Reliability Assessment of Hybrid Cable Laying Configurations in Urban Dense Cable Channels Based on Modified Weibull Distribution

Abstract

With the acceleration of urbanization and the increasing demand for aesthetics, cable laying is progressively transitioning into urban dense cable channels. The internal environment of these channels is complex, and arbitrary cable laying poses significant threats to normal cable operation. Therefore, this paper proposes a reliability assessment method for hybrid cable laying configurations in urban dense cable channels based on a modified Weibull distribution. Firstly, a Weibull proportional hazards model is constructed by incorporating channel operational risk factors as covariates. Then, Bayesian inference is employed to update the Weibull parameters by integrating expert experience with cable channel O&M data. Subsequently, we select the parameter distribution and reliability evaluation indicators suitable for the operating environment of the cable channel and analyze the influence of the overcrowding rate of cable laying, the operating temperature of the cable, and the distance between cables in the channel on the reliability of the cable channel. Finally, a case study is conducted on a cable channel in a region of the China Southern Power Grid utilizing its actual O&M data to perform a reliability assessment. The effectiveness of the proposed modified Weibull distribution assessment method is validated through model comparison. Furthermore, this study provides differentiated maintenance strategies for specific cables within the channel and proposes a set of highly applicable O&M guidelines.

1. Introduction

With the continuous development of urban construction in China, traditional overhead lines not only affect aesthetics but also lack safety. Power transmission and distribution lines are gradually being buried underground, forming urban dense cable channels [1,2,3,4]. Compared with overhead lines, after the construction of urban dense cable channels, it is necessary to focus on monitoring the operation of the channels to prevent the lines from being shut down due to reasons such as dampness, fire, and line damage, which would cause serious economic losses [5,6,7], as shown in Figure 1. Maintenance personnel regularly inspect and replace faulty cables. However, as the density of urban power grids continues to increase, the pressure on maintenance work also grows. Complex detection data makes it difficult for maintenance personnel to accurately assess the operational status of cables, necessitating a versatile and reliable assessment method for urban dense cable channels.

The operational lifespan of cables in China is generally around 30 to 40 years [8]. However, cables in urban dense cable channels that were constructed earlier are nearing the end of their service life and require reliability assessments. The safety of cable channels is affected by factors such as the overcrowding rate of cable laying in dense cable channels; the operating temperature of the cables themselves; and whether high-, medium-, and low-voltage cables are laid together [9,10]. Maintenance personnel need to check the relevant data and use their cable maintenance experience to maintain or replace the cables in the channels. However, development of cable channel in China is relatively late, and there is a lack of relevant operation and maintenance experience. Therefore, research on the maintenance of dense cable channels is needed.

Currently, domestic and foreign researchers have conducted in-depth studies on the reliability assessment of cables in urban dense cable channels from multiple dimensions. The common method is to conduct accelerated aging tests on cables in the laboratory and evaluate the operation of the cables by analyzing the aging results. Ref. [11] analyzes the dielectric spectra in the high-voltage frequency domain of XLPE cables with different aging degrees and evaluates the aging degree of the cables through the characteristics of the spectra. Ref. [12] analyzes the depolarization charge and current curves of cables using the polymer trap theory and studies the aging factors of cables under different laying methods. The results shows that this method could effectively determine the degree of cable aging. Although these experiments can reveal some of the aging mechanisms of materials, their experimental environments are quite different from the actual complex operating conditions of cables in urban dense cable channels, which reduces the direct applicability of the relevant assessment results. With the development of artificial intelligence technology, machine learning algorithms have been widely applied in the classification of engineering risk levels and state assessment [13,14,15]. Ref. [16] proposes using K-means clustering and the random forest algorithm to classify risk levels, analyze risk weights, and classify cable aging, but it requires a large number of fault samples for model training. However, the lifespan of cables in dense channels is generally longer, with less fault data, and overfitting problems are prone to occur during the model training process.

Therefore, reliability assessment methods based on statistical models have become the mainstream research direction in this field at present [17,18]. The standard Weibull distribution model is widely used to assess the overall fault trend of power cables due to its flexibility. Ref. [19] conducts a breakdown voltage test on the cable to obtain the Weibull distributed parameters and reveal the variation law of the distributed parameters with the breakdown voltage but fails to take into account the narrow space of the cable channel, making on-site experiments impossible. Ref. [20] analyzes the reliability of cable joints by combining the Weibull distribution and the Croke-AMSAA model but does not take into account the mutual influence between cables within the channel. The standard Weibull distribution model can only describe the changing trend of the overall failure probability of cables over time, and it is difficult to quantify the combined impact of multiple risk factors in dense cable channels. Therefore, researchers began to introduce risk factors as covariates into the Weibull distribution model to form the Weibull proportional hazards model. Ref. [21] adopts the Weibull proportional hazards model to analyze the monitoring data and deterioration threshold of optical cables, optimizing the maximum availability of optical cables. However, it lacks attention to the heat generation of cables and is not suitable for scenarios with dense cable channels.

