A Formal Approach for Efficient Navigation Management of Hybrid Electric Vehicles on Long Trips

1 A F ormal Approach for Ef ﬁcient Na vigation Management of Hybrid Electric V ehicles on Long T rips Mohammad Ashiqur Rahman, Md Hasan Shahriar , Ehab Al-Shaer , and Quanyan Zhu Abstract Plug-in Hybrid Electric V ehicles (PHEVs) are gaining popularity due to their economic ef ﬁciency as well as their contrib ution to green management. PHEVs allo w the dri ver to use electric power exclusi vely for driving and then switch to gasoline as needed. The more gasoline a vehicle uses, the higher cost is required for the trip. Howe ver , a PHEV cannot last for a long period on stored electricity without being rechar ged. Thus, it needs frequent rechar ging compared to traditional gasoline-powered vehicles. Moreo ver , the battery recharging time is usually long, which leads to longer delays on a trip. Therefore, it is necessary to pro vide a ﬂexible navigation management scheme along with an efﬁcient recharging schedule, which allo ws the driv er to choose an optimal route based on the fuel-cost and time-to-destination constraints. In this paper, we present a formal model to solve this PHEV navigation management problem. The model is solved to pro vide a dri ver with a comprehensive routing plan including the potential recharging and refueling points that satisfy the given requirements, particularly the maximum fuel cost and the maximum trip time. In addition, we propose a price-based navigation control technique to achiev e better load balance for the trafﬁc system. Evaluation results show that the proposed formal models can be solv ed efﬁciently ev en with lar ge road netw orks. Index T erms Hybrid electric vehicle; navigation management; navigation control; formal model; satisﬁability . I . I N T RO D U C T I O N Plug-in Hybrid Electric V ehicles (PHEV) ha ve the potential of addressing contemporary and future en vironmental and economic challenges. In addition, they provide new possibilities for smart grid management and the integration of renewable energy sources into electricity networks [1]. A PHEV is typically equipped with a battery that can be fully recharged by connecting it to the power grid with an extension cord. The stored electric energy is used as alternati ve to traditional gasoline, providing economic and en vironmental beneﬁts. The cost for electricity to po wer a PHEV has been estimated to be less than one fourth of the cost of gasoline [2]. According to the typical battery capacity of an Electric V ehicle (EV), a driver is allowed to use electric energy exclusi vely for 30 to 50 miles of driving before switching to gasoline for longer trips [1]. Thus, an EV needs frequent recharging in longer trips. M. A. Rahman (marahman@ﬁu.edu) and M. H. Shahriar (mshah068@ﬁu.edu) are with the Department of Electrical and Computer Engineering, Florida International University , USA. E. Al-Shaer (ealshaer@uncc.edu) is with the Department of Software and Information Systems, University of North Carolina at Charlotte, USA. Q. Zhu (quanyan.zhu@nyu.edu) is with the Department of Computer and Electrical Engineering, New Y ork University , USA. 2 Moreov er , recharging takes more time than refueling. As a result, EVs suffer with excessi ve delays in long trips. Howe ver , unlike EVs, a PHEV can use gasoline when stored ener gy is fully consumed instead of frequently recharging the battery . Therefore, it is necessary to select the optimal route to the destination that provides an ef ﬁcient recharging and refueling schedule plan for a PHEV in order to satisfy different constraints. V ehicle dri vers usually ha ve constraints on driving time as well as on the fuel cost for a trip. There can be additional requirements like needing to pass through some speciﬁc locations ( i.e. , intermediate points of interest or via points). In this work, we model the PHEV navig ation management as a constraint-satisfaction problem. Satisﬁability Modulo Theories (SMT) [3] to implement this model and obtain a satisﬁable solution that includes the routing plan and the recharging and refueling points, satisfying the constraints on fuel cost, dri ving time, and intermediate points of interest. This work assumes that a service provider can execute this formal model to compute a navigat ion plan for a PHEV based on its driv er’ s requirements and the information about the road network, the gas and char ging stations, the gas and char ging prices, etc. A PHEV can obtain a navigation plan from the service provider at the beginning of its trip. That plan can be updated on-demand according to the latest information on traf ﬁc and charging stations. W e also propose a navigation control technique that adjusts the charging prices of the stations at each time slot and (indirectly) distribute the PHEVs, being moti vated by the lo wer prices, on the roads and in the charging stations. Unlike our previous work [4], where we addressed the PHEV navigation management problem on highways with only the battery recharging, in this paper we extend the model to provide a comprehensiv e navig ation management. This extended design models a generic road system with numerous distinguishing characteristics: location point as the unit of the road system, refueling and recharging requirements, ﬂexible amounts of recharging and refueling, time-varying queue lengths for waiting vehicles in charging stations, and time-v arying a verage trafﬁc speeds. W e also devise a mechanism of executing sev eral instances of the proposed model parallelly to solv e a problem. This parallelism signiﬁcantly increases the scalability of the solution. The e valuation results sho ws that it takes few seconds to solve the proposed model for a road system consisting of 1,000 location points and a proportional number of charging and gas stations. Our work contributes to environmental preservation, as it inherently minimizes gasoline use by efﬁciently managing the use of rechar ging and refueling installations. This paper is organized as follo ws: W e discuss the state of the art of PHEVs and the PHEV navigation management problem in Section II. W e present the formal model of the navigation problem along with an illustrativ e example in Section III. In the following section, we propose the navigation control technique and the corresponding model. Section V presents ev aluation results. In the following section, we brieﬂy discuss sev eral aspects of the proposed solution. In Section VII, we brieﬂy discuss the related work. Section VIII concludes the paper . I I . B A C K G RO U N D A N D M O T I V AT I O N This section presents necessary background of the navigation management problem for PHEVs. A. PHEVs for Long Distance T ravels EVs are gaining slo wly but steady popularity in the mark et. An estimated number of 361,307 of plug-in EVs , including hybrids, were sold in the United States (U.S.) in 2018 [5], [6]. Different models of PHEVs exist on the roads [1], [2]. These cars usually can operate in the range of 50 miles after a full charge as a pure EV . Due to the recent dev elopment in PHEV technology , few 3 well-advanced PHEVs are released to the market. For example, the Hyundai IONIQ Electric [7] can go more than 120 miles per charge and can be fully recharged in around 30 minutes. Though most of the cars run trips within short distances, around 20% of the cars in the U.S. travel more than 50 miles [8]. United States has a very large road network. According to the statistics recently published by the Federal Highway Administration, the U.S. interstate highway system alone has a total length of 47,714 miles [9]. 8 / 7 / 2 0 1 4 G o o g l e M a p s h t t p s: / / w w w . g o o g l e . co m / m a p s/ @ 3 5 . 6 5 5 3 1 1 , - 8 0 . 6 6 2 2 4 5 4 , 1 0 z.. 1 / 1 M a p d a t a © 2 0 1 4 G o o g l e 5 m i T r a f f i c , B i c y c l i n g , T e r r a i n , D i r e c t i o n s Figure 1. The map shows part of U.S. routes, particularly between two cities: Gastonia (Point 1) and W inston-Salem (Point 2) in North Carolina. The thick lines represent the roads. PHEVs are used for long-distance and short- distance trips. Since PHEVs cannot run for long distances with electric power only , the battery needs to be recharged to use electric power instead of using gasoline for further operation on long trips. Again, PHEVs need long rechar g- ing times, which leads to long waiting times at charging stations. Figure 1 shows a map of the major U.S. routes/roads ( i.e. , interstate highways along with country roads) between two cities, Gastonia and W inston-Salem, in North Carolina [10]. As it can easily be seen in the ﬁgure that there are se veral alternati ve navigation paths from the source (point 1) to the destination (point 2). Therefore, there is always a need for choosing a suitable route. For travel efﬁcienc y and comfortable dri ving, it is necessary to opti- mally plan a trip, i.e. , the routing path and the recharging and refueling schedule. There are few existing works related to the PHEV navigation management problem. For example, a fuel-efﬁcient navigation mechanism for con ventional gasoline-powered vehicles is dev eloped in [11], while an optimal rechar ging scheduling for EVs is proposed in [12]. Ho wev er , neither of these model the uses of alternativ e fuels (electricity and gasoline) by PHEVs to satisfy the constraints for cost-effecti veness and time-efﬁciency . In this paper , we assume a centralized system that provides a na vigation plan to a PHEV for a long trip that satisﬁes the PHEV’ s ( i.e. , its user’ s/driv er’ s) requirements. The service provider will ex ecute our navigation management model for synthesizing the navigation plan. Ho wev er , a PHEV can also run the model by itself, if it has a processor for executing an SMT solver . For navigation planning within the requirements, our model needs different static and dynamic information about the road network and corresponding trafﬁc. W e assume that all the charging stations and the PHEVs are connected to the service provider through a wired or wireless, infrastructure-based or infrastructure-less communication model ( e.g . , as shown in Figure 2). The stations and PHEVs provide necessary local information to the service provider . The provider updates the dynamic part of the information system frequently with the collected data. This dynamic information includes the current (real) status and future (predicted) status of the charging stations and the trafﬁc on the road (especially , the number of waiting vehicles