Hongping Wang

Yi-Ping Fang
email: yiping.fang@centralesupelec.fr

Enrico Zio

Resilience-based optimal post-disruption reconfiguration for traffic-power systems

Due to the growing the penetration of electric vehicles (EV), road networks (RNs) and power networks (PNs) more rely on the normal functioning of each other. However, only a few of studies have discussed the potential negative consequences of adopting EVs during disaster. Moreover, the negative impact of EVs on the interdependent traffic-power systems has been rarely considered during restoration period. In this study, restoration plans for independent RNs and PNs, and interdependent traffic-power systems are investigated, respectively. For the restoring the traffic-power systems, a two-stage mixed integer optimization model is proposed to provide system optimal reconfiguration and operational solutions after disruptions. The objective of this model is to minimize the total system performance loss cost. The total system performance loss cost is quantified by cumulative unsatisfied gasoline vehicles (GVs) and EVs traffic demand for RNs, and shedded load cost for PNs. In tactical level, reconfiguration solutions (i.e., links reversing for RNs and line switching for PNs) are determined. In the operational level, the integrated system optimal dynamic traffic assignment and optimal power flow problem is solved to obtain the optimal traffic-power flow. In this model, RNs and PNs are coupled through EVs and the coordinately allocated spatiotemporal charging demand. A partial highway network in North Carolina (NC), USA and modified IEEE-14 bus system are used to illustrate the developed methods. The results show the added value of coordinately planing restoration for traffic-power systems. When EV penetration increases from

0% to 100%, the total performance loss costs increase 46.7%. This result also hints that more FCSs should be deployed in this area.

Keywords: Reconfiguration, Restoration, Rraffic-power systems, Optimal traffic-power flow, Electric vehicles, Fast

Introduction

High-impact and low-probability (HILP) incidents usually cause severe negative consequences on road networks (RNs) and power networks (PNs). 2003 North America blackout caused 50 million customers affected by this outages [START_REF] Fang | Optimum post-disruption restoration under uncertainty for enhancing critical infrastructure resilience[END_REF]. 65% of New Jersey's customers were experienced the disconnections from the power systems during Hurricane Sandy in 2012 [START_REF] Ma | Resilience enhancement strategy for distribution systems under extreme weather events[END_REF]. In July 2021, heavy rainfall and floods hit Henan, China [3]. This HILP event devastated the local critical infrastructures, including the transportation and the power systems. The estimated direct economic loss is on the order of RMB 88.5 billion. This has highlighted an urgent need for strengthen the resilience of the RNs and PNs against such HILP incidents. The concept of resilience is defined in various explanations [START_REF] Ganin | Resilience and efficiency in transportation networks[END_REF][START_REF] Zhang | Resilience-based network design under uncertainty[END_REF][START_REF] Zhu | Cyber-physical resilience modelling and assessment of urban roadway system interrupted by rainfall[END_REF], however, many of them share the general idea that resilience is the ability of a system to prepare for, absorb, recover from, and adapt to disturbances [START_REF] Wang | Resilience assessment of electrified road networks subject to charging station failures[END_REF]. Restoration as one of the most effective manners to enhance the system resilience after disruptions has received increasing attention. A large body of work have been done to investigate the optimal restoration after disruptions for RNs and PNs, sparately.

For PNs, reconfiguration as an effective strategy to restore the service and to enhance the system resilience against emergencies has been intensively investigated in the literature [START_REF] Aziz | Review on optimization methodologies in transmission network reconfiguration of power systems for grid resilience[END_REF][START_REF] Wang | On microgrids and resilience: A comprehensive review on modeling and operational strategies[END_REF][START_REF] Fan | Restoration of smart grids: Current status, challenges, and opportunities[END_REF]. Generally, maximizing the network resilience/minimizing system performance loss and minimizing the number of line switches/minimizing restoration cost are included in the objectives of these models. For example, Sekhavatmanesh and Cherkaoui [START_REF] Sekhavatmanesh | Distribution network restoration in a multiagent framework using a convex opf model[END_REF] developed the concept of multiagent automation in smart grids, which was applied to restore a maximum of loads with minimum switching operations service after disruptions; Sabouhi et. al [START_REF] Sabouhi | Electricity distribution grids resilience enhancement by network reconfiguration[END_REF] presented an operational network reconfiguration strategy during a high wind event to maximize network resilience and minimize the number of line switches. Sometimes, islanding or not after disruptive events are also treated differently. Agrawal et. al [START_REF] Agrawal | Resiliency in active distribution systems via network reconfiguration[END_REF] developed a self-healing algorithm to restore maximum priority loads by reconfiguring network without intentional islanding during blackouts. Guimaraes et. al [START_REF] Guimaraes | A decomposition heuristic algorithm for dynamic reconfiguration after contingency situations in distribution systems considering island operations[END_REF] proposed a three-stage algorithm for the dynamic reconfiguration of distribution networks with islanding. The three stages include calculating the network reconfiguration solutions in each hour, reducing the number of configurations and generating the optimal sequence of topologies. Li et. al [START_REF] Li | A full decentralized multi-agent service restoration for distribution network with dgs[END_REF] presented a concept of fully decentralized multi-agent system to build a restoration service framework for distribution network. Based on this concept, a network reconfiguration algorithm is proposed for restore service, where reconfiguration with intentional islanding was considered. Except reconfiguration, other corrective actions, such as generator re-dispatch, control of distributed energy storage systems (ESSs), and on-load tap changers, also can be considered as supplementary strategies to enhance the resilience of the power systems. Liberati et. al [START_REF] Liberati | Efficient and risk-aware control of electricity distribution grids[END_REF] proposed control system optimizes grid operations through network reconfiguration, control of distributed energy storage systems, and on-load tap changers. Sekhavatmanesh and Cherkaoui [START_REF] Sekhavatmanesh | Analytical approach for active distribution network restoration including optimal voltage regulation[END_REF] developed an analytical and global optimization model to find the most efficient restoration plan minimizing the number of de-energized nodes with the minimum number of corrective actions. The considered corrective actions included network reconfiguration, the tap setting modification of voltage regulation devices, the nodal load-rejection, and the active/reactive power dis-patch of distribution generators. Zhang et. al [START_REF] Zhang | Risk-neutral and risk-averse transmission switching for load shed recovery with uncertain renewable generation and demand[END_REF] introduced two-stage stochastic models to deal with the uncertainty in generation and demand during the recovery process. Switching transmission lines and generator redispatch strategies were used to maximize load shed recovery in the bulk transmission network. Nazemi and Dehghanian [START_REF] Nazemi | Seismic-resilient bulk power grids: Hazard characterization, modeling, and mitigation[END_REF] introduced a framework for modeling and characterization of seismic hazards, vulnerability assessment of electric systems to the earthquake. The generation re-dispatch strategy and corrective network topology control were considered to maximize the load outage recovery after disasters.

For the RNs, short-term and long-term recovery periods usually are investigated separately. For short-term recovery period, reconfiguring network topology and controlling traffic lights are frequently discussed manner to increase the resilience of RNs after disruptive events. Wang and Wang [START_REF] Wang | Integrated reconfiguration of both supply and demand for evacuation planning[END_REF] developed an integrated reconfiguration strategy that reconfigures both the supply and demand sides of transportation systems. The traffic demand was reconfigured using a heterogeneous fleet of vehicles and the network topology was reconfigured through a heterogeneous contraflow control. Later on, they further presented a framework [START_REF] Wang | Measuring and maximizing resilience of transportation systems for emergency evacuation[END_REF] for resilience analysis including measurement and improvement. Two strategies were used to maximize the system resilience. They were integrated reconfiguration of both traffic supply and demand by reducing traffic demand through combining different traffic modes and a contraflow control. Chiou [START_REF] Chiou | A traffic-responsive signal control to enhance road network resilience with hazmat transportation in multiple periods[END_REF] proposed a period-dependent traffic responsive signal control model to enhance resilience of urban RNs. Koutsoukos et. al [START_REF] Koutsoukos | Sure: A modeling and simulation integration platform for evaluation of secure and resilient cyber-physical systems[END_REF] developed a modeling and simulation integration platform for experimentation and evaluation of resilient transportation systems. Resilient traffic signal control in the presence of denial-of-service attacks was studied in the case studies. For long-term recovery period, scheduling repair crew, allocating resources and determining restoration priority of components in the RNs are generally the main focuses. Wu et. al [START_REF] Wu | Post-earthquake resilience assessment and long-term restoration prioritization of transportation network[END_REF] proposed a methodology to assess the resilience of transportation networks and a restoration priority measure to support post-earthquake restoration of damaged bridges. Zhao and Zhang [START_REF] Zhao | Transportation infrastructure restoration optimization considering mobility and accessibility in resilience measures[END_REF] proposed a bi-objective bi-level optimization framework to determine an optimal transportation network restoration plan. The lowerlevel problem uses elastic user equilibrium to model the imbalance between demand and supply. The upper-level problem, formulated as bi-objective mathematical programming, determines optimal resource allocation for roadway restoration.