To address the issue of sparse cable fault data in dense cable channels, some scholars have utilized the Analytic Hierarchy process or fuzzy evaluation method, integrating expert experience into the assessment model to analyze the weights of risk factors and conduct subjective and objective evaluations of cable reliability levels [22,23,24]. However, such methods are highly subjective in their evaluations, and the assessment results show regional differences with limited generalization capabilities. Therefore, Bayesian inference needs to be introduced. Bayesian inference can systematically integrate expert operation and maintenance experience with a small number of cable fault samples [25], overcome the small sample problem in reliability assessment, and has a better assessment effect on complex problems with multiple risk factors.

Against this backdrop, the cable reliability assessment method that combines Bayesian inference with the Weibull proportional hazard model has become one of the more advanced research routes in this field. At present, relevant papers have applied this method to the reliability analysis of general high-voltage cables [26]. However, the covariates selected in this study are whether the cable is directly buried and the load rate of the cable operation. The focus is on the operation status and environment of the cable itself, and it does not delve into the complex cable operation environment of urban dense cable channels. There are multiple risk factors and complex interwoven phenomena in dense cable channels. The accumulation of local hotspots caused by excessive cable laying in the channels; the dynamic changes in cable operating temperatures; and the cross-thermal effects resulting from the mixed laying of high-, medium-, and low-voltage cables will all have a significant impact on the normal operation of the cables. At present, for the analysis of multiple risk factors within urban dense cable channels, there are few unified models and comprehensive quantifications in the existing literature.

In response to the above-mentioned challenges, this paper focuses on the engineering scenario of urban dense cable channels, conducts reliability assessment on them, and substitutes the influencing risk factors as covariates into the Weibull distribution function to form a Weibull proportional hazard model. A reliability assessment method for urban dense cable channels that combines Bayesian parameter estimation with the Weibull proportional hazard model is proposed, quantifying and simulating the synergistic effects of three key risk factors within dense cable channels. By combining expert experience with limited cable channel O&M data through the Bayesian model, the problem of difficulty in modeling dense cable channels due to the scarcity of fault samples has been solved, avoiding the occurrence of overfitting, and making the reliability assessment results more objective and accurate. Furthermore, we transform the reliability assessment results into differentiated maintenance strategies to guide on-site work, providing a scientific basis for the maintenance of dense cable channels. Compared with previous modeling strategies, this method can update the prior information and model parameters based on the newly added fault samples and changes in the operating environment in order to obtain continuously optimized evaluation results.

2. Reliability Assessment Model of Urban Dense Cable Channels

2.1. Weibull Distribution Function

The Weibull distribution is a continuous probability distribution that can effectively simulate a variety of life cycle behaviors. It is often used to analyze and test the reliability and lifespan of random variables in various engineering and natural phenomena and is currently widely used in the lifespan testing of urban dense cable channels. Common Weibull distributions include two-parameter and three-parameter forms. A commonly used two-parameter Weibull distribution probability density function is shown in Equation (1).

$$f(t)={\displaystyle \frac{\beta }{\eta }}{\left({\displaystyle \frac{t}{\eta }}\right)}^{\beta -1}\exp \left[-{\left({\displaystyle \frac{t}{\eta }}\right)}^{\beta }\right], t\ge 0$$

(1)

where β is the shape parameter. When β < 1, it is applicable to the early stage of dense cable operation, and the failure rate is in a decreasing state. When β = 1, it indicates that the dense cable is in the middle stage of failure, and the cable is failing randomly. When β > 1, it indicates that the dense cable is in the equipment wear stage, the frequency of cable failures in the channel is increasing, and the failure rate is increasing. η is the scale parameter, which is used to specify the scale and position of the distribution. t is the time for cable operation in a dense cable channel, with the default unit being days.

The Weibull distribution function can be used to analyze the normal operation and failure data of cables in dense cable channels and to conduct risk assessments and reliability tests on the cables in the channel. However, the disadvantage of using the Weibull distribution function alone is that it can only handle the overall failure trend of the data and cannot directly introduce specific influencing factors such as temperature, humidity, and overcrowding rate. Therefore, it is necessary to use the Weibull proportional hazards model to introduce covariates and combine the operational status information of cables in dense cable channels with reliability.

