Robust Optimal Eco-driving Control with Uncertain Traffic Signal Timing
LL A TEX CLASS FILES, ACC 2018 1
Robust Optimal Eco-driving Controlwith Uncertain Traffic Signal Timing
Chao Sun, Xinwei Shen, and Scott Moura,
Member, IEEE
Abstract —This paper proposes a robust optimal eco-drivingcontrol strategy considering multiple signalized intersections withuncertain traffic signal timing. A spatial vehicle velocity profileoptimization formulation is developed to minimize the globalfuel consumption, with driving time as one state variable. Weintroduce the concept of effective red-light duration (ERD),formulated as a random variable, to describe the feasible passingtime through signalized intersections. A chance constraint isappended to the optimal control problem to incorporate ro-bustness with respect to uncertain signal timing. The optimaleco-driving control problem is solved via dynamic programming(DP). Simulation results demonstrate that the optimal eco-drivingcan save fuel consumption by 50-57% while maintaining arrivaltime at the same level, compared with a modified intelligent drivermodel as the benchmark. The robust formulation significantlyreduces traffic intersection violations, in the face of uncertainsignal timing, with small sacrifice on fuel economy compared toa non-robust approach.
Index Terms —Eco-driving, Optimal , Robust control, Trafficsignal, Stochastic.
I. I
NTRODUCTION
Connected and automated vehicle (CAV) technology isrevolutionizing the automotive industry. In particular, CAVsmay significantly improve safety, energy economy, and con-venience. CAVs are able to realize autonomous driving, ve-hicle to infrastructure (V2I) communication and/or intelligentpath/velocity planning [1]. Optimal eco-driving control – anovel technology brought by CAVs – is defined as a velocitycontrol method to achieve the most economical fuel, energyor cost performances [2]. Intuitively speaking, optimal eco-driving seeks the best velocity profile, in some sense, overa specific driving mission. Fig. 1 illustrates the optimal eco-driving concept through V2I communication with a numberof traffic signals incorporated.In the literature, optimal eco-driving is also known asecological driving, speed trajectory planning, driving advisoryor driver assistance systems. Over the past 10 years, in par-ticular, the optimal eco-driving problem has been intensivelystudied in the published literature. A heuristic optimal eco-driving strategy is proposed in [3] to minimize the vehiclefuel consumption based on instantaneous fuel performance.With velocity constraints derived from real driving data, an-other dynamic programming (DP) based optimal eco-drivingcontrol is developed in [4] for trajectory optimization of aninternal combustion engine (ICE) vehicle. Similar approaches
C. Sun and S. Moura are with the Department of Civil and EnvironmentalEngineering, University of California, Berkeley, CA, 94704 USA. e-mail:[email protected], [email protected]. Shen is with the Tsinghua-Berkeley Shenzhen Institute, Shenzhen,China. email: [email protected] received September 17, 2017.
