When a distribution network fault occurs, it is necessary to adjust the operation mode immediately to isolate the fault area and reduce the scope of the fault propagation. This section formulates the distribution network scheduling strategy based on the principle of ensuring the power supply of important loads and meeting the power supply of coupling points. This section does not consider the involvement of distributed power sources or other mobile emergency recovery measures to ensure that the path planning solution is still feasible in adverse environments. The objective function is as follows: max ∑ i ∈ V P τ i C S P i C S y i C S + τ i l o a d P i l o a d y i l o a d (1) where τ i C S and τ i l o a d are the weighting coefficients of the charging station load and other loads respectively excluding the charging station; the coefficients of the first-level, second-level, and third-level loads are 1, 0.1, and 0.01, respectively; in order to restore the charging station load as soon as possible, τ i C S is set to 1; P i C S and P i l o a d are the transmission power of the charging station load and other loads, respectively; y i l o a d and y i C S are the restoration status of the charging station load and other loads, respectively.

After making the scheduling decision, the distribution network provides the flow distribution results. The intelligent network system then recommends the optimal charging station for the EVs based on the original load and node voltage information of each coupled node, and calculates the road traffic flow distribution. Because this study plans the optimal charging route for EVs from a dynamic perspective, static traffic distribution models are insufficient to describe the time-varying characteristics of the network state and cannot provide accurate traffic and charging information for the planning stage. Therefore, in this section, a dynamic traffic distribution model is established as follows: min ∫ 0 T 1 ϑ ∑ r ∈ R ∑ k ∈ K r q r k t ln q r k t d t + ∫ 0 T ∑ a ∈ E r T ∫ 0 υ a o u t t U a q a t , z d z d t (2) q r k t = q r t ⋅ P r k t P r k t = exp − ϑ ⋅ C P T r k t ∑ k ∈ K r exp − ϑ ⋅ C P T r k t (3) s . t .   q a t = q a t − 1 + υ a o u t Δ t (4) q a t − q a t − 1 = Δ t υ a i n t − υ a o u t t ∀ a , t (5) q a t = ∑ r ∈ R ∑ k ∈ K r ∑ i , j ∈ V T x i j ⋅ q r k t ∀ a , k , t (6) ∑ υ a o u t t + τ a t = ∑ υ a i n t 1 + τ a t − τ a t − 1 / Δ t ∀ a , t (7) υ a i n t ≥ 0 , υ a o u t t ≥ 0 , q a t ≥ 0 (8) where ϑ is the dispersion coefficient, which characterizes the degree of familiarity of car owners with the network state; q r k t is the traffic demand of path k at time t ; q r t is the traffic demand of OD pair at time t ; E r T is the set of road segments between OD pairs; υ a i n t and υ a o u t t are the inflow and outflow rates of road segment a at time t ; q a t and q a t − 1 are the traffic flow of road segment a at time t and the previous time, respectively; Δ t is the simulation step size, which is set to 5 min; U a is the travel cost of road segment a ; P r k t is the probability that car owners choose path k at time t ; C P T r k t and C P T a t are the cumulative prospect values of path k and road segment a , respectively, at time t ; x i j is the selection variable for path i j ; τ a t and τ a t − 1 are the travel times of road segment a at time t and the previous time, respectively. The dynamic distribution model is solved using the iterative weighted method (Yang et al., 2022) to obtain the flow distribution of the traffic network quickly.

When an EV arrives at a charging station, if the number of vehicles waiting in the queue is fewer than the number of available charging piles, the waiting time for all EVs in the queue is zero, and the arrival time is considered the start time of service. However, if the number of vehicles waiting in the queue is greater than the number of available charging piles, the waiting time for an EV is determined by the charging time of the EVs that are currently receiving service. The model can be expressed as follows: t b w a i t = 0   N b w a i t t ≤ N b − N b c h a r t ∑ c = 1 N b c h a r t S O C b , c c − S O C b , c 0 P ¯ b c h a r ⋅ η c h a r   N b w a i t t > N b − N b c h a r t (9) H = N b w a i t t − N b − N b c h a r t (10) where t b w a i t is the waiting time of the EV at the charging station b ; b and c are the identification numbers of the EV and charging station, respectively; N b w a i t t and N b c h a r t are the number of EVs in the queue and the number of EVs currently being charged at the charging station in the current period, respectively, based on real-time monitoring data; N b is the number of available charging piles at charging station b ; S O C b , H c and S O C b , H 0 are the expected and initial state of charge of the c -th EV, respectively; P ¯ b c h a r is the rated charging power of the charging pile; η c h a r is the charging efficiency of the charging station, which is set at a constant value of 0.95.