2.2. Weibull Proportional Hazards Model with Covariates

Taking into account the service life of the cable and its current status information, different influencing factors are converted into covariates and introduced into the hazards model to obtain the proportional hazards model as shown in Equations (2) and (3).

$$\eta =\exp \left({\gamma }_0+{\displaystyle \sum _{i=1}^n{\gamma }_i{X}_i}\right)$$

(2)

$$h_0\left(t\right)={\displaystyle \frac{\beta }{\eta }}{\left({\displaystyle \frac{t}{\eta }}\right)}^{\beta -1}$$

(3)

where h₀(t) is the basic failure rate function based on the Weibull distribution and related only to time; X_i represents the covariate formed after quantifying the i-th influencing factor; and γ_i is the state covariate parameter. γ greater than 0 indicates that the covariate has a positive correlation with cable life, while γ less than 0 indicates that the covariate has a negative correlation with cable life. The larger the absolute value of γ, the more significant the impact on cable life. n represents the number of covariates. Combining Equations (1)–(3), we obtain the reliability function of the Weibull proportional hazards model as shown in Equation (4).

$$R\left(t\right)=\exp \left[-{\displaystyle {\int }_0^t\frac{\beta }{\eta }{\left(\frac{t}{\eta }\right)}^{\beta -1}\exp \left({\displaystyle \sum _{i=1}^n{\gamma }_i{X}_i}\right)dt}\right]$$

(4)

The failure probability density function is shown in Equation (5).

$$\begin{array}{l}f\left(t\right)=\exp \left({\gamma }_0+{\displaystyle \sum _{i=1}^n{\gamma }_i{X}_i}\right)\beta t^{\beta -1}\cdot \\ \exp \left(-\exp \left({\gamma }_0+{\displaystyle \sum _{i=1}^n{\gamma }_i{X}_i}\right)t^{\beta }\right)\end{array}$$

(5)

The Weibull proportional hazards model can be used to detect risks in urban dense cable channels, but it requires the application of maximum likelihood estimation to estimate the unknown parameters in the formula in order to perform reliability assessments. Therefore, this section selects the Bayesian method for reliability assessment of cables in dense channels, which can effectively utilize prior information to update parameters even when there are few samples and is widely used in life prediction and fault probability judgment.

3. Update the Prediction Model Based on Bayesian Estimation

3.1. Bayesian Parameter Estimation

Due to the generally long service life of cables in densely populated urban cable channels and the relatively well-developed maintenance measures, cable fault information is characterized by limited samples and small quantities. To avoid overfitting, large-sample reliability calculation methods are not applicable.

The principle of parameter estimation using Bayesian models is to treat any unknown parameter as a random variable and use prior information and historical operating data to update and improve existing information. First, determine the prior distribution based on the prior information of the unknown parameters. According to Bayes’ theorem, by introducing the likelihood function, we obtain the joint posterior density function expression containing dense cable channel fault sample information as shown in Equation (6).

$$f\left(\theta \left|D\right.\right)={\displaystyle \frac{L\left(D\left|\theta \right.\right)\pi \left(\theta \right)}{{\displaystyle \int L\left(D\left|\theta \right.\right)\pi \left(\theta \right)d\left(\theta \right)}}}$$

(6)

where θ represents unknown parameters such as β, γ₁, and γ₂ in the Weibull distribution; D is the sample of fault in the dense cable channel; L(D|θ) is the joint conditional density function that integrates sample information, which is the likelihood function; π(θ) represents the prior distribution of unknown parameters; the denominator is a marginal density function independent of unknown parameters, used for normalization; and f(θ|D) is the posterior distribution of the unknown parameter. When there are a large number of unknown parameters, set the remaining parameters to fixed values and update only the current parameters to infer the overall reliability index.

3.2. Dense Cable Channel Prior Distribution Selection

The key to parameter updating using Bayesian models is to determine the prior distribution of the parameters to be sought. In dense cable channels with low cable failure rates, the accuracy of the estimate depends on the reliability of the prior information. Currently, prior information mainly includes maintenance personnel repair experience, historical repair data, relevant reference materials, etc.

In the Weibull proportional hazards model, parameters such as β, γ₁, and γ₂ are independent of each other. Due to the complexity of the likelihood function, it is difficult to find the corresponding conjugate prior distribution. Therefore, the corresponding prior distribution can only be found based on the characteristics of the parameter values. For the shape parameter β, selecting Gamma distribution can meet the requirements of actual engineering applications. Its basic form is shown in Equation (7).

$$\pi \left(\beta \right)={\displaystyle \frac{b^a}{\Gamma \left(a\right)}}{\beta }^{a-1}{e}^{-b\beta }$$

(7)

where a and b are hyperparameters of the prior distribution, both greater than 0; π(β) is the prior distribution corresponding to β; and Γ(a) is Gamma function.

