Powering and mobility through disasters: a dial-a-ride problem for electric taxis with charging and discharging regulations

Abstract

This study proposes an integrated strategy to effectively utilize electric taxis (ETs) for post-event recovery of power distribution networks (DNs) in disaster scenarios. The approach addresses a dial-a-ride problem (DARP) by coordinating charging and discharging regulations to improve grid security. It focuses on optimal taxi routes, considering charging, and limited range. The model highlights proactive strategies such as pre-positioning electric taxis (ETs) with vehicle-to-grid (V2G) capabilities and reconfiguring the power distribution network. A network interdiction strategy is implemented to reduce cyberattacks on the power distribution network. The results demonstrated the importance of a holistic approach considering various factors, including routing costs, cybersecurity implications, and optimal power flow constraints. When tested with New York taxi data, the hybrid planning approach emerged as the most effective model. It resulted in the lowest total tour length, reduced power load shedding, and fewer vulnerable system nodes to cyberattacks compared to other cases examined.

Introduction

The resilience of transportation systems is crucial during disaster events, as reliable mobility plays a vital role in effective emergency response, safe evacuation of affected populations, and the continuity of essential services. The ability to quickly mobilize and transport people and resources becomes even more critical when disasters occur. These disasters can impact all modes of transportation, including traditional gasoline/diesel vehicles and electric vehicles (EVs), along with their charging infrastructure¹. In the case of ET vehicles, power outages affecting charging stations can render them immobile unless they have sufficient battery charge to reach an operational charger. Consequently, transportation systems often face challenges in meeting the heightened demands and disruptions caused by such crises.

Similarly, the resilience of DNs is crucial in power outages caused by disasters such as typhoons^2,3, wildfires^4,5, extreme heat and cold⁶, earthquake⁷, and cyberattacks^8,9. These disasters have the potential to cause significant disruptions in electricity distribution, resulting in prolonged power outages and inconvenience for consumers. For instance, the digital era has brought significant advancements to the power sector, enabling efficient monitoring and control of distribution systems (DSs). However, this increased connectivity has also exposed power grids to the risk of cyberattacks. In recent decades, cyberattacks have targeted the DNs, leading to disruptions in electricity supply and potential damage to critical infrastructure. These attacks emphasize the need for robust strategies to protect power grids against cyber threats¹⁰. Therefore, there is a growing demand to enhance the resilience of DNs, ensuring a continuous and reliable supply of electricity to end-users.

The increasing vulnerability of modern power grids and charging infrastructure to cyberattacks poses a significant threat. This vulnerability results from the widespread use of digital technologies and interconnected systems¹¹. Malicious actors can exploit weak points in the DN, such as smart meters, substation controllers, and power lines, to infiltrate control systems and disrupt grid operations, including EV charging¹². The consequences of these cyberattacks can be severe, leading to equipment damage, voltage fluctuations, and widespread blackouts that impact critical infrastructure, essential services, and EV charging networks¹³. Real-world incidents, like the 2016 attack on Ukraine’s electric utility, highlight the tangible impact of these threats¹⁴. As the frequency and complexity of cyberattacks continue to rise, power utilities, EV charging providers, and policymakers must develop effective strategies to prevent, detect, and respond to these threats.

Deploying backup sources within DNs is a practical approach to enhance their resilience¹⁵. These backup sources, including distributed generators and energy storage systems, can provide additional power supply during outages, minimizing the impact of disaster events^16,17. The use of mobile power sources (MPSs) has emerged as a promising solution to enhance the resilience of distribution systems (DS) against extreme weather events¹⁸. MPSs encompass EV fleets, truck-mounted mobile energy storage systems, and emergency generators. Their flexibility and mobility make them valuable in mitigating the impacts of such events on critical loads.

The development of transportation electrification offers an opportunity to enhance the resilience of DNs through the use of ETs and Vehicle-to-Grid (V2G) technology^19–21. Projections indicate that by 2030, EVs will account for 7.7% of China’s electricity consumption, and V2G technology will play a key role in optimizing load curves and improving grid efficiency²². Implementing V2G can reduce power system costs by approximately 2% and decrease carbon emissions by 2-3%²³. Intelligent charging management can yield over 20% savings in power purchase costs for operators and users²⁴. Moreover, by 2030, EVs could offer flexible regulation capacity equivalent to more than 10% of China’s annual renewable electricity generation²⁴. However, the dispatch of ETs within DSs needs to be adequately investigated.

The growing number of ETs in urban transit, driven by the increasing adoption of EVs and the demand for sustainable transportation, creates an opportunity to utilize these taxis for the recovery of DNs after events^25,26. ETs, equipped with bidirectional charging capabilities, not only offer on-demand mobility services but also serve as distributed energy storage resources that can support the stability and resilience of the local power grid during emergencies. By dispatching ETs to charge stations according to a strategic plan, power can be fed back to the grid during disaster events, supplementing existing backup sources and facilitating the restoration of power supply²⁷. For instance, a study on 19,900 ETs in Shenzhen demonstrated their ability to supply a minimum of 50 MW for 1 hour during peak periods without disrupting regular operations²⁶.

Reconfiguring a DN by incorporating backup power sources, particularly through V2G systems using electric taxis, is essential for enhancing resilience. These ETs can serve as mobile backup power sources, feeding energy back into the grid during outages or disruptions, thereby ensuring a continuous electricity supply when primary sources are compromised. To optimize the utilization of ETs for the recovery of DNs after events, this study proposes an innovative approach called the electric DARP. Figure 1 displays an integrated system of taxicab dispatching system and the power grid facing cyber attack threats. This approach incorporates routing, charging, and discharging plans for ETs. It aims to enhance the coordination between taxicab dispatching and power grid security, specifically in cyberattack disasters. This integrated approach encompasses four key features. Firstly, it focuses on efficient DARP for ET fleets, ensuring reliable and on-demand transportation services for customers. Secondly, it integrates the V2G capabilities of ETs, enabling them to contribute to grid resilience by feeding power back to the grid when needed. Thirdly, it formulates a network interdiction problem to strategically disrupt the propagation of cyberattacks across power grids by identifying and removing critical network elements. Lastly, it proactively reconfigures the DNs into a less impacted or stressed state. To establish a robust foundation that can withstand the challenges posed by extreme events, the study applies an optimal power flow problem to manage power dispatch between generators and electric buses effectively. This comprehensive approach aims to enhance the resilience and efficiency of DNs during post-event recovery.

Fig. 1.

Integrated electric taxi dispatching system and power grid network under cyber attack threats.

This study proposes a novel approach to tackle the challenges associated with the DARP for ETs, with a particular focus on power flow and minimizing the impact of disruptions, such as cyberattacks, on the power network. The proposed strategy is visually represented in Fig. 2. The DARP enables customers to request transportation services from their current locations to their desired destinations²⁸. ETs are dynamically routed to effectively fulfill these requests, considering various factors, including passenger preferences for travel costs and pickup/drop-off locations, as well as driver considerations such as battery levels and charging requirements (red box). We calculate the energy consumption of electric taxis while servicing customer requests, ensuring efficient travel between specified pickup and drop-off points.

Fig. 2.

The proposed electric DARP for mitigating cyberattacks on power grids.

A key component of the proposed model is its consideration of charging and discharging regulations for electric taxis, which is crucial for balancing power supply and demand (green box). This aspect ensures that the energy utilized by the ETs aligns with the available power grid capacity, leading to a more sustainable and reliable on-demand service. To enhance the stability of the DNs, the model leverages the V2G capabilities of ETs. These taxis can strategically discharge their batteries and feed electricity back into the grid during periods of high demand or grid instability, providing crucial support and stabilization. The DARP manages this bidirectional energy flow, ensuring that the taxis maintain sufficient charge to continue serving customers while contributing to the power grid’s resilience. Therefore, the proposed DARP considers factors such as the locations of charging and discharging stations, ET battery capacities, anticipated electricity demand during disasters, and prioritization of passenger requests based on urgency and vulnerability.