in the stations and the a verage 4 W A N /I nte r ne t S e r vic e P r ovide r V e hic le C ha r ging S ta tion W a iting V e hic le s B a s e In fras t ru ct u r e Figure 2. The communication model for the navigation management system. trafﬁc speeds on the roads). The information speciﬁc to a future time is predicted from the past records and current status. Since trafﬁc can be very randomly distributed, spatially and temporally , the exact knowledge about the future may be different from the forecasted information. Ho we ver , a PHEV can frequently update the navigation plan from the service provider based on the latest information and the same or modiﬁed requirements. B. PHEV Navigation Management Pr oblem The goal of the navigation management problem is to ﬁnd an ef ﬁcient navigation plan on a road system under a number of constraints. Along with the basic objectiv e of reaching the destination, on the way to the destination, a vehicle may require passing through one or more intermediate points of interest, i.e. , via-points. There are usually dif ferent time and cost constraints. There can be a time-to-reach-destination (or simply time) constraint, which means the vehicle should reach the destination on or before a speciﬁed time. There may also be time constraints to reach the via-points. The fuel-cost (or simply cost) incurred in a trip depends on the type and amount of the consumed fuel. A hybrid vehicle can use either electricity or gasoline as fuel. W e assume that the vehicle does not use both kinds of fuel simultaneously , but rather sequentially . Hence, the cost is the summation of the price of electricity and that of gasoline consumed during the trip. The constraint on fuel-cost speciﬁes that the cost cannot exceed a given v alue. W e assume that a PHEV can recharge its battery at the char ging stations only . Charging prices and waiting queues are usually different at dif ferent stations. Due to the time-varying price model of power -grids, a particular station may have different prices at different time slots. The objectiv e of the PHEV na vigation management problem is to ﬁnd a dri ving routing plan from the source that potentially enables the vehicle to reach the destination, including the intermediate points of interest, within time and fuel-cost constraints. It was pro ven in [4] that the PHEV Navigation Management problem is NP-complete. 5 Link R oad Highway Location Point Figure 3. A route is modeled as a collection of small link roads, where a link road connects two location points. C. SMT Logic and the Solver W e use SMT to formalize our proposed navigation management models. SMT is a powerful logic tool that can solve constraint satisfaction problems arising in many div erse areas, such as software and hardware veriﬁcation, test-case generation, scheduling, planning, etc. [13]. SMT in volves determining whether a formula is satisﬁable or not. For example, the SMT instance with the following two constraints is satisﬁable with the assignments of x = 1 and y = 0 : ( x + y < 2) ∨ ( x − 2 y > 0) and x ≤ 1 An SMT instance is a ﬁrst-order logic formula, where some functions and predicate symbols hav e additional interpretations. In SMT , complex logics are replaced by ﬁrst order predicates/functions using a v ariety of underlying theories, including the theory of equality , linear arithmetic, difference logic, etc. An SMT solver searches for the solution(s) following an extended DPLL backtracking algorithm [14], [15]. Modern SMT solvers can check formulas with hundreds of thousands of variables and millions of clauses [13]. In this work, we solve the proposed model using Z3, an ef ﬁcient SMT solver , de veloped and managed by Microsoft Research [3], [16]. I I I . N A V I G A T I O N M A N AG E M E N T M O D E L In this section, we formalize the PHEV navigation management problem as a satisfaction of a number of constraints. A. Road Network The roads used for long trips often are highway and country routes. These routes connected to each other directly (through ramps) or through other highways, country , or local roads and form the road network. W e will use the term “routes” frequently to represent both highways and country/local roads in the road network. A route usually consists of a large number of location points (simply locations or points), at which it connects with other routes/roads, or there are establishments of gas stations or charging stations, including the source, destination, and various places of interest. Therefore, we can consider a route/road as a sequence of small roads, where each of these small roads links two points as shown in Figure 3. W e assume that all the roads are bidirectional. W e do not consider the overhead of any trafﬁc jam or signaling system on the routes, rather the effects of their existence are reﬂected in the a verage vehicle speeds on the roads. The gas and charging stations are deployed at different points. A PHEV can charge its battery at a point if there is a charging station, and then continue moving toward its destination. Similarly , if there is a gas station at a point, the vehicle can refuel. 6 The road network ( i.e. , the routes, locations, and the placements of the stations) is known. Dif ferent gas or char ging stations often have dif ferent prices for refueling or rechar ging. Different times of the day also hav e different prices for recharging. W e assume that a day is di vided into 24 hourly slots. The charging price remains the same within a slot while it may vary between the slots. Usually recharging a PHEV takes a substantially long time and, thus, there are often vehicles waiting for recharging in a station. The queue size of the waiting cars can be different at dif ferent times of the day . For example, the number of waiting vehicles at the end of the day is often larger than the number at midnight. It is worth mentioning that if a point has multiple charging stations, we consider a single charging station, av eraging their charging prices, queue lengths, and other parameters. W e do the same for the gas station. B. System Model In this subsection, we present the modeling of routes/roads, stations, the vehicle ( i.e. , a PHEV), and its user’ s requirements ( i.e. , user requirements). T able I shows the notations used in the modeling. Road Network. As already mentioned earlier , we deﬁne the road netw ork as a collection of routes/roads consisting of a number of location points and the connecting roads. Thus, a route/road is modeled as a sequence of points and the roads connecting them. A location point (also, simply , location or point) is denoted by a number , l . L ˆ l,l represents the road between location ˆ l to location l . D ˆ l,l is the distance (miles) between location ˆ l to location l . The average speed at which a vehicle mov es on the road is S ˆ l,l,t (miles/minute) during time slot t . The average speed comes from the speed limit of the road and/or the trafﬁc on the road. Char ging Stations. S l is a Boolean parameter denoting whether there is a charging-station at point l . It has different properties. The price of charging (per kWh) at a time slot t is denoted by P s l,t (dollars). The expected queue length of the waiting vehicles at time slot t is Qs l,t . If a particular vehicle is considered, it takes time T s l minutes to recharge its battery for each kWh charge. In order to estimate the waiting time in queue, it is assumed that each vehicle in queue takes ˆ T s l minutes on an average to recharge its battery . W e do not consider the number of charging outlets, since this can easily be reﬂected in the av erage time required for each waiting vehicle. Gas Stations. Similar to a char ging station, we also deﬁne se veral parameters for a gas station. G l denotes whether point l has a gas station ( G l ). W e deﬁne P g l as the price of each gallon of gasoline at this station. Since there are often more than one gas stations at a point and they often hav e different prices although very close to each other, we can consider all of these gas stations as a single station and the price as the average of their prices. W e deﬁne ˆ T g l as the av erage time of taking the gasoline. Since the gasoline pouring time per gallon is very small considering other overheads, we only consider an av erage time. Moreov er , we do not consider any waiting queue for a gas station, because such a queue is rare to occur at a gas station. The V ehicle. A plug-in hybrid vehicle V has different properties: the current location S v (an entrance point), the destination Dv (a point), the stored electric charge E v (kWh) and its price P v ($/kWh), the battery capacity C v (kWh), the electric energy consumption rate Re (miles/kWh), the stored gasoline Gv (gallon) and its price ˆ P v ($/gallon), the gasoline capacity ˆ C v (gallon), and the gasoline consumption rate Rg (miles/gallon). W e assume that an y vehicle driv es on a road at the speed limit of the road ( i.e. , S ˆ l,l,t ). 7 T able I M O DE L I N G P A R AM E T E RS Notation Deﬁnition T ype l A location point on the road network. Integer L ˆ l,l If there is a road between l and ˆ l . Boolean D ˆ l,l Length of road L ˆ l,l . Real S ˆ l,l,t A verage speed of a vehicle on road L ˆ l,l during time slot t . Real S l If there is a charging station at point l . Boolean P s l,t Price of charging (per kWh) at time slot t at station S l . Real Qs l,t Queue length of waiting vehicles at time slot t at station S l . Integer T s l Time for recharging of each kWh charge at station S l . Real ˆ T s l A verage time for recharging at station S l . Real G l If there is a gas station at point l . Boolean P g l Price of gasoline (per gallon) at station S l . Real ˆ T g l A verage time for taking the gasoline at station G l . Real S v Starting (or current) location of the vehicle. Integer Dv Destination point of the vehicle. Integer C v Capacity of the vehicle’ s battery . Integer ˆ C v Capacity of the vehicle’ s gasoline tank. Real E v (Initially) stored electric charge of the vehicle. Real P v Price of each kWh stored electric charge of the vehicle. Real Gv (Initially) stored gasoline of the vehicle. Real ˆ P v Price of each stored gallon gasoline of the vehicle. Real Re The number of miles that a vehicle can get per kWh electric charge. Real Rg Gas consumption of the vehicle for each mile. Real P g Price of gasoline per gallon. Real C p Cost constraint of the vehicle. Real C t Time constraint of the vehicle to reach the destination. Real C t l Time constraint of the vehicle to reach via point l ∈ I , I is the set of via points. Real X l Whether the vehicle reaches point l . Boolean S c l Whether station S l is selected for charging. Boolean C e l Stored electricity of the vehicle when it reaches point l . Real ˆ C e l (Effecti ve) stored electricity of the vehicle when it leaves point l . Real C g l Stored gasoline of the vehicle when it reaches point l . Real ˆ C g l (Effecti ve) stored gasoline of the vehicle when it leaves point l . Real P e l A verage price of each kWh of electric charge stored in the vehicle at point l . Real P g l A verage price of each gallon gasoline stored in the vehicle at point l . Real P l Cost spent by the vehicle to reach point l . Real T l Time spent by the vehicle to reach point l . Real S t l Time slot at which time T l falls. Integer User Requir ements. W e consider that the vehicle is required to satisfy two major constraints. The ﬁrst constraint is on the cost C p (dollars); that is, the price of the fuel consumed by the vehicle should be within a cost bound. The second constraint is on the trav eling time C t Dv (minutes); that is, the vehicle has to reach the destination D v within a time limit. There can be more constraints. The vehicle might need to go via some intermediate points denoted as a set I . Moreover , a vehicle might be required to reach a via-point within a time constraint ( C t l for l ∈ I ). There can be other arbitrary user requirements, e.g. , minimum electricity stored when the destination is reached. 