Host Vehicle Traffic
Signals
Red
Light i i+1 i+2 Distance
Time Speed
ControlVehicle to Infrastracture
Communication
Fig. 1. Car optimal eco-driving based on V2I communication with multipletraffic signals incorporated, with vehicle speed control as the main task. are found in [5], [6], for the optimal energy management aswell as speed control of electric vehicles (EV). Experimentalresults showed a significant increase in energy efficiency. Morecomprehensively, a cloud-based velocity profile optimizationapproach is designed in [7], under a spatial domain formula-tion. Historical velocity data is gathered for speed advising.Spatial domain optimization is further adopted by [8] forecological driving. Uniquely, a short-term adaptation level isadded to avoid traffic congestion. Optimal eco-driving hasalso been integrated into the energy management strategy ofhybrid electric vehicles, with interactive Pontryagins MinimalPrinciple used to solve the optimization problem in [9].Signal phase and timing (SPaT) information is critical inaddressing the optimal eco-driving problem. In [10] and [11],hierarchical model predictive control (MPC) is employed foreco-driving in varying traffic environments. Assume the SPaTinformation is known a priori , [12] solved the optimal eco-departing problem at signalized intersections. Furthermore,[13] developed a sophisticated on-board driver assistance,which is able to calculate the optimal speed profile withdeterministic traffic signals. By considering each signalizedintersection as one stage, a multi-stage pseudospectral controlmethod is proposed by [14] in an arterial road structure. Hi-erarchical MPC is also adopted in [15], and has demonstratedeffective online eco-driving control capabilities. Reference[16] considers the car waiting queue in a multi-lane road sce-nario, and designed an eco-cooperative adaptive cruise controlscheme. With a simplified powertrain model and assumingthe engine mainly operates along the optimal brake specificfuel consumption (BSFC) line, sequential convex optimizationtherefore is applied to speed trajectory planning [17].In the aforementioned studies, signalized intersections areeither not considered, or the SPaT information is assumed tobe deterministic in the optimal eco-driving control. Ideally,when CAVs have realized V2I communication, SPaT canbe communicated to vehicles for optimal eco-driving. This a r X i v : . [ m a t h . O C ] F e b A TEX CLASS FILES, ACC 2018 2
Uncertainty 1:
Car Waiting Queue
Uncertainty 3:
Pedestrian Crossing Uncertainty 2:
Traffic Light Time Variation
Fig. 2. Uncertain factors when passing the road intersections. future, however, would require significant penetration of V2I-equipped intersections, which may take decades to realize.Even with V2I-equipped intersections, uncertainty exists dueto car waiting queue, pedestrians, bicyclists, varying patternsof traffic lights and other factors, as demonstrated in Fig. 2.Moreover, an optimal eco-driving approach assuming deter-ministic SPaT will often pass through intersections exactlyat the phase transitions, and thus risks collision. The issue ofSPaT uncertainty in optimal eco-routing is significant, and notfully addressed in the existing literature.This paper investigates a fuel-minimizing eco-driving ap-proach that is robust to uncertain feasible vehicle passingtimes through multiple signalized intersections. The goal is tosimultaneously achieve energy economy and safety. The maincontributions include: • Effective red-light duration (ERD) is proposed to describethe stochastic feasible passing time of vehicles at sig-nalized intersections, composed of a deterministic basered-light duration and a random delay; • Signalized intersections are modeled and integrated intothe spatial optimal eco-driving formulation, which elim-inates the requirement for prior knowledge of accuratearrival time; • A robust optimal eco-driving control variant is developedand solved via the dynamic programming. The controllerrobustness – and therefore safety – is significantly im-proved with little sacrifice of fuel economy.The remainder of the paper is organized as follows. Section IIdescribes the vehicle, traffic signal and driver models. SectionIII introduces the spatial optimal eco-driving control strategy.Section IV details a robust formulation that considers uncertainfeasible passing time at signalized intersections. Section Vexhibits the main results, and Section VI draws the mainconclusions and future work.II. V
EHICLE , T
RAFFIC S IGNAL AND D RIVER M ODELING
A. Vehicle dynamics
