Dynamic Programming Resolution and Database Knowledge for Online

Predictive Energy Management of Hybrid Vehicles

Rustem Abdrakhmanov and Lounis Adouane

Institut Pascal/IMobS3, UCA/SIGMA - UMR CNRS 6602, Clermont-Ferrand, France

Keywords:

Hybrid Electric Vehicle, Energy Management Strategy, Dynamic Programming, Online Sub-optimal Opti-

mization, SOC Prediction, Multi-dimensional Database Knowledge.

Abstract:

This paper presents a sub-optimal strategy, based on Dynamic Programming (DP) approach, for online energy

(electric battery and fuel) optimization of a Hybrid Electric Vehicle (HEV). An optimal ofﬂine optimization is

ﬁrst proposed in this work, permitting to have simultaneous speed proﬁle optimization and optimal power split

strategy of a series-parallel hybrid bus. The aim of this optimization is mainly to reduce the fuel and electrical

energy consumption of the studied HEV while maintaining smooth bus navigation to ensure the passengers’

comfort. It is assumed in this ﬁrst proposal that current road proﬁle (slope, geometry, etc.) and the overall

bus trip (time at the stations) are known in advance. Afterward, the basis of the ofﬂine optimal strategy has

been adapted in order to deal online with the current road proﬁle and driver velocity demand. The proposed

sub-optimal online strategy uses mainly an appropriate speed proﬁle and power-split database, obtained ofﬂine

with DP, in order to cope with the current bus situations, and this is carried out by using a multi-dimensional

interpolation method. The present work is conducted on a dedicated high-ﬁdelity model of the hybrid bus that

was developed on MATLAB/TruckMaker software.

1 INTRODUCTION

The problem of reducing the environment pollution in

order to save the planet became one of the most im-

portant challenges in the world. Besides, the world-

wide crisis of the fossil fuel resources, which dimin-

ish at high rate, aggravates it. These two global as-

pects made the big industrial companies and the state

governments invest increasingly into the alternative

energy sources. The hybrid electric vehicles (HEV)

promise a relevant solution with regard to the objec-

tives of reducing the fuel consumption, as well as

the decrease of the exhaust gazes emission (Murphey,

2008).

Concerning the energy optimization in the HEV

and pure electric vehicles (EV), the researches mainly

deal with two kind of problems: 1) energy power

management for a given velocity proﬁle (Rousseau,

2008) (Chen et al., 2014) (Kitayama et al., 2015);

2) velocity proﬁle optimization for EV or conven-

tional vehicles (Ozatay et al., 2014) (Tokekar et al.,

2014) (Dib et al., 2014). The application of the

optimal control theory to power management on

HEV has been the most popular approach, which in-

cludes linear programming, optimal control and espe-

cially Dynamic Programming (DP) (Rousseau, 2008)

(Chen et al., 2014) (Pei and Leamy, 2013) (Song

et al., 2015)(Kamal et al., 2017) (Abdrakhmanov and

Adouane, 2017). These techniques have been widely

studied and applied to a broad range of vehicles.

The authors in (Pei and Leamy, 2013) propose

an approach for determining the State Of the Charge

(SOC)-dependent equivalent cost factor in HEV su-

pervisory control problems using DP. (Song et al.,

2015) use the DP approach to deal with the global op-

timization problem for deriving the best conﬁguration

for the drivetrain components sizes and energy split

strategies of a hybrid energy storage system, includ-

ing a battery and a supercapacitor, for an electric city

bus. (Zhang et al., 2014) proposed a DP-rule based

(DP-RB) algorithm to solve the global energy opti-

mization problem in a real time controller of a plug-

in hybrid electric bus (PHEB). A control grid (a set of

deterministic rules) is built for a typical city route ac-

cording to the station locations and discrete SOC lev-

els. An ofﬂine DP with historical running information

of the driving cycle is used to deduce optimal control

parameters of RB on all points of the control grid. For

a RB control strategy, control parameters are selected

according to the current position and battery SOC of

132

Abdrakhmanov, R. and Adouane, L.

Dynamic Programming Resolution and Database Knowledge for Online Predictive Energy Management of Hybrid Vehicles.

DOI: 10.5220/0006437301320143

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 132-143

ISBN: 978-989-758-263-9

Copyright © 2017 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

PHEB. The trajectory is divided into N segments and

the control parameters between station st

n

and station

st

n+1

are selected as a Rule

n

, which is determined by

the SOC S

n

at station st

n

.

Other authors as (Tokekar et al., 2014) studied

the problem of the velocity proﬁles search for a car-

like robots in order to minimize the energy consumed

while traveling along a given path, whereas (Dib et al.,

2014) tackle an energy management problem for an

electric vehicle compliant with online requirements

for “eco-driving application”. The main difference

between two last papers cited above is that the robot

is fully autonomous, and the electric vehicle is con-

trolled by a driver, but the driver receives the velocity

proﬁle proposed by an eco-driving system. (Ozatay

et al., 2014) propose an optimization of the speed tra-

jectory to minimize the fuel consumption and com-

municate it to the driver. In their approach the driver

sends the information of the intended travel destina-

tion to the server. The server generates a route, col-

lects the associated trafﬁc and geographical informa-

tion, and solves the optimization problem by a spa-

tial domain DP algorithm that utilizes accurate vehi-

cle and fuel consumption models to determine the op-

timal speed trajectory along the route. (Kim et al.,

2009) use model predictive control for the velocity

and power split optimization in HEVs. A given ve-

locity proﬁle is optimized by setting the constraints on

the velocity and the acceleration of the vehicle. This

allows to smooth the current velocity proﬁle without

generating a new one. The authors (Van Keulen et al.,

2010a) proposed a method that solves the velocity op-

timization problem for HEVs, based upon informa-

tion from Global Navigation Satellite-based Systems,

assuming that the velocity trajectory has a predeﬁned

shape. Although this method is used for HEVs, the

authors do not deal with the energy management op-

timization aspect. An optimal control problem was

proposed to determine the potential of a HEV opera-

tional strategy controlling gear shift, torque split, and

velocity at the same time (Heppeler et al., 2014). It

is proposed in this paper a discrete dynamic program-

ming approach to ﬁnd a global optimal solution for

fuel efﬁciency potential analysis, optimizing torque

split, gear shifting and velocity trajectory.

In their latter works (Heppeler et al., 2016), as

well as (Shen et al., 2015) and (Sun et al., 2015), deal

with the problem of prediction of the battery SOC.

These articles use the ofﬂine global optimal control

to generate the desired SOC trajectory, later this val-

ues are used as an input in Model Predictive Control

(MPC). It is proved that prediction of the future tra-

jectories, based upon either past or predicted vehicle

velocity and road grade trajectories, could help in ob-

taining a solution close to the optimal (Van Keulen

et al., 2010b).

Unlike the previous publications, the present pa-

per proposes an optimal ofﬂine optimization based

on DP permitting to have simultaneous speed pro-

ﬁle optimization and optimal power split strategy of

a series-parallel hybrid bus, aiming to reduce the fuel

and electrical energy consumption. For an urban bus

the route is normally known in advance, so the opti-

mization is performed for given road proﬁle. One of

the most important aspects of the public transporta-

tion is the passengers comfort. For that purpose the

maximal permitted acceleration and deceleration are

taken into account. As the hybrid bus’ electric mo-

tor and engine have different dynamic characteristics

(the power supplied, the response time, etc.), the dy-

namic constraints linked to the dynamic motors are

taken into account. An online sub-optimal speed pro-

ﬁle and related power split generation using a multi-

dimensional interpolation is developed to deal online

with the current road proﬁle and driver velocity de-

mand. This is carried out using the Optimal Proﬁle

Database based on DP (OPD-DP). The battery SOC

is estimated and predicted, using Kalman Filter esti-

mation, in order to guarantee that at the end of op-

erational cycle (in the end of a course of a day) the

electric battery charge is not below its permitted min-

imum level.

The rest of the paper is organized as follows. In

section 2, the studied bus powertrain and its dynami-

cal model are presented. Section 3 presents the pro-

posed ofﬂine DP algorithm and its constraint set. Sec-

tion 4 describes DP modeling of ofﬂine strategy, as

well as the use of the obtained results for online im-

plementation of the sub-optimal strategy and SOC

prediction. In section 5, several simulation results

are presented showing the efﬁciency of the proposed

velocity proﬁle optimization and energy management

strategies. Finally, conclusions and some prospects

are given in the last section.

2 MODELING OF THE HYBRID

BUS

The aim of this section is to illustrate the architecture

and the mathematical model of the studied system,

i.e., BUSINOVA hybrid bus, developed by SAFRA

company (cf. Figure 1)

1

. This bus is composed of an

electric motor, a hydraulic motor, an internal combus-

tion engine and battery as the propulsion powertrain

system of the vehicle.

1

http://www.businova.com

Dynamic Programming Resolution and Database Knowledge for Online Predictive Energy Management of Hybrid Vehicles

133

Figure 1: BUSINOVA hybrid bus.

2.1 Hybrid Bus Powertrain

Architecture

The model of the studied hybrid bus is based

on a series-parallel power-split hybrid architecture

(Bayindir et al., 2011). A simple block diagram of

the power ﬂows in the bus is shown in Figure 2.

Figure 2: Block diagram of the powertrain power ﬂows.

(ICE: internal combustion engine, HP: hydraulic pump,

HM: Hydraulic motor, EM: electric motor).

The electric (EM) and hydraulic (HM) motors are

both directly connected to the transmission and can

ensure simultaneously or independently the traction

of the bus. On the other hand, the internal combustion

engine (ICE) is coupled to a hydraulic pump (HP) for

driving the HM, and therefore allowing the ICE load

shifting.

The rotational speeds of the HM and the EM are

imposed by the wheels speed in proportion to the re-

duction ratios of HM and EM respectively. Moreover,

the rotational speed ω

HM

and the torque T

HM

are ex-

pressed as follows:

ω

HM

(T

ICE

,D

HM

) =

D

HP

.η

v

HM

.ω

ICE

D

HM

.η

v

HP

T

HM

(T

ICE

,D

HM

) =

D

HM

.η

m

HM

.T

ICE

D

HP

.η

m

HP

(1)

where ω

ICE

, T

ICE

are respectively rotational speed

and torque of the ICE, and D

HM

, D

HP

, η

m

HM

, η

m

HP

,

η

v

HM

, η

v

HP

are respectively displacement, mechanical

efﬁciency and volumetric efﬁciency of the HM and

the HP.

The BUSINOVA can operate according to the

modes described below:

1. the propulsion is fully supplied by the electric mo-

tor (mode I),

2. the bus is actuated by the hydraulic motor via the

ICE (mode II),

3. the mode III implies the hybrid operation of the

EM and the HM via ICE,

4. the regenerative braking (mode IV) - the part of

the kinetic energy during braking phase is recu-

perated to charge the electric battery.

2.2 Dynamical Model

This part is dedicated to the dynamical equations de-

scribing the bus. The purpose of the dynamical model

is to have a realistic global behavior of the bus in or-

der to validate the proposed energy management tech-

niques. To describe it in a generic manner, assume

that the bus is moving up the slope of θ degree (cf.

Figure 3). The origin of the coordinates is situated in

the Center of Mass (CoM). It is supposed that CoM

of the bus is in its geometric center. The dynamical

equation of the bus is given as follows:

~

F

tr

+

~

F

rr

+

~

F

ad

+

~

F

g

+

~

F

brake

= (M + M

eq

)~a (2)

where

~

F

tr

traction force,

~

F

rr

rolling resistance,

~

F

ad

aerodynamic force,

~

F

g

gravity force,

~

F

brake

mechan-

ical brake force, M bus weight, M

eq

equivalent mass

of rotating parts, ~a bus acceleration.

To produce the bus acceleration, it is necessary to

take into account the moments of inertia of the rotat-

ing components (e.g., rotor of the EM, crankshaft of

the ICE, driving axle, etc.). It is done by introducing

the equivalent mass M

eq

of the rotating components:

M

eq

=

i

g

η

pt

J

rot

r

2

(3)

where i

g

gear ratio, η

pt

powertrain efﬁciency, J

rot

total

inertia of the rotating components in the transmission,

and r the wheel radius (Cheng et al., 2007).

The traction force F

tr

is linked to the torque pro-

duced by the powertrain T

pt

via gear ratio i

g

, power-

train efﬁciency η

pt

. Expanding the dynamical equa-

tion (2), the following relation is obtained:

a =

dv

dt

=

1

M + M

eq

H (4)

with

H =

i

g

η

pt

T

pt

r

− µ

rr

F

N

sign(v) −

1

2

ρAC

d

(v + v

wind

)

2

− Mg sin(θ) −

T

brake

r

(5)

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

134

where:

• T

pt

: output powertrain torque from the gearbox,

• µ

rr

: rolling resistance coefﬁcient, F

N

= Mgcos(θ)

normal force, g gravity acceleration, θ slope an-

gle, v bus speed,

• ρ: the air density, A the frontal area of the bus, C

d

drag coefﬁcient, v

wind

wind speed,

• T

brake

: the brake torque provided by the bus me-

chanical brake system.

Figure 3: Forces acting on the bus.

3 OVERALL MULTI-CRITERIA

OPTIMIZATION

FORMULATION

The objective of the optimal control problem is to ﬁnd

the optimal bus velocity proﬁle and energy split be-

tween the actuators for a given trajectory D. The opti-

mization is performed in the spatial domain by means

of the following transformation (conversion from time

domain into spatial domain) (Ozatay et al., 2014):

a =

dv

dt

=

dv

dD

dD

dt

=

dv

dD

v (6)

In that case the dynamics of the BUSINOVA given

in equation (4) could be re-written with the following

equation:

dv

dD

=

1

(M + M

eq

)v

H (7)

As any problem of optimization, it is important to

deﬁne optimization criteria. In our case, the criteria is

deﬁned in order to optimize the fuel mass consumed

˙m

f uel

(Zeng, 2009) and the electric power P

EM

con-

sumed by the electric motor during the trip D:

J

D

= αJ

1

+ (1 − α)J

2

+ βJ

3

(8)

where:

J

1

=

Z

D

f

0

˙m

f uel

(P

ICE

(D),P

HM

(D),D

HM

)

v(D)

dD

(9)

with ˙m

f uel

fuel mass consumed is a function of

ICE power P

ICE

, as well as of HM power P

HM

and HM displacement D

HM

(Zeng, 2009);

J

2

=

Z

D

f

0

P

EM

(D)

v(D)

dD (10)

with P

EM

(D) EM power.

J

3

=

Z

D

f

0

|a|

v(D)

dD (11)

which is introduced to avoid abrupt velocity

changes and make the speed proﬁle smoother with

a vehicle acceleration,

D

f

is the total traveled distance;

v the bus speed;

α is a constant weight coefﬁcient such as α ∈ [0 1]

and β is a scale factor.

In order to have adequate results, each part of the cost

function (8) is normalized. The fuel consumption rate

˙m

f uel

is transformed into equivalent consumed engine

power P

engine

:

P

engine

= ˙m

f uel

Q

LHV

(12)

where Q

LHV

is lower heating value of a used fuel. For

diesel Q

LHV

= 43MJ/kg.

The minimization of the cost function (8) is the

subject to the bus dynamical model (7), as well as to

the constraints imposed on the control input and state

during the performed optimization.

3.1 Boundary Conditions

To set the boundary conditions for optimization pro-

cedure, we take into consideration a normal operation

for an urban bus, traveling from one stop to another

with the appropriate velocity proﬁle v(D) (cf. Figure

4), which should be obtained according to the pro-

posed optimization algorithm based on DP (cf. sec-

tion 4).

To simplify the notation, initial v(0) and ﬁnal

v(D

f

) speeds are called v

0

and v

f

. Generally for the

developed DP-based algorithm (cf. section 4), they

Figure 4: An example of a bus trip from one bus stop to

another.

Dynamic Programming Resolution and Database Knowledge for Online Predictive Energy Management of Hybrid Vehicles

135

can have any non-negative value. But in practice, the

bus trip to run from one bus stop to another implies

that:

v

0

= v

f

= 0 (13)

The trip duration T to arrive to the next bus stop is im-

posed by the schedule that must be respected. As the

optimization is computed with ﬂoating-point num-

bers, a constant τ is introduced (cf. equation (14)) to

calculate the range of the permitted ﬁnal time t

f

ﬂex-

ibility. But introduction of τ is also justiﬁed in real

systems, as the trip duration T cannot be absolutely

precise during the trip from D

0

to D

f

due to the trafﬁc

jams, trafﬁc lights, pedestrian crossing, etc.

t

f

= t

0

+ (1 ± τ)T (14)

where T is the pre-set trip duration, t

0

is the initial

time, τ is a very small value (τ → 0).

4 OFFLINE AND ONLINE

OPTIMIZATION STRATEGIES

BASED ON DP ALGORITHMS

This section is dedicated to the development of a

simultaneous speed proﬁle optimization and an en-

ergy management strategy in ofﬂine and online mode.

The procedure of the decision tree construction is de-

scribed in details in section 4.1. The developed of-

ﬂine algorithm permits to move to the online imple-

mentation by means of the Optimal Proﬁles Database

(cf. section ??), which generates the speed proﬁle

and its energy management strategy, depending on the

road proﬁle, the bus current bus state and velocity set-

point. The battery SOC estimation and prediction pro-

cedure is described in section 4.1.2.

4.1 Ofﬂine Optimization Strategy

DP Formulation

The optimization method used to solve the given op-

timal control problem is based on DP (Bellman and

Dreyfus, 2015) (Bertsekas, 1995), which provides the

global optimal solution over a given trip. The algo-

rithm proceeds from 0 to K steps in order to minimize

the following cost function:

J

k

(v

k

) = min

u

k

∈U

D

g

k

(v

k

,u

k

,∆D

k

) + J

k−1

( f (v

k

,u

k

))

(15)

where

• J

k

(v

k

) is the cost-to-go function from step 0 to

step K starting from v

0

with initial cost J

0

(v

0

) =

g

0

(v

0

) = 0;

• g

k

(v

k

,u

k

,∆D

k

) is the cost-to-go from state i to

state j.

• J

k−1

( f (v

k

,u

k

)) is the total cost starting from the

initial state to the state i.

• u

k

∈ U

D

is the control input determining the ve-

locity to go from the state i to the next state j.

The given optimization is aimed to solve two main

problems simultaneously: (i) Find the optimal speed

proﬁle from D

0

to D

f

in time t

f

, minimizing the elec-

tric energy and fuel consumption; (ii) ﬁnd the optimal

power split strategy to the speed proﬁle in order to

provide the optimal functioning mode (cf. section 2)

and the percentage of the contribution of each motor

(EM and HM) in order to move the hybrid bus.

The paragraphs below detail the resolution of the

nonlinear optimization control problem formulated in

the spatial domain by using DP algorithm. A set of

points deﬁnes the route. Namely, the route consists

of the points P = [p

0

, p

1

, p

2

, ..., p

K

]. Every

point p

k

∈ P for k = 0,1, 2,...,K has its own charac-

teristics: p

k

= [x

k

,y

k

,θ

k

], where x

k

longitudinal po-

sition, y

k

lateral position, θ

k

road slope angle. The

given route of the length D is divided into K segments

of the sample length ∆D. Depending on the accelera-

tion/deceleration limits and on the ∆D segment length

(cf. Figure 4), the velocity v(D

k

) for a given segment

D

k

can increase or decrease with a ﬁxed step ∆v. The

maximum number of ∆v is equal to 2N

v

+ 1:

V = {−N

v

∆v, ..., 0, ∆v, .. ., N

v

∆v} (16)

where N

v

∈ N.

DP-SEO Algorithm Flowchart

Figure 5 shows the proposed DP based Speed and

Energy Optimization (DP-SEO) algorithm. The pro-

posed DP-SEO has the following main steps.

Action 1. It initializes the road proﬁle P, dis-

tance discretization step ∆D (cf. Figure 4), K num-

ber of ∆D, a set of maximum permissible values of

speed increment ∆v. To perform a simultaneous bus

speed proﬁle and its power split optimization, the spe-

ciﬁc vector Λ = {0, 0.1,0.2,... ,1} is assigned with

Card(Λ) = m cardinal number of the set Λ, which

corresponds to the contribution of the EM and HM in

traction. Each state ν

j

≡ [P

k

,v

j

,t

j

,Parent(v

j

)], with

v

j

speed, t

j

time, Parent(v

j

) previous state.

Action 2. It attributes an acceleration ∆v.

Action 3. It calculates the required torque T

set point

.

Decision 1. If T

set point

is negative, the braking is

applied (cf. Action 7).

Decision 2. If T

set point

less than the maximum

torque that can be produced by EM and HM, then go

Action 4.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

136

Figure 5: Block diagram of the Dynamic Programming

based Speed and Energy Optimization (DP-SEO).

Action 4. It calculates the possible torque ratio

combinations according to Λ.

Decision 3. The EM and the engine, which drives

the HM, have different dynamic characteristics. EM

is capable of delivering a high starting torque. Unlike

EM, engine cannot provide torque at zero speed and

it produces maximum power at a certain speed (Pisu

and Rizzoni, 2007). The efﬁciency of the engine is

very much dependent on the operating point in the en-

gine’s performance map. Thus, to go from one state

to another not all the ∆T are feasible. To take this

constraint into account, the maximum values ∆T

EM

max

and ∆T

HM

max

are introduced. If the demanded torque

is higher that these values, then the development of

the decision tree in that direction is stopped. Other-

wise, go to Action 5.

Action 5. For the feasible combinations the cost-

to-go function g

k

(v

k

,u

k

,∆D

k

) is calculated (cf. equa-

tion (15)).

Decision 4. If the ﬁnal conditions are reached,

then go to Action 6. Otherwise, go to Action 2.

Action 6. The minimum cost function is calculated

according to equation (15). Optimal speed proﬁle and

related power split are calculated. In this work, DP is

used to obtain the minimal cost leading the hybrid bus

from its initial state to the ﬁnal one.

Action 7. A part of the kinetic energy is regener-

ated to charge the battery, and the rest is dissipated via

mechanical brake. If the vehicle speed is high enough

then the kinetic energy can be regenerated (limited to

the maximum torque T

regenmax

produced by EM and

depending on the current vehicle speed). No regener-

ation is possible under v < 10 km/h. In future work,

the regenerative aspect and dynamic will be studied in

more details.

DP-SEO Results Analysis

In order to validate the proposed ofﬂine optimization,

it is performed several DP-SEO in different elemen-

tary conditions of slope and desired velocity progress

(constant speed (v

0

= v

f

), acceleration (v

0

< v

f

) and

deceleration (v

0

> v

f

) phases) for a ﬁxed distance D,

K number of ∆D and Λ. Figures 6-7 show the av-

erage energy consumption for different slopes θ for

aforementioned phases. It can be seen that the energy

consumption increases with a bigger θ. A bigger bus

weight also increases the fuel consumption. To keep

the cruise speed on the downhill slope (negative θ),

the acceleration produced by the bus’ own dynamics

is enough (sometimes even excessive and it is neces-

sary to apply a negative torque). So on the downhill

slope the EM part is 100% for deceleration and cruise

speed phases. When accelerating, the EM torque goes

to its peak with the increase of the uphill slope, and it

is complemented by the HM (via ICE), so we can see

that HM ratio increases.

Figure 6: Average energy consumption for different slopes.

Even if the proposed strategy of the DP-SEO gives

good optimal results (all the possible states are ex-

plored and the global minimum is found), neverthe-

less it is too time consuming. However, from the

results of the optimization presented above, the op-

Dynamic Programming Resolution and Database Knowledge for Online Predictive Energy Management of Hybrid Vehicles

137

Figure 7: Average power split for different slopes.

timal ofﬂine solutions can be used online in order to

generate the sub-optimal solutions. For that purpose,

the optimal proﬁles were generated ofﬂine for the dis-

tance of ∆d = 21 m (chosen prediction horizon), with

different ∆v for several masses and slopes. Different

scenarios (for acceleration, deceleration, maintaining

the same speed) were simulated and for each simula-

tion conﬁguration an optimal speed proﬁle and related

power split are obtained.

The Optimal Proﬁles Database based on DP

(OPD-DP) was built for all the possible combination

of velocities from 0 to 15 m/s with ∆v = 1 m/s, the

bus weight varies from 13 to 15 tonnes with ∆M = 0.5

tonne, and the slope angle value ranges from -5

◦

to 5

◦

.

Based on the ofﬂine optimal solutions, OPD-DP will

be used to generate the speed proﬁles and the related

power split in a real-time system.

The multi-dimensional OPD-DP is used for on-

line speed proﬁle and its power split generation.

The scheme of the online control implementation

is presented in Figure 8. The inputs of the multi-

dimensional OPD-DP are the road proﬁle (x,y, θ), the

bus weight, the current speed and the driver’s refer-

ence (desired) speed, and estimated SOC value (this

block is detailed later). Each 21 meters this block

generates the sub-optimal speed proﬁle reference and

torque split depending on the current state of the vehi-

cle and the road proﬁle. The bus weight is considered

constant to travel from one bus station to another, thus

the weight is susceptible to change only after each bus

station. The driver’s command must be followed as

precise as possible due to the safety measures, ensur-

ing the collision avoidance.

The OPD-DP generates the sub-optimal proﬁle

references, as well as T

EM

and T

HM

set points. The set

point torques sent to EM and HM local controllers,

which generate the control input to the motors (I

EM

EM current, D

HM

HM displacement).

Knowing that the multidimensional OPD-DP has

only ﬁnite set of values for the parameters (slope θ,

weight M, etc.), it is very important to have a mean to

use this Database even for values not belonging to it.

To solve this problem, below it is proposed a Multidi-

mensional interpolation method.

4.1.1 Multidimensional Interpolation

There are several possible ways to approximate the

actual optimal proﬁle to follow. In our approach,

the Linear Multidimensional (LMD) interpolation

method was used to approximate the optimal speed

proﬁle and the powersplit vector (LaValle, 2006). As

a desired velocity value v lies between two values in

the OPD-DP, the weighted sum of the lower bound

speed v

UB

and the upper bound speed v

LB

are applied

to generate the speed proﬁle:

v

opt

= ζ

v

v

LB

+ (1 − ζ

v

)v

UB

(17)

The weight coefﬁcient ζ

v

is obtained as follows:

ζ

v

= 1 −

v − v

LB

v

UB

− v

LB

(18)

The weight coefﬁcients applied to calculate the cor-

responding power split Λ

opt

, depend on the current

speed v, the bus weight M, and the road slope θ. This

results into three dimensional interpolation function

given as follows:

Λ

opt

(v, M, θ) = ζ

v

ζ

M

ζ

θ

Λ(v

LB

,M

LB

,θ

LB

)+

(1 − ζ

v

)ζ

M

ζ

θ

Λ(v

UB

,M

LB

,θ

LB

)+

ζ

v

(1 − ζ

M

)ζ

θ

Λ(v

LB

,M

UB

,θ

LB

)+

ζ

v

ζ

M

(1 − ζ

θ

)Λ(v

LB

,M

LB

,θ

UB

)+

(1 − ζ

v

)(1 − ζ

M

)ζ

θ

Λ(v

UB

,M

UB

,θ

LB

)+

(1 − ζ

v

)ζ

M

(1 − ζ

θ

)Λ(v

UB

,M

LB

,θ

UB

)+

ζ

v

(1 − ζ

M

)(1 − ζ

θ

)Λ(v

LB

,M

UB

,θ

UB

)+

(1 − ζ

v

)(1 − ζ

M

)(1 − ζ

θ

)Λ(v

UB

,M

UB

,θ

UB

) (19)

where Λ

opt

is the power split vector, and weight coef-

ﬁcients ζ

M

and ζ

θ

are calculated as follows:

ζ

M

= 1 −

M − M

LB

M

UB

− M

LB

(20)

ζ

θ

= 1 −

θ − θ

LB

θ

UB

− θ

LB

(21)

with indexes LB - lower bound and U B - upper bound

of the corresponding parameter.

To illustrate how this method works in 3D, let us

ﬁx speed parameters, and suppose that the powersplit

λ change depends only on the variation of the bus

weight and the road slope. Figure 9 illustrates the case

when for different combinations of weight and slope

angle, we have four known values of λ stored in the

OPD-DP. Now let us address the case when weight

and slope angle values lie in between the known data.

This method permits to calculate the corresponding

powersplit.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

138

Figure 8: Scheme of the online sub-optimal speed proﬁles and its power split generation.

4.1.2 SOC Estimator based on Kalman Filter

The BUSINOVA bus is a plug-in hybrid electric vehi-

cle and its standard functioning time is 8 hours a day

(so called “course of a day”). Figure 10 illustrates the

spatial bounds of a bus running cycle. The bus travels

from its starting location to another route terminus,

stopping at bus stations (BS) along the route to allow

passengers to board and to alight. This movement is

called a trip. By the end of a day, the bus reaches

its SOC

min

value and can be recharged during all the

night long to ensure the service the next day.

In this work, the principle idea is to consider that

a better usage of the electric energy is such that it is

available until the end of the day (during 8 hour op-

erational cycle), and this is considered as an optimal

functioning of the bus. The working hypothesis be-

hind this assumption is to use the maximum amount

of energy that can be consumed from the battery in

one day driving so that the battery energy is spread

as uniformly as possible in one working day. This

implies the smooth battery discharging rate (C-rate),

avoidance of the high or low SOC and excessive depth

of charge, which lead to a high rate of battery capacity

loss (Tang et al., 2015) (Choi and Lim, 2002) (Brous-

sely et al., 2005). As Li-ion batteries represent a big

Bus Weight [ton]

13

13.2

1

Road Slope [°]

0

0.63

0.75

0.79

0.92

Powersplit λ

Weight: 13.2

Slope angle: 1

Powersplit: 0.63

Weight: 13.2

Slope angle: 0

Powersplit: 0.75

Weight: 13

Slope angle: 0

Powersplit: 0.92

Weight: 13

Slope angle: 1

Powersplit: 0.79

Weight: 13.14

Slope angle: 0.8

Powersplit: 0.72

Figure 9: Linear Multi-dimensional Interpolation method

illustration.

part of a vehicle cost, the clear interest is to prolon-

gate the battery life. For that purpose a SOC Estima-

tor based on Kalman Filter (Welch and Bishop, 1995)

(Grewal, 2011) is proposed.

It is assumed that the trafﬁc data are measured and

collected, so that for each Trip, and BS distance there

is an amount of ∆SOC = f (E

EM

) that is permitted to

consume according to the statistical analysis, depend-

ing on an average congestion level, travel distance,

average velocity, etc. (Sun et al., 2015) (Van Keulen

et al., 2010b). The Kalman ﬁlter addresses the general

problem of trying to estimate the state x of a discrete-

time controlled process that is governed by the linear

stochastic difference equation:

x

k

= Ax

k−1

+ Bu

k

+ w

k

(22)

with measurements z:

z

k

= Hx

k

+ v

k

(23)

Matrix A in the difference equation 22 relates the state

at the previous step k − 1 to the state at the current

step k. The matrix B relates the optional control in-

put u to the state x. The matrix H in the measurement

equation 23 relates the state x to the measurement z

k

.

The random variables w

k

and z

k

represent the process

and measurement noise, respectively. In our case, the

state vector x = [E

EM

P

EM

]

T

, which is a total elec-

tric energy consumed during a trip and instantaneous

electric motor power, respectively. The control input

Figure 10: Spatial bounds of a bus running cycle.

139

u = T

EM

ω

EM

is a product of electric motor torque and

its speed.

A predicted state is calculated according to the

equation 22:

x

k

p

= Ax

k−1

+ Bu

k

+ w

k

(24)

A Predicted Process Covariance Matrix is calculated

as follows:

P

k

p

= AP

k−1

A

T

+ Q

k

(25)

where P

k−1

is previous step Process Covariance Ma-

trix, Q

k

stands for process Noise Covariance Matrix

associated with noisy control inputs.

To update the measurements, the following three

steps are carried out:

1. Calculation of the Kalman Gain KG:

KG =

P

k

p

H

T

HP

k

p

H

T

+ R

(26)

where R is Observation Errors Matrix.

2. Calculation of the current state:

x

k

= x

k

p

+ KG(z

k

− Hx

k

p

) (27)

3. Next step is updating a Process Covariance Ma-

trix:

P

k

= (I − KG · H)P

k

p

(28)

with I identity matrix.

In this manner,

ˆ

∆SOC

k

= f (E

EM

) is estimated at each

step k in order to respect SOC

min

at the end of the bus

day work. The value of

ˆ

∆SOC

k

is sent to the OPD-

DP block to ﬁnd the solution which respects the given

constraint and choose an appropriate powersplit. The

simulation results for a speciﬁc scenario are demon-

strated in the next section.

5 SIMULATION RESULTS

It is presented below several results of the use of the

OPD-DP based real time predictive energy optimiza-

tion. The Kalman Filter is used to predict the SOC

and to constrain the use of the battery. Two scenarios

are proposed. For both of them the bus travels a trip

consisting of ﬁve Bus Station distances.

1. The bus arrives at all the Bus Stations without

stopping at the trafﬁc lights and without being

stuck in a trafﬁc jam.

2. In this case, at some points it had to stop because

of the trafﬁc lights or before the vehicle ahead,

which entailed underspeciﬁed stops along the trip.

Distance [m]

0 500 1000 1500 2000

Road Slope [°]

-2

-1

0

1

2

Road Profile

Distance [m]

0 500 1000 1500 2000

Weight [kg]

× 10

4

1.3

1.35

1.4

1.45

Bus Weight

Figure 11: Road slope and bus weight.

Road proﬁle and the weight of the bus change dur-

ing the trip (cf. Figure 11). However, the weight

changes only after each bus stop when people get on

or get off the bus. It is considered in this simula-

tion that the road-tire frictional coefﬁcient, which de-

pends on the surface where the bus moves and on the

weather conditions, is constant and equal to 1, which

corresponds to the dry pavement (Ming, 1997).

The value of the SOC

min

in the end of the trip re-

mained the same. Basically, the SOC

initial

= 90%,

and in the end of the trip must be around SOC

min

=

87.79%. Total trip distance is 2 km (more precisely

2058m).

Figure 12 represents the results of the ﬁrst sce-

nario. The ﬁrst ﬁgure shows the obtained speed pro-

ﬁle. Red crosses indicate the driver’s demanded speed

at each ∆d. The speed tracking of the driver’s com-

mand is well followed. The ﬁgure below shows bat-

tery SOC trajectory. It can be seen that by the end of

the trip we tend to the SOC

min

. The last graph shows

the power split value λ. The applied notation is as fol-

low: λ = 1 corresponds to 100% electric mode, λ = 0

corresponds to 100% hydraulic mode (via ICE), and

λ = −0.1 corresponds to regenerative mode.

Figure 13 represents the second scenario. In this

case, we can see that at the distance ≈500m there is a

non-planned stop, as well as at the distance ≈1200m,

etc. Although the speed proﬁle changed, the online

energy optimization strategy is adapted in such a man-

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

140

Table 1: Comparison of the ODP-DP online strategy and the ofﬂine DP.

SOC

min

= 87.79 %

OPD-DP DP DP vs OPD-DP

SOC

f inal

, % Fuel, l SOC

f inal

, % Fuel, l SOC

f inal

, % Fuel, l

Scenario 1 87.85 0.195 88.02 0.178 -0.02 0.017

Scenario 2 87.84 0.228 87.88 0.189 -0.03 0.039

ner that it will always tend to SOC

min

at the end of

the trip. We can see this adaptation, according to the

change in λ.

The OPD-DP based real time predictive energy

optimization is compared to the ofﬂine DP. The of-

ﬂine DP based optimization was carried out for the

same speed proﬁles. Table 1 shows that the obtained

results are close to the ofﬂine DP solution.

6 CONCLUSIONS AND

PROSPECTS

In this paper, the DP technique is used to simulta-

neously generate the optimal velocity proﬁle and its

power split strategy in order to ensure the electric en-

ergy and fuel economy, respecting passengers com-

fort (by limiting the acceleration/deceleration). As the

Distance [m]

0 500 1000 1500 2000

Speed [km/h]

0

20

40

Speed profile

LMD Interpolation

Driver command each ∆ d

Distance [m]

0 500 1000 1500 2000

SOC [%]

87

88

89

90

Battery SOC

LMD Interpolation

SOCmin

Distance [m]

0 500 1000 1500 2000

Powersplit λ

0

0.5

1

Powersplit ratio

Figure 12: First scenario: 1) speed proﬁle 2) actual SOC

and the desired value in the end of the trip SOC

min

3) power

split λ.

Distance [m]

0 500 1000 1500 2000

Speed [km/h]

0

20

40

Speed profile

LMD Int

Driver command each ∆ d

Distance [m]

0 500 1000 1500 2000

SOC [%]

87

88

89

90

Battery SOC

LMD Int

SOCmin

Distance [m]

0 500 1000 1500 2000

Powersplit λ

0

0.5

1

Powersplit ratio

Figure 13: Second scenario: 1) speed proﬁle 2) actual SOC

and the desired value in the end of the trip SOC

min

3) power

split λ.

dynamics of the electric motor and engine are differ-

ent, not all the energy management conﬁgurations are

feasible. This aspect is taken into account in the DP-

SEO algorithm.

Thereafter, the ofﬂine optimal solutions of the

DP based optimization were collected into a multi-

dimensional OPD-DP for the online implementation

for a priori unknown trafﬁc conditions for instance.

To cope with the problem of the ﬁnite set of sample

points in the database, a Linear Multidimensional in-

terpolation was used to obtain the values at all other

points. The SOC estimator based on Kalman Filter

was used to restrict the use of the battery through-

out the cycles and allows us to have smooth bat-

tery discharge rate, guaranteeing a regular electric en-

ergy consumption without falling below the minimum

SOC value. The results obtained by Online Predictive

Energy Management strategy were compared to DP

141

solutions obtained ofﬂine, and it was shown that near

optimal results were obtained in real-time application.

The inﬂuence of road-tire frictional coefﬁcient on the

online energy management strategy will be studied in

near future. Later, the given approach will be imple-

mented in the real bus.

ACKNOWLEDGEMENTS

This project is supported by the ADEME (Agence De

l’Environnement et de la Matrise de l’Energie) for the

National French program “Investissement d’Avenir”,

through BUSINOVA Evolution project.

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