Towards an Analytical Eco-Driving Cycle Computation for Conventional Cars

9th IFAC International Symposium on Advances in Automotive 9th 9th IFAC IFAC International International Symposium Symposium on on Advances Advances in in Automotive Automotive Control 9th IFAC International Symposium onAvailable Advancesonline in Automotive Control at www.sciencedirect.com Control 9th IFAC International Symposium on Advances in Automotive Orléans, France, June 23-27, 2019 Control Orléans, France, France, June June 23-27, 23-27, 2019 2019 Orléans, Control Orléans, France, June 23-27, 2019 Orléans, France, June 23-27, 2019

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Towards an Analytical Eco-Driving Cycle Towards an Analytical Eco-Driving Cycle Towards an Analytical Eco-Driving Cycle Computation for Conventional Cars Towards an Analytical Eco-Driving Cycle Computation for Conventional Cars Computation for Conventional Cars Computation for Conventional Cars ∗∗ ∗ ∗ ∗ ∗

Emmanuel Emmanuel Nault Nault ∗∗ Guillaume Guillaume Colin Colin ∗∗ Kristan Kristan Gillet Gillet ∗∗ Emmanuel Nault Guillaume Colin Kristan Gillet ∗ ∗∗ ∗ Madjid ∗∗ Nicolas ∗∗ Emmanuel Nault Guillaume Colin Gillet ∗∗∗ Yann Chamaillard Zerar Dollinger ∗ ∗∗ ∗∗ Yann Chamaillard Madjid Zerar Nicolas Dollinger ∗ Madjid ∗ Kristan Yann Chamaillard Zerar Nicolas Dollinger Emmanuel Nault Kristan Gillet ∗∗ ∗ Guillaume Colin ∗∗∗∗ ∗∗ Yann Chamaillard C´ Madjid Zerar Nicolas Dollinger e dric Nouillant ∗∗ C´ e dric Nouillant ∗ edric Nouillant Yann Chamaillard C´ Zerar ∗∗∗∗Nicolas Dollinger ∗∗ C´ eMadjid dric Nouillant ∗∗ C´ e dric Nouillant ∗ ∗ Orl´ e ans, PRISME, EA 4229, F45072, ∗ Univ. Univ. Orl´ e ans, PRISME, EA 4229, F45072, Orl´ ans, France (e-mail: eans, PRISME, EA 4229, F45072, Orl´ Orl´eeeans, ans, France France (e-mail: (e-mail: ∗ Univ. Orl´ eans, PRISME, EA 4229, F45072, Orl´eans, France (e-mail: [email protected]). [email protected]). ∗ Univ. Orl´ [email protected]). Univ.∗∗ Orl´ e ans, PRISME, EA 4229, F45072, Orl´ e ans, France (e-mail: ∗∗ PSA [email protected]). ∗∗ PSA Peugeot Citro¨ Citro¨eeen, n, Direction Direction Recherche Recherche Innovation Innovation & & Citro¨ n, Direction Recherche Innovation & ∗∗ PSA [email protected]). Citro¨en, Direction Recherche Innovation & (DRIA), France Technologies (DRIA), France ∗∗ PSA PeugeotTechnologies Technologies (DRIA), France PSA PeugeotTechnologies Citro¨en, Direction Recherche (DRIA), France Innovation & Technologies (DRIA), France Abstract: This paper proposes an analytical methodology to obtain fuel optimal vehicle Abstract: Abstract: This This paper paper proposes proposes an an analytical analytical methodology methodology to to obtain obtain aaa fuel fuel optimal optimal vehicle vehicle Abstract: This paper proposes an analytical methodology to obtain a fuel optimal vehicle speed trajectory in order to realize a trip with a conventional vehicle with an internal speed trajectory in order to realize a trip with a conventional vehicle with an internal speed trajectory in order to realize a trip with a conventional vehicle with an internal Abstract: This paper proposes an analytical methodology to obtain a fuel optimal vehicle speed trajectory orderIn realize a trip with a conventional with Pontryagin’s an internal combustion engine (ICE). In order to give an analytical formulation of the issue, Pontryagin’s combustion engine order to an formulation of combustion enginein (ICE). Into order to give give an analytical analytical formulation vehicle of the the issue, issue, Pontryagin’s speed trajectory in(ICE). orderIn towas realize aThe trip with a conventional vehicle with an internal combustion engine (ICE). order to give an analytical formulation of the issue, Pontryagin’s Maximum Principle (PMP) used. expression considers the driving constraints (e.g. Maximum (PMP) was used. The considers the constraints (e.g. Maximum Principle Principle (PMP) was used. Theanexpression expression considers theofdriving driving constraints (e.g. combustion engine (ICE). In order to give analytical formulation the issue, Pontryagin’s Maximum Principle (PMP) was used. The expression considers the driving constraints (e.g. consumption, distance, duration) with respect to control variables (e.g. time, position, vehicle consumption, distance, duration) with respect to control variables (e.g. time, position, vehicle consumption, distance, duration) with respect to controlconsiders variablesthe (e.g. time, constraints position, vehicle Maximum Principle (PMP) was used. The expression driving (e.g. consumption, duration) with of respect to controlis position,solution. vehicle speed, engine torque). The knowledge of an optimality is needed to find the optimal solution. speed, torque). knowledge an needed find the speed, engine engine distance, torque). The The knowledge of an optimality optimality is variables needed to to(e.g. findtime, the optimal optimal solution. consumption, distance, duration) with of respect to control variables (e.g. time, position, vehicle speed, engine torque). The knowledge an optimality is needed to find the optimal solution. The reference solution used to compare results was Dynamic Programming (DP). The analytical The solution used to compare results was Programming (DP). The The reference reference solution used to compare of results was Dynamic Dynamic Programming (DP). The analytical analytical speed, engine torque). The knowledge an optimality is needed to find the optimal solution. The reference developed solution used results Dynamic (DP). The analytical methodology developed in this paper led to +1.1% error in consumption compared to DP methodology in this paper to +1.1% error in compared to methodology developed in to thiscompare paper led led to aaawas +1.1% errorProgramming in consumption consumption compared to DP DP The reference solution used to compare results was Dynamic Programming (DP). The analytical methodology developed in this paper led to a +1.1% error in consumption compared to DP simulations for the same trip with the same constraints of rendez-vous. simulations for the same trip with the same constraints of rendez-vous. simulations fordeveloped the same in trip with the same constraints of rendez-vous. methodology this paper led to a +1.1% error in consumption compared to DP simulations for the same trip with the same constraints of rendez-vous. simulations the same trip with the same constraints of rendez-vous. © 2019, IFACfor (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Keywords: Eco-driving, Analytical method, conventional vehicle, Pontryagin’s Maximum Keywords: Eco-driving, Eco-driving, Analytical Analytical method, method, conventional conventional vehicle, vehicle, Pontryagin’s Pontryagin’s Maximum Maximum Keywords: Eco-driving, Analytical method, conventional vehicle, Pontryagin’s Maximum Principle Principle Principle Keywords: Principle Eco-driving, Analytical method, conventional vehicle, Pontryagin’s Maximum Principle 1. INTRODUCTION (Dib et al., 2014) defined it as an optimal control problem. 1. (Dib 1. INTRODUCTION INTRODUCTION (Dib et et al., al., 2014) 2014) defined defined it it as as an an optimal optimal control control problem. problem. 1. INTRODUCTION (Dib et al., 2014) defined it as an optimal control problem. (Sciarretta et al., 2015) and (F. Mensing, 2013) developed (Sciarretta et al., 2015) and (F. Mensing, 2013) developed (Sciarretta et al., 2015) and (F. Mensing, 2013) developed 1. INTRODUCTION (Dib et al., 2014) defined it as(F. an optimal control problem. (Sciarretta et al.,on 2013)an methods based on the PMP. In this paper, an analytimethods the PMP. In this analytimethods based based on2015) the and PMP. InMensing, this paper, paper, andeveloped analytiThis paper deals with energy use in transport, and in in (Sciarretta et al., 2015) and (F. Mensing, 2013) developed This This paper paper deals deals with with energy energy use use in in transport, transport, and and in methods based on the PMP. In this paper, an analytical formulation of the Pontryagin’s Maximum Principle cal formulation of the Pontryagin’s Maximum Principle cal formulation of the Pontryagin’s Maximum Principle This paper deals with energy use in transport, and in particular with fuel savings. For ecological, economic or methods based on the PMP. In this paper, an analytiparticular with fuel savings. For economic or particular with fuel savings. Foruseecological, ecological, economic or formulation of the Pontryagin’s Principle (PMP) will be explored. We propose Hamiltonian forThis paper deals with energyFor in transport, and in cal (PMP) will We aaa Hamiltonian for(PMP) will be be explored. explored. We propose proposeMaximum Hamiltonian forparticular with fuel savings. ecological, economic or resource reasons (B. Saerens, 2012), transport has to use cal formulation of the the Pontryagin’s Maximum Principle resource reasons (B. Saerens, 2012), transport has to use resource reasons (B. Saerens, 2012), transport has to use (PMP) will be explored. We propose a Hamiltonian formulation and describe different assumptions needed particular with fuel savings. For ecological, economic or (PMP) mulation and the different assumptions needed mulationwill andbedescribe describe the different assumptions needed resource Saerens, one 2012), has energy to use less fuel. In the literature, one of the optimal energy explored.the Wedifferent proposeassumptions a Hamiltonian forless In literature, of the less fuel. fuel.reasons In the the (B. literature, one of transport the optimal optimal energy and describe needed to develop the model. resource reasons (B. Saerens, 2012), transport has energy to use mulation to develop the model. to develop the model. less fuel. In the literature, one of the optimal approaches proposed for vehicles is eco-driving. mulation and describe the different assumptions needed approaches proposed for vehicles is eco-driving. approaches proposed for vehicles is eco-driving. to develop the model. less fuel. Inaims the to literature, one is ofeco-driving. the optimal energy approaches for vehicles Eco-driving determinean economical or ecological Eco-driving aims economical or Eco-driving proposed aims to to determinean determinean economical or ecological ecological to develop the model. approaches proposed for vehicles is eco-driving. Eco-driving aims to economical or ecological This This paper proposes an analytical methodology to be used driving strategy. It consists in an Energy Management driving It consists Energy This paper paper proposes proposes an an analytical analytical methodology methodology to to be be used used driving strategy. strategy. It determinean consists in in an an Energy Management Management Eco-driving aimsenergy to determinean economical or ecological This paper proposes an analytical methodology to be used driving strategy. It consists in an Energy Management on-line in order to determine the best driving strategy System to save during a trip. Depending on the on-line in order to determine the best driving strategy System to save energy during a trip. Depending on the on-line in order to an determine themethodology best drivingtostrategy System to save energy duringinaan trip. Depending on the This paper proposes analytical be used driving strategy. It consists Energy Management in order to determine strategy System to save energy during a trip. Depending the on-line considering the evolution of the trip. The aim adapt technology of the vehicle architecture, the energy saved considering the of trip. The aim is to adapt technology of vehicle architecture, the saved considering the evolution evolution of the the the trip.best Thedriving aim is is to to adapt technology of the the vehicle architecture, the energy energyon saved in speed order to determine the best driving strategy System to save energy during aEven trip. ifDepending on the on-line considering the evolution of the trip. The aim is to adapt technology of the vehicle architecture, the energy saved the vehicle according to changes in the environment. can be fuel, electricity or both. some basic rules the vehicle speed according to changes in the environment. can be fuel, electricity or both. Even if some basic rules the vehicle speed according to changes in the environment. can be fuel, electricity or both. Even if some basic rules the evolution oftothe trip. The aim isontoan adapt technology ofeconomically, the vehicle architecture, the energy saved the speed according changes environment. can or both. Even if to some basic speed rules considering The new methodology must be able to work unexist to drive it is is possible possible to optimize speed The new must be able to work on exist to drive economically, it optimize Thevehicle new methodology methodology must be able in to the work on an an ununexistbe to fuel, driveelectricity economically, it is possible to optimize speed vehicle speed according to be changes in the environment. can be fuel, electricity or both. Even if to some basic speed rules the The new methodology must able to work on an unexist to drive economically, it is possible optimize known cycle and enable faster calculations. trajectories with mathematical approaches. known cycle and enable faster calculations. trajectories with mathematical approaches. known cycle and enable faster calculations. trajectories with mathematical approaches. The new methodology must be able to work on an unexist to drive economically, it is possible to optimize speed cycleis enable faster calculations. trajectories The paper is structured as follows. Section presents Eco-driving issues concern kinds The structured as Section Eco-driving issues concern two kinds of transport probThe paper paper isand structured as follows. follows. Section 222 presents presents Eco-driving with issuesmathematical concern two twoapproaches. kinds of of transport transport probprob- known known cycleis and enable faster calculations. trajectories with mathematical approaches. The paper structured as follows. Section 2 presents Eco-driving issues concern two kinds of transport probthe vehicle model and the asumptions made about its lems: fuel consumption and pollutant emissions. In order the vehicle model and the asumptions made about its lems: and pollutant emissions. In order the vehicle model and the asumptions made 2about its lems: fuel fuel consumption consumption and two pollutant emissions. In proborder The paper is structured as section follows.describes Section presents Eco-driving issues concern kinds some of transport the vehicle model and the asumptions made about its lems: fuel consumption and pollutant emissions. In order environment. The following the analytto reduce consumption and pollution, mathematical environment. The following section describes the analytto reduce consumption and pollution, some mathematical environment. The following section describes analytto reduce consumption and pollution, some mathematical vehicle model and the section asumptions madethe about its lems: fuel consumption consumption and pollution, pollutant emissions. In 2018) order the The developed. following to reduce some mathematical ical methodology developed. Section reports the global approaches have been developed. (Khalik et al., ical Section 444 reports the global approaches have developed. (Khalik et ical methodology methodology developed. Sectiondescribes reportsthe theanalytglobal approaches have been been and developed. (Khalik et al., al., 2018) 2018) environment. environment. The following section describes the analytto reduce consumption and pollution, some mathematical methodology developed. Section 4 reports the global approaches been based developed. et Quadratic al., 2018) ical results comparing analytical simulation compared with established strategy based on Sequential Quadratic results comparing the analytical simulation compared with established strategy on Sequential results comparing the the analytical simulation compared with established aaahave strategy based on aaa (Khalik Sequential Quadratic ical methodology developed. Section 4 reports the global approaches have been developed. (Khalik et al., 2018) results comparing the analytical simulation compared with established a strategy based on a Sequential Quadratic DP simulations. Programming (SQP) algorithm to work on a complex forDP simulations. Programming (SQP) algorithm to work on a complex forDP simulations. Programming (SQP) algorithm to work on a complex forcomparing the analytical simulation compared with established athestrategy based onproblem awork Sequential Quadratic DP simulations. Programming algorithm on a complex for- results mulation of optimal control and decomposed mulation optimal control problem and decomposed mulation of of the the(SQP) optimal control to problem and decomposed Programming (SQP) algorithm to work on a complex for- DP simulations. mulation of the optimal control problem and decomposed it into several SQP subproblems. it SQP subproblems. it into into several several SQP subproblems. mulation of the optimal control problem and on decomposed 2. MODEL 2. it into several SQP subproblems. Another way to solve the problem is based Dynamic 2. MODEL MODEL Another way to solve the Another way to solve the problem problem is is based based on on Dynamic Dynamic 2. MODEL it into several SQP subproblems. Another way to solve the problem is based on Dynamic Programming (DP) (Bellman and Dreyfus, 1962). This Programming (DP) (Bellman and Dreyfus, 1962). This 2. MODEL Programming (DP) (Bellman and Dreyfus, 1962). This 2.1 Constraints and Modeling Another way to solve the based on Dynamic Asumptions Programming (DP) (Bellman and is Dreyfus, This 2.1 method is very useful forproblem off-line problems. (Maamria method useful for off-line problems. (Maamria 2.1 Constraints Constraints and and Modeling Modeling Asumptions Asumptions method is is very very useful for off-line problems.1962). (Maamria Programming (DP) (Bellman and Dreyfus, 1962). This 2.1 Constraints and Modeling Asumptions method is very useful for off-line problems. (Maamria et al., 2016) explored different methods and compared et explored different methods and et al., al., 2016) 2016) explored different methods and compared compared 2.1 Constraints and Modeling Asumptions method is very useful for off-line problems. et al., 2016) explored methods and (Maamria compared the computation times different required. They compared results The The vehicle simulated in this paper is conventional car the computation times required. They compared results the computation times required. They compared results The vehicle vehicle simulated simulated in in this this paper paper is is aaa conventional conventional car car et al., 2016) explored different methods andet compared the computation times required. They compared results The vehicle simulated in this paper is a conventional car given by time and space approaches. (Bouvier al., 2015) with an internal combustion engine (ICE). It is assumed given by time and space approaches. (Bouvier et al., 2015) with an internal combustion engine (ICE). It is assumed given by time and space approaches. (Bouvier et al., 2015) with an internal combustion engine (ICE). It is assumed the computation times They compared vehicle simulated in this paper is a conventional car given by time space required. approaches. (Bouvier et al.,results 2015) The with an trajectory internal combustion is assumed compared two computational strategies, one off-line based that the trajectory includes no slopes and no It turns. All the the compared two computational strategies, one based that includes slopes and turns. All compared two and computational strategies, one off-line off-line based that the the trajectory includes no noengine slopes (ICE). and no no turns. All the given by time and space approaches. (Bouvier et al.,on-line 2015) with an trajectory internal combustion engine (ICE). It is assumed compared two computational strategies, one off-line based that the includes no slopes and no turns. All the on the PMP (Pontryagin et al., 1962) and another notations used are defined in Table 1. The vehicle model on (Pontryagin et 1962) another on-line notations used defined Table 1. The vehicle model on the the PMP PMP (Pontryagin et al., al., 1962) and and another on-line notations used are are includes defined in in Table 1. The vehicle model compared two computational strategies, one off-lineSystem based that thedetailed trajectory noone slopes and no vehicle turns. All the on the PMP (Pontryagin et al., 1962)Management and another on-line notations used are defined in Table The model suboptimal Equivalent Consumption Management will be in two two parts, for 1. the vehicle dynamics suboptimal Equivalent Consumption System will be detailed in parts, one for the vehicle dynamics suboptimal Equivalent Consumption Management System will be detailed in two parts, one for the vehicle dynamics on the PMP (Pontryagin et al., 1962) and another on-line notations used are defined in Table 1. The vehicle model suboptimal detailed two one for vehicle dynamics (ECMS). and another one for the ICE and its the consumption model. (ECMS). and another one the ICE its consumption model. (ECMS). Equivalent Consumption Management System will and be another oneinfor for theparts, ICE and and its consumption model. suboptimal will be detailed two one for vehicle dynamics (ECMS). Equivalent Consumption Management System and another oneinfor theparts, ICE and its the consumption model. (ECMS). and another oneLtd. for All therights ICEreserved. and its consumption model. 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier

Copyright © 2019 IFAC 550 Copyright ©under 2019 responsibility IFAC 550Control. Peer review© of International Federation of Automatic Copyright 2019 IFAC 550 Copyright © 2019 IFAC 550 10.1016/j.ifacol.2019.09.087 Copyright © 2019 IFAC 550

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Table 1. Definitions of parameters Notation ci

Definition Coefficients of the road load equation (i ∈ {0, 1, 2})

m m ˙f Peng Plim

Equivalent vehicle mass Fuel mass flow rate Engine power Engine power with the best efficiency called limit power Idle fuel mass flow Optimal fuel mass flow Gear box ratio Differential ratio Wheel radius Time Engine torque Vehicle speed Position Transmission efficiency Engine speed

Q0 Qlim RGB Rt rtire t Teng v x η ωeng

d0 (t) =

Unit N N.(m.s−1 ) N.(m.s−1 )2 kg g.s−1 W W

551

RGB (t).Rt rtire

(4)

The resistant forces, which consider friction, can be defined by the road load equation: Fr (t) = c0 + c1 .v(t) + c2 .v(t)2

(5)

Finally, the expression of the acceleration is:

g.s−1 g.s−1 − − m s N.m m.s−1 m − rad.s−1

a=

1 dv(t) = .(η.d0 .Teng (t) − c0 − c1 .v(t) − c2 .v(t)2 ) (6) dt m

2.3 ICE Modeling

Driving behaviour on a journey depends on the contraints that govern it. The first constraint to consider is the legal speed limit. Secondly, on a journey, a driver does not want to take too long to reach his destination. Thirdly, there are physical constraints that concern the engine capabilities and are given by the engine map. The vehicle limitations are due to Newton’s second law and driving comfort.

The ICE used in this study is a Diesel engine, but it could easily be extended to a gasoline one. In order to consider the consumption of the ICE, the goal is to model the consumption depending on the load power. From an engine map, the ICE consumption will be modeled thanks to a two-slope regression with (Hadj-Said et al., 2017):

m ˙f =



(7)

With the parameters:  

To constrain the engine and the vehicle behaviour, the following assumptions were made: • no slopes on the journey, • no turns on the journey, • no reverse gear (position is positive and increasing and speed is positive), • during slow down the engine runs in engine braking mode (no consumption), • at stops, the engine is off (no consumption).

α1 .Peng + Q0 if Peng ≤ Plim α2 .(Peng − Plim ) + Qlim if Peng > Plim

2 Q0 = q0 + q1 .ωeng + q2 .ωeng P = p + p .ω  Qlim = α0 .P 1 +eng Q0 lim 1 lim

(8)

The ICE power is defined by:

Peng = Teng .ωeng

(9)

Moreover, passing by the driveline, the engine speed is directly related to the vehicle speed:

2.2 Vehicle and Transmission Modeling To determine the acceleration of a conventional car on a flat road, the Fundamental Principle of Dynamics in a longitudinal study gives: m.

dv(t) = Ft (t) − Fr (t) dt

(1)

where m is the equivalent vehicle mass containing the rotational parts. We can define the traction forces Ft (t) thanks to the relation between the input and the output of the transmission, the engine torque Teng and the wheel torque Twh : Twh (t) = rtire .Ft (t) = η.RGB (t).Rt .Teng (t)

(2)

We deduce: Ft (t) = η.d0 (t).Teng (t)

(3)

With the definition of the transmission ratio:

551

ωeng = d0 .v

(10)

Accordingly, the fuel mass flow can be defined as the following system:

m ˙f =



σT .Teng .v + σ0 + σ1 .v + σ2 .v 2 if Peng ≤ Plim τT .Teng .v + τ0 + τ1 .v + τ2 .v 2 if Peng > Plim (11)

With the coefficients shown in Table 2. Table 2. Coefficients of fuel mass flow rate Index T 0 1 2

σ α1 .d0 q0 q1 .d0 q2 .d20

τ α2 .d0 (α1 − α2 ).p0 + q0 [(α1 − α2 ).p1 + q1 ].d0 q2 .d20

The model defined for the ICE is split in two parts but for modeling we only consider the polynomial equation from the vehicle speed and the engine torque. For the sake of simplicity, the rest of the paper will consider the first expression with σ.

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m ˙ f = σT .Teng .v + σ0 + σ1 .v + σ2 .v 2

(12)

2.4 Validation of the Model

Fig. 2. Specific fuel consumption versus engine speed and torque. For confidential reasons, all the values are normalized between the idle engine speed and the max engine speed. Fig. 1. Fuel consumption versus engine torque for different engine speeds (data in blue and model in red). Fig. 1 shows the fitted models for different engine speeds. The fuel consumption results are close for different cycles so the model can be considered correct to test the command and compare fuel consumptions.

H=m ˙ f + λx .v + λv .a

Table 3. Error made with the ICE model of consumption Cycle

NEDC Artemis Urban Artemis Highway WLTC

Fuel Consumption [g] Reference Model look-up table 501.7 491.5 215.8 224.1 1743.1 1744.4 973.3 985.7

Error [%]

  λ˙ x (t) = − ∂H ∂x

-2 +3.9 +0.1 +1.3

Fig. 2 shows the engine map built with the two-slope model defined. Tmax is the maximum torque possible and Tlim refers to the torque reached at the power Plim defined in (7) and (11). 3. RESOLUTION METHODOLOGY 3.1 Definition of the Hamiltonian (Maamria et al., 2016) showed that the problem can be solved using two different methods, temporal or spatial. This paper considers the temporal one, so: J=

tf in

m ˙ f dt

(14)

According to (Pontryagin et al., 1962), we can define the co-state variable λx and λv by the differential equations with their respective temporal derivatives:

Table. 3 reports the results from a simulation tuned to follow the maximum speeds. In each case, the comparison is with the same settings. The following results are therefore those of the continuous model which shows an error below 4% compared to the initial look-up table.



function made of a cost function and constraints functions. We define the cost function J in (13) from the fuel consumption and the constraints functions are multiplications of the constraint variables Xi and their associated co-state variables λi .

(13)

tinit

The mathematical methodology is based on PMP. In this case, the aim is to find the minimum of a Hamiltonian 552

(15)

˙ λv (t) = − ∂H ∂v

After some calculations, assuming that the vehicle position and speed depend on the time only and that the engine torque is an independent variable because it is the manipulated variable, we deduce: 

λ˙ x (t) = 0 λ˙ v (t) = −σT .Teng − σ1 − 2.σ2 .v − λx +

λv m .(c1

+ 2.c2 .v) (16)

The main problem is the resolution of the differential equation of λv , because it depends on time and speed depends also on time but the expression of v according to t is unknown. Linearization avoids the use of quadratic equations which are time-dependent and speed-dependent. The terms to be linearized in the Hamiltonian are fuel mass and acceleration from (12) and (6) respectively. These linearizations have no impact on λx : 

m ˙ f (v) = (ψ1 + 2.ψ2 .v0 ).v + ψ0 − ψ2 .v02 a(v) = (g1 + 2.g2 .v0 ).v + g0 − g2 .v02

We can express (17) as:

(17)

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m ˙ f (v) = Ψ1 .v + Ψ0 a(v) = G1 .v + G0

∂H =0 ∂Teng

(18)

With:

553

(26)

The convexity is verified in subsection 3.3.  Ψ   1 Ψ0 G   1 G0

= ψ1 + 2.ψ2 .v0 = ψ0 − ψ2 .v02 = g1 + 2.g2 .v0 = g0 − g2 .v02

(19)

With the coefficients ψ and g defined in Table 4. Table 4. Coefficients ψ and g Index

ψ

0

σ0

1

σT .T + σ1

− cm1

σ2

− cm2

2

g 1 .(η.d0 .Teng m

− c0 )

According to (6) and the definition of d0 in (4), we note that the acceleration depends on the gear box ratio. (14) shows that the Hamiltonian can have different values changing the tansmission ratio. To simulate the analytical simulation considering a driving strategy close to the real conditions, we calculate the minimum Hamiltonian for different gear box ratio. Because we have to simulate real driving conditions, we consider the current gear box ratio k RGB for the k-th gear and calculate H k . We compare k H to H k−1 and H k+1 and choose the lower. Thus we look for an optimality according to several gear box ratio considering progressive gear changes. 140 Speed Trajectory Maximal Speed

120

H = Ψ1 .v + Ψ0 + λx .v + λv .(G1 .v + G0 )

(20)

Speed [km/h]

100

Grouping the factors in the following expression, the Hamiltonian can be used to determine the expression λv :

80

60

40

According to (15) and (14) developed with the expressions (12) and (6), we obtain the simplified expression:

20

λ˙ v + G1 .λv = −(Ψ1 + λx )

(21)

0 0

2000

4000

Solving the differential equation (Appendix B), we obtain: λv = −

Ψ1 + l x + lv G1

(22)

6000

8000

10000

12000

Position [m]

Fig. 3. Analytical Simulation proposed with respect to a theoretical maximal speed corresponding with the NEDC cycle.

Identifying terms from (17) and (18): λv = F. e−(g1 +2.g2 .v0 ).t +

ψ1 + 2.ψ2 .v0 g1 + 2.g2 .v0

6

(23)

5.5

To sum up, the expressions of λx and λv are:

5

λx (t) = lx c1 +2.c2 .v0 σ .T +σ1 +2.σ2 .v0 +lx .t m λv (t) = F. e +m. T engc1 +2.c 2 .v0

(24)

3.2 Minimization of the Hamiltonian

Gear Box Ratio

4.5



4 3.5 3 2.5

In order to get the optimal control formulation, it is important to respect the optimal criterion:

2 1.5

∗ H(t, x, v, Teng ) ≤ H(t, x, v, Teng )

(25)

This criterion stipulates that the Hamiltonian linked to ∗ the optimal command Teng is the minimum Hamiltonian. Assuming that the Hamiltonian is a convex function with respect to the command variable, we can find its minimum with: 553

1 0

2000

4000

6000

8000

10000

12000

Position [m]

Fig. 4. Gear box ratio strategy corresponding with the speed trajectory in Fig. 3.

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3.3 Expressions

420

The expression of the Hamiltonian, according to (6), (12) and (24), is a second degree polynomial depending on Teng with the expressions of the coefficients in Table 5: (27)

Table 5. Coefficients h Index

h

340

σ .v i + lx .v i=0 i

2 .v0 +lx −( σ1c+2.σ + +2.c .v 1

2

0

σT .(v −

F. e−G1 .t ). m

2

i=0

ci .v i

c1 +2.c2 .v0

2 .v0 +lx +η.d0 .( σ1c+2.σ + +2.c .v 1

2

360

320

2

0

1

380

Consumption [g]

2 H = h2 .Teng + h1 .Teng + h0

Pareto DP Analytic Simulation

400

2

0

2

c .v i=0 i

300 i

280 1000

1040

1060

1080

1100

1120

1140

1160

1180

Time [s]

)

Fig. 5. Pareto of the consumption depending on the time, without stops, for an assumed theoretical maximal speed on the NEDC cycle.

F. e−G1 .t ) m

σT .η.d0 c1 +2.c2 .v0

140

(28)

This proves the convexity of the Hamiltonian. We can find the expression of the engine torque Teng . According to (26) and (27), we can write the equation: h1 2.h2

100

Speed [km/h]

h2 > 0

DP 1 DP 2 DP 3 DP 4 DP 5 DP 6 Analytical Simulation Maximum Speed

120

The linearization of the Hamiltonian is available locally. So, v0 = v. Considering a positive velocity, the quadratic coefficient is positive:

Teng = −

1020

80

60

40

20

(29) 0 0

4. SIMULATION RESULTS

2000

4000

6000

8000

10000

12000

Position [m]

Tunable parameters lx and F play a role in the vehicle speed trajectory. If accelerations are parametrized by the Hamiltonian, we assume an economical deceleration using engine braking. Decelerations are therefore realized by calculating the distance needed to slow down and considering the remaining distance. The following simulations show results for a theoretical maximal speed trajectory corresponding with the NEDC cycle. On the defined cycle, we can compare the consumption allowed by the analytic simulation with the DP simulations in Fig. 5 and see the respect of the template with the parameters chosen Fig. 3. Analytical simulation shows only a +1.1% error compared to DP simulations for a configuration given. Change the configuration enables to change speed trajectory, therefore time and consumption on the cycle change. 5. CONCLUSION The analytical method could help to perform on-line calculations. This method implies knowing some basic rules about the effects of the speed and the acceleration on 554

Fig. 6. Speed trajectories, given by DP simulations and an analytical simulation, compared to the theoretical maximal speed assumed for a NEDC cycle. The analytical simulation corresponds with the ‘Analytical Simulation’ in Fig. 5. the consumption. Changing the engine torque has direct effects on the engine power and thereby on the consumption. The consumption, which is a mass flow rate, is linked to the engine speed because of the increasing number of explosions per minute and to the acceleration because the increase in torque requires a greater fuel mass for each explosion. The method could be based on a compromise between the need for acceleration to save time and fuel economy keeping a constant speed. Most of the computation would involve evaluating the most economic solution considering a phase and its constraints to reach a reasonable speed without unnecessary accelerations. Computational methodologies to determine eco-driving cycles need road data to update the trajectory continuously taking into account turns and direction changes or slopes of the road which impact the vehicle speed. In order to show the efficiency of the methodology it is necessary

2019 IFAC AAC Orléans, France, June 23-27, 2019

Emmanuel Nault et al. / IFAC PapersOnLine 52-5 (2019) 550–555

555

to use it on driving cycles and consider obstacles on the road with slowdowns with or without stops.

Appendix B. DIFFERENTIAL EQUATION WITH RESPECT TO λV

REFERENCES

(21) can be solve with the sum of the solution of the homogeneous equation λvh in one hand and a particular solution λvp in another hand:

B. Saerens (2012). Optimal Control Based Eco-Driving. Ph.D. thesis, Katholieke Universiteit Leuven, Louvain. Bellman, R. and Dreyfus, S. (1962). Applied dynamic programming. Princeton Univ. Press, Princeton, NJ. Bouvier, H., Colin, G., and Chamaillard, Y. (2015). Determination and comparison of optimal eco-driving cycles for hybrid electric vehicles. European Control Conference. Dib, W., Chasse, A., Moulin, P., Sciarretta, A., and Corde, G. (2014). Optimal energy management for an electric vehicle in eco-driving applications. Control Engineering Practice 29, Elsevier., 299–307. F. Mensing (2013). Optimal Energy Utilization in Conventional, Electric and Hybrid Vehicles and its Application to Eco Driving. Ph.D. thesis, INSA Lyon, Lyon. Hadj-Said, S., Colin, G., Ketfi-Cherif, A., and Chamaillard, Y. (2017). Analytical solution for management of parallel hybrid electric vehicles. 20th IFAC World Congress, Toulouse. Khalik, Z., Padilla, G., Romijn, T., and Donkers, M. (2018). Vehicle energy management with ecodriving: A sequential quadratic programming approach with dual decomposition. Annual American Conference (ACC), Milwaukee, USA. Maamria, D., Gillet, K., Colin, G., Chamaillard, Y., and Nouillant, C. (2016). Which methodology is more appropriate to solve eco-driving optimal control problem for conventional vehicles? IEEE Conference on Control Applications (CCA), IEEE Multi-Conference on Systems and Control, Buenos Aires, Argentina. Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., and Mishchenko, E. (1962). The Mathematical Theory of Optimal Processes. New York: Interscience. Sciarretta, A., De Nunzio, G., and Leon Ojeda, L. (2015). Optimal ecodriving control energy-efficient driving of road vehicles as an optimal control problem. IEEE Control Systems Magazine., 71–90. Appendix A. CALCULATIONS OF LINEARIZED FUNCTION

(A.1)

With A, B and C coefficients which do not depend on v: ∂f (v) = 2.A.v + B ∂v

(A.2)

The linearized expression gives:  ∂f (v)  y(v) ≈ .(v − v0 ) + f (v)|v0 ∂v v0 y(v) ≈ (B + 2.A.v0 ).v + C − A.v0 2

(A.3) (A.4) 555

(B.1)

λvh is the solution of the equation: λ˙ vh + G1 .λvh = 0

(B.2)

From (21), it is possible to deduce the solution of this homogeneous equation λvh = F. e−G1 .t

(B.3)

Considering the expression of λvh to get λvp varying the constant F : 

λvp (t) = F (t). e−G1 .t λ˙ vp (t) = F  (t). e−G1 .t −G1 .F (t). e−G1 .t

(B.4)

The latter system of equations and (21) give: λ˙ v (t) + G1 .λv = F  (t). exp(−G1 .t)

(B.5)

F  (t). e−G1 .t = −(Ψ1 + λx )

(B.6)

F  (t) = −(Ψ1 + λx ). eG1 .t

(B.7)

We deduce F (t) by an integration of F  (t):

F (t) = −

Ψ1 + λx G1 .t .e G1

(B.8)

Thus: λvp = −

For a given function F , which is a quadratic polynomial depending on v, we can identify its tangent to a value v0 as: y(v) = f (v) = A.v 2 + B.v + C

λv = λvh + λvp

Ψ1 + λx G1 .t −G1 .t Ψ1 + λ x =− .e .e G1 G1

(B.9)

Therefore: λv (t) = F. e−G1 .t −

Ψ 1 + λx G1

(B.10)