Document [original]





Citation: Geng, S.; Schulte, T.; Maas,

J. Model-Based Analysis of Different

Equivalent Consumption

Minimization Strategies for a Plug-In

Hybrid Electric Vehicle. Appl. Sci.

2022,12, 2905. https://doi.org/

10.3390/app12062905

Academic Editor: Daniela Anna

Misul

Received: 14 February 2022

Accepted: 8 March 2022

Published: 11 March 2022

Publisher’s Note: MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

applied

sciences

Article

Model-Based Analysis of Different Equivalent Consumption

Minimization Strategies for a Plug-In Hybrid Electric Vehicle

Stefan Geng 1,2,*, Thomas Schulte 1and Jürgen Maas 2

1iFE—Institute for Energy Research, OWL University of Applied Sciences and Arts, Campusallee 12,

32657 Lemgo, Germany; [email protected]

2Mechatronic Systems Laboratory, Faculty of Mechanical Engineering and Transport Systems,

Technical University of Berlin, Hardenbergstraße 36, 10623 Berlin, Germany; juer[email protected]

*Correspondence: [email protected]

Abstract:

Plug-in hybrid electric vehicles (PHEVs) are developed to reduce fuel consumption and the

emission of carbon dioxide. Common powertrain configurations of PHEVs (i.e., the configuration of

the combustion engine, electric motor, and transmission) can be operated either in series, parallel,

or power split hybrid mode, whereas powertrain configurations with multimode transmissions

enable switching between those modes during vehicle operation. Hence, depending on the current

operation state of the vehicle, the most appropriate mode in terms efficiency can be selected. This,

however, requires an operating strategy, which controls the mode selection as well as the torque

distribution between the combustion engine and electric motor with the aim of optimal battery

depletion and minimal fuel consumption. A well-known approach is the equivalent consumption

minimization strategy (ECMS). It can be applied by using optimizations based on a prediction of the

future driving behavior. Since the outcome of the ECMS depends on the quality of this prediction,

it is crucial to know how accurate the predictions must be in order to obtain acceptable results.

In this contribution, various prediction methods and real-time capable ECMS implementations are

analyzed and compared in terms of the achievable fuel economy. The basis for the analysis is a holistic

model of a state-of-the-art PHEV powertrain configuration, comprising the multimode transmission,

corresponding powertrain components, and representative real-world driving data.

Keywords: PHEV; ECMS; multimode transmission; optimization; powertrain modeling

1. Introduction

Today, vehicle manufacturers are forced to reduce the fuel consumption of their

products because of enhanced environmental regulations. A promising solution is the

development of plug-in hybrid electric vehicles (PHEV), as they combine an extended

electric cruising range and the possibility of propelling the vehicle by an internal com-

bustion engine (ICE) when the battery is depleted or when high performance is required.

Common powertrain configurations of PHEVs are the series, parallel, and power split

configurations. The efficiencies of these configurations vary depending on the distance and

the power demand of the intended trip [

]. Increased efficiency can be obtained by using

so-called multimode or dedicated hybrid transmissions (DHTs). This type of transmission

enables switching between different powertrain configurations during vehicle operation

and combining the individual advantages of these configurations. A general property of a

DHT is that the electric motors are an integral and indispensable part of the transmission [

which is the case when only the electric motor is able to propel the vehicle in a certain speed

range. Since the part of the transmission for the ICE can be designed for the remaining

and, in general, smaller speed range, a DHT requires fewer speeds and is less complex in

comparison with a conventional automatic transmission. This saves a part of the additional

weight and installation space caused by the electrification of the vehicle. Examples of

available PHEVs equipped with a DHT can be found in [3–7].

Appl. Sci. 2022,12, 2905. https://doi.org/10.3390/app12062905 https://www.mdpi.com/journal/applsci

Appl. Sci. 2022,12, 2905 2 of 17

Figure 1shows the basic concept of a hybrid electric powertrain with a DHT, which is

abstractly considered a configuration of two clutches (C), a transmission (T) with a constant

gear ratio

iED

, a planetary gear (PG) with a stationary gear ratio

, and a two-speed

transmission (2ST) with the gear ratios

i1/2

. The concept requires only one electric drive

and enables a continuous variable transmission mode (CVT), a parallel hybrid mode (PAR),

and an electric driving mode (EM), where each mode can be driven in two speeds due to the

two-speed transmission at the output. Clutch C

connects the ICE to the DHT, and clutch

blocks the planetary gear (i.e., all shafts of the planetary gear rotate with the same speed).

In this case, the speed ratios between the transmission’s output and the two input shafts

are constant and depend on the state of C

when either the parallel hybrid or the electric

driving mode are active. Due to the chosen transmission ratios, the parallel hybrid mode

can only be driven at higher vehicle speeds. Otherwise, the rotational speed of the ICE will

fall below its lower limit, and the engine will stall. To operate the vehicle at lower speeds,

the electric driving mode or, if the battery is discharged, the CVT mode must be activated.

In this mode, the rotational speeds of the ICE and electric motor are superimposed by a

planetary gear, whereby the speed of the ICE can be adjusted continuously as a function of

the speed of the electric motor and the final drive. The design of the transmission ratios

ensures that the electric motor operates as a generator in CVT mode until an appropriate

vehicle speed is reached. A more detailed description can be found in a previous work [

where the basic concept was used for the development of a new DHT including a specific

configuration of gears and clutches and the corresponding transmission ratios.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 2 of 18

fewer speeds and is less complex in comparison with a conventional automatic trans-

mission. This saves a part of the additional weight and installation space caused by the

electrification of the vehicle. Examples of available PHEVs equipped with a DHT can be

found in [3–7].

Figure 1 shows the basic concept of a hybrid electric powertrain with a DHT, which

is abstractly considered a configuration of two clutches (C), a transmission (T) with a

constant gear ratio ED

i, a planetary gear (PG) with a stationary gear ratio 0

i, and a

two-speed transmission (2ST) with the gear ratios 1/2

i. The concept requires only one

electric drive and enables a continuous variable transmission mode (CVT), a parallel

hybrid mode (PAR), and an electric driving mode (EM), where each mode can be driven

in two speeds due to the two-speed transmission at the output. Clutch C1 connects the

ICE to the DHT, and clutch C2 blocks the planetary gear (i.e., all shafts of the planetary

gear rotate with the same speed). In this case, the speed ratios between the transmission’s

output and the two input shafts are constant and depend on the state of C1 when either

the parallel hybrid or the electric driving mode are active. Due to the chosen transmission

ratios, the parallel hybrid mode can only be driven at higher vehicle speeds. Otherwise,

the rotational speed of the ICE will fall below its lower limit, and the engine will stall. To

operate the vehicle at lower speeds, the electric driving mode or, if the battery is dis-

charged, the CVT mode must be activated. In this mode, the rotational speeds of the ICE

and electric motor are superimposed by a planetary gear, whereby the speed of the ICE

can be adjusted continuously as a function of the speed of the electric motor and the final

drive. The design of the transmission ratios ensures that the electric motor operates as a

generator in CVT mode until an appropriate vehicle speed is reached. A more detailed

description can be found in a previous work [8], where the basic concept was used for the

development of a new DHT including a specific configuration of gears and clutches and

the corresponding transmission ratios.

Figure 1. Basic concept of a hybrid electric powertrain with a DHT.

The operation of the hybrid electric powertrain requires an appropriate operating

strategy, which determines the operation mode of the DHT and the torque distribution

between the ICE and electric motor with the aim of maximal fuel economy. According to

[9–11], numerous approaches for implementing such strategies are already known and

classified into heuristic and optimization-based methods. Figure 2 shows an overview of

the commonly applied methods.

Heuristic operating strategies are implemented as simple rules, maps [12–14], state

machines, or fuzzy controllers [15–17], where the parametrization is frequently carried

out based on the experience of the powertrain’s developers and adjusted by means of

empirical studies and test driving. It often aspires to reduce fuel consumption by shifting

all operating points of the vehicle to an efficient operating area of the ICE. Since the op-

eration of the electric motor is not taken into account optimally, optimal operation in

terms of minimal fuel consumption is not possible. A further approach for parametriza-

CVT

PAR

Mode

DHT

Final

Drive

2ST

Internal

Combustion

Engine

(ICE)

Battery Electric Drive

1/2

Figure 1. Basic concept of a hybrid electric powertrain with a DHT.

The operation of the hybrid electric powertrain requires an appropriate operating

strategy, which determines the operation mode of the DHT and the torque distribution

between the ICE and electric motor with the aim of maximal fuel economy. According

to [

–

], numerous approaches for implementing such strategies are already known and

classified into heuristic and optimization-based methods. Figure 2shows an overview of

the commonly applied methods.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 3 of 18

tion is to incorporate the results from an optimization-based method, which is performed

using representative driving cycles [18–20] (see dotted arrow in Figure 2).

Figure 2. Classification of operation strategies for hybrid electric vehicles.

Usually, heuristic methods provide only suboptimal results, since the parametriza-

tion is based on predefined assumptions without considering real driving behavior. Due

to this disadvantage, it is generally not possible to obtain the minimal possible fuel con-

sumption. However, the main advantage is the simplicity of the method’s implementa-

tion, which makes it suitable for real-time application on a vehicle’s electronic control

units. Moreover, no information on future driving behavior must be known in advance in

order to operate the vehicle.

Optimization-based methods provide optimal results in terms of fuel economy.

Here, fuel consumption is defined to be a cost function, which is minimized for a given

driving cycle by using a mathematical optimization. In general, these methods require a

powertrain model in order to describe the fuel consumption, depending on the vehicle’s

operating point. For this purpose, map-based models are frequently used, where the fuel

consumption of the ICE and the energy efficiency of the electric drive are considered by

means of corresponding maps. When using map-based powertrain models, only numer-

ical optimization methods can be applied. Typical numerical methods for optimiza-

tion-based operating strategies proposed in the literature are Dynamic Programming

(DP) [21,22], Stochastic Dynamic Programming (SDP) [23], Pontryagin’s Maximum Prin-

ciple (PMP) [24,25], Particle Swarm Optimization (PSO) [26], and Sequential Quadratic

Programming (SQP) [27]. Another approach is to use an analytic powertrain model,

where the maps are approximated by convex functions [28]. These functions are often

simple polynomials and enable solving the optimization problem of the operating strat-

egy analytically by means of PMP [29–31]. However, the application is restricted to con-

tinuous control variables (e.g., the torque distribution between ICE and the electric

drive). In [32], the analytical optimization with PMP was combined with DP in order to

consider the gear-shifting command as a discrete control variable. For the same power-

train configuration and control problem, in [33], a combination of DP and an optimiza-

tion based on interior point methods using the SeDuMi tool was applied on a convex

powertrain model.

The advantage of an operating strategy based on mathematical optimization is that

it provides minimal fuel consumption. However, a driving cycle must be known in ad-

vance, and some of the methods require high computational effort. Therefore, these

methods are inapplicable for real vehicle operation but are appropriate for powertrain

analyses and optimizations.

For real driving operation, real-time capable operating strategies are required. An

overview of various methods can be found in [34,35]. Optimization-based methods are

preferred in general, since they consider information about the future driving behavior

and consequently provide better results than heuristic methods. Since the exact driving

behavior is unknown, and due to unforeseeable driving styles and traffic situations,

•Rules

•Maps

•State-Machines

•Fuzzy-Controller

Analytic Numeric

Parametrization

•PMP

Combined

•DP-PMP

•DP-SeDuMi

Heuristic

Methods

Optimization-Based

Methods

Operating Strategies

•DP, SDP

•PMP

•PSO, SQP

Figure 2. Classification of operation strategies for hybrid electric vehicles.

Appl. Sci. 2022,12, 2905 3 of 17

Heuristic operating strategies are implemented as simple rules, maps [

–

], state

machines, or fuzzy controllers [

–

], where the parametrization is frequently carried out

based on the experience of the powertrain’s developers and adjusted by means of empirical

studies and test driving. It often aspires to reduce fuel consumption by shifting all operating

points of the vehicle to an efficient operating area of the ICE. Since the operation of the

electric motor is not taken into account optimally, optimal operation in terms of minimal

fuel consumption is not possible. A further approach for parametrization is to incorporate

the results from an optimization-based method, which is performed using representative

driving cycles [18–20] (see dotted arrow in Figure 2).

Usually, heuristic methods provide only suboptimal results, since the parametrization

is based on predefined assumptions without considering real driving behavior. Due to this

disadvantage, it is generally not possible to obtain the minimal possible fuel consumption.

However, the main advantage is the simplicity of the method’s implementation, which

makes it suitable for real-time application on a vehicle’s electronic control units. Moreover,

no information on future driving behavior must be known in advance in order to operate

the vehicle.

Optimization-based methods provide optimal results in terms of fuel economy. Here,

fuel consumption is defined to be a cost function, which is minimized for a given driving

cycle by using a mathematical optimization. In general, these methods require a powertrain

model in order to describe the fuel consumption, depending on the vehicle’s operating point.

For this purpose, map-based models are frequently used, where the fuel consumption of the

ICE and the energy efficiency of the electric drive are considered by means of corresponding

maps. When using map-based powertrain models, only numerical optimization methods

can be applied. Typical numerical methods for optimization-based operating strategies

proposed in the literature are Dynamic Programming (DP) [

], Stochastic Dynamic

Programming (SDP) [

], Pontryagin’s Maximum Principle (PMP) [

], Particle Swarm

Optimization (PSO) [

], and Sequential Quadratic Programming (SQP) [

]. Another

approach is to use an analytic powertrain model, where the maps are approximated by

convex functions [

]. These functions are often simple polynomials and enable solving

the optimization problem of the operating strategy analytically by means of PMP [

–

However, the application is restricted to continuous control variables (e.g., the torque

distribution between ICE and the electric drive). In [

], the analytical optimization with

PMP was combined with DP in order to consider the gear-shifting command as a discrete

control variable. For the same powertrain configuration and control problem, in [

], a

combination of DP and an optimization based on interior point methods using the SeDuMi

tool was applied on a convex powertrain model.

The advantage of an operating strategy based on mathematical optimization is that it

provides minimal fuel consumption. However, a driving cycle must be known in advance,

and some of the methods require high computational effort. Therefore, these methods

are inapplicable for real vehicle operation but are appropriate for powertrain analyses

and optimizations.

For real driving operation, real-time capable operating strategies are required. An

overview of various methods can be found in [

]. Optimization-based methods are

preferred in general, since they consider information about the future driving behavior

and consequently provide better results than heuristic methods. Since the exact driving

behavior is unknown, and due to unforeseeable driving styles and traffic situations, pre-

dictions are used (e.g., based on telemetry data). Well-known real-time capable operating

strategies are the Equivalent Consumption Minimization Strategy (ECMS) [

–

] and

various Model Predictive Control (MPC) approaches [

–

], while the optimization-based

implementation of the ECMS is equivalent to PMP [41].

In this contribution, different implementations of the optimization-based EMCS al-

gorithm are analyzed in terms of the theoretically achievable fuel economy. These imple-

mentations are based on different approaches for predicting the future driving cycle, while

the accuracy of the predictions increases with the prediction effort and the quality of the

Appl. Sci. 2022,12, 2905 4 of 17

predicted information. For the analysis, the hybrid electric powertrain given by the basic

concept shown in Figure 1is used. All implementations of the ECMS are evaluated by

means of corresponding powertrain simulations, which are carried out with representative

real-world driving data as the input.

Therefore, in Section 2, the powertrain model used for implementing and evaluating

the optimization-based ECMS is presented. The control of the hybrid electric powertrain by

means of the ECMS requires the definition of an optimization problem and a method for its

solution. Both are described based on the previously defined powertrain model in Section 3

and are applied in Section 4, comprising the ECMS algorithm and the different prediction

methods. Finally, in Section 5, the results of the powertrain simulations considering various

ECMS implementations are evaluated and discussed.

2. Powertrain Model

The ECMS implementation is based on a powertrain model, which is used to perform

a local optimization in order to determine the powertrain’s control signals. Since the

optimization needs to be applied in real time, a powertrain model with low computational

effort and yet a sufficient level of detail is required. This is obtained by the so-called

backward approach [

]. Starting with the requested acceleration

aveh

and speed

vveh

the vehicle, the operating states of each powertrain component are determined backwards

(see Figure 3). A vehicle dynamics model determines the required torque

TFD

and angular

velocity

ωFD

at the final drive. In order to satisfy these requirements, the DHT model

determines the corresponding torques and angular velocities of the ICE and electric drive

considering the transmission’s control variables

and

uED

. It is always assumed that

the ICE and the electric drive are capable of generating the required torque and satisfying

the requested vehicle acceleration, respectively. The submodels of the DHT, ICE, electric

drive, and battery consider stationary states only, as no controllers are required to operate

these components. Only the operation mode of the powertrain must be controlled via

the DHT. Therefore,

defines the mode (CVT, PAR, or EM mode) and

uED

the torque

or angular velocity of the electric drive, definable due to the resulting degree of freedom

in the PAR and CVT modes, respectively. Both control variables are determined by the

ECMS and are considered here as inputs of the powertrain model. Since no dynamic

behavior is considered, the operation mode changes immediately without any transition

(e.g., continuously changing the state of a clutch from engaged to disengaged). The output

of the powertrain simulation is the mass flow rate of the fuel

and the state of charge of

the battery SoC.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 4 of 18

predictions are used (e.g., based on telemetry data). Well-known real-time capable oper-

ating strategies are the Equivalent Consumption Minimization Strategy (ECMS) [35–37]

and various Model Predictive Control (MPC) approaches [38–40], while the optimiza-

tion-based implementation of the ECMS is equivalent to PMP [41].

In this contribution, different implementations of the optimization-based EMCS al-

gorithm are analyzed in terms of the theoretically achievable fuel economy. These im-

plementations are based on different approaches for predicting the future driving cycle,

while the accuracy of the predictions increases with the prediction effort and the quality

of the predicted information. For the analysis, the hybrid electric powertrain given by the

basic concept shown in Figure 1 is used. All implementations of the ECMS are evaluated

by means of corresponding powertrain simulations, which are carried out with repre-

sentative real-world driving data as the input.

Therefore, in Section 2, the powertrain model used for implementing and evaluating

the optimization-based ECMS is presented. The control of the hybrid electric powertrain

by means of the ECMS requires the definition of an optimization problem and a method

for its solution. Both are described based on the previously defined powertrain model in

Section 3 and are applied in Section 4, comprising the ECMS algorithm and the different

prediction methods. Finally, in Section 5, the results of the powertrain simulations con-

sidering various ECMS implementations are evaluated and discussed.

2. Powertrain Model

The ECMS implementation is based on a powertrain model, which is used to per-

form a local optimization in order to determine the powertrain’s control signals. Since the

optimization needs to be applied in real time, a powertrain model with low computa-

tional effort and yet a sufficient level of detail is required. This is obtained by the

so-called backward approach [42]. Starting with the requested acceleration veh

a and

speed veh

v of the vehicle, the operating states of each powertrain component are deter-

mined backwards (see Figure 3). A vehicle dynamics model determines the required

torque FD

T and angular velocity FD

at the final drive. In order to satisfy these re-

quirements, the DHT model determines the corresponding torques and angular veloci-

ties of the ICE and electric drive considering the transmission’s control variables m

and

u. It is always assumed that the ICE and the electric drive are capable of generating the

required torque and satisfying the requested vehicle acceleration, respectively. The

submodels of the DHT, ICE, electric drive, and battery consider stationary states only, as

no controllers are required to operate these components. Only the operation mode of the

powertrain must be controlled via the DHT. Therefore, m

defines the mode (CVT, PAR,

or EM mode) and ED

u the torque or angular velocity of the electric drive, definable due

to the resulting degree of freedom in the PAR and CVT modes, respectively. Both control

variables are determined by the ECMS and are considered here as inputs of the power-

train model. Since no dynamic behavior is considered, the operation mode changes im-

mediately without any transition (e.g., continuously changing the state of a clutch from

engaged to disengaged). The output of the powertrain simulation is the mass flow rate of

the fuel f

 and the state of charge of the battery SoC .

Figure 3. Powertrain model according to the backward approach and the basic concept shown in

Figure 1.

ICE

Driving Cycle

Electric

Drive

DHT

Battery

Vehicle

Dynamics

veh

ICE

TSoC



Figure 3.

Powertrain model according to the backward approach and the basic concept shown in

Figure 1.

Figure 4shows the submodels of the powertrain and their most significant parameters.

The models of the ICE and electric drive are map-based, whereby only the state behavior of

the components is considered. To determine the specific fuel consumption

and power

loss PED,loss, the corresponding torques and speeds are used.

Appl. Sci. 2022,12, 2905 5 of 17

Appl. Sci. 2022, 12, x FOR PEER REVIEW 5 of 18

Figure 4 shows the submodels of the powertrain and their most significant param-

eters. The models of the ICE and electric drive are map-based, whereby only the state

behavior of the components is considered. To determine the specific fuel consumption e

and power loss ED,loss

P, the corresponding torques and speeds are used.

Figure 4. Models of the powertrain components and the most important parameters.

The vehicle dynamics are considered in a longitudinal direction only. The torque

T and angular velocity FD

at the final drive are determined according to the fol-

lowing equation:

()

FD w drag veh roll downhill veh veh ,TrFv FF ma=⋅ + + + ⋅ (1)

with the radius of the wheels w

r, the vehicle mass veh

m, the vehicle speed veh

v, and the

vehicle acceleration veh

a (extracted from the driving cycle). The forces represent the

driving resistances comprising the drag force drag

F, the rolling friction force roll

, and the

downhill force downhill

F. In order to consider the inertias of the wheels and the flywheel

mass of the ICE, equivalent masses are added to the overall vehicle mass. In the case of the

flywheel mass, the average transmission ratio is used for parameter conversion.

By means of the torque FD

T and the rotational speed FD

at the final drive, the

model of the DHT is evaluated (i.e., the torques and rotational speeds of the ICE and the

electric drive are determined by considering the operation mode):

BatteryVehicle Dynamics

x,drag veh veh,x

(,, )FfAcdv=



veh

veh veh,x

ma⋅

x,roll

x,downhill

mg=⋅⋅ w

DHT

ICE

1/2

Vehicle mass

Projected

frontal area

Drag coefficient

Rolling friction

coefficient

veh 1920kgm=

veh 2.6mA=

r0.01f=

w0.25c=

Internal cell

voltage

Battery

capacity

Number of

cells

Internal cell

resistance

0,nom 3.75VV=

cell,nom 40AhQ=

i 2.6mΩR=

cell 80z=

Stationary gear

ratio plan. gear

Gear ratio for

electric motor

Gear ratio

first gear

Gear ratio

second gear

02.2i=

ED 1.9i=

23.39i=

16.79i=

number of cells

i ()

SoC

Batt

cell

0()VSoC

cell

Electric Drive

ICE

min.

spec. fuel consumption b

g/kWh

ICE,max

ICE,min

nICE,1

min

max power losses

ED,loss

in W

ED,max

ED,1

Maximal

torque

Rotational

speed @

Minimal rota-

tional speed

Maximal rota-

tional speed

ICE,max 190 NmT=

ICE,1

4100minn=

-1

ICE,min 6400minn=

-1

ICE,min 800minn=

ICE,max

Maximal

torque

Rotational speed

@ max. power

Maximal rota-

tional speed

ED,max 210NmT=

ED,1

5700minn=

-1

ED,max 15,000minn=

Figure 4. Models of the powertrain components and the most important parameters.

The vehicle dynamics are considered in a longitudinal direction only. The torque

TFD

and angular velocity

ωFD

at the final drive are determined according to the following

equation:

TFD =rw·Fdrag(vveh)+Froll +Fdownhill +mveh·aveh, (1)

with the radius of the wheels

, the vehicle mass

mveh

, the vehicle speed

vveh

, and the

vehicle acceleration

aveh

(extracted from the driving cycle). The forces represent the driving

resistances comprising the drag force

Fdrag

, the rolling friction force

Froll

, and the downhill

force

Fdownhill

. In order to consider the inertias of the wheels and the flywheel mass of the

ICE, equivalent masses are added to the overall vehicle mass. In the case of the flywheel

mass, the average transmission ratio is used for parameter conversion.

By means of the torque

TFD

and the rotational speed

ωFD

at the final drive, the model

of the DHT is evaluated (i.e., the torques and rotational speeds of the ICE and the electric

drive are determined by considering the operation mode):

sm=





jfor CVT-Mode in j-th gear,

j+2 for PAR-Mode in j-th gear,

j+4 for EM-Mode in j-th gear,

(2)

where

denotes the gear in which the corresponding mode is driven. Since a two-speed

transmission is considered, each operation mode can be driven in two speeds

(j∈{1, 2})

Appl. Sci. 2022,12, 2905 6 of 17

Furthermore, the control variable for the electric drive

uED

is defined according to current

operation state smand the corresponding degree of freedom:

uED =









ωED for sm=1 . . . 2,

TED for sm=3 . . . 4,

f(TFD,ωFD)for sm=5 . . . 6.

(3)

In the case of the EM mode, no degree of freedom exists, and consequently,

TED

and

ωED

are completely defined by means of the torque and angular velocity at the final drive.

The model of the DHT comprises a simple planetary gear model with the stationary gear

ratio

. Its sun gear is connected to the ICE, its ring gear to the electric drive, and its carrier

to the two-speed transmission. Considering this configuration as well as the transmission

ratios of the two-speed transmission

and the electric drive

iED

, the torques and angular

velocities in the CVT mode are determined according to the following equation:

TED =−i0·(1+i0)·iED·ij−1·TFD,

TICE =(1+i0)·ij−1·TFD,

ωICE = (1+i0)·ij·ωFD +i0·i−1

ED·ωED,

(4)

with the angular velocity of the electric drive

ωED

as the degree of freedom (

uED =ωED

Analogously, the angular speeds and torques in the PAR mode can be derived, yielding

the following:

ωED =−iED·ij·ωFD,

ωICE =ij·ωFD,

TICE =i−1

j·TFD +iED·TED,

(5)

where the torque of the electric drive

TED

needs to be chosen (

uED =TED

). For the EM

mode, only the torque and angular velocity of the electric drive must be determined:

TED =−iED·ij−1·TFD,

ωED =−iED·ij·ωFD.(6)

The ICE is represented by a map, which describes the specific fuel consumption

a function of the torque

TICE

and rotational speed

ωICE

. The mass flow rate of the fuel is

determined according to the following equation:

mf=be(TICE,ωICE)·TICE·ωICE

3.6·106, (7)

with

g·s−1

. Similarly, the required electric power

Pel

of the electric drive is deter-

mined by

Pel =TED·ωED +PED,loss(|TED|,|ωED|), (8)

where the power loss

PED,loss

represents the result of the corresponding map, shown in

Figure 4. This map also includes the power losses of the inverter. Since the electric power

Pel must be provided by the battery, the battery current is expressed as

q=−iBatt =−V0(SoC)

2·Ri(SoC)+sV0(SoC)

2·Ri(SoC)2

−Pel

zcell·Ri(SoC). (9)

This follows from the simplified equivalent circuit in Figure 4, where

zcell

is the

number of cells,

is the internal cell voltage, and

is the internal cell resistance. In

Appl. Sci. 2022,12, 2905 7 of 17

this representation,

and

are functions of the state of charge, considering the nominal

battery capacity Qcell,nom:

SoC =100

3600·Z.

zcell·Qcell,nom

dt. (10)

Since the powertrain model is executed in discrete time steps, the integration of

can

be solved explicitly (i.e., no algebraic loop occurs).

Next, the powertrain model described above is used for the optimal control of the

considered hybrid electric powertrain. Furthermore, it is part of the ECMS implementation

presented in Section 4.

3. Optimal Control of the Hybrid Electric Powertrain

The criterion for optimal control of the hybrid electric powertrain is minimal fuel

consumption (i.e., for a given driving cycle, the electric drive and the ICE must be operated

in such a way that the consumed fuel is minimal). Therefore, the mass flow rate of the

fuel

in Equation (7) is defined to be a cost function, while the overall costs and fuel

consumption are obtained by integrating .

mfover the duration of the driving cycle te:

mf=

mf(TICE,ωICE)dt. (11)

Additionally, the battery current is expressed as

q=−iBatt(q,Pel), (12)

with the following initial and end conditions:

q0=SoC0

100 ·zcell·Qcell,nom,

qe=SoCe

100 ·zcell·Qcell,nom,(13)

which are related to the battery’s state of charge at the beginning

SoC0

and the end

SoCe

of the driving cycle that must be considered. The evaluation of the state of Equation (12)

requires the electrical power

Pel

, which is consumed or generated by the electrical drive.

According to Equation (8), it can be summarized as

Pel =fED(TED,ωED). (14)

The torques and angular velocities of the ICE and electric drive in

Equations (11) and (14)

are determined by the transmission model of Equations (2)–(6). This can be summarized

as follows:

[TICE TED ωICE ωED]T=fTrans(TFD,ωFD,u), (15)

with the torque

TFD

and angular velocity

ωFD

at the final drive resulting from the vehicle

dynamics model in Equation (1) and the control variable of the powertrain:

u=sm

uED . (16)

It must be taken into account that the control variable as well as the state variable are

limited due to the technical boundaries of the powertrain components:

sm∈ UT, with UT={sm∈N|1≤sm≤6},

uED ∈ UED, with UED =uED ∈RuED ∈ UTED or uED ∈ UωED ,

q∈ Xq, with Xq={q∈R|0≤q≤zcell·Qcell,nom}.

(17)

Appl. Sci. 2022,12, 2905 8 of 17

The sets

UTED

and

UωED

correspond to the torque and angular velocity boundaries,

respectively, given by the maps of the ICE and electric drive shown in Figure 4. In order to

obtain the optimal control of the hybrid electric powertrain, the minimal fuel consumption

m∗

f=min

umf(u)(18)

must be determined with respect to Equations (11)–(17). The optimal control variable

u∗

a result of Equation (18).

In general, the various methods for the solution of the optimization problem in

Equations (11)–(18) are known. One approach is Pontryagin’s Maximum Principle (PMP).

It is based on variational calculus and the enhancement of considering the boundaries of

the control variables in Equation (17). For obtaining the optimal control variable

u∗

means of PMP, the Hamilton function

H(λ,u,q)=−.

mf(u)+λ·.

q(u,q)(19)

must be evaluated while considering the necessary conditions given in [

]. In

Equation (19)

is the Lagrange multiplier. If it is known in advance, the optimal control variable

u∗

will be obtained by finding the maximum of the Hamilton function with respect to

Equations (11)–(17):

H∗=max

u−.

mf(u)+λ·.

q(u,q). (20)

Usually, the Lagrange multiplier

is a function of time and the vehicle’s position such

that Equation (20) can be carried out as a local optimization for a certain position of the

vehicle. In the EMCS algorithm, the control variables of the powertrain are determined by

means of this local optimization, where λis obtained by a prediction (see Section 4).

Another method for solving the optimization problem in Equations (11)–(18) is based

on Dynamic Programming (DP). Here, the driving cycle is separated into discrete instants

of time

tk=k·T

, with the step size

and

. . . N

. Additionally, the state and input

variables are quantized by means of the state variable grid

qg= [qg

1. . . qg

n]T

and the input

variable grids

m= [sg

m,1 . . . sg

m,o]T

and

ED = [ug

ED,1 . . . ug

ED,m]T

. A cost matrix

with

rows and

columns defines the accumulated costs for all possible state transitions from

one time step to another. Each row is assigned to the corresponding state within

and each

column to the instant of time

. In order to obtain

, a backward computation beginning

with

k=N−

1 and ending with

0 is carried out. In each time step

, the minimal costs

must be determined for all i=1 . . . nas follows:

Ji,k=min

uknJp,k+1+T·.

mf(uk)o. (21)

The minimum is determined by evaluating all possibilities for the control variable

(i.e., all combinations of the elements within the grid vectors

and

with respect to

the system boundaries defined in Equation (17). For the determination of the costs

Jp,k+1

the state equation is evaluated:

qp=qg

i+T·.

quk,qg

i, (22)

where the index

denotes the row in

assigned to

. Since

describes a continuous

variable, it might not match an element within the predefined state grid

. In this case,

Jp,k+1is interpolated based on the surrounding elements within qg[21].

Once the cost matrix

is known, the optimal control variable

u∗

is obtained by means

of a forward computation beginning with k=0 and ending with k=N−1:

u∗

k=arg min

uknJp,k+1+T·.

mf(uk)o, (23)

Appl. Sci. 2022,12, 2905 9 of 17

where

Jp,k+1

is determined analogously to the backward computation based on the follow-

ing state equation:

q∗

p=q∗

k+T·.

q(uk,q∗

k). (24)

Here, the optimal electric charge

q∗

and beginning with

q∗

relate to the initial state of

charge

SoC0

(see Equation (13)). Since DP can only be applied on a driving cycle, which is

known in advance, it cannot be used as a real-time capable operating strategy. Furthermore,

a very high computation effort is required when choosing a high resolution for the state

and input variable grids. Nevertheless, DP can be used to predict

for the ECMS and to

compute the theoretically minimal fuel consumption for a given driving cycle. The latter is

used as an optimum serving as reference for the evaluation of the results obtained by the

ECMS algorithm.

4. Equivalent Consumption Minimization Strategy (ECMS)

In this section, the ECMS algorithm will be presented. It is based on the powertrain

models presented in Section 2and the optimization methods presented before. Figure 5

shows the algorithm of the ECMS. The input variables are the current state of charge

SoCk

the vehicle speed

vveh,k

, and the vehicle acceleration

aveh,k

. Considering an implementation

adapted to the real vehicle operation,

SoCk

and

vveh,k

are measured values, and

aveh,k

is a request from the driver. However, for the intended analysis,

aveh,k

as well as

vveh,k

were obtained from a driving cycle and the

SoCk

from a powertrain model. The ECMS

algorithm is based on PMP and carries out a local optimization of the Hamilton function

(Equation (19)) in each time step. In order to evaluate the Hamilton function, the powertrain

model (Equations (1)–(10)) was used to determine the fuel consumption and battery current

for a given control variable

. Furthermore, the Lagrange multiplier

λk

was required,

representing a function of the current state of charge

SoCk

and position of the vehicle

zveh,k

This function was obtained from a prediction of the future driving behavior, which had

to be carried out in advance. For the optimization of the Hamilton function, the control

variable ukwas quantized by means of the grid vectors sg

mand ug

ED (see Section 3).

Figure 5

Figure 6

Figure 7

Evaluation of the Lagrange multiplier



Prediction of the driving behavior and

data processing (offline)

ECMS-Algorithm

Computation of the control variables for the transmission, ICE and

electric drive

Optimization of the Hamilton function (Equation (19))

Powertrain Model (Equations (1)–(10))

()

=veh,

kkk

fSoC z

veh,k

SoC veh,k

veh,k

() ( )

{

}

=−+⋅



aar m x ,g

kkkkkk

mqq

uuu



=





*m,

ED,

()

ωω ω



 *

ICE, ED, ICE, ED, Trans FD, FD,

kk k k k kk

TT f T u

Evaluation of the fuel consumption and state of charge

Equation (1)

+f, 1k



f

Equation (7)

+1k

SoC

Equations (8)‒(10)

2.2

2.4

2.6

2.8

State of Charge

SoC in %

Lagrange Multi-

plier λin 10

-5

·l/As

010 20 30 40 50 60 70 80

2.3

2.5

2.7

Position z

veh

in km

Lagrange Multi-

plier λin 10

-5

·l/As

SoC

veh

)

(SoC

veh

) for:

SoC

= 80%

SoC

= 30%

PI-Controller

Prediction

SoC

veh,k

zSP

SoC

Figure 5.

Implemented ECMS algorithm based on PMP with a prediction of the future driving

behavior and the evaluation of the overall fuel consumption and the state of charge.

Appl. Sci. 2022,12, 2905 10 of 17

This enabled finding the maximum by testing all possible combinations given by

the grid vectors. In general, it is possible to apply other methods to obtain the optimum,

but it must be ensured that the correct solution is found and the computation time of the

ECMS algorithm is still adequate. Once the control variable

u∗

was known, it was used

to evaluate the torques and angular velocities of the ICE and electric drive by means of

the transmission model (Equations (4)–(6)). For real vehicle operation, these values as

well as the transmission control variable

s∗

were the setpoints for the controllers of the

corresponding powertrain components. Here, these values were used to determine the fuel

consumption

mf,k+1

and state of charge

SoCk+1

by means of the same powertrain model,

which was already applied to the optimization.

Crucial for obtaining good results with the ECMS is a well-chosen Lagrange multiplier

λk

matching the real driving behavior as well as possible. Since the real driving behavior

was unknown in advance,

λk

was obtained by means of a prediction. For this purpose, two

already known methods from the literature were considered in this contribution. These

methods are presented in the following subsections.

4.1. Method 1: Hamilton–Jacobi–Bellman Equation

The Lagrange multiplier

λk

required for evaluating the ECMS algorithm can be ob-

tained by means of DP in Equations (21)–(24). Due to the high computational effort of DP,

it must be precomputed based on a driving cycle. This driving cycle can only be predicted

and needs to approximate the future driving behavior as well as possible. In order to

determine λk, the Hamilton–Jacobi–Bellman equation is used [43]:

∂J∗(q,t)

∂t=max

u{− .

mf(u)−∂J∗(q,t)

∂q

| {z }

·.

q(u,q)}. (25)

This equation describes the connectedness between the DP and PMP, where the right

side represents the Hamilton function (Equation (19)). Accordingly, the partial derivative

of the optimal accumulated costs J* from the battery charge

corresponds to

. Due to

the discrete functional principle of DP, the partial derivative must be approximated by the

following means for k=0 . . . N−1 and i=1 . . . n:

λi,k=−Ji+1,k−Ji,k

i+1−qg

. (26)

The states

correspond to the elements within the state variable grid

qg= [qg

1. . . qg

n]T

used for carrying out DP and the costs

Ji,k

for the elements within the cost matrix

describing the result of the backward computation (Equation (21)). Since each index

is assigned to an instant of time

, it can be also assigned to a vehicle position

zveh,k

integrating the vehicle speeds

vveh,k

given by the considered driving cycle. Moreover, the

elements in the state variable grid

can be converted into a state of charge according to

Equation (10). Consequently, from Equation (26),

is the result of a function of

zveh,k

and

SoCk, as shown in Figure 5.

Figure 6shows an example for the Lagrange multipliers

determined according to

Equation (26). The upper figure shows

as a function of

zveh

and

SoC

. It also contains

the optimal characteristics of the state of charge

SoC∗

for the initial values

SoC0=

80%

(red line) and

SoC0=

30% (green line). Both characteristics were the result of DP, which

was applied on the same driving cycle used to determine

. The lower figure shows the

Lagrange multipliers

belonging to

SoC∗

. Both characteristics were obtained from the

upper diagram. If the battery is constantly discharged during vehicle operation, the values

for

will be lower than if the state of charge is kept constant or the battery is charged. In

the algorithm of the ECMS shown in Figure 5, the function for

is implemented as a map,

which corresponds to the upper part of Figure 6.

Appl. Sci. 2022,12, 2905 11 of 17

Appl. Sci. 2022, 12, x FOR PEER REVIEW 11 of 18

. Due to the discrete functional principle of DP, the partial derivative must be ap-

proximated by the following means for 01kN=− and 1in=:

1, ,

,gg

ik ik

−

=− − (26)

The states g

q correspond to the elements within the state variable grid

ggg

[]

qq=q



used for carrying out DP and the costs ,ik

for the elements within the

cost matrix J, describing the result of the backward computation (Equation (21)). Since

each index k is assigned to an instant of time k

t, it can be also assigned to a vehicle

position veh,k

z by integrating the vehicle speeds veh,k

v given by the considered driving

cycle. Moreover, the elements in the state variable grid g

q can be converted into a state

of charge according to Equation (10). Consequently, from Equation (26),

is the result

of a function of veh,k

z and k

SoC , as shown in Figure 5.

Figure 6 shows an example for the Lagrange multipliers

determined according to

Equation (26). The upper figure shows

as a function of veh

z and SoC . It also con-

tains the optimal characteristics of the state of charge *

SoC for the initial values

080%SoC = (red line) and 030%SoC = (green line). Both characteristics were the result

of DP, which was applied on the same driving cycle used to determine

. The lower

figure shows the Lagrange multipliers

belonging to *

SoC . Both characteristics were

obtained from the upper diagram. If the battery is constantly discharged during vehicle

operation, the values for

will be lower than if the state of charge is kept constant or

the battery is charged. In the algorithm of the ECMS shown in Figure 5, the function for

is implemented as a map, which corresponds to the upper part of Figure 6.

Figure 6. The Lagrange multiplier

as a function of the state of charge SoC and the vehicle’s

position veh

z, determined by the Hamilton–Jacobi–Bellman equation (Equation (25)).

4.2. Method 2: PI Controller

The second investigated method for obtaining the Lagrange multiplier is a feedback

control, where

is used to control the SoC [35,44] (see Figure 7). A PI controller is

used, since the integrating part enables initializing

to a value within an appropriate

range (see Figure 6). In addition, the dynamic behavior of

can be influenced by means

of the time constant and gain of the controller. If the current state of charge k

SoC falls

below the setpoint SP

SoC , the PI controller increases k

, and the battery charges. Other-

wise, k

will be decreased, and the battery discharges (compare Figure 6). The parameters

of the PI controller are determined empirical in such a way that the initial 0

is chosen

2.2

2.4

2.6

2.8

State of Charge

SoC in %

Lagrange Multi-

plier λin 10

-5

·l/As

010 20 30 40 50 60 70 80

2.3

2.5

2.7

Position z

veh

in km

Lagrange Multi-

plier λin 10

-5

·l/As

SoC

veh

)

(SoC

veh

) for:

SoC

=80%

SoC

=30%

Figure 6.

The Lagrange multiplier

as a function of the state of charge

SoC

and the vehicle’s position

zveh, determined by the Hamilton–Jacobi–Bellman equation (Equation (25)).

4.2. Method 2: PI Controller

The second investigated method for obtaining the Lagrange multiplier is a feedback

control, where

is used to control the

SoC

[

] (see Figure 7). A PI controller is used,

since the integrating part enables initializing

to a value within an appropriate range

(see Figure 6). In addition, the dynamic behavior of

can be influenced by means of the

time constant and gain of the controller. If the current state of charge

SoCk

falls below the

setpoint

SoCSP

, the PI controller increases

λk

, and the battery charges. Otherwise,

λk

will

be decreased, and the battery discharges (compare Figure 6). The parameters of the PI

controller are determined empirical in such a way that the initial

λ0

is chosen appropriately,

and an oscillating behavior of

λk

is almost avoided. Due to this, frequent changes in the

transmission speeds and modes during vehicle operation are avoided.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 12 of 18

appropriately, and an oscillating behavior of k

is almost avoided. Due to this, frequent

changes in the transmission speeds and modes during vehicle operation are avoided.

Figure 7. Control of the SoC with the control variable

and predicted setpoints SP

SoC .

The setpoints for the controller SP

SoC are the result of a predicted function, which

describes the state of charge as a function of the current vehicle position veh,k

z. For this

purpose, the three approaches shown in Figure 8 are considered. Figure 8a shows a linear

discharge characteristic, beginning with the initial state of charge at the start position and

ending with the lower boundary of the state of charge at the end position of the trip. For

this kind of prediction, only the distance of the intended trip to be driven is required. In

Figure 8b, the linear discharge characteristic is adapted when the state of charge exceeds

the current characteristic due to recuperating the braking energy. This adaptation is car-

ried out in every execution step of the ECMS algorithm. The last approach for the pre-

diction, shown in Figure 8c, is based on an optimized discharge characteristic, which is

obtained by applying DP on a predefined driving cycle. As in the case of the Hamilton–

Jacobi–Bellmann equation presented in the previous section, this kind of prediction re-

quires a representative driving cycle and time-consuming optimization in advance.

Figure 8. Prediction of SP

SoC : (a) linear characteristic, (b) linear characteristic with adaptation to

the recuperated braking energy, and (c) optimized characteristic obtained from DP for a predicted

driving cycle.

The method based on the Hamilton–Jacobi–Bellmann equation as well as the con-

trol-based method can be used for the determination of the Lagrange multiplier required

for the ECMS algorithm shown in Figure 5. In order to apply one of these methods, either a

driving cycle or the distance of the intended trip must be predicted. In the following sec-

tion, the ECMS is applied by means of both methods. Each ECMS implementation is ap-

plied on different predictions and is analyzed in terms of the resulting fuel consumptions.

5. Analysis of Different ECMS

For the analysis of the ECMS, the fuel consumptions obtained by applying the dif-

ferent prediction methods described in Section 4 were compared to each other. The

powertrain model was parametrized according to Figure 4, while the evaluation of the

fuel consumptions was carried out according to the ECMS algorithm shown in Figure 5.

In the simulation, the vehicle braked by means of recuperation, which was limited to a

maximal electrical power of 35 kW. Furthermore, the battery’s initial state of charge was

defined to be 80%, and the lower boundary was set to 30%.

In order to consider the difference between the predictions and the real driving be-

havior, various driving cycles of the same route were simulated. For this purpose, a daily

commute in both directions was measured 39 times, including sections on a German in-

PI-Controller

Prediction

SoC

veh,k

zSP

SoC k

a) b) c)

max

min

max

min

max

min

max

SoC

veh

zveh

SoC SP

SoC

Figure 7. Control of the SoC with the control variable λand predicted setpoints SoCSP.

The setpoints for the controller

SoCSP

are the result of a predicted function, which

describes the state of charge as a function of the current vehicle position

zveh,k

. For this

purpose, the three approaches shown in Figure 8are considered. Figure 8a shows a linear

discharge characteristic, beginning with the initial state of charge at the start position and

ending with the lower boundary of the state of charge at the end position of the trip. For

this kind of prediction, only the distance of the intended trip to be driven is required. In

Figure 8b, the linear discharge characteristic is adapted when the state of charge exceeds

the current characteristic due to recuperating the braking energy. This adaptation is carried

out in every execution step of the ECMS algorithm. The last approach for the prediction,

shown in Figure 8c, is based on an optimized discharge characteristic, which is obtained by

applying DP on a predefined driving cycle. As in the case of the Hamilton–Jacobi–Bellmann

equation presented in the previous section, this kind of prediction requires a representative

driving cycle and time-consuming optimization in advance.

Appl. Sci. 2022,12, 2905 12 of 17

Figure 8

Figure 9

Figure 10

(a) (b) (c)

max

min

max

min

max

min

max

SoC

veh

zveh

SoC SP

SoC

Measured driving

cycles:

Min/Max

Mean

Driving cycle

obtained from

navigation data

Vehicle Position z

veh

in km

Vehicle Speed

veh

in km/h

010 20 30 40 50 60 70 80

120

150

180

1 3

7 9 11 13 1

17 19 21 23 25 27 29 31 33 35 37 39

2.8

3.2

3.4

3.6

3.8

4.2

offline optimization (DP)

CD-CS-Operation (control-based method)

Map obtained from the Hamilton-Jacobi-Bellmann equation (Figure 6):

Optimization based on mean driving cycle

Optimization based on navigation data

Driving Cycle No.

Average Fuel Consumption

in l/100km

Control-based method with setpoints defined by

Optimized characteristic, Figure 8c)

Linear characteristic with adaption, Figure 8b)

Linear characteristic, Figure 8a)

veh

()SoC f z

veh

(,)SoC z

Figure 8.

Prediction of

SoCSP

: (

) linear characteristic, (

) linear characteristic with adaptation to

the recuperated braking energy, and (

) optimized characteristic obtained from DP for a predicted

driving cycle.

The method based on the Hamilton–Jacobi–Bellmann equation as well as the control-

based method can be used for the determination of the Lagrange multiplier required for the

ECMS algorithm shown in Figure 5. In order to apply one of these methods, either a driving

cycle or the distance of the intended trip must be predicted. In the following section, the

ECMS is applied by means of both methods. Each ECMS implementation is applied on

different predictions and is analyzed in terms of the resulting fuel consumptions.

5. Analysis of Different ECMS

For the analysis of the ECMS, the fuel consumptions obtained by applying the different

prediction methods described in Section 4were compared to each other. The powertrain

model was parametrized according to Figure 4, while the evaluation of the fuel consump-

tions was carried out according to the ECMS algorithm shown in Figure 5. In the simulation,

the vehicle braked by means of recuperation, which was limited to a maximal electrical

power of 35 kW. Furthermore, the battery’s initial state of charge was defined to be 80%,

and the lower boundary was set to 30%.

In order to consider the difference between the predictions and the real driving be-

havior, various driving cycles of the same route were simulated. For this purpose, a daily

commute in both directions was measured 39 times, including sections on a German inter-

state. Each of these driving cycles had a distance of 80 km and was unique due to varying

driving behavior and traffic. Figure 9shows the minimal, maximal, and mean vehicle

speeds resulting from the measured driving cycles. Additionally, a driving cycle of the

same route obtained from the navigation data is shown. This driving cycle as well as the

mean driving cycle was used for the predictions, while the measured driving cycles were

used to evaluate the fuel consumptions.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 13 of 18

terstate. Each of these driving cycles had a distance of 80 km and was unique due to

varying driving behavior and traffic. Figure 9 shows the minimal, maximal, and mean

vehicle speeds resulting from the measured driving cycles. Additionally, a driving cycle

of the same route obtained from the navigation data is shown. This driving cycle as well

as the mean driving cycle was used for the predictions, while the measured driving cy-

cles were used to evaluate the fuel consumptions.

Figure 9. Minimal, maximal, and mean value of 39 measured driving cycles of the same route and

the route’s driving cycle obtained from navigation data [45].

Figure 10 shows the simulation results obtained from the ECMS for all considered

prediction methods and driving cycles. For comparison, the results of an offline optimi-

zation with DP (black marker) are shown, representing the theoretically minimal fuel

consumption for each of the measured driving cycles. Furthermore, the results of a sim-

ple CD-CS operating strategy (charge depleting–charge sustaining) are shown, too (grey

marker). It was first driven with the electric motor until the battery reached its lower

boundary of the state of charge. Then, it was mainly driven by the ICE, keeping the state

of charge constant. The CD-CS strategy was applied by means of the control-based ap-

proach, where SP 30%SoC = was a constant setpoint. The results of the 39 driving cycles

were arranged in ascending order of energy for propulsion so that the fuel consumptions

on the left corresponded to a conservative driving style and those on the right corre-

sponded to an agile driving style.

The lowest fuel consumptions were obtained by the ECMS based on the Hamilton–

Jacobi–Bellmann equation (red marker), where the mean driving cycle was used for para-

metrization. However, since such a driving cycle is generally unknown, it can only be de-

termined when the driving behavior of frequently driven routes is measured (e.g., daily

commute). More common is the parametrization according to the navigation data (green

marker), which does not need any measurements and causes slightly higher fuel con-

sumptions. For the application of both ECMS variants, a representative driving cycle must

be determined, and in particular, a time-consuming optimization needs to be carried out in

advance. Therefore, it is rather appropriate for trips, which can be scheduled in advance.

The fuel consumptions obtained by applying the control-based methods were gen-

erally higher compared with the consumptions obtained with the method based on the

Hamilton–Jacobi–Bellmann equation. As expected, the best results were obtained with an

optimized state of charge characteristic for the setpoints of the controller (purple marker).

This variant also required a predefined driving cycle and optimization in advance, which

was carried out here by means of DP. However, it was basically possible to apply another

optimization method, which enabled faster computation of the state of charge character-

istic (e.g., the combination of DP and PMP [32]. In comparison with the approach based

on the Hamilton–Jacobi–Bellmann equation, the control-based approach had the poten-

tial to be initialized faster. The other variants (blue and cyan marker) only required the

distance of the intended trip in order to define the characteristic for the setpoints of the

controller. Adapting this characteristic to the current state of charge when recuperating

(blue marker) enabled slightly lower fuel consumptions. This, however, depended on how

Measured driving

cycles:

Min/Max

Mean

Driving cycle

obtained from

navigation data

Vehicle Position z

veh

in km

Vehicle Speed

veh

in km/h

010 20 30 40 50 60 70 80

120

150

180

Figure 9.

Minimal, maximal, and mean value of 39 measured driving cycles of the same route and

the route’s driving cycle obtained from navigation data [45].

Figure 10 shows the simulation results obtained from the ECMS for all considered pre-

diction methods and driving cycles. For comparison, the results of an offline optimization

with DP (black marker) are shown, representing the theoretically minimal fuel consumption

for each of the measured driving cycles. Furthermore, the results of a simple CD-CS operat-

ing strategy (charge depleting–charge sustaining) are shown, too (grey marker). It was first

driven with the electric motor until the battery reached its lower boundary of the state of

Appl. Sci. 2022,12, 2905 13 of 17

charge. Then, it was mainly driven by the ICE, keeping the state of charge constant. The

CD-CS strategy was applied by means of the control-based approach, where

SoCSP =

30%

was a constant setpoint. The results of the 39 driving cycles were arranged in ascending

order of energy for propulsion so that the fuel consumptions on the left corresponded to a

conservative driving style and those on the right corresponded to an agile driving style.

Appl. Sci. 2022, 12, x FOR PEER REVIEW 14 of 18

much energy could be recuperated. Since these variants did not require a driving cycle or

optimization in advance, it was also possible to apply them to instantaneous trips.

Figure 10. Simulation results for the ECMS, showing average fuel consumptions for all driving cy-

cles and prediction methods.

Table 1 shows an overview of the considered operating strategies and the corre-

sponding information required for its applications. Furthermore, it contains the mean

percentage increase of the fuel consumptions obtained by each operating strategy related

to the theoretically minimal fuel consumption obtained with DP. The results show that

only minor information about the driving cycle enabled the ECMS to achieve signifi-

cantly lower fuel consumptions compared with the non-optimization-based CD-CS op-

eration. In addition, the more information about the future driving cycle was available,

the lower the fuel consumptions obtained by the ECMS were. Optimization-based pre-

dictions led to an improvement in the results, while the best results were obtained when

the real driving behavior was used for optimization. The control-based method with a

linear characteristic required the same input information as the control-based method

with the adaption to the SoC but caused higher fuel consumption. For this reason, it

was more beneficial to apply the variant with the adaption, since the additional imple-

mentation effort was only slightly increased. The variant with the optimized characteris-

tic further reduced the fuel consumption but required optimization in advance. The ad-

vantage in comparison with the method based on the Hamilton–Jacobi–Bellmann equa-

tion is that the optimization does not need to be carried out with DP. Another optimiza-

tion method can be chosen which computes the required characteristic significantly faster

(e.g., the method presented in [32]).

1 3

7 9 11 13 1

17 19 21 23 25 27 29 31 33 35 37 39

2.8

3.2

3.4

3.6

3.8

4.2

offline optimization (DP)

CD-CS-Operation (control-based method)

Map obtained from the Hamilton-Jacobi-Bellmann equation (Figure 6):

Optimization based on mean driving cycle

Optimization based on navigation data

Driving Cycle No.

Average Fuel Consumption

in l/100km

Control-based method with setpoints defined by :

Optimized characteristic, Figure 8c)

Linear characteristic with adaption, Figure 8b)

Linear characteristic, Figure 8a)

veh

()SoC f z=

veh

(,)SoC z

Figure 10.

Simulation results for the ECMS, showing average fuel consumptions for all driving cycles

and prediction methods.

The lowest fuel consumptions were obtained by the ECMS based on the Hamilton–

Jacobi–Bellmann equation (red marker), where the mean driving cycle was used for

parametrization. However, since such a driving cycle is generally unknown, it can only be

determined when the driving behavior of frequently driven routes is measured (e.g., daily

commute). More common is the parametrization according to the navigation data (green

marker), which does not need any measurements and causes slightly higher fuel consump-

tions. For the application of both ECMS variants, a representative driving cycle must be

determined, and in particular, a time-consuming optimization needs to be carried out in

advance. Therefore, it is rather appropriate for trips, which can be scheduled in advance.

The fuel consumptions obtained by applying the control-based methods were generally

higher compared with the consumptions obtained with the method based on the Hamilton–

Jacobi–Bellmann equation. As expected, the best results were obtained with an optimized

state of charge characteristic for the setpoints of the controller (purple marker). This

variant also required a predefined driving cycle and optimization in advance, which was

carried out here by means of DP. However, it was basically possible to apply another

optimization method, which enabled faster computation of the state of charge characteristic

(e.g., the combination of DP and PMP [

]. In comparison with the approach based on the

Hamilton–Jacobi–Bellmann equation, the control-based approach had the potential to be

initialized faster. The other variants (blue and cyan marker) only required the distance

of the intended trip in order to define the characteristic for the setpoints of the controller.

Adapting this characteristic to the current state of charge when recuperating (blue marker)

Appl. Sci. 2022,12, 2905 14 of 17

enabled slightly lower fuel consumptions. This, however, depended on how much energy

could be recuperated. Since these variants did not require a driving cycle or optimization

in advance, it was also possible to apply them to instantaneous trips.

Table 1shows an overview of the considered operating strategies and the correspond-

ing information required for its applications. Furthermore, it contains the mean percentage

increase of the fuel consumptions obtained by each operating strategy related to the theo-

retically minimal fuel consumption obtained with DP. The results show that only minor

information about the driving cycle enabled the ECMS to achieve significantly lower fuel

consumptions compared with the non-optimization-based CD-CS operation. In addition,

the more information about the future driving cycle was available, the lower the fuel

consumptions obtained by the ECMS were. Optimization-based predictions led to an

improvement in the results, while the best results were obtained when the real driving

behavior was used for optimization. The control-based method with a linear characteristic

required the same input information as the control-based method with the adaption to

the

SoC

but caused higher fuel consumption. For this reason, it was more beneficial to

apply the variant with the adaption, since the additional implementation effort was only

slightly increased. The variant with the optimized characteristic further reduced the fuel

consumption but required optimization in advance. The advantage in comparison with the

method based on the Hamilton–Jacobi–Bellmann equation is that the optimization does

not need to be carried out with DP. Another optimization method can be chosen which

computes the required characteristic significantly faster (e.g., the method presented in [

]).

Table 1.

Overview of the considered operating strategies and summary of the simulation results

shown in Figure 10.

Operating Strategy Prediction Method Input Information Deviation to Minimal Fuel

Consumption 1(

Appl. Sci. 2022, 12, x FOR PEER REVIEW 15 of 18

Table 1. Overview of the considered operating strategies and summary of the simulation results

shown in Figure 10.

Operating

Strategy

Prediction

Method

Input

Information

Deviation to Minimal Fuel

Consumption 1

ECMS based on

the Hamil-

ton-Jacobi-Bell-ma

nn equation

Optimization results

from DP (map)

Averaged driving cy-

cle of the real driving

behavior

Driving cycle from

navigation data 3.8 (

ECMS with con-

trol-

ased method

Optimization results

from DP 2 (Optimized

characteristic SoC*)

Averaged driving cy-

cle of the real driving

behavior

6.9%

Linear characteristic

with adaption to the

current SoC Distance of the in-

tended route

8.7%

Linear characteristic 10.1%

CD-CS-

Operation -

- 17.4% )

1 Mean percental increase of the fuel consumptions obtained with each operating strategy related to

the theoretically minimal fuel consumptions obtained with the DP. 2 Not necessarily DP, as the

optimized characteristic can also be obtained by means of other optimization methods.

Since the ECMS was applied for the same model used for the ECMS algorithm itself,

some of the resulting fuel consumptions were very close to the theoretical minimum. In

practice, however, the powertrain model used for the ECMS algorithm differed from the

real powertrain (e.g., due to neglecting the powertrain dynamics) so that, in reality,

higher fuel consumptions were to be expected. Nevertheless, it can be assumed that the

previously identified tendencies of reducing the fuel consumptions by improving the

prediction methods will still remain.

The required input information can be obtained by using the intended route infor-

mation of the navigation system, providing the driver’s needs. Furthermore, automatized

approaches are recommended for recognizing and logging frequently driven routes.

6. Conclusions

This contribution described a model-based analysis of different ECMS implementations

for a plug-in hybrid electric powertrain, which can be operated in different operation modes

and each of them operating at two speeds. A corresponding powertrain model was devel-

oped based on the backward simulation approach. This model neglects the system dynamics

except for the vehicle mass and describes the fuel and energy consumptions by means of

maps. The powertrain model is part of the ECMS algorithm, which is an optimization-based

operating strategy based on PMP. In order to apply the ECMS on the considered powertrain,

an appropriate description of the Lagrange multiplier is required. Two methods for the de-

termination of this description were presented: the Hamilton–Jacobi–Bellmann equation,

which requires a predefined driving cycle and a corresponding optimization with DP in ad-

vance, and a control-based method, where the state of charge is controlled based on a prede-

fined characteristic for the controller’s setpoints. Both methods require a prediction of the

future driving behavior. Either a driving cycle or the distance to be driven must be predicted.

The analysis of the ECMS and its prediction methods was carried out based on various

driving cycles, which were measured for the same route. For the predictions, either the mean

driving cycle obtained from the measurements, the corresponding navigation data, or the

distance to be driven was considered. As it turned out, the ECMS based the Hamilton–

Jacobi–Bellmann equation enabled the best results. However, this method requires a driving

)

ECMS based on the

Hamilton-Jacobi-Bell-

mann equation

Optimization results from

DP (map)

Averaged driving cycle of the real

driving behavior 0.6% (