The subject vehicle is equipped with a gasoline ICE anda 6-speed gearbox. Since speed control is the main objectiveof optimal eco-driving, we consider longitudinal vehicle dy-namics and disregard the lateral dynamics. The longitudinalacceleration is calculated by ma = r gb T eng R whl − mgcos ( θ ) C r − mgsin ( θ ) − ρAC d v − T brk (1) C r = C r + C r v (2)where m is the vehicle mass, a is the acceleration, r gb isthe integrated ratio of gearbox and final drive, T eng is theICE output torque, R whl is the rolling radius of wheel, g isthe gravitational acceleration, θ is the road grade, and C r isthe rolling resistance coefficient. Parameters ρ , A , C d are theair density, frontal area, and air-dragging resistance coefficientrespectively. Variable v is the vehicle velocity, T brk is thebraking force enforced on the wheels, C r and C r are rollingresistance constants. The longitudinal velocity is computed by v = ω eng r gb (3)where ω eng is the ICE rotation speed. The ICE fuel consump-tion is modeled as a nonlinear map φ ( · , · ) that depends on theengine torque and speed: ˙ m fuel = ψ ( T eng , ω eng ) (4)where ˙ m fuel is the instantaneous fuel consumption, and ψ isthe pre-stored fuel map (e.g. a look-up table). The transmissionefficiency is ignored in this study. Assume r fd is the finaldrive ratio. The integrated transmission ratio is formulated asa function of the gear number N gb , r gb = f ( N gb ) r fd , N gb ∈ { , , , , , } (5) B. Traffic signal model
The traffic signal at an intersection is a spatial-temporalsystem in the optimal eco-driving control problem. Assumethe total length of the target driving route is D f . The positionof the i th traffic signal is noted as D i if we treat the signalizedintersection as a single point on the road. Therefore, D i ∈ [0 , D f ] , i = { , , , , ...I } (6)where I is the total number of traffic signals along the route.Each traffic signal is modeled with an independent signal-cycling clock in this paper. The universal traveling time of thevehicle is denoted as t ∈ R , and the periodic cycling clocktime of the i th traffic signal has a period of c if ∈ R (clock timezero denotes the beginning of the red light phase). Normally,the period c if varies at different intersections. The red-lightduration is denote by c ir . Then we have c ir ∈ [0 , c if ] (7)Consider the time when the vehicle departs from its origin.Denote by c i the periodic signal clock time at this moment.Suppose t ip is the time at which the subject vehicle passesthrough the i th intersection in the universal time domain. Wecan compute the corresponding time in the periodic trafficsignal clock timing by c ip = ( c i + t ip ) mod c if (8)where c ip is the vehicle passing time in the signal-cyclingclock. The modulo operator allows for conversion from theuniversal time domain to the periodic traffic signal clock timedomain. Note that un-signalized intersections or crossings canalso be integrated into the model above, which might requireon-board cameras or radars to detect the passing conditions. A TEX CLASS FILES, ACC 2018 3
C. Modified intelligent driver model
A modified intelligent driver model (IDM) is introduced forcomparison with the optimal eco-driving, by imitating humandriving behaviors. IDM is originally developed by Treiber et.al. , based on the computation of desired distance between thesubject vehicle and the vehicle in front or speed limit [18]. Weenhanced the driver model with an ability to preview trafficsignals and adjust speed accordingly. Assume the desireddistance between the subject vehicle and front vehicle is D des ,then D des = D mindes + v · t hw − vD sf √ a max a c (9)where D mindes is the minimal vehicle distance, t hw is the desiredtime headway to the preceding vehicle, D sf is the real distancebetween the subject vehicle and preceding vehicle, a max is themaximal vehicle acceleration ability, and a c is the preferreddeceleration for comfort.The vehicle acceleration at each time step is computed bycomparing the desired gap distance with the current distance.An additional speed limit term is added to ensure safety, a = a max (cid:34) − (cid:16) vv max (cid:17) − (cid:18) D des D sf (cid:19) (cid:35) (10)To interact with traffic signals or stop signs, we modify theIDM by enabling the driver model to preview the traffic signalor stop line status at a human-vision distance D v . Assume thecurrent location of the vehicle is D , the vehicle longitudinalvelocity dynamics in (10) thus becomes a = a max [1 − (cid:0) vv max (cid:1) − (cid:16) D des D sf (cid:17) ] , if S tss ( D + D v ) = 0 . − v D sf , if S tss ( D + D v ) = 1 . (11)where S tss ( D + D v ) is the traffic signal and stop sign status D v in front of the vehicle, with the value of 1 meaning thetraffic signal is red or there is a stop sign in front, with thevalue of 0 meaning the traffic signal is green or there is nostop sign. Variable D sf here indicates the distance to the trafficlight or stop sign when no vehicle is in front.III. D ETERMINISTIC O PTIMAL E CO - DRIVING
