A Review of Aging Models for Electrical Insulation in Power Cables
Abstract
:1. Introduction
2. Insulation Degradation in Solids and Aging Models
3. SingleStress Aging Models
3.1. ElectricalAging Models
3.2. ThermalAging Models
 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.
4. Multistress Aging Models
 Phenomenological Models.
 Thermodynamic Models.
 Physical Models.
4.1. Phenomenological Models
 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.
 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).
4.2. Physical Models
4.3. Thermodynamic Models
5. Discussion
 It has been observed that most of the models are materialoriented, 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 ethylenepropylene 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 nonthreshold 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 (peaktopeak value of voltages) at nominal frequency (50 Hz) and the impulse frequency expressed as
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
References
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Model No.  Model Equation  Model Type and No. of Parameters  Validity Tested On  References 

1.  $\mathrm{L}={\mathrm{B}}_{1}[{\mathrm{e}}^{\left({\mathrm{B}}_{2}\mathrm{E}\right)}{\mathrm{e}}^{\frac{{\mathrm{C}}_{1}}{\mathrm{T}}}{\mathrm{e}}^{\frac{{\mathrm{C}}_{2}\mathrm{E}}{\mathrm{T}}}]$  Phenomenological 4  Poly propyleneoil system  [56] 
2.  $\mathrm{L}={\mathrm{K}}_{{\mathrm{n}\mathrm{L}}_{0}}\frac{{\mathrm{e}}^{\left(\mathrm{hE}\mathrm{BT}+\mathrm{bET}\right)}}{\frac{{\mathrm{E}}_{\mathrm{G}}}{{\mathrm{E}}_{\mathrm{G}}{}_{\mathrm{to}}}+\frac{\Delta \mathrm{T}}{\Delta {\mathrm{T}}_{\mathrm{to}}}1}$  Phenomenological 6  Composite (NomexMylarNomex), cycloaliphatic epoxy resin and polyurethane resin  [31] 
3.  $\mathrm{L}={\mathrm{L}}_{0}\frac{{\mathrm{e}}^{\left(\mathrm{hE}\mathrm{BT}+\mathrm{bET}\right)}}{{(\frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{to}}}+\frac{\mathrm{T}}{{\mathrm{T}}_{\mathrm{to}}}\left({\mathrm{bTt}}_{\mathrm{o}}\mathrm{ET}\right)/\left({\mathrm{h}\mathrm{E}}_{\mathrm{to}}{\mathrm{T}}_{\mathrm{to}})\right)1)}^{\mathsf{\beta}}}$  Phenomenological 7  XLPE Cables and (Composite NMN)  [33] 
4.  $\mathrm{F}\left(\mathrm{t},\mathrm{E};\mathrm{T}\right)=1{\mathrm{e}}^{{\left[\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{s}}}{\left(\frac{\mathrm{E}}{{\mathrm{E}}_{\mathrm{s}}}\right)}^{\mathrm{n}}\right]}^{\mathrm{q}\left(\mathrm{E},\mathrm{T}\right)}}$  Phenomenological 5  XLPE Cables  [59] 
5.  ${\mathrm{L}}_{\mathrm{E}}=\frac{1}{{\mathrm{fb}}_{1}[{\mathrm{e}}^{{\mathrm{b}}_{2}\left(\mathrm{E}{\mathrm{E}}_{\mathrm{tac}}\right)}1]{\mathrm{e}}^{{\mathrm{b}}_{3}\left({\mathrm{E}}_{\mathrm{b}}\right)}+{\mathrm{b}}_{4}}$  Physical 5  XLPE, PE (Polyethylene) and (EPR) Cables  [60] 
