorsingh, S.; Rowland, S. A Framework Linking Knowledge of Insulation Aging to Asset Management. IEEE Electr. Insul. Mag.; 2008; 24, pp. 38-46. [DOI: https://dx.doi.org/10.1109/MEI.2008.4591433]

14. Shafiq, M.; Kauhaniemi, K.; Robles, G.; Isa, M.; Kumpulainen, L. Online condition monitoring of MV cable feeders using Rogowski coil sensors for PD measurements. Electr. Power Syst. Res.; 2019; 167, pp. 150-162. [DOI: https://dx.doi.org/10.1016/j.epsr.2018.10.038]

15. Hovonen, P. Prediction of Insulation Degradation of Distribution Power Cables Based on Chemical Analysis and Electrical Measurements. Ph.D. Dissertation; Aalto University: Espoo, Finland, 2008.

16. Shafiq, M.; Kiitam, I.; Kauhaniemi, K.; Taklaja, P.; Kütt, L.; Palu, I. Performance Comparison of PD Data Acquisition Techniques for Condition Monitoring of Medium Voltage Cables. Energies; 2020; 13, 4272. [DOI: https://dx.doi.org/10.3390/en13164272]

17. Ahmed, Z. Development of a Continuous Condition Monitoring System Based on Probabilistic Modelling of Partial Discharge Data for Polymeric Insulation Cables. Ph.D. Dissertation; Mississippi State University: Starkville, MS, USA, 2019.

18. Choudhary, M.; Suwanasri, T.; Kumpalavalee, S.; Fuangpian, P.; Suwanasri, C. Condition Assessment of Synchronous Generator Using Fuzzy Logic. IEET Int. Electr. Eng. Trans.; 2018; 4, 27.

19. David, E.; Parpal, J.-L.; Crine, J.-P. Aging of XLPE cable insulation under combined electrical and mechanical stresses. Proceedings of the 1996 IEEE International Symposium on Electrical Insulation; Montreal, QC, Canada, 16–19 June 1996; [DOI: https://dx.doi.org/10.1109/elinsl.1996.549445]

20. Zhou, C.; Yi, H.; Dong, X. Review of Recent Research Towards Power Cable Life Cycle Management. High Volt.; 2017; 2, pp. 179-187. [DOI: https://dx.doi.org/10.1049/hve.2017.0037]

21. Higinbotham, W. Review of Medium-Voltage Asset Failure Investigations. Prem. Electr. Maint. Saf. Event; 2018; 1, 17.

22. Toshikatsu, T.; Greenwood, A. Advanced Power Cable Technology: Present and Future; CRC Press Inc.: Boca Raton, FL, USA, 1983; Volume 2, pp. 57-94.

23. Su, J.; Du, B.; Li, J.; Li, Z. Electrical tree degradation in high-voltage cable insulation: Progress and challenges. High Volt.; 2020; 5, pp. 353-364. [DOI: https://dx.doi.org/10.1049/hve.2020.0009]

24. Helleso, S.; Benjaminsen, J.T.; Selsjord, M.; Hvidsten, S. Water tree initiation and growth in XLPE cables under static and dynamic mechanical stress. Proceedings of the 2012 IEEE International Symposium on Electrical Insulation; San Juan, PR, USA, 10–13 June 2012; pp. 623-627. [DOI: https://dx.doi.org/10.1109/ELINSL.2012.6251546]

25. IEEE 1407 IEEE Guide for Accelerated Aging Tests for Medium-Voltage (5 kV–35 kV) Extruded Electric Power Cables Using Water-Filled Tanks; IEEE: Piscataway, NJ, USA, 2017.

26. IEEE 98 IEEE Standard for the Preparation of Test Procedures for the Thermal Evaluation of Solid Electrical Insulating Materials; IEEE: Piscataway, NJ, USA, 2016; pp. 1-32.

27. Montanari, G.; Motori, A. Short-term thermal endurance characterization of polymeric cable insulating materials. Use of oxidative stability measurements. IEEE Trans. Dielectr. Electr. Insul.; 1996; 3, pp. 561-566. [DOI: https://dx.doi.org/10.1109/94.536736]

28. IEC-60270:2000+AMD1 High-Voltage Test Techniques-Partial Discharge Measurements; IEC: Geneva, Switzerland, 2015.

29. Hussain, G.A.; Shafiq, M.; Lehtonen, M.; Hashmi, M. Online Condition Monitoring of MV Switchgear Using D-Dot Sensor to Predict Arc-Faults. IEEE Sens. J.; 2015; 15, pp. 7262-7272. [DOI: https://dx.doi.org/10.1109/JSEN.2015.2474122]

30. Simoni, L. A General Approach to the Endurance of Electrical Insulation under Temperature and Voltage. IEEE Trans. Electr. Insul.; 1981; 16, pp. 277-289. [DOI: https://dx.doi.org/10.1109/TEI.1981.298361]

31. Simoni, L. General Equation of the Decline in the Electric Strength for Combined Thermal and Electrical Stresses. IEEE Trans. Electr. Insul.; 1984; 19, pp. 45-52. [DOI: https://dx.doi.org/10.1109/TEI.1984.298732]

