Fuel and Pollutant Efficient Vehicle Speed Optimization in Real Driving Conditions⁎

5th IFAC Conference on 5th IFAC Conference on Engine Powertrain Simulation and Modeling 5th IFACand Conference onControl, Engine and Powertrain Control, Simulation and Modeling Available at www.sciencedirect.com Changchun, China, September 2018 and online 5th IFACand Conference onControl,20-22, Engine Powertrain Simulation Modeling Changchun, China, September 20-22, 2018 Changchun, China, September 2018 and Modeling Engine and Powertrain Control,20-22, Simulation Changchun, China, September 20-22, 2018

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Fuel and Pollutant Efficient Vehicle Speed Fuel and Pollutant Efficient Vehicle Speed Fuel and Pollutant Efficient Vehicle Speed Optimization in Real Driving Conditions Fuel and Pollutant Efficient Vehicle Speed Optimization in Real Driving Conditions Optimization in Real Driving Conditions  Optimization in Conditions Jos´ e Manuel Luj´ an, Real Carlos Driving Guardiola, Benjam´ ın Pla,

Jos´ e Manuel Luj´ an, Carlos Guardiola, Benjam´ın Pla, Jos´ e Manuel Luj´ an, Alberto Carlos Guardiola, Benjam´ın Pla, Reig Reig Jos´ e Manuel Luj´ an, Alberto Carlos Guardiola, Benjam´ın Pla, Alberto Reig Alberto Reig CMT-Motores T´ e rmicos, Universitat Polit` e cnica de Val` Val`eencia, ncia, Spain Spain CMT-Motores T´ermicos, Universitat Polit`ecnica de (e-mail: {jlujan,carguaga,benplamo,alreiber}@mot.upv.es). CMT-Motores T´ e rmicos, Universitat Polit` e cnica de Val` e ncia, Spain (e-mail: {jlujan,carguaga,benplamo,alreiber}@mot.upv.es). CMT-Motores T´ermicos, Universitat Polit`ecnica de Val`encia, Spain (e-mail: {jlujan,carguaga,benplamo,alreiber}@mot.upv.es). (e-mail: {jlujan,carguaga,benplamo,alreiber}@mot.upv.es). Abstract: Abstract: This This paper paper analyzes analyzes the the potential potential benefits benefits in in terms terms of of fuel fuel consumption consumption and and NO NOx Abstract: This paper analyzes the potential benefits in terms of fuel The consumption and NOx emissions of an optimized speed trajectory in a real driving situation. speed management x emissions ofThis an optimized speed the trajectory in abenefits real driving situation. The speed management Abstract: paper analyzes potential in terms ofon fuel consumption andmodel NOx emissions of an optimized speed trajectory in a real driving situation. The speedvehicle management of the vehicle is approached as an optimal control problem, based a simplified of the vehicle is optimized approachedspeed as antrajectory optimal control problem, based on a The simplified model emissions of an in a real driving situation. speedvehicle management supported by experimental This control problem is twice on of the vehicle approached measurements. as an optimal control problem, based on a simplified vehicle model supported by is experimental measurements. This optimal optimal control problem is addressed addressed twice on of the vehicle is approached as an optimal control problem, based on a simplified vehicle model a daily commuting route: one for minimum fuel consumption and another for minimum NO supported by experimental measurements. This optimal control problem is addressed twice on x asupported daily commuting route: one for minimum fuel consumption and another for minimum NO x by experimental measurements. This control problem is addressed twice a daily commuting route: one for minimum fueloptimal consumption and another for minimum NOon emissions. These speed trajectories are followed in a vehicle test bench, simulating the actual x emissions. These speed trajectories are followed in a vehicle test bench, simulating the actual a daily commuting route: oneinstrumented for minimum fuel in consumption andbench, another forare minimum NOto emissions. These with speed trajectories are followed aThe vehicle test simulating the actual road conditions, a fully vehicle. experimental results compared x road conditions, with a fully instrumented vehicle. The experimental results are compared to emissions. These speed trajectories are followed in aThe vehicle testamount bench, simulating the actual the way two different drivers perform the same route in the same of time with their own road conditions, with a fully instrumented vehicle. experimental results are compared to the two different the same route The in the same amount of time their own roadway conditions, withdrivers aOptimal fullyperform instrumented vehicle. experimental results arewith compared to natural driving styles. results that optimal speed trajectory is strongly the way two different drivers perform thedemonstrate same route in the same amount of time with their own natural driving styles. Optimal results demonstrate that optimal speed trajectory is strongly the way driving twoondifferent drivers perform thedemonstrate same route in the same amount of time with their own natural styles. Optimal results that optimal speed trajectory is strongly dependent the minimization objective (either fuel or NO ), and that reductions around 4% x dependent on the minimization objective (either fuel that or NO that reductions around 4% x ),inand natural driving styles. Optimal results demonstrate optimal speed trajectory is strongly in fuel consumption and 35% in NO emissions possible the testing route compared to dependent on the minimization objective (eitherwere fuel or NO that reductions around 4% x x ),inand in fuel consumption and 35% in NO emissions were possible the testing route compared to x dependent on thedriving minimization objective (eitherwere fuel or NOx ),inand that reductions around 4% route compared to intuitive human styles. in fuel consumption and 35% in NO emissions possible the testing x intuitive human driving styles. in fuel consumption and styles. 