Predictive planning of optimal velocity and state of charge trajectories for hybrid electric vehicles

Control Engineering Practice ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Predictive planning of optimal velocity and state of charge trajectories for hybrid electric vehicles$ Gunter Heppeler a,n, Marcus Sonntag a, Uli Wohlhaupter b, Oliver Sawodny a a b

University of Stuttgart, Institute for System Dynamics, Waldburstraße 17/19, 70569 Stuttgart, Germany Daimler AG, Stuttgart, Germany

art ic l e i nf o

a b s t r a c t

Article history: Received 15 May 2015 Received in revised form 1 July 2016 Accepted 2 July 2016

The combination of electric motors and internal combustion engines in hybrid electric vehicles (HEV) can considerably improve the fuel efficiency compared to conventional vehicles. In order to use its full potential, a predictive intelligent control system using information about impending driving situations has to be developed, to determine the optimal gear shifting strategy and the torque split between the combustion engine and the electric motor. To further increase fuel efficiency, the vehicle velocity can be used as an additional degree of freedom and the development of a predictive algorithm calculating good choices for all degrees of freedom over time is necessary. In this paper, an optimization-based algorithm for combined energy management and economic driving over a limited horizon is proposed. The results are compared with results from an offline calculation, which determine the overall fuel savings potential through the use of a discrete dynamic programming algorithm. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Hybrid vehicles Dynamic programming Sequential quadratic programming Energy management Potential analysis Velocity optimization Predictive control Economic driving

1. Introduction Over the last few decades, reducing fuel consumption for vehicles has become a more and more important field of research. Reasons for this development are increasing environmental awareness accompanied by stricter regulations and rising fuel costs. One technology emerging from this research is the hybrid electric vehicle (HEV), combining electric batteries and motors with the internal combustion engine (ICE) in one powertrain. There exists a wide variety of HEVs with different structures, e.g. parallel hybrid vehicles or series hybrid vehicles, and different degrees of hybridization, e.g. micro-hybrids, which are characterized by small batteries and electric motors (EM) whose main purpose is to implement an automatic engine start/stop function. At the other end of the scale plug-in hybrid vehicles with large batteries and motors facilitate long all-electric range and the possibility to charge the battery from the power grid (Guzzella and Sciarretta, 2013). To achieve the best fuel efficiency, an optimal energy management system (EMS) is necessary to coordinate both power ☆ This work is part of the “Promotionskolleg Hybrid”, a cooperation between science and industry, funded by the Ministry of Science, Research and the Arts of the State of Baden-Württemberg, Germany and the Daimler AG. n Corresponding author. E-mail address: [email protected] (G. Heppeler).

sources. Since an optimal control strategy depends on the driving cycle, this poses two main challenges: firstly, the parameters of the driving cycle are not necessarily known and, secondly, finding a global, optimal solution for the resulting nonlinear optimal control problem is numerically challenging. Therefore, this is a demanding field of research. For previous work in this field, see Sciarretta and Guzzella (2007), Pisu and Rizzoni (2007), Johannesson and Egardt (2008), Bender, Kaszynski, and Sawodny (2013) and Panday and Bansal (2014). Another approach is economic driving, i.e., using the velocity as an additional degree of freedom to reduce fuel consumption. In recent years, work for different types of vehicles, such as conventional cars and trucks (e.g. Hellström, Åslund, and Nielsen, 2010; Hooker, 1988; Kamal, Mukai, Murata, and Kawabe, 2013; Llamas, Eriksson, and Sundström, 2013; Terwen, Back, and Krebs, 2004), fuel-cell cars (Sciarretta, Guzzella, and van Baalen, 2004) and electric cars (Petit and Sciarretta, 2011) has been published. The next step to increasing fuel efficiency is to combine both economic driving and predictive EMS. In van Keulen et al. (2009, 2010), the velocity profiles for a hybrid electric truck are optimized. Therefore, the driving cycle is partitioned into segments of constant power request and each of these segments is divided into four phases: max. power acceleration, constant velocity, coasting and max. power deceleration. The parameters of these phases are then optimized to maximize power recovery and minimize fuel consumption. In Kim, Manzie, and Sharma (2009), a model

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Please cite this article as: Heppeler, G., et al. Predictive planning of optimal velocity and state of charge trajectories for hybrid electric vehicles. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.07.003i

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Nomenclature discretization of the short horizon optimization stage transition costs for the short horizon optimization phase length Δp maximum phase length Δpmax minimum phase length Δpmin maximum allowed velocity exceedance Δvex offset between reference and maximum velocity Δvoff Δvsafety security offset to the maximum possible curve velocity ΔxSoC,max maximum SoC change for a use case in long horizon battery planning ΔxSoC,min minimum SoC change for a use case in long horizon battery planning fuel consumption per second of the internal combusṁ f tion engine fuel saved per second for a certain engine and motor ṁ f , save operation point γ road grade κ road curvature λ equivalence factor of the ECMS use case in the long horizon battery planning < μ friction coefficient between road and wheels rolling coefficient μR speed of the electric motor ωEM ωICE speed of the internal combustion engine speed of the propshaft ωprop speed of the wheel ωwhl ςi weighting factors in the cost functions of the short horizon optimization ξi weighting factors in the cost functions of the long horizon battery planning a vehicle acceleration state of the clutch (open or closed) bclt engine on/off state bICE air drag coefficient CAir air drag resistance force Fdrag grade resistance force Fgrade rolling resistance force Froll position derivative of the states x SHO fSHO fs vector of time derivatives for the vehicle states xs ft vector of time derivatives for the vehicle states xt g gravitational acceleration h physical constraints of the short horizon optimization problem Ibat,max maximum battery current minimum battery current Ibat,min battery current Ibat gear transmission ratio iG rear axle transmission ratio iRA effective vehicle inertia, inertias of all rotating parts Jeff inertia of the electric motor JEM inertia of the gear box JG inertia of the internal combustion engine JICE inertia of the rear axle JRA inertia of the wheels Jwhl JLHBP,fuel cost during use cases in LHBP except recuperation final costs for the long horizon battery planning JLHBP,f JLHBP,recup cost during recuperation use cases in LHBP cost functions for the long horizon battery planning JLHBP, k final costs for the short horizon optimization JSHO,f phase transition stage j kP, j

Δd ΔJSHO

m meff NLHBP

NP NS, j NS p Paux Pbat Pbrk Pel PEM,el PEM,max PEM,min PICE,max PICE,min Plim Preq Q bat REM RICE Ri rwhl s sP, j sSBS t Tbrk Tdes Tdrag TEM,high TEM,low