Based on the analysis in Section 2.2, this section adopts the prospect theory to describe the limited rationality of EV owners in decision-making (Wu et al., 2020). The charging station decision-making model is constructed by establishing a search direction and remaining power constraint to ensure the correctness of the path search direction. The specific model is presented as follows: C P T c b c g o a l = max C P T c 1 , C P T c 2 , ⋯ , C P T c b ⋯ , C P T c n C S , (11) s . t .   E ¯ c ⋅ l i j , min E − S t < E c l t θ C S < 90 ∘ (12) where c is the EV number; the physical meaning of C P T c b c g o a l is that the c -th EV selects the charging station with the maximum cumulative prospect value as the target charging station, and the calculation process of the cumulative prospect value is shown in Appendix D; b c g o a l is the node number of the target charging station for the c -th EV; n C S is the number of charging stations; y c is the decision variable for charging station selection; E ¯ c is the average power consumption per kilometer of the EV; l i j , min E − S t and E c l t are the shortest distance from the current time to the nearest charging station and the remaining power of the EV; θ C S is the angle between the line connecting the node i where the EV is located to the charging station and the line connecting the node i to the destination.

The classic BPR model is based on regression analysis of low-saturation highways in the United States. While simple and easy to solve, it cannot be applied to urban road networks under fault conditions for two reasons. First, the model is suited to road sections, such as highways, where traffic flow exhibits continuous characteristics. However, traffic flow in a faulty road network exhibits pulse characteristics due to the influence of intersections and congested traffic flow (Yuan and Tang, 2021). As a result, the BPR function, which is a monotonic increasing function, cannot reflect the dynamic fluctuations of EV flow velocity that increase first and then decrease after a surge in traffic flow (Zhao et al., 2020). Second, the model always assumes that traffic demand is less than traffic capacity. When a road segment approaches saturation or oversaturation, the time curve approaches an asymptote parallel to the y-axis (Jiang et al., 2010). However, actual traffic networks exhibit high and stable travel times on road segments due to the existence of signal controls. If the BPR function, calibrated with uniform parameters, is used continuously, the calculated results will significantly deviate from actual values. Therefore, the traditional BPR function is no longer applicable to congested urban road sections with complex and variable traffic flow distributions.

Unlike the traditional static BPR model, the time-time occupancy rate model proposed in this paper considers traffic density as the research object instead of traffic volume. This is because traffic volume is only a temporal observation of a road section and cannot describe the interaction trends among vehicles in high-density road sections. In contrast, traffic density is a spatial observation of the number of vehicles in a road section, which can be used to analyze the relationship among vehicles through the density changes of the road section at different time periods, and thus describe the dynamic development process of traffic flow. Traffic wave theory describes the energy flow generated when traffic density changes due to a sudden event. In cases where the coupling between the power distribution network and the traffic network fails, the congestion effect causes some sections of the traffic network to reach saturation or even oversaturation. This causes vehicles to constantly switch between stagnant and low-speed driving states, and the queue of vehicles to move forward in a traffic wave form. In addition, the cascading propagation characteristics of coupling faults cause vehicles to constantly gather and form a congested traffic flow, which dissipates after passing through signal controls at intersections. In the dynamic process of convergence and dissipation, the traffic flow exhibits intermittent flow characteristics. Traffic wave theory studies the relationship between the three parameters of traffic flow and the length of the vehicle queue, which can describe the process of gathering and dissipating of queued vehicles from a macro perspective (Ma et al., 2015). In addition, when calculating the number of queuing vehicles using the methods of traffic wave theory such as the accumulation wave and dissipation wave of traffic flow, the change in traffic density parameter is the research object. This is consistent with the use of traffic load parameters (i.e., traffic density on a road section) in the analysis of dynamic urban road network state. Therefore, this section adopts the traffic wave theory to model and analyze the three types of road conditions in the faulty traffic network, accurately depicting the charging behavior of EVs in the faulty network, and providing modeling and solving ideas for the path planning model to solve the coupled fault between the power distribution network and the traffic network.

The equation for the three-parameter relationship of traffic flow is: q = k v (14) where q is the traffic flow; k is the traffic density; v is the vehicle speed on the road.

Density k is obtained by fixed-point measurement through loop detectors installed at road intersections. The value of k is the ratio of the number of vehicles passing through the detector during the observation period to the observation time. When the observation time is short, k is more closely related to the traffic attributes around the detector. When the traffic flow states upstream and downstream of the road section are different, it will lead to deviations in the calculation results.