The optimal eco-driving control problem is formulated as anonlinear spatial trajectory optimization problem to minimizevehicle fuel consumption. The cost function J is defined asminimize J = (cid:90) D f ˙ m fuel ( T eng ( D ) , ω eng ( D )) dD (12)The engine torque, wheel braking torque and transmissiongear number are chosen as the control variables. u = [ T eng ( D ) , T brk ( D ) , N gb ( D )] (13)The vehicle velocity and traveling/driving time are chosen asthe state variables. x = [ v ( D ) , t ( D )] (14) dv ( D ) dD = a ( D ) v ( D ) ; dt ( D ) dD = 1 v ( D ) (15) Subject to the following vehicle physical constraints, T mineng ≤ T eng ( D ) ≤ T maxeng , ∀ D [0 , D f ] T minbrk ≤ T brk ( D ) ≤ T maxbrk , ∀ D [0 , D f ] N gb ( D ) ∈ { , , , , , } , ∀ D [0 , D f ] v (0) = v ( D f ) = 0 a min ≤ a ( D ) ≤ a max , ∀ D [0 , D f ] v min ( D ) ≤ v ( D ) ≤ v max ( D ) , ∀ D [0 , D f ] (16)Subject to the following final arrival time and traffic signalpassing constraints, t ( D f ) ≤ t f (17) c ip ≥ c ir (18)A key beneficial feature of a spatial trajectory formulation(as opposed to temporal) is that the signal and final destinationarrival times do not need to be known a priori . A pre-setmaximal arrival time constraint t f is imposed on the finalstate variable t ( D f ) , to balance fuel economy and travelingspeed. The traffic signal constraint in (18) enforces vehiclesto pass through signalized intersections only at green lights.The above nonlinear optimization problem is solved via dy-namic programming adopted from [19]. Detailed formulationsare omitted here. Interested readers please refer to [20], [21].IV. R OBUST O PTIMAL E CO - DRIVING
In Section III, it is assumed the SPaT information isdeterministic and perfectly known. Mathematically, c ir in (18)is known and deterministic. However, as illustrated in Fig. 2,the feasible passing time through signalized intersections orcrossings is usually uncertain and random. Here, an effectivered-light duration (ERD) variable is defined to describe thefeasible passing time, denoted as c iERD : c iERD = c ir + α (19)Fig. 3 exhibits the ERD concept. Parameter c ir is the base red-light duration, which is the minimal red-light time. Randomvariable α is a stochastic time of delay, caused by signaluncertainties or vehicle waiting queue. In this paper, weassume the total signal cycling-time is not affected by theseuncertain factors, meaning c if is deterministic and known.Intuitively, α is a random variable over time 0 to ( c if − c ir ) , whose distribution could be (truncated) Poisson, Gaussian,Beta or completely non-parametric. Assume the probabilitydensity function of α is f ( α ) . Therefore, the traffic signalpassing constraint in (18) can be modified to c ip ≥ c iERD = c ir + α, ∀ α (20)However, enforcing the constraint above for all values in thesupport of α is too restrictive. Consequently, we relax thisconstraint via chance constraints.Denote by η a required reliability for the subject vehicleto pass through a specific signalized intersection, and F ( α ) indicates the cumulative distribution function (CDF) of α .Equation (20) can be relaxed into the following chance con-straint, Pr ( c ip ≥ c ir + α ) ≥ η (21) A TEX CLASS FILES, ACC 2018 4
Host Vehicle Traffic SignalsDeterministicSituation ProbabilisticSituation i i+1 i+2
Stochastic Part of ERD Effective Red-light Duration (ERD)
Base Red-light Time
Fig. 3. Effective red-light duration, meaning the feasible passing time of avehicle through an intersection. Pr ( α ≤ c ip − c ir ) = F ( c ip − c ir ) ≥ η (22)We assume the CDF F ( · ) is bijective, and therefore has an in-verse function F − ( · ) . Thus, we can solve for the optimizationvariable c ip to obtain c ip ≥ c ir + F − ( η ) (23)Again, c ip is the passing time of subject vehicle through the i th intersection in the signal-cycling clock, which is a functionof the control and state variables, c ip ( x, u ) .It should be noted that the real world probability distributionof ERD might vary at different times of day, seasons orlocations, and may not be accurately modeled by a parametricdistribution. Investigating the actual probability distribution of α via measured data is planned as future work. DP is also usedto solve the above robust optimal eco-driving control problem.V. S IMULATION
The vehicle parameters and engine fuel map used for simu-lation are extracted from Autonomie [22], and summarized inTable I. The maximum and minimal velocity limits are set as16 and 0m/s, respectively. The acceleration constraints are notactivated here, because the engine output torque constraint willrestrict the vehicle acceleration within the feasible domain.Three different cases are considered for comparison in thesimulation: • Modified IDM. with the human preview-vision distance D v set as 100 meters; • Optimal eco-driving with traveling time as cost, denotedas “Op-time”. Equation (12) is re-formulated as J = (cid:90) D f t ( D )) dD (24) • Optimal eco-driving with fuel consumption as the cost,denoted as “Op-fuel”.