35% in NOx emissions were possible in the testing route compared to intuitive human driving © 2018, IFAC (Internationalstyles. Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. intuitive human Keywords: Driverdriving Assistance; Powertrain Powertrain Control; Control; Hardware-in-the-loop Hardware-in-the-loop Simulation. Simulation. Keywords: Driver Assistance; Keywords: Driver Assistance; Powertrain Control; Hardware-in-the-loop Simulation. Keywords: Driver Assistance; Powertrain Control; Hardware-in-the-loop Simulation. 1. INTRODUCTION INTRODUCTION Passenberg 1. Passenberg et et al. al. (2009); (2009); Reig Reig (2017). (2017). In In this this sense, sense, RDE RDE Passenberg etthis al. factor (2009); Reig (2017). In thishomologation sense, RDE tests address as long as traditional 1. INTRODUCTION tests address this factor as long as traditional homologation 1. INTRODUCTION Passenberg etthis al.the (2009); (2017). Indriving, thishomologation sense, RDE cyclesaddress neglected effect of anas efficient so ADAS Pollution factor as Reig long traditional Pollution is is nowadays nowadays aa major major problem problem for for the the automotive automotive tests cycles neglected effect of anasefficient driving, so ADAS tests address thisthe factor as long traditional homologation such as intelligent cruise controls or driving advisor systems industry. Current regulation such as Euro 6d reflect cycles neglected the effect of an efficient driving, so ADAS Pollution is nowadays a major problem for the automotive asneglected intelligentthe cruise controls or driving advisorsosystems industry. regulation such asforEuro 6d reflect such cycles effect of an term efficient driving, ADAS Pollution isCurrent nowadays a majorpollutant problem the automotive such asbe intelligent cruise controls or driving advisor systems could important in a short Lindgren and Chen this problem with stringent and future industry. Current regulation such aslimits, Euro 6d reflect could be important in a short term Lindgren and Chen this problem with stringent pollutant limits, and future such asbe intelligent cruise or driving advisor systems industry. Current regulation suchthis aslimits, Euro Payri 6d reflect could important in (2016). acontrols short term Lindgren and Chen (2006); Ziebinski et al. this problem with stringent pollutant and future regulations are expected to follow trend et al. (2006); Ziebinski et al. (2016). regulations are expected to follow this trend Payri et al. could be important in (2016). a short term Lindgren and Chen this problem with stringent pollutant limits, and future (2015). Moreover, major cities have plans to ban diesel (2006); Ziebinski et al. regulations are expected to follow this trend Payri et al. (2015). Moreover, major to cities have plans to Payri ban diesel How drive order et al. in (2016). regulations follow this trend et al. (2006); How to to Ziebinski drive aa vehicle vehicle in order to to reduce reduce fuel fuel consumption consumption vehicles dueare to expected their increased NO emissions M¨odiesel hner (2015). Moreover, major cities have plans to ban x How to drive a that vehicle in been order present to reduce fuel consumption a question has in literature since vehicles due to their increased NO emissions M¨odiesel hner is x (2015). Moreover, major cities have plans to ban is a question that has been present in literature since (2018). Although this is expected to significantly reduce the vehicles due to their increased NO emissions M¨ o hner x How toSchwarzkopf drive a that vehicle in been order present to (1977). reduce fuel consumption (2018). Although this is expected to significantly reduce the 1970s and Leipnik During the last is a question has in literature since vehicles due to their increased NO emissions M¨ o hner 1970s Schwarzkopf and Leipnik (1977). During the last (2018). Although this is expected to significantly reduce the NO concentration in urban areas, it is counterproductive x x is a question thatbeen has been present in approaches literature since NO concentration in urban areas, it is counterproductive decades there has a wide variety of that x 1970s Schwarzkopf and Leipnik (1977). During the last (2018). Although thisinother isurban expected to significantly reduce the decades there has been a wide variety of approaches that in terms of CO and pollutants that may be present NO concentration areas, it is counterproductive x 2 Schwarzkopf and ato Leipnik (1977). During