In recent years, treating the independent RNs and PNs as a whole and the problems how to model and operate them together have gained much of attention [START_REF] Wei | Network equilibrium of coupled transportation and power distribution systems[END_REF][START_REF] Wang | Risk assessment of an electrical power system considering the influence of traffic congestion on a hypothetical scenario of electrified transportation system in new york state[END_REF]. However, only a few studies have investigated the problem of how to restore them after disruptions to minimize the whole system performance loss subjecting to physical constraints from both networks. Moreover, most of these studies only focused on minimizing the performance loss of PNs and only considered the constraints of RNs. Their topics generally are about how to route and schedule of mobile energy storage systems (MESSs)/mobile energy sources [START_REF] Nazemi | Uncertainty-aware deployment of mobile energy storage systems for distribution grid resilience[END_REF][START_REF] Lei | Resilient disaster recovery logistics of distribution systems: Co-optimize service restoration with repair crew and mobile power source dispatch[END_REF][START_REF] Taheri | Distribution system resilience enhancement via mobile emergency generators[END_REF][START_REF] Anokhin | Mobility-as-a-service for resilience delivery in power distribution systems[END_REF], sometimes coordinating with repair crews [START_REF] Lei | Resilient disaster recovery logistics of distribution systems: Co-optimize service restoration with repair crew and mobile power source dispatch[END_REF][START_REF] Taheri | Distribution system resilience enhancement via mobile emergency generators[END_REF][START_REF] Anokhin | Mobility-as-a-service for resilience delivery in power distribution systems[END_REF] and line switch strategies [START_REF] Lei | Resilient disaster recovery logistics of distribution systems: Co-optimize service restoration with repair crew and mobile power source dispatch[END_REF][START_REF] Taheri | Distribution system resilience enhancement via mobile emergency generators[END_REF]. One of the key issues of restoring traffic-power networks is to figure out and properly model the objects that connect and make influence on the both networks. Wang et. al [START_REF] Wang | Dynamic load restoration considering the interdependencies between power distribution systems and urban transportation systems[END_REF] considered the PNs and urban RNs coupled through traffic lights and mobile emergency resources (i.e., mobile sources and repair crews). The availability of mobile emergency resources relates to their dispatch in the RNs, while the effect of traffic lights is also considered. They developed a service restoration method to maximize the efficiency of both PNs restoration and RNs. Yao et. al [START_REF] Yao | Rolling optimization of mobile energy storage fleets for resilient service restoration[END_REF] proposed a rolling integrated service restoration strategy to minimize the total system cost by coordinating the scheduling of MESS fleets, resource dispatching of microgrids, and network reconfiguration of PNs. The integrated strategy takes into account damage and repair to both the roads in RNs and the branches in PNs. Li et. al [START_REF] Li | Resilient restoration of distribution systems in coordination with electric bus scheduling[END_REF] presented an optimization model for joint post-disaster DS restoration, considering coordinated dispatching with electric buses. By assuming that the PN can rent some electric buses. Idle buses are placed at designated areas and feed power back to the grid via charging equipment in case of need. The schedule of the remaining buses should meet the passenger transport demand. The objective was to maximize the total benefits and minimize the bus rental cost. Different from the existing work, this paper consider that the RNs and PNs are coupled through electric vehicles (EVs) and their charging demand. This is because EVs and fast-charging stations (FCSs) are being increasingly deployed around the world [START_REF]International Energy Agency (IEA), Global ev outlook 2020[END_REF]. It makes PNs and RNs inevitably accelerate the coupling together. On the other hand, EVs probably threatened by outages in FCSs because of the natural disaster have raised a few of researchers' concerns [START_REF] Adderly | Electric vehicles and natural disaster policy implications[END_REF][START_REF] Feng | Can we evacuate from hurricanes with electric vehicles?[END_REF]. However, this problem and its solution approaches have not been solidly investigated yet, especially in the context of enhancing the resilience of the interdependent traffic-power networks. In this paper, we focus on the problem of how to jointly reconfigure and operate the trafficpower networks so that the total system performance loss can be minimized for short-term recovery period. Specifically, road links in RNs and lines in PNs may be damaged simultaneously after a natural disaster or a malicious attack. In RNs, vehicles in RNs may need to bypass and the RNs' performance therefore decrease. Due to the extra distance, both the number of EVs and the amount of charging demand for each EV may increase in FCSs. Such charging demand can become a burden for PNs during the restoration after the disruptive event. On the other hand, the PNs may need to shed the EV charging load due to the reduced flow and generation capacity after the disruptions. Consequently, the unavailability of the service in FCSs can further influence the charging demand patterns and decrease the performance of the RNs. To fill the above mentioned research gaps, in this paper, we develop a two-stage mixed integer optimization model to minimize the system performance loss after disruptions. In the first stage, reconfiguration solutions over the whole studied horizon for the traffic-power systems are provided to enhance the system resilience after a disruptive event. In the second stage, an integrated traffic-power systems model is developed to model the dynamic interaction between RNs and PNs through EVs and spatial and temporal charging demand. The results show the added value of coordinately reconfiguring traffic-power systems and managing EV charging demand.

The main contributions of this paper are summarized as follows: 1. To best our knowledge, it is the first work that consider the negative impact of EVs and interdependent traffic-power systems in terms of restoration and network topology control.

2. The strategies of link directions reversing in the RN and line switching in the PN are mathematically formulated in an independent traffic-power system model, and in independent RN and PN models, respectively.

3. For the interdependent traffic-power systems, a two-stage mixed integer optimization model is presented to minimize the system performance loss after disruptions. The tactical level problem is formulated as an integrated system optimal dynamic traffic assignment and DC power flow problem considering EVs. The optimal topology of traffic-power systems are served as the operational level problem.

4. Different response source levels, EV penetration levels and decisionmaking environments are compared in the case study.

The remainder of the article is structured as follows. Section 2 formulated the reconfiguration problems in independent RN and PN models, as well as in the interdependent traffic-power system model. Section 3 illustrates a case study to show the application of the proposed models and compares the solutions under different response resource levels, EV penetration levels and decision-making environments. Finally, Section 4 provides some concluding remarks and future research directions.

Reconfiguring infrastructures

In this section, reconfiguration problem in independent RNs, independent PNs and interdependent traffic-power systems are formulated.

Reconfiguring electrified highway networks

In this subsection, we first present an electrified traffic system model considering the critical characteristics of EVs and FCSs, which is based on link transmission model. Then, reconfiguration problem after disruption is formulated based on this electrified traffic model.

Electrified traffic model with EVs and FCSs

In this model we assume that the electricity consumed by an EV is linearly related to the distance traveled. The electricity amount charged by an EV is linearly related to the charging time. All EV batteries have the same energy consumption efficiency, similar to Ref. [? ].

A RN with multiple sources (origins) and sinks (destinations) is denoted as G(N , A), where N and A are the sets of nodes and links, respectively. Links in the RN are classified into four types: source A R , sink A S , general A G and charging A C links. Nodes are classified into two types: source-sink N SR and general N G nodes. Within the RN, each source-sink node attaches only one source and one sink link. All charging, source and sink links are dummy with lengths 0. All source and sink links are with infinite outflow, inflow and storage capacities. For SO-DTA problems, the outflow capacity of all sink links are assumed to be 0, which means that all vehicles are collected upon their arrival. The time horizon H is discretized into a finite set of periods

T = {t = 1, 2, • • • , T }. T is calculated according to T = H/δ,
where δ is the period length. The period length should be equal to or smaller than the smallest link travel time so that vehicles take at least one time unit to traverse a link [START_REF] Yperman | The link transmission model for dynamic network loading[END_REF].

A triangular fundamental diagram is used in LTM, which is an approximation and describes a macroscopic property of roads [START_REF] Yperman | The link transmission model for dynamic network loading[END_REF]. The diagram is defined by three parameters: a jam density (k jam ), a maximum flow (q max ) and a fixed-free flow speed (v f ). The backward shock-wave speed w can be obtained by

w = q max • v f /(q max -k jam • v f ).
Given a certain class of EV denoted as c, its battery capacity is B c kWh and the energy consumption efficiency is η kWh/mile, then, the mileage of this class EV is L max c = B c /η miles. We discretize its mileage into integer energy levels (ELs). When this EV has full battery, it has the maximum EL

E c = L max c /(δ • v f ). Once this EV traveled δ • v f miles, its ELs decrease one EL, i.e., 1 EL = δ • v f miles. Assuming there are C EV classes represented as C = {E 1 , E 2 , • • • , E C }. Each element E c in set C is a set, which contains the energy levels that EV of class c could have, denoted as E c = {1, 2, • • • , E c }.
The LTM updates the traffic flow evolution by calculating the cumulative number of vehicles at entry and exit of each link in each period.