A. Deterministic optimal eco-driving
Two sample driving routes with 3 and 7 signalized inter-sections, named route 1 and route 2 respectively, are studiedin this paper. All of the full cycling periods c if and red-lightdurations c ir are intentionally set as 60s and 30s, respectively,for easier analysis of the results. The beginning time c i isarbitrarily selected between 0 to 30s. However, other realistic TABLE IS
UBJECT V EHICLE P ARAMETERS
Parameter(Unit) Value Parameter(Unit) Value m (kg) 1745 r fd R whl (m) 0.3413 ω maxeng (rad/s) 600 A (m ) 2.841 T maxeng (Nm) 240 ρ (kg/m ) 1.1985 gearbox ratios 4.584,2.964, C d C r selections of the full cycling time and red-light durationcan also be incorporated in the proposed optimal eco-drivingcontrol strategy. The position and timing information for allthe two sample routes are shown in Table II. TABLE IIP
OSITION AND SP A T INFORMATION OF SAMPLE ROUTES
Route 1 Route 2 1&2No. Type D i (m) c i * Type D i (m) c i * c if c ir For sample route 1, the vehicle velocity and traveling timeresults derived from the three driving strategies are plottedin Fig. 4. It can be seen in the modified IDM approach, thedriver started decelerating the vehicle at D =100m when the itsees a red traffic signal in front. Because of the lack of fullSPaT information, the modified IDM is not able to previewthe future signal dynamics. About 10 seconds later, it had toswitch to accelerate the vehicle again at D =170m, as the signalturned to green. This behavior wastes fuel.At the 2nd traffic signal, the red light blocks the intersection.Modified IDM waits for 20 seconds until the light turns green.A similar scenario happened at the 3rd signalized intersection,but with a shorter waiting time. The vehicle eventually arrivedat the stop sign (also its destination) at t =117s.The Op-time strategy accelerates the vehicle wheneverpossible until the velocity hits the maximal boundary. Underthis strategy, the vehicle commonly meets red lights. In the800-meter-long route assumed in this section, the vehiclecumulatively waited for 50s at all three intersections. Thevehicle arrived at the final stop sign at t =107s, which is theshortest time among the three driving strategies.In the Op-fuel case, t f is set as 115s in order to makesure the vehicle arrives to the destination at the same timescale as the other two cases. The Op-fuel strategy smoothlypasses through the first two intersections, by adjusting thevelocity between 7 and 9m/s. A deeper deceleration happenedjust before driving through the third signalized intersection A TEX CLASS FILES, ACC 2018 5 S peed ( m / s ) Modified IDM 0 200 400 600 80005101520 Op-time Case 0 200 400 600 80005101520 Op-fuel Case0 200 400 600 800
Distance (m) T i m e ( s ) red light stop sign 0 200 400 600 800 Distance (m)
Distance (m)
Fig. 4. Modified IDM, Op-time and Op-fuel results with deterministic formulation of route 1. S peed ( m / s ) Modified IDM 0 500 1000 150005101520 Op-time Case 0 500 1000 150005101520 Op-fuel Case0 500 1000 1500
Distance (m) T i m e ( s ) red light stop sign 0 500 1000 1500 Distance (m)
Distance (m)
Fig. 5. Modified IDM, Op-time and Op-fuel results with deterministic formulation of route 2. ( D =600m) to wait for a green light. After that, the vehiclevelocity restored to about 12m/s to ensure it can arrive thefinal destination within the time limit. The total driving timein the Op-fuel case is t =110.5s, which is 3.5s longer than theOp-time case.The vehicle velocity and traveling time results for route2 are shown in Fig. 5, where similar trends are observed.The modified IDM avoided complete stops at 3 signalizedintersections out of 7, with a final arrival time of 226s. Thisindicates that even without any future information of the trafficsignals, the vehicle can still catch green light at normal drivingpattern. However, in the Op-time case, the vehicle encountered6 red lights in order to minimize the arrival time. The finalarrival time is 217.9s, which is about 8s (3.5%) smaller thatthe modified IDM. As expected, the Op-fuel controller refusedto aggressively accelerate the vehicle, and crossed most of the intersections at lower speeds without any complete stops.Eventually, the Op-fuel arrived at the destination at t =228.5s.The vehicle velocity, acceleration, engine speed, and enginetorque trajectories for route 1 are shown in Fig. 6. The enginespeed is restricted between 200 and 300 rad/s in the Op-fuelcase, and the engine torque is relatively smaller than the othertwo approaches, forming a milder driving style.The operating points on the brake specific fuel consumption(BSFC) map for route 1 are shown in Fig. 7. The engine op-erating point results in Fig. 7 may seem counter-intuitive, butare actually very interesting. The engine operating points fromthe Op-time case are located more in the high efficiency areaof the BSFC map (the lower value the better), compared withthe Op-fuel and modified IDM approaches. This is usuallypreferable in the operation of engines, and often results inbetter fuel economy. However, the simulated fuel consumption A TEX CLASS FILES, ACC 2018 6 S peed ( m / s ) Time (s) -4-202468 A cc e l e r a t i on ( m / s ) Modified IDMOp-timeOp-fuel E ng i ne S peed (r ad / s ) Time (s) E ng i ne T o r que ( N m ) Bounday in Op-time
Fig. 6. Vehicle velocity, acceleration, engine speed, torque results versus timeof route 1.
Engine Speed (rad/s) E ng i ne T o r que ( N m ) BSFCModified IDMOp-timeOp-fuel
Fig. 7. Engine operating points on the BSFC map of route 1. in the Op-time case is, in fact, the highest one and much higherthan the Op-fuel result. The main reason is that although theaverage engine fuel efficiency in the Op-fuel approach is lower,its total engine power requirement is much less. Thus, betterfuel economy is achieved with modified IDM and Op-fuel.The arrival time, average BSFC value, and total engine fuelconsumption results of route 1 and 2 are reported in Table III.The average BSFC of Op-fuel is 16.9-21.2% higher than thatof Op-time, while the overall fuel consumption is 51.9-59.5%less. This significant fuel economy improvement is achievedby sacrificing 2.8-4.9% of the arrival time, which is trivial indaily driving.
B. Robust optimal eco-driving
In this section, we assume α is a truncated Gaussian randomvariable for illustrative purposes, with the PDF and CDF drawnin Fig. 9. The true probability distribution for the ERD isgenerally unknown. Future work will focus on this challenge.Three possible scenarios are shown in Fig. 9: light, moderateand heavy traffic situations. When traffic is light, the high-probability values of α are around 0 to 6s, indicating smalltime delays. For heavy traffic scenarios (for example, at rushhours), the average α value is much larger at around 15s. It TABLE IIIA
RRIVAL TIME , AVERAGE
BSFC
AND FUEL CONSUMPTION RESULTS OFROUTE AND WITH DETERMINISTIC SP A TRoute Method t ( D f ) (s) B (cid:5) avg (g/kWh) Fuel (g)Modified IDM 117 478.26 88.24800m Op-time 107 485.94 91.493 lights Op-fuel 110 568.19 43.95Change* +2.8% +16.9% -51.9%Modified IDM 226 477.31 172.841600m Op-time 217.9 486.62 182.157 lights Op-fuel 228.52 586.53 73.79Change* +4.9% +21.2% -59.5%* Indicates the performance change of Op-fuel compared with Op-time; (cid:5) B avg is the average BSFC value. is even possible that the vehicle would have to wait for thenext green signal, which is normal in real life. The moderatetraffic situation is adopted for simulation, where α ∼ N [0 , (6 , ∈ [0 , (25)The vehicle traveling trajectories of Op-fuel control alongroute 1, with various chance reliability η are illustrated in Fig.8. As can be seen in Fig. 8(a), the arrival time increases asreliability parameter η increases. For η = 0 . , the optimalvelocity trajectory has very little robustness to delay beyondthe base red-light time. This can be clearly seen from thevehicle trajectories at the 600-meter traffic signal, where thevehicle passed the intersection immediately after the lightswitched to green. As η increases, the passing time graduallyincreases as the solution becomes more cautious to delayedERD. Yet at the 200-meter and 400-meter signalized intersec-tions, the originally planned passing time has already avoidedmost of the possible delays. Thus, the trajectories with chancereliability η from 0.1 to 0.8 are quite identical.Fig. Fig. 8(b) shows the normalized fuel consumptionand arrival time results of Op-fuel with different reliabilitiesenforced along route 1. When η is 0.9, the fuel consumptionincreased 16% compared with η = 0 . . The arrival timechange is much smaller, with an increase of only 5%. At the η =1.0 point, the arrival time is raised by nearly 8%, yet thefuel consumption decreased by 2%. Fig. 10 is the comparisonof Op-time and Op-fuel results under the robust optimal eco-driving control formulation, when η =0.3,0.6,0.9 respectively.The main difference is that the Op-time control is less tolerantto the delay uncertainty, and thus its performance decline ismore severe than the Op-fuel control.Table IV summaries the arrival time, average BSFC andfuel consumption results of robust optimal eco-driving andmodified IDM with both driving route 1 and 2. Comparingdeterministic control with robust control across the threemethods, we find that the final arrival time grows as thereliability η increases. However, the fuel consumption increaseis not as significant. Obviously, the fuel consumption of Op-fuel is less than that of Op-time and modified IDM. Its arrivaltime is slightly longer than Op-time, but mostly shorter thanmodified IDM. A TEX CLASS FILES, ACC 2018 7
Time (s) D i s t an c e ( m ) red light stop sign Op-fuel Case(a) Reliability =1.0Reliability =0.9Reliability =0.8Reliability =0.7Reliability =0.6Reliability =0.5Reliability =0.4Reliability =0.3Reliability =0.2Reliability =0.1PDF of delay
Reliabity N o r m a li z ed P e r f o r m an c e (b) Fuel ConsumptionArrival Time
Fig. 8. (a) Vehicle traveling trajectories of Op-fuel with different chance reliability η used along route 1 under the moderate traffic scenario. (b) Normalizedfuel consumption and arrival time results of Op-fuel control with different reliabilities enforced along route 1. Time Delay (s) CD F P D F Light TrafficModerate TrafficHeavy Traffic
Fig. 9. Probability density function and cumulative density function of α . VI. C
ONCLUSIONS AND F UTURE W ORK
This paper proposes a novel robust optimal eco-drivingcontrol strategy to solve the vehicle velocity planning problemwith multiple signalized intersections, based on a spatial opti-mization formulation. The requirement for prior knowledge ofthe destination arrival time is eliminated. We propose a noveltraffic signal modeling approach. Effective red-light duration(ERD) is proposed to capture the random feasible passingtime at signalized intersections. The optimal control problemis solved via dynamic programming (DP). Simulation resultsindicate that the developed optimal eco-driving strategy is ableto reduce fuel consumption by approximately 50-57%, whilemaintaining the arrival time at the same level compared withthe modified intelligent driver model. The controller robustnessto signal timing uncertainty is greatly improved with slightsacrifices to vehicle fuel economy.Future work includes real-world traffic SPaT probabilitydistribution study. We also plan to develop methods to reducethe optimal eco-driving control computation complexity. D i s t an c e ( m ) red light stop sign (a) Op-time Case0 20 40 60 80 100 120 140 Time (s) D i s t an c e ( m ) red light stop sign (b) Op-fuel Case Reliability =0.3Reliability =0.6Reliability =0.9Uncertain Red-light Delay Fig. 10. Comparison of robust Op-time and robust Op-fuel control resultswhen η =0.3,0.6,0.9, respectively. R EFERENCES[1] A. Sciarretta, G. De Nunzio, and L. L. Ojeda, “Optimal ecodrivingcontrol: Energy-efficient driving of road vehicles as an optimal controlproblem,”
IEEE Control Systems , vol. 35, no. 5, pp. 71–90, 2015.[2] Q. Jin, G. Wu, K. Boriboonsomsin, and M. J. Barth, “Power-basedoptimal longitudinal control for a connected eco-driving system,”
IEEETransactions on Intelligent Transportation Systems , vol. 17, no. 10, pp.2900–2910, 2016.[3] Y. Saboohi and H. Farzaneh, “Model for developing an eco-drivingstrategy of a passenger vehicle based on the least fuel consumption,”
Applied Energy , vol. 86, no. 10, pp. 1925–1932, 2009.[4] F. Mensing, E. Bideaux, R. Trigui, and H. Tattegrain, “Trajectoryoptimization for eco-driving taking into account traffic constraints,”
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TABLE IVA
RRIVAL TIME , AVERAGE
BSFC
AND FUEL CONSUMPTION RESULTS OFROUTE AND WITH STOCHASTIC SP A TRoute Method Type t ( D f ) B (cid:5) avg FuelModified Deterministic 117 478.26 88.24IDM Robust η =0.3 121 479.33 89.48Robust η =0.6 125 478.10 91.09Robust η =0.9 129 481.93 90.63Op-time Deterministic 107.2 485.94 91.49800m Robust η =0.3 111.1 485.52 92.113 lights Robust η =0.6 114.0 485.31 92.22Robust η =0.9 118.2 485.38 91.78Op-fuel Deterministic 110.4 568.19 43.95Robust η =0.3 112.3 557.59 48.14Robust η =0.6 114.6 562.23 50.43Robust η =0.9 118.4 567.61 53.16Modified Deterministic 226 477.31 172.84IDM Robust η =0.3 229 478.90 172.72Robust η =0.6 233 480.97 172.71Robust η =0.9 237 482.98 172.70Op-time Deterministic 217.9 486.62 182.151600m Robust η =0.3 221.5 486.66 181.247 lights Robust η =0.6 224.6 487.08 181.80Robust η =0.9 227.7 485.87 183.18Op-fuel Deterministic 228.52 586.53 73.79Robust η =0.3 229.6 579.84 72.66Robust η =0.6 230.8 576.53 72.98Robust η =0.9 232.4 564.48 72.85 (cid:5) B avg is the average BSFC value.[6] M. Kuriyama, S. Yamamoto, and M. Miyatake, “Theoretical study oneco-driving technique for an electric vehicle with dynamic program-ming,” in Electrical Machines and Systems (ICEMS), 2010 InternationalConference on . IEEE, 2010, pp. 2026–2030.[7] E. Ozatay, S. Onori, J. Wollaeger, U. Ozguner, G. Rizzoni, D. Filev,J. Michelini, and S. Di Cairano, “Cloud-based velocity profile optimiza-tion for everyday driving: A dynamic-programming-based solution,”
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