In the analysis of dense cable channels, for the state covariate parameter γ, let it follow a normal distribution whose basic form is shown in Equation (8).

$$\pi \left(\gamma \right)={\displaystyle \frac{1}{\sqrt{2\pi }\sigma }}\exp \left(-{\displaystyle \frac{\left(\gamma -\mu \right)}{2{\sigma }^2}}\right)$$

(8)

where π(γ) is the prior distribution corresponding to γ; σ is the standard deviation corresponding to γ; and μ is the mean value of γ.

In practical applications, the posterior expectations E(β) and E(γ) are often used as Bayesian estimates, reflecting the importance of this parameter for the reliability of dense cable channels. The posterior variances D(β) and D(γ) are used to measure the accuracy of Bayesian estimates and reflect the uncertainty of parameters. After updating the parameters using the Bayesian model, a comparison with the original data can be made to directly determine the extent to which different influencing factors affect the reliability of dense cable channels.

3.3. Update Prior Distributions with Bayesian Models

A city in southern China can provide corresponding risk values by monitoring the operating status of dense cable channels. In the Weibull proportional hazards model, low risk corresponds to the reliability function R(t), which indicates that the probability of failure of the cable channel within time t is low; medium risk and high risk correspond to the failure probability density function f(t), indicating that the probability of failure occurring within time t is high. Based on the joint density function, the likelihood function can be constructed as Equation (9).

$$L\left(D\left|\beta ,{\gamma }_1,{\gamma }_2,...,{\gamma }_n\right.\right)={\displaystyle \prod _{i=1}^{m}f{\left(t\right)}^{p}R{\left(t\right)}^{1-p}}$$

(9)

where m is the number of fault samples, and p is the risk value identifier. When the dense cable channel is in a low-risk state, p is set to 0; when it is in a medium- or high-risk state, p is set to 1. Substituting Equations (7)–(9) into Equation (6) yields the complete posterior distribution expression.

Due to the complexity of the high-dimensional marginal density function in the posterior distribution expression and the inclusion of double integrals, it is not possible to obtain the updated parameters through direct solution methods. Therefore, the Markov Chain Monte Carlo (MCMC) algorithm is typically used for calculations. For the update of parameter β, in the Weibull model, the posterior distribution of the Gamma distribution cannot be obtained through a conjugate prior, so it is suitable to use the M-H sampling method to construct a Markov chain, design an appropriate acceptance probability, generate random samples from the target distribution, and update the parameters.

3.4. Reliability Index Assessment of Urban Dense Cable Channels

For reliability assessment and risk analysis of urban dense cable channels, it is necessary to select multiple indicators and conduct a comprehensive analysis using the Weibull probability density function in order to evaluate the operational status of cables within the channels.

R(t|s), known as conditional residual reliability (CRR), can represent the reliability of cables in urban dense channels after operating for s years in the future t years. In the Weibull proportional hazards model, conditional residual reliability can be expressed as Equation (10).

$$R\left(t\left|s\right.\right)=\exp \left(-\exp \left({\gamma }_0+{\displaystyle \sum _{i=1}^n{\gamma }_i{X}_i}\right)\left[{\left(s+t\right)}^{\beta }-s^{\beta }\right]\right)$$

(10)

H(t), known as the risk intensity integral (CHI), represents the cumulative risk amount of the cable in the channel from the start of operation to time t and can be used to assess the overall risk exposure of the cable. In the Weibull proportional hazards model, it can be expressed as Equation (11).

$$H\left(t\right)={\displaystyle {\int }_0^{t}h\left(u\right)}du=\exp \left({\gamma }_0+{\displaystyle \sum _{i=1}^n{\gamma }_i{X}_i}\right)\cdot t^{\beta }$$

(11)

T_0.85 is the total operating time when the reliability of the cable in the channel drops to 85%. Relevant standards of the International Electrotechnical Commission stipulate that when the reliability of high-voltage cables falls below 0.85, they enter a high-risk stage and require risk assessment and replacement. In the Weibull proportional hazards model, Equation (12) can be obtained based on the failure probability density function.

$$T_{0.85}={\left(-{\displaystyle \frac{\ln 0.85}{\exp \left({\gamma }_0+{\displaystyle \sum _{i=1}^n{\gamma }_i{X}_i}\right)}}\right)}^{1/\beta }$$

(12)

Based on the above three indicators, combined with the failure data of urban dense cable channels, it is possible to calculate and analyze the operating conditions of cables under different circumstances.

