An Optimization and Analysis Framework for TCO Minimization of Plug-in Hybrid Heavy-duty Electric Vehicles

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

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An Optimization and Analysis Framework An Optimization and Analysis Framework An Optimization and Analysis Framework for TCO Minimization of Plug-in Hybrid An Optimization and Analysis Framework for TCO Minimization of Plug-in Hybrid for TCO Minimization of Plug-in Hybrid Heavy-duty Electric Vehicles for TCO Minimization of Plug-in Hybrid Heavy-duty Electric Vehicles Heavy-duty Electric Vehicles Heavy-duty Electric Vehicles ∗ ∗∗ D.P.H. Kolodziejak ∗∗ T.H. Pham ∗∗ ∗∗ T. Hofman ∗ S. Wilkins ∗∗

D.P.H. Kolodziejak ∗ T.H. Pham ∗∗ T. Hofman ∗ S. Wilkins ∗∗ D.P.H. Kolodziejak ∗ T.H. Pham ∗∗ T. Hofman ∗ S. Wilkins ∗∗ D.P.H. Kolodziejak T.H. Pham T. Eindhoven Hofman S. Wilkins ∗ of ∗ Department of Mechanical Mechanical Engineering, Engineering, Eindhoven University University of of ∗ Department Technology, Netherlands, ([email protected]). Department of Mechanical Engineering, Eindhoven University of ∗∗∗ Technology, Netherlands, ([email protected]). Department of Mechanical Engineering, Eindhoven University of TNOTechnology, Powertrains Department, Helmond, Netherlands, ([email protected]). ∗∗ Powertrains Department, Helmond, Netherlands Netherlands (e-mail: (e-mail: ∗∗ TNO Technology, Netherlands, ([email protected]). [email protected]) TNO Powertrains Department, Helmond, Netherlands (e-mail: [email protected]) ∗∗ TNO Powertrains Department, Helmond, Netherlands (e-mail: [email protected]) [email protected]) Abstract: Abstract: This This paper paper develops develops an an optimization optimization framework framework to to minimize minimize the the Total Total Cost Cost of of Ownership (TCO) for Plug-in Hybrid Electric Vehicles (PHEVs). In this paper, TCO is Abstract: This paper develops an optimization framework to minimize the Total Cost of Ownership (TCO) for Plug-in Hybrid Electric Vehicles (PHEVs). In this paper, TCO is Abstract: Thisofpaper develops anmain optimization framework to minimize thepaper, Totaldeveloped Cost of the summation operational and vehicle powertrain components cost. The Ownership (TCO) for Plug-in Hybrid Electric Vehicles (PHEVs). In this TCO is the summation of operational and main Electric vehicle powertrain components cost. paper, The developed Ownership (TCO) for Plug-in Hybrid Vehicles (PHEVs). In this TCO is the summation of operational and main powertrain components cost. The developed optimization framework is formulated formulated viavehicle combining convex optimization and Dynamic Proand optimization framework is via combining convex optimization Dynamic Prothe summation of operational and main vehicle powertrain components cost. The developed gramming technique. This framework aims at minimizing TCO by optimizing not only the optimization framework is formulated via combining convex optimization and Dynamic Programming technique. This framework via aimscombining at minimizing by optimizing not onlyProthe optimization framework is formulated convexTCO optimization and Dynamic sizing of of the thetechnique. main powertrain components but at also the powertrain powertrain topology. Using the developed gramming This framework aims minimizing TCO by optimizing notdeveloped only the sizing main powertrain components but also the topology. Using the gramming Thisthis framework aims minimizing TCOtopology. by optimizing notdeveloped only bus the optimization framework, paper elaborates elaborates relevant design factors for considered sizing of thetechnique. main powertrain but at also the powertrain Using the optimization framework, thiscomponents paper relevant design factors for aa considered bus sizing of thenamely: main powertrain components but aalso thewith powertrain topology. Using the developed optimization framework, this paper elaborates relevant design factors for a considered bus application i) the value of equipping HEV plug-in functionality; ii) the effect of application namely: i) the this valuepaper of equipping a HEV with design plug-in factors functionality; ii) the effectbus of optimization framework, elaborates relevant for a cost; considered battery aging and replacement cost; iii) the sensitivity to fuel and electricity Simulation application namely: i) the value of equipping a HEV with plug-in functionality; ii) the effect of battery aging and replacement cost; iii) the sensitivity to plug-in fuel andfunctionality; electricity cost; Simulation application namely: i)TCO the can value ofreduced equipping a HEVplug-in with ii) the effect of results show show that the becost; by having having functionality in the the cost; HEVs. However, battery aging andthe replacement iii) the sensitivity to fuel and electricity Simulation results that TCO can be reduced by plug-in functionality in HEVs. However, battery aging and replacement cost; iii) the sensitivity to fuel and electricity cost; Simulation this may may notthat holdtheif if TCO the electricity electricity price by (inhaving Euros/kWh) is higher higher than than certain times of the the results show can be reduced plug-in functionality in certain the HEVs. However, this not hold the price (in Euros/kWh) is times of results show that theif TCO can be2.25 reduced by plug-in functionality in certain the HEVs. However, this may not hold the electricity price (inhaving Euros/kWh) is in higher than times of the fuel price (in Euros/kWh), e.g. for the simulated cases this paper. Simulation results fuel price (in Euros/kWh), e.g. 2.25 for the simulated cases in this paper. Simulation results this may not hold electricity (in simulated Euros/kWh) is in higher certain times of the also price suggest that itifis isthe more profitable to the equip the vehicle vehicle with bigthan enough battery toresults avoid fuel (inthat Euros/kWh), e.g. 2.25price for cases paper. Simulation also suggest it more profitable to equip the with aathis big enough battery to avoid fuel price (in Euros/kWh), e.g. 2.25 for the simulated cases in this paper. Simulation results replacing it during during the vehicle economical life. the vehicle with a big enough battery to avoid also suggest that itthe is more profitable to equip replacing it vehicle economical life. also suggest that itthe is more profitable to equip replacing it during vehicle economical life. the vehicle with a big enough battery to avoid © 2019, IFAC (International Federation of Automatic replacing it during the vehicle economical life. Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: plug-in plug-in hybrid hybrid electric electric vehicles, vehicles, optimization, optimization, total total cost cost of of ownership, ownership, component component Keywords: sizing, energy management. Keywords: plug-in hybrid electric vehicles, optimization, total cost of ownership, component sizing, energy management. Keywords: plug-in hybrid electric vehicles, optimization, total cost of ownership, component sizing, energy management. sizing, energy management. 1. INTRODUCTION INTRODUCTION those decision decision variables variables are are not not convex convex Tobias Tobias N¨ N¨ uesch esch et et al. al. 1. those u 1. INTRODUCTION those decision variables are not convex Tobias N¨ uesch et al. (2014). (2014). 1. INTRODUCTION those decision variables are not convex Tobias N¨ uesch et al. (2014). Hybrid The authors authors in in Murgovski Murgovski et et al. al. (2012), (2012), have have overcome overcome this this Hybrid electric electric vehicles vehicles (HEVs) (HEVs) are are able able to to reduce reduce fuel fuel The (2014). Hybrid electric (HEVs) are ablecontrol to reduce fuel The consumption by:vehicles (i) exploiting exploiting additional freedom authors inbyMurgovski etdiscrete disadvantage solving the optimization variables al. (2012), have overcome this consumption by: (i) additional control freedom solving theetdiscrete optimization variables Hybrid electric (HEVs) are able torecuperating reduce fuel disadvantage consumption The inby al.by (2012), haveheuristic overcome this to optimize optimize the ICEexploiting operating point, (ii) by:vehicles (i) additional control freedom disadvantage byMurgovski solving the discrete optimization variables priorauthors to convex convex optimization applying rules to the ICE operating point, (ii) recuperating prior to optimization by applying heuristic rules consumption by: (i) exploiting additional control freedom to optimize disadvantage by solving the discrete optimization variables brake energy, and (iii) downsizing the ICE Pourabdollah the ICE operating point, (ii) recuperating prior to convex optimization by applying heuristic rules in an outer loop; for a Series and Parallel PHEV with brake energy, andICE (iii) downsizingpoint, the ICE Pourabdollah in an to outer loop;optimization for a Seriesbyand Parallelheuristic PHEV rules with to optimize (ii) recuperating brake and (iii) operating downsizing the ICE Pourabdollah prior convex applying et Compared to Hybrid Vehicles energy,the an outer loop; for aThe Series Parallel with variable battery sizing. studyand showed that PHEV the adopted adopted et al. al. (2016). (2016). Compared to HEVs, HEVs, Plug-in Plug-in Hybrid Vehicles in variable battery sizing. The study showed that the brake energy, and (iii) downsizing the ICE Pourabdollah et al. (2016). to HEVs, Plug-in battery Hybrid Vehicles in an outer loop; for aThe Series and Parallel with (PHEVs) areCompared able to to charge charge the vehicle vehicle directly variable battery sizing. study showed that PHEV the approach achieves solution close to the the brute forceadopted nested (PHEVs) are able the battery directly approach achieves aa solution close to brute force nested et al. the (2016). toreduce HEVs, Plug-in Hybrid Vehicles (PHEVs) areCompared variable battery sizing. The study showed that the adopted from electrical grid to fuel consumption further. able to charge the vehicle battery directly approach achieves a solution close to the brute force nested Dynamic Programming Programming approach, approach, applied applied on on the the origiorigifrom the electrical to reduce consumption further. (PHEVs) are improved ablegrid to charge thefuel vehicle battery directly Dynamic from the electrical grid to reduce fuel consumption further. approach achieves a solution close to applied the brute nested However, the fuel reduction benefits are achieved achieved Dynamic Programming approach, onforce themodel. original nonlinear, non-convex, mixed-integer vehicle However, the improved fuel reduction benefits are nal nonlinear, non-convex, mixed-integer vehicle model. from the electrical grid tocharging reduce fuel consumption further. Dynamic Programming approach, applied on the origiHowever, improved reduction achieved at electric grid and possibly additional nal nonlinear, etnon-convex, mixed-integer model. Pourabdollah al. extended extended the work work of of vehicle Murgovski by at cost cost of ofthe electric gridfuel charging andbenefits possiblyare additional etnon-convex, al. the Murgovski by However, improved fuel reduction benefits areof achieved at cost ofthe electric gridmay charging and possibly additional nal nonlinear, mixed-integer model. component cost. This increase Total Cost Owner- Pourabdollah Pourabdollah et al. extended the work of vehicle Murgovski by adding ICE and EM sizing to the convex optimization component cost. This may increase Total Cost of OwnerICE and EMextended sizing tothe thework convex optimization at of electric gridto charging and possibly additional component cost. This may increase Total Costinof OwnerPourabdollah et al. of Murgovski by shipcost (TCO) compared conventional vehicle the same adding adding and EM sizing the convex optimization problemICE in Pourabdollah Pourabdollah et al. al.to(2016, (2016, 2014). Additionally, ship (TCO) compared to aa conventional vehicle inofthe same problem in et 2014). Additionally, component cost. This may increase Total Cost Ownership (TCO) compared to a conventional vehicle in the same adding ICE and etEM sizing toan the convex optimization class. Therefore, for a strong marked position of PHEVs problem in Pourabdollah et al. (2016, 2014). Additionally, in Pourabdollah al. (2014) optimization method is class. Therefore, for atostrong marked position ofthe PHEVs Pourabdollah et al. (2014) an optimization