Optimal Sizing of a Series PHEV: Comparison between Convex Optimization and Particle Swarm Optimization

4th IFAC Workshop on 4th Workshop on 4th IFAC IFACand Workshop on Control, Simulation and Modeling Engine Powertrain 4th IFAC Workshop on Engine and Powertrain Control, Simulation and Modeling Available at www.sciencedirect.com Engine and Powertrain Control, Simulation and Modeling August 23-26, 2015. Columbus, USA Engine and Powertrain Control, OH, Simulation and online Modeling August 23-26, 2015. Columbus, OH, USA August 23-26, 2015. Columbus, OH, USA August 23-26, 2015. Columbus, OH, USA

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Optimal Sizing of a Series PHEV: Comparison Optimal Optimal Sizing Sizing of of a a Series Series PHEV: PHEV: Comparison Comparison Optimal Sizing of a Series PHEV: Comparison between Convex Optimization and Particle Swarm between Convex Optimization and Particle Swarm between and Particle Swarm between Convex Convex Optimization Optimization and Particle Swarm Optimization Optimization Optimization Optimization

Mitra Pourabdollah11 , Emilia Silvas22 , Nikolce Murgovski11 , Maarten Steinbuch22 , Bo Egardt11 Mitra Mitra Pourabdollah Pourabdollah11 ,,, Emilia Emilia Silvas Silvas22 ,,, Nikolce Nikolce Murgovski Murgovski11 ,,, Maarten Maarten Steinbuch Steinbuch22 ,,, Bo Bo Egardt Egardt11 Mitra Pourabdollah Emilia Silvas Nikolce Murgovski Maarten Steinbuch Bo Egardt 1 1 Chalmers University of Technology, Gothenburg, Sweden, (e-mail: 1 Chalmers University of Technology, Gothenburg, Sweden, (e-mail: 1 Chalmers University of of Technology, Technology, Gothenburg, Gothenburg, Sweden, Sweden, (e-mail: (e-mail: {mitrap,nikolce.murgovski,bo.egardt}@chalmers.se), Chalmers University {mitrap,nikolce.murgovski,bo.egardt}@chalmers.se), 2 University of Technology, Eindhoven, The Netherlands, (e-mail: {mitrap,nikolce.murgovski,bo.egardt}@chalmers.se), 2 Eindhoven{mitrap,nikolce.murgovski,bo.egardt}@chalmers.se), University of Technology, Eindhoven, The Netherlands, (e-mail: 2 2 Eindhoven Eindhoven Technology, e.silvas,[email protected]) Eindhoven University University of of Technology, Eindhoven, Eindhoven, The The Netherlands, Netherlands, (e-mail: (e-mail: e.silvas,[email protected]) e.silvas,[email protected]) e.silvas,[email protected]) Abstract: Building a plug-in hybrid electric vehicle that has a low fuel consumption at low hybridization Abstract: Building aa plug-in hybrid electric vehicle that has a low fuel consumption at low hybridization Abstract: Building hybrid vehicle that has fuel at cost requires detailed design optimization studies. investigates optimization of ahybridization PHEV with Abstract: Building a plug-in plug-in hybrid electric electric vehicleThis thatpaper has aa low low fuel consumption consumption at low low cost requires detailed design optimization studies. This paper investigates optimization of aahybridization PHEV with cost requires detailed design optimization studies. This paper investigates optimization of PHEV acost series powertrain configuration, where plant andThis control parameters areoptimization found concurrently. Inwith this requires detailed design optimization studies. paper investigates of a PHEV with aa series powertrain configuration, where plant and control parameters are found concurrently. In this series plant and control parameters found concurrently. In this work twopowertrain often usedconfiguration, methods are where implemented to find optimal energyare management with component awork series powertrain configuration, where plant and control parameters are found concurrently. In this two often used methods are implemented to find optimal energy management with component work two two often used methods are implemented implemented to find find optimal optimal energy management with with the component sizes. In the firstused method, the optimal energy management is found simultaneously optimal work often methods are to energy management with component sizes. In the first method, the optimal energy management is found simultaneously with the optimal sizes. In method, the optimal energy is simultaneously optimal design of the the first vehicle by using to minimize the sum of operationalwith and the component sizes. In the first method, the convex optimaloptimization energy management management is found found simultaneously with the optimal design of the vehicle by using convex optimization to minimize the sum of operational and component design of the vehicle by using convex optimization to minimize the sum of operational and component costs over a given driving cycle. To find the integer to variable, i.e.,the engine on-off, dynamicand programming design of the vehicle by using convex optimization minimize sum of operational component costs over a given driving cycle. To find the integer variable, i.e., engine on-off, dynamic programming costs over driving To the variable, i.e., engine dynamic programming and heuristics are used. Incycle. the second