4. Case Study

There are a large number of cable channels and distribution network cables in the power grid of a certain region in southern China. According to the local channel risk investigation and management records, some cables have been laid beyond the design capacity of the channels. At the same time, high-, medium-, and low-voltage cables are laid in the same channel within the dense cable channel, which is difficult to maintain and poses a significant fire risk. In addition, there are important cable channels for key power users in the local area, which require increased maintenance frequency. By introducing the above influencing factors as covariates into Equation (5), we can obtain the Weibull proportional hazards model at this time as Equation (13).

$$\begin{array}{l}f(t)=\exp \left({\gamma }_0+{\gamma }_1{X}_1+{\gamma }_2{X}_2+{\gamma }_3{X}_3\right)\beta t^{\beta -1}\cdot \\ \exp \left(-\exp \left({\gamma }_0+{\gamma }_1{X}_1+{\gamma }_2{X}_2+{\gamma }_3{X}_3\right)t^{\beta }\right)\end{array}$$

(13)

where X₁ represents the overcrowding rate of cable laying in dense cable channels; X₂ represents the temperature of the cable itself in the channel, with 0 representing normal cable temperature and 1 representing excessive cable temperature; X₃ indicates whether high-, medium-, and low-voltage cables are laid in the same channel, with 0 indicating that they are laid in different channels and 1 indicating that they are laid in the same channel; and γ₀, γ₁, γ₂, and γ₃ are the covariate parameters for each influencing factor.

4.1. Prior Distribution Parameter Selection

In the Weibull proportional hazards model, the prior distribution parameter selection for the shape parameter β is performed first. Based on the O&M data of the cable channels in the research area and the experience of experts, the local cables have been in service for many years and are in a Wear-out Period with β > 1. To reasonably quantify the stable aging rate of the cables in this area, the initial expected value that can be selected is 3, indicating that the cable is in a normal aging state. Due to environmental differences, climatic conditions, geographical locations, and other reasons within different cable channels, the actual aging rate of local cables may deviate from the expert knowledge judgment, and the reliability of the expert’s prior information will be reduced, so the variance of β is set to 1, indicating that the variance of the prior distribution at this time is relatively large, in order to enhance the uncertainty of the prior information. Therefore, the prior distribution of the shape parameter β is shown in Equation (14).

$$\beta \sim G\mathrm{amma}\left(3,1\right)$$

(14)

For covariate parameters γ₀, γ₁, γ₂, and γ₃, since the ledger and the actual maintenance process did not reflect the correlation between the various influencing factors, the three covariate parameters are independent of each other and do not have a conjugate distribution. The selection of covariate parameters must satisfy the cable operating time required by Equation (12).

According to existing literature and research, when the cable capacity utilization rate in a cable channel reaches 100%, the temperature of the cables accumulated in the channel will rise rapidly, greatly affecting their service life. When a cable channel is damaged, the infiltrating liquid will corrode the outer sheath of the cables, affecting their normal operation.

Therefore, the collection of prior information on cable life should focus on three situations: whether the cable channel is overloaded, whether the temperature of the cable itself is normal, and whether the high- and low-voltage cables are touching each other. By collecting lifespan information for cables of various voltage ratings in dense urban cable channels, and by utilizing cable O&M and inspection data, we analyzed the impacts of the channel overcrowding rate, cable operating temperature, and the mixed laying of high-, medium-, and low-voltage cables on the lifespan of cables within the channel. This data was then integrated with expert knowledge and related information as prior information for a comprehensive analysis. The resulting lifespan estimations for cables within the channel under different operating conditions are presented in

Table 1. The life of the cable in the channel under different operating conditions.

Operating Conditions	Service Life T0.85
No overcrowding, normal temperature, segregated installation	65–70
No overcrowding, cable overheating, segregated installation	40–50
100% overcrowding, normal temperature, segregated installation	25–30
100% overcrowding, normal temperature, mixed installation	15–20
100% overcrowding, cable overheating, mixed installation	10–15

.

For the benchmark covariate parameter γ₀, it represents the inherent risk within the channel when all risk factors are at an ideal level. Based on expert assessment of cable lifespan under ideal operating conditions, its initial expected value is set to −32.3. Furthermore, because the operating conditions of cables vary among different channels, the baseline risk has a high degree of uncertainty, therefore requiring the selection of a large variance. For the covariate parameter γ₁, which represents the impact of the overcrowding rate, its initial expected value is set to 2.7. This value is based on an analysis of O&M data concerning the effect of channel overcrowding on cable lifespan, combined with expert experience. A variance of 16 is selected to reflect the experts’ moderate confidence in this risk assessment. For the covariate parameter γ₂, which represents the impact of the cable’s operating temperature, its initial expected value is set to 0.9 based on expert experience. Since excessive cable temperature is known to accelerate insulation aging, this risk factor has a low degree of uncertainty. For covariate parameter γ₃, it represents the impact of mixed cable laying at different voltage levels. By analyzing multiple accidents caused by mixed cable laying in historical O&M data, it is identified as a significant risk factor with relatively low uncertainty. Combining the above information, we can obtain the prior distribution of the four covariate parameters as shown in Equation (15).