method is ship (TCO) compared aTCO conventional vehicle inof same in class. Therefore, for the a strong problem in Pourabdollah et al. (2016, 2014). Additionally, it is important that of those vehicles are minimarked position PHEVs in Pourabdollah et al. (2014) an optimization method is introduced that iteratively combines Dynamic Programit is important that the TCO of those vehicles are minithat iteratively combines Dynamic method Programclass. Therefore, for the arequires strong ofare PHEVs it is important that TCO marked of optimal thoseposition vehicles mini- introduced in Pourabdollah etOptimization al. (2014) anfor optimization is mized. This objective objective an designed PHEV introduced that iteratively combines Dynamic aProgramming and Convex Convex optimizing parallel mized. This requires an optimal designed PHEV ming and Optimization for optimizing a parallel it is important that the TCO of those vehicles are minimized. introduced that iteratively combines Dynamic Programpowertrain, considering; topology, component sizing, and This objective requires an optimal designed PHEV ming and Convex Optimization for optimizing a parallel topology with gear shifting and engine on/off switching as powertrain, considering; topology, component sizing, and topology with gear shifting and engine on/off switching as mized. objective requires an(EMS) optimal designed PHEV powertrain, considering; topology, component sizing, and topology ming andoptimization Convex for optimizing parallel EngergyThis Management Strategy Silvas (2015). with gear Optimization shifting and engine on/off switching as discrete variables. This method cana find find the Engergy Management Strategy (EMS) Silvas (2015). optimization variables. This method can the powertrain, considering; topology, component sizing, and discrete Engergy Management Strategy topology with gear shifting and engine on/off switching as (EMS) Silvas (2015). discrete optimization variables. This method can find the optimal sizing parameters with good accuracy after few A promising promising method to toStrategy find the the (EMS) optimalSilvas component sizing optimal sizing parameters with This goodmethod accuracy after few Engergy Management (2015).sizing discrete optimization variables. can find the A method find optimal component accuracy after few iterationssizing whenparameters component with sizes good are chosen chosen appropriately A promising method to find optimal componentdue sizing and EMS simultaneously simultaneously is the Convex optimization, to optimal iterations when component sizes are appropriately optimal sizing with good accuracy after few and EMS is Convex optimization, due to whenparameters component sizes size are chosen appropriately e.g, the engine size and battery may not severely be A promising method to find optimal component sizing and EMS simultaneously is the optimization, due to iterations its higher computational efficiency, compared to optimal Convex e.g, the engine size and battery size may not severely be iterations when component sizes size are chosen appropriately its higher computational efficiency, compared to optimal e.g, the engine size and battery may not severely be over-sized initially. and EMS simultaneously is Convex optimization, due to its higher computational efficiency, compared to(DP) optimal control methods such as Dynamic programming and over-sized initially. size and battery size may not severely be control methods such as Dynamic programming (DP) and e.g, the engine initially. its higher computational efficiency, compared optimal control methods such as Dynamic programming Equivalent Consumption Minimization Strategyto(DP) (ECMS); and over-sized To the best of our knowledge, over-sized initially. Equivalent Consumption Minimization Strategy (ECMS); To the best of our knowledge, TCO TCO minimization minimization of of SeriesSeriescontrol methods such as Dynamic programming (DP) and Equivalent Consumption Minimization Strategy (ECMS); who cannot handle component sizing directly, and require To the best of our knowledge, TCO minimization of Parallel HEVs by Convex optimization is still anSeriesopen who cannot handle component sizing directly, and require Parallel HEVs by Convex optimization is still an open Equivalent Minimization Strategy (ECMS); who cannotConsumption component sizingoptimization. directly, and require To the best of our TCOTCO minimization an outer-loop outer-loop for component component sizing On the Parallel handle HEVs byknowledge, Convex optimization is still of anSeriesopen research topic. This paper studies minimization for aa an for sizing optimization. On the topic. This paper studies TCO minimization for who cannot component sizingoptimization. directly, and On require an outer-loop component Parallel HEVstopology by Convex optimization is still an open other hand, handle a for disadvantage ofsizing Convex optimization is the its research research topic. This paper studies TCObattery minimization for a Series-Parallel with different replacement other hand, a disadvantage of Convex optimization is its with different replacement an outer-loop component optimization. Onis the other disadvantage ofsizing Convex optimization its Series-Parallel research topic. topology This paper studies TCObattery minimization for a incapability tofor optimize discrete optimization variables hand, ato Series-Parallel with replacement strategies, compares the results to incapability optimize discrete optimization variables strategies, and andtopology compares thedifferent results battery to aa Conventional, Conventional, other hand, atoshifting disadvantage of Convex optimization is its Series-Parallel incapability optimize discrete optimization variables topology with different battery replacement such as gear and ICE on/off switching because strategies, and compares the results to a Conventional, such as gear shifting and ICE on/off switching because incapability optimize optimization such as gear toshifting anddiscrete ICE on/off switching variables because strategies, and compares the results to a Conventional, such as gear shifting and ICE on/off switching because 2405-8963 © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.