The Newell's simplified theory is used in LTM to calculate sending S a (t) and receiving R a (t) capacities of link a:

S a (t) = min{U a (t -ν a ) -V a (t -1), Of a (t)} (1a) R a (t) = min{V a (t -β a ) + L a • k jam -U a (t -1), If a (t)} (1b)
where U a (t)/V a (t) denotes the cumulative number of vehicles that enter/leave link a by the end of period t. If a (t) and Of a (t) are the inflow capacity at the entering point and outflow capacity at the leaving point of link a during period t. They can be obtained by δ • q max at the corresponding location and period. L a is the length of link a. ν a is the free-flow travel time on link a and β a is the travel time required by the backward shock wave from the exit to the entry of link a. They can be obtained by

ν a = L a /(δ • v f ) and β a = L a /(δ • w), respectively.
The inflow and outflow of link a during interval t are constrained by its corresponding sending and receiving capacities:

U a (t) -U a (t -1) ≤ R a (t), ∀a ∈ A\{A C }, ∀t (2a) V a (t) -V a (t -1) ≤ S a (t), ∀a ∈ A\{A C }, ∀t (2b)
Substituting Eqs. ( 1) and (1b) into the system of inequality (2), we obtain the following system of linear LTM-based flow constraints:

V a (t) ≤ U a (t -ν a ), ∀a ∈ A\{A C }, ∀t (3) V a (t) -V a (t -1) ≤ Of a (t), ∀a ∈ A\{A C }, ∀t (4) U a (t) -U a (t -1) ≤ If a (t), ∀a ∈ A\{A C }, ∀t (5)
U a (t) -V a (t -β a ) ≤ L a k jam , ∀a ∈ A\{A C }, ∀t (6)
In eLTM-based model, both EVs and conventional vehicles are considered as follows :

U a (t) = ∑ s∈N S U G s a (t) + ∑ s∈N S ∑ c∈C ∑ e∈Ec U E s,e a,c , ∀a ∈ A\{A C }, ∀t (7a) V a (t) = ∑ s∈N S V G s a (t) + ∑ s∈N S ∑ c∈C ∑ e∈Ec V E s,e a,c , ∀a ∈ A\{A C }, ∀t (7b)
where U E s,e a,c (t)/V E s,e a,c (t) denotes the cumulative number of EVs that belong to type c with EL e that enter/leave link a to destination s by the end of interval t under scenario o; U G s a (t)/V G s a (t) denotes the cumulative number of GVs that enter/leave link a to destination s by the end of interval t under scenario o.

Substituting Eq. ( 7) into the inequalities in Eqs. (3) -( 6), we can have the following constraints for the mixed traffic of EVs and GVs:

∑ s∈N S [V G s a (t) -V G s a (t -1)] + ∑ s∈N S ∑ c∈C ∑ e∈Ec [V E s,e a,c (t) -V E s,e a,c (t -1)] ≤ Of a (t), ∀a ∈ A\{A C }, ∀t, s (8)
∑ s∈N S [U G s a (t) -U G s a (t -1)] + ∑ s∈N S ∑ c∈C ∑ e∈Ec [U E s a (t) -U E s a (t -1)] ≤ If a (t), ∀a ∈ A\{A C }, ∀t, s (9)
∑ s∈N S ∑ c∈C ∑ e∈Ec [U E s,e a,c (t) -V E s,e a,c (t -β a )] + ∑ s∈N S [U G s a (t) -V G s a (t -β a )] ≤ L a k jam , ∀a ∈ A\{A C }, ∀t, s (10)
For GVs, the cumulative outflow disaggregated by destinations and scenarios should also be constrained by the boundary condition at the inflow. Hence, we have

∑ s∈N S V G s a (t) ≤ ∑ s∈N S U G s a (t -ν a ), ∀a ∈ A\{A C }, ∀t (11)
For EVs, the disaggregated cumulative outflow should also be constrained by the battery condition at the inflow. Hence, we have

V E s,e a,c (t) ≤ U E s,e+ρa a,c (t -ν a ), ∀a ∈ A\{A C }, e ∈ E c ∩ {e ≤ E c -ρ a }, ∀s, c, t (12a) V E s,e a,c (t) = 0, ∀a ∈ A\{A C }, e ∈ E c ∩ {e > E c -ρ a }, ∀s, c, t (12b)
where, ρ a is the ELs required to traverse link a and it is calculated by

ρ a = L a /(δ • v f ).
Eq. (12a) guarantees that outflow should be less than or equal to the inflow. It also guarantees that the outflow ELs are updated from inflow after EVs traversed the corresponding links. Eq. (12b) ensures that all EV ELs should be less than their maximum ELs.

The traffic demand is satisfied by letting the cumulative inflows of source links equal the cumulative demands:

U G s a (t) = DG s a (t), ∀a ∈ A R , ∀s, t (13a) U E s,e a,c (t) = DE s,e a,c (t), ∀a ∈ A R , e ∈ E c , ∀s, c, t (13b)
where DG s a (t)/DE s,e a,c (t) represents the cumulative GVs/EVs travel demand between the entry of origin link a and destination s at the end of period t.

The inflow and outflow of a general node should be restricted by the following flow conservation constraints:

∑ a∈B(i) V G s a (t) = ∑ b∈A ( i) U G s a (t), ∀i ∈ N /{N SR }, ∀s, t (14a) ∑ a∈B(i) V E s,e a,c (t) = ∑ b∈A (i) U E s,e a,c (t), ∀i ∈ N /{N SR }, ∀e ∈ E c , ∀s, c, t (14b)
where A(i)/B(i) represents the set of links whose tail/head node is i.

For EVs, the current occupancy on charging link a should be limited by the maximum number of chargers on this link:

∑ s∈N S ∑ c∈C ∑ e∈Ec [U E s,e a,c (t) -V E s,e a,c (t)] ≤ N C a (t), ∀a ∈ A C , ∀t (15)
where N C a (t) is the physical number of type a chargers on charging link a during period t.

The following equations are used to update the current occupancy and their ELs on a charging link:

xs,e a,s (t) = x s,e a,s (t -

1) + [U E s,e a,c (t -1) -U E s,e a,c (t -2)]- [V E s,e a,c (t -1) -V E s,e a,c (t -2)], ∀a ∈ A C , ∀e ∈ E c , ∀s, c, t (16)
where xs,e a,s (t) and x s,e a,s (t) are the number of EVs before and after their ELs have been updated on charging link a.

Based on the obtained occupancies, the following equations are used to model their charging process where ELs of EVs linearly increase with time on charging links:

x s,Ec a,c (t) = α t a ∑ l=0 xs,Ec-l a,c (t), ∀a ∈ A C , ∀s, c, t (17a) x s,e a,c (t) = xs,e-α t a a,c (t), ∀a ∈ A C , ∀e ∈ {α t a ≤ e < E c }, ∀s, c, t (17b)
x s,e a,c (t

) = 0, ∀a ∈ A C , ∀e ∈ {e < α t a }, ∀s, c, t (17c)
where α t a represents the average charging speed for charging link a during period t, which translates to how many energy levels can be supplied using type a charger during a period δ. Assuming the charging power of charging link a is p ev a , then, α t a can be calculated by

p ev a •δ η•δ•v f = p ev a η•v f .
Eqs. (17a) and (17c) constrain the upper and lower boundaries of the updated ELs. Eq. (17b) describe the process of linear increase in ELs.

Additionally, the outflow disaggregated by each EL on charging link a should be less than its occupancy, as formulated in Eq. ( 18):

V E s,e a,c (t) -V E s,e a,c (t -1) ≤ x s,e a,c (t), ∀a ∈ A C , ∀e ∈ E c , ∀s, c, t (18)
The occupancies on charging links are nonnegative, which is formulated as follows:

x s,e a,c (t) ≥ 0, xs,e a,c (t) ≥ 0, ∀a ∈ A C , ∀e ∈ E c , ∀s, c, t, (19)
The cumulative flows should be nonnegative and nondecreasing:

V G s a (t) -V G s a (t -1) ≥ 0, ∀a ∈ A, ∀s, t (20a) V E s,e a,c (t) -V E s,e a,c (t -1) ≥ 0, ∀a ∈ A, ∀e ∈ E c , ∀s, c, t (20b) U G s a (t) -U G s a (t -1) ≥ 0, ∀a ∈ A, ∀s, t (21a) U E s,e a,c (t) -U E s,e a,c (t -1) ≥ 0, ∀a ∈ A, ∀e ∈ E c , ∀s, c, t (21b)
The following constraints force the initial cumulative flows to be 0:

U G s a (0) = V G s a (0) = 0, ∀a ∈ A, ∀s (22a) U E s,e a,c (0) = V E s,e a,c (0) = 0, ∀a ∈ A, ∀e ∈ E c , ∀s, c (22b)

Modeling reconfiguration strategy in traffic-power system model

To mitigate the impacts after disruptions, we consider the strategy of contraflow to reconfigure the topology of the highway networks. Contraflow can be easily implemented by reversing the direction of lanes of highway networks. . However, if we reverse the direction of the link from nodes D to O, the total number of arrivals can be increased from 40 to 50 vehicles after the disruption, as shown in Fig. 1(c). This example shows that reconfiguring the highway network after disruption could effectively reduce the system performance loss. Another example in Ref. [START_REF] Zhang | Network reconfiguration for increasing transportation system resilience under extreme events[END_REF] can be found illustrating how contraflow strategy increases the network outbound capacity and mitigates the congestion.

To model the contraflow strategy, we define that each link in the highway network has and only has one unique opposite link corresponding to it. For example, there are two links a 1 and a 2 from nodes O to D in Fig. 2.1.2. Their corresponding opposite links are â1 and â2 , respectively. Meanwhile, links a 1 and a 2 are the opposite links of links â1 and â2 . Mathematically, we use variable h a to denote whether link a is changed to the opposite direction or not. â represents the unique opposite link of link a. If the direction of the link is reversed, the outflow capacity, the inflow capacity and the maximum number of vehicles that can be present on that link of direction will be correspondingly reconfigured. Therefore, Eqs. ( 8) -( 10) are reformulated as follows:

∑ s∈N S [V G s a (t) -V G s a (t -1)] + ∑ s∈N S ∑ c∈C ∑ e∈Ec [V E s,e a,c (t) -V E s,e a,c (t -1)] ≤ (1 -h a ) • Of a (t) + h â • Of â(t), ∀a ∈ A\{A C }, ∀t, s (23)
∑ s∈N S [U G s a (t) -U G s a (t -1)] + ∑ s∈N S ∑ c∈C ∑ e∈Ec [U E s a (t) -U E s a (t -1)] ≤ (1 -h a ) • If a (t) + h â • If â(t), ∀a ∈ A\{A C }, ∀t, s (24)
∑ s∈N S ∑ c∈C ∑ e∈Ec [U E s,e a,c (t) -V E s,e a,c (t -β a )] + ∑ s∈N S [U G s a (t) -V G s a (t -β a )] ≤ (1 -h a )L a k jam + h âL âk jam , ∀a ∈ A\{A C }, ∀t, s (25)
Eq. ( 23) states that the outflow on link a are constrained by the status of links a and â. If h a = 0 and h â = 0, no link is reversed and the outflow capacity on the direction of original link a is unchanged, i.e., outflow capacity of link a; If h a = 1 and h â = 1, both links are reserved and the outflow capacity is modified to the outflow capacity of link â; If h a = 1 and h â = 0, the direction of link a is reversed and the outflow capacity becomes 0; If h a = 0 and h â = 1, the direction of the opposite link â is reversed and the outflow capacity increase to the sum of outflow capacities of links a and â. Similarly, we can have Eqs. ( 24) and ( 25) to constrain the inflow and maximum occupancies on the direction of original link a after reconfiguration.

h a , h â = {0, 1}, ∀a, â ∈ A\{A C } (26) ∑ a h a ≤ N h , ∀a ∈ A\{A C } (27)
Eq. ( 26) guarantees that h a and h â are binary variables. Eq. ( 27) constrains the total number of links that can be reversed. This constraint reflects the limited resources that can be used to emergency response to the disruption, in practice.

The emergency response problem is formulated as follows:

min x∈Ψ P T = ∑ s∈N S ∑ t∈T ∑ a∈A S [DG s a (t) -U G s a (t) + ∑ c∈C ∑ e∈Ec (DE s,e a,c (t) -U E s,e a,c (t))] • ϕ ( 28
)
where Ω = {x| Subject to constraints ( 11)-( 27)}. ϕ denotes the time value. The objective of the transportation operator is to minimize the system performance loss cost after disruptions within a certain period. The system performance loss is measured by the unsatisfied traffic demand. In this paper, it is calculated by the cumulative difference between the target demand (i.e., DG s a (t) and DE s,e a,c (t)) and the number of vehicles arrived at their des-tinations (i.e., U G s a (t) and U E s,e a,c (t), a ∈ N S ). In Eq. ( 28)The first term is the cumulative unsatisfied GVs travel demand and the second term is the cumulative unsatisfied EVs travel demand.

Reconfiguring power system

We consider a PN G P (P N , P L ), where P N and P L represent the sets of buses and branches, respectively. P L represents the set of damaged transmission lines after a disruption, P L ⊂ P L . Γ -(j) and Γ + (j) denote the sets of predecessors and successors of bus j, respectively.

After the disruption, the objective of independent system operator is to minimize the cost of shedding unsatisfied load, which is formulated as follows:

min P P = ∑ j ∑ t c b j • LS b j,t + c dc j • LS dc j,t • p dc j (t) (29)
where c b j and c dc j are cost of shedding base load and EV charging load, respectively; LS b j,t is a continuous variable representing the amount of unsatisfied base demand at bus j in period t; LS dc j,t is a binary variable denoting where the charging demand p dc j (t) at bus j in period t is shedded or not. The power flow in the transmission systems are subjected to the following constraints:

p g j,t + ∑ i∈Γ -(j) P i,j,t - ∑ k∈Γ + (j) P j,k,t = p b j,t -LS b j,t +(1-LS dc j,t )•p dc j (t), ∀j ∈ P N (30) -Pi,j • u i,j ≤ P i,j,t ≤ Pi,j • u i,j , ∀(i, j) ∈ P l \{ P L }, ∀t (31)
P i,j,t = 0, ∀(i, j) ∈ { P L }, ∀t (32)
B i,j • (θ i,t -θ j,t ) -P i,j,t + (1 -u i,j ) • M i,j ≥ 0, ∀(i, j) ∈ P L \{ P L }, ∀t (33) B i,j • (θ i,t -θ j,t ) -P i,j,t -(1 -u i,j ) • M i,j ≤ 0, ∀(i, j) ∈ P L \{ P L }, ∀t (34) -p ramp j ≤ p g j,t -p g j,t-1 ≤ p ramp j , ∀j ∈ P N , ∀t ∈ T (35) 0 ≤ LS b j,t ≤ p b j,t , ∀j ∈ P N ( 36
)
0 ≤ P g j,t ≤ pg j , ∀j ∈ P j (37) ∑ (i,j)∈P L (1 -u i,j ) ≤ N u ( 38
)
u i,j = 0, 1, ∀(i, j) ∈ P L (39)
LS dc j,t = {0, 1}, ∀j ∈ P N [START_REF] Washington | Power systems test case archive[END_REF] Constraint [START_REF] Taheri | Distribution system resilience enhancement via mobile emergency generators[END_REF] relaxes the power flow balance constraint at each bus by allowing to shed unsatisfied demand. Constraint [START_REF] Anokhin | Mobility-as-a-service for resilience delivery in power distribution systems[END_REF] guarantees that the power flow on transmission lines do not exceed their capacities if they function. Constraint (32) enforces the amount of power flow on the damaged lines to be 0. Constraints ( 33)-( 34) denote Kirchhoff's power flow equations, where power flow are limited by lines' susceptance and the phase angle difference between the two end buses. It is necessary to include the big-M in the equations. If the constraint is directly written as

B i,j • (θ i,t -θ j,t ) = P i,j,t • (1 -u i,j ):
when the line's status is not switched and in service (i.e., u i,j = 1), this equation works normally; when the line is switched off (i.e., u i,j = 0), the phase angle between the two end buses of this line will be enforced to be 0, which is not logical for the power flow in the network. Constraint [START_REF]International Energy Agency (IEA), Global ev outlook 2020[END_REF] limits the generator ramp between two successive periods. Constraint [START_REF] Adderly | Electric vehicles and natural disaster policy implications[END_REF] gives the lower and upper boundaries of the amount of base load that can be shedded at each bus. Constraint [START_REF] Feng | Can we evacuate from hurricanes with electric vehicles?[END_REF] ensures that the flow generated by generators is within their capacity. Constraint [START_REF] Yperman | The link transmission model for dynamic network loading[END_REF] limits the number of lines that can be switched. Constraints ( 39) -( 40) state u i,j and LS dc j,t are binary decision variables.

Reconfiguring traffic-power system

In this subsection, we assume that the independent transportation network and PN are taken over by an emergency response department. Thus, it integrally operates and reconfigures the traffic-power networks to minimize the total system performance loss.

In this situation, the EV charging load p dc j (t) at each bus becomes a decision variable, which can be calculated by the following equation:

p dc j (t) = ∑ a∈M (j) ∑ s∈N S ∑ c∈C ∑ e∈Ec p ev a [U E s,e a,c (t) -V E s,e a,c (t)] (41)
where M (j) is a mapping from bus set P N to charging links set A C , which specifies the connection between buses in a power system and charging links in a RN. Since the traffic-power system is integrally operated, the charging locations and times of EVs can be flexibly arranged to assist minimizing the objective. Therefore, it is not necessary to have variable LS dc j,t anymore to control whether the EV charging load is shedded or not. Eq. ( 30) is rewritten as follows:

p g j,t + ∑ i∈Γ -(j) P i,j,t - ∑ k∈Γ + (j) P j,k,t = p b j,t -LS b j,t + p dc j,t , ∀j ∈ P N ( 42
)
The whole problem is formulated as follows:

min z∈Ψ P I = ∑ s∈N S ∑ t∈T ∑ a∈A S [DG s a (t) -U G s a (t) + ∑ c∈C ∑ e∈Ec (DE s,e a,c (t) -U E s,e a,c (t))] • ϕ + ∑ t∈T c b j • LS b j,t (43)
where Ψ = {z| s.t. ( 11) -( 27), ( 31) -( 39) and ( 41) -( 42)}.