the that last in terms of CO2 andinother pollutants that may be present 1970s decades therean has been wide variety of approaches tried to give answer that question. Some examples NO concentration urban areas, it is counterproductive to give an answer thatvariety question. Some examples on gasoline engines, the problem still continues. x in terms of CO other pollutants that may be present tried 2 and so decades there has been ato wide of approaches that on gasoline engines, so the problem still continues. are works based on MPCs Li et al. (2011), using looktried to give an answer to that question. Some examples in terms of CO other pollutants that may be present are works based on MPCs Li et al. (2011), using look2 and so on gasoline engines, theautomotive problem still continues. triedworks toinformation give an answer to that question. Somefuzzy-logic examples ahead Stanger and del Re (2013), During recent years, industry has based on MPCs Li et lookon gasoline engines, theautomotive problem still continues. ahead information Stanger and delal. Re(2011), (2013), using fuzzy-logic During recent years,sothe the industry has move move are are works based on MPCs Li et al. (2011), using lookcontrollers Naranjo et al. (2003) and SDP McDonough towards complex technologies that allow to reduce the ahead information Stanger and del Re (2013), fuzzy-logic During recent years, the automotive industry has move NaranjoStanger et al. (2003) and SDP McDonough towards complex technologies that allow to reduce the controllers ahead information and del Re (2013), fuzzy-logic During recent years, the automotive industry has move controllers Naranjo et al. (2003) and SDP McDonough et al. (2012). However, if fuel or pollutant minimization is towards complex technologies that allow to reduce the pollution footprint Payri These systems aim et al. (2012).Naranjo However, ifal.fuel or pollutant minimization is pollution footprinttechnologies Payri et et al. al. (2015). (2015). These aim et (2003) and SDP McDonough towards complex that pollutant allow tosystems reduce the controllers to be addressed, only optimal control (OC) can provide the pollution footprinttargets: Payri etavoiding al. (2015). These systems aim two fundamental generation et al. (2012). However, if fuel or pollutant minimization is to be addressed, only optimal control (OC) can provide the two fundamental targets: avoiding pollutant generation al. addressed, (2012). However, if fuel or pollutant minimization is pollution footprint Payri etavoiding al. (2015). These systems aim et mathematical guarantee of best vehicle operation under (EGR, HEV, new combustion concepts, NO and λ sensors, to be only optimal control (OC) can provide the two fundamental targets: pollutant generation x guarantee of best vehicle operation under (EGR, HEV, new combustion concepts, NOx andgeneration λ sensors, mathematical to be addressed,scenario only optimal control (OC) canFrom provide the two fundamental targets: avoiding pollutant the considered Mensing et al. (2011). an OC in-cylinder pressure control) and capturing/transforming mathematical guarantee of best vehicle operation under (EGR, HEV, new combustion concepts, NO and λ sensors, x the considered scenario Mensing etvehicle al. (2011). From an OC in-cylinder control) concepts, and capturing/transforming mathematical guarantee of besthave operation under (EGR, HEV,pressure new combustion NOthe λ sensors, perspective, the considered scenario Mensing et al. experience (2011). From an OC works in literature a natural in-cylinder pressure control) and capturing/transforming the pollutants prior to their release to atmosphere x and perspective, works in literature have experience a natural the pollutants prior to their release to the atmosphere the considered scenario Mensing et al. experience (2011). From an OC in-cylinder pressure control) and capturing/transforming evolution over the time. In 1977 Schwarzkopf and Leipnik the pollutants priordevices to their release to the atmosphere with aftertreatment (DOC, DPF, SCR, LNT) Xie perspective, works in literature have a natural over the time. In 1977have Schwarzkopf anda Leipnik with aftertreatment (DOC, DPF, SCR,atmosphere LNT) Xie evolution perspective, works in optimal literature experience natural the pollutants priordevices to their release to the calculated speed profile onand a hill; then et al. (2011). Unfortunately, the benefits of any of these evolution over thethe time. In 1977 Schwarzkopf Leipnik with aftertreatment devices (DOC, DPF, SCR, LNT) Xie (1977) et al. (2011). Unfortunately, the benefits of any of these (1977) calculated the optimal speed profile on a hill; then evolution thethe time. Inthe 1977 Schwarzkopf and Leipnik with aftertreatment devices (DOC, DPF, SCR, LNT) Xie (1977) Hookercalculated et over al. (1983) found optimal speedontrajectory on technologies is limited in terms of pollution reduction, and optimal speed profile a hill; then et al. (2011). Unfortunately, the benefits of any of these technologies is limited in terms of pollution reduction, and Hooker et al. (1983) found the optimal speed trajectory