In a radial topology, electricity flows from a single source through a series of branches to various loads. Reconfiguring a DN by establishing alternative transmission lines creates redundant pathways for electricity flow, allowing the network to reroute power when specific areas are under threat or experiencing failures. This study also introduces a strategic solution by utilizing a formulated network interdiction problem (blue box). Network interdiction involves disrupting the spread of cyberattacks across power grid infrastructure by identifying and removing or disabling critical network connections or components. The goal is to prevent the cascading effect of an attack by strategically isolating the power network lines and substations that the attacker relies on for propagation. To achieve this, the optimal power flow problem (OPF) is applied (yellow box). OPF determines the optimal generation dispatch and power flow across the grid to meet demand while minimizing load-shedding costs. Using OPF, compromised grid components can be quickly identified and isolated, power flows can be rerouted, and grid stability and resilience can be maintained in the face of a cyberattack. Furthermore, OPF optimization can be employed to manage the charging and discharging of ET batteries, enabling them to provide grid services such as voltage support and frequency regulation. This can effectively stabilize the grid and counteract the impacts of a cyberattack. In summary, this study’s contributions can be summarized as follows:

• We propose a DARP for dispatching ETs that considers the limited battery capacity by directing drivers to stations for charging and discharging while serving customers. We integrate ETs and V2G technology into a DARP, offering a promising pathway to strengthen their resilience and crisis response capabilities.

• We conduct a comprehensive analysis and optimization of power flows in a DN, considering various disaster scenarios and their impact on grid stability and robustness.

• We implement a proactive network interdiction strategy to mitigate cyberattacks on the DN. This involves strategically pre-positioning ETs and reconfiguring the network by adding backup power sources and alternative transmission lines to prevent the spread of infections.

Results

The study analyzed and optimized power flows in a DN, considering cyberattack scenarios and their impact on grid stability and robustness. This analysis proved valuable for incorporating ET fleets into the power grid. By leveraging this analysis, taxicab dispatching systems were able to identify optimal locations for charging infrastructure to minimize grid impacts and quantify the precise amounts of power the taxis could supply back to the grid during peak demand periods. The analysis also optimized the dispatch and scheduling of the vehicles to support overall grid operations and evaluated any necessary infrastructure upgrades required to enable the integration. The comprehensive power injection details provided in the analysis guided the strategic placement of charging infrastructure and the efficient dispatch of ETs. The model helped quantify the valuable grid services that ETs could provide, strengthening the business case for their large-scale adoption and seamless V2G participation.

Furthermore, we implemented a proactive network interdiction strategy to mitigate cyberattacks on the DN, instilling confidence in the effectiveness of their cybersecurity measures. This multifaceted approach sought to develop reliable and resilient electric mobility systems that balanced the needs of transportation users and power grid operators. As the threat of cyber threats continues to grow, power utilities must take a proactive approach to strengthening their cybersecurity measures. Implementing this strategy can instill confidence in the effectiveness of the utility’s cybersecurity efforts, assuring customers and stakeholders that the power grid is well-protected against malicious actors.

The proposed model was tested using NYC taxicab data as a case study. This comprehensive benchmarking analysis examined the optimization of ET routing and power distribution system operations. The results demonstrated the importance of a holistic approach considering various factors, including routing costs, cybersecurity implications, and power system constraints. The proposed hybrid planning approach represented in Case 5 was the most effective solution, achieving the lowest total tour length, minimizing power load shedding, and reducing the number of system nodes vulnerable to cyberattacks compared to the other cases examined. Additionally, the results illustrated the total costs associated with varying numbers of ETs, combining both user and system costs. As the number of taxis grew from 2 to 8, total costs consistently dropped, indicating economies of scale and improved operational efficiencies with a more extensive fleet. The analysis also explored cyberattack scenarios across various fleet sizes, showing that as the fleet size increased, the number of removed lines and susceptible nodes decreased, indicating improved network resilience and enhanced cybersecurity measures.

Literature review

This section emphasizes the importance of integrating charging and discharging plans into taxicab routing and dispatching systems to enhance the resilience of DSs. In recent years, the taxicab dispatching and routing field has gained significant attention due to the growing demands of urban transportation. Managing a fleet of taxis to meet customer demands efficiently is a vital focus of this research. ETs require frequent charging to maintain an adequate range, significantly affecting their availability and routing. Several approaches, including mathematical models and dynamic routing algorithms, have been proposed to address this challenge^29,30. These approaches aim to optimize taxi assignment while considering the ETs’ charging constraints.

Routing and dispatching strategies play a crucial role in the efficient operation of EVs. Previous studies have focused on optimizing these strategies to improve various aspects such as fuel efficiency, travel time, and customer satisfaction. For example,³¹ presented a dynamic inventory routing and pricing problem involving a mixed fleet of electric and conventional vehicles to maximize social welfare.³² introduced an approach to optimize the battery swapping infrastructure and scheduling for an urban taxi system, addressing peak demand, service quality, and energy efficiency. ³³proposed a dispatching strategy considering taxi demand, remaining power, and charging station availability to reduce charging wait times and increase client opportunities. In this study, we emphasize the potential of optimized routing and dispatch to improve ET efficiency, profitability, and sustainability. Furthermore, by adjusting to the energy needs and real-time availability of charging stations, taxis can effectively meet customer demands. ³⁴emphasized the importance of routing for electric drones in vaccine delivery, addressing logistical challenges such as flight range, payload capacity, and recharging plans to ensure efficient and equitable distribution of vaccines in underserved regions.

V2G technology allows EVs to consume power from the grid and return excess power back to the grid. This bidirectional flow of electricity enables EVs to act as distributed energy resources, contributing to the stability and resilience of DNs²⁰. Several studies have explored the potential benefits of V2G technology in mitigating the impact of disasters. For instance,³⁵ demonstrated that V2G enhances the reliability of DSs by utilizing EV energy during emergencies.³⁶ investigated the use of EVs to improve resilience in active DNs, particularly during power outages caused by natural disasters. ³⁷discussed the advantages and challenges of V2G implementation, including its potential for ancillary services and the associated battery degradation issues. In this study, we highlight the considerable potential of V2G technology to improve grid stability, reliability, and resilience. Additionally, we explore how optimizing routing and dispatch systems can facilitate smooth trips without anxiety.

Various models explore innovative strategies for enhancing the resilience of DSs using mobile power sources and EVs. For example, a study by³⁸ investigated the dispatch of mobile power sources in coordination with dynamic network reconfiguration, focusing on seismic disasters. ³⁹considered a framework for strategically placing electric vehicle charging stations (EVCSs) in distribution systems, optimizing their locations and scheduling to enhance efficiency and reduce operational costs. Another study by⁴⁰ proposed a two-stage framework for routing and scheduling mobile power sources to improve system survivability and recovery. In a comprehensive review by⁴¹, resilience strategies were examined, highlighting the importance of preparation and resource allocation. Additionally⁴², presented a two-stage EV coordination framework that enhances network resilience and includes proactive pre-positioning and spatial-temporal dispatching. Our study demonstrates the potential of mobile power sources, especially ETs, to enhance the resilience of power systems in the face of extreme events.

The growing dependence on technology in power distribution systems has heightened their vulnerability to cyberattacks, threatening their stability and resilience. To counter these risks, innovative frameworks are being developed to reduce the impact of such attacks. For example, the framework proposed in⁴³ focused on the optimal deployment of static volt-ampere reactive (VAR) compensators to mitigate voltage violations resulting from false data injection (FDI) cyberattacks in smart distribution grids, covering both planning and operational phases. Another study by⁴⁴ considered a voltage-based relay scheme to enhance protection systems against cyberattacks on adaptive overcurrent relays. ⁴⁵investigated the integration of cyber technologies into traditional energy systems, showing potential for improved energy performance and cost-efficiency in smart grids. Additionally,⁴⁶ explored a graph-based model for interdependent cyber-physical risk analysis to assess device criticality and the resilience of networks to cyber threats. Addressing vulnerabilities stemming from the integration of physical and cyber systems⁴⁷, proposed a distributed compressive sensing (CS) state estimation method for unobservable grids. ⁴⁸developed models aimed at minimizing the spread of infections in general networks. ⁴⁹provided a comprehensive review of cyberattacks on power systems, emphasizing impact analysis, detection techniques, and security measures.

Recent studies have increasingly focused on the vulnerabilities of power grids in the context of cyberattacks, particularly as they relate to emerging technologies and distributed energy resources. For instance⁵⁰, examined the impact of cyberattacks on power grids with distributed energy storage, suggesting that widespread storage could increase vulnerabilities if not managed properly. ⁵¹focused on enhancing the security and efficiency of interconnected power grids and EV charging stations specifically for private vehicles, not for shared vehicles like taxis. ⁵²explored resilient operation that incorporates renewable energy sources and battery storage systems in the face of physical and cyberattacks. ⁵³proposed advanced distributed control algorithms to manage distributed energy resources efficiently, along with anomaly detection and mitigation techniques to minimize the impact of cyberattacks. Moreover, mobile energy resources have the potential to enhance power grid stability and resilience. ⁹examined the capacity of V2G to maintain the stability of the grid and facilitate the mobility of electric vehicles during disasters. A review by⁵⁴ focused on mobile energy storage systems, including truck-mounted and towable batteries, for supporting critical loads during outages. In this study, we introduce regulations for charging and discharging, along with considerations for routing, passenger load, and energy consumption estimation.