In each time period, there are expected demand E(t) and unsatisfied demand ∆E(t) in the system. The following equation is employed to measure the system performance P (t) [START_REF] Fang | Optimum post-disruption restoration under uncertainty for enhancing critical infrastructure resilience[END_REF]:

P (t) = E(t) -∆E(t) E(t) ( 44
)
where 0 ≥ ∆E ≥ E. This equation can be understood as the percentage of demand that can be satisfied in the system in period t.

In the studied traffic-power system, the expected demand includes the all vehicle types traffic demand over all OD pairs and base electricity demand over all buses, which is formulated as follows:

E(t) = ∑ s∈N S ∑ a∈A S [DG s a (t) + ∑ c∈C ∑ e∈Ec DE s,e a,c (t)] • ϕ + ∑ j∈P N c b j • p b j,t (45)
where time value ϕ and shedding load cost c b j are used so that the system performance of PNs and RNs have the same physical dimension, thus, additivity. It can be understood as if these demands are satisfied, the corresponding amount of money can be saved.

The only difference between ∆E and Eq. ( 43) is without summation over time. Eq. ( 46) is rewritten as follows:

P (t) = ∑ s∈N S ∑ a∈A S [U G s a (t) + ∑ c∈C ∑ e∈Ec U E s,e a,c (t)] • ϕ + ∑ j∈P N [p b j,t -LS b j,t ] • c b j ∑ s∈N S ∑ a∈A S [DG s a (t) + ∑ c∈C ∑ e∈Ec DE s,e a,c (t)] • ϕ + ∑ j∈P N c b j • p b j,t (46)

Case study

The modified IEEE 14-bus system is used as the PN part in this study. It is preventive of small-size power systems and the original IEEE 14-bus test case is a portion of the American electric power system (in the Midwesten US) [START_REF] Washington | Power systems test case archive[END_REF]. There are 14 buses and 20 transmission lines, and the detailed data can be found in Ref. [START_REF] Fang | Optimizing power system investments and resilience against attacks[END_REF]. The transportation network part is illustrated by a partial highway network in North Carolina (NC), U.S., which is shown in Figure 3. Figure 2(a) shows the locations of EV charging stations within this area and the geographic data of the highway network are collected from Google map. This partial highway network is abstracted into an approximated topology network as shown in Figure 2(b). The number along the link is the link ID. In spite of its relatively small scale, the system is complete enough to illustrate the proposed models. There are 9 fast-charging stations in the studied highway network and their connections to the served buses are listed in Table 1. The data used in this studied is detailed in Appendix A.

The proposed model is illustrated by solving and analyzing the following hypothetical scenario: it is reported that links 4, 17, 19 in the highway network and lines 2-3, 2-4, 7-8 in the PN are destroyed, and they cannot provide services normally. The problem that the emergency response department face is how to reconfigure and operate both the traffic and power systems, so that their performance loss caused by the disruption can be minimized during the peak hours (i.e., 17:00-18:59).

All of the experiments have been run on a computer with an Intel Core i7-8700 3.2-GHz CPU with 32 GB of RAM. All of the problems have been solved by the commercial software IBM ILOG CPLEX (version 20.1.0.0).

The impact of the different response resource level

In this subsection, 5 different resource levels are investigated: N h = N u = 0, 1, 2, 3, 4, 5. Figure 3.1 shows the system performance level evolution over the studied time horizon under different resource levels. Time step = 0 denotes the occurrence of the reconfiguration. In this period, it can be found that the system performance levels are different under different resource levels. It is because that once the transmission network topology is reconfigured, the effects (i.e., the shedded load) are influenced immediately. Whereas, the effects of reconfiguring highway networks are taken into account in the following time steps, since the delay is resulted from the time required from origins to destinations for vehicles. The performance level denotes the percentage of the total demand is satisfied. As expected, the system performance increase with the resource levels. If the response resource level increase from 0 to 2, the system performance can be largely increased from 76.58% to 86.26%. After that, the marginal economic benefit of additional response resource reduces as the number of links reversing and lines switching raises. This is also can be found in Figure 3.1. When the resource level is 2, the nominal cost of both highway networks and transmission networks reduces largely.

It also shows the effectiveness of reconfiguring network topology during the restoration period. Table 3.1 shows the reconfiguration solutions of links in the RNs and that of lines in the PNs. From the third to the fifth columns, they represent the number of vehicles, GVs and EVs arrived at destinations in the end of the studied horizon. The last column represents the total charging demand during the studied horizon. The Third column in Table 3.1 shows that the optimal set of the switched lines in low resource level scenarios is not necessarily a subset of the switched lines in high resource level scenarios. For instance, line 4-7 is switched off when resource level is 1, whereas lines 4-9 and 7-9 are switched off when resource level increase to 2. However, this is not applied in RNs in this example. There could be two reasons resulted from the traffic demand distribution: 1) The used gravity model generates high traffic demand between two cities where their distance is short and pop- ulation is large. This may cause high traffic volume on some certain two-way road sections; 2) To model the directional differences of traffic volumes, the direction of traffic demand between two cities is randomly selected. It could make the bidirectional high traffic volumes become one-way high traffic volumes on some road sections. Therefore, once these links with high traffic volumes are damaged (e.g., links 17 and 19), they may alway have priorities to be restored so that the system loss can be minimized. Moreover, when there are large volume differences between two opposite links, the link capacity can be greatly improved by reversing the link with less volumes (e.g., links 5 and 26). Less nominal cost of system performance loss and higher resource level do not mean more vehicles can arrive the destinations during the studied period. For example, Figure 3.1 shows the nominal loss cost of resource level being 2 is less than that of resource level being 1. However, Table 3.1 shows that there are also less arrivals when the resource level is 2 than when the resource level is 1. It is because the vehicles arrive their destinations earlier when the resource level is 2 than when the resource level is 1. There is a trade-off between the number of arrivals and their travel time for RNs.

Different EV penetration levels and decision environments

Without loss of generality, both the maximum number of lines can be switched and links can be reversed are set to be 3 (i.e., N u = 3 and N h = 3). When RNs and PNs independently optimize their restoration plans, we assume that the operators of RNs share their temporal and spatial charging demand with the PNs operators first at the beginning of the restoration horizon and they no longer change their plans. This situation can be regarded as the unmanaged charging demand scenarios from the PNs operator's perspective. In this case, the PNs operators have to satisfy all EV charging demand and only the base electricity load can be shedded when they optimize their * The solution is same to the interdependent environment restoration plans. Figure 3.2 shows the traffic-power systems performance evolution over restoration horizon under different EV penetration levels. Table 3 shows the benefit of line switching and link reversing in terms of system performance loss for different EV penetration levels and different decision-making environments. As shown in Figure 3.2, the traffic-power system performance decreases as the EV penetration increases. When EV penetration increase from 0% to 100%, the nominal total costs of the traffic-power system increases from $773300 to $1133009, leading to a 46.7% increase in costs. Extra charging time needed for EVs comparing to GVs and the limited chargers are the main reasons of this result. When the EV penetration is equal to or less than 50%, the reconfiguration solutions are stable for both the traffic-power systems and the independently optimized RNs. In this situation, links 5, 104 and 117 in RNs are reversed. Lines 4-7, 4-9 and 6-13 are always switched off, when the restoration plans of PNs are coordinately optimized. When there are no EV in RNs, no matter interdependently or independently plan the restoration of RNs and PNs, both of them have the same nominal total system performance costs. When EVs arise in RNs, the nominal total costs of interdependently planing the traffic-power system are lower than that of independently planing them. It shows the added values of coordinately optimize the two networks. The total charging demand over the studied horizon in the last column in Table 3 shows the difference between coordinately managing the EV charging demand and independently managing them from the perspective of the RNs' operator. The coordinately scheduled spatiotemporal charging demand is a trade-off between RNs and PNs in terms of their individual system performance loss.