on calculated thetwo optimal profile a similar hill; then et al.have (2011). any of these et al. (1983) found the speed optimal speedon trajectory on aHooker flat road between stops–whose results are to technologies isUnfortunately, limited of benefits pollutionofreduction, and (1977) they a high cost. in termsthe aHooker flat road between two stops–whose results are similar to they have a high cost. etWan al. (1983) found the optimal speed trajectory on technologies is limited in terms of pollution reduction, and athose in et al. (2016)–, and lately they included some flat road between two stops–whose results are similar to they have a high cost. those in Wan et al. (2016)–, and latelyresults they included some However, both vehicle efficiency and pollution are strongly a flat road between two stops–whose are similar to they haveboth a high cost. efficiency and pollution are strongly those road gradients in Hooker (1988). During the last decade However, vehicle in Wan et al. (2016)–, and lately they included some road gradients inal.Hooker (1988). During the last decade However,by both vehicle efficiency and pollution and are strongly affected the driving pattern Schwarzkopf Leipnik those in Wan et (2016)–, and lately they included some this topic has the many with affected by thevehicle drivingefficiency pattern and Schwarzkopf and Leipnik road gradients in Hooker (1988).of During the last decade this topic has become become the interest interest many researchers, researchers, with However, both pollution are strongly (1977). point out that the driving style might affected Studies by the driving pattern Schwarzkopf and Leipnik road gradients in use Hooker (1988).of the last decade this topic has become the interest ofDuring many researchers, with works that make of numerous techniques in order to (1977). Studies point out that the driving style might affected by the driving pattern Schwarzkopf and Leipnik works that make use of numerous techniques in order to be responsible of up to 30% of the fuel consumption (1977). Studies point out that the driving style might this topic has become the interest of many researchers, with find optimal speed trajectories at different scenarios. Some be responsible of up to 30% of the fuel consumption works that make use of numerous techniques in order to find optimal speeduse trajectories at different scenarios. Some (1977). Studiesand point out thatfound the driving style might works Hooker (1988), it has been that a sophisticated be responsible of up to 30% of the fuel consumption that make of numerous techniques in order to of these approaches are Pontryagin minimum principle Hooker (1988), and it has been found that a sophisticated find optimal speed trajectories at different scenarios. Some of these approaches are Pontryagin minimum principle be responsible of up to been 30% of consumption thethat fuela consumption Hooker (1988), could and it reduce has found sophisticated cruise control fuel by about find optimal speed trajectories at different scenarios. Some of these approaches are Pontryagin minimum principle cruise control could fuel consumption by about (PMP) Fr¨oberg et al. (2006); Dib et al. (2014); Petit et al. Hooker (1988),to and it reduce hasconstant been found thathighway a sophisticated ooberg (2006); Dib et (2014); Petit et 5% compared typical speed cruise control could reduce fuel consumption bydriving about (PMP) of theseFr¨ approaches are Pontryagin principle (PMP) Fr¨ berg et et al. al. (2006); Dib Dynamic et al. al.minimum (2014); Petit et al. al. (2011); Sciarretta et al. (2015), Programming 5% compared to typical constant speed highway Sciarretta et al. (2015), Dynamic Programming cruise control to could reduce fuel consumption bydriving about (2011); 5% compared typical constant speed highway driving  (PMP) Fr¨ o berg et al. (2006); Dib et al. (2014); Petit et al. (DP) Hellstr¨ o m et al. (2006); Hellstr¨ o m et al. (2009); The authors acknowledge the support of Spanish Ministerio de (2011); Sciarretta et al. (2015), Dynamic Programming  The 5% compared to typical the constant highway driving authors acknowledge supportspeed of Spanish Ministerio de (DP) Hellstr¨ om etetal.al.(2006); om Programming et al. (2009); (2011); Sciarretta (2015), Hellstr¨ Dynamic  The authors Econom´ ıa, Industria y Competitividad through project TRA2016(DP) Hellstr¨ o m et al. (2006); Hellstr¨ o m et al. (2009); acknowledge the support of Spanish Ministerio de Econom´ ıa, Industria y Competitividad through project TRA2016 The authors 78717-R. (DP) Hellstr¨om et al. (2006); Hellstr¨om et al. (2009); acknowledge the supportthrough of Spanish Ministerio de Econom´ ıa, Industria y Competitividad project TRA201678717-R. 78717-R. Econom´ıa, Industria y Competitividad through project TRA201678717-R. 78717-R. 2405-8963 © © 2018 2018, IFAC IFAC (International Federation of Automatic Control) Copyright 248 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2018 IFAC 248 Control. Peer review responsibility of International Federation of Automatic Copyright © 2018 IFAC 248 10.1016/j.ifacol.2018.10.041 Copyright © 2018 IFAC 248