32. Simoni, L.; Mazzanti, G.; Montanari, G.; Lefebre, L. A general multi-stress life model for insulating materials with or without evidence for thresholds. IEEE Trans. Electr. Insul.; 1993; 28, pp. 349-364. [DOI: https://dx.doi.org/10.1109/14.236212]

33. Dissado, L.; Mazzanti, G.; Montanari, G.C. The incorporation of space charge degradation in the life model for electrical insulating materials. IEEE Trans. Dielectr. Electr. Insul.; 1995; 2, pp. 1147-1158. [DOI: https://dx.doi.org/10.1109/TDEI.1995.8881933]

34. IEEE Dielectrics and Electrical Insulation SocietyMultifactor Stress Committee. IEEE Guide for Multifactor Stress Functional Testing of Electrical Insulation Systems; IEEE: Piscataway, NJ, USA, 1991.

35. Edwards, V. Electrons Theory; Edtech Press: London, UK, 2018; Chapter 1 pp. 1-15.

36. Fisher, L.E.; Haries, C.D.; Lehr, J.M. Initiation and Growth of the Secondary Electron Avalanche along High-Gradient Insulators in Vacuum. IEEE Trans. Dielectr. Electr. Insul.; 2021; 28, pp. 972-980. [DOI: https://dx.doi.org/10.1109/TDEI.2021.009400]

37. IEC 61251 Electrical Insulating Materials and Systems-AC Voltage Endurance Evaluation; IEC: Geneva Switzerland, 2015.

38. IEEE 930-1987 IEEE Guide for the Statistical Analysis of Electrical Insulation Voltage Endurance Data; IEEE: Piscataway, NJ, USA, 1995.

39. Stone, G.C.; Lawless, J.F. The Application of Weibull Statistics to Insulation Aging Tests. IEEE Trans. Electr. Insul.; 1979; 14, pp. 233-239. [DOI: https://dx.doi.org/10.1109/TEI.1979.298226]

40. Cygan, P.; Laghari, J.R. Models for insulation aging under electrical and thermal multistress. IEEE Trans. Electr. Insul.; 1990; 25, pp. 923-934. [DOI: https://dx.doi.org/10.1109/14.59867]

41. Pattini, G.; Simoni, L. Discussion on modeling of voltage endurance. Proceedings of the 1980 IEEE International Conference on Electrical Insulation; Boston, MA, USA, 9–11 June 1980; [DOI: https://dx.doi.org/10.1109/icei.1980.7470860]

42. Dakin, T.W. The Endurance of Electrical Insulation. Proceedings of the 4th Symposium Electrical Insulation Materials, JIEE; Saclay, France, 21–23 April 1971.

43. Montanari, G.C.; Simoni, L. Aging phenomenology and modeling. IEEE Trans. Electr. Insul.; 1993; 28, pp. 755-776. [DOI: https://dx.doi.org/10.1109/14.237740]

44. Hirose, H. A Method to Estimate the Lifetime of Solid Electrical Insulation. IEEE Trans. Electr. Insul.; 1987; 22, pp. 745-753. [DOI: https://dx.doi.org/10.1109/TEI.1987.298936]

45. Montsinger, V.M. Loading Transformers by Temperature. Cond. Affect. Success Main Line Electrif.; 1930; 49, pp. 776-790. [DOI: https://dx.doi.org/10.1109/T-AIEE.1930.5055572]

46. Brancato, E. Insulation Aging a Historical and Critical Review. IEEE Trans. Electr. Insul.; 1978; 13, pp. 308-317. [DOI: https://dx.doi.org/10.1109/TEI.1978.298079]

47. IEEE Std 101-1987 IEEE Guide for the Statistical Analysis of Thermal Life Test Data; IEEE: Piscataway, NJ, USA, 1988.

48. Endicott, H.S.; Hatch, B.D.; Sohmer, R.G. Application of the Eyring Model to capacitor aging data. Proceedings of the Annual Report 1962 Conference on Electrical Insulation; Hershey, PA, USA, 15–17 October 1962; pp. 47-50.

49. Simoni, L. Application of a new geometrical approach to determination of combined stress endurance of insulating materials. IEEE Trans. Electr. Insul.; 1988; 23, pp. 489-492. [DOI: https://dx.doi.org/10.1109/14.2391]

50. IEC 60216–3 Electrical Insulating Materials—Thermal Endurance Properties—Part 3: Instructions for Calculating Thermal Endurance Characteristics; IEC: Geneva, Switzerland, 2021.