$$\begin{array}{l}{\gamma }_0\sim Normal\left(-32.3,900\right)\\ {\gamma }_1\sim Normal\left(2.7,16\right)\\ {\gamma }_2\sim Normal\left(0.9,9\right)\\ {\gamma }_3\sim Normal\left(1,4\right)\end{array}$$

(15)

Substituting this prior distribution into Equation (12) for verification, the calculation results are shown in Equation (16), which conform to the prediction of cable life based on expert experience in Table 1, indicating that this prior parameter can be used as the prior distribution of covariate parameters.

$$\left\{\begin{array}{l}{T}_{0.85}(X_1=0,X_2=0,X_3=0)=25039\\ T_{0.85}(X_1=0,X_2=1,X_3=0)=18579\\ T_{0.85}(X_1=1,X_2=0,X_3=0)=10183\\ T_{0.85}(X_1=1,X_2=0,X_3=1)=7195\\ T_{0.85}(X_1=1,X_2=1,X_3=1)=5402\end{array}\right.$$

(16)

4.2. Selection of Cable Channel Maintenance Data

Since the maintenance data provided by the power grid company was quite large, 15 sets of data were selected from the records for this calculation based on the principle of proportionality. The selected data includes cable operating time within the channel; channel overload rate; cable operating temperature conditions; and the installation of high-, medium-, and low-voltage cables in the same channel. The organized data is shown in

Table 2. Cable operation and maintenance data within the channel.

Channel ID	Operating Days	Risk Value	Overcrowding Rate/%	Temperature Value	Mixed Value
1	4577	1	88.00	1	1
2	4892	1	100.00	0	1
3	5420	1	100.00	1	0
4	6981	0	16.67	0	0
5	7200	0	33.33	0	1
6	7396	0	40.00	1	0
7	7452	0	25.00	1	0
8	7863	0	10.00	0	0
9	7944	0	40.00	0	1
10	8307	0	25.00	0	1
11	8340	0	40.00	1	1
12	8588	0	33.00	1	1
13	8803	0	40.00	0	0
14	8896	0	33.30	1	0
15	9124	0	13.30	1	0

.

4.3. Update Prior Distributions with Maintenance Data

Based on the maintenance data in the ledger, combining the likelihood function and substituting it into Equation (6) yields the joint posterior density function expression containing the maintenance sample information of the dense cable channel.

The M-H sampling method was used to sample the target parameters, and five Markov chains were constructed. The iteration count was set to 70,000, and the first 20,000 iterations produced 100,000 samples, which were discarded as dissipation samples. The last 50,000 iterations produced 250,000 samples, which were used for analysis. We set the proposed standard deviation for β and γ, initialized the sampling matrix, set the upper limit of the sampling acceptance rate to 40%, and updated the parameters. The posterior distributions of each parameter obtained after iterative updating using the data are shown in

Table 3. Parameter posterior distribution.

Param.	Mean	SD	Median	LCL (5%)	UCL (5%)	Accept. Rate Mean	$\widehat{\mathit{R}}$
β	4.07	1.40	3.90	2.03	6.42	19.48%	1.0205
γ0	−43.95	12.66	−42.43	−67.63	−25.24	31.76%	1.0221
γ1	7.99	2.30	7.86	4.42	11.96	20.53%	1.0074
γ2	0.91	1.64	0.88	−1.70	3.67	28.76%	1.0003
γ3	1.41	1.47	1.33	−1.91	2.91	27.18%	1.0006

.

After the iteration, the acceptance rates of M-H sampling for each parameter are all below 40%, and the Gelman–Rubin statistics of each posterior parameter are all around 1. Moreover, it can be seen from Figure 2 that the trace graphs of each parameter have good convergence, indicating that the iterative data are valid. The analysis of the posterior distribution of the parameters obtained in Table 3 shows that the covariance parameter γ₁ corresponding to the overcrowding rate of cable laying in dense channels varies from 2.7 to 7.99, and the standard deviation decreases from 4 to 2.3. This indicates that the overcrowding rate of cable laying has a significant impact on the reliability of cable channels, and that maintenance data provides effective information on γ₁, increasing the certainty of this influencing factor. For the covariate parameter γ₂ corresponding to the temperature of the cable body, the degree of change is not significant, indicating that the prior information on this influencing factor is basically accurate, and the certainty has increased. For the covariate parameter γ₃ corresponding to the mixed laying of high-, medium-, and low-voltage cables, the mean value has changed significantly, and the standard deviation has decreased, indicating that the mixed laying of cables has a significant impact on the operational reliability of channel cables and increases determinism.