Discussion

The study introduced a DARP for dispatching ETs, considering the limited battery capacity during customer service and taxi drivers’ detours to stations for charging and discharging purposes. The model can help taxi companies better manage their fleets, ensuring efficient utilization of the vehicles and minimizing downtime due to battery depletion. This can lead to cost savings, improved service reliability, and a more sustainable transportation solution. Furthermore, we integrated ETs and V2G technology into a DARP, offering a promising pathway to strengthen their resilience and crisis response capabilities. Power utilities can prioritize investments, implement targeted reinforcement measures, and develop more effective emergency response plans by identifying the potential impacts of different events on grid stability and robustness. Furthermore, the proposed model can be utilized between the power grid and electric vehicle aggregators to determine the optimal EV fleet size and balance power supply with demand using charging and discharging regulations. EV aggregators act as a link between the grid and EVs, facilitating a more flexible and efficient energy system. This can ultimately lead to a more reliable and secure power supply, benefiting residential and commercial consumers.

Integrating taxicabs with V2G systems presents an opportunity to create a more efficient and resilient DN. This dynamic interaction between taxicabs and the DN significantly contributes to its stability and resilience. Recent research has explored the potential of ETs to enhance mobility and power resilience during disasters⁵⁵. V2G technology has been introduced to allow ETs to function as mobile energy storage, participating in load restoration alongside traditional emergency power vehicles²⁷. For instance,⁵⁶ studied the integration of energy services from shared plug-in electric vehicles into a mobility-on-demand platform. Our study aims to integrate the power grid and transportation systems by aligning the incentives of electric taxi drivers, power system operators, and taxi dispatching systems. This approach seeks to balance the competing objectives of meeting essential transportation needs while facilitating energy transfer across the network.

Future research could explore the interplay between various disaster scenarios that simultaneously impact the DN and the transportation network. This could involve combining natural disasters (e.g., hurricanes, earthquakes, wildfires) with cyber-physical attacks to understand the cascading effects and develop more robust mitigation strategies⁵¹. Digital twins combined with blockchain technology and machine learning would allow power utilities to develop real-time virtual replicas of their grid infrastructure. This innovation would enable precise simulations of cyberattacks and power outages. By continuously monitoring for anomalies, these simulations would improve the ability to quickly detect and respond to potential threats^57–60. Additionally, the potential role of shared autonomous vehicles (AVs) in enhancing post-disaster resilience could be investigated, particularly in terms of coordinating charging/discharging plans and optimizing routing to support power grid restoration⁶¹.

The integration of multimodal transit systems, including ETs, buses, bikes⁶² and other modes, and their collective response to disasters also merits further study. The disaster recovery framework could be enhanced by incorporating renewable energy sources such as wind turbines (WT) and photovoltaic (PV) systems, improving overall sustainability and self-sufficiency^63–68. Advanced solution algorithms could be developed to tackle the complex optimization problems inherent in these integrated transportation-power network recovery strategies, particularly when dealing with medium and large-scale problems^69–72. Moreover, incorporating an electricity pricing policy and battery degradation costs into the proposed model for bidirectional energy flow between electric vehicles and the grid could provide valuable insights for future research⁷³. Finally, Future studies should investigate V2G strategies in parking lots to optimize energy management and support sustainable urban mobility^67,74.

Methods

The proposed framework

When passengers submit their pickup and drop-off locations to the dispatch center, the center must determine the most optimal routes for the taxis to follow to pick up and drop off these passengers. This routing problem considers not only the passengers’ locations but also the locations of charging stations and discharging nodes. Taxi drivers need to visit charging stations periodically to recharge their batteries. Additionally, they may need to visit discharging nodes to stabilize the power network during disasters by supplying power back to the grid (V2G). The dispatch center must consider all these requirements when planning the taxi routes.

The transportation network is formally defined using a subgraph that includes several key elements. These elements consist of the locations of idle ETs (), a dummy node representing the end of a taxi’s path (), the set of passenger pickup points (), and the set of passenger drop-off points (). The taxis must serve n passengers, each with specific pickup and drop-off locations. The dispatch center’s challenge is determining the most optimal routes for the taxis that meet the passengers’ needs while also considering the operational constraints of the ET system. These constraints include managing battery levels through strategic charging and discharging stops to support the power grid during disasters. It is assumed that all nodes are capable of both charging and discharging ETs. These locations, called charging and discharging nodes, serve as physical sites where EVs can be charged and discharged.

The problem considers the interdependencies between the transportation and power networks. The power network includes electric buses (), power lines (), and generators () that supply electricity to the ETs and other EVs. Furthermore, the growing automation of power grid control systems has raised security concerns, especially regarding vulnerabilities susceptible to cyberattacks. Numerous studies suggest algorithms designed to detect and analyze cyberattacks on network parameter data within electric power systems, with an emphasis on optimal power flow issues^57,75–78. The proposed model introduces the concept of infection nodes () and susceptible nodes () within the power network. These nodes can be vulnerable to cyberattacks, potentially disrupting the power supply. In this study, we assume that the attacker targets specific nodes to disrupt electricity flow and cause power outages, and we are informed about which nodes have been infected. For a comprehensive list of notations used in the model, readers refer to Table 1.

Table 1.

Nomenclature.

Notations	Description
N	Set of nodes (pickup, drop-off, start node, and end node)
i, j	Nodes
	Pickup nodes,
	Drop-off nodes,
v	Set of ETs,
L	Power transmission lines
B	Set of power buses,
G	Buses with thermal generators,
I	Infected nodes
S	Susceptible nodes

Parameter	Description
s	Starting node of the path
e	Dummy node for path termination
	Energy consumption between nodes i and j
M	Large positive constant
	Lower and upper bounds of the time window at node i
	Service time at node i
	Travel times
	Distances
	Battery capacity
K	Passenger capacity of the taxi
	Initial number of passengers in the taxi
	Initial charge
	Minimum charge at the end of the path
	Load shedding cost at bus station j
	Cost parameter for susceptible nodes
	Resistance between buses i and j
	Reactance between buses i and j
	Maximum allowed number of power line removals
	Set of distinct node pairs with an edge
	Set of neighbors of node i
	Total reactive power consumed at bus j
	Power demand of other consumers
	Power factor angle
	Voltage limits at bus j
	Active power limits at generator bus j
	Minimum and maximum reactive power limits at generator bus j
	Maximum power demand at bus j
	Minimum active power flow between buses i and j
	Maximum active power flow between buses i and j
	Minimum reactive power flow between buses i and j
	Maximum reactive power flow between buses i and j

Decision Variables	Description
	Binary variable indicating whether the power line between nodes i and j is connected (1) or not (0)
	Binary variable indicating whether the power line between nodes i and j is removed (1) or not (0)
	Binary variable indicating whether susceptible node i is at risk of infection (1) or not (0)
	If the taxi v is charged or discharged at node i, the value is 1; otherwise, it is 0
	Binary variable indicating whether taxi v travels from node i to node j (1) or not (0)
	Battery level of taxi v at node i
	Number of passengers in the taxi v when it leaves node i
	Time at which taxi v starts servicing node i
	Amount of electricity discharged at node i for taxi v
	Amount of charged at node i for taxi v
	Charge adjustment at node i for taxi v
	Voltage magnitudes at buses i and j
	Total active power output of the generator at bus j
	Total reactive power output of the generator at bus j
	Total active power demand at bus j
	Total power demand for charging ET at bus station j
	Active power load shedding at bus j
	Reactive power load shedding at bus j
	Active power flow between buses i and j
	Reactive power flow between buses i and j

Energy consumption

This section introduces a function that calculates the expected energy consumption () based on⁷⁹. The function incorporates different parameters associated with a vehicle’s power system to determine the energy consumption. It considers three prominent cases based on the signs of the traction power () and battery power (). When both and are negative, indicating energy generation or recuperation, the function calculates the energy consumption using the traction power, gear ratio (), generator efficiency (), output power (), motor power (), a normalization factor (), auxiliary power consumption (), time step ( is 1/60 (minute)), and round-trip efficiency (RTE). This case represents situations where the vehicle generates or recovers energy, such as during braking. When is negative but is positive, the function adjusts the calculation to account for the battery’s energy consumption. In this case, the energy consumption is calculated using the same parameters as the first case, but the term involving the square root of RTE is replaced with . This reflects that the energy is consumed from the battery rather than being generated or recovered. For more comprehensive details about this function, readers are encouraged to refer to⁷⁹.