6.  $\mathrm{F}\left(\mathrm{t},{\mathrm{Q}}_{\mathrm{i}},\mathrm{E}\right)=1{\mathrm{e}}^{{\left[\left(\frac{{\mathrm{tk}}_{5}(\mathrm{E}{\mathrm{E}}_{\mathrm{T})}{}^{\mathrm{n}}}{\mathrm{ln}{\left[\left(\frac{{\mathrm{Q}}_{\mathrm{i}}}{{\mathrm{k}}_{2}}\right)+1\right]}^{\mathrm{d}}}\right)\right]}^{\mathsf{\beta}}}$  Physical 4  XLPE Cables  [61] 
7.  ${\mathrm{t}}_{\mathrm{b}}\cong \frac{\mathrm{h}}{{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{\mathrm{e}}^{\left(\frac{\Delta \mathrm{G}\mathrm{E}\u019b\mathrm{F}}{{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}\right)}$  Thermodynamic 1  Polyethene (HMWPE), Cross linked PE, XLPE and EPR  [63] 
8.  $\mathrm{t}={{\displaystyle \int}}_{\mathrm{N}}^{{\mathrm{n}}_{\mathrm{c}}}\{\frac{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}{\mathrm{h}}{[({\mathrm{e}}^{\left(\frac{{\mathrm{U}}_{\mathrm{r}}\left(\mathrm{E}\right)}{{\mathrm{k}}_{\mathrm{B}}\mathrm{T}}\right).\left(\mathrm{N}\mathrm{n}\right)}{\mathrm{e}}^{\left(\frac{{\mathrm{U}}_{\mathrm{b}\left(\mathrm{E}\right)}}{\mathrm{kT}}\right).\left(\mathrm{n}\right)}]\}}^{1}\mathrm{dn}$  Thermodynamic 2  XLPE Cables  [65] 
9.  $\mathrm{L}\left(\mathrm{E},\mathrm{T}\right)=\frac{\mathrm{h}}{2{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}{\mathrm{e}}^{\left[\frac{\frac{\Delta \mathrm{H}}{{\mathrm{K}}_{\mathrm{B}}}\frac{{{\mathrm{C}}^{\prime}\mathrm{E}}^{2\mathrm{b}}}{2}}{\mathrm{T}}\frac{\Delta \mathrm{S}}{\mathrm{k}}\right]\mathrm{ln}\left[\frac{{\mathrm{A}}_{\mathrm{e}}\left(\mathrm{E}\right)}{{\mathrm{A}}_{\mathrm{e}}\left(\mathrm{E}\right){\mathrm{A}}^{*}}\right]{\left[\mathrm{Cos}\mathrm{h}\left(\frac{\frac{\Delta}{\mathrm{k}}{{\mathrm{C}}^{\prime}\mathrm{E}}^{2\mathrm{b}}}{2\mathrm{T}}\right)\right]}^{1}}$  Thermodynamic 8  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]$  Phenomenological 1  Enameled Wires  [69] 
Model No.  Pros  Cons 

1. 
 Only valid for nonthreshold materials. 
2. 


3. 
 Increased number of parameters can increase the complexity of model. 
4. 
 The model doesn’t represent the impact of individual stress. There is no separate term for thermal or electrical stress. 
5. 


6. 
 Model parameters rely on occurrence of electrical tree. Thus, life prediction prior to tree is not possible. 
7. 


8. 


9. 


10. 


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Choudhary, M.; Shafiq, M.; Kiitam, I.; Hussain, A.; Palu, I.; Taklaja, P. A Review of Aging Models for Electrical Insulation in Power Cables. Energies 2022, 15, 3408. https://doi.org/10.3390/en15093408
Choudhary M, Shafiq M, Kiitam I, Hussain A, Palu I, Taklaja P. A Review of Aging Models for Electrical Insulation in Power Cables. Energies. 2022; 15(9):3408. https://doi.org/10.3390/en15093408
Chicago/Turabian StyleChoudhary, Maninder, Muhammad Shafiq, Ivar Kiitam, Amjad Hussain, Ivo Palu, and Paul Taklaja. 2022. "A Review of Aging Models for Electrical Insulation in Power Cables" Energies 15, no. 9: 3408. https://doi.org/10.3390/en15093408