particle swarm optimization is used to find the optimal costs over aa given given driving cycle. To find findmethod, the integer integer variable, i.e., engine on-off, on-off, dynamic programming and heuristics are used. In the second method, particle swarm optimization is used to find and heuristics heuristics are used. used. In the the second method, particle swarm swarm optimization isenergy used to to find the the optimal optimal component sizing, together with dynamic programming to findoptimization the optimalis management. The and are In second method, particle used find the optimal component sizing, together with dynamic programming to find the optimal energy management. The component sizing, together with dynamic programming to find the optimal energy management. The results showsizing, that both methods converge toprogramming the same optimal design, achieving a 10.4% fuel reduction component together with dynamic to find the optimal energy management. The results show that both methods converge to the same optimal design, achieving a 10.4% fuel reduction results show that both methods converge to the same optimal design, achieving a 10.4% fuel reduction from the initial powertrain design. Additionally, it is optimal highlighted that achieving the usage aof10.4% each of thereduction method results show that both methods converge to the same design, fuel from the initial powertrain design. Additionally, it is highlighted that the usage of each of the method from the initial design. it that of of poses challenges, such as computational time (where convex optimization outperforms swarm from the initial powertrain powertrain design. Additionally, Additionally, it is is highlighted highlighted that the the usage usage of each each particle of the the method method poses challenges, such as computational time (where convex optimization outperforms particle swarm poses challenges, challenges, such asofcomputational computational time effort (wherefor convex optimization outperforms particle swarm optimization by a such factoras 20) and the tuning the particle swarm outperforms optimization particle and the swarm ability poses time (where convex optimization optimization by aa factor of 20) and the tuning effort for the particle swarm optimization and the ability optimization by factor of 20) and the tuning effort for the particle swarm optimization and the ability to handle integer variables of convex optimization. optimization by a factor of 20) and the tuning effort for the particle swarm optimization and the ability to handle integer variables of convex optimization. to handle integer variables of convex optimization. to handle integer variables of convex optimization. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Plug-in Hybrid Electric Vehicles, Optimal Sizing, Energy Management, Convex Keywords: Plug-in Hybrid Electric Vehicles, Optimal Sizing, Energy Management, Convex Keywords: Plug-in Hybrid Electric Optimal Energy Optimization, Dynamic Programming, Particle SwarmSizing, Optimization. Keywords: Plug-in Hybrid Electric Vehicles, Vehicles, Optimal Sizing, Energy Management, Management, Convex Convex Optimization, Dynamic Programming, Particle Swarm Optimization. Optimization, Dynamic Programming, Particle Swarm Optimization. Optimization, Dynamic Programming, Particle Swarm Optimization. 1. INTRODUCTION Since the problem of sizing and control of a PHEV is non1. INTRODUCTION INTRODUCTION Since the the problem problem of of sizing sizing and and control control of of aa PHEV PHEV is is nonnon1. Since linear, mixed integer, has and several states, poses is signifi1. INTRODUCTION Since the problem ofand sizing control of this a PHEV nonlinear, mixed integer, and has several states, this poses signifilinear, mixed integer, and has several states, this poses significant challenges on the algorithms used to solve it. Researchers linear, mixed integer, and has several states, this poses signifiHybrid electric vehicles (HEV) have an electric propulsion sys- cant challenges on the algorithms used to solve it. Researchers Hybrid electric electric vehicles vehicles (HEV) (HEV) have have an an electric electric propulsion syssys- have cant challenges on solve used exhaustive search, i.e.,used the to evaluation of the cost cant on the the algorithms algorithms solve it. it. Researchers Researchers Hybrid tem, in electric additionvehicles to a conventional engine. HEVs Hybrid (HEV) havecombustion an electric propulsion propulsion sys- have have challenges used exhaustive exhaustive search, i.e., i.e.,used the to evaluation of the the cost tem, in addition to a conventional combustion engine. HEVs used search, the evaluation of function for different combinations of the design variables, to have used exhaustive search, i.e., the evaluation of the cost cost tem, in to aa conventional engine. HEVs can reduce the fuel by combustion downsizing the engine, re- function for different combinations of the design variables, tem, in addition addition to