1

The optimal tours are determined using the proposed DARP described in Section 3.2. This approach employs the energy consumption values between each pair of nodes to calculate the most efficient routes.

The proposed DARP for ETs with charging and discharging

This section explores an electric DARP where vehicles can charge and discharge their batteries at charging stations. The model establishes a connection between the dispatch center and the power utilities, enabling taxis to generate power and contribute it back to the network (V2G) during emergencies, such as cyberattacks, while ensuring that taxis can complete their trips. The optimization process considers various factors, including pickup and delivery locations and the number of passengers. The ETs have limited battery capacity and can charge or discharge at stations.

The first term of objective function (2) represents the costs associated with reducing electrical load at different locations. The variable represents the cost coefficient, and indicates the amount of load shed at each location j. The second term considers the expenses related to the connections between infected and susceptible nodes. Parameters and likely influence the connectivity-based costs. The last component of the objective function represents the costs associated with ET routing and dispatching. In this term, likely represents the distance or travel cost between locations i and j, and is a decision variable that determines the assignment of taxi v to the route between locations i and j.

2

Constraints (3) through (7) maintain flow equivalence in the network flow problem under investigation. These constraints establish that the flow through the network is evenly distributed, ensuring that all vehicles visit each node precisely once. Additionally, they guarantee that each vehicle commences and concludes its tour at the designated source and sink nodes while maintaining flow conservation at intermediate nodes. Constraint (3) requires that all vehicles visit each node exactly once. Constraint (4) ensures that each node is departed from exactly once by all vehicles. Constraint (5) maintains flow conservation at each node, excluding the source and sink nodes.

3

4

5

Constraint (6) guarantees that each vehicle starts its tour from the source node. Constraint (7) considers that each vehicle ends its tour at the sink node. Constraint (8) ensures that each edge in the network is visited exactly once by all the vehicles combined.

6

7

8

The model includes constraints (9) and (10) to manage the vehicle’s battery charge level as it travels through the network. Constraint (9) calculates the charge level at node j, denoted as , is determined by summing the charge levels at the previous nodes i (), adding the amount of recharge of the vehicle at node i (), subtracting the charge depleted () the energy consumed along the edge from i to j (), over all the incoming edges .

9

Constraint (10) imposes an upper bound on the amount of recharge allowed for each vehicle v at each node i. The recharge amount is limited by the difference between the maximum battery capacity and the current charge level of the vehicle, plus the charge depleted due to traveling along the edges. This ensures that the vehicle’s battery charge level after recharging does not exceed the maximum capacity, helping to manage the battery charge of the vehicles as they travel through the network.

10

Constraint (11) aims to determine the charging level achieved at each node by calculating the difference between the maximum possible charge and any charge adjustments or discharge that may occur. To prevent overcharging beyond the battery’s available capacity, a parameter is introduced to regulate the replenishment amount at each location. This is implemented through the equation , where in Constraint (11) is replaced by in Constraint (10). The optimal model would use the inequality , with representing a recharging efficiency between 0-1 dependent on each stop’s available battery space. The coefficient governs the replenishment at each site, as described in⁵¹. However, this approach results in a non-linear equation. Therefore, an adjustable factor is introduced to linearize the model while approximating the desired behavior.

11

The constraint (12) guarantees that the vehicle can only recharge at a node if it is visiting that node as part of its route.

12

Constraint (13) ensures that the charge level at node i, denoted as , must be greater than or equal to the total energy consumption, as represented by the sum of over all the outgoing edges j to nodes i.

13

The problem also involves restricting the initial and final charge levels of the vehicle’s battery to predefined values. Constraint (14) specifies that the initial charge level at the source node s, denoted as , must be set to a fixed value for all vehicles v. This ensures the vehicles start their journeys with a known, consistent battery charge level. Constraint (15) stipulates that the final charge level at the destination node e, denoted as , must be greater than or equal to a minimum value for all vehicles v. This constraint ensures that the vehicles arrive at their destinations with at least a certain level of battery charge, preventing them from being depleted to an unacceptable level.

14

15

In addition to the constraints related to the vehicle’s battery management, the problem also includes a set of constraints (16)–(19) that calculate and limit the number of passengers in the vehicle at each node along the route. Constraint (16) specifies that the number of passengers in the vehicle at the starting node s, denoted as , must be set to a predetermined initial value for all vehicles v. This ensures that the vehicles depart with a known, consistent number of passengers.

16

Constraint (17) calculates the number of passengers in the vehicle at each intermediate node j that is a drop-off node (), denoted as .

17

Constraint (18) calculates the number of passengers in the vehicle at each intermediate node j that is a pickup node (). In this case, the number of passengers is incremented by one to account for the passenger being picked up, again multiplied by the binary decision variable .

18

Finally, Constraint (19) limits the maximum number of passengers K that can be transported in the vehicle at any node i and for any vehicle v. This ensures that the vehicle’s capacity is not exceeded during the dispatching service.

19

Constraint (20) specifies the beginning of service time at each node j, denoted as , for each vehicle v. This constraint ensures that the service time at node j is greater than or equal to the sum of the service time at the previous node i (), the service time required at the previous node (), and the travel time from node i to node j (), multiplied by the binary decision variable that indicates whether the vehicle travels from node i to node j. Constraint (21) then imposes time windows for the service at each node i.

20

21

Constraints (22), (23), and (24), are applied to incorporate the origin-destination precedence of the nodes into the model. This is necessary because the original model, which includes a dummy node to remove the necessity of returning to the initial node, does not consider the order in which the nodes are visited. Constraint (22) sets the order variable for the dummy node s to 0. This is because the dummy node is the last node that must be visited, and we want to establish it as the reference point for the order of visits. Constraint (23) defines the order variable for each non-dummy node j and each vehicle v. This ensures that the order of the nodes increases sequentially as the vehicles visit them. Constraint (24) ensures that the order of the nodes is maintained across the origin-destination pairs (i, j) of trips in the set PD.

22

23

24

The constraints (25), (26), and (27) is used to linearize the nonlinear model by introducing a new variable , which represents the product of the binary decision variable and a continuous variable . Constraint (25) ensures that if is 0, then must also be 0. This is achieved by setting an upper bound on as , where M is a large positive constant. Constraint (26) sets the upper bound of to be , the continuous variable. For example, Constraint (27) is the most crucial for the linearization. It ensures that when is 1, is equal to . This is achieved by setting a lower bound for as . By introducing these three constraints, the nonlinear product term is effectively linearized, as the new variable is forced to take the appropriate value based on the values of and .

25

26

27

The variable represents a continuous value, and its trivial upper bound can be denoted as . This means that the value of is always less than or equal to . Similarly, the variable is when vehicle v starts servicing node i, and its trivial upper bound can be depicted as maximum . The variable is an integer variable, and its trivial upper bound can be represented as K. This implies that the value of is always less than or equal to K. Lastly, the variable represents the order of the visited nodes, and its trivial upper bound is the total number of nodes, |N|. This means that the value of is always less than or equal to |N|.

Cyber-attacks and power flow balance

Integrating the power and transportation networks is essential to accurately estimate the charging demands at the charging stations, reflecting the total power load in the overall network. This study utilizes a commonly used linearized distribution flow model to analyze power flows in electrical grids⁸⁰. The amount of electricity recharged for each vehicle v at each charging station i that is connected to the bus station j, representing the total power load at the bus station j as shown in Equation (28):

28

where is the total power load at bus station j and is the amount of electricity recharged for vehicle v at the charging node i.

The constraint represented by Equation (29) specifies that the total power demand at a bus node j is the sum of two components: the power demand for EV charging and the power demand from other consumers . The power demand from other consumers encompasses a variety of loads, including residential, commercial, and industrial demands. This can be expressed mathematically as:

29

Constraint (30) calculates the total power generated by discharging vehicles within a specific bus node j. The equation can be expressed as:

30

where represents the total power output at bus node j due to the discharging of vehicles and is the amount of power discharged by vehicle v at node i.

Constraint 31 represents the active power balance at each bus j in the power system. It ensures that the total active power injected into bus j (generated power, minus demand, plus load shedding and discharged power by ETs) is equal to the net active power flow out of the bus (the summation of active power flows on the lines connected to bus j).

31

Constraint (32) ensures that the reactive power supplied by generators () minus the reactive power demand () plus the reactive power from load-shedding () is equal to the net reactive power flowing into and out of bus j.

32

Constraint (33) calculates the reactive power demand () at bus j based on the active power demand () and the power factor () at that bus.

33

Constraint (34) determines the voltage difference between buses i and j based on the resistance (), reactance (), active power flow (), and reactive power flow () between the two buses.