8 C. V ehicle Routing Model The routing of a vehicle from a source to a destination is a sequence of moves from one point to another from the source to the destination. The cost and time incurred due to this sequence of moves must be within the cost and time constraints. W e assume that the v ehicle’ s primary energy source is the battery , i.e. , the electric char ge, as it costs signiﬁcantly less than the gasoline. Therefore, if the battery is out of charge, then gasoline is used as the energy source. 1) Modeling P arameters: The following parameters are used to model a vehicle’ s routing to a destination: • V isited point. X l denotes that the vehicle reaches point l , i.e. , it mov es to point l on route h . It is a boolean value. If the vehicle reaches the destination, the goal is achiev ed ( i.e . , no more movement is required). X ˆ l,l denotes that the vehicle mov es from ˆ l to l . • Char ging station selected for rec har ging. S c l denotes whether the vehicle recharges at the charging station located at point l . • Gas station selected for r efueling gasoline. Gc l denotes if the vehicle refuels at the gas station located at point l . • Stor ed electricity . C e l represents the stored electric charge of the vehicle at point l . • Effective stored electricity . ˆ C e l is the effecti ve stored electricity at point l . Obviously , ˆ C e l ≥ C e l . If the vehicle is recharged at the charging station at point l , it is greater than C e l . • Stor ed gasoline. C g l is the stored gasoline of the vehicle at point l . • Effective stor ed gasoline. ˆ C g l is the effecti ve stored gasoline at point l . Obviously , ˆ C g l ≥ C g l . If the vehicle is refueled from the gas station at l , it is greater than C g l . • A verage electricity price . P e l represents the av erage price of each kWh of electric charge stored in the vehicle at point l . • A verage gasoline price . P g l represents the av erage price of each gallon gasoline stored in the vehicle at point l . • T rip-cost. P l is the cost that the vehicle has spent to reach point l , starting from the source. • T ime-to-reac h. T l is the time at which the vehicle has reached point l . • Slot of a time. S t l represents the time slot in which the time T l falls. 2) Modeling V ehicle Movement: A vehicle can move to point l from point ˆ l , if the v ehicle is in ˆ l ( i.e. , it is already reached) and there is a road from ˆ l to l . The vehicle can move to l from any of the neighboring points ( i.e. , the points connected to l ). This requirement is encoded as follows: X ˆ l,l → X ˆ l ∧ L ˆ l,l (1) X l → _ ˆ l X ˆ l,l (2) In this model, we assume that no point is revisited. Due to this assumption, it is obvious that, if there are some intermediate destinations ( i.e . , via-points), they have to be on the way toward the ﬁnal destination. If there is a requirement to backtrack a path to reach the ﬁnal (or an intermediate) destination, which could be mandatory due to the road system, then the model will fail to ﬁnd a routing plan. The following constraints ensure that no point is re visited: X ˆ l,l → ^ ˆ l 0 6 = ˆ l ¬ X ˆ l 0 ,l (3) X ˆ l,l → ^ l 0 6 = l ¬ X ˆ l,l 0 (4) 9 3) Modeling Battery Rechar ging: The battery of the vehicle can be recharged at a location if the vehicle have reached the point plus there is a char ging station, as shown in Equation (5). If the battery is rechar ged, the stored charge of the battery increases. If it is not recharged, then the effecti ve charge remains the same. W e assume that when a battery is recharged, it is always rechar ged in an integer amount (kWh) b ut no more than its capacity . S c l → X l ∧ S l (5) S c l → ( ˆ C e l > C e l ) ∧ ( ˆ C e l ≤ C v ) (6) ¬ S c l → ( ˆ C e l = C e l ) (7) The formalization of the stored electricity of the vehicle when it reaches a point is shown in Equation (8). The stored electricity at point l depends on the stored electricity at the last visited point ( i.e . , ˆ l ) and the char ge required to cross the distance between these points. X ˆ l,l → ( B → (( C e l = ˆ C e ˆ l − C ) ∧ ( ˆ D ˆ l,l = 0))) ∧ ( ¬ B → (( C e l = 0) ∧ ( ˆ D ˆ l,l = D ˆ l,l − D ))) (8) In Equation (8), C and D denote ( D ˆ l,l /Re ) and ( ˆ C e ˆ l × Re ) , respectively . B stands for the Boolean result of ( ˆ C e ˆ l ≥ C ) , and ˆ D ˆ l,l represents the distance trav eled by electric power from ˆ l to l . If the battery is recharged at point l , then the average cost of each kWh of electric charge stored in the battery needs to be updated as follows: X ˆ l,l → ( S c l → ( P e l = ( P e ˆ l × C e l + P s l, ˆ t × ( ˆ C e l − C e l )) / ˆ C e l ) ∧ ( ¬ S c l → ( P e l = P e ˆ l )) (9) In this equation, ˆ t stands for S t l . 4) Modeling Gasoline Refueling: The vehicle can refuel at the gas station at a location pro vided that the vehicle reaches that point, as shown in (10). If the tank is refueled, the stored gasoline of the vehicle increases. If it is not refueled, then the effecti ve stored gasoline remains the same. W e assume that when the gas tank is refueled, it is always refueled in an integer amount up to its capacity . Gc l → X l ∧ G l (10) Gc l → ( ˆ C g l > C g l ) ∧ ( ˆ C g l ≤ ˆ C v ) (11) ¬ Gc l → ( ˆ C g l = C g l ) (12) The formalization of the stored electricity of the vehicle when it reaches a point is sho wn in Equation (13). The stored gasoline at point l depends on the stored gasoline at the pre vious visited point ( ˆ l ) and the gasoline consumed to trav el the distance between these points. Remember that the gasoline is used as the energy source only when the battery is out of charge. X ˆ l,l → (( ˆ C g ˆ l ≥ ˆ C ) ∧ ( C g l = ˆ C g ˆ l − ˆ C )) (13) In Equation (13), ˆ C represents (( D ˆ l,l − ˆ D ˆ l,l ) /Rg ) . If the vehicle tak es gasoline at point l , then the a verage cost of each gallon of electric charge stored in the battery needs to be updated as follows: X ˆ l,l → ( Gc l → ( P g l = ( P g ˆ l × C g l + P g l × ( ˆ C g l − C g l )) / ˆ C g l ) ∧ ( ¬ Gc l → ( P g l = P g ˆ l )) (14) 10 5) Modeling T ime and Cost Spent: The computation of the cost of traveling from one point to another depends on the energy type, i.e. , electric charge and gasoline, that has been used by the vehicle. In a trip, the vehicle can use an y one of them or both based on the availability of the battery charge. The cost, P l , to reach point l is modeled in Equation (15) according to our assumption that the vehicle’ s primary energy source is the battery . In the equation, P G l represents ((( D ˆ l,l − ˆ D ˆ l,l ) /Rg ) × P g ˆ l ) that denotes the price of the gasoline consumed during trav eling from ˆ l to l , and P E l stands for (( ˆ C ˆ l − C l ) × P e ˆ l ) which denotes the price of the electric charge used during trav eling from ˆ l to l . X ˆ l,l → ( P l = ( P ˆ l + P G l + P E l )) (15) The time required for trav eling does not depend on the energy source (fuel) type (refer to Section III-B). It depends on the av erage vehicle speed on the road. Equation (16) sho ws the modeling of time computation. X ˆ l,l → ( S c ˆ l → ( T l = ( T ˆ l + T + T Q ))) ∧ ( ¬ S c ˆ l → (( T l = ( T ˆ l + T )))) (16) In the abov e equation, T stands for ( D ˆ l,l /S ˆ l,l, ˆ t ) and T Q represents ( Qs ˆ l, ˆ t × ˆ T s ˆ l + ( ˆ C ˆ l − C ˆ l ) × T s ˆ l ) . 6) Modeling User Requir ements: The main objecti ve of the model is to ﬁnd a route from the source to the destination. There are three more user requirements on the na vigation plan: limited cost ( C p ), time boundary ( C t ), and the intermediate points of interest ( I ) with/without associated time boundaries ( C t l for l ∈ I ). The following equations represent these constraints: X S v ∧ X Dv ^ l ∈ I X l (17) P Dv ≤ C p (18) T Dv ≤ C t (19) ∀ l ∈ I T l ≤ C t l (20) It is easy to understand that, in the na vigation plan, a point cannot be reached by the v ehicle taking more cost or time than the ov erall cost or time boundaries ( i.e. , C p or C t ), respectiv ely . W e incorporate these constraints as follows: X l → P l ≤ C p (21) X l → T l ≤ C t (22) The user may have other constraints to satisfy . For example, when the vehicle reaches the destination, the stored electricity in the battery may need to be at least to some threshold v alue, which can be used for the next trip. This constraint is formalized as follows: ˆ C Dv ≥ C e (23) Here, C e is the threshold v alue. 11 7) Modeling Initial Constraints: W e need to consider the constraints that initialize the model. These constraints are about the stored electric capacity , the time and the cost at the source point. The starting time of trav el ( i.e. , time at source) is initialized with the current time. Equation (24) models the initialization constraints. ( C e S v = E v ) ∧ ( C g S v = Gv ) ∧ ( T S v = T curr ent ) ∧ ( P S v = 0) (24) W e need to initialize the prices for each kWh of stored electric charge and each gallon of stored gasoline at the source point ( S v ). The initialization depends on whether the battery is recharged at this point. This is formalized as follows: ( S c S v → ( P e S v = ( P v × C e l + P s S v, ˆ t × ( ˆ C e S v − C e S v )) / ˆ C e S v ) ∧ ( ¬ S c l → ( P e S v = P v )) (25) Similarly , the price of each gallon of stored gasoline at point S v is initialized as follo ws: ( Gc S v → ( P g S v = ( ˆ P v × C g l + P g S v × ( ˆ C g S v − C g S v )) / ˆ C g S v ) ∧ ( ¬ Gc S v → ( P g S v = ˆ P v )) (26) D. A Case Study Figure 4. The topology of the road network considered in the example illustrated in Sec- tion III-D2. This topology resembles the major routes from Gastonia to W inston-Salem. The locations are numbered from 1 to 16, while the source and destination of the trip are speciﬁed using two arrows. Here, we ﬁrst brieﬂy discuss the implemen- tation of the proposed formal model. Then, we present a synthetic case study demonstrating the ex ecution of the model. 