Considering that the charging mode allows EVs to charge without exceeding the battery capacity and that the constant current process in a complete slow charging cycle is very short, the entire slow charging process is approximated as constant voltage charging. Assuming that the charging current for slow charging is 0.1C (C refers to battery capacity), the charging power Pc of a typical EV falls within the range of 2–3 kW and follows a uniform distribution, with its probability distribution function as (Eq. 1) f P c x = 1   x ∈ 2,3 0  otherwise (1)

The time required for each EV’s slow charging is calculated as (Eq. 2) T c = S W 100 100 P c (2)

Considering two types of slow charging modes: dispersed and centralized, this study focuses on dispersed slow charging for both pure electric and hybrid vehicles. Assuming that all slow-charging EVs finish their last daily trip and return to their parking spots at the time which marks the beginning of charging. Based on a survey of American household vehicles, the start time of charging is also approximated as a normal distribution, with the specific probability density function as (Eq. 3): f s x = 1 σ s 2 π exp − x − μ s 2 2 σ s 2   μ s − 12 < x ≤ 24 1 σ s 2 π exp − x + 24 − μ s 2 2 σ s 2   0 < x ≤ μ s − 12 (3)

With the initial scale of EVs in the region, combining the EV driving mileage and charging start time, the remaining battery charge of each vehicle can be calculated.

In rapid charging mode, the initial phase of EV charging generally involves constant current charging, following an exponential distribution. When the battery is charged to 80% of its rated capacity, the process is as (Eq. 4): P f c t = P N e − P N 0.9 − S O C 0 C t (4) For instance, P N = 100 kW, for a battery capacity of 60 kW and a remaining charge of 40%, an EV would require approximately 0.67 h to charge.

The extent of EV charging and battery swapping demand is primarily influenced by user behavior, battery capacity, and the technical level of charging equipment. At a certain stage of development, constrained by the level of technology, battery capacity and the power rating of charging equipment are generally fixed, hence EV charging and swapping demand mainly depends on daily driving mileage.

To obtain the daily load curve of EV charging and swapping, it is first necessary to know the daily driving mileage S of EV users and the electric energy W 100 consumed per 100 km by the EV. The product of these two factors gives the daily power consumption of the EV. Based on relevant statistical survey data, the daily mileage is approximated to fit a log-normal distribution, with the corresponding probability density function as (Eq. 5): f d = 1 d σ D , i 2 π exp − ln ⁡ d − μ D , i 2 2 σ D , i 2 , i = 1,2 , . . . , 7 (5)

Compared to the dispersed slow charging mode, the rapid charging mode sees a higher degree of charging demand, mainly affected by the power consumption during various time periods. This paper assumes that the distribution of start times for rapid charging is consistent with the swapping time distribution in the swapping mode. It is also assumed that when pure electric and hybrid EVs undergo rapid charging, the remaining battery charge follows a normal distribution of (0.4, 0.133), leading to a specific probability density function for the remaining battery capacity (Eq. 6). f K x = 1 σ K 2 π exp − x − μ K 2 2 σ K 2 (6)

The introduction of nonlinear loads by AC charging of EVs leads to the generation of harmonics. These harmonic signals affect both voltage and current, encompassing harmonic voltage and current. According to the definition by IEC 61000, these are represented as (Eq. 7) and (Eq. 8): σ THD u = ∑ 2 n U n 2 U 1 (7) σ THD i = ∑ 2 n I n 2 I 1 (8)

As per the IEC 60287 standard, the core temperature θ of a 10 kV three-core cable can be expressed as θ = θ c + θ 0 , the values of related parameters can be found in Wang et al. (2019) θ _c can be calculated by (Eq. 9). θ c = ∑ i = 1 n I i 2 R AC i T 1 + c i T 2 + c i 1 + λ 2 i T 3 + T 4 (9)

The harmonic currents generated during the charging process of EVs can cause the total current carrying capacity of the cable to exceed its design value, potentially leading to overload and safety issues. The current carrying capacity of 10 kV three-core cable can be calculated using (Eq. 10) I h , rated = Δ θ c ∑ i = 1 n I i 2 I 1 2 R A C i T 1 + 1 + λ 1 i T 2 + 1 + λ 1 i + λ 2 i T 3 + T 4 (10)