34

Maintaining voltages within acceptable ranges is crucial for adequately operating electrical equipment and preventing damage to the grid infrastructure. Constraint (35) ensures that the voltage magnitude at each bus j is within the specified minimum () and maximum () limits.

35

Constraint (36) and (37) helps to maintain the grid’s active and reactive power balance, respectively.

36

37

Constraint (38) examines that the active power load shedding cannot exceed the maximum active power demand, as it would be physically impossible to shed more load than what is demanded. Similarly, Constraint (39) confirms that reactive power load shedding cannot exceed the maximum reactive power demand, as it would be physically impossible to shed more reactive power than what the loads require.

38

39

Constraint (40) determines the consistency between the binary variables and for each branch (i.e., edge) in the set L⁴⁸. The variable is a binary indicator that denotes whether node i is connected to node j (with a value of 1 if connected and 0 otherwise). The variable represents the removal of the power line (i, j) (with a value of 1 if the line is removed and 0 otherwise). This constraint guarantees that for each branch in the system, at least one of the two binary variables, or , must be greater than or equal to 1. This means that if a branch is present in the system (i.e., ), the corresponding line cannot be removed (). Conversely, if the line is removed (i.e., ), then the branch cannot be present (). This constraint verifies the logical consistency between the connection and removal of lines in the power system network, preventing contradictory or infeasible scenarios.

40

Constraint (41) presents the logical consistency between the binary connection and line removal variables in the power grid network. It states that for each pair of connected nodes (k, i) and their common neighbor j, where j is not a critical infrastructure node, either both nodes k and j must be connected to i ( and ), or the power line between j and i must be removed (). This constraint prevents the creation of infeasible closed loops in the network when a node is disconnected from one of its neighbors, maintaining the overall connectivity and topology of the power grid while considering the potential removal of lines to enhance the system’s resilience against cyber attacks.

41

Constraint (42) limits the number of lines removals that can be made in the power grid network to enhance its resilience against cyber attacks. The parameter represents the maximum budget or number of power lines that can be removed from the system. This constraint guarantees that the sum of the binary variables , which indicate the removal of the power line between nodes i and j for all branches in the set L, does not exceed the specified limit .

42

Constraint (43) establishes a relationship between the binary variables and to ensure that susceptible nodes are correctly connected to infected nodes in the power grid network. The parameter is a binary variable that indicates whether a susceptible node i (i.e., ) is at risk of infection. At the same time, represents the binary connection variable between nodes i and j, where j is an infected node (i.e., ). This constraint stipulates that for any susceptible node i, if is equal to 1, indicating that the node is at risk of infection, then there must exist at least one pairwise connection between node i and an infected node j, where . This secures that the model correctly identifies the susceptible nodes that are at risk of infection based on their connectivity to the infected nodes in the power grid.

43

Constraints (44) and (45) regulate the active power and reactive power flow on the transmission lines of the power grid, respectively, considering the potential removal of lines due to cybersecurity measures. These constraints validates that the power flow on the remaining transmission lines in the interdicted network respects the system’s physical limits while accounting for the impact of line removals on the power flow. These limits are modulated by the binary variable , indicating whether the line between nodes i and j has been removed. If the line is not removed (), the active power flow is constrained within the normal operational limits. However, if the line is removed (), the active power flow is automatically set to zero, effectively disconnecting the line.

44

45

Constraints (46), (47), and (48) define the domains of the binary decision variables used in the optimization model for the power grid cybersecurity problem.

46

47

48

Stage-by-stage framework for the proposed DARP model

The model optimizes the DARP for ETs by integrating charging and discharging regulations to enhance power resilience (see Algorithm 1). It initializes key parameters such as battery characteristics, taxi dispatching strategies, and cyberattack risks. The algorithm defines variables for routing, charging, V2G regulations, and power flow, while establishing constraints for power balance and service times. After solving the optimization problem with CPLEX in Julia 1.7.1, it identifies decision variables such as optimal taxi routes and energy delivered to the grid, ultimately improving the efficiency of the transportation and power system. The calculations utilized a laptop with a 2.50 GHz processor and 12 GB of RAM.

Algorithm 1.

Proposed DARP with Charging and Discharging Regulations for Power Resilience.

Findings and analysis

This section illustrates examples to demonstrate the performance of our model. The central aspect under examination is how taxis can effectively manage the charging and discharging of passengers within a single tour or route during a cyberattack scenario that disrupts normal operations. We utilize the parameters listed in Table 2 to solve our model. Furthermore, the analysis explores a DN scenario using the widely recognized IEEE 39-bus system. This model represents a realistic and comprehensive power grid, with 10 generators and 39 interconnected busbars designed to study power flow, stability, and optimization strategies. The lower bound for the voltage magnitude is and the upper bound is .

Table 2.

Input parameters.

Parameter	Description	Value
g	Acceleration of gravity ()	9.8066
	Air density ()	1.25
A	Frontal area ()	2.19
	Drag coefficient	0.29
	Mass correction factor	0.05
	Rolling coefficient	0.008
u	Vehicle speed (km/h)	40
a	Acceleration ()	0
	Road slope angle from node i to node j	[2%]
	Vehicle mass + driver mass (kg)	1633
	Gear efficiency	0.97
	Power consumption of accessories (W)	300
	The rated power of motor (KW)	75
norm	Normalization factor of motor efficiency	0.987
RTE	Battery round trip efficiency	0.95
	The average mass of user (kg)	80
	Load shedding cost	$100081
	Cost for susceptible nodes	$100081
	power angle	0.99
	Maximum active power(kW)	1000
	Minimum active power(kW)	1000
	Maximum reactive power(kVAr)	1000
	Minimum reactive power(kVAr)	1000

Figure 3 visually represents the intricate interconnections between the generators and busbars in the IEEE-39 bus distribution system. Given the limited dataset of the power system, this study assumes that the power grid is integrated into the transportation network. The data of the IEEE 39-bus system is publicly available at the GitHub repository https://github.com/homase2003/HamidSayarshad.git. We also assume a cyberattack on the IEEE39 network for our numerical study. The set of infected nodes is represented by variable , and the set of susceptible nodes is represented by variable S. In this case, the set of infected nodes, , is defined as I = [10, 12, 15, 18, 25]. This means the nodes with indices 10, 12, 15, 18, and 25 are initially infected in the network. Conversely, the remaining power nodes are defined as susceptible nodes, denoted as . In this study, our primary objective is to develop a novel model that considers optimal power flow, routing, charging and discharging plans for ETs, along with cybersecurity conditions, and we present its performance. Future research may benefit from focusing on solution algorithms for medium and large-scale problems.

Fig. 3.

The single-line diagram of the IEEE-39 bus system.

Illustration example

This example includes a set of locations that need to be visited, with some locations designated as pickup points and others as drop-off points. The set consists of 12 nodes, represented by , each corresponding to a different location. Specifically, there are 5 pickup nodes [2, 3, 4, 5, 10] and 5 drop-off nodes [6, 7, 8, 11, 12]. The route commences at node 1 and concludes at node 9. This problem also includes information about the EVs used for serving customers. The charging capacity of the vehicle battery is (kW), and the initial charge on the battery is (kW). The lower bound on the charge at the end of the path is set to . The number of passengers at the start node is , and the maximum capacity of the taxi is . Finally, the number of ETs available is . The taxicab pickup/drop-off locations numbered 1 through 12 are directly connected to the corresponding electric buses in the power grid. The electric buses with the indices [2, 3, 5, 8, 9, 12, 15, 20, 24, 26, 27, 29] are explicitly associated with and positioned at taxicab locations 1 through 12, respectively. The proposed model’s runtime for the test case is approximately 5 s.

Table 3 displays the battery levels of two ETs, Taxi 1 and Taxi 2, alongside the number of passengers at each node. Figure 4 illustrates the regulations for charging and discharging taxis. For taxi 1, the route taken is 1 3 4 7 8 10 12, with charging stations at nodes 4, 8, and 10, where 19.69 (kW), 19.69 (kW), and 0.17 (kW) of electricity was recharged, respectively, and discharging station at nodes 3, 7, and 10, where 19.37 (kW), 19.36 (kW), and 19.83 (kW) of electricity were discharged, respectively. For taxi 2, the route is 1 2 5 6 11, with discharging station at node 2, where 19.70 (kW) of electricity were discharged, and charging stations at nodes 5, and 6, where 0.13 (kW), and 19.87 (kW) of electricity were charged, respectively.

Table 3.

Results of the vehicles’ movements and energy usage.