1) Implementation: W e implement our model by encoding the system conﬁguration and the constraints into SMT using the Z3 SMT solver [16]. The encoding requires to use Boolean, integer , and real terms. Although many parameters like distance, speed and queue size usually take integer values, we encode them as real terms, since some di vision operations that generate fractions are needed to perform. The system conﬁgurations and the constraints are giv en in a text ﬁle. The inputs are parsed into the model. Executing the model (in Z3), we obtain the veriﬁcation result as either satisﬁable (SA T) or unsatisﬁable (UNSA T). If the result is UNSA T , it means that the problem has no route from the source to the destination satisfying the constraints. In the case of SA T , we get the navigation plan from the assignments of the variables, X l , Sc l , and Gc l . X l shows the route, while Sc l and Gc l represent the stations that are selected for recharging and refueling, respectiv ely . 2) An Example Case Study: W e demonstrate our model through a synthetic case study . Figure 4 sho ws the topology of the road network considered in this study . The topology resembles the major routes between Gastonia and W inston-Salem, two cities in North Carolina. The input ﬁle regarding this example is shown in T able II. The objective of the PHEV is to ﬁnd a routing plan from Gastonia (location 1) to Winston-Salem (location 10) that satisﬁes the giv en constraints. The v ehicle starts its trip at time 0 and, according to the constraints, the vehicle needs to reach the destination by time 130 (130 minutes) and with no more 12 T able II I N PU T T O T H E E X A MP L E # The numbers of locations and time slots 16 8 # Links (location pair, length, avg trafﬁc speed) 20 1 2 23 1 2 3 21 1 3 4 15 1 4 5 19 1 6 7 15 1 . . . . . . . . . # Charging stations (location, charging time/kWh, charging # time/waiting vehicle, queue size, price/kWh at each time slot) 6 1 2 20 4 1 3/2 3/2 5/4 5/4 5/4 3/4 3/4 12 2 15 1 1 3/2 3/2 3/2 5/4 5/4 3/4 3/4 13 2 18 2 3/4 1 3/2 1 5/4 5/4 3/4 3/4 9 2 16 1 3/4 1 3/2 3/2 1 5/4 3/4 3/4 4 2 18 2 5/4 3/2 3/2 3/2 5/4 5/4 3/4 3/4 16 2 16 2 1 3/2 3/2 5/4 5/4 5/4 3/4 3/4 # Gas stations (location, price/gallon, avg fueling time) 16 1 4 8 2 39/10 10 3 38/10 10 4 38/10 10 5 39/10 10 6 39/10 8 . . . . . . . . . # PHEV’s properties (source, destination, stored energy and its price/kWh, battery capacity , # stored gas and its price/gallon, gas capacity, kWh/mile, gallon/mile, trip start time) 1 10 2 5/4 8 0 0 10 1/8 1/12 0 # Constraints (time, price) 130 21 # Via points 0 than $21 of energy cost. The vehicle has no via-point to pass by . The execution of the model corresponding to the example returns a SA T result. According to the solved result, the assignments of values to different v ariables of the model, we ﬁnd that a possible route for the vehicle is { 1, 11, 12, 6, 7, 8, 9, 10 } . The battery needs to be recharged at the station at location 12, and the v ehicle is refueled at location 1. The energy cost required through this route is $20.83, while the time cost is $129.5 minutes, both of which satisfy the constraints. It is worth mentioning that if the trip start time is changed to 60, then there is no routing path from the source to the destination under the abov e properties and constraints. In this case, if we can increase the time-to-reach constraint to time 220 ( i.e. , 160 minutes of traveling time considering the start time) and increase the trip cost to $22, then there is a solution, wherein the vehicle needs to use more electric charge compared to the pre vious case in order to keep the cost low . As a result, the vehicle needs to recharge for twice (at locations 12 and 9). The reason for this dif ference is 13 Algorithm 1 Optimal Conﬁguration Determination 1: C p min & C p max are the minimum and maximum values of C p . 2: K max is the maximum number of executions of the loop-body . 3: if Solver returns SA T then 4: Get Model, M . 5: Update C p max according to the v alue of P Dv in M 0 . 6: K = 0 . 7: repeat 8: C p = ( C p min + C p max ) / 2 . 9: Update the constraints (Equations (18) and (21)) associated to C p . 10: if Solver returns SA T then 11: Get Model, M 0 . 12: Update C p max according to the v alue of P Dv in M 0 . 13: else 14: C p min = C p . 15: end if 16: K = K + 1 . 17: until ( C p max − C p min ≈ 0 ) or ( K = K max ). 18: end if that the charging price is higher during the trip time. Let us go back to the ﬁrst scenario, except with the following changes, as the vehicle is now more sensiti ve to cost compared to time. In this case, the trip cost cannot be more than $18 while the vehicle must reach the destination by time 160 (160 minutes). W e receiv e a satisﬁable answer with the following results: the routing path is { 1, 11, 12, 6, 7, 8, 9, 10 } . The vehicle needs to recharge at locations 12 and 9, while refueled at location 1. The vehicle uses electric charge to travel the largest part of the route in this trip. E. Optimal Navigation Plan Determination The veriﬁcation result comprehensi vely represents a consistent PHEV navigation plan for the network, satisfying the user requirements. Usually , there are more than one model that satisfy the constraints. These models also take dif ferent time and cost, though all of them take time and cost less than or equal to the time constraint ( C t ) and the cost constraint ( C p ). Observing these models, one can choose the most cost- (or time-) efﬁcient route among all alternativ e satisﬁable models for the same set of constraints. W e propose Algorithm 1 that considers two values: C p max and C p min , and ﬁnds the optimal navigation plan based on the cost. T ypically , C p max is the user’ s given constraint C p , while C p min is zero. The algorithm utilizes a binary search method to ﬁnd the optimal v alue. Algorithm 1 usually takes a longer time compared to the time for ﬁnding a satisﬁable model only , since the algorithm requires several in vocations of the model synthesis. The complexity of the algorithm is O ( T v er if y × l og 2 D ) , where T v er if y is the synthesis time and D is the difference between T h A,max and T h A,min . Since T v er if y is very high in unsatisﬁable cases as well as in tight constraint-based cases (see Section V for details), the time for ﬁnding the optimal would be very high for a lar ge number of vehicles. Howe ver , if the na vigation management service is provided online, where the formal model is executed by an SMT solv er running in a centralized serv er , the navigation 14 Navigation Servi ce Pr ovid er & Con tr oll er A New Vehicle to th e Sy ste m Req u est f o r a n ew Plan with M o d if ied Data Nav ig atio n Plan Ch argin g Statio n Ch argin g Price at eac h tim e slo t An Exis tin g Vehicle in th e Sy stem Necessary Data Nav ig atio n Plan No tif icatio n o f Pr ice Ch an g es Figure 5. The steps of the navigation control technique. management service provider can utilize powerful machines to compute this optimization. The algorithm also lets the provider control the number of steps ( K max ) to reduce the optimization time. In this case, a vehicle may receiv e a semi-optimal navigation plan. I V . P R I C E - B A S E D N A V I G AT I O N C O N T RO L The model we hav e discussed in Section III ﬁnds a satisﬁable navigation plan for a PHEV under a number of giv en constraints. The main constraints are time and cost. The tra veling cost depends not only on the electricity price but also on the a vailability of the charging stations, the queue lengths of waiting vehicles in the stations, the time constraint to reach the destination, and the vehicle speeds on dif ferent roads. The queue lengths at charging stations and the vehicle speeds on dif ferent road segments (which incorporate the trafﬁc congestion) at different time slots are predicted from historical/past data and current/present status. Since we consider long trips, the initially predicted trafﬁc scenario may change during the trip period. Moreov er , due to non- speculated incidents, the number of vehicles choose a route or route segments can be larger than estimated. While heavy trafﬁc can slow do wn the v ehicles than the speed limits, the queue sizes of the charging stations on these routes have much higher possibility of being longer compared to that of other stations. While some routes may become busier than forecasted, many other routes can become freer than expected. Therefore, to deal with the uncertainty in predicted data, it is advantageous to control the (unexpected) trafﬁc ﬂow by distrib uting the unexpected loads among the roads, so that the roads hav e lesser trafﬁc jams and the charging stations have smaller waiting queues. In this section, we present a simple moti vation-based load control technique that motiv ates the cost-sensitive users to choose the roads with fe wer loads. The navigation management service provider can play the role of a controller in this technique. Ho wev er , the success of the model depends on the e ven distribution of the charging stations throughout the system. 15 A. Overview of Navigation Contr ol T echnique The basic idea we follow in our control technique is to adjust the charging prices of the stations at each time slot to indirectly distribute the vehicles in the roads and the charging stations. The control technique is shown in Figure 5, which works as follows: When a PHEV enters the road system, it connects to the controller with necessary information necessary to get the na vigation route. The controller computes a satisﬁable navigation plan (based on the model discussed in Section III-B) and returns it to the vehicle. Hence, the controller has the navigation plans of all the existing PHEVs on the roads. The controller can follo w a decentralized framework of computing systems to deal with an excessi ve number of PHEVs. W ith the time, the routing data about the PHEVs gets updated. The forecasted trafﬁc (including non-PHEV vehicles) on the roads can also be changed due to updated situations. Therefore, the controller hav e an updated information about the traf ﬁc on each road segment, and it adjusts the charging price (for a time slot) when the update is different than the estimation. The price adjustment technique considers that the routing paths of the e xisting PHEVs are expected to remain unchanged. Hence, the technique considers the cost requirement of each v ehicle as a constraint, so that the na vigation plan for the vehicle will remain valid e ven after any price adjustment. The controller can notify each existing vehicle in the system if there is a price adjustment. Then, a v ehicle may request the controller for a new routing plan. In order to adjust the charging prices to achie ve better load balance, the controller gi ves a weight for each point l associated to a charging station during a time slot t based on the number of vehicles that have the point on its routing path during t . The busier the point is during t , the higher weight gi ven to it at t . The control chooses the prices according to these weights − higher weights get higher prices, while lower weights hav e lo wer prices. The prices during a time slot ( t ) should be within some minimum ( P min,t ) and maximum ( P max,t ) bounds. Moreov er , the av erage of the char ging prices of all the char ging stations during a time slot t is constrained to be equal to P av g ,t , where P av g ,t is roughly equal to ( P min,t + P max,t ) / 2 . B. Navigation Contr ol Model W e deﬁne a list of parameters to model the na vigation control technique. A solution to this model will provide the char ging prices at the charging stations. Parameters for Modeling Navigation Control: W e deﬁne S as the set of the charging stations, i.e. , the set of the points where the stations are located. W e also deﬁne V as the set of the vehicles currently existing in the system. At the beginning of time slot ¯ t , the controller updates the charging prices for the stations for a number of future time slots ( i.e . , time t ≥ ¯ t ) by modeling a constraint satisfaction problem. W e use the following parameters in the model: Char ging Cost . ¯ P E v denotes the total cost that the vehicle v ∈ V spent for charging the battery during the trip. On the way from the source to the destination the vehicle may need to charge its battery sev eral times, here, m times. W e deﬁne ¯ P E v as follows: ¯ P E v = ( P E l 1 ) v + · · · + ( P E l m ) v Here, l i is the i ’th point ( 1 ≤ i ≤ n ) on the routing path from the source to the destination and P E l i is already deﬁned in Section III-C5. Remember that P E l i depends on P e l i , while P e l i depends on P s l ¯ i ,t ¯ i ( ¯ i ≤ i ), the charging price of the charging station located on point l ¯ i , where the vehicle char ges its battery during time slot t ¯ i . In the case when time slot t ¯ i < ¯ t , the 16 vehicle has already charged its battery , while in the case when times slot t ¯ i ≥ ¯ t , the vehicle will charge its battery . Hence, the the updated prices hav e an impact on ¯ P E v for the time slots equal or higher than t . Gasoline Cost . ¯ P G v denotes the total cost that the vehicle spent for gasoline during the trip. ¯ P G v is computed as: ¯ P G v = ( P G l 1 ) v + · · · + ( P G l n ) v Here, l i is the i ’th point ( 1 ≤ i ≤ n ) on the routing path from the source to the destination and P G l 1 is already deﬁned in Section III-C5. Since the gasoline price is kept unchanged, ¯ P G v remains the same, irrespecti ve of the change in the charging prices. W eight . W l,t denotes the weight of the vehicles passing through point h, l ∈ S during slot t ( t ≥ ¯ t ), i.e. , the ratio of the number of vehicles passing (actually expected to pass according to the v ehicles’ na vigation plans) through point l o ver the total number of vehicles passing all the points in the set S during t . W l,t = N v l,t P l 0 ∈ S N v l 0 ,t Here, N v l,t represents the total number of v ehicles passing through point l , which is computed from X ˆ l,l for any ˆ l and S t l associated with all vehicles. Modeling Na vigation Contr ol: The gasoline cost ¯ P G v for a vehicle and the weight W l,t for each gasoline station during a time slot are tak en as computed value which are constant in the model. W e synthesize the charging prices for each station at upcoming time slots ( i.e . , for each time slot t ≥ ¯ t ) while the charging prices during previous time slots are inv ariable. The charging price of a gas station during a slot t is proportional to the weights of the vehicles passing through the point associated to the station. This is formalized by the following relation: ( W l,t > W l 0 ,t ) → ( P s l,t > P s l 0 ,t ) (27) The prices cannot be less than P min,t and more than P max,t during time slot t . If the weight is zero, i.e. , no vehicles passing through the point corresponding to the charging station, the price is taken as the minimum. W e consider the similar constraint for maximum weight ( i.e. , 1). These constraints are formalized as follows: ( P s l,t ≥ P min ) ∧ ( P s l,t ≤ P max,t ) (28) ( W l,t = 0) → ( P s l,t = P min,t ) (29) It is also ensured that the average of the charging prices at different char ging stations during time slot t are equal to the (av erage) electricity price set for this time slot ( P av g ,t ) (Equation (30)). This price can be identiﬁed based on the electricity price(s) set by the associated energy provider(s) and the proﬁt mar gin set for the charging stations. P h,l,S l P s l,t P l ∈ S 1 = P av g ,t . (30) The crucial constraint is that the vehicles already in the system have to be able to reach their destination within the cost limit. Since the prices may change (for the upcoming time slots), we need to verify whether the cost for a v ehicle v remains within the limit C p v . ¯ P G v + ¯ P E v ≤ C p v . (31) 17 0 0.5 1 1.5 2 8 9 10 11 12 13 14 15 16 Dollar Time Slot Charging Prices Price/kWh of the station at point 2 Price/kWh of the station at point 5 (a) 0 0.5 1 1.5 2 8 9 10 11 12 13 14 15 16 Dollar Time Slot Charging Prices Price/kWh of the station at point 2 Price/kWh of the station at point 5 (b) Figure 7. Change of charging prices with the time: (a) the gas stations hav e the same price rate, and (b) the gas stations have different price rates. C. A Case Analysis Figure 6. The topology of the road network used in the case analysis of Section IV -C. In Figure 7, we sho w the change of the charging prices with the time slots. W e consider the road network as sho wn in Figure 6, where each location point has a charging station and a gas station. In this e xample, we take 100 vehicles, which are arbitrarily distributed during 24 time slots. For each vehicle, the source and destination, and the starting time of the trip are chosen randomly follo wing uniform distrib ution. The maximum and minimum charging prices per kWh are taken as $1.5 and $0.5, respectively . The initial price for each charging station is taken randomly . In the ﬁrst scenario, the gas price is ﬁxed and the same for all gas stations ($4.00/gallon) while in the second scenario, the gas stations have dif ferent prices. These prices are ﬁxed throughout the day . The cost and time constraints are taken from the ranges of 10-50 dollars and 100-200 minutes, respectiv ely . The results of the ﬁrst case are presented in Figure 7(a), which shows the charging prices for two stations: the charging stations located at point 2 and point 5. W e see that the prices change with the time. It is interesting to observe that the charging price of the station at location 5 is most often less than the charging price of the station at point 2, except for a fe w cases. The reason behind this behavior is that the connecting roads to point 2 are shorter compared to the connecting roads to point 5. That is why the navigation management plans contain point 2 more than point 5, especially due to the longest road segment between 4 and 5. Hence, to div ert the vehicles to point 5, most of the time the charging price of the station at point 2 is kept high. The results associated with the second case are shown in Figure 7(b). Here, we observe a different behavior in price changes because the gas prices of the stations on points 1, 2, and 3 are higher ($4.00, $3.90, and $3.80, respecti vely) than that of the stations on points 4, 5, and 6 ($3.90, $3.80, and $3.70, respectiv ely). Therefore, the PHEVs hav e incentiv es to choose routes through points 4, 5, and 6 compared to the case in Figure 7(a). 18 V . E V A L UATI O N W e e valuate the scalability of our proposed na vigation management model and control model. Due to the unav ailability of the real-life data, we analyze the models using different synthetic data. A. Methodology W e ev aluate the scalability of our proposed models by analyzing the time and memory required for synthesizing the outputs. W e apply efﬁcient techniques for increased scalability of solving the navigation management model. 1) Evaluation Data Overview: The problem size depends mainly on the number of location points, i.e. , roads, as well as charging and gas stations. The problem size is primarily speciﬁed by the total number of location points. The location points are taken (or abstracted) based on the real road maps (particularly , motorways/highways in the Eastern United States). W e download the map from OpenStreetMap [17] in the OSM (Open Street Map) JSON ﬁle format and process the data to con vert it to our input template. Since we consider long trips in this research, we do wnload and process the roadways with the “highway” key values as “motorway , ” “trunk, ” and “primary . ” W e take the longitude and latitude of each location point, the points on a highway (using the “ref ” and “tiger/cfcc” tags), and speed limit of a road from the data. The distance between two consecutiv e points on a highway is calculated considering the coordinates of the points. W e consider points on a highway with a minimum distance apart from each other . The distances are kept within a range of 1 − 40 miles. The intersections/links between the roads are computed from the coordinates and the tags like “motorway link” and “motorway junciton. ” In the case of v arying the problem size in terms of the number of links (or branch roads) while keeping the number of location points constant, we arbitrarily consider these links. The charging stations at dif ferent locations are considered according to the Alternate Fuels Data Center , managed by the Ofﬁce of Energy Ef ﬁciency and Renewable Energy , U.S. Department of Energy [18]. The gas station locations can be found from many sources. W e get them OpenStreetMap (using the “amenity” tag) [17]. Ho wev er , we consider arbitrary locations for the charging or gas stations when we v ary their number to observe the impact of their av ailability on the scalability . The rest of the ev aluation data is synthetic. The number of time slots in a day is considered as 24, where each time slot covers 60 minutes. For the gasoline and electricity prices, the ranges are taken from the ranges of $3 − $5/gallon and $0.5 − $1.5/kWh, respectiv ely . The capacity of a battery is taken randomly from the range of 10 − 20 KWh. W e also randomly choose the mileage achiev ed by a vehicle from the range of 12 − 24 miles/gallon and the range of 8 − 16 miles/kWh. W e encoded our model using Z3 .NET API and ran the veriﬁcation of the model on an Intel Core i7 Processor with 16 GB memory . 2) Efﬁcient Execution Mec hanism: a) P arallel Execution of Multiple Instances: Since the navigation management model often needs to deal with numerous parameters and arithmetic constraints, ﬁnding an optimal navigation plan in a “near real-time” period is o verwhelming for a lar ge road network. Therefore, we de vise a scalable mechanism for a controlled execution of sev eral instances of the navigation management model parallelly so that a satisﬁable navigation plan can be achiev ed within a practically acceptable time. Algorithm 2 presents the mechanism that executes multiple processes simultaneously . All these processes explore the navigation space together . Each process runs the navigation management model with a random seed such that SMT searches in an arbitrary order (Lines 4 and 5 in Algorithm 2). As we look for one satisﬁable navig ation route, the parent process keeps checking if a process ends (Lines 8 and 9). A process is successful at ﬁnding a path when it receives a SA T result (Lines 12). When the 19 Algorithm 2 Parallel Search for the Na vigation Plan Require: F := The navigation management model;  According to the giv en inputs Require: P is the set of n processes;  Can be threads 1: for Each Process P ∈ P do 2: R P := A random number;  T o randomize the na vigation space exploration in F 3: Call ﬁndNavigationPlan ( F P , R P ) of Process P; 4: end for 5: R oute := NULL;  A navigation route is yet to recei ve. 6: Flag := TR UE;  Whether the loop should continue. 7: while Flag do 8: W ait for an arbitrary small period; 9: for Each Process P ∈ P do 10: if Done P = TR UE then  Process P completes its execution. 11: Flag := F ALSE; 12: if R esult P 6 = NULL then  A SA T result is recei ved and sav ed in R esult P . 13: R oute := R esult P ; 14: Break; 15: end if 16: end if 17: end for 18: if Flag = F ALSE then 19: for Each process P ∈ P do 20: if Done P 6 = TR UE then 21: T erminate process P ; 22: end if 23: end for 24: end if 25: end while result is UNSA T , there is no navigation route. When a navigation route is found, the rest of the processes (which are yet to be completed) are terminated since we look for a single navigation route. On the other hand, if a process ( i.e . , the model instance running by this process) fails to ﬁnd a path, there is no w ay other processes can be successful, as the solv ed model has already explored the whole search space. Therefore, the rest of the processes are terminated (Lines 18- 23). b) Sear ch Space Pruning: The efﬁciency of the navigation route synthesis can be further improved by pruning the search space with respect to the source and destination. The idea is explained here. The navigation route depends on the source and the destination, along with the road network. Although the given road network can be large, according to the source and the destination there are often man y points inc the network, which will not practically be advantageous to traverse to reach the 20 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 100 200 300 400 500 600 700 800 900 1000 Time (Second) Number of Points Efficient Route Synthesis Time w.r.t. Number of Location Poin ts 12 Slots 24 Slots (a) 0 2 4 6 8 10 12 15 20 25 30 35 40 45 50 Time (Second) Percentage of Points that Create Branches Efficient Route Synthesis Time w.r.t. Connectivity (Branches) 300 Points 500 Points (b) Figure 8. The navigation management plan synthesis time with respect to: (a) the number of points and (b) the number of links (particularly , the number of points connected with the neighboring route). destination. Those points can easily be pruned from the search space by initializing corresponding X l s as false. Such a constraint can be a user’ s preference. For example, a simple pruning condition can be as follows. If the summation of the shortest distance from the starting point to a point l (let it be ˆ D Sv ,l ) and that from the point to the destination point ( ˆ D Sv ,l ) is larger than r times ( e.g . , twice) the shortest distance between the source and the destination ( ˆ D Sv , Dv ), the point should not be tra versed to reach the destination. The corresponding formalization is presented below: ( ˆ D Sv ,l + ˆ D l, Dv ) ≥ r × ˆ D Sv , Dv → ¬ X l Since the shortest distances are expected to be already known, the time for space pruning is insigniﬁcant. 3) Evaluation Approac h: W e e v aluate the scalability of the proposed navigation management models on v arious scenarios in terms of the problem size as well as the na vigation requirements. W e mainly present the ev aluation results for the navigation management model, since we observe similar behaviors for the control model. W e do not e valuate the model by performing a comparison with a related work because to the best of our knowledge there is no existing work that considers the same problem like ours. W e consider a comprehensiv e list of routing properties for navigation planning. GreenGPS [11], [19], a closest work to this research, provides fuel-efﬁcient routes, considering stop lights and the trafﬁc congestion. The technique does not consider recharging (or refueling) requirements. In highways where there are usually no stop light or trafﬁc congestion, this tool performs like traditional GPS services. B. Evaluation Results Impact of the Problem Size: Figure 8(a) sho ws the model veriﬁcation time, i.e. , the navigation management plan synthesis time with respect to the problem size. As shown in Figure 8(a), we observ e that the analysis time is super -linear with respect to the number of location points. W e consider two scenarios in this analysis. In one scenario the number of time-slots is 12, while in another scenario, this number is 24. W e also observe that when the number of time-slots is larger , it takes a longer time. The increase in the problem size expands the model space, i.e. , the search space for the potential solution. As a result, we observe 21 20 40 60 80 100 120 140 160 180 200 220 100 200 300 400 500 600 700 800 900 1000 Time (Second) Number of Points Route Synthesis Time w.r.t. Number of Location Points 12 Slots 24 Slots (a) 0 100 200 300 400 500 600 700 800 900 1000 15 20 25 30 35 40 45 50 Time (Second) Percentage of Points that Create Branches Route Synthesis Time w.r.t. Connectivity (Branches) 300 Points 500 Points (b) Figure 9. The navigation management plan synthesis time, when neither parallelism nor space pruning mechanism is applied, with respect to: (a) the number of points and (b) the number of links (particularly , the number of points connected with the neighboring route). greater synthesis time for larger problems. Figure 8(b) shows the impact of the number of connecting roads on the navigation plan synthesis time. Unlike the experiment regarding Figure 8(a), we vary the percentage of location points that are connected with a neighboring point ( i.e. , the branch roads). W e consider 24 time slots and perform experiments in two scenarios − one with 300 points and another with 500 points. As we see in Figure 8(b), the analysis time increases rapidly when the number of branch roads increases. Due to more connecting roads, there are man y options to reach a speciﬁc location, which in turn increases the search space. It is worth mentioning that the devised efﬁcient mechanism of executing the synthesis model ( i.e. , the parallelism and the pruning techniques) signiﬁcantly improv es the scalability . Figures 9(a) and 9(b) show the na vigation path synthesis time when neither parallelism nor pruning is applied. The results demonstrate the obvious improvement, when we compare them with that in Figures 8(a) and 8(b). The cost incurred in a trip depends on the distance driv en and the type of fuel used by the vehicle. The more electricity (stored in the battery) is used as fuel instead of gasoline, the less the cost. As we know a battery requires frequent recharging, the av ailability of the charging stations is important. Howe ver , if there is high av ailability of charging stations, there are many alternativ es av ailable for recharging the battery . If there are too many alternativ es, the search space becomes larger and the synthesis time increases accordingly . On the other hand, if the number of charging stations becomes very small ( e.g . , less than 40-50% of the points ha ve charging stations), there are fewer alternati ves for satisﬁable solutions. As a result, the solver needs to cov er a larger space to ﬁnd a solution. Figure 10(a) justiﬁes this argument. A vehicle may also need refueling to reduce the time requirement − av ailability of charging stations is less while battery rechar ging takes a longer time). Therefore, there is an impact from the a vailability of gas stations as well, which is reﬂected in Figure 10(a) as well. In this analysis, we consider 500 location points and two scenarios, one with gas stations at 50% of the points while another has them at 90% of the points. Impact of the T rip Distance: The distance (expressed in terms of location points) between the source and the destination has an impact on the navigation synthesis time. The analysis result is sho wn in Figure 10(b) with respect to three qualitativ e distances: low (10-20 points), medium (50-60 points), and high (90-100 points), in a road system of 500 location points. W e take three different arbitrary scenarios (in terms of source and destination) for each qualitativ e distance. The farther the destination from 22 1.4 1.6 1.8 2 2.2 2.4 2.6 20 30 40 50 60 70 80 Time (Second) Number of Stations (in Percentage of Points) Route Synthesis Time w.r.t. Number of Charging Stations 50% Gas Stations 90% Gas Stations (a) 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 Low Medium High Time (Second) Distance Route Synthesis Time w.r.t. Trip Distance (b) Figure 10. The navigation management plan synthesis time with respect to: (a) the number of stations and (b) the trip distance. 