Conclusion

In this paper, the mathematic models of independently reconfiguring road networks (RNs) and power networks (PNs) are proposed to minimize the system performance loss during the restoration period. In both networks, system performance loss was measured by the unsatisfied demand, i.e., cumulative unsatisfied gasoline vehicles (GVs) and electric vehicles (EVs) traffic demand for RNs and cumulative shedded electricity load for PNs over the studied period. For RNs, the proposed model was to solve the system optimal dynamic traffic assignment problem considering the characteristics of EVs and fastcharging stations (FCCSs). These characteristics include driving range (battery capacity) and state of charge (SoC) of EVs, and physical constraints in FCSs, such as number of chargers and charging power. Moreover, a two-stage mixed integer optimization model was proposed to minimize the integrated system performance loss duration the restoration period. In the tactical level, reconfiguration solutions for the traffic-power systems are solved to enhance the system resilience after a disruptive event. In the operational level, an integrated traffic-power systems model was proposed to describe the dynamic interdependency between the PNs and RNs networks. The two networks were coupled through the spatial and temporal EV charging demand, which is coordinately managed in the proposed model. A partial highway network in North Carolina (NC), USA and modified IEEE 14-bus system were used to illustrate the proposed methods. The results showed that: 1) applying reconfiguration strategies on PNs and RNs during restoration period could effectively improve the system performance after disruptions; 2) the system performance loss could be largely mitigated by coordinately reconfiguring the traffic-power systems and managing EV charging demand, comparing to independently plan the reconfiguration solutions for PNs and RNs; 3) with the increasing of EV penetrations, the efficiency of the RNs decreased significantly, which hinted that more FCSs needed to be deployed in this area. The proposed models could be employed to provide effective reconfiguration solutions (i.e., links reversing in RNs and lines switching in PNs) for trafficpower systems during restoration period to enhance the system resilience. Operational solutions (i.e., system optimal dynamic traffic assignment and optimal power flow distribution) could be served as a benchmark to mange thee traffic-power flow and EV charging demand.

This work can be extended in several directions: 1) Replacing system optimal by user equilibrium in dynamic traffic assignment problem is valuble but challenging for the traffic-power systems model, although such model may not be suitable to apply into emergency response problem. It is because satisfying user equilibrium conditions mean more complicated optimization model and extremely expensive computational cost. 2) Including mobile energy storage systems (MESSs) into emergency response strategies could be an effective way to improve the resilience of RNs. However, how to integrate MESSs into the traffic-power system model still need more efforts; 3) EVs are assumed to only replenish batteries in FCSs, in this paper. Bidirectional transmission electricity service in FCSs can be considered to more efficiently operate the traffic-power systems and strengthen their resilience.

Appendix A. Data discription

A partial highway network in NC, USA is shown in Figure 3. The used parameters of this studied network are listed in Tables A. [START_REF] Ganin | Resilience and efficiency in transportation networks[END_REF] and A.5. The node ID, its corresponding town or city name and its population within this area are listed in Table A.6. The cities or towns attached source-sink nodes are those whose population is more than 11000. According to their geographic distances among these nodes and their population, the gravity model is used to generate the daily traffic demand. The generic form of gravity model [START_REF] Ren | Predicting commuter flows in spatial networks using a radiation model based on temporal ranges[END_REF] is usually written as f od = P α o P β d /D γ od , where P o and P d are the population sizes of origin a and destination d, D od is the shortest distance between them, α, β and γ are fitting parameters. We set α = β = 0.92 and γ = 1, in this study. To consider the worst-case scenario, the traffic volumes at 17:00 and 18:00 are adopted, which are the peak and account for approximately 15.3% of the daily traffic, in the basic time-of-day patterns [START_REF] Hallenbeck | Vehicle volume distributions by classification[END_REF]. The traffic volumes usually show the directional differences and it is difficult to get the applicable statistics for time-of-day travel by direction for each O-D pair [START_REF] Hallenbeck | Vehicle volume distributions by classification[END_REF]. For simplicity, only one direction is randomly selected for each O-D pair and traffic volumes in the other direction is ignored. The obtained traffic demand is shown in Table . A.7. According to Ref. [START_REF] Administration | Hourly electric grid monitor[END_REF], the electricity demand in U.S. has the similar peak hours with traffic volumes and their demand does not change a lot during this period. For simplicity, it is assumed that the base load at each bus is constant during this period and follows the standard test data [START_REF] Fang | Optimizing power system investments and resilience against attacks[END_REF].

Fig. 2 .

 2 1.2 shows how the contraflow assists increasing the throughput of the network after disruptions. Assuming that there are 20 vehicles per minute starting from nodes O to D and 10 vehicles per minute from nodes D to O. Node M serves as transshipment. The number along each link represents the time required to traverse the link at a free-flow speed. Fig. 1(a) shows that there are 20 and 40 vehicles arriving at nodes O and D, respectively, after 6 minutes, when each link works normally. If the link from nodes O to D fails, the arrivals on node D decrease to 20 vehicles, as shown in Fig. 1(b)

Figure 1 :

 1 Figure 1: Contraflow illustration: (a) Normal condition (b) After disrutption (c) After reconfiguration

Figure 2 :Figure 3 :

 23 Figure 2: The studied highway network 34

Figure 4 :Figure 5 :

 45 Figure 4: Nominal cost for the studied traffic-power systems under different resource level

Table 1 :

 1 Connections between charging links and Buses

	Charging link Bus N C a (t)
	301	2	30
	302	3	45
	303	4	30
	304	5	30
	305	6	30
	306	7	15
	307	8	30
	308	9	15

Table 2 :

 2 Solutions under different resource levels

	Resource levels	ha = 1	u i,j = 0	Vehicles	GVs	EVs	Charging demand (MW)
	0			19091	17336 1755	182.8
	1	117	4-7	22142	20322 1820	148.8
	2	104,117	4-9,7-9	21573	19512 2061	165.68
	3	5,104,117	4-7,4-9,6-13	21987	19656 2331	207.68
	4	5,26,114,117	4-7,4-9,1-2,9-14	22005	19669 2336	222.88
	5	5,25,26,114,117 4-7,4-9,1-2,6-12,13-14 21999.5	19656 2343.5 201.28

Table 3 :

 3 Solutions under different EV penetration levels and decision environments

	EV Penetration	Environments	ha = 1	u i,j = 0	Total cost ($)	Cost for RNs ($)	Cost for PNs ($)	Vehicles	GVs	EVs	Charging demand (MW)
	0%	Interdependent Independent	5,104,117 -*	4-7,4-9,6-13 4-9,7-9,13-14	773300 773300	729300 729300	44000 44000	30540 22142	30540 30540	0 0	0 0
	25%	Interdependent Independent	5,104,117 -	4-7,4-9,6-13 4-7,4-9,9-14	821260.15 824392.15	775340.15 770192.15	45920 54200	26933.5 21987	25346 19656	1587.5 2331	166.4 255.2
	50%	Interdependent Independent	5,104,117 -	4-7,4-9,6-13 4-7,4-9,13-14	894274.2 897389.5	845226.2 843069.5	49048 54320	21987 21999.5	19656 19656	2331 2343.5	207.68 294
	75%	Interdependent Independent	11,109,117 -	4-7,4-9,6-13 4-7,4-9,13-14	994669.5 998,262	945509.5 943462	49160 54800	13980 14430	11325 10437	2655 3105	218.4 318
	100%	Interdependent Independent	22,102,117 117	4-7,4-9 4-7,4-9,13-14	1133009 1134561.5	1083849 1081801.5	49160 52760	2822.5 3272.5	0 0	2822.5 3272.5	210 243.6

Table A .

 A 4: Parameters of the studied highway network Link ID Start End ν a β a ρ a Type L a k jam If a /Of a Lanes Table A.5: Parameters of the studied traffic-power system Table A.6: Population of the towns and cities

		Table A.7: O-D pairs and their traffic demand
	Parameters Link ID Node ID Demand Link ID Node ID Demand Values
	1/101 2/102 3/103 4/104 5/105 6/106 7/107 8/108 9/109 10/110 4/7 2/1 130 2/3 129 3/8 130 1/5 134 2/5 130 2/6 131 3/4 5/6 4/6 11/111 4/8 136	1/2 203 3/2 202 8/3 206 5/1 202 5/2 205 6/2 202 4/3 6/5 6/4 7/4 8/4	5 10 5 G v f (m/h) 6460 136 3 6 3 G k jam (veh/m) 5700 129 4 8 4 G δ (min) 5500 133 3 6 3 G q max (veh/h/lane) 2500 65 204 13910 8346 214 208 5564 6 203 8346 5380 131 205 3 6 3 G 8346 p ev a (kW) 80 3720 133 204 3 6 3 G 8346 η (kMh/mile) 0.4 3560 131 201 1 2 1 G 1391 1 2 1 G 2782 3 6 3 G 4173 2 4 2 G 2782 3 6 3 G 4173 Initial EL of EV 3 α t a (ELs/δ) 3 E c 10 C 1 ϕ ($/h) 13	500 620 500 620 250 460 500 460 500 460 500 400 250 500 250 250 250	2 2 1 2 2 2 1 1 1 2 1
	12/112 6/7		7/6		5 10 5 G	13910	500	2
	13/113 6/7		7/6		5 10 5 G	6955	250	1
	14/114 7/8 Node ID Name	8/7	2 4 2 G Population Node ID Name 5564	500	2 Population
	15/115 7/8 1 Zebulon	8/7	4526	2 4 2 G 2	2782 Rocky Mount 250	1 56650
	16/116 1/10 3 Tarboro	10/1 11255 4 8 4 G 4	11128 Pinetops	500	2 1351
	17/117 10/14 14/10 3 6 3 G 5 & 6 Wilson 49436 7	8346 Farmville	500	2 4695
	18/118 5/15 8 Greenville 86142 14/5 5 10 5 G 9	6955 Kenly	250	1 1344
	19/119 11/14 14/11 2 4 2 G 10 Raleigh 418099 11	5564 Selma & Smithfield 17901 500 2
	20/120 5/9 12 Goldsboro 35609 9/5 2 4 2 G 13	5564 Snow Hill	500	2 1611
	21/121 6/9 14 Clayton	9/6	2 4 2 G 16529	5564	500	2
	22/122 9/11		11/9	2 4 2 G	5564	500	2
	23/123 11/9		9/11	2 4 2 G	5564	500	2
	24/124 11/12 12/11 4 4 4 G	11128	500	2
	25/125 6/12		12/6	4 4 4 G	11128	500	2
	26/126 12/13 13/12 3 6 3 G	4173	250	1
	27/127 7/13		13/7	2 4 2 G	2782	250	1
	29/129 2/201 201/2 0 0 0 S/R inf	inf
	30/130 10/202 202/10 0 0 0 S/R inf	inf
	36/136 5/203 203/5 0 0 0 S/R inf	inf
	31/131 11/204 204/11 0 0 0 S/R inf	inf
	32/132 12/205 205/12 0 0 0 S/R inf	inf
	33/133 14/206 206/14 0 0 0 S/R inf	inf
	34/134 8/207 207/8 0 0 0 S/R inf	inf
	35/135 3/208 208/3 0 0 0 S/R inf	inf


Hui Peng
email: davidpeng1749@gmail.com

Double Slit Experiment Still Has Much to Offer (1) ---2D-Cross-Double Slit Experiments

Keywords: double slit experiment, cross-double slit experiments, wave description, wave interpretation

Young's one-dimension-double-slit experiment represents the basic mystery of quantum world.

Although it was performed over 200 years ago, the double slit experiment still has much to offer. To explore the mystery, in this article, the 1D-double slit experiments have been extended to 2D-cross-double slit experiments. It is a challenge to explain those experiments consistently.

Introduction

What is light made of, waves or particles or kind of both? This basic question has fascinated physicists since the early days of science. There was an intense debate between the particle and wave hypothesis of nature of light. Descartes (1637) and Newton (1704) developed the particle theory of light. Huygens (1678) proposed the wave theory of light. Young (1801) performed the first double slit experiment [START_REF] Ananthaswamy | Through Two Doors at Once[END_REF] [START_REF] Robinson | The Last Man Who Knew Everything[END_REF], which supported Huygens' wave theory of light. The concept of nature of light changed dramatically again. Hertz observed the photoelectric effect (1887). In 1902, Lenard observed that the energy of individual emitted electrons increased with the frequency of light. In 1905, Einstein showed theoretically that the photoelectric effect could be explained by using the concept that light is made of particles, photons. Quantum physicists postulated the wave-particle duality to try to interpret and unify those "wave" and "particle" phenomena of light. In 1928, Bohr postulated complementarity principle [START_REF] Bohr | Complementarity: Bedrock of the Quantal Description[END_REF], which states: when a photon is observed, it behaves either as a particle or as a wave, but both aspects are never observed simultaneously. The nature of photons really puzzled Einstein. He wrote (1951) to M. Besso: "All these 50 years of conscious brooding have brought me no nearer to the answer to the question: What are light quanta?" [START_REF] Rashkovskiy | Is a rational explanation of wave-particle duality possible[END_REF]. Feynman (1956) summarized the significance of Young's double slit experiment as that it "a phenomenon […] has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]" [START_REF] Feynman | The Feynman Lectures on Physics[END_REF]. A. Jogalekar explained (2013): "Yet we have no clue how any of the fundamental facts of quantum mechanics including wave-particle duality, […], or the double-slit experiment, that disarmingly simple setup which […]-actually work."

Young's double-slit apparatus is one-dimension and creates 1D-interference pattern. The experimental result was interpreted as that each photon has arrived by both slits at the same time and created interference pattern. However, the light is always found to be absorbed at the screen at discrete points, as individual particles; the interference pattern appears via the varying density of these particle hits on the screen. There are arguments about the mechanism behind the phenomena [START_REF] Gottlied | [END_REF]. A variety of experiments have been proposed and performed, such as using entangled photons [START_REF] Kim | [END_REF] [START_REF] Kaiser | [END_REF], and placing a light source on a satellite and measuring on the ground [START_REF] Vedovato | [END_REF]. The double-slit experiment has also been performed with varying numbers of slits, e.g., parallel-multi-slits [10].

The double-slit experiment still has much to offer (Pinner).

To study the double slit experiments, we present a series articles showing varieties of the novel double slit experiments and including the following topics:

(1) 2D-cross-double slit experiments (this manuscript);

(2) which way 2D-cross-double slit experiments: violating complementarity principle [START_REF] Peng | Double Slit Experiment Still Has Much to Offer (2): Extended-Which-Way Experiments---Violating Complementarity Principle[END_REF];

(3) delayed choice 1D-double slit/2D-cross-double slit experiments: supporting causality [START_REF] Peng | Double Slit Experiment Still Has Much to Offer (3): Extended-Delayed-Choice Experiments---Supporting Causality[END_REF];

(4) comprehensive 1D-double slit/2D-cross-double slit experiments with shields/blockers contacting/near detector: violating wave description;

(5) comprehensive 1D-double slit/2D-cross-double slit experiments with shields/blockers contacting/near diaphragm of double slit: violating trajectory theory; [START_REF] Gottlied | [END_REF] experiments: rotating 1D-double slit/2D-cross-double slit around one axis: showing dynamically curved, expanded and inclined patterns; [START_REF] Kim | [END_REF] experiments: rotating 1D-double slit/2D-cross-double slit around two axes sequentially: particle nature, straight pattern and curved pattern co-exist. The observations of above experiments provide the comprehensive phenomena/information/data to study the mystery of the double slit experiments and the nature of light/photons. Furthermore, some phenomena may be applicable in daily life.

2D-Cross-Double Slit Apparatuses

The diaphragms (Figure 1) contain double slit crossing double slit and double slit crossing triple-slit, with either the equal spacing or different spacing between slits. Hereafter, we referred all as the "cross-double slit". The configurations of experiments are: a laser source points to a slide two feet away, and a detection screen is 20 feet away from the slide. We draw briefly illustrations of cross-double-slits on the left of Figure 3 to Figure 9, and experimental observations on the right correspondingly.

Experiment-1 (Figure 3): 2D-cross-double-slit and its 2D-cross-interfernce pattern [START_REF] Peng | Cross-Double-Slit Experiment and Extended-Mach-Zehnder Interferometer[END_REF] [START_REF] Peng | Cross-Double-Slit Experiment and Which-Way-Cross-Double-Slit Experiment[END_REF]. The experiments may be performed with two kinds of sources respectively. Observations (Figure 3b): a 2D-cross-interference pattern.

(1) 1D-horizontal interference pattern and 1D-vertical interference pattern.

(2) Two 1D-interference patterns are created independently.

Discussion-1 (Figure 3c): For this aim, let us enlarge the intersection of double-slit-AB and double-slit-CD, as shown in Figure 3c. The intersection has the shape of square.

Four small dots, denoted as "a/c", "b/c", "b/d" and "a/d", represent the four corners of the intersection. Except the double-slit-AB and double-slit-CD, all of other area (including the shadow square) is the opaque area.

The intersection of slit-A and slit-C is denoted by a "Red square".

We refer the portion of slit-A, slit-B, slit-C and slit-D inside the intersection as "inside-portion", while the portion outside the intersection as "outside-portion".

Let's assume the cross-section of the laser beam is a circular with a diameter that is slightly larger than the dimension of the intersection area, which is summation of both the spacing, which is 0.25 mm (for our cross-double-slit), between two slits of a double-slit, and two times of the width of a slit, which

. (a) (b) (c)
.

is 0.04 mm. The majority of photons will land on the opaque area, mainly the shadow square, while minority of photons will pass through both double-slit-AB and double-slit-CD.

According to the regular interpretation of 1D-double-slit experiment, photons passing through the inside-portion of slit-A (slit-B) will interfere with photons passing through the inside-portion of slit-B (slit-A), while photons passing through the inside-portion of slit-C (slit-D) will interfere with photons passing through the inside-portion of slit-D (slit-C).

Since emitted photons have the same optical parameters (such as wavelength, phase, polarization), the challenge to the regular interpretation of 1D-double-slit experiments is to explain the following questions:

(1) Why do photons passing through the inside-portion of slit-A (slit-B) interfere with photons passing through the inside-portion of slit-B (slit-A), but not interfere with photons passing through the inside-portion of slit-C or of slit-D, even when which are closer. Similarly, why do photons passing through the inside-portion of slit-C (slit-D) interfere with photons passing through the inside-portion of slit-D (slit-C), but not with photons passing through the inside-portion of slit-A or of slit-B, even when which are closer.

(2) When the photons pass through the "Red square", there are two questions:

(2-1): how the photons determine whether they are passing through slit A or slit C?

(2-2): how the photons determine which photons they should interfere with, the photons passing through slit-B or the photons passing through slit-D.

(2-3): whither the photons passing through the "Red square" would interfere with each other?

If one treats photon as a "person" as in the interpretation of delayed-choice-double-slit experiment, then the challenges are to explain the following questions:

(1) How do photons "sense" which slit they would pass through;

(2) How do photons "sense" which portion, i.e., the inside-portion or the outside-portion, they would pass through;

(3) how do the photons passing through the inside-portion of a slit "find" photons passing through the inside-portion of its "paired slit" to interfere and create the interference-pattern accordingly;

Discussion-2: About the interpretation of the behavior of photons emitted one at a time:

An emitted photon has a larger probability landing on the opaque area, while has a smaller probability landing on the cross-double-slit, i.e., double-slit-AB and double-slit-CD.

In this experiment, a photon needs to play as "photons" of twelve: 4 of them pass through the inside-portions of slit-A, slit-B, slit-C, and slit-D respectively; and 8 of them pass through the outside-portions of slit-A, slit-B, slit-C, and slit-D respectively, and create simultaneously different interference pattern correspondingly.

How does a photon "sense" how many slits it would pass through when it arrives the inside-portions of an apparatus with multi-slits simultaneously? For this experiment, there are 4 slits, however, for the following experiments, there are more than four slits. The most difficult part is to explain: how does the photon passing through the inside-portion of slit-A interfere with itself passing through the inside-portion of its paired slit-B, but not with itself passing through the inside-portions of slit-C and or slit-D, to create an interference pattern.

According to the regular wave interpretation, in regular 1D-double-slit experiments interfering waves are in the same plane; on the other hand, in 2D-cross-double-slit experiment interfering waves are in two perpendicular planes. The regular interpretation seems impossible to explain how a photon travels through both the double-slit-AB and the double-slit-CD at the same time and somehow interferes with itself accordingly.

It is a challenge to interpret how a photon "senses": (1) which slit it passes through; (2) which photons it will interfere with, then to create interference patterns accordingly.

The above discussion may be applied to all of the cross-double slit experiments.

Experiment-2 (Figure 4): Tilt double slit crossing double slit and pattern. Discussion: The discussion of experiment-1 is applicable for this experiment. In addition, the photons have to "measure" the angle of the tilt-double slit to form the pattern accordingly. And the photons need to judge which slit they will pass through and behave accordingly.

Experiment-3 (Figure 5): Three double slits crossing and pattern.

The angle between the tilt double slit and the vertical double slit is 45 degrees. Experiment-4 (Figure 6): Three double slits crossing and pattern. The angle between the tilt double slit and the vertical double slit is 15 degrees.

Experiment-5 (Figure 7): Three double slits crossing and pattern.

Figure 7 Three double slits crossing (3) Figure 8 Three double slits crossing (4)

Experiment-6 (Figure 8): Three crossing double slits and pattern. The tilt angles are 15 and 30 degrees respectively.

Experiment-7 (Figure 9): Four crossing double slits and pattern. The intersection is octagon. Experiment-10 (Figure 12): Rotation Invariance of double slit/cross-double slit.

The combination of Figures 9, 10 and 11 suggests that the double slit and cross-double slit have rotation invariance around their normal vectors. If we increase the number of double slits that intersect at the same spot, the shape of the intersection will approach a circular disc. To rotate either a double slit or cross-double slit around its normal vector, each slit is tangent to the intersection and forms disc-3, which is surrounded by ring-2.

The final pattern can be obtained by applying an apparatus of disc-3 with a diameter that is equal to the spacing between two slits, and ring-2 of the width that is equal to the width of a single slit.

Figure 12 indicates that double slit and cross-double slit have rotation-invariance around their normal vector.

The "task" of the photon becomes more and more difficult when the number of the double slits crossing to each other increases. Even, the photon has to "know" whether it pass through a circle (Figure 12) or a regular cross-double slit.

Double Slit Crossing Double Slit: Different Spacing

Experiment-11 (Figure 13). Double slit crossing double slit of different spacing and pattern. The photons need to "Know" the spacing of the double slit they pass through to behave.

Double slit Crossing Triple-Slit Experiments: equal spacing

A regular triple slit and its pattern are shown in Figure 15.

Figure 15 Triple slit and its pattern Experiment-13 (Figure 16): Double slit crossing triple slit and pattern. All of spacing between slits are 0.25 mm.

Figure 16

Figure 17 Experiment-14 (Figure 17): Double slit crossing triple slit and pattern. Spacings between slits are 0.75 mm.

Experiment-15 (Figure 18): Double slit crossing tilt triple slit and pattern. Spacings between slits are 0.25 mm.

Experiment-2.16 (Figure 19): Double slit crossing tilt triple slit. Spacings between slits are 0.5 mm.

The photons need to "know" whether they are passing through the double slit or the triple slit and the angle of the double slit and/or the triple slit to "decide" how to behave.

Double slit crossing triple-slit experiments: different spacing.

Experiment-17 (Figure 20): The spacing between two vertical slits is 0.5 mm, and the spacing between three horizontal slits is 0.25 mm.

Figure 20 Figure21

Experiment-18 (Figure 21): The spacing between two vertical slits is 0.75 mm, and the spacing between three horizontal slits is 0.25 mm.

Experiment-19 (Figure 22): The spacing between two vertical slits is 0.5 mm, and the spacing between two of three tilt slits is 0.25 mm.

Figure 22

Figure 23 Experiment-20 (Figure 23): The spacing between two vertical slits is 0.75 mm, and the spacing between three tilt slits is 0.25 mm.

The photons need to "know" the following to "decide" how to behave: (1) whether they are passing through the double slit or the triple slit; (2) the angle of the double slit and/or the triple slit; [START_REF] Bohr | Complementarity: Bedrock of the Quantal Description[END_REF] the spacing of the double slit and/or the triple slit.

4.Discussion and Conclusion

The regular double slit experiment is the mystery of the quantum mechanics. The cross-double slit experiments are more mysterious to be described and interpretated. The cross-double slit experiments not only challenge the regular description of the double slit experiments, but also provide comprehensive phenomena/information/data for exploring the mystery of the double slit experiments, studying the nature of light/photons and developing a theoretical model. Furthermore, some phenomena may be applicable in daily life.

Figure 1

 1 Figure 1 Diaphragms of 2D-cross-double slit

Figure 2

 2 Figure 2 Regular 1D-double-slit and its 1D-interference pattern The classical interpretation is that light wave passes through two slits simultaneously, interferes to each other and creates an 1D-interference pattern.

Apparatus

  (Figure3a): slit-A and slit-B pair up to form double-slit-AB, while slit-C and slit-D pair up to form double-slit-CD that intersects to the double-slit-AB at the angle of 90 0 .

Figure 3 .

 3 Figure 3. Two double slits crossing and its 2D interference pattern (1)Source-1: A laser source aims at the intersection and emits a beam of photons. The majority of photons will shine at the intersection, while the minority of photons will shine at outside of the intersection, including the portions of 1D-double-slit-AB and 1D-double-slit-CD outside of the intersection.Source-2:A laser source aims at the intersection and emits photons one at a time. The probability of a photon passing through the portions of double-slit-AB and double-slit-CD at the intersection is higher than that of a photon passing through the portion of double-slit-AB and double-slit-CD outside of the intersection.

Figure 4

 4 Figure 4 Two double slits crossing and its 2D interference pattern (2) Observation: (1) The vertical double-slit-AB creates the horizontal interference patter, while tilt-double-slit-CD creates the tilt-interference pattern; (2) two interference patterns intersect at an angle that is the same as the angle between 1D-double-slit-AB and 1D-double-slit-CD.

Figure 5 Figure 6

 56 Figure 5 Three double slits crossing (1)

Figure 9 Four double slits crossing Figure 10 Five double slits crossing Experiment- 8 (Experiment- 9 (

 91089 Figure 9 Four double slits crossing Figure 10 Five double slits crossing Experiment-8 (Figure 10): Five crossing double slits and 2D-pattern. The photon needs to "know" which double slit it would pass through to behave accordingly.Experiment-9 (Figure11): Six crossing double slits and 2D-pattern. The intersection is dodecagon.

Figure 11

 11 Figure 11 Six double slits crossing Figure 12 Rotation symmetry

Figure 13

 13 Figure 13 Two double slits crossing Figure 14 Three double slits crossing Experiment-12 (Figure 14): Double slit crossing double slit of different spacing and pattern.