Nodes	(kW)
	Taxi 1	Taxi 2	Taxi 1	Taxi 2
1	20	20	0	0
2	0	19.91	0	1
3	19.72	0	1	0
4	0.31	0	2	0
5	0	0.15	0	2
6	0	0.13	0	1
7	19.69	0	1	0
8	0.04	0	0	0
9	0	0	0	0
10	19.96	0	1	0
11	0	19.87	0	0
12	0.25	0	0	0

Fig. 4.

Charging and discharging strategies for taxis across nodes.

Optimal power flow analysis allows the ET fleet to be incorporated into the detailed power system model. Table 4 displays the voltage and power values for a 39-bus system. This 39-bus system is closely associated with an optimal power flow analysis for an electric taxi V2G system. This enables analysis to identify the optimal locations for charging infrastructure to minimize grid impacts. It can also quantify the precise amounts of power the taxis can supply back to the grid during peak demand periods. In addition, the analysis can optimize vehicle dispatch and scheduling to support overall grid operations. The model can even evaluate any necessary infrastructure upgrades to enable integration. Finally, analysis can help monetize the grid services provided by ETs.

Table 4.

Voltage and power values for 39 bus system.

Bus Number
1	0	0	0	0	0	0.88	0	0
2	0	0	0	0	0	1.12	0	0
3	19.70	0	64.40	0	0	0.89	9.18	0
4	0	0	100.00	0	0	0.89	14.25	0
5	19.37	0	0	0	0	1.03	0	0
6	0	0	0	0	0	1.03	0	0
7	0	0	46.76	0	0	1.01	6.66	0
8	0	19.69	124.09	0	0	0.89	17.68	0
9	0	0.13	0.13	0	0	1.00	0.02	0
10	0	0	0	0	0	1.02	0	0
11	0	0	0	0	0	0.88	0	0
12	0	19.87	21.37	0	0	0.97	3.05	0
13	0	0	0	0	0	1.12	0	0
14	0	0	0	0	0	0.90	0	0
15	19.36	0	64.00	0	0	0.90	9.12	0
16	0	0	65.80	0	0	0.98	9.38	0
17	0	0	0	0	0	0.93	0	0
18	0	0	31.60	0	0	0.88	4.50	0
19	0	0	0	0	0	1.02	0	0
20	0	19.96	155.96	0	0	0.94	22.22	0
21	0	0	54.80	0	0	1.03	7.81	0
22	0	0	0	0	0	1.12	0	0
23	0	0	49.50	0	0	1.12	0	0
24	0	0	61.72	0	0	1.02	8.79	0
25	0	0	44.80	0	0	1.12	6.38	0
26	19.83	0.17	27.97	0	0	0.93	3.99	0
27	0	0	56.20	0	0	0.91	8.01	0
28	0	0	41.20	0	0	0.88	5.87	0
29	0	0	56.70	0	0	0.92	8.08	0
30	0	0	0	211.49	0	1.12	0	0
31	0	0	1.84	172.30	2.19	1.12	0	0
32	0	0	0	0	2.71	1.12	0	0
33	0	0	0	155.23	3.99	1.12	0	0
34	0	0	0	0	5.07	1.12	0	0
35	0	0	0	270.76	8.39	0.88	0	0
36	0	0	0	367.35	11.16	0.88	0	0
37	0	0	0	0	0	1.12	0	0
38	0	0	0	34.25	4.69	1.12	0	0
39	0	0	220.80	0	47.66	1.12	0	0

The result of the details of the power injection is crucial to understanding the overall operation and behavior of the power system. Table 5 provides a comprehensive overview of the power flows within the system, with active and reactive power injections listed for 39 different pairs of buses. The sign of values shows the power flow direction: positive for injection, negative for extraction. The optimal power flow analysis can effectively guide the integration of ET fleets into the grid. For instance, the analysis can identify the optimal bus locations where the taxi fleet can provide the most beneficial active and reactive power support, enabling the strategic placement of charging infrastructure and the efficient dispatch of the vehicles. Furthermore, the power injection results can help quantify the valuable grid services that ETs can provide, bolstering the business case for their large-scale adoption and seamless V2G participation.

Table 5.

Power Injection between buses.

Bus Pair		Power Injection
From Bus	To Bus	Reactive Power ()	Active Power ()
1	2	11.64	170.95
1	39	11.64	170.95
2	3	3.25	50.85
2	30	0	211.49
2	25	8.39	10.31
3	4	9.20	152.37
3	18	12.45	146.22
4	5	33.31	447.59
4	14	24.11	395.21
5	6	77.22	1000.00
5	8	43.92	533.04
6	7	48.47	753.88
6	11	26.82	416.58
6	31	1.93	170.46
7	8	48.47	707.12
8	9	4.56	49.98
9	39	4.56	49.85
10	11	7.37	85.32
10	13	4.66	85.32
10	32	2.71	0
12	11	19.45	501.90
12	13	19.45	480.54
13	14	24.11	395.21
15	16	0	44.64
16	17	25.53	44.64
16	19	1.08	0.72
16	21	53.97	945.20
16	24	27.36	473.12
17	18	12.45	177.82
17	27	13.08	183.10
19	20	5.07	155.96
19	33	3.99	155.23
20	34	5.07	0
21	22	53.97	1000.00
22	23	45.58	729.24
22	35	8.39	270.76
23	24	27.36	411.40
23	36	11.16	367.35
25	26	8.39	55.11
26	27	13.08	126.90
26	28	90.18	1000.00
26	29	85.49	936.35
28	29	90.18	958.80
29	38	4.69	34.25

The model also assists power utilities in identifying and eliminating infected or susceptible lines within the DN. By analyzing the susceptibility and connection status of various elements in the power grid, as shown in Table 6, the model helps utilities pinpoint the vulnerable points that could potentially lead to the spread of infections or disruptions. The result shows susceptible nodes, with nodes 2, 3, and 17 at risk of infection (). Additionally, the model indicates that the power line between nodes 14 and 15 is the only one that can be removed (). Integrating the taxicab dispatching system with the V2G capabilities further enhances the model’s ability to maintain a reliable and secure power supply for the broader energy ecosystem. Therefore, the result allows the power utilities to take targeted action to remove or isolate these problematic lines, effectively strengthening the overall resilience and reliability of the power infrastructure.

Table 6.

Susceptibility and connection status in the power distribution system.

Susceptible Nodes
1	0	(1, 2)	0	1
2	1	(1, 39)	0	1
3	1	(2, 3)	0	1
4	0	(2, 25)	0	1
5	0	(2, 30)	0	1
6	0	(3, 4)	0	1
7	0	(3, 18)	0	1
8	0	(4, 5)	0	1
9	0	(4, 14)	0	1
11	0	(5, 6)	0	1
13	0	(5, 8)	0	1
14	0	(6, 7)	0	1
16	0	(6, 11)	0	1
17	1	(6, 31)	0	1
19	0	(7, 8)	0	1
20	0	(8, 9)	0	1
21	0	(9, 39)	0	1
22	0	(10, 11)	0	1
23	0	(10, 13)	0	1
24	0	(10, 32)	0	1
26	0	(12, 13)	0	1
27	0	(13, 14)	0	1
28	0	(14, 15)	1	0
29	0	(15, 16)	0	1
30	0	(16, 17)	0	1
31	0	(16, 19)	0	1
32	0	(16, 21)	0	1
33	0	(16, 24)	0	1
34	0	(17, 18)	0	1
35	0	(17, 27)	0	1
36	0	(19, 20)	0	1
37	0	(19, 33)	0	1
38	0	(20, 34)	0	1
39	0	(21, 22)	0	1
		(22, 23)	0	1
		(22, 35)	0	1
		(23, 24)	0	1
		(23, 36)	0	1
		(25, 26)	0	1
		(25, 37)	0	1
		(26, 27)	0	1
		(26, 28)	0	1
		(26, 29)	0	1
		(28, 29)	0	1
		(29, 38)	0	1

The model identifies and eliminates vulnerable lines in the DN while using V2G technology to stabilize the grid. Table 7 displays cybersecurity results for various scenarios based on the fuel capacity of the vehicle’s battery, represented by . It indicates that the power line between nodes (14, 15) can be removed at kW. With an increased charging range, more power lines can be removed without requiring power shutdowns. Additionally, nodes 2, 3, and 17 are susceptible to infection at kW, with only node 3 remaining vulnerable at kW or higher.

Table 7.

Cybersecurity results across different energy levels.