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Mass [kg] Drag coefficient [-] Frontal area [m2 ] Friction coefficient [-] Engine type Displacement [cm3 ] Max power [kW] Max torque [Nm] Breathing method Emissions control

1580 0.35 2.2 0.01 Euro 6, inline-four, CI 1598 96 @ 4000 rpm 320 @ 1750 rpm turbocharger, intercooler, VGT HP-EGR, LP-EGR, DOC, DPF, LNT

Table 1. Experimental vehicle specifications. Hellstr¨om et al. (2010); Ozatay et al. (2014) and Direct Methods (DM) Saboohi and Farzaneh (2009). There are many promising results in literature but, unfortunately, most of these works are purely theoretical. Few include experimental data on real vehicles Hellstr¨om et al. (2009); Hellstr¨om et al. (2010); Ozatay et al. (2014), but still only took into consideration fuel consumption as the optimization target, leaving aside pollutant emissions on their algorithms. In this context, the contribution of the present work is to analyze the effect of both fuel consumption and NOx emissions minimization on a real route by addressing an optimal control problem (OCP), and examine how it translates in terms of fuel and pollution to an actual vehicle with experiments on a vehicle test bench. The potential benefits of an optimal speed management will be compared to the performance of a natural driving style of several drivers. In order to achieve the above targets, first the tree necessary elements for an OCP are described in section 2: experimental facilities and vehicle information (section 2.1), a vehicle model (section 2.2) and the optimization algorithm (section 2.3). Then, several runs on the same route are performed experimentally in section 3, both tracking optimal speed trajectories and with arbitrary driving styles, followed by a discussion of the presented results. Finally, main conclusions and potential future works are stated in section 4. 2. TOOLS 2.1 Experimental facilities The experimental tests, required to fit the vehicle model and validate optimal speed trajectories, have been performed with a factory standard SUV. It is a FWD vehicle equipped with a turbocharged Euro 6 compression ignited engine, featuring HP-EGR and LP-EGR (they are used sequentially at the factory calibration), and a combination of DOC, DPF and LNT. The main specifications of this vehicle can be found in table 1. The powertrain has been instrumented with an additional NOx probe and a rapid prototyping dSpace system in order to acquire fuel, emissions and performance measurements. This setup gathers information from both the factory in-vehicle sensors–by accessing to the ECU variables in real time through ETK port–and custom instrumentation at additional CAN networks. 249

Fig. 1. Measured operating points as steady state engine characterization. The model characterization and validation experiments are performed in a dynamic vehicle test bench equipped with torque meters at each wheel. This system allows to simulate road conditions such as track grade and vehicle dynamics, while keeping the vehicle standstill in a laboratory under controlled conditions with additional sensors and peripherals. This vehicle test bench features two different modes: (i) constant speed, where the braking torque is modulated such that vehicle runs at constant speed, and (ii) road load simulation, where a speeddependent braking torque is applied at the wheels in order to recreate the vehicle dynamics in a predefined road profile (road friction, aerodynamic drag, road grade and vehicle inertia). This system is capable to provide ±250 kW at the wheels, so it can simulate both accelerations and overrun phases. Test campaign The vehicle powertrain has been characterized with a large set of experimental data. These measurements include a wide region of the engine operating map, more particularly between 1000 and 2500 rpm and from overrun to approximately 65% load, covering typical operating conditions of road driving. Figure 1 represents all the operating points that have been measured. These data has been gathered at nominal engine temperature and in steady state conditions by using the constant speed mode at the vehicle test bench, keeping constant injection rate by bypassing ECU variables. The recorded variables include wheel torque, engine speed, airpath temperatures and pressures, injection parameters, coolant and oil temperature, EGR valves position, VGT position and NOx emissions. 2.2 Model A vehicle model is required by the optimization problem in order to properly define the optimization objective–fuel, emissions or any other minimization target–and the vehicle mission–to cover a particular route. The vehicle model must include dynamic effects as well as the powertrain behavior. For the scope of this work, a simple model approach has been chosen in order to improve the optimization algorithm efficiency, as long as more complex models usually mean harder optimization problems and increased computation time.

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227

At vehicle dynamics level, the model considers typical longitudinal forces acting on the chassis, namely the aerodynamic drag Milliken and Milliken (1995): Fa =

1 ρACd v 2 2

(1)

with ρ the air density, A the front area of the vehicle, Cd the drag coefficient and v the vehicle speed; the road friction Waltermann (1996): Fr = cr mg cos α

(2)

with cr the road friction coefficient (typically between 0.010 and 0.015 Rajamani (2011)), m the vehicle mass, g the gravity acceleration and α the road gradient; the effect of gravity: Fg = mg sin α

(3)

Fi = mv˙

(4)

the vehicle inertia:

the engine torque contribution to the wheels: Tw Ft = rw

(5)

with Tw the engine torque at the wheels and rw the wheel radius; and the effect of the brakes: Fb = ub Fb

(6)

with ub the brakes actuation (0 is released position and 1 is fully braking) and Fb the maximum braking force. Summing up all the above terms results in the following ordinary differential equation (ODE): Ft − Fa − Fr − Fg − Fb v˙ = m

(7)

which defines the longitudinal dynamics of the vehicle. The powertrain is composed of a transmission system and an engine. On the one hand, the transmission is modeled as an ideal set of gears Guzzella and Sciarretta (2005); its inefficiencies will be implicitly considered at the engine model. On the other hand, the engine is modeled with a quasi-steady approach. This is a strong simplification for a nonlinear system such as an internal combustion engine, but there are several reasons for this choice: (i) a fully dynamic engine model increases in several orders of magnitude the complexity of the optimization problem with its corresponding increase in computational time, (ii) vehicle dynamics are much slower than engine dynamics, (iii) optimal speed profiles use to be smooth in order to avoid inefficient transients and therefore the operation of the engine is expected to be pretty steady, and (iv) the engine control is performed by the factory ECU and no control over its variables is available in this work. The engine is therefore represented by its equivalent operating maps Guzzella and Onder (2004). Fuel consumption 250

Fig. 2. Fuel consumption (in kg/h) and NOx emission (in mg/s) engine maps used for the powertrain model. ˙ nox (the main variables of interest m ˙ f and NOx emissions m in this study) are tabulated as a function of engine speed N and pedal position up , as shown in figure 2. These maps are constructed from the experimental tests described in section 2.1.1 at the operating points denoted in figure 1. Torque output is calculated directly from the pedal position and the maximum and minimum torque curves (Tmax (N ) and Tmin (N ) respectively): T = up (Tmax − Tmin ) + Tmin

(8)

Note that the maximum torque curve considered in this work is the corresponding to the top points at figure 1. The kinetic relations between engine and wheel torques as well as vehicle speed and engine speed are given by the following expressions Guzzella and Sciarretta (2005): Tw = Rgb (ugb )T N=

30Rgb (ugb )v πrw

(9) (10)

where Rgb is the total transmission ratio–includes both gearbox and differential ratios–which is a function of the selected gear ugb . The gear is selected as a function of vehicle speed with the policy shown in figure 3. 2.3 Optimization method There are mainly three families of optimization methods to address an optimal control problem (OCP) Diehl et al. (2006)–dynamic programming (DP), direct methods and indirect methods –each of them with their own advantages and drawbacks. Due to the particular characteristics of

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has been issued by introducing v, which allows to express the fuel consumption (originally as an increment per unit of time) in distance.

Fig. 3. Gear selection policy as used both at the model and actual vehicle. the OCP in this work, DP is the algorithm of choice for the following reasons: (i) the model described before is simple, featuring only one state (i.e. a single ODE, shown in equation (7)), and DP handles simple problems easily, (ii) DP accepts discrete quantities (such as gearbox ratios) and non-explicit functions (such as engine maps), while other algorithms do not, and (iii) the solution achieved by DP is guaranteed to be the global solution to the problem instead of a local minimum. DP is based on the Hamilton-Jacobi-Bellman (HJB) equation Lewis and Syrmos (1995) and it is a direct application of Bellman’s Principle of Optimality (BPO) Bellman (1954). The main idea is to split the complete problem into a finite number of small problems that can be easily solved. Hence, the time domain of the problem is divided into a number of discrete time instants ti , where the following sub-problem is to be addressed Bryson and Ho (1975): min u



ti+1 ti

L(x, u, t)dt + J (x, ti+1 )



(11)

In the above single-step problem, L stands for the cost of the minimization objective (fuel consumption or NOx emissions in this work), x are the states of the problem (vehicle speed), u the controls (pedal position and brakes), and J is called the cost-to-go function. This function contains the cost for the transition from any state value, from ti+1 to the end of the problem, as calculated and stored in previous steps of the global problem. Therefore, the idea is to minimize the above expression step-by-step, starting at the last time step of the problem and moving backwards to t = 0 while accumulating the costs in J . The only requisite is that the chosen control set does not violate any of the problem constraints, which can be easily addressed by assigning J = ∞ for those infeasible cases. Further information about DP can be found in Geering (2007); Vinter (2010). Optimal control problem statement The main target in this work is to minimize the fuel consumption for a vehicle covering a prescribed route. This idea can be translated into an OCP by stating the following objective function: J=



S 0

m ˙f ds v

(12)

that must be minimized at the whole route length S. Note that although the typical domain in OCPs is time, it has been shifted above by distance s, as long as the route characteristics (speed limits and track grade) are defined in distance and not in time Hellstr¨om et al. (2006); Hellstr¨om et al. (2009); Hellstr¨om et al. (2010). This domain change 251

As stated before, the problem features just one state, the vehicle speed. Regarding controls, originally they are pedal up , brakes ub and gear number. The last is controlled according to the gear policy shown in figure 3. Pedal and brakes have been merged into a single control u as long as simultaneous actuation of both is counterintuitive for an optimal driving. Merging the controls also avoids unnecessary calculations for the DP algorithm such as cases where pedal and brakes are actuated simultaneously, avoiding what Bellman calls the curse of dimensionality Bellman (1956). In order to properly define the rest of requisites for the vehicle to perform the route, several constraints must be defined. First of all, the vehicle must start and finish the route at standstill: v(0) = 0 v(S) = 0

(13)

The vehicle speed has to be kept below road limits: v(s) ≤ v(s)

(14)

And also the route has to be driven within a given time limit (such that the average speed is 75 km/h for the results in this study): 

S 0

1 ds ≤ T v

(15)

The above constraints fully define the OCP to be solved. Unfortunately, DP is not able to include integral constraints such as equation (15). However, there is a workaround consisting in adjoining this constraint to the cost function Monastyrsky and Golownykh (1993) as a direct consequence of the calculus of variations Pontryagin et al. (1962); Vinter (2010). In this situation, the integral constraint vanishes from the problem definition, and the augmented objective function to be minimized results:

J=



S 0



1 m ˙f +µ v v



ds

(16)

with µ a constant multiplier whose appropriate value is such that the OCP solution fulfills the former integral constraint. This multiplier is found by shooting. In addition to this OCP described with the equations above, a similar OCP has been also stated for this study: minimization of NOx emissions with the same vehicle and route requisites. All previously described constraints apply for NOx minimization, and only the objective function changes. Therefore, equation (16) is shifted for:

J=



S 0



1 m ˙ nox +µ v v



ds

(17)

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urban

highway

229

rural

Fig. 4. Speed limits (up) and track elevation profile (down) at the real driving route. Gray areas denote the urban, highway and rural parts of the road. The blue elevation profile comes from cartographic data, while the red one is a simplified profile used in the vehicle test bench. Note that most of it remains the same, and that the integral constrain is still adjoined. 3. CASE STUDY 3.1 Case description The target of this study is to explore the potential of optimal speed profiles in real driving conditions. To do so, the aforementioned optimization method has been used to calculate in advance the optimal speed sequence that should be followed in a known route, based on the described vehicle model. The output of this optimization problem is a sequence of target speed vs distance pairs to be followed in order to minimize fuel, NOx or a weighted combination of both. The chosen route is a 33 km daily commute between two cities in Spain, consisting of urban sections, highway driving and rural roads. Track grade and speed limits (shown in figure 4) are introduced as constraints to the optimization problem, while traffic conditions and traffic lights are neglected as long as their characterization and pseudorandom nature would exceed the scope of this study. Firstly, two human drivers did this route several times with a natural driving style on the vehicle test bench. They were asked to reach at all tests a target average speed of 75 km/h in order to get comparable results. Only speed limits and current track grade were shown to the drivers during the cycle. These results are used as a benchmark to the optimized speed profiles, in order to quantify their advantage compared to typical human driving. Then, a trained driver was asked to follow an optimal speed trajectory with an average speed of 75 km/h as well. An ad-hoc graphical interface was set up showing a plot with the current speed and the target speed within the next 250 meters. This was repeated twice, one for a minimum fuel consumption speed profile and another for minimum NOx emissions. The ability of the driver to follow the target speed profile was measured with the diagram represented 252

Fig. 5. Comparison between vehicle speed with a human driver and the target speed shown to the driver. The color scale is the time in percentage spent at a point in the diagram. in figure 5, which shows the correlation between target and actual vehicle speed. According to this, most of the time the driver satisfactorily followed the target and, in fact, the correlation mostly collapses to the diagonal which is a good indicator. During all road simulations the vehicle kept fully instrumented, including additional NOx probe and full ECU access. In order to provide fair comparisons, all tests were performed with engine at working temperature and all driving aids deactivated. The vehicle test bench was set to road load simulation mode, introducing the current road grade profile. Note that the elevation data used in the vehicle test bench is a simplification of the actual elevation profile (the red profile at figure 4) due to test bench limitations. 3.2 Experimental results The minimum fuel OCP shows an quite constant speed trajectory as shown in figure 6. It may be appreciated that, in general, the target speed (in blue) tries to reach the speed limits in both urban and rural segments, as long as these speeds are not too high to produce excessive losses and allow the vehicle to run slower in highway (recall that the vehicle must meet an average speed of 75 km/h at the end of the route). There are few exceptions, such as the 80 km/h speed limit at s = 2 km or the 90 km/h speed limit at s = 31 km, where vehicle speed kept far from the limit as long as there are more restricting speed limits in the short term (60 and 40 km/h respectively) and the use of brakes is avoided by the optimal solution unless strictly necessary–it wastes energy. During highway phase speed is low–always below 90 km/h, even with 120 km/h limits–in order to reduce losses, specially those because of aerodynamic reasons. It is also remarkable the effect of track grade, resulting in a lower speed as the grade increases that allows to keep an efficient engine operating point. As a side note, the human driver trajectory is also plotted in figure 6 (in yellow) to give an idea of its accuracy and to complement the analysis shown before in figure 5. The minimum NOx emissions solution shows a completely different speed trajectory, as shown also at figure 6 (in red). The main idea to strongly reduce NOx emissions by