TEM TG,in TG,out Tgrade TICE Tloss,clt Tloss,G Tloss,RA Tresist Troll Twhl u U0 Ubat uclt uG uICE uSHO v vcorr vcurve,max vdes vlaw vlim,min vmax vmin

vehicle mass effective vehicle mass, including all inertias number of use cases in the long horizon battery planning number of phases in the short horizon optimization number of stages in phase j number of stages in short horizon optimization parameter vector electric power consumed by the auxiliaries battery power braking power power demand of power electric loads electric power of the electric motor maximum EM power minimum EM power maximum ICE power minimum ICE power power limits for long horizon battery planning predicted power demand for long horizon battery planning battery capacity torque derivatives of the electric motor torque derivatives of the internal combustion engine battery resistance wheel radius position starting position of phase j battery power substitution benefit time service brake torque desired torque from driver model air drag resistance torque highest possible EM torque for SoC change limit calculation in LHBP lowest possible EM torque for SoC change limit calculation in LHBP electric motor torque torque transmitted from the motor side to the gearbox torque transmitted from the gear box to the propshaft grade resistance torque internal combustion engine torque torque loss in the clutch torque loss in the gear box torque loss in the rear axle driving resistance torque rolling resistance torque accumulated torque at the wheels vector of model inputs open circuit voltage of the battery battery voltage clutch open/close command desired gear engine on/off command input vector for short horizon optimization vehicle velocity merged velocity limits curve velocity limit desired velocity speed limit lowest curve velocity limit maximum velocity minimum velocity

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vref xclt,pred xG,pred xG xSHO xSoC,des xSoC,pred xSoC xs xt

xclt, iP xG, iP xSoC,f,des zk

reference velocity predicted clutch trajectory predicted gear sequence current gear state vector for short horizon optimization desired battery state of charge predicted battery state of charge trajectory battery state of charge vector of the model states for position dependent differential equations vector of the model states for time dependent differential equations clutch state in phase iP gear state in phase iP desired state of charge at the end of the prediction horizon normalized position of stage k in its phase

predictive controller (MPC) with costs similar to an adaptive “equivalent consumption minimization strategy” (ECMS, Musardo, Rizzoni, Guezennec, and Staccia, 2005) is developed. However, this work does not take into account road grades. The publication (Ngo, Hofman, Steinbuch, and Serrarens, 2010) proposes an algorithm combining discrete dynamic programming (DDP, see Bellman, 1957) with an adaptive ECMS, calculating torque split and gear shifting strategy. The equivalence factor of the ECMS is adapted iteratively, if charge sustainment is not achieved. In Mensing, Trigui, and Bideaux (2012), a dynamic programming approach is used in combination with the identified EMS of a Toyota Prius. In Wahl, Bauer, Gauterin, and Holzapfel (2013), a real-time capable dynamic programming approach is presented. To achieve realtime capability the state and input space is reduced by heuristics. Yu, Yang, and Yamaguchi (2015) extend the optimal control problem by taking into account traffic lights, using two switching MPCs to separate the problems of calculating the cruising velocities and the optimal speed to pass the traffic lights. Their study considered a vehicle with a continuously variable transmission. In other recent work (Johannesson, Murgovski, Jonasson, Hellgren, and Egardt, 2015), a three-layered approach for hybrid long-haul trucks is presented, where the algorithm in the uppermost layer generates optimal velocity and state of charge trajectories. To solve this problem, the discrete states are eliminated and the problem is either formulated as a quadratic or a second order cone program. The middle layer optimizes gear and powertrain modes using the information from the higher level control. In the lowest layer, the instantaneous torque split is calculated. In previous work (Heppeler, Sonntag, and Sawodny, 2014), we proposed a DDP approach optimizing torque split, gear shifting and the velocity trajectory to analyze the potential of combining EMS and economic driving. This work shows that even for comparably small batteries, the dynamics of the state of charge (SoC) are slow compared to the dynamics of the vehicle's kinetic energy, represented by the vehicle's velocity. Therefore, it is reasonable to split the problem into two sub-problems: the planning of the battery's state of charge over a long prediction horizon and a combined planning of velocity and SoC over a short horizon, utilizing information of the long horizon battery planning. The structure of this approach is shown in Fig. 1. A navigation system provides speed limits vlaw , road curvature κ and the road grade γ . Speed limits and curvature are used to calculate the maximum velocity vmax and the reference velocity for both long and short horizon. Information regarding the calculation of vref and vmax is given in Appendix A. The long horizon battery

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Abbreviations DDP ECMS EM EMS HEV ICE LHBP MPC OCP SHO SoC SQP

Discrete dynamic programming Equivalent consumption minimization strategy Electric motor Energy management system Hybrid electric vehicle Internal combustion engine Long horizon battery planning Model predictive control Optimal control problem Short horizon optimization Battery state of charge Sequential quadratic programming

Fig. 1. Structure of the two-layered planning approach.

planning calculates an optimal SoC trajectory xSoC,pred and defines corresponding all-electric driving sections by evaluating the power demand of the vehicle needed over the horizon. This power demand is calculated using the reference velocity vref , the road grade γ and a driver model including heuristics to determine the gear sequence. This information is used to formulate an optimal control problem (OCP) for the short horizon, yielding fuel optimal results for combined velocity and SoC optimization. Subsequently, the calculated optimal velocity, SoC and gear trajectories vdes, xSoC,des and uG are used as inputs for an intelligent cruise controller. Thus the influence due to the driver is negligible. In order to track the SoC trajectory, an additional SoC controller is added to manipulate the EMS of the vehicle. By utilizing the predicted gear and clutch trajectories, xclt,pred and xG,pred , a sequential quadratic programming (SQP) solver can be used to solve the OCP. The long horizon battery planning (LHBP) algorithm for calculating a SoC trajectory was introduced in previous work (Heppeler, Sonntag, and Sawodny, 2015), therefore, this work focuses on the OCP for the short horizon and the combination of both algorithms. This paper is organized as follows: in Section 2, the vehicle model is shown. In Section 3, the optimal control problems for long horizon and short horizon optimization are introduced. Section 4 gives an overview of the simulation framework and the considered configurations. Simulation results are shown in Section 5 and Section 6 gives concluding remarks.

2. Vehicle model In this section, the model for the hybrid electric vehicle is introduced. The vehicle considered here is a parallel hybrid electric

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Fig. 4. External forces on the vehicle.

(

)

Troll = − rwhl·μ R ·m·g ·cos γ (s ) = rwhl·Froll,

Fig. 2. Powertrain configuration.

passenger vehicle with a 225 kW homogenous gasoline engine, a 20 kW electric motor, a 0.9 kWh lithium-ion battery and an automatic gearbox. Due to the small electric components, it is classified as a mild hybrid vehicle (Guzzella and Sciarretta, 2013). The structure of the powertrain is shown in Fig. 2. Because the clutch is located between the ICE and the EM, the vehicle can be operated in all-electric driving and recuperation modes. Also, it can coast in neutral gear with the ICE turned off. In order to create operational strategies and evaluate the fuel consumption, a simple quasi-stationary longitudinal model similar to the one proposed in Guzzella and Sciarretta (2013) is sufficient. The model used here is described in the following subsections.

( )

Tgrade = − rwhl·m·g ·sin γ (s ) = rwhl·Fgrade,

(4)

(5)

where CAir is the air resistance coefficient, v the vehicle velocity, rwhl the wheel radius, μR the rolling coefficient, γ(s ) the position s dependent slope, m the vehicle mass and g the gravitational acceleration. The corresponding forces are illustrated in Fig. 4. The differential equation for the velocity v of the vehicle is given by

Twhl d v=a= . dt meff ·rwhl

(6)

Here a is the vehicle acceleration and

Jeff

2.1. Longitudinal dynamics

meff = m +

The structure of the powertrain is given in Fig. 3. The main components with their inertias Ji, external torques Ti and ratios ii are shown. The losses in clutch, gearbox and rear axle are given by torque loss maps Tloss,clt ωICE − ωEM , Tloss,G ωEM, Tgear,in, xG and

is the effective mass, where the effective inertia Jeff includes all the inertias from the rotating parts and the transmission, as shown in Fig. 3. The gasoline consumption per second of the ICE ṁ f (ωICE , TICE ) is given by a quasi-stationary map. All necessary constants and maps have been provided by Daimler AG.

(

(

)

(

)

)

Tloss,RA ωprop, TG,out , where ωICE , ωEM and ωprop are the speeds of ICE, EM and prop-shaft respectively. xG is the current gear, TG,in is the torque applied to the gearbox from the motor side and TG,out the torque transmitted to prop-shaft from the gearbox. Hence, the torque at the wheel is calculated by

Twhl =

(((1 − b

clt

)·TICE + bclt ·Tloss,clt + TEM)·iG+Tloss,G)

·iRA + Tloss,RA + Tbrk + Tresist,

(1)

^ closed, 1 = ^ open}, gear with the discrete clutch state bclt ∈ {0 = transmission iG(xG ), rear axle transmission iRA , ICE torque TICE , EM torque TEM, service brake torque Tbrk and the torque induced by the driving resistance Tresist , consisting of air drag Tdrag , rolling resistance Troll and grade resistance Tgrade . The driving resistances are calculated as

Tresist = Tdrag + Troll + Tgrade,

(2)

Tdrag = − rwhl·CAir·v2 = rwhl·Fdrag,

(3)

2 rwhl

(7)

2.2. Electric components The model for the battery is derived from the equivalent circuit model shown in Fig. 5. The dynamics for the battery state of charge xSoC are given by

U0 − U02 − 4·R i·Pel I d xSoC = − bat = − , dt Q bat 2·R i·Q bat

(8)

where Ibat is the battery current, Q bat the battery capacity, U0 the open circuit voltage as a function of SoC and Ri the battery resistance. Ri is dependent on the battery temperature and the sign of Ibat . In the vehicle considered, the battery is water cooled, therefore the battery temperature is assumed to be constant. The usable range of the battery SoC is xSoC ∈ [40%…90%]. Pel = PEM,el + Paux is the battery power consisting of the electric power of the EM PEM,el and the power Paux consumed by the auxiliaries. As a simplification, Paux is considered to be constant in this work. PEM,el(ωEM, TEM) is given by a map, including the inverter. The

ωprop Fig. 3. Translations, inertias and external torques in the powertrain.

Fig. 5. Equivalent circuit model of the lithium-ion battery pack.

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maps and constants used here have also been provided by Daimler AG. The proposed vehicle model has six states T x t = ⎡⎣ s , v, xSoC , bICE , bclt , x G⎤⎦ ,

(9)

^ off, 1 = ^ on}, bclt and xG where the engine on/off-state bICE ∈ {0 = are discrete states. The model also has six inputs T u = ⎡⎣ TICE, TEM, Tbrk, uICE , uclt , uG⎤⎦ ,

(10)

which leads to nonlinear differential equations of the form

d x t = ft (x t , u). dt

Fig. 6. Vehicle interface.

(11)

The discrete inputs are the engine on/off command uICE , the clutch open/closed command uclt and the desired gear uG . ft is the function of time derivatives of xt . Changes to the discrete states are assumed to immediately take effect at the next time step. If the ICE or the EM change their speed due to a discrete state transition, the necessary energy has to be provided by the ICE or the EM respectively. The energy is calculated by determining the change in rotational energy and using an average efficiency factor for either the ICE or the electrical system. This way it is incorporated in the fuel consumption or the SoC change. The model also contains physical constraints of the system, such as maximum and minimum torques for the ICE and EM, battery voltage and current limits, speed limits, and admissible combinations of the discrete states. For the sake of brevity these are omitted here.

are measured and accessible.

2.3. Position dependent differential equations


Prediction of the power demand
Use case classification
SoC optimization

In order to use the position instead of time as the independent variable in the optimization, the time derivative (11) is transformed into the position derivative of the states

d 1 xt = f (x t , u). ds v(s ) t

(12)

This transformation allows us to calculate the integral over position instead of time and the position can be eliminated from the state vector, leading to T x s = ⎡⎣ v, xSoC , bICE , bclt , x G⎤⎦ ,

(13)

and

d x s = fs (x s, u), ds

(14)

where xs and fs are the reduced state space and the corresponding function of the time derivatives respectively. Note that the division by the vehicle speed v(s) leads to a singularity in the system equations at standstill. 2.4. Vehicle interface In the prototype vehicle it is not possible to set the torques TICE , TEM and Tbrk freely. However, the cruise controller and an ECMS as the energy management system can be used to set them indirectly. This setup is shown in Fig. 6 and is also used for the simulations in this work. The cruise controller tracks the desired velocity vdes and determines the required torque Tdes from the powertrain. The ECMS calculates the torque split, engine-off phases and the clutch position based on the vehicle states and the torque demand. The equivalence factor λ can be set externally and is used to influence the torque split. The desired gear uG can also be set externally and it is assumed that all relevant vehicle states

3. Optimal control problem In this section, a brief introduction of the algorithm for long horizon battery planning and the optimal control problem for the combined SoC and velocity planning on a shorter horizon is given. 3.1. Long horizon battery planning The algorithm for the long horizon battery planning was already shown in Heppeler et al. (2015). For completeness, a brief introduction is given in the following. The algorithm is separated into three parts:

The structure of the algorithm is shown in Fig. 7 and is explained in the following sections. 3.1.1. Power demand prediction Based on the reference velocity and road grade, given by map data, a driver model in combination with a simplified longitudinal model is used to determine the combined power demand Preq from the ICE and EM as well as the power limits Plim of these components. The driver model consists of a PI controller and a feedforward part compensating the driving resistances as given by Eq. (2). The gear sequence xG,pred is determined by simple heuristics. 3.1.2. Use case classification Depending on the power demand, the horizon is separated into a sequence of four different use cases

Recuperation: Negative power demand, the torque request is split between the EM and service brakes.


Electric driving: Propulsion can be provided by the EM alone;

Fig. 7. Structure of the long horizon battery planning.

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⎧ ⎪ JLHBP,recup (⋯), if
(22)

The recuperation costs penalize missed recuperation potential

(

)

JLHBP,recup (⋯) = ξ2· ΔxSoC,max, k − ΔxSoC, k .

Fig. 8. Use case separation.

the ICE is turned off.


Partial load: The ICE is either assisted by the EM or the battery is


actively charged. Full load: The ICE is assisted by the EM for high loads.

An example of this separation of power demand is given in Fig. 8. The use cases have minimum durations. Each use case is characterized by an average power demand and ICE speed. Since the efficiency of the powertrain varies for different speeds and power demands, long use cases are split if the variation of these parameters exceeds a predefined threshold. Instead of positions, the sequence of use cases is now used as the optimization horizon, greatly reducing the number of stages. For each use case
min

Δx SoC, k

∑

JLHBP, k (
k=0

(15)

s.t.

xSoC, k + 1 = xSoC, k + ΔxSoC, k ,

∀ k = 0, …, NLHBP − 1,

xSoC,0 = xSoC(t0),

(16)

(17)

xSoC,min, k ≤ xSoC, k ≤ xSoC,max, k

∀ k = 0, …, NLHBP,

(18)

(23)

Note that, as stated before, the maximum SoC change ΔxSoC,max, k depends on the SoC. The costs for electric driving, partial load and full load use cases are calculated by estimating the fuel saved for a certain amount of battery energy usage

JLHBP,fuel (⋯) = sBSB(Pbat, k,
(24)

where sBSB is the fuel saved per battery energy depending on the load point in the use case. For details on how to derive these costs see Appendix C. To solve the problem online, a discrete dynamic programming algorithm is used. 3.2. Short horizon optimization The goal of the short horizon optimization (SHO) is to generate a velocity trajectory and a state of charge trajectory, with the following objectives:


minimization of the fuel consumption,
small deviations from the desired reference velocity,
compliance with the physical constraints. The objective of having only small deviations from the reference speed is partially conflicting with the objective to minimize the fuel consumption, however, it is necessary to avoid high downward deviations for longer times, as they will considerably increase trip time, reduce driver acceptance and impede following vehicles, when driving slower than the speed limit for longer times. High upward deviations, on the other hand, present safety issues and can lead to serious legal consequences. Therefore, it is advisable to keep these deviations reasonably small. In order to reduce the complexity of the OCP and to use a gradient based optimization algorithm, the discrete states bclt and xG are replaced by the clutch and gear trajectories from the LHBP, xclt,pred(s ) and xG,pred(s ), which can be seen as position-dependent parameters. Utilizing these parameters, the OCP can be formulated as a multiphase optimal control problem, separating it into NP phases with constant parameters xclt and xG . The starting position of each new phase is, therefore, determined by a change in xclt,pred(s ) or xG,pred(s ): for each phase starting position sP, j , 0 < j < NP

xclt,pred(sP, j ) = xclt, j ≠ lim xclt,pred(ϵ) − ϵ→ sP, j

(25)

or

ΔxSoC,min, k ≤ ΔxSoC, k ,

∀ k = 0, …, NLHBP − 1,

(19)

x G,pred(sP, j ) = x G, j ≠ lim x G,pred(ϵ), − ϵ→ sP, j

ΔxSoC, k ≤ ΔxSoC,max, k , The final cost

∀ k = 0, …, NLHBP − 1.

(20)

JLHBP,f penalizes divergence from a desired SoC

xSoC,f,des at the end of the horizon

(

2

)

JLHBP,f (xSoC, NLHBP ) = ξ1· xSoC, NLHBP − xSoC,f,des .

(21)

For the simulations in this paper xSoC,f,des is set to 70%, the starting SoC of all simulations, to achieve charge sustainability. Use case costs JLHBP, k differ for the mentioned use case types,

(26)

applies. The sequence of phase changing positions is

s0 = sP,0 < sP,1 < ⋯ < sP, NP − 1 < sP, NP = sf ,

(27)

where sP, NP is the final position of the optimization. Here, the phases are directly related to the use cases of the LHBP: the clutch trajectory xclt,pred is defined by the all-electric driving and recuperation use cases predicted in the LHBP. The connection between the LHBP use cases and the gear trajectory xG,pred is not as close, since use cases can include several gear shifts. Note that the phase length is also optimized by using sP, j , 0 < j < NP as additional optimization inputs. The phase start and end positions form the

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parameter vector pk for stage k together with the gear and clutch parameters xG and xclt : T pk = ⎡⎣ sP, j , sP, j + 1, x G, j , xclt, j ⎤⎦ ,

(28)

assuming that k is in phase j. The number of total stages NS, j of each phase j depends on the initial phase length and the position discretization Δd . It is calculated by

⎛ NS, j = max⎜3, ⎝

⎞ ⎡ sP, j + 1 − siP, j ⎤ ⎢ ⎥ + 1⎟ , ⎢ ⎥ Δd ⎠

∀ j ≤ NP − 1.

(29)

Subsequently, the indices kP, j of each phase transition are calculated:

For the sake of brevity, xSHO(sk ) is written as xk in the following. The input vector is written as

⎧⎡ T ⎪ ⎣ R , R , T ⎤⎦ , k ≠ kP,j, uk = ⎨ ICE EM brk ⎪ k = kP, j, ⎩ sP, j + 1,

(30)

kP, j = kP, j − 1 + NS, j − 1,

0 < j ≤ NP − 1

NS− 2

minJSHO,f (x NS− 1) +

(32)

Since sP, j is not fixed for 0 < j < NP , the position sk at stage k cannot be calculated beforehand. It can be expressed in dependence to the parameter pk :

sk = (sP, j + 1 − sP, j )·zk + sP, j , ∀ k ∈ {kP, j + 1, …, kP, j + 1} , j ≤ NP − 1,

s0 = s(0),

(34)

z 0 = 0,

x 0 = x SHO(0),

(43)

⎡ s0 ⎤ ⎢ s ⎥ 0 p0 = ⎢ x ⎥, ⎢ G,0 ⎥ ⎢x ⎥ ⎣ clt,0 ⎦

(44)

x k + 1 = x k,

(45)

∀ k = kP, j, j ≤ NP − 1,

⎡ sj ⎤ ⎥ ⎢ uk ⎥ pk + 1 = ⎢ , ⎢ x G,j ⎥ ⎥ ⎢ ⎣ xclt,j⎦

xk + 1 = xk +

∫s

∀ k = kP, j, j ≤ NP − 1, (46)

sk + 1

k

k − kP, j − 1

,

∀ k ∈ {kP, j + 1, …, kP, j + 1} , j ≤ NP − 2,

zk =

NS − kP, j − 1,

∀ k ∈ {kP, j + 1, …, kP, j + 1} , j = NP − 1,

∀ k ≠ kP, j, j ≤ NP − 1,

(37)

As stated above, the discrete quantities are eliminated from the vehicle model. Additionally, the torque derivatives RICE and REM are used as inputs instead of the EM and ICE torques, in order to generate smoother trajectories by limiting and penalizing the changing rates. This yields the adapted state and input vectors T

x SHO = ⎡⎣ TICE, TEM, v, xSoC⎤⎦ ,

(38)

T uSHO = ⎡⎣ RICE, REM, Tbrk⎤⎦

(39)

(47)

∀ k ≠ kP, j, j ≤ NP − 1,

0 ≤ h(xk, uk , pk , sk ),

∀ k ≤ NS − 1.

(48)

(49)

The final costs

JSHO,f (⋯) = ς1·(vNS− 1 − vref (sf ))2 + ς2·(xSoC, NS− 1 − xSoC,pred(sf ))2

(50)

ensure that the deviations from the reference velocity and SoC are small at the end of the horizon. The transition costs between the sampling points are given by

ΔJSHO (⋯)

and the resulting differential equations for x SHO with position s as free variable

⎡ RICE ⎤ ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎢ REM ⎥ ⎢ TICE ⎥ d d ⎢ TEM ⎥ 1 ⎢ Twhl ⎥ ⎥ = fSHO (x SHO, uSHO, p, s ). ⎥= ⎢ ⎢ x SHO = ds ds ⎢ v ⎥ v ⎢ meff ·rwhl ⎥ ⎥ ⎢ ⎢ xSoC ⎥ ⎢ − Ibat ⎥ ⎥ ⎢ ⎢⎣ Q bat ⎥⎦ ⎥⎦ ⎢⎣

fSHO (x SHO(s ), uk , pk , s )ds ,

(36) pk + 1 = pk ,

k − kP, j − 1

(42)

s.t.

(35)

kP, j + 1 − kP, j − 1

ΔJSHO (xk, uk , pk , k ),

(33)

where zk is a normalized parameter describing the position of stages k within its phase j:

zk =

∑

(31)

and the total number of stages is

NS = kP, NP − 1 + NS, NP − 1 + 1.

(41)

for all j ≤ NP − 1. The distinction of cases for uk is necessary since it is assumed that the discrete states change instantaneously and therefore, in the phase transitions kP, j only the length of the phases is adapted. This leads to the following OCP:

k=0

kP,0 = 0,

7

(40)

=

1 ⎡ 2 ⎢ ς 3·(v(s ) − vref (s )) + ς4·ṁ f (x SHO(s )) v(s ) ⎣ k ⎤ + ς5·RICE(s )2 + ς6·REM(s )2 + ς 7·Tbrk(s )2⎥ds . ⎦

∫s

sk + 1

(51)

By penalizing the torque changing rates, smooth trajectories are achieved. Due to the penalty on the brake torque, the service brakes are only used if necessary. The reference velocity is the same as used for the LHBP. The constraints in Eqs. (43)–(48) represent the system dynamics and the change of the parameters p . Eq. (49) represents

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8

the constraints of the system, e.g. physical limitations, such as SoC limits, torque limits, or velocity limits. For brevity these are omitted here. Note that Eq. (49) also includes limits on the phase length Δpj = sP, j + 1 − sP, j , for j ≤ NP − 1:

0 ≤ Δpj − Δpmin, j ,

(52)

0 ≤ Δpmax, j − Δpj .

(53)

These constraints are necessary to avoid big changes in the discretization, which can lead to numerical problems. Also, the final position is constrained by sP, NP = sf . The solver MSQNLP (Bender, Sonntag, and Sawodny, 2015; Sonntag, 2014), which is based on sequential quadratic programming (Boggs and Tolle, 1995) is used to solve the OCP. The necessary initialization of the OCP is done by a PI controller, acting as a model for the driver and an ECMS for the torque split.

4. Simulation setup The simulation setup used in this work is shown in Fig. 9. It mainly consists of the vehicle interface from Fig. 6, as well as the long horizon battery planning, short horizon optimization and driving cycle prediction, shown in Fig. 1. As previously stated, modern navigation systems provide detailed maps, including data pertaining to legal speed, road grade and curvature. In the case of predefined trips, this data can be used for a prediction. In this work, it is assumed that the trip is known beforehand and, therefore, also the desired velocity, the maximum speed due to curvature and the road grade. The road data is used to predict the gear sequence, electric driving phases and a SoC trajectory using the long horizon battery planning by considering the horizon given by the navigation system. Due to practical system limitations, this horizon is restricted to 7 km. As stated above, these results are used in the short horizon optimization in order to calculate optimal velocity and more detailed SoC trajectories as well as optimizing the gear shifting positions. High computational effort restricts the short horizon to 1 km. Since the vehicle interface for the prototype vehicle already includes a velocity input for the cruise controller and an input for gear command, only the torque split from the optimized SoC trajectory has to be considered separately. In the ECMS algorithm, which calculates the final torque split and the driving mode, the equivalence factor λ can be set externally. By adding a PI controller for the SoC, this input is used to follow the desired state of charge.

The controller has the form λ(t ) = λ 0 − KP·(xSoC(t ) − xSoC,des(t )) − KI

tf

∫t0

(xSoC(t ) − xSoC,des(t ))dt ,

(54)

where λ0 is the initial equivalence factor, KP and KI are the gains for the proportional and the integral part of the controller. λ(t ) is limited to the range of [0.5, … , 5] and an additional anti-windup is used to avoid integrator wind-up. The torque changing rates RICE and REM, which are the inputs of the model used in the SHO, only indirectly influence the final torque split by allowing the calculation of smoother reference trajectories vdes and xSoC,des , resulting in smoother control inputs for the vehicle. The simulations in the next section show that using the cruise controller, lambda controller and ECMS with these inputs leads to torque splits and driving modes reasonably close to the ones calculated in the short horizon optimization. Utilizing the vehicle controllers has the advantage that all functional safety mechanisms of the vehicle control units can be maintained. It also increases robustness against model uncertainties and varying gear demands from the gearbox control unit. However, the cruise control might be slow and lead to higher deviations from the desired trajectories in practical applications. 4.1. Configurations In order to evaluate the results, a purely ECMS based baseline configuration is used. For this baseline configuration the long and short horizon algorithms are omitted, the desired velocity is taken directly from the map data, the gears are determined by heuristics and a fixed λ achieving charge sustainment is used. The simulation framework allows the comparison of several configurations with different algorithms in the long and short horizon. To compare the energy management only in configurations 1 and 2, the short horizon is omitted, predicted gears and the desired state of charge trajectory are taken from the long horizon. The desired speed is fixed and taken directly from the map data. In configuration 1 the long horizon results are taken from a global solution by offline optimization with fixed velocity, as presented in previous work (Heppeler et al., 2014). Configuration 2 uses the online capable long horizon battery planning algorithm proposed in Section 3.1. The configurations 3–5 include the combined velocity and SoC optimization in the short horizon. In configuration 3, the global offline solution for free velocity from Heppeler et al. (2014) is used. Configuration 4 consists of the same offline solution for the long horizon as configuration 1 in combination with the online capable algorithm from Section 3.2 in the short horizon. For configuration 5, both online capable algorithms, the LHBP

Fig. 9. Simulation framework.

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of the reference velocity see Appendix A. The plot indicates that the road is hilly with its grade varying between  8.1% and 9.3%.

Table 1 Configuration overview. Configuration

Baseline Config. 1 Config. 2 Config. 3 Config. 4 Config. 5

Free vel.

– – – × × ×

9

Fixed λ

× – – – – –

Long horizon

Short horizon

Offline

Online

Offline

Online

– × – × × –

– – × – – ×

– – – × – –

– – – – × ×

introduced in Section 3.1 and the short horizon optimization from Section 3.2, are joined. A summary of the combinations is given in Table 1. In all scenarios, the vehicle starts at a speed of 30 km/h, with an SoC of 70% and a closed clutch. The vehicle is supposed to be charge sustaining, hence the SoC at the end of the driving cycle should be 70%. Due to the SoC controller, this cannot be guaranteed, therefore the fuel consumption is corrected by a value equivalent to the final SoC error. A solution to the online optimization problem is generated in a real-time manner every 4 s. The LHBP calculates a horizon of up to 7 km, restricted by the prediction data length of the navigation system, the short horizon optimization only calculates a horizon of up to 1 km due to the much higher computation time.

5.2. Controller performance In order to show that the simulation setup can be used to compare results of different configurations, the performance of the controllers has to be studied. Therefore, results from offline optimization via discrete dynamic programming are set as a reference: the gear trajectory, velocity trajectory and SoC trajectory are used as desired values for the respective controllers in the simulation setup in Fig. 9. Fig. 11 shows the resulting plots for offline results with fixed velocity. The controllers manage to track both the velocity and the SoC trajectories very well. These results are also presented in Table 2. The mean velocity deviation in both scenarios is about 0.06 km/h and the worst case tracking error is less than 1.00 km/h. The combination of the SoC controller and the ECMS algorithm manages to track the SoC trajectory very well and the mean error is less than 0.5%. The instantaneous nature of the ECMS leads to changes in the engine-on/off events and differences in engine-off time. Also, the necessary discretization of torques for DDP leads to small discrepancies. The difference in the consumption predicted by DDP and that calculated by the simulation using the proposed simulation setup with the offline reference is very small, see Table 2. In summary, these results indicated that the setup is suitable for the following investigations. 5.3. Comparison without velocity optimization

5. Results In this section, the results for the different configurations are shown and compared. Firstly, the considered driving cycle is briefly described. In the following, the performance of the velocity and the torque split controllers are shown. Finally, the results for the different configurations are presented.

As a first step to evaluating the performance of the proposed control setup, the configurations without velocity optimization are analyzed. A characteristic section for the comparison of the purely

5.1. Driving cycle The driving cycle considered to study the fuel efficiency is an existing country road cycle close to the city of Ulm, Germany. It consists of sections with speed limits of 100 km/h and 30 km/h as shown by the black, dashed line in Fig. 10. The desired speed is further limited by the maximum velocities derived from the road curvature. Between these speed limits, transition sections with acceleration and deceleration limits of ±1 m/s2 are added to avoid jumps in the velocity profile. The solid vdes trajectory is the resulting reference velocity. For further information on the planning

Fig. 10. Velocity and road grade used for simulation.

Fig. 11. Comparison of results from the discrete dynamic programming approach, with and without simulation framework and fixed velocity.

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Table 2 Controller performance: comparison between optimization and simulation results of the DDP solution. Velocity Consumption difference (normalized) (%) Fixed Free

 0.10 þ0.54

Trip time difference (normalized) (%)

Mean velocity error (km/h)

Maximum velocity error (km/h)

Mean SoC error (%)

Maximum SoC error (%)

Engine-off to trip time ratio difference (%)

þ 0.01  0.02

0.05 0.05

0.60 0.36

0.44 0.43

1.38 0.87

þ 1.02  1.19

configurations and is in general higher. This leads to a reduced recuperation potential due to battery limitations. The lower recuperation torque has to be compensated by additional usage of the service brakes e.g. 14.0–15.0 km , 20.0–20.5 km and 23.0– 23.5 km. The state of charge trajectory from the LHBP algorithm has a greater similarity to the DDP results and therefore a higher recuperation potential than the baseline configuration. However, additional braking cannot always be avoided, e.g. 14.5–15.0 km. The all-electric driving sections are very similar in all three configurations, only configuration 1 has a slightly higher percentage. One main difference on how electrical energy is consumed during engine assist phases e.g. 22.0–23.0 km: the simulation of the DDP solution uses higher torques to assist the ICE, while the EM torque in the baseline configuration is much lower. The LHBP algorithm supports the ICE with a medium EM torque, compared to the one of DPP and baseline configuration. Also, the DDP based solution uses electrical energy to avoid down shifts, see for example 18.0– 19.0 km. This is not possible for other configurations since the gear shift heuristics they use are not predictive which leads to a much higher number of gear shifts. Altogether, the offline based solution configuration 1 is able to save 2.97% fuel compared to the baseline configuration. The online capable algorithm in configuration 2 still saves about 1.43% compared to the ECMS with fixed equivalence factor. The main results are shown in Table 3. 5.4. Comparison with combined velocity and state of charge optimization

Fig. 12. Comparison of configurations without velocity optimization.

ECMS baseline configuration with a fixed equivalence factor λ, the simulated offline solution from DDP in configuration 1 and the online battery planning algorithm in configuration 2 is shown in Fig. 12. The SoC trajectories planned by the three approaches are qualitatively very similar. The SoC trajectory of the baseline configuration has a high offset in comparison to the other

In configurations 3–5, the additional degree of freedom for the velocity is used for combined velocity and SoC planning. The results in Table 3 indicate that offline planning in configuration 3 can save up to 9.49% in fuel consumption compared to the baseline configuration and still up to 6.54% compared to the offline optimized EMS in configuration 1 for this driving cycle. In configuration 4, where the upper layer gear, SoC and electrical driving phase planning is solved offline, the short horizon algorithm proposed in Section 3.2 is used and results in savings of 8.48% and 5.51% respectively are compared to the baseline and configuration 1. With fuel savings of 6.78% and 3.71% for the combination of LHBP and short horizon optimization the fuel efficiency is still very close to the global solution from configuration 3. The increase in trip time due to variable velocity is less than 0.70% in all three configurations. A comparison of the configurations is shown in Fig. 13. For the SoC, again, the trajectories are qualitatively very similar, only configuration 3 diverges between 15.5 and 18.0 km. The reason for this difference is that the velocity, SoC and electrical driving phases are optimized for the whole horizon. Therefore, additional electrical driving phases are found in sections where the EM, alone, cannot maintain the velocity. These additional phases require a considerable amount of electrical energy and are partially compensated by slight active charging, see 16.5–17.0 km. Also, in this case the acceleration back to the reference velocity is done by the ICE resulting in a slight increase of fuel efficiency. In configurations 4 and 5, the electric driving phases are identified from the fixed reference velocity from the map and, therefore, these

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Table 3 Simulation results: comparison between different configurations. Configuration

Fuel consumption (normalized) (%)

Trip duration (normalized) (%)

Gear shifts

Engine-off to trip time ratio (%)

Mean velocity deviation (km/h)

Max. velocity deviation (km/h)

Average C rate

Baseline Config. 1 Config. 2 Config. 3 Config. 4 Config. 5 Config. 5 (sub.)

100.00 97.03 98.57 90.51 91.52 93.22 93.34

100.00 100.00 100.00 100.71 100.51 100.53 100.53

140 23 140 21 28 152 152

49.23 49.41 49.51 59.47 55.02 52.39 52.52

0.05 0.05 0.05 0.99 0.96 0.91 0.91

0.36 0.60 0.36 7.97 9.25 9.66 9.66

10.27 12.37 11.60 11.47 11.52 11.16 11.31

possibilities for electric driving cannot be detected. The SoC trajectories in these two configurations are close to the ones in configuration 2 respectively 3, since they use the same algorithms to calculate the SoC prediction xSoC,pred for the long horizon. The velocity trajectories show a very similar behavior. One common characteristic of all three configurations is the conversion of potential energy to kinetic energy: at the end of a downhill slope the vehicle accelerates and benefits from additional kinetic energy at the beginning of the next uphill grade. The trajectories in Fig. 13 show this behavior at several positions, e.g. 15.0 and 21.0 km. The optimized trajectories also smooth the velocity profile by reducing small peaks and therefore avoiding short high accelerations and decelerations, e.g. 23.5 km. The mean deviation of all three configurations is less than km/h, while the maximum deviation for the configurations with SHO is slightly higher than that in the offline DDP based solution, due to a fixed limit of 8 km/h in the DDP. In Fig. 14, the results for set speed changes are shown. Again, the behavior for all 3 configurations is very similar. The speed is reduced in advance in order to extend the period of deceleration and lessen its magnitude. High deceleration requires significant negative torques. Once these torques exceed the recuperation limit, service brakes must be used. Hence, during slower deceleration, less braking is necessary and more kinetic energy can be transformed into electrical energy. Configurations 4 and 5 are slightly more sensitive to the road grade during the velocity

Fig. 13. Comparison of the configurations with velocity optimization.

Fig. 14. Speed profiles during velocity changes.

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change: the velocities are reduced earlier during the high uphill grades (26.4 km) and the speed difference from configuration 5 is recovered during the deceleration and the change to lower grades. A similar behavior can be seen at the end of the acceleration at 27.5 km: in both cases, the set speed is achieved with slightly lower acceleration during the high grade section. The slight differences in fuel consumption and SoC trajectories of configurations 3–5, show that the estimation of an optimal SoC trajectory as a reference for fixed velocity leads to very good results for the combined optimization, if only small velocity deviations are allowed. Regarding the gearshifts, a similar behavior as for the configurations without velocity optimization can be seen: the predictive gear shifting strategies in configurations 3 and 4 lead to a considerably lower number of gear shifts, by using higher power assists and double shifts. Velocity optimization enables configuration 3 to further reduce the number of gearshifts. The optimization of phase lengths in the short horizon algorithm, in combination with the computational delay, can lead to gear toggling, e.g. 23.0 km. Therefore, the numbers of gear shifts in configurations 4 and 5 are higher compared to the ones in configuration 1 and the baseline configuration. This could be avoided by warm starting the short horizon algorithm in future work. The high C-rates (normalized current value necessary to fully charge the battery in one hour) in Table 3 show that the stress on the battery is extensive in all considered scenarios. Since high battery currents decrease the state of health of the battery (see Peterson, Apt, and Whitacre, 2010), it is desirable to reduce the C-rate. One possibility to achieve this in future work would be to add a battery current weight to the cost function in Eq. (51). This

technique reduces the C-rate considerably, when the velocity deviations are increased, without having a big impact on the fuel consumption (Heppeler et al., 2014). 5.5. Computation time Fig. 15 shows the histogram of the short horizon optimization computation times on an Intel Core i5- 2.5 GHz processor. More than 93.6% of the optimal control problems were solved in less than 4 s. This ratio is increased to 100.0% by using suboptimal solutions, generated by stopping the algorithm before the calculation time reaches 4 s. This is fast enough for use in prototype vehicles and still leads to results very close to the optimization without time limit: The fuel consumption only increases by about 0.1%, while maintaining the same trip time, see Table 3. Nevertheless, several options exist to further decrease computation time: the warm start characteristics of the solver can be used, which has not been done so far. Furthermore, the optimal control problem can be tuned to allow for faster convergence. The LHBP algorithm is much faster and calculates the 7 km horizon in less than 100 ms using an online discrete dynamic programming algorithm. The LHBP was also already tested on a dSpace MicroAutoBox where the computation time was less than 250 ms.

6. Conclusion In this work, a method to calculate optimal velocity and SoC trajectories for HEV is introduced. The proposed approach has a two layered structure where the state of charge is optimized for a long horizon using prediction data from a navigation system. The obtained information from this calculation is used for a combined optimization of SoC and velocity trajectories on a shorter horizon. Both algorithms are presented in this paper. The method is tested in simulations and compared with the results calculated by an offline DDP algorithm which was developed in a previous work. The results given by the proposed method are very close to the DDP reference. A considerable amount of fuel is saved compared to the ECMS baseline strategy. In future work the algorithm will be transferred to prototype hardware and tested in a prototype vehicle.

Appendix A. Velocity reference planning In order to avoid jumps in the velocity profile and consider curves on the driving cycle, a smooth velocity trajectory needs to be planned. Firstly, velocity limitations due to curves are calculated:

⎛ g ·μ ⎞ vcurve,max(s ) = max⎜ − Δvsafety , vlim,min⎟, ⎝ κ (s ) ⎠

Fig. 15. Computation time for the short horizon optimization.

(A.1)

where g is the gravitational acceleration, μ the friction coefficient of the road, κ the road curvature, Δvsafety an additional offset for safety reasons and vlim,min a limit on the lowest bend velocity. This is necessary due to singularities in the position dependent system equations at standstill, see Eq. (12), and numerical problems close to the singularity. The maximum curve velocity vcurve,max avoids exceeding of the maximum centrifugal acceleration 2 vcurve,max ·κ (s ) ≤ g ·μ. For this work the road friction factor is set to μ = 0.6, which is a safe estimate for wheels on dry asphalt streets. ^ 5 km/h . The additional safety margin is set to Δvsafety = 1.39 m/s = ^ 12 km/h . The lower limit is set to vlim,min = 3.33 m/s = Next, the maximum bend velocity and the speed limits are

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TEM,high(t ) = max(Preq(t ) − PICE(t )),

13

s. t. ∀ t ∈ {tk, tk + 1}

PICE,min(t ) ≤ PICE(t ) ≤ PICE,max(t ),

(B.3)

PICE,min(t ) ≤ TEM,max(t )·ω EM(t ) ≤ PICE,max(t ),

(B.4)

xSoC(t ) = xSoC, k

∫t

t

Ibat(MEM,high(τ ), ω EM(τ ))dτ ,

k

Ibat,min(xSoC(t )) ≤ Ibat(⋯) ≤ Ibat,max(xSoC(t )),

merged to

(

(

)

)

(B.5)

(B.6)

where TEM,high is the highest possible torque satisfying the request Preq at time t and the power limits PICE,max , PICE,min for the ICE and PEM,max , PEM,min for the EM, as well as the battery limits Ibat,min(xSoC ) and Ibat,max(xSoC ). tk and tk + 1 are the starting and ending time of the use case
Fig. A1. Smoothed velocity trajectory and curvature.

vcorr = max min vlaw, vbend,max − Δvoff , vmin ,

(B.2)

ΔxSoC,min, k = (A.2)

^ 10 km/h is the minimum velocity for where vmin = 2.78 m/s = calculating the reference velocity trajectory. Δvoff = vlim,min − vmin ensures a slightly lower reference velocity than the hard limit vcurve,max . Note that vlim,min > vmin has to hold. The smoothed veloamax = 1 m/s2 and city trajectory with acceleration limits 2 amin = − 1 m/s is calculated by two loops, see Algorithm 1. The trajectory starts from position s0, has N points and a discretization of Δs . Algorithm 1. Velocity smoothing.

∫t

tk + 1

Ibat(TEM,low(t ), ω EM(t ))dt ,

k

(B.7)

with

TEM,low(t ) = min(Preq(t ) − PICE(t ) − Pbrk(t )), s. t. ∀ t ∈ {tk, tk + 1}

(B.8)

PICE,min(t ) ≤ PICE(t ) ≤ PICE,max(t ),

(B.9)

PEM,min(t ) ≤ TEM,low(t )·ω EM(t ) ≤ PEM,max(t ),

(B.10)

Pbrk ≤ 0,

(B.11)

vref,0 = vcorr(s0) for i¼ 1 to N − 1 do

(

vref, i = max vcorr, i, vref, i − 1 + ∫ s

si

1

i − 1 v (s )

)

amaxds

end for for i = N − 2 to 0 do

(

xSoC(t ) = xSoC, k

)

si − 1 1 a ds v(s ) min i

vref, i = max vref, i, vref, i + 1 − ∫ s

∫t

t

Ibat(TEM,low(τ ), ω EM(τ ))dτ ,

k

(B.12)

end for

Ibat,min(xSoC(t )) ≤ Ibat(⋯) ≤ Ibat,max(xSoC(t )). Subsequently, the maximum velocity is given by

(

)

vmax(s ) = min vref (s ) + Δvex , vcurve,max(s ) ,

(A.3)

Here TEM,low is the lowest possible EM torque satisfying the constraints. Note that a negative braking power Pbrk(t ) is needed to avoid infeasibility for very small power requests.

^ 10 km/h is the allowed excess of the rewhere Δvex = 2.78 m/s = ference velocity. The results are presented in Fig. A1. This approach guarantees that the planned reference velocity does not exceed the speed limits and stays below the maximum drivable curve velocities.

Appendix C. Fuel saving cost derivation for LHBP

Appendix B. Calculation of SoC change constraints for use cases in LHBP

TEM(t )

∫t

tk + 1 k

Ibat(TEM,high(t ), ω EM(t ))dt ,

The goal of the battery planning is to minimize the fuel consumption over the prediction horizon for fixed ωICE(t ) and Treq :

min

∫t

tf

ṁ f (ω ICE(t ), Treq(t ) − TEM(t ))dt ,

0

(C.1)

This goal can also be expressed by maximizing the fuel savings mf,save compared to not using the electric components at all:

The maximum SoC change is calculated by

ΔxSoC,max, k =

(B.13)

(B.1)

max TEM

∫t

tf 0

ṁ f,save(ω ICE(t ), Treq(t ), TEM(t ))dt

(C.2)

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=max TEM

∫t

tf

ṁ f (ω ICE(t ), Treq(t )) − ṁ f (ω ICE(t ), Treq(t ) − TEM(t ))dt

0

(C.3)

NLHBP − 1

=min P bat

=min TEM

∫t

tf

ṁ f (ω ICE(t ), Treq(t ) − TEM(t )) − ṁ f (ω ICE(t ), Treq(t ))dt

0

(C.4)

Since the fuel consumption without electric components does not ⁎ depend on the torque TEM from the EM, the resulting TEM (t ) is identical for both (C.1) and (C.3). The torque request Treq can be expressed by the power request Preq = Treq·ωICE . A map calculating the EM torque from the electric power of the EM and its speed TEM(ωEM, PEM,el) is obtained by inverting the map PEM,el(ωEM, TEM). Using this map, it is assumed that the power consumption of the auxiliaries Paux(t ) is known for all times and that in the simple equivalent circuit battery model shown in Fig. 5, the EM torque can be expressed by the battery power Pbat = U0·Ibat :

⎛ ⎞ P2 TEM⎜⎜ ω EM, Pbat − Ri · bat + Paux⎟⎟. 2 U0 ⎝ ⎠

(C.5)

Due to the configuration, the EM speed satisfies ωEM(t ) = ωICE(t ). Since in (C.5) all quantities except for Pbat are given trajectories, (C.2) can be written as

max P bat

∫t

tf

ṁ f,save(ω ICE(t ), Preq(t ), Pbat(t ))dt .

0

(C.6)

Separating the horizon in NLHBP use cases with starting time tk,0 and tk,f leads to NLHBP − 1

∑ ∫ t

max P bat

k=0

tk + 1

ṁ f,save(ω ICE(t ), Preq(t ), Pbat(t ))dt .

k

(C.7)

With the assumptions

ω ICE(t ) = ω ICE, k

∀ t ∈ [tk, tk + 1],

(C.8)

Preq(t ) = Preq, k

∀ t ∈ [tk, tk + 1],

(C.9)

Pbat(t ) = Pbat, k

∀ t ∈ [tk, tk + 1],

(C.10)

and Δtk = tk + 1 − tk , Eq. (C.7) simplifies to NLHBP − 1

∑

max P bat

ṁ f,save(ω ICE, k , Preq, k, Pbat, k )Δtk.

k=0

(C.11)

Since ωICE, k and Preq, k are given by the use case
max P bat

∑

ṁ f,save(
NLHBP − 1

=max P bat

∑

P bat

ṁ f,save(
k=0

NLHBP − 1

=max

∑

ṁ f,save(
k=0

NLHBP − 1

=min P bat

(C.12)

k=0

∑ k=0

−

·Pbat, k·Δtk

·ΔxSoC, k ·Q bat

ṁ f,save(
·ΔxSoC, k ·Q bat

(C.13)

(C.14)

(C.15)

∑ k=0

sBSB(Pbat, k,
(C.16)

This shows that the battery substitution benefit sBSB(Pbat, k,
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Please cite this article as: Heppeler, G., et al. Predictive planning of optimal velocity and state of charge trajectories for hybrid electric vehicles. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.07.003i

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Please cite this article as: Heppeler, G., et al. Predictive planning of optimal velocity and state of charge trajectories for hybrid electric vehicles. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.07.003i