Lines
(2, 25)	0	1	1	1	1
(14, 15)	1	0	1	1	1
(17, 18)	0	1	1	1	1

Susceptible Nodes
2	1	0	0	0	0
3	1	1	1	1	1
14	0	1	0	0	0
17	1	0	0	0	0

New York Taxicab

We evaluate the proposed model using taxicab data from New York City. Our analysis determines the optimal charging, discharging, and routing plans for 4 taxis. These plans were generated by considering a maximum budget of for removing power branches. The pickup and drop-off locations for passengers are depicted in Figs. 5 and 6. We assumed that all taxis would begin their routes with an initial battery charge of 100 kW. The energy consumption for each vehicle on every route is calculated using the energy consumption function.

Fig. 5.

Start node and passenger pickups locations.

Fig. 6.

Passenger drop-offs locations.

We solve the case study using four taxis. We also utilized distances and travel times derived from the pickup and drop-off data in the NYC taxicab dataset. The taxicab pickup/drop-off spots are numbered from 1 to 30, each directly connected to an electric bus in the grid. The electric buses at indices 1 through 30 are linked to locations 1 through 30, respectively. For more information and specific parameters about our model, readers can refer to the publicly accessible data at the following URL: https://github.com/homase2003/HamidSayarshad.git. The proposed model’s runtime for the test case is approximately 300 s.

Model evaluation

This section investigates the effectiveness of our approach, which we have accomplished through five distinct case studies.

1. Case 1: This analysis focused on routing factors and the costs incurred by ET users. It specifically examined the third component of the objective function, as detailed in Equation (2).

2. Case 2: We delved into the cybersecurity aspects and implications in this case. Our evaluation centered around the second component of the objective function, as described in Equation (2).

3. Case 3: We explored factors related to the power distribution system, such as load-shedding costs. Our analysis primarily considered the first term of the objective function, as specified in Equation (2).

4. Case 4: This case combined the factors from the power distribution system (Case 3) and the cybersecurity considerations (Case 2).

5. Case 5: We conducted a comprehensive analysis by incorporating a combination of Cases 1 through 4 factors. This approach allowed us to account for the power distribution system conditions, routing, and cybersecurity considerations.

We consider two levels of charging and discharging strategies: coordinated and uncoordinated. In the coordinated strategy, such as Cases 3 and 4, the charging and discharging of the ETs are planned and scheduled to enhance the synchronization between the ET dispatching and the power grid security, aiming to improve the overall resilience of the system. In contrast, the uncoordinated approach, such as Case 1, does not explicitly consider the impact on the power grid, with ETs charging or discharging their batteries based on their energy requirements, which may lead to suboptimal utilization of the ETs and potential conflicts between the taxicab dispatching needs and the power grid’s requirements.

Evaluating these two charging and discharging regulations helps identify the most effective method for utilizing ETs in the post-event recovery of DNs. Table 8 illustrates the impact of power load shedding across five distinct cases. The total power shutdown for Case 1 and Case 2 is 648.981 kW and 877.274 kW, respectively, significantly higher than the zero load shedding observed in Cases 3, 4, and 5. Conversely, Cases 1 and 2 necessitate power load-shedding interventions. Furthermore, Table 9 is a benchmark for evaluating the performance of different cases involving vehicle tour lengths and routing strategies. Each case represents a unique scenario, and the total tour length is used to assess their effectiveness. The results show that Cases 1 and 5 have the lowest tour length of 349.801 min, suggesting that Case 5 present a more efficient strategy compared to the other cases.

Table 8.

Shutdown of power in different case studies.

Bus number	Case 1	Case 2	Case 3	Case 4	Case 5
1	0	0	0	0	0
2	0	0	0	0	0
3	0	0	0	0	0
4	100	100	0	0	0
5	0	0	0	0	0
6	1.36	0	0	0	0
7	46.98	46.76	0	0	0
8	104.40	104.40	0	0	0
9	0	0	0	0	0
10	0	0	0	0	0
11	0.48	0	0	0	0
12	1.5	0	0	0	0
13	0	0	0	0	0
14	0	0	0	0	0
15	64	64	0	0	0
16	66.21	65.80	0	0	0
17	0	0	0	0	0
18	31.6	31.6	0	0	0
19	0	0	0	0	0
20	0	136	0	0	0
21	4.64	54.8	0	0	0
22	0	0	0	0	0
23	0	0	0	0	0
24	61.78	0	0	0	0
25	44.88	0	0	0	0
26	27.8	27.8	0	0	0
27	50.32	56.2	0	0	0
28	41.2	0	0	0	0
29	0	0	0	0	0
30	0	0	0	0	0
31	1.84	0	0	0	0
32	0	0	0	0	0
33	0	0	0	0	0
34	0	0	0	0	0
35	0	0	0	0	0
36	0	0	0	0	0
37	0	0	0	0	0
38	0	0	0	0	0
39	0	189.91	0	0	0

Table 9.

Vehicle tour length and routing under different cases.

Taxi 1	Taxi 2	Taxi 3	Taxi 4	Total tour length
Case 1
1, 29, 30	1, 21, 20, 23, 24, 25, 26	1, 3, 2, 7, 6, 10, 12, 13, 16, 27, 28	1, 5, 4, 8, 11, 15, 14, 17, 19, 18, 22	349.801
Case 2
1, 4, 8, 13, 16	1, 5, 11, 20, 21, 23, 24, 27, 28	1, 3, 7, 10, 12, 14, 15, 18, 17, 25, 26, 29, 30	1, 2, 6, 19, 22	513.795
Case 3
1, 25, 26	1, 3, 7, 10, 12, 19, 21, 20, 22, 24, 23	1, 2, 5, 4, 6, 8, 11, 13, 16, 27, 28, 29, 30	1, 14, 15, 18, 17	429.413
Case 4
1, 3, 2, 5, 7, 6, 11, 27, 29, 28, 30	1, 10, 12, 13, 16	1, 4, 8, 14, 15, 18, 17, 21, 25, 24, 26	1, 20, 19, 23, 22	483.782
Case 5
1, 29, 30	1, 3, 2, 7, 6, 10, 12, 13, 16, 27, 28	1, 21, 20, 23, 24, 25, 26	1, 4, 5, 8, 11, 15, 14, 17, 19, 18, 22	349.801

The DN Configuration involves different approaches to setting up the power distribution network (DN) for post-event recovery, categorized into Baseline and Proactive levels. The baseline configuration (such as Case 1) represents the default setup without any pre-positioning of ETs or network reconfiguration. In contrast, the proactive configuration (such as Case 5) involves strategically pre-positioning the ETs and reconfiguring the network by adding backup power sources and alternative transmission lines. Table 10 shows the cybersecurity results under five different cases. In Cases 1 and 3, where cybersecurity was ignored, the analysis identified a larger number of nodes (4 nodes) as being susceptible to potential cyberattacks. This represents a concerning vulnerability, as a successful attack on any of these nodes could have cascading impacts on the overall power system. In contrast, Cases 2, 4, and 5, which included the cybersecurity component, were able to identify and mitigate these vulnerabilities, reducing the number of susceptible nodes to just one. These insights highlight the varied outcomes of each case and underscore the reliability and efficiency of Case 5 over other cases.

Table 10.

Cybersecurity results in different cases.

Lines	Case 1	Case 2	Case 3	Case 4	Case 5
(2, 25)	0	1	0	1	1
(3, 18)	0	0	0	1	1
(14, 15)	0	1	0	0	1
(17, 18)	0	1	0	1	0

Susceptible Nodes	Case 1	Case 2	Case 3	Case 4	Case 5
2	1	0	1	0	0
3	1	1	1	0	0
14	1	0	1	1	0
17	1	0	1	0	1

Furthermore, we investigate the voltage behavior of a power grid under various operational scenarios. Table 11 depicts the voltages under five cases. The total voltages for Cases 1 to 5 across all buses are 39.971, 38.038, 41.079, 38.369, and 38.741, respectively. These results illustrate the overall voltage levels for each case, highlighting variations in voltage distribution across the buses. When examining the averages for each case, we find results of 1.025(p.u), 0.975(p.u), 1.053(p.u), 0.984(p.u), and 0.993(p.u). Case 3 exhibits the highest average voltage, while Case 2 has the lowest. This fluctuation in average voltages suggests differing operational conditions or configurations that impact voltage levels. Notably, Cases 1, 3, and 5 demonstrate higher average voltages, indicating enhanced performance or stability in those scenarios. In contrast, while voltage levels in the power shutdown cases (Cases 1 and 2) remained relatively elevated, this stability was achieved at the cost of power outages affecting specific nodes within the grid. Conversely, scenarios without power shutdowns (Cases 3 and 5) exhibited a more stable and resilient voltage profile across the entire system.

Table 11.

Voltage (p.u) profiles in various case studies.