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Fig. 6. Optimal target speed trajectory for minimum fuel consumption ( ), and the vehicle speed at the test bench with a human driver following this reference ( ), and optimal target speed for minimum NOx emissions ( ). Road elevation profile ( ) and road speed limits ( ) are shown as well. changing the way the vehicle is driven is keeping within the EGR zone and avoiding strong transients, and this explains most of the particularities of this solution. First of all, this trajectory is much smoother than the minimum fuel solution. Speed is no longer reduced in the highway section and, in fact, it reaches the road speed limits several times. Contrarily to the minimum fuel situation, in this case the vehicle has to keep a high pace at highway to compensate the slower speed at the rural section. During the rural phase speed is much slower because of two factors: (i) speed limits alternate making impossible for the vehicle to reach higher speeds while keeping this steady driving style, and (ii) track grade increases so the vehicle has to reduce the average speed in order to stay at the EGR zone. Track grade effect is even stronger for minimum NOx emissions. Whenever track grade increases, speed experiences a transition to a lower value. This contrasts with the effect appreciated in the minimum fuel solution, where speed moved between levels much faster. In the case of minimum NOx emissions is not the speed that is reduced for efficiency reasons but the occurrence of two factors that are pretty obvious between s = 18 km and s = 21 km: on the one hand the engine load is increased in order to keep a fast pace and save some time to spend driving even slower at the rural section while, on the other hand the load cannot be increased much more without getting out of the EGR zone. The balance between those two factors result in a lower torque output than that necessary to keep the speed and, therefore, speed slowly decreases. Probably with a wider EGR zone, speed might be kept at the road limits. Some other examples of reduced speed as a consequence of the track grade–because higher load is detrimental for NOx emissions–are s = [3, 6] km and specially s = [32, 33] where grade is 7%. Another interesting analysis is inspecting how the engine operating points shifts from one region to another depending on the minimization objective. In this sense, figure 7 shows the cumulative distribution function for fuel consumption and NOx emissions at both OCPs, i.e. for minimum fuel consumption (blue) and minimum NOx emissions (red). Therefore, as it can be expected, when fuel consumption is minimized, the engine operates more time in lower fueling rate zones–about 80% of the time fuel consumption is lower than 4.1 kg/h, while it is lower than 5.7 kg/h if NOx is minimized. NOx emissions show a similar behavior: the engine operates more time in the EGR zone when NOx emissions are minimized (91%) than when fuel is minimized (81%). This translates into the fact that 80% 253

Fig. 7. Cumulative distribution function for fuel consumption (up) and NOx emissions (down) at the engine operating points resulting from the optimal speed profiles. : optimization results for minimum fuel consumption, : optimization results for minimum NOx emissions. The amount of time the engine was running within the EGR zone is shown in the lower plot for both optimization results. of the time NOx emissions are lower than 6.4 mg/s when NOx is minimized, while they are below 7.6 mg/s for the fuel minimization case. A last relevant indicator is the comparison between the experiments following optimal speed targets and the arbitrary driving cycles performed at the same route under the same constraints. Figure 8 show the total fuel consumption and NOx emissions for all these experiments. As it can be appreciated, driving following a fuel minimization speed trajectory (blue bars) could reduce fuel consumption by 4% compared to natural driving styles (yellow bars), while reaching nearly the same NOx emissions level. This is a significant reduction in fuel consumption with almost no impact on emissions, showing up the importance of an appropriate speed management. Similarly, the minimum NOx emissions speed trajectory (red bars) achieved a reduction of 35% in NOx emissions compared to typical human driving styles. Intermediate solutions between those

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Fig. 8. Accumulated fuel and NOx emission values resulting from the experimental tests. : optimal speed for minimum fuel consumption, : optimal speed for minimum NOx emissions, : average of arbitrary cycles. two optimal cases are also possible by combining fuel consumption and NOx emissions in the objective function. 4. CONCLUSIONS Calculation of optimal speed trajectories are possible with simple vehicle models–which means a quick computation– being of application to real driving situations and replicating results in a vehicle test bench. Significant fuel and NOx reductions are possible by improving the vehicle speed sequence compared to a natural human driving style, while keeping the same average speed. Particularly, in the studied route, a 4% reduction in fuel consumption and 35% in NOx emissions were achieved with the presented methodology. These results show the importance of an appropriate speed management in vehicles, as long as it can bring more benefits in terms of CO2 and NOx emissions than any engine technology alone with an insignificant cost. However, the true challenge with speed optimization is the availability of information Winstead and Kolmanovsky (2005). In this work the route was considered to be known in advance, but that is not necessarily the case in a real world situation. Without look-ahead information a true optimization is not possible, and only heuristic solutions can be used. This fact really limits the application of the presented methodology, but there are still some cases where route information is already available. This is the situation of intelligent cruise controllers in combination of a navigation system with elevation data or a semi-autonomous vehicle driving in a given route. Also, a side application may be an advising system–such as the one developed in the present work to carry out the tests–that not only suggests but also educates the driver in order to improve its driving style during the route. A further extension of this work could be the incorporation of engine control to the speed optimization problem. A custom engine calibration or even an optimal control would probably maximize the engine efficiency for the optimal driving style as shown in simulations in Reig (2017). Also, it is known that engine control can benefit from the availability of a speed profile in advance Luj´an et al. (2018). In this sense, the optimal speed target could be forwarded to a low level engine controller such that the ECU may be aware of future load changes. Among other effects, this information allows the engine to build pressure upon future request, eliminating the necessity to keep a constant torque reserve that spoils engine efficiency. 254

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