51. Nelson, W. Analysis of Accelerated Life Test Data—Part I: The Arrhenius Model and Graphical Methods. IEEE Trans. Electr. Insul.; 1971; 6, pp. 165-181. [DOI: https://dx.doi.org/10.1109/TEI.1971.299172]

52. Mazzanti, G.; Montanari, G. A comparison between XLPE and EPR as insulating materials for HV cables. IEEE Trans. Power Deliv.; 1997; 12, pp. 15-28. [DOI: https://dx.doi.org/10.1109/61.568220]

53. Montanari, G.C.; Pattini, G.; Simoni, L. Long—Term Behavior of XLPE Insulated Cable Models. IEEE Trans. Power Deliv.; 1987; 2, pp. 596-602. [DOI: https://dx.doi.org/10.1109/TPWRD.1987.4308151]

54. Ramu, T.S. On the estimation of life of power apparatus insulation under combined electrical and thermal stress. IEEE Trans. Electr. Insul.; 1985; EI-20, pp. 70-78. [DOI: https://dx.doi.org/10.1109/TEI.1985.348759]

55. Montanari, G.C. Insulation Aging Models. Encyclopedia of Electrical and Electronics Engineering; J. Wiley & Sons: Hoboken, NJ, USA, 1999.

56. Fallou, B.; Burguiere, C. First Approach on Multiple Stress Accelerated Life Testing of Electrical insulation. Annu. Rep. Proc. CEIDP; 1979; pp. 621-628.

57. Cacciari, M.; Montanari, G. Optimum design of life tests for insulating materials, systems and components. IEEE Trans. Electr. Insul.; 1991; 26, pp. 1112-1123. [DOI: https://dx.doi.org/10.1109/14.108148]

58. Cacciari, M.; Montanari, G.C. Probabilistic models for life prediction of insulating materials. J. Phys. D Appl. Phys.; 1990; 23, pp. 1592-1598. [DOI: https://dx.doi.org/10.1088/0022-3727/23/12/016]

59. Montanari, G.; Cacciari, M. A probabilistic life model for insulating materials showing electrical thresholds. IEEE Trans. Electr. Insul.; 1989; 24, pp. 127-134. [DOI: https://dx.doi.org/10.1109/14.19877]

60. Bahder, G.; Garrity, T.; Sosnowski, M.; Eaton, R.; Katz, C. Physical Model of Electric Aging and Breakdown of Extruded Pplymeric Insulated Power Cables. IEEE Trans. Power Appar. Syst.; 1982; 101, pp. 1379-1390. [DOI: https://dx.doi.org/10.1109/TPAS.1982.317185]

61. Montanari, G. Aging and life models for insulation systems based on PD detection. IEEE Trans. Dielectr. Electr. Insul.; 1995; 2, pp. 667-675. [DOI: https://dx.doi.org/10.1109/94.407031]

62. Zhang, Y.; Lewiner, J.; Alquie, C.; Hampton, N. Evidence of strong correlation between space-charge buildup and breakdown in cable insulation. IEEE Trans. Dielectr. Electr. Insul.; 1996; 3, pp. 778-783. [DOI: https://dx.doi.org/10.1109/94.556559]

63. Crine, J.-P.; Parpal, J.-L.; Lessard, G. A model of aging of dielectric extruded cables. Proceedings of the 3rd International Conference on Conduction and Breakdown in Solid Dielectrics; Trondheim, Norway, 3–6 July 1989; pp. 347-351.

64. Sanche, L. Electronic aging and related electron interactions in thin-film dielectrics. IEEE Trans. Electr. Insul.; 1993; 28, pp. 789-819. [DOI: https://dx.doi.org/10.1109/14.237742]

65. Griffiths, C.; Freestone, J.; Hampton, R. Thermoelectric aging of cable grade XLPE. Proceedings of the Conference Record of IEEE International Symposium on Electrical Insulation; Arlington, VA, USA, 7–10 June 1998; Volume 2, pp. 578-582.

66. Dissado, L.A.; Mazzanti, G.; Montanari, G.C. The role of trapped space charges in the electrical aging of insulating materials. IEEE Trans. Dielectr. Electr. Insul.; 1997; 4, pp. 496-506. [DOI: https://dx.doi.org/10.1109/94.625642]

67. Mazzanti, G.; Montanari, G.; Dissado, L. A space-charge life model for ac electrical aging of polymers. IEEE Trans. Dielectr. Electr. Insul.; 1999; 6, pp. 864-875. [DOI: https://dx.doi.org/10.1109/94.822029]

68. Gubanski, S.M. Analysis of lifetime estimation methods based on data for electro-thermally aged PET films. Proceedings of the Conference Record of IEEE International Symposium on Electrical Insulation; Toronto, ON, Canada, 3–6 June 1990; pp. 27-30. [DOI: https://dx.doi.org/10.1109/elinsl.1990.109701]

69. Cavallini, A.; Fabiani, D.; Montanari, G.C. Power electronics and electrical insulation systems—Part 2: Life modeling for insulation design. IEEE Electr. Insul. Mag.; 2010; 26, pp. 33-39. [DOI: https://dx.doi.org/10.1109/MEI.2010.5511187]

70. Frigione, M.; Rodríguez-Prieto, A. Can Accelerated Aging Procedures Predict the Long Term Behavior of Polymers Exposed to Different Environments?. Polymers; 2021; 13, 2688. [DOI: https://dx.doi.org/10.3390/polym13162688] [PubMed: https://www.ncbi.nlm.nih.gov/pubmed/34451228]