Copyright © 2019 IFAC 484 Copyright 2019 IFAC 484 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 484 10.1016/j.ifacol.2019.09.077 Copyright © 2019 IFAC 484

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SICE , TICE (k)

γT r (k)

ICE

SEM 1 , TEM 1 (k)

SICE , TICE (k), µICE (k)

Tr

485

γT r (k)

ICE

Tp (k)

Tr

FD

Tp (k)

EM1

FD

PEM 1 (k) Pt (k)

TBr (k)

SBat , Pb (k), Pg (k)

Tw (k)

Fig. 1. Conventional Topology for which the optimization variables are in bold. The torque balance at the transmission input shaft is given by Tp = TICE SEM 2 , TEM 2 (k)

SICE , TICE (k), µICE (k)

SEM 1 , TEM 1 (k)

SEM 2 , TEM 2 (k)

SICE , TICE (k), µICE (k)

EM2

EM1

FD

Tw (k)

Fig. 3. Plug-in Parallel Topology for which the optimization variables are in bold. The electrical power balance at the battery terminal when driving is given by Pt = PEM 1 , while the torque balance at the transmission input shaft is expressed by Tp = µICE · TICE + TEM 1 .

ICE Tp (k)

TBr (k)

Bat

EGU

µm (k)

SEM 1 , TEM 1 (k)

γT r (k)

∗

ICE PEM 2 (k)

Tr EM2

PEM 1 (k) Pt (k)

Bat

EM1

Tp (k)

FD

TBr (k)

SBat , Pb (k), Pg (k)

PEM 2 (k)

Tw (k)

SBat , Pb (k), Pg (k)

Fig. 2. Plug-in Series Topology for which the optimization variables are in bold. The electrical power balance at the battery terminal when driving is given by Pt = PEM 1 + PEM 2 , while the torque balance at the drive shaft is expressed by Tp = TEM 1 . Series, and Parallel topology; with similar performance requirements i.e. acceleration and gradeability. The main contribution of this paper are: (i) TCO analysis with respect to a conventional powertrain; for a Series, Parallel and Series-Parallel topology on different battery replacement strategies, and (ii) Effect of plug-in charging analysis including energy cost sensitivity.

PEM 1 (k) Pt (k)

Bat

TBr (k) Tw (k)

* ICE and EM2 together form the EGU in Series Mode

Fig. 4. Plug-in Series-Parallel Topology for which the optimization variables are in bold. The electrical power balance at the battery terminal when driving is given by Pt = PEM 1 + PEM 2 , while the torque balance at the transmission input shaft is expressed by Tp = TEM 1 + µm · (TEM 2 + µICE · TICE ). Table 1. Control inputs per topology. In which S, P, S-P, and C refer to Series, Parallel, SeriesParallel, and Conventional Topology, respectively Control Input

2. PROBLEM FORMULATION

TICE TEM 1 TBr TEM 2 Pb Pg

Discrete

This paper investigates three PHEV topologies, namely a Series, Parallel and, Series-Parallel topology, as depicted in Fig. 2, Fig. 3, and Fig. 4, respectively. The conventional topology shown in Figure 1 acts as a benchmark for TCO comparison. In these figures ’Bat’,’Tr’,and ’FD’ represent the Battery, Transmission, and Final Drive, respectively. A description for all time dependent control inputs in Figures 1-4 is shown in Table 1. The control inputs are divided in two types (i) Continuous inputs; and (ii) Discrete inputs. Discrete input µICE ∈ {0, 1} with 0 indicating that the engine is off, and µm ∈ {0, 1} with 0 indicating series mode. The control inputs together form the Energy Management Strategy (EMS) that aims to minimize the operational cost. The component sizes to be optimized are the ICE, Battery, and both EMs. These components are scaled with scaling factor Si with i ∈ {ICE, Bat, EM 1, EM 2}. The resulting TCO minimization problem is discussed in 2.1.

Continuous

Type Symbol Parameter

µICE γT r µm

Topology S

P

SP

C

ICE Torque EM1 Torque Friction Brake Torque EM2 Torque Internal Battery Power Grid Charging Power

  

  

  



 



 







ICE on/off Engaged Gear Ratio Selected mode



 

 







gradeability requirements. This is expresses mathematically as in Silvas (2015) by [xp , xc ] = arg min Jo (xp , xc (k), Λ(k)) + Jc (xp ) (1) xp ,xc (k)

2.1 TCO Minimization Problem The minimization problem is defined as follows: given predefined vehicle transportation mission and a set of topologies, find the: optimal topology, main component sizes and optimal EMS to minimize TCO, while satisfying predefined system performance e.g. acceleration and 485

where the cost function consists of operational cost Jo and component cost Jc . Operational cost consist of fuel and electricity cost and component cost are the weighted cost of the topology main components. In (1) design vector xp contains all design variables of the plant which are the topology with associated component sizing, while control

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vector xc contains all control variables discussed in Table 1. Both vectors are optimized for vehicle mission Λ and will be discussed in 2.2. The minimization problem is subjected to system dynamics consisting of the battery State of Charge (SoC) and State of Health (SoH). Both states have a predefined initial value and a specified final value at the end of the vehicle mission. Operational cost The operational cost consists of fuel and electrical grid charging cost calculated over a discretized vehicle mission with length N and sampling interval ∆h. The operational cost is denoted as J o = wf

N 

k=1

Pf (k) ∆h + wg

N 

Pg (k) ∆h

(2)

k=1

where Pf [W ] and Pg [W ] are the fuel power and electrical grid power, respectively. Parameter wf is a weight factor for converting fuel energy to fuel cost in e /100km calφ ·100 culated as wf = (ρf ·Υf LHV ·d) . In this φf , ρf , ΥLHV , and d are fuel cost [e /l], fuel density [kg/l], Lower Heating Value [J/kg], and vehicle mission distance [km], respectively. The baseline diesel cost are set to 1.30 e /l. Weight factor wg that converts electric energy to electricity cost φe ·100 in e /100km, is expressed by wg = (3.6e6·d) with electricity cost φe in e /kWh set to a baseline value of 0.2 e /kWh. Component cost Component cost is a function of the selected topology in combination with the sizing of the main components, being selected as the Transmission, ICE, EM including power electronics, and the battery pack. The battery pack may have to be replaced nr ∈ N times during vehicle economical life, and there is assumed that the replaced battery has similar specifications and price as the former battery. The component cost Jc [e /100km] is expressed by   wc · (ΨICE + (1 + nr )ΨBat +    +ΨEM 1 + ΨEM 2 ) for S Top     w · (Ψ + (1 + nr )ΨBat + ICE  c Jc = +ΨEM 1 + ψT r ) for P Top   wc · (ΨICE + (1 + nr )ΨBat +     +ΨEM 1 + ΨEM 2 + ψT r ) for S-P Top    wc · (ΨICE + ψT r ) for C Top (3) with weight factor wc = dy1·lT · 100 converting component cost from e into e /100km by taking into account the average yearly vehicle mileage dy [km/year] and the total vehicle lifetime Lt [years]. The component cost of the baseline main components are expressed by ψ as shown in Table 2, while the sizable main components are expressed by Ψ. For component scaling cost a linear relation has been assumed. Herewith, the scalable component cost are calculated by Ψi = Si · ψi ,

i ∈ {ICE, Bat, EM 1, EM 2}.

(4)

The baseline ICE cost ψICE is obtained by linear scaling of the provided ICE cost in Silvas (2015). The specific EM cost ψEM including inverter cost is obtained from Kailasam (2014), and the battery cost are provided by an OEM. 486

Table 2. Baseline component cost Component

Power/capacity

Specific Cost

Cost (ψi )

ICE EM 1 EM 2 Bat Tr

210 kW 90 kW 90 kW 12 kWh -

25.7 e /kW 46.6 e /kW 46.6 e /kW 200 e /kWh -

e 5397 e 4194 e 4194 e 2400 e 3500

2.2 Vehicle mission profile Topology selection and component size optimization for PHEVs requires knowledge about the lifetime driving of the vehicle Pourabdollah et al. (2016). This includes the velocity- and slope profile of the trip, as well as, the distance driven between plug-in possibilities, the charging time, and maximum available charging power. Therefore, powertrain design requires a vehicle mission that is representable for real world driving. The design of such a mission is not within the scope of this study. This study uses a single measured bus route that includes a 30km long zero-emission zone where the ICE is not allowed to be switched on. Figure 5 (a), shows the 217 minute long measured vehicle mission profile. This vehicle mission consists of two 30 min stops at bus stations where it is possible to charge the battery directly from the electrical grid. After the second stop the bus enters the zero-emission zone where the vehicle is propelled pure electrically. Remark 1. This paper assumes that the bus will drive the measured vehicle mission over its entire vehicle economical life for TCO minimization. Remark 2. For analyzing TCO sensitivity to fuel and electricity cost the SORT cycle based vehicle mission shown in Figure 5 (b) has been used. Similar as in the measurement based vehicle mission the mission contains two charging opportunities and a zero emission zone. For the sensitivity analysis a shorter vehicle mission has been used to reduce the computational burden. This is reasonable since there is observed that both vehicle missions showed similar trends in component sizing and TCO for the results presented in 5. When using a laptop with Intel Core i7 processor, 2.8GHz and 16GB RAM, it takes about 90 and 730 seconds for the SORT and measured cycle, resepectively. Both vehicle missions are captured by their time dependent vehicle mission matrix of the form T

Λ(k) = [v(k) a(k) θ(k) ζ(k) Θ(k)] ,

k ∈ [1, N ]

(5)

where v, a and θ defines the velocity, acceleration, and slope profile, respectively. While ζ is a zero emission zone flag for which a high value indicates that the vehicle is driving in a zero emission zone and Θ is a flag for which a high value indicates that the vehicle has a grid charging opportunity. The measurement based vehicle mission does not contain any extreme conditions where the vehicle is loaded to its maximum payload. To make sure that the vehicle is able to operate under these heavy conditions some performance requirements for acceleration and gradeability are defined.

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487

Table 3. Vehicle resistance parameters Parameter

Value

Wheel radius Vehicle mass Glider mass Payload mass Gravitational constant Rolling resistance coefficient Inertia Wheels Aerodynamic drag coefficient Air density Vehicle frontal area

Rw = 0.491m mv (7) mg = 10715kg mp = 3850kg ∗ g = 9.81m/s2 fr = 0.005 Jw = 55kgm2 cd = 0.55 ρa = 1.204kg/m3 Ad = 7.956m2

∗

based on a constant number of 50 passengers

Table 4. Mass of baseline components Fig. 5. Vehicle Missions (a) representing a real measured bus route, and (b) is a sort cycle based vehicle mission. In both vehicle missions grid charging possibilities are indicated by red and a zero-emission zone is indicated by green ∗∗ Initial: SBat , SICE , SEM 1 , SEM 2

Dynamic Programming

Power/capacity

Specific mass

Mass (mi )

ICE EM 1 EM 2 BAT T rn

210 kW 90 kW 90 kW 12 kWh -

2.55 kg/kW 2 kg/kW 2 kg/kW 9.17 kg/kWh -

536 180 180 110 350

Tw (k) = Rw SBat SICE SEM 1 ∗∗ SEM 2

µICE γT∗ r µ∗∗∗ m

Component

+

Jw a(k) + 2 Rw

 1 2

kg kg kg kg kg

  mv g fr cos(θ(k)) + g sin(θ(k)) + a(k) +  ρa Ad cd v 2 (k)

(6)

Convex Optimization * Not for Series topology ** Not for Parallel topology *** Series-Parallel topology only

Fig. 6. Schematic overview of the optimization framework. 3. OPTIMIZATION FRAMEWORK The TCO minimization problem is a mixed-integer non convex optimization problem. The authors of Pourabdollah et al. (2014) studied the use of an iterative optimization framework that combines dynamic programming and convex optimization for a parallel topology. This study extends the method Pourabdollah et al. (2014) to the Parallel, Series, and Series-Parallel topology by adding the SoH as an extra state. Figure 6 shows a schematic overview of the optimization framework with the required inputs/outputs for DP and Convex optimization. Both blocks are solved iteratively until component sizing and TCO converge. SoH is taken into account within Convex optimization. For Convex optimization, the open source CVX package is used, which is a modeling system for constructing and solving disciplined convex programs (DCPs), CVX Research (2012). The CVX solver requires the components to be convex. The section below presents the convex models for the hybrid powertrain components. 4. SYSTEM MODEL 4.1 Vehicle road load

for which an overview of symbols is given in Table 3. Vehicle mass mv is expressed by  mg + mp + mT r + MBat +    +MICE + MEM 1 + MEM 2 if S and S-P mv =  + m p + mT r + m   g +MBat + MICE + MEM 1 if P (7) where mg , mp , and mT r are the baseline masses of the glider, payload, and transmission respectively. While masses indicated with a capital Mi are effected by a linear component scaling law given by Mi = Si · mi , i ∈ {ICE, Bat, EM 1, EM 2} (8) The baseline component masses mi are given in Table 4 and are provided by OEMs. 4.2 Powertrain component model The powertrain components are modelled with quasi-static approach (low fidelity) to comprise the computational burden and accuracy of the developed optimization framework. Internal Combustion Engine For convex optimization and sizing the fuel power Pf,ICE is approximated by second order strictly convex polynomial functions de Jager et al. (2013) with speed dependent coefficients and scaling factor SICE Pf,ICE = SICE · µICE · α0 +

+α1 · TICE (k) + α2 ·

The torque demand at the wheels is calculated by 487

TICE (k) 2 SICE

(9)

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Angular velocity dependent coefficients α0 , α1 , and α2 are determined by fitting the fuel power with the measured engine fuel map.

Electric Machine The electrical power PEM including power electronic losses is modeled as a function of the electric machine torque TEM , angular velocity ωEM , and scaling factor SEM . Moreover, an angular velocity dependent second-order polynomial function is used to approximate the electric machine efficiency map. The inclusion of SEM in the model is based on similar assumptions as in the ICE model. Herewith, EM electrical power flow PEM with sizing factor SEM , is estimated by 2

TEM (k) (10) SEM with angular velocity dependent coefficients β0 , β1 , and β2 . EM torque limits T EM and T EM are included by linear scaling of the maximum and minimum admissible torque of the baseline EM. PEM (k) = SEM β0 + β1 TEM (k) + β2

Engine Generator Unit The Engine Generator Unit (EGU) converts fuel energy to electrical energy by propelling an EM by an ICE. The control input of the model is selected to be TEGU and the scaling factors are SEM and SICE . The highest possible efficiency is obtained when the EGU is operated at the economy line. However, for simplicity since the EGU needs to be convex in TEGU , SEM and SICE the angular velocity is fixed on a velocity near the sweet spot of the baseline EGU. This approach is justified since in the optimal solution it is expected that the EGU is operated in its highest efficiency range where both the e-line based model and the fixed angular velocity based model are nearly equal as depicted in Figure 7. By using TEGU as the control variable it is possible to use the ICE and EM model as afore-mentioned. Herewith, EGU fuel power Pf,EGU and electrical power Pe,EGU are expressed by Pf,EGU = SICE α0 + α1 TEGU (k) + α2

TEGU (k) SICE

2

(11) 2

TEGU (k) (12) SEM The EGU torque is limited either by the ICE or the EM depending on the machine with the smallest admissible torque. Both machines use a linear scaling factor for the torque limit. Pe,EGU = SEM β0 − β1 TEGU (k) + β2

Battery energy model The battery pack consists of lithium-iron-phosphate cells (ANR26650m1A). Each cell is modeled by a resistive equivalent circuit model. By assuming equally distributed power over each individual battery cell Pourabdollah et al. (2016); Murgovski et al. (2012); Hu et al. (2015); Johannesson et al. (2013), the power at the battery pack terminal Pt [W ] as a function of the number of battery cells SBat , is Pt (k) = Pb − R ·

Pb2 (k) (SBat · Vn2 )

(13) 488

e-line fixed angular velocity

45 40 35

η EGU [%]

The torque limit of the scaled ICE model T ICE is included by assuming a linear relation between SICE and the maximum admissible baseline ICE torque T ICE,b

EGU efficiency 50

30 25 20 15 10 5 0 0

20

40

60

80

100

120

P EGU [kW]

Fig. 7. Baseline EGU efficiency according to e-line operation and fixed angular velocity in sweet-spot In which Pb [W ] is the internal battery power. From expression (13), and utilization of conversion of energy; the stored electrical energy in the battery is Eb (k + 1) = Eb (k) − Pb (k) ∆h(k)

(14)

where by definition the battery is discharged if a positive internal battery power Pb is applied, and vice-versa for charging the battery. With Eb (k + 1) ∈ [Eb , Eb ], and Eb (0) = Eb (N ). Battery wear model Battery wear is expressed by its capacity degradation as a function of charge/discharge rate Cr , lumped cell temperature, and tolerated Ah throughput IAh,t Wang et al. (2011). From this, assuming battery end of life at 20% capacity degradation, the allowable number of discharge cycles Nc are calculated, by Nc (Cr ) =

IAh,t (Cr ) · 3600 . Qb,c

(15)

The State of Health (SoH), as a function of Nc and energy throughput is expressed by Ebbesen et al. (2012) 1 SoH(t) = 1 − (2 · Nc (Cr ) · Qb,c · VOC )

t 0

|Pb,c (τ )|dτ(16)

where Pb,c is the internal cell power. By definition the SoH ∈ [0, 1] with SoH = 0 indicating battery end of life.

The resulting derivative of (16) has been approximated by a quadratic piecewise function for concavity  2  h0,1 Pb,c + h1,1 if |Pb,c | ≤ 43   h P 2 + h if 43 < |Pb,c | ≤ 55 0,2 b,c 1,2 ˙ = SoH (17) 2  h P + h if 55 < |Pb,c | ≤ 61 0,3 b,c 1,3    2 h0,4 Pb,c + h1,4 if 61 < |Pb,c |

Figure 8 (b) shows the Original SoH model derivative and its convex approximation. Furthermore, Figure 8 (a) indicates that the battery in the lowest internal power range can withstand less cycles than in the medium range. This is a result of not decoupling battery cycle-life and calendar-life in (15). Due to the assumption that each cell is accountable for a similar internal power throughput, the scalable battery wear model at pack level is obtained as 2 ˙ b (t) = h0,j · Pb (t) + SBat · h1,j , SoH SBat

j ∈ {1, 2, 3, 4} (18)

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a) Cycles untill end of life

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-2 -3

Original SoH model Convex SoH model 2 h0,1 Pb,cell + h1,1 2 + h1,2 h0,2 Pb,cell 2 h0,3 Pb,cell + h1,3 2 h0,4 Pb,cell + h1,4

-4 -5

CV

-6 0

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20

30

40

50

60

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PHEV

0 CV

HEV

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|Pb,cell |[W ] 40

Fig. 8. State of Health model illustration: a) Number of discharge cycles till end of life as a function of internal cell power, and b) Original State of Health model derivative and the piece-wise quadratic convex approximation of the SoH model derivative as a function of internal cell power b after substitution of Pb,c = SPBat and variable change SoHb = SBat · SoH. To take into account the piecewise approximation of SOH (17) in the optimization framework, the equality constraints (18) is relaxed to inequality ones 2 ˙ b (t) ≤ h0,j · Pb (t) + SBat · h1,j for j ∈ {1, 2, 3, 4} and SoH SBat battery power Pb is within its power limit. It can be seen that the constraints will be satisfied with equality at the optimum. The maximum SoH degradation over a single r +1) vehicle mission is calculated by ∆SoH = d·(n with dy ·Lt an assumption that, a linear SoH degradation over vehicle life distance is present. The battery SOH constraint at the end of a single vehicle mission is herewith defined as SoH(N ) ≤ ∆SoH.

Transmission The relation between the torque delivered at the vehicle wheels Tw and the torque at the input shaft of the transmission Tg (k) is given by  Tw (k)   if Tw (k) ≥ 0  (η · γ Tr T r (k) · γF D ) (19) Tg (k) = Tw (k) · ηT r   if T (k) < 0  w (γT r (k) · γF D ) where γF D is the final drive ratio andηT r is the transmission efficiency. The relation between the rotational speed of the wheel ωw and the rotational input speed of the gearbox is ωg = γT r · γF D · ωw . For the transmission model there is assumed that there are no power losses during gear shifting. 5. COMPARATIVE RESULTS & DISCUSSION This section analyzes two design choices: (i) effect of plugin functionality, and (ii) effect of battery replacement strategy. Effect of Plug-in functionality & Topology This section investigates how plug-in charging effects TCO for the 489

Series Operational Cost

40

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40

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Diesel Electricity

30

30

30

20

20

20

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0 CV

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PHEV

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PHEV

Fig. 9. Effect of having a Plug-in functionality on TCO, Component Cost, and Operational Cost for the SeriesParallel topology with different charging powers compared to the conventional topology. hybrid topologies discussed in section 2, and compares them to the conventional topology. Figure 9 shows how TCO, Component cost, and Operational Cost are effected by plug-in charging for the hybrid topologies optimized for the Measurement based vehicle mission with diesel cost of 1.30 e /l and electricity cost of 0.2 e /kWh. For this analysis the maximum grid charging power is limited to 120kW. As shown, although component cost increases when using plug-in charging, operational cost decreases for all topologies. As can be seen, reduction of TCO are achieved by enabling plug-in charging since operational cost reduction is larger than component cost increasement. The conventional topology is significantly outperformed by all hybrid topologies even when no Plug-in functionality is present. This is mainly driven by the assumption that all brake energy is available for regenerative braking as shown in Figure 10. This figure shows TCO reduction for the Parallel Topology in comparison to the conventional topology with different regenerative brake energy potentials. The brake energy potential indicates the percentage of the total brake energy that is available for regenerative braking. As shown, there is better TCO reduction performance when the regenerative brake energy potential is higher. In case no regenerative braking is applied there is only 11.35% TCO reduction in comparison to a reduction of 52.9% when all energy is available for regenerative braking. Effect of battery replacement strategy This section presents the influence of battery replacement strategy on the TCO (Figure 11). As seen, all topologies show a similar trend. For each topology the TCO of the ’ideal’ case are the lowest since operational cost and component cost are not

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Sensitivity to brake energy recovery potential (Measured Cycle)

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o

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Fig. 11. Effect of the selected battery replacement Strategy on TCO, Absolute Component Cost, and Battery Capacity for the selected hybrid topologies. For which ’Ideal’ represents an ideal scenario where the battery does not wear, and ’No pres’ indicates that SoH preservation is not taken into account. While ’0 rep’, ’1 rep’, and ’2 rep’ are scenarios where the SoH is preserved according to 0, 1, and 2 battery replacements, respectively. Battery Wear (Measured Cycle) SoH trajectory over vehicle mission

1.0000e+00

SoH trajectory over vehicle life

1 0.9

9.9995e-01 0.8 9.9990e-01

0.7 0.6

9.9985e-01

SoH

The Electricity price and Diesel price has an impact on vehicle design choices such as plug-in charging functionality and therefore on component sizing as well. This section analysis the TCO sensitivity, and sizing sensitivity to energy cost for the Parallel topology for demonstration purpose. Figure 13 shows the sensitivity of vehicle design to three different Diesel prices with various electricity cost φe for the SORT cycle based vehicle mission. As shown, TCO increases with increasing energy price but saturates from a certain electricity price for each Diesel pricing scenario. This saturation is a result of a breakpoint in electricity cost from which it is no longer beneficial to apply plug-in charging. As can be seen, this brake point moves towards a higher electricity price when the diesel price increases. For analyzing the breakpoint the φe ·ρf ·ΥLHV energy cost ratio  [-] is defined as  = (φ . Figure 6 f ·3.6e ) 13 shows that for each diesel price plug-in charging is no longer beneficial when the Energy Cost ratio exceeds approximately 2.25, i.e. as long as Diesel Energy is 2.25 times as cheap as electricity it is not beneficial in terms of TCO to apply plug-in charging. Additionally, the vehicle acts as a full electric vehicle when the Energy Cost Ratio is lower than approximately 1.5. Moreover, when Energy Cost ratio ∈ [1.5 2.25] operational cost consists Diesel cost and Electricity cost. A similar trend is expected for the Measurement based vehicle mission , although the value of the energy cost ratio is uncertain.

5

SoH

6. SENSITIVITY TO ENERGY COST ANALYSIS

10

5

o

C To p te nt ia l po 10 t 0% enti po al te nt ia l po

%

Fig. 10. Sensitivity of TCO, Component Cost, and Operational Cost to brake energy potential in relation to the Conventional Topology. The 0% potential indicates that all brake energy is dissipated in the friction brake, while for the 100% potential all energy is available for regenerative braking. compromised by battery wear limitation. However, TCO increases significantly when for the ’ideal’ case the battery wear is post calculated as in the ’No pres’ case, where the battery needs to be replaced ones, as shown in Figure 12. This confirms the need for battery wear model inclusion as indicated in literature. When battery wear is preserved for the different replacement strategies there is seen that it is most beneficial to not replace the battery at all, but rather select a higher capacity battery that requires less energy throughput per cell to preserve the battery life. The additional advantage of this strategy is that battery power losses are reduced for equal terminal battery power since the power is divided over more cells.

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2019 IFAC AAC 490 Orléans, France, June 23-27, 2019

9.9980e-01

0.5 0.4 0.3

9.9975e-01 No SoH preservation SoH preservation for 0 battery replacements SoH preservation for 1 battery replacement SoH preservation for 2 battery replacements

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Fig. 12. SoH trajectory for different battery replacement strategies for the Series-Parallel Topology. The graph on the left shows the SoH trajectory for the first ever vehicle mission of the vehicle, while the graph on the right indicates the SoH trajectory over the entire vehicle life. SoH 1 indicates that the battery is unused while SoH 0 indicates that the battery needs to be replaced. Component sizing shows a similar trend for all Diesel prices over the Electricity Cost ratio interval. This trend is similar as observed in section 5.0.1; the battery size and EM size increase, while the ICE size decreases when plugin charging is profitable. As can be seen, the vehicle is still

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

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into account start-up costs as well as cost reduction / inflation.

Energy Cost Sensitivity

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REFERENCES

240

70

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This work is supported by ORCA project addressing topic GV-03-2016, “System and cost optimised hybridisation of road vehicles” of the Green Vehicle work programme.

φe 16

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ACKNOWLEDGEMENTS

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Fig. 13. TCO and component sizing sensitivity to different energy pricing scenarios equipped with a relative large EM when plug-in charging is not beneficial, because regenerative braking efficiency has a part in EM sizing. 7. CONCLUSIONS & RECOMMENDATIONS In this study, an optimization framework combining Convex Optimization and Dynamic Programming is developed to minimize TCO for heavy duty plug-in hybrid electric vehicles. The framework optimizes component sizing, energy management strategy including battery wear, in order to minimize the sum of component- and operational cost for a Series, Parallel, and Series-Parallel topology. Two design decisions for TCO minimization were analyzed for a Heavy Duty Transit Bus application: (i) the value of equipping a HEV with a plug-in functionality, and (ii) the effect of the battery replacement strategy. Adding a plug-in functionality appeared to be beneficial for all topologies in the case study. Although the initial powertrain investment went up, the additional investment is earned back over the course of the vehicle life time. Energy cost sensitivity analysis were performed for the plug-in Parallel topology, there is concluded that plug-in charging is no longer beneficial when the electricity prise becomes approximately 2.25 times the diesel price for the Parallel topology and SORT cycle based vehicle mission. Battery replacement for TCO reduction is not beneficial for the studied scenario. It is more profitable to equip the vehicle with a sufficiently large battery for SoH preservation. For future work, sensitivity analysis are recommended to study the effect of the battery wear model on the optimal battery replacement strategy. It is relevant to refine the cost model of powertrain components to take 491

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