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 Time (Second) Number of Via-Points Route Synthesis Time w.r.t. Number of Via-Points 300 Points 500 Points (a) 0 5 10 15 20 25 400 500 600 700 800 900 1000 Time (Second) Number of PHEVs Control Analysis Time w.r.t. Number of PHEVs 100 Charging Stations 200 Charging Stations (b) Figure 11. The analysis time, i.e. , the navig ation management plan synthesis time, with respect to: (a) the via-points, and (b) the impact of the number of vehicles on executing the navigation control model. the source, the more alternati ves there are to reach the destination satisfying the constraints. These alternativ es mean navigation routes and recharging/refueling choices. As a result, the synthesis time increases with the increase in the trip distance. Impact of the Points of Interest: W e observe that the number of intermediate points of interest ( i.e. , via-points) impacts the synthesis time. The results are shown in Figure 11(a) for two different problem sizes. The ﬁgure shows that time increases with the increase in the number of via-points. Since the number of intermediate destinations increases and each destination is associated with a time constraint to be reached, a larger time is required to ﬁnd a solution. Ho we ver , the execution time is also dependent on ho w tight time and cost constraints are associated with these via points and the ultimate destination. This impact is illustrated later in Figure 12. Impact of the Pr oblem Size on Control Mechanism Synthesis: W e also ev aluate the scalability of our control model, and the results are presented in Figure 11(b), which shows the synthesis time with respect to the number of vehicles. In these 23 1 1.5 2 2.5 3 3.5 4 500 600 700 800 900 1000 Time (Second) Time Constraint (Minute) Route Synthesis Time w.r.t. Time Constraint 300 Points 500 Points (a) 0 1 2 3 4 5 6 45 50 55 60 65 70 Time (Second) Cost Constraint (Dollar) Route Synthesis Time w.r.t. Cost Constraint 300 Points 500 Points (b) Figure 12. The impact of constraints on the navigation management plan synthesis time: (a) the time constraint and (b) the cost constraint. 0 2 4 6 8 10 12 14 16 1 2 3 4 5 6 Time (Second) Number of Parallel Processes Route Synthesis Time w.r.t. Number of Parallel Processes 300 Points 500 Points (a) 0 50 100 150 200 250 300 100 200 300 400 500 600 700 800 900 1000 Memory (MB) Number of Points Memory Requirement w.r.t. Number of Location Points Charging Stations at 20% Points Charging Stations at 40% Points (b) Figure 13. (a) The navigation route synthesis time with respect to the number of processes executing the synthesis model parallelly and (b) the memory requirement with respect to the number of points. experiments, we consider 500 location points for two different numbers of char ging stations (100 and 200). W e observe that the time for synthesizing the charging prices increases almost linearly with the increase in the number of vehicles. Impact of the Constraint Tightness: W e analyze the impact of the tight/ relaxed constraints on the model synthesis time. T ightening (relaxing) a time/cost constraint means decreasing (increasing) the time/cost constraint value. The analysis result is shown in Figure 12(a) where we vary the time constraint v alue. W e observe that the execution time increases with the reduction of the time constraint v alue because tightening a constraint reduces the number of possible solutions to the model; as a result, more searches are usually required to ﬁnd a solution. W e also observe the same behavior in the case of the cost constraint (Figure 12(b)). Impact of Number of Parallel Processes: In the abov e mentioned analyses, four processes are executed parallelly to synthesize the navigation route according to Algorithm 2. Figure 13(a) presents the impact of the number of processes on the generation 24 time. As the ﬁgure sho ws, the more is the number of processes, the smaller the ex ecution time. Since we run the processes on a single physical host, the processes need to share the resources, and thus the overhead goes up when the number of sharing processes is high as well as the problem size, compared to the number of processing cores of the host (which is four in our case). Hence, there is a trade-off between the ef ﬁciency and the o verhead. Memory Complexity: The memory requirement of the SMT solver [16] for our model is e valuated by changing the number of points. The ev aluation is done considering the memory required for encoding the problem in two scenarios with respect to the charging stations. In scenario 1, 20% location points have char ging stations, while in scenario 2, they exist at 40% points. The required memory for a model synthesis is the sum of the memory for modeling the system properties and the memory for modeling the constraints. The analysis result is shown in Figure 13(b). W e observ ed that the memory requirement increases linearly with the increase in the number of location points. An increase in the model size depends on the problem size, mainly the number of location points and that of the charging and gas stations. V I . D I S C U S S I O N Here we discuss some concerns or aspects of the proposed navigation management solution. A. Optimal Solution W e model the na vigation planning logically as a constraint satisfaction problem and solve using an efﬁcient SMT solver . The solver provides a solution if there is one satisfying the constraints on the giv en road system. The solution may not be the optimal one. Howe ver , the model can provide all possible routes to the destination, including the optimal one(s), if it is solved for all solutions ( i.e. , complete exploration of the solution space). The tighter are the constraints ( e.g . , the time and cost to reach the destination constraints), the smaller the solution space ( i.e. , the number of alternative solutions) and the closer a solution to the optimal. The constraints can be too tight to get a solution ( i.e. , no feasible route). Howe ver , a smaller solution space often takes a larger ex ecution time. The impacts of time and cost constraints on the running time are shown in Figures 12(a) and 12(b). The tight constraints usually lead into larger ex ecution times. B. Usability The navigation route synthesis needs to be done within a reasonable time, which can be expected to be in run-time. Although the e xecution of the formal model follows an exponential scale of growth, the applications of parallel ex ecution and routing path pruning, as we hav e presented in Section V -A, signiﬁcantly reduce the computation time. W ithin our limited computing capacity (i.e., limited processing power and memory), our ev aluation results sho w a fe w seconds of execution time for a large problem space. W e present the impact of the number of parallel processes in Figure 13(a). In practical uses, particularly for run-time planning of the navigation paths, the computing capacity can easily be extended to further reduce the e xecution time. Since parallelism is used, a powerful multi-core processor is required to get the desired performance. It can be argued that a PHEV is unexpected to be equipped with an IoT device (deployed in the vehicle or the user’ s smart device) which has sufﬁcient processing ability for such a synthesis. Since clouds can provide high-performance computing platforms, the navigation management can be implemented as a powerful cloud-based service. Such an implementation will be able to perform the na vigation planning al ways in a feasible time by of fering necessary computing resources according to the size of the navig ation management problem and its constraints. 25 C. Extendability The proposed formal framework for the navigation management synthesis is generic enough to consider further constraints if required. As long as the road network is represented using a set of links connecting the dif ferent locations, the model can consider all the destination constraints ( e.g . , delays at the via points or intermediate point of interests, av oiding the toll roads, highway preference, etc.) to ward the destination. V I I . R E L A T E D W O R K In the last few years, many researchers addressed the efﬁcient management of plug-in electric vehicles considering the vehicle- to-grid (V2G) aspects, especially the V2G services like charging, discharging, and frequency regulation services. For e xample, the works in [20] and [21] proposed mechanisms for maximizing the proﬁts from vehicle-to-grid (V2G) services by selling stored electricity to the grid or by participating in frequency regulation. Some other works, such as [22], [23], and our work in [24], dev eloped control algorithms and models for the optimal V2G management. Ho wev er , none of these works addressed the problem of driving EVs on long trips and scheduling the charging of batteries. The authors in [12] addressed the issue of minimizing the EV recharging waiting time through intelligently scheduling recharging activities. The authors theoretically formulated the minimum waiting time for the problem of recharging scheduling. Based on the analysis, they proposed a distributed scheme to optimize the recharging schedule. Howe v er , their model is limited only to EVs with a mere objective of minimizing the waiting time in the charging stations only . The model only considers a single highway road and cannot ﬁnd the navigation/routing path for a complex road network. Moreov er , the work did not consider the alternativ e use of fuels in the case of hybrid EVs. In this paper, we address the navigation management problem for hybrid electric vehicles for long trips, considering a broader and real-life aspect. Zhang and V ahidi [25] proposed a dynamic programming (DP)-based technique to ﬁnd strategies to operate the powertrain- based energy management system (EMS) ef ﬁciently with the optimal use of (stored) electric charge and gasoline until the next charging