Bus number	Case 1	Case 2	Case 3	Case 4	Case 5
1	1.12	0.88	1.12	0.90	0.92
2	0.88	0.92	1.11	0.90	0.92
3	1.01	0.92	1.12	0.93	0.90
4	0.99	0.97	1.02	0.92	0.88
5	0.93	1.05	1.05	0.89	1.04
6	0.96	1.09	1.12	0.98	1.12
7	1.11	0.96	1.12	0.88	0.89
8	0.99	1.10	0.98	0.88	1.00
9	0.90	1.12	1.12	1.12	1.12
10	1.02	1.01	1.11	1.05	1.06
11	0.92	1.12	1.09	1.04	0.96
12	1.01	1.00	1.11	1.04	1.01
13	1.09	0.88	1.12	1.04	1.07
14	1.06	0.92	1.12	1.12	0.88
15	0.88	0.92	0.88	0.91	0.88
16	1.02	0.92	0.96	0.95	0.97
17	0.94	0.92	0.93	0.93	0.95
18	0.96	0.92	0.99	0.93	0.90
19	0.98	0.98	0.99	0.96	1.01
20	0.93	0.94	0.94	0.92	0.95
21	1.09	0.93	1.02	1.02	1.11
22	1.02	0.88	1.12	1.12	1.12
23	1.01	0.89	1.12	0.93	1.07
24	1.05	1.00	1.05	0.99	1.04
25	1.12	0.92	1.12	0.90	0.92
26	1.05	0.92	0.90	0.89	0.91
27	0.99	0.92	0.88	0.88	1.00
28	1.12	0.91	0.88	0.88	0.88
29	1.09	0.97	0.93	1.06	0.93
30	1.10	0.94	1.12	0.88	0.93
31	1.12	1.12	0.97	1.12	0.88
32	1.12	1.03	1.12	1.12	1.12
33	1.12	1.06	1.12	1.12	1.12
34	1.12	0.88	1.12	1.12	1.12
35	1.12	0.98	1.12	0.88	0.88
36	1.12	1.12	1.12	0.90	1.12
37	0.88	0.97	1.12	1.12	1.12
38	0.98	1.12	1.12	1.12	0.91
39	1.01	0.93	1.05	0.98	1.09

Figure 7 compares the total power requirements for charging and discharging in different study cases. Cases 1, 2, and 3 show low charging and discharging needs. Cases 1 and 2 involve non-zero load shedding, while Case 3 has a high number of susceptible nodes. In Case 3, with four susceptible nodes and no removed lines, the system doesn’t address power requirements to reduce cyberattack risks, resulting in low total power needed for charging and discharging. Conversely, Case 3 has the highest power generation at 1131.37 kW, as shown in Table 12. Cases 1 and 2 yield 184.85 kW and 253.92 kW for power generation, respectively. The results suggest that power dispatching is inefficient in Cases 1, 2, and 3. Additionally, Cases 4 and 5 generate 933.043 kW and 835.782 kW, respectively. While charging amounts are similar in Cases 4 and 5, Case 5 has higher discharging at 513.53 kW compared to 414.348 kW for Case 4. However, Case 4 experiences a longer tour length than Case 5, highlighting the reliability and efficiency of Case 5.

Fig. 7.

Power consumption for electric vehicles and vehicle-to-grid systems in different case studies.

Table 12.

Generation of power by generators in various case studies.

Bus number	Case 1	Case 2	Case 3	Case 4	Case 5
30	0	0	80.993	0	0
31	184.856	0	452.205	277.168	395.039
32	0	0	0	0	0
33	0	0	71.169	0	0
34	0	0	0	0	105.775
35	0	0	142.525	441.032	270.142
36	0	111.220	100.911	0	0
37	0	44.800	0	0	0
38	0	97.900	61.873	214.844	0
39	0	0	221.699	0	64.827

Elasticity evaluation

This section evaluates the total cost of the fleet size by solving the proposed model (Case 5) while keeping all other factors remaining unchanged. The results are graphically summarized in Fig. 8. These results show the total costs in minutes associated with varying numbers of ETs, considering both sojourn time (user delay) and system cost (vehicle tour length). Total costs consistently decrease from 2 to 7, then increase when the number of electric taxis reaches 8, coinciding with an increase in dispatched passengers at that fleet size. This suggests economies of scale and improved operational efficiencies with a fleet of 7 ETs for the case study.

Fig. 8.

Tour length and user delay across various fleet sizes.

Furthermore, we explore cyberattack scenarios across various fleet sizes. Table 13 illustrates the intricate relationship among fleet size, network security vulnerabilities (shown by power line removals), and node susceptibility to cyber threats. As the fleet size increases up to , there is a notable increase in the number of removed lines. This result suggests enhancements in network resilience and security measures within the power network. The optimal fleet size with ETs and V2G capabilities seems to enhance network resilience and address power needs for line removals due to cyber threats, highlighting their potential in strengthening network security.

Table 13.

Cybersecurity results under various fleet sizes.

Lines
(2, 25)	1	1	1	1	1	1	1
(3, 18)	1	1	1	0	1	1	1
(14, 15)	1	0	1	1	1	1	1
(17, 18)	0	1	0	1	0	0	0

Susceptible Nodes
2	0	0	0	0	0	0	0
3	0	0	0	1	0	0	0
14	0	1	0	0	0	0	0
17	1	0	1	0	1	1	1

Energy efficiency and fleet size

This section describes a test scenario assessing the performance of an ET service in serving customers over longer distances and during peak periods of power demand. Taxis begin their routes from John F. Kennedy (JFK) airport, picking up and dropping off passengers at different locations, illustrated in Fig. 9. Furthermore, we analyze the model under heightened power demand. The power demand for other consumers is raised by 30%, as shown in Equation (29), while all other parameters remain constant.The purpose is to assess the ETs’ ability to handle longer routes and distances, providing insights into their range, efficiency, and logistical capabilities under more challenging conditions. For more information and specific parameters about our model, readers can refer to the publicly accessible data at the following URL: https://github.com/homase2003/HamidSayarshad.git.

Fig. 9.

Pickup and drop-off locations for passengers.

The study aims to determine the optimal fleet size and total power required to address a cyberattack while ensuring electric taxis complete their trips with minimal travel costs. investigate the performance of the proposed taxicab dispatching system during load-shedding scenarios. Load-shedding refers to temporarily reducing the electrical load on a power grid, often implemented to prevent the grid from becoming overloaded and experiencing a blackout. This analysis aims to understand how the percentage of power shed is affected as the size of the taxicab fleet, which is part of the DN, increases from 2 to 6 taxis. Figure 10 presents the power shutdown concerning the fleet size. The percentage of power shed is measured while the taxicab fleet size increases from 2 to 6 taxis. The results show that the load-shedding values decreased from 220.20 (kW) for 2 taxis to 126.94 (kW) for 6 taxis. This load shedding occurs at electric bus nodes 15, 26, and 28.

Fig. 10.

Power outage based on different fleet sizes.

Furthermore, we measure both the power charge and power discharge of the taxis across different fleet sizes, which helps to assess their impact on load-shedding. Figure 11 illustrates the relationship between fleet size and the charging and discharging of the taxicab fleet. The amount of active power generation by generators is also investigated as the fleet size of ETs increases from 2 to 6. Table 14 illustrates the relationship between fleet size and power generation, showing decreased active power output. With 2 taxis, the active power generation is 2772.62 kW, which diminishes to 2471.87 kW as the fleet expands to 6 taxis. Notably, the power shutdown decreases by 42% as the fleet size expands from 2 to 6. Hence, expanding the ET fleet increases mobile storage capacity, aiding power utilities in reducing the impact of cyberattacks and power outages.

Fig. 11.

The power charge and discharge levels differ with various fleet sizes.

Table 14.

Generation of active power by generators for various fleet sizes.

Bus number	V=2	V=3	V=4	V=5	V=6
30	482.632	489.207	494.493	489.313	489.976
31	646	646	646	646	646
32	253.058	273.456	274.413	176.267	63.045
33	436.079	422.252	409.150	172.114	173.003
34	0	0	0	0	0
35	341.289	304.939	247.617	396.266	296.361
36	417.063	244.769	235.756	244.216	356.021
37	0	0	0	0	0
38	168.109	350.435	352.646	444.812	447.465
39	28.393	0	0	0	0

Supplemental materials

The supplemental materials are available at the GitHub repository https://github.com/homase2003/HamidSayarshad.git.

Author contributions

The single author (HS) provided all contributions.

Funding

The author received no funding for this work. This research did not receive any grant from funding agencies in the public, commercial, or not-for-profit sectors.

Data availability

All dataset sources are cited in the article. Additional data is available upon request to the corresponding author.

Declarations

Competing interests

The authors declare no competing interests.