The harmonic currents produced during the EV charging process cause additional heating in the cable, affecting its thermal life. The cable’s life per hour is given by (Eq. 11): L h t = 8760 ⋅ L 0 e − Δ w k B T t (11)

The daily life degradation of the cable is expressed as (Eq. 12): L d = ∑ t = 1 24 1 L h t (12)

Cable fault sample data follow a Weibull distribution with the sought parameters. Such a function can simulate various fault factors by selecting appropriate parameters and can also assess the systemic reliability when the cable fault rate dynamically changes. The mathematical model for cable fault rate is shown as (Eq. 13): λ t = β η t η β − 1 (13)

To account for the difference in cable fault rates before and after maintenance, this paper introduces an age regression factor α , representing the actual change in cable fault rate. The actual service age of the cable is then given by (Eq. 14): T real = T − α T = 1 − α T (14)

This allows for the determination of the fault rate after the k‐th maintenance. As shown in (Eq. 15). λ k t = λ t + 1 − α k T = β η t + 1 − α k T η β − 1 (15)

The FHO algorithm is a meta-heuristic algorithm that simulates the foraging behavior of fire hawks, considering processes like setting fires, spreading, and capturing prey. Initially, several candidate solutions are determined as the location vectors of the fire hawks and prey, using a random initialization process as per the given formula to identify these vectors’ initial positions in the search space. As shown in (Eq. 16) and (Eq. 17). X = X 1 ⋮ X i = x 1 1 x 1 2 ⋯ x 1 j ⋮ x i 1 x i 2 ⋯ x i j , i = 1,2 , … , N j = 1,2 , … , D (16) x i j 0 = x i , ⁡ min j + r a n d ⋅ x i , ⁡ max j − x i , ⁡ min j (17)

The distance between a fire hawk m and prey n is calculated to determine the nearest prey around each fire hawk and define their territories. As shown in (Eq. 18). D n m = x n − x m 2 + y n − y m 2 (18)

Then, positions are updated: fire hawk m collects burning sticks from the main fire and throws them into a specific area to force the prey to flee. The detailed process of updating positions can be referred to in the cited literature (Azizi et al., 2023).

Based on the solution process of the aforementioned FHO optimization algorithm, the steps to optimize the BiGRU model hyperparameters are as follows:

Step 1

Define the Hyperparameter Search Space. Identify the hyperparameters of the BiGRU model to be optimized and establish a range or a set of possible values for each hyperparameter.

Step 2

Initialize Solution Candidates. Generate initial solution candidates of hyperparameter combinations randomly, in accordance with the defined search space.

Step 3

Evaluate Initial Solution Candidates. Employ cross-validation or other suitable methods to train the model and assess the performance of the initial solution candidates, calculating their fitness values (such as accuracy, loss function values, etc.).

Step 4

Set the Global Optimum. Designate the best solution among the initial candidates as the global optimum.

Step 5

Iterative Optimization. Continuously update the hyperparameter combinations by generating falcons and prey. The distance between falcons and prey is computed, and territories of the falcons are determined. Falcons update their hyperparameter combinations based on their positions, and prey both within and outside the territories also update their combinations. By evaluating the fitness values of the updated hyperparameter combinations and updating the global optimum to the current best solution, the FHO algorithm progressively searches for improved hyperparameter combinations, thereby enhancing the BiGRU model’s performance through iterative optimization.

Step 6

Return the Global Optimum. Upon completion of the iterations, the global optimum is returned as the optimized set of hyperparameters for the BiGRU model.

Step 1

Data Preprocessing. Undertake preparatory processing of the monitoring data for cable equipment, involving data cleansing and normalization among other procedures, to ready the input data.

Step 2

Feature Extraction. Utilize a CNN to extract salient features from the cable equipment monitoring data, capturing key temporal sequence characteristics.

Step 3

Sequence Modeling. Employ a BiGRU to model the extracted time series features, considering both historical and prospective state information.

Step 4

Predictive Output. Utilize the terminal temporal step’s BiGRU hidden state as the vector representation of the equipment’s health status, and input this into a fully connected layer for the prediction of the health state.

Step 5

Model Optimization. Apply the FHO algorithm to refine hyperparameters such as the convolutional kernel size, GRU’s hidden unit count, learning rate, and other factors pertinent to model performance, thereby enhancing the accuracy of health status prediction.

Step 6

Model Evaluation. Assess the model’s performance using appropriate metrics (like accuracy, recall, F1 score, etc.), comparing it against the actual health states of the equipment.

Step 7

Health Status Prediction. Utilize the optimized FHO-CNN-BiGRU model to predict the health status of cable equipment based on new monitoring data, thus obtaining prognostications regarding the health state of the cable apparatus.

Combining the FHO-CNN-BiGRU prediction model, the GPR model is utilized for its superiority in handling nonlinear problems to obtain nonlinear mappings between measurements and state quantities. The integration of FHO-CNN-BiGRU and GPR algorithms allows for weighting the predicted values from FHO-CNN-BiGRU and the state estimates from GPR, leading to more accurate predictions of the cable equipment’s state.

The Gaussian Regression model maps input features to a high-dimensional space through a set of basis functions, denoted as ψ · , thereby facilitating the discovery of linear relationships among data in this elevated dimensional space. By representing the measured quantities Z, with these basis functions ψ · , one can derive the probability associated with new data.

The GPR uses confidence judgment to estimate results, which can mitigate noise interference, thereby improving the accuracy of the estimates. Offline training is first conducted based on the nonlinear relationship between historical measurement and state data of cable equipment, followed by state prediction using the trained model based on new measurement data.

Aging and deterioration are phenomena that span the entire lifecycle of equipment. Cables, especially those operating in humid environments for extended periods, are prone to water tree aging. The operational state of the cable, maintenance strategies, construction quality, and operating environment significantly impact cable deterioration. Maintenance can enhance the performance of components and extend the equipment’s life to a certain extent, but it cannot restore the equipment to its original healthy state, as the rate of deterioration remains unchanged. Assessing the health status is an effective way to monitor the condition of cable equipment. Based on the cable status prediction results, this paper introduces the concept of a Cable Health Index and constructs a cable health status assessment framework based on this index. The cable health status assessment process, as illustrated in Figure 3, involves the following detailed steps:

Step 1

Data Collection and Processing. Utilize distributed fiber optic sensors to collect relevant data of the cable, including its lifespan, insulation thermal resistance, and other parameters. Combine these with predicted parameters like current carrying capacity, temperature, and lifespan to construct a health status assessment dataset. Analyze and process the collected data, employing methods like Principal Component Analysis to remove redundant features and repair missing data.

Step 2

Determination of Cable Fault Factor Weights. Considering the operational conditions of the cable, use the Analytic Hierarchy Process to calculate the weights of factors that could potentially cause cable faults, obtaining the weight w i for the ith type of fault factor. Factors include heavy overload, lightning strikes, high temperatures, humidity, chemical corrosion, human-induced damage, animal damage, and insulation aging.

Step 3

Calculation of Cable Fault Rate. Calculate the cable fault rate λ t and the fault rate λ k after kth maintenance using the designated (Eq. 19). Modify the fault rate based on the cable’s service age using the appropriate (Eq. 20) to obtain the real-time fault rate. λ t = 0.0198 t ≤ 20 6.849 35.536 × t 35.536 5.849 t > 20 (19) λ * = w i ⋅ λ k (20)

Step 4

Calculation of Cable Health Index. Utilize the real-time fault rate to calculate the cable’s real-time health index H * . As shown in (Eq. 21). H * = 1 ξ ln λ * K (21)

Step 5

Assessment of Cable Health Status. Based on the calculated real-time health index, assess the health status of the cable. Categorize the cable into different health statuses, including healthy, sub-healthy, early warning, and damaged, according to predefined standards and thresholds.

Step 6

Early Warning and Decision-Making. Set warning thresholds based on the real-time fault rate and health index. When the fault rate exceeds the threshold or the health index declines, the system can issue an alert, prompting maintenance personnel to take appropriate actions such as repair, replacement, or upgrade of the cable.

To assess the impact of different scales of EV integration on cable temperature rise, we analyzed cable temperatures under various charging demands, setting EV penetration rates at 0%, 10%, 20%, 30%, 40%, and 50%. The simulation tested the cable temperature variations in each scenario, as shown in Figure 6; Table 1.

From Figure 6, it is evident that with the increase in the number of EVs, the rising charging demand leads to a continuous increase in cable temperature. Especially when the EV penetration rate reaches half of the total number of private cars in the community, the cable temperature exceeds the permissible limit. At peak demand times, the cable temperature reaches 91.67°C, exceeding the 90°C limit even without considering the impact of harmonics. Therefore, reducing harmonics and improving cable thermal management are crucial for the reliable operation of cables and the entire distribution system. Implementing rational orderly charging strategies or incentivizing users to charge during off-peak hours through time-of-use tariffs could mitigate the risk of cable current exceeding limits due to the cumulative demand of charging and base electrical loads.

To verify the impact of different charging strategies on cable life, the temperature variations under slow charging demands at different EV penetration rates are shown in Figure 7. Additionally, considering the peak and off-peak periods as defined in (Sun et al., 2019), Figure 8 shows the temperature variation curves for two scenarios at a 50% penetration rate under time-of-use charging.

From the figures, it is seen that in uncoordinated charging, all EV users start charging immediately upon returning home, with charging concentrated between 6 p.m. and 2 a.m. the next day. At 6 p.m., the cable temperature reaches 91.67°C due to the peak in total electricity demand from both charging and base loads. Under the influence of time-of-use tariffs, EV users tend to choose lower-priced periods for charging, mostly during the peak hours of solar power generation. Although the surge in charging demand increases the cable temperature, it effectively avoids the temperature rise caused by “peak upon peak” by occurring during the low-demand period of the grid. The cable temperature remains at 88.28°C, not exceeding its maximum tolerance, thereby reducing the fluctuation range of cable temperature. This strategy not only avoids impact on the cable but also enhances the utilization of solar power in the grid.

An analysis is conducted on the operational status of cables under the two charging scenarios mentioned above, and a health status assessment is performed using the health index. The criteria for classifying health levels are shown in Table 2, and the future health status of the cables is predicted using the model proposed in this paper.

As indicated in Table 2, S1 represents equipment that meets all performance standards, with good foundational conditions, intact technical performance, and stable operational conditions, capable of withstanding external environmental risks. S2 denotes equipment with intact functionality and performance, though slightly below the healthy state, still meeting standard operational requirements. S3 indicates equipment in basically good condition, capable of performing specified functions, but with some performance degradation, having a minor overall impact and needing maintenance. S4 signifies equipment with severe technical performance deficiencies, unable to meet operational standards, and posing a threat to system safety and stability, urgently requiring replacement.

Figure 9 shows an increase in the fault rate of cables over time. Considering the climate environment of the Xiong’an New Area, 100 cables are selected, and the weights of various fault factors are calculated using the Analytic Hierarchy Process, as shown in Table 3.

The cable fault rate and health index are calculated based on the weights of the fault factors and used as initial parameters to predict future changes in the cable’s health status. Table 4 presents the fault rates and health information of 10 selected cables and predicts their future health status.

To validate the superiority of the method proposed in this study, its prediction results were compared with those of the Back Propagation Neural Network (BPNN) and Support Vector Machine (SVM) models, as shown in Table 5. S1 and S2 respectively correlate to the health grades outlined in Table 2. The amalgamation of S3 and S4, denoted as S34, indicates that the cable is in a state of malfunction or pre-alert.

The analysis of the provided data in Table 5 reveals a clear superiority of the proposed method in accuracy rates across all categories. It surpasses both BPNN and SVM, achieving 92.5% in S1, 93.6% in S2, and 90.3% in the combined S34 category, with a comprehensive accuracy of 92.5%. This consistently higher performance across all metrics establishes the “Proposed Method” as the most effective and accurate among the methods evaluated.