station. Bader et al. [26] dev eloped a DP-based mechanism for predicti ve real-time ener gy management with the help of precalculated lookup tables for dif ferent points of the po wertrain. Larsson et al. proposed an energy-optimal route selection mechanism based on historical driving data (logged GPS data) about commuter routes and the EMS optimization [27]. The mechanism also precomputes the optimal solutions to the energy management control problems using DP and provide them to a user as a lookup table. T ribioli et al. proposed a heuristic methodology , utilizing Pontryagin’ s minimum principle-based optimal control theory , to solv e the real-time energy management problem [28]. Later , this real-time energy management problem is solved using the estimation distrib ution algorithm by Qi et al. [29] and the particle swarm optimization algorithm by Chen et al [30]. Li and Chen [31] presented a solution for the optimal, real-time energy management of a parallel PHEV (here, the electric charge can be used to dri ve one axle while the gasoline to drive the other one). Y onggang et al. [32] developed an EMS control technique utilizing the GPS/GIS data and the knowledge of road e vents (trip distance, road grade, altitude, v elocity , etc.) to predict the road slope grade and the corresponding electric energy consumption, and thus control the battery usage so that energy is not discharged during uphill. It is worth mentioning that this strategic operation of the EMS or powertrain for optimal energy management of PHEVs is out of the scope of this work. Raghu K. Ganti et al. developed a na vigation service, named GreenGPS [11], for traditional vehicles. GreenGPS pro vides the most fuel-efﬁcient route between two points, unlike the shortest and fastest routes provided by the traditional navigation tools 26 like Google Maps [10] and MapQuest Maps [33]. GreenGPS collects the necessary data, which is continuously updated by the participating v ehicles and answers queries on the most fuel-ef ﬁcient route. Later, further results are presented in [19] according to an adv anced real-world phone-based implementation and deployment of GreenGPS, which considers reliable end-to-end data collection and route recommendation. Ho wev er , GreenGPS cannot work for PHEVs as it does not consider the rechar ging of these vehicles. It considers mainly the trafﬁc congestion and the stop lights to ﬁnd out the most fuel-efﬁcient path, which is suitable in urban areas. Moreover , for trav eling on highways, GreenGPS will perform almost similarly to a traditional GPS, as there is no stop light and often there is no or minor congestion. Nejad et al. [34] proposed routing algorithms for PHEVs that account vehicle operating modes (either electric, gasoline, or hybrid) and recommend the most fuel-efﬁcient path by choosing the optimal mode of operation for each road segment during route planning. Howe ver , the solution requires to choose a single mode of operation for each segment and to segment the roads appropriately based on the relati vely uniform energy efﬁciency conditions. Moreov er , the work does not consider recharging/refueling of these vehicles and thus inapplicable for long trips. Our proposed model for long trips is comprehensiv e (considering PHEV properties), ﬂexible (based on user requirements), and practical (allowing recharging and refueling). The formal modeling of a navigation problem presented in this paper is also unique. Our model is extendable for broader aspects with nov el and further requirements. V I I I . C O N C L U S I O N Plug-in hybrid vehicles require switching to gasoline or recharging their batteries for long trips. Recharging batteries takes a longer time compared to the refueling of gasoline. Due to these characteristics, a ﬂexible navigation management scheme is required. In this paper , we hav e presented an SMT -based formal modeling of the PHEV navigation management problem. Our model of fers an optimal management plan that includes the route, along with the potential charging points. This satisﬁes all the constraints on the fuel cost, the tra veling time, and the intermediate points of interest. W e hav e also presented a price- based navigation control technique to achiev e better load balancing for the system. W e implemented both of our models, ran simulation experiments using the Z3 SMT solver and ev aluated their scalability . W e observed that the running time of our navigation management model usually lies within few seconds for a road network of a thousand location points, while that of our control model is around 25 seconds for a problem of 20% charging stations on a road network with 1000 points and 1000 PHEVs. R E F E R E N C E S [1] A comprehensiv e guide to plug-in hybrids. http://www .hybridcars.com/plug- in- hybrid- cars. 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[24] Mohammad Ashiqur Rahman, Fadi Mohsen, and Ehab Al-Shaer. A formal model for sustainable vehicle-to-grid management. In Fir st ACM W orkshop on Smart Ener gy Grid Security (SEGS) , pages 81–92, 2013. [25] C. Zhang and A. V ahidi. Route preview in energy management of plug-in hybrid vehicles. IEEE T ransactions on Control Systems T echnology , 20(2):546– 553, March 2012. [26] B. Bader , O. T orres, J. Orte ga, G. Lux, and J. Romeral. Predictive real-time energy management strategy for phev using lookup-table-based dynamic programming. In W orld Electric V ehicle Symposium and Exhibition , pages 1–11, November 2013. [27] V . Larsson, L. Johannesson Mrdh, B. Egardt, and S. Karlsson. Commuter route optimized energy management of hybrid electric vehicles. IEEE Tr ansactions on Intelligent Tr ansportation Systems , 15(3):1145–1154, June 2014. [28] L. Tribioli, M. Barbieri, R. Capata, E. Sciubba, E. Jannelli, and G. Bella. A real time energy management strategy for plug-in hybrid electric v ehicles based on optimal control theory . Energy Procedia , 45:949 – 958, 2014. [29] Xuewei Qi, G. Wu, K. Boriboonsomsin, and M. J. Barth. An on-line energy management strategy for plug-in hybrid electric vehicles using an estimation distribution algorithm. In 17th International IEEE Confer ence on Intelligent T ransportation Systems (ITSC) , pages 2480–2485, Oct 2014. [30] Zeyu Chen, Rui Xiong, Kunyu W ang, and Bin Jiao. Optimal energy management strategy of a plug-in hybrid electric vehicle based on a particle swarm optimization algorithm. Energies , 8(5):3661–3678, 2015. [31] Jichao Liu and Y angzhou Chen. An online energy management strategy of parallel plug-in hybrid electric buses based on a hybrid vehicle-road model. In 19th International IEEE Conference on Intelligent T ransportation Systems (ITSC) , pages 927–932, Nov 2016. [32] Y onggang Liu, Jie Li, Ming Y e, Datong Qin, Yi Zhang, and Zhenzhen Lei. Optimal energy management strategy for a plug-in hybrid electric vehicle based on road grade information. Ener gies , 10(4):1–20, March 2017. [33] Mapquest. http://www .mapquest.com. Last accessed on January 15, 2019. [34] Mark M. Nejad, Lena Mashayekhy , Daniel Grosu, and Ratna Babu Chinnam. Optimal routing for plug-in hybrid electric vehicles. T ransportation Science , 51(4):1304–1325, 2017. 28 Mohammad Ashiqur Rahman is an Assistant Professor in the Department of Electrical and Computer Engineering (ECE) at Florida International University (FIU), USA, and leading the Analytics for Cyber Defense (A CyD) Lab at FIU. Before joining FIU, he was an Assistant Professor at T ennessee T ech Univ ersity . He recei ved the BS and MS degrees in Computer Science and Engineering from Bangladesh Univ ersity of Engineering and T echnology (BUET), Dhaka, in 2004 and 2007, respectiv ely , and obtained the PhD degree in computing and information systems from the University of North Carolina at Charlotte (UNC Charlotte) in 2015. Rahman’ s research focus primarily includes computer and information security , risk analysis and security hardening, secure and dependable resource allocation and optimal management, and distributed and parallel computing. He has already published over 50 peer -re viewed journals and conference papers. He has also served as a member in the technical programs and organization committees for v arious IEEE and A CM conferences. Md Hasan Shahriar is a PhD student in the Department of ECE at FIU. Earlier, he received his BS in Electrical and Electronics Engineering from BUET , Dhaka, in 2016. After the graduation, he joined Uttara University , Dhaka as a Lecturer in the Department CSE. He also worked as an Assistant Engineer at Electricity Generation Company of Bangladesh (EGCB). Shahriar is interested in the formal analysis of security and resiliency for cyber-ph ysical systems (CPS)/ internet of things (IoT) His current focus areas include smart grid/microgrid and industrial IoT . He is also a member of the ACyD Lab at FIU. Ehab Al-Shaer is a Professor and the Director of the Cyber Defense and Network Assurability (CyberDN A) Center in the College of Computing and Informatics at UNC Charlotte. He receiv ed his MSc and Ph.D. in Computer Science from the Northeastern Univ ersity (Boston, MA) and Old Dominion Univ ersity (Norfolk, V A) in 1998 and 1994 respectively . His primary research areas are network security , security management, fault diagnosis, and network assurability . Prof. Al-Shaer edited/co-edited more than 10 books and book chapters, and published about 200 refereed journals and conferences papers in his area. Prof. Al-Shaer also served as a Conference/W orkshop Chair and Program Co-chair for a number of well-established conferences/workshops in his area including IM 2007, POLICY 2008, ANM-INFOCOM 2008, A CM CCS 2009-2010. Quanyan Zhu is an Assistant Professor in the Department of Electrical and Computer Engineering at New Y ork Univ ersity . He receiv ed the B. Eng. in Electrical Engineering from McGill University in 2006, the M.Sc. from Uni versity of T oronto in 2008, and the Ph.D. from the Uni versity of Illinois at Urbana- Champaign (UIUC) in 2013. He is a recipient of man y awards including NSERC Canada Graduate Scholarship (CGS), Ma vis Future Faculty Fellowships, and NSERC Postdoctoral Fellowship (PDF). He spearheaded the INFOCOM workshop on Communications and Control on Smart Ener gy Systems (CCSES), the Midwest W orkshop on Control and Game Theory (WCGT), and NYU W orkshop on Control and Optimization of Network Systems (CONES).

A Formal Approach for Efficient Navigation Management of Hybrid Electric Vehicles on Long Trips

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