The time occupancy ratio is determined by the vehicle speed. It represents the ratio of the time that vehicles pass through the detector to the total observation time. Because the vehicle speed is the same at both the intersection and the midstream of the road segment, the calculation results are independent of the observation time and can accurately reflect the operational status of the road segment. Therefore, the time occupancy ratio is used instead of density. The conversion relationship between the two is as follows: k = o c k (15) where c k is the sum of the length of the vehicle and the length of the detector.

The traditional Greenshields model is limited in its suitability for low-density road sections and its reliance on subjective parameter determination. To address these limitations, the logistics speed-density model proposed by Ma Xiaolong et al. (Ma et al., 2015) is introduced. In addition, v m − v f / v f and v 0 / v m − v 0 are fixed constants, represented by c f − 1 and c 0 , respectively. Substituting the quantified time occupancy rate ω = o / o f , a model for traffic flow, parking wave, and starting wave is established as below:

The expression of traffic flow q is as follows: q t = o t v m c k − 1 v m − v f v f o t o f v 0 v m − v 0 o t o f − 1 + 1 − 1 = o t c 1 c 2 ω + c 0   o t > o t p v 0 v 0 − v f v f o t − o t p o f − o t p + 1 − 1   o t ≤ o t p (16) where v m , v 0 , and v f are the maximum speed that vehicles can reach under ideal conditions, free-flow speed of the road, and traffic flow speed in congested conditions, respectively, and all three speeds are parameters in the Logistic model; v f is set as 2 km/h; o f is the time occupancy rate in congested conditions; o t p is the turning time occupancy rate, the physical meaning is the point on the speed-density curve with the maximum absolute value of the slope, which is also the turning point from free flow to congested flow of traffic. v a w s , − t = q a s t a r t t k a s t a r t t − k f = q a s t a r t t o a s t a r t t − o f v a w u , − t = q a e n d t k a e n d t − k f = q a e n d t o a e n d t − o f (17) where c 1 = c 0 ⋅ v m / c k ; c 2 = c 0 / c f ; v a w s , − is the parking wave, defined as the energy wave generated when entering the stationary area from the non-zero velocity area; v a w u , − is the start-up wave, defined as the energy wave generated when entering the non-zero velocity area from the stationary area; q a s t a r t t and q a e n d t are the upstream and downstream traffic flows of section a at the current time, respectively; k a s t a r t t and k a e n d t are the upstream and downstream road densities of section a at the current time, respectively; k f is the road density under congested conditions; o a s t a r t t and o a e n d t are the upstream and downstream time occupancy rates of section a at the current time, respectively; ω a s t a r t t and ω a e n d t are the upstream and downstream quantified time occupancy rates of section a at the current time, respectively.

Three road resistance models have been established for three road queuing conditions. Situation 1 and situation 2 correspond to the EV passing the whole road in the first green signal cycle. Situation 3 corresponds to the vehicle not passing the road in the first green signal cycle. Figure 1 is the main characteristic diagram of the traffic wave model.

The objective function is established from two aspects: the benefits of the vehicle owners and the coupled network state. F = γ 1 min C E V ′ C E V t + γ 2 min T E V ′ T E V t + γ 3 min S E V ′ S E V t + γ 4 min R E V ′ R E V t (35) where C E V and T E V are the optimization objectives for EV charging; S E V is the optimization objective for power grid operation; R E V is the optimization objective for traffic network state; γ 1 , γ 2 , γ 3 and γ 4 are the weighting coefficients of the objectives.

This section analyzes four EVs (No. 2, 5, 7, and 9) located on congested and uncongested road segments.

Figure 5 shows the topology of the distribution network after scheduling, and Figure 6 depicts the load variations of the charging stations. The upper and lower surfaces of the surface plots correspond to the load variations after adopting the uncoordinated and coordinated charging strategies, respectively.

In Scenario 1, branch 2-S3 is faulted, triggering the closure of contact switch 2-S37 and restoring the system connectivity. The minimum node voltage of the system is 11.75 kV, and the maximum power transmission capacity of the line is 7.41 MW, which exceeds the threshold by 5.86%. To address this issue, third-level loads 10, 14, 19, and 21 of the No. 1 distribution network and third-level loads 9, 27, and 29 of the No. 2 distribution network are disconnected.

In Scenario 2, branch 1-S25 is added as a fault, triggering contact switch 1-S37. In Scenario 3, branch 1-S3 is added as a fault, triggering contact switch 1-S35. The minimum system voltages are 11.962 kV and 12.159 kV, respectively, and the line transmission powers are both less than 7 MW, with no line overload or voltage violations observed. The scheduling process is completed.

