A Bellman-Ford Approach to Energy Efficient Routing of Ev

8/20/2019 A Bellman-Ford Approach to Energy Efficient Routing of Ev

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A Bellman-Ford Approach to Energy Ef cient Routing oElectric Vehicles

R m b usle m n nd s m h R w shdehDepartment of Electrical and Computer Engineering

Oakland UniversityRochester, chigan, 48326, USA

{abousleiman rami@gmail com and rawashd2@oakland edu}

Abstract-Most ex perts foresee more demand for electric andplug in electric vehicles. This demand is triggered byenvironmental concerns, energy de pendency, and unstable fuelprices. Available vehicle routing algorithms are designed forfossil fuelled vehicles. These algorithms o ptimize for the shortestdistance or the shortest travel time between points. Dijkstra orDijkstra ike algorithms are mostly used for solving suchoptimization problems. Energy e cient routing for electricvehicles, on the other hand, requires di erent a pp roaches as itcannot be solved using Dijkstra or Dijkstra ike algorithms.Negative path costs generated by regenerative braking, batterypower and energy limits, and vehicle parameters that are onlyavailable at query time, make the task of electric vehicle energye cient routing a challenging problem. In this paper, we presenta solution a pp roach to the electric vehicle energy e cient routingproblem using Bellman Ford. Bellman Ford is a deterministicoptimization method that is ca pable of solving routes withnegative paths. A model re presenting electric vehicles ispresented. Bellman Ford search is then a pp lied on the model andis used to nd the most energy e cient route. The generatedsolution is then used to guide the electric vehicle through thedesired path. The performance of the Bellman - Ford algorithmis then studied by a pp lying the im p lemented algorithm ondi erent ma p sizes.

dex Terms - De erm s p m z Elec r c eh cleseh cle R g

NTRO UCTION

Automotive vehicle drivers usually have hundreds or even thousands of route options to pick om when travelling om astart to a destination point on a prede ned map. The routes that are mostly assumed practical are the ones that yield theleast travel time or the shortest travel distance. Sometimes acompromise between these 2 constraints is considered a viablesolution, as optimizing for both at the same time might not befeasible. In general, optimizing for either the travel time or the travel distance will indirectly lead to energy ef cient routeselection or a "close enough solution. This is true because theless time traveled or the shorter distance traveled will yieldless engine run time. The less time the engine is run, the less

el it will consume.This is not necessarily true for an electric vehicle (EV)

mainly because they are capable of regenerating energy. EVsare powered by high-e cient electric motors that can nctionas generators when the EV is in coast mode, travellingdownhill, or when the driver steps on the brake pedal todecelerate the vehicle. The kinetic and/or the potential

97 -1-4673-6741-7/1 /$31 00 ©201 IEEE

energies are recovered and stored back onboard the vehicle in the high-voltage battery pack [1]. This stored energy can be used at a later time to propel the vehicle. The range of an EVcan increase by 20% or more due to regenerative braking [2].

Some studies have tackled the energy ef cient routing problem for electric vehicles. These studies can be split into 2categories: deterministic and stochastic optimization methods. Previously we have done some work in the stochastic eld [3].The energy ef cient routing problem was solved using Particle Swarm Optimization [4], Ant Colony Optimization[5], and Tabu Search [6]. Other work tackled the problem

om the deterministic side. Artmeier et al. [7] studied energyef cient routing for an EV om a graph theory context. Theauthors developed a generic shortest path algorithm using potential shi ing to discard the negative edge costs and tosolve the EV routing problem. They analysed the shortest path

problem and adapted the method to t speci c EVcharacteristics. Furthermore, they identi ed hard constraints

which restrict the battery from discharging below certainlimits and so constraints which impose that the batterycannot store more energy than its maximum capacity. Thedeveloped algorithm had a worst case complexity of n3Sachenbacher et al. [8] presented an A * search that considers regenerative energy and the battery parameters which yields aconsistent heuristic function of ncomplexity problem.The authors developed the routing algorithm claiming that itexpands each node at most once. The presented algorithmadjusted for the battery limits by modi ing the weight

nction that the authors presented. The weight function isgenerated by the potential energy and the propulsion system inef ciencies. In [9], Baum et al. presented a fast energyef cient routing algorithm for an EV and they tested their results on European and Japanese road networks. Although the model was based on an actual vehicle (Peugeot iOn) theauthors also did not prove their results on an actual EV. Furthermore, their developed algorithm does not handle

negative cycles. The authors also did not consider traf cconditions. The effects of traf c can be substantial as a roadsegment might be completely blocked. That will substantiallydelay the developed algorithm's processing time by a factor of thousands. Severe traf c might completely block a roadsegment thus yielding to a road topology change. Finally, theauthors did not consider the vehicle travel time. Vehicle traf c time has a lot of impact not only on the passengers' comfort but also on the EV energy ef cient routing problem. Ambient



temperatures might yield high loads on the EV's battery thussubstantially affecting the routing problem.