The reliability indicators of urban dense cable channels can be analyzed by applying posterior distribution parameters. To quantify the uncertainties in the analysis process, this paper calculates all 250,000 samples obtained from MCMC sampling, generates the posterior prediction distribution of each reliability indicator, and selects the median of the posterior prediction as the central trend metric of the prediction results. This value is robust to the skew of the predicted distribution. Substituting the posterior distributed parameters into Equation (12) and Equation (10) can yield the pre-retirement operating time and reliability of the channel under different cable laying conditions, as shown in Figure 3 and Figure 4. To demonstrate the prediction results and their uncertainty, Figure 3a, Figure 4a and Figure 5a analyze the posterior predictive median, posterior predictive mean, and 95% credible interval for each index, using the benchmark operating conditions (normal cable body temperature and separate cable laying in the channel) as an example. Figure 3b, Figure 4b and Figure 5b analyze the posterior predictive medians under four operating conditions to compare the impact of different operating conditions within the channel on typical performance.

As can be seen from Figure 3a, the shaded area representing the 95% credible interval is quite wide. This is because the presence of a nonlinear term in Equation (12) causes a few parameter combinations to produce extreme values during the MCMC sampling process. Additionally, the model parameters exhibit uncertainty under a small dataset, leading to a large gap between the upper and lower bounds of the credible interval. At the same time, the posterior predictive mean is also affected by these extreme values, causing it to deviate from the posterior predictive median. This indicates that the prediction results are influenced by low-probability extreme values, which can lead to outcomes that are far beyond expectations. As can be seen from Figure 3b, as the overcrowding rate of cables laid in the channel increases, the total operating time at which cable reliability drops to 85% becomes shorter. When the cable overcrowding rate approaches 100%, the service life of cables in the channel is only 8–10 years. This indicates that as the overcrowding rate of cables in the channel increases, the heat generated by cable current carrying capacity exacerbates the impact on surrounding cables. Furthermore, the heat generated by high-voltage cables can affect surrounding medium- and low-voltage cables that are not heat-resistant, causing the insulation materials of these cables to age more quickly and significantly reducing their operational lifespan. The impact of mixed installation of cables of different voltage levels is greater than the impact of overloading and temperature rise of the cables themselves. Therefore, when conducting reliability analysis of cable channels, it is important to focus on whether there is mixed installation of cables.

From Figure 4a, it can be seen that due to the limited training dataset, the model’s prediction of reliability has high uncertainty. As the channel overcrowding rate increases, the posterior predictive mean and the posterior predictive median intersect. This indicates that low-probability risks cause the predictive uncertainty to change with the operating conditions. At low overcrowding rates, the uncertainty is primarily dominated by a few risks where the reliability is much lower than expected. Conversely, at high overcrowding rates, the uncertainty is mainly dominated by a few possibilities that are better than expected.

The cable channels are located in a hot and humid southern region, which affects the service life of the cables. They are generally replaced after approximately 25 years of operation. Substituting the posterior parameters into Equation (12) yields the cumulative risk exposure over the 25-year operational period of the cable, which is shown in Figure 5. Combining this with the curve in Figure 4 showing the change in cable reliability with channel overcrowding rate, it can be observed that as the overcrowding rate increases, the reliability of the cable decreases rapidly. When the cable body temperature is excessively high, and high-, medium-, and low-voltage cables are mixed in the same channel, the cumulative risk exposure of the cable increases gradually with the channel overcrowding rate, reaching 10.43. When the reliability drops to 0.85, the channel overcrowding rate is only 39%, necessitating enhanced management of the channel. When cables are laid reasonably within the channel, their reliability is high and the cumulative risk is within a reasonable range, so their service life should be appropriately extended.

To evaluate the performance of the improved Weibull distribution assessment method used in this paper, the Weibull Accelerated Failure Time model with maximum likelihood estimation (MLE) and the semi-parametric Cox proportional hazards model were selected for comparison, and the consistency index (C-index) and the time-dependent Area Under the Curve (AUC) of different models were analyzed. The comparison results are shown in

Table 4. Quantitative performance comparison.

Performance Indicators	Proposed Model	Weibull-MLE	Cox PH
C-index	0.882	0.815	0.793
AUC	0.865	0.798	0.771

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From the comparison results, it can be seen that the modified Weibull proportional hazards model used in this paper has higher accuracy when dealing with small sample problems and can better integrate expert experience with O&M data.

To verify the validity of the posterior distribution parameters, two new samples were selected from the ledger. The two cable channels selected both belong to important power users or power supply users, and it is necessary to focus on analyzing the operating conditions and reliability of the power supply cable channels. The channel operating condition data is shown in

Table 5. New sample cable channel operation and maintenance data.