consumption conventional combustion engine. HEVs to can reduce the fuel consumption by downsizing the engine, refunction for different combinations of the design variables, to roughly estimate the shape of the cost function and choose a defunction for different combinations of the design variables, to can reduce the fuel consumption by downsizing the engine, recovering braking energy, eliminating engine idling, and having can reduce the fuel consumption by downsizing the engine, reroughly estimate the shape of the cost function and choose a decovering braking energy, eliminating engine idling, and having roughly estimate the shape of the cost function and choose a deThis is boththe sub-optimal and time inefficient, which alead roughly estimate shape of the cost function and choose decovering braking energy, eliminating engine idling, and extra power control freedom by the two power sources. Plug- sign. covering braking energy, eliminating engine idling, and having having sign. This This is both both sub-optimal sub-optimal and time inefficient, which lead lead extra power power control freedom by the the two two power sources. Plug- to sign. is and time inefficient, which theThis usage of optimization-based algorithms. Among these, sign. is both sub-optimal and time inefficient, which lead extra control freedom by power sources. Plugin hybrid electric vehicles (PHEV) are the next generation of extra power control freedom by the are twothe power Plugto the the usage usage of optimization-based optimization-based algorithms. algorithms. Among Among these, these, in hybrid hybrid electric vehicles (PHEV) nextsources. generation of evolutionary to algorithms, such as particle swarmAmong optimization to the usage of of optimization-based algorithms. these, in electric vehicles (PHEV) are the of hybrid electric vehicles that have the to generation store energy in hybrid electric vehicles (PHEV) are ability the next next generation of evolutionary evolutionary algorithms, such as as particle particle swarm optimization optimization hybrid electric vehicles that have the ability to store energy algorithms, such swarm (PSO), genetic algorithms or simulated annealing, have shown evolutionary algorithms, such as particle swarm optimization hybrid electric vehicles that have the ability to store energy from theelectric electrical grid, using largethe capacity hybrid vehicles that have abilitybatteries. to store PHEVs energy (PSO), (PSO), genetic genetic algorithms algorithms or or simulated simulated annealing, annealing, have have shown shown from the the electrical electrical grid, using using large capacity capacity batteries. PHEVs results. calculate the fuel consumption some these (PSO), geneticTo algorithms or simulated annealing, haveof shown from grid, large batteries. PHEVs may drive short trips entirely on stored electrical energy, thus good from the electrical grid, using large capacity batteries. PHEVs good results. To calculate the fuel consumption some of these may drive short trips entirely on stored electrical energy, thus good results. To calculate the fuel consumption some of these methods use rule-based control strategies e.g.,Gao et al. [2007]. good results. To calculate the fuel consumption some of these may drive short trips entirely on stored energy, thus decreasing the vehicle’s dependency onelectrical petroleum. However, may drive short trips entirely on stored electrical energy, thus methods use rule-based control strategies e.g.,Gao et al. [2007]. decreasing the vehicle’s dependency on petroleum. However, methods use rule-based control strategies e.g.,Gao et al. [2007]. Besides the heuristic algorithms, dynamic programming methods use rule-based control strategies e.g.,Gao et al. [2007]. decreasing the dependency on petroleum. However, the main challenge with PHEVs is their cost which moti- Besides the heuristic algorithms, dynamic programming (DP) decreasing the vehicle’s vehicle’s dependency onhigh petroleum. However, (DP) the main main challenge challenge with PHEVs PHEVs is their their high cost which which moti- has Besides heuristic dynamic (DP) beenthe used together with programming evolutionary algoBesides theextensively heuristic algorithms, algorithms, dynamic programming (DP) the with is high cost motivates the study of optimal design is fortheir these vehicles. the main challenge with PHEVs high cost which motihas been extensively used together with evolutionary algovates the the study study of of optimal optimal design design for for these these vehicles. vehicles. has been extensively used together with algorithms, e.g., by Li et al. [2012], Ebbesen etevolutionary al. [2012], Ravey has been extensively used together with evolutionary algovates vatestotal the study optimal design for these vehicles. rithms, e.g., e.g., by by Li et et al. [2012], [2012], Ebbesen et et al. [2012], [2012], Ravey rithms, The cost ofofownership of a PHEV depends directly on the et al. [2012]. e.g., by Li Li et al. al. [2012], Ebbesen