This paper will address the EV energy ef cient routing problem in its simplest and most straightforward form.Bellman-Ford approach is applied to an EV model. Thegenerated results are compared to the shortest routes and the routes with the least travel time.

This paper is organized as follows: the next section will

discuss the Bellman-Ford algorithm, followed by a section thatdescribes the EV model. The developed algorithm will then be presented and analysed. The paper is then ended with the results' discussions and the conclusion sections.

II. ELLM OR LGORITHM

There are numerous deterministic methods that are used tosolve for the shortest path problem. Table I lists some of thesealgorithms along with their complexity, negative edge support,and the year they were introduced. One of the most signi cantDynamic Programming methods is the Bellman-Fordalgorithm [10, 11]. It is mostly used to solve for problems with

a single source vertex that might include negative routes Thisalgorithm can detect negative nodes but it cannot nd theshortest path if a negative cycle is reachable from the source. Furthermore, it is slower than Dijkstra's algorithm and it is mostly used in applications where the target to be sought is notat a negative cycle, this is tr e because of the fact that it terminates upon nding a negative cycle.

T LE IMajor developed shortest path algorithms

Year Algorithm

1957 Bellman, et al.Complexity Negative Edge

upport

··

i·

958···-·

·s················-··

O·

(·

V"·······

·· i·962···-·· F oy: ···-·· O·(·V3 ············

-...............�.............. 19 4 Williams

1974 Suurballe

o

+ o

1984 Fredman, et al. + Vo

The pseudo code that describes how the Bellman - Fordalgorithm operates is shown in Fig. 1. The key idea of theBellman - Ford algorithm is the relaxation step or operation.This operation takes 3 inputs which include 2 nodes ( , B) and the edge length connecting the 2 nodes and (B). If thedistance between the source and the rst node plus theedge length is less than the distance to the second node (B), then the rst node is marked as the predecessor for the second node (B) (line 11) . In this case, the distance om the source to the second node (B) is calculated according to (1). The throughout the paper refers to the signed distance or length

between 2 points. If the distance between the source and therst node plus the edge length is greater than the distance

to the second node (B), then no changes are applied.

D t B=II AII+ Iledge BIl (1)

Bellman - Ford Algorithm0 function redecessor) Bell anFord(vertices, edges, source)00040007OS09

0

7IS90

for i = to total number of verticesdo

I distance[i] =

predecessor[i]= nullend

distance[ source]= 0

for i = to length(vertices)- dofor each edge edo

distance[e_ om]+ length(e) < length(e_to)do

end

end

Idistance[e_to]= distance[e_ om]+ length(e)predecessor[e_to]= e_ om

end

for each edge edo

I distance[e_ om]+ length(e) <distance[e_to]doreturn E ROR

endend

return predecessorFig : Bellman - Ford Algorithm Pseudo Code

III. LECTRIC EHICLE O EL

For an EV the theoretical drive line energy consumption isan integration of the power output at the battery terminals. The power output (in watts) can be expressed as shown in (2) .Where M is the vehicle mass in kilograms, g is thegravitational acceleration, is the tire rolling resistancecoef cient, is the air mass density, is the aerodynamicdrag coef cient, is the frontal area of the vehicle in m , is the vehicle speed in m/ ,8 is the rotational inertia factor, is the road grade, and is the vehicle acceleration in m/ •

Equation (3 ) shows the regenerative braking power at the battery terminals, where 0 < < 1 is the percentage of the total brake energy that can be regenerated by the electric motor [12]. The total energy shown in Equation (4) is the netenergy consumption at the battery terminals. The goal of the proposed optimization process is to minimize (4), given that the electric vehicle has to be driven between 2 points.

(2)

(3)

(4)



In general, the terms in (2) and (3) can take values asshown in Table II. Furthermore, these equations can be rthersimpli ed to make the calculations less tedious. Equation (5)shows a simpler form representing (2). K, A,B, and Care allconstants.

out= K+A. i + B. + C.a)Where:

K=M g rA=M.gB =O.5 .Pa.CD.A fC=M o

(5)

Similarly, (3) can be expressed in the same way as shown in (6).

in= u. K+A. i + B. + C.a)

T BLE IITypical Vehicle parametersDescription

Aerodynamic rag Coef cientFrontal reaRolling Resistant Coef cientVehicle Mass

Symbol

Gravitational Acceleration GAir Mass ensity PaRotational ertia Factor (mass factor) 8Regenerative Braking Factor a

Value0.252.250.0051,5009.8

1.275

0.4-0.

(6)

Unit

IV. ELLM N R PPROACH PLEMENTATION

To be able to implement the Bellman-Ford searchalgorithm on a source to destination problem the energyconsumption between the nodes needs to be calculated. Thiscalculated energy might be negative. To better illustrate the process a simple routing example is presented. Fig. 2 shows asimpli ed map with 7 nodes and 10 edges. The variables between the nodes marked as [d x' x] represent the distanceand the speed limit respectively on that speci c edge. Table IIIshows the values of these variables. Furthermore Table IVshows the potential energy level of each node i.e. the relativeelevation. This is needed to calculate the term in (5) and (6).

Using (5) and (6) along with the parameters listed inTables II, III, and IV a new map is generated. This new map is

shown in Fig. 3. The numbers listed on the path between the nodes represents the energy consumption needed for the EV to travel between these nodes in the direction of the arrow.Observing Fig. 2 the shortest route between A and G is: A toC to F to G with a total distance of 1740 meters. The route that

will lead the fastest travel time between A and G is: A to B to E to G with a total travel time of 101.2 seconds this yields anaverage driving speed of �80 km h or �49 mph. Applying the

Bellman-Ford Algorithm onto Fig. 3 , the most energy ef cient route between A and G is: A to C to D to G.

Fig. 2: Simple Map used to Illustrate the Bellman-Ford Implement(Map not to scale)

T BLE III

Ed Pge roper esEdge Distance (meters) Speed Limit (Km Hr)[d , s ][d2, s2][d3, s3][d4, s4][d5, s5][d , s ][d7, s7][d8, s8][d9, s9][d , s ]

d =2500d2 =750d3 =500d4= 50d5 =100d = 50d7 =500d8 =900d9=1100d10=340

T BLE IVNode Elevation

Node ElevationA 500B 550C 350

00E 550F 450G 400

9.792 11.91

100.59- -� 0

s = 120s2 = 50s3 = 50s4 =75s5 =10s =50s7 =50s8 =100s9 =120s =40

5 5

2.525 G

1.1SS_

�:765

2 66

16.169 Fig. 3: Energy Consumption(x 06) in Joules Needed to Move the EV

between the Nodes



V. DISCUSSIONS

It can be observed om the results that neither the shortest route nor the route with the least travel time yielded the mostenergy ef cient route. This concludes that applying theoptimization process on either the distance or the travel time will not yield optimal results.

The Bellman-Ford algorithm was applied to different mapsizes. The simulation was run using a quad core desktopcomputer running Matlab R2009b at 2.27 GHz of processingspeed. Table V shows different results using different mapsizes. These results show that the implementation is practicaland can be used to generate real-time results if the number of vertices and edges is reasonable. Whenever the number of vertices and edges become high Bellman Ford becomes impractical and cannot be used to solve for the EV routing problem. Other suggested techniques include preprocessing of the map, but that might be also impractical because of thedynamic characteristics of the battery and the routes. Anotherapproach is to use metaheuristic methods as we used in [3 6].These optimization techniques are slower for smaller maps, but for more complex maps where Bellman Ford startslacking, they produce very promising results as they can be used in real time routing problems.

Table VBellman-Ford Computational Time to Generate a Solution

Number of Vertices Number of Ed�es Time to Generate Solution

20 37 0.054 sec29 3 0.138 sec294 1,152 25.80 sec988 10,02 774.8 sec

1 ,340 270,780 203 hours

V. ONCLUSIONIn this paper, we presented a Bellman-Ford solution to the

energy ef cient routing problem for electric vehicles. An EV model was presented and the algorithm was applied on the model. The developed algorithm proved to run fast and can be used in real-time to provide drivers the most energy e cient route.

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A Bellman-Ford Approach to Energy Efficient Routing of Ev