Channel ID	Operating Days	Risk Value	Overcrowding Rate/%	Temperature Value	Mixed Value
1	5047	0	40	1	0
2	7356	1	66.7	1	1

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The cables in Channel 1 have been in operation for approximately 14 years, and the operating environment is suitable. Long-term high-load operation has led to the aging of the insulation outer skin, and it is approaching half of the normal operating life, requiring maintenance and repair. Due to the high electricity demand of the affiliated unit, the cables in Channel 2 are densely laid. The operation time is close to the normal service life. The cables have been running at full load for a long time, resulting in a relatively high body temperature. There is also a situation where medium- and low-voltage cables such as transmission cables and communication cables are mixed and laid. To analyze the reliability of the cable channel, it is necessary to calculate the reliability indicators of both.

Analysis shows that the overcrowding rate in Channel 1 is 40%, and the current reliability of the cable is 0.941, with low cumulative risk. This indicates that the reliability of the cable in the channel is currently high and does not require replacement. The reliability will only drop to 0.85 after 30.17 years of operation, with a service life far exceeding 25 years. Under the condition that the channel remains undamaged, it can maintain stable operation for an extended period.

The overcrowding rate in Channel 2 is 66.7%, and the reliability of the cable is only 0.392. At this point, the reliability has fallen below 0.85, indicating a high-risk state. Prompt maintenance and repairs should be conducted, with high-, medium-, and low-voltage cables laid separately, and a new channel installed to reduce the overcrowding rate.

In order to more intuitively represent the reliability of the cables in the two channels, it is necessary to calculate their conditional remaining reliability R(50|s) and analyze the situation of the two channels after another 50 years of operation in their current state, thereby providing guidance for cable replacement and maintenance, as shown in Figure 6.

As shown in Figure 6, the reliability of the cables in Channel 1 remains at a normal level even after 15 years of operation. Therefore, the frequency of manual inspections can be appropriately reduced to cut labor costs, and the replacement of cables can be postponed. During the 15- to 25-year operational period of the cables, regular inspections of their operational status should be gradually intensified, with close monitoring of cable aging and consideration given to replacing the cables; the cables in Channel 2 require strict monitoring of their operational status, and the channel should be renovated as soon as possible to replace the aging cables.

By analyzing the quantitative indicators T_0.85 and R(25|0) output by the reliability assessment model, they can be transformed into a set of risk-based cable operation and maintenance guidelines. By setting reliability index thresholds, it provides a scientific basis for the asset management of power companies, as shown in

Table 6. Cable asset management guidelines based on reliability modeling.

O&M Level	Reliability Metric Threshold	O&M Action
Planned Replacement	T0.85 < 15 years	Execute standard inspection cycles; add the cable to the replacement list.
Early Warning	R(t) < 0.85	Increase inspection frequency; implement derating for the cable.
Critical Risk	R(1|s) < 0.75	Immediately decommission the relevant cable and perform overhaul/replacement.

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The parameter selection for the model in this study is based on cable O&M data and expert experience from a specific region with unique cable operating conditions and climate. To ensure the research results are applicable to other regions, this paper proposes a universally applicable reliability assessment framework. Relevant O&M units first need to set the prior distribution for the parameters, guided by expert knowledge and in accordance with local environmental and technical conditions. Subsequently, they should use the O&M and inspection data from cables in their local channels to update this prior distribution, thereby obtaining posterior parameters that accurately reflect the local situation. Furthermore, the O&M units can add relevant covariates based on the operating environment of their local dense cable channels to conduct a more comprehensive reliability assessment.

5. Conclusions

In order to analyze the impact of cable laying conditions in urban dense cable channels on the reliability of cable channels, this paper proposes a reliability assessment method for mixed cable laying in urban dense cable channels based on an improved Weibull distribution. Risk factors were introduced as covariates into the Weibull proportional hazards model. A Bayesian model was used to combine expert experience information with cable channel operation and maintenance data. The MCMC algorithm was used to obtain updated Weibull parameters. Three cable channel reliability indicators were established to analyze the impact of cable channel overcrowding, cable operating temperature, and cable spacing within the channel on cable channel reliability. By analyzing the operating environment of cable channels in actual cases, we compared and assessed the reliability of four typical cable channel operating conditions. The assessment results were consistent with actual conditions, verifying the feasibility and accuracy of the method. A subsequent comparison with other models further demonstrated the accuracy of the model selected in this study. Finally, two important lines within the cable channel were selected for analysis. Based on their operational status, a differentiated maintenance strategy was proposed, such as increasing the frequency of inspections of high-risk cable channels and promptly improving the operational environment of cable channels. A set of generally applicable cable O&M guidelines was also presented. This will provide maintenance personnel with more scientific and effective cable channel maintenance guidelines, which is of great significance for the stable operation of urban underground power transmission lines.