In Scenario 1, after adopting the uncoordinated strategy, charging stations 1, 2, 6, and 7 exceed the threshold in the 6th time window, with loads of 6.7687 MW, 6.8642 MW, 6.369 MW, and 6.5315 MW, respectively. At this time, the minimum loads of the four stations are 2.76 MW, 2.98 MW, 2.62 MW, and 2.32 MW, respectively, with significant differences in load peaks and valleys. After adopting the coordinated strategy, only charging station 6 approaches the threshold in the 10th time window, with a load of 5.98 MW, and the minimum load increases to 3.1438 MW. The peak loads of the other charging stations are all less than 4.2 MW, with a minimum load of 3.221 MW. All charging station loads decrease to within the threshold, and the peak-to-valley difference decreases, which is beneficial for the stable operation of the charging stations.

In Scenario 2, charging stations 1, 4, and 6 have peak loads in the 8th and 9th time windows, with loads of 5.7546 MW, 5.7981 MW, and 5.6495 MW, respectively. In Scenario 3, charging stations 2, 6, 7, 8, and 11 approach the threshold in the 6th to 12th time windows, with charging station 7 having a peak load of 5.998 MW in the 10th time window.

Figure 7 illustrates the voltage variation of the nodes for the unordered and ordered charging strategies. The voltage deviation rate of nodes 14–18 exceeds the limit under the unordered strategy, which poses a threat to the safe operation of the distribution network. However, under the ordered strategy, the average voltage deviation rates of the distribution network for the three scenarios are 3.03%, 3.23%, and 3.53%, respectively, all of which are within the deviation limit of 7%. Therefore, the planning scheme can guarantee the safe operation of the distribution network.

To describe the level of congestion, the TTI index is used as the evaluation indicator in this section, and the conversion relationship is shown in Table 5. In Scenario 1, after adopting the disordered charging strategy, 21.77% of the road segments were in a congested state, with a maximum TTI value of 2.02. After adopting the ordered charging strategy, even during the peak charging period from 16:20 to 16:35, only 11 road segments experienced slight congestion, accounting for 7.48% of the total, and the maximum TTI value was only 1.64. Although there are still a few lightly congested road segments, the traffic flow distribution is more even, which helps alleviate traffic congestion.

Figure 8 shows the distribution of road traffic in the three scenarios. In Scenario 2, 27 road sections experienced minor congestion between 16:20 and 16:40, accounting for 11.2% of the total. The average TTI was 1.41, with a maximum of 1.7. In Scenario 3, 53 road sections experienced congestion between 16:15 and 16:55, accounting for 36.1% of the total. The average TTI was 1.55, with a maximum of 1.812. As the fault range expanded, the number of congested roads increased significantly; however, the overall traffic flow remained relatively smooth. Therefore, the planning scheme can ensure the safe and continuous operation of the traffic network.

As shown in Figure 9, the satisfaction degree of the path planning results is all above 75%. Among them, when the weight ratio exceeds 3:7, the satisfaction degree exceeds 90%. This indicates that under the fault state, the travel time has a greater impact on the planning results, and the car owners tend to choose the path plan with the shortest travel time.

When the utilization rate of charging piles is less than 1, the satisfaction of car owners is above 94%. When the utilization rate reaches 1, the satisfaction drops to 93.1%. This indicates that in a fault state, driving time has a greater impact on the planning results, when there are available charging spots in the charging station, the satisfaction is higher, and the impact of the charging station’s operational status on the planning results is smaller. When there is a queue in the charging station, the satisfaction of car owners is affected by the waiting time, and the impact of operational status of the charging station on the planning results deepens.

To compare the propagation range of coupling failures in the transportation network caused by the power outage of coupling nodes with different importance levels, this section introduces two new scenarios: Scenario 4 and Scenario 5.

(1) Scenario 1: Node 4 in distribution network 2 fails, and charging station 5 connected to this node terminates service, with a power outage capacity ratio of 7.14%.

The number of road vehicles in Scenario 4 and Scenario 5 are shown in Figure 10. Comparing Scenario 1 and scenario 5, it can be observed that when a heavily affected charging station fails, there are significantly more congested road sections than when a lightly affected charging station fails, indicating a more pronounced impact on the transportation network. Comparing Scenario 1 and scenario 4, it can be seen that station 5 can distribute the affected charging flows to nearby stations such as stations 1, 2, 4, and 6, as well as the more remote station 10. Although the moderately congested road sections are more numerous than in scenario 4, the flow peak-to-valley ratio is smaller, indicating a more even distribution.

Therefore, after a fault occurs, the overall state of the transportation network is affected not only by the capacity of the failed node, but also by the geographical location of the failed node within the transportation network and the distribution of nearby charging stations.