Ebbesen et al. al. [2012], Ravey Ravey The total total cost cost of of ownership ownership of of aa PHEV PHEV depends depends directly directly on on the the rithms, et al. [2012]. The et al. [2012]. size of thecost powertrain components. Moreover, energy manageThe total of ownership of a PHEV depends directly on the et al. [2012]. size of of the the powertrain powertrain components. components. Moreover, Moreover, energy energy managemanage- Evolutionary algorithms require parameter tuning, large comsize ment, is the control strategy Moreover, that determines power Evolutionary algorithms require parameter tuning, large comsize ofwhich the powertrain components. energythe management, which is the control strategy that determines the power putation Evolutionary require large comtimesalgorithms and have no proofparameter of globaltuning, optimality require large(Silvas comment, which is the control strategy that determines split between engine and additional energy sourcethe at power every Evolutionary ment, which is the control strategy that determines the putation times timesalgorithms and have have no no proofparameter of global globaltuning, optimality (Silvas split between between engine engine and and additional additional energy energy source source at at power every et putation and proof of optimality (Silvas al. [2014]). As an alternative to evolutionary algorithms, putation times and have no proof of global optimality (Silvas split every time instant, affects To exclude influence on et al. [2014]). As an alternative to evolutionary algorithms, split enginethe anddesign. additional energy its source at every time between instant, affects affects the design. To exclude exclude its influence on the et al. [2014]). As an reformulated alternative evolutionary algorithms, problem a convex optimization [2014]).can Asbe alternative to toas algorithms, time instant, design. To its influence component sizing, thethe control strategy should ideally be part on of et time instant, affects the design. To exclude its influence on theal.problem problem can bean reformulated reformulated asevolutionary a convex convex optimization optimization component sizing, the control strategy should ideally be part of the can be as a problem as shown by Murgovski et al. [2011], Egardt et al. the problem can be reformulated as a convex optimization component sizing, the control strategy should ideally be part of the optimal design process (Sundstr¨ o m et al. [2008]). Hence, component the control strategy should ideally be Hence, part of problem problem as as shown shown by by Murgovski Murgovski et et al. al. [2011], [2011], Egardt Egardt et et al. al. the optimal optimal sizing, design process (Sundstr¨ o m et al. [2008]). [2014], Pourabdollah et al. [2013]. Both component sizes problem as shown by Murgovski et al. [2011], Egardt al. process et Hence, the optimal problemdesign of optimizing the totaloom cost should be [2014], Pourabdollah et al. [2013]. Both component sizesetand design process (Sundstr¨ (Sundstr¨ mvehicle et al. al. [2008]). [2008]). Hence, and the problem of optimizing the total vehicle cost should be [2014], Pourabdollah et al. [2013]. Both component sizes and the complete control trajectory of the continuous variables can [2014], Pourabdollah et al. [2013]. Both component sizes and the problem of optimizing the total vehicle cost should be approached optimization of both the problemby optimizing the totalenergy vehiclemanagement cost shouldand be the the complete complete control control trajectory trajectory of of the the continuous variables variables can can approached byofoptimization optimization of both both energy management and be included optimization variables. Convex problems have completeas control trajectory of the continuous continuous variables canaa approached by of energy component sizes. The general procedure of management sizing PHEVsand is the approached by optimization of both energy management and be included as optimization variables. Convex problems have component sizes. sizes. The The general general procedure procedure of of sizing sizing PHEVs PHEVs is is unique be included as optimization problems have a optimum and can be variables. solved fastConvex and reliably. However, be included as optimization variables. Convex problems have component performed by optimizing total cost of vehicle ownership for a component sizes. The general procedure of sizing PHEVs isa unique unique optimum optimum and and can can be be solved solved fast fast and and reliably. reliably. However, However,a performed by optimizing total cost of vehicle ownership for the drawback with convex optimization is that integer variables, unique optimum and can be solved fast and reliably. However, performed by total of ownership for set of driving cycles. To reflect lifetime driving and charging performed by optimizing optimizing total cost cost of vehicle vehicle ownership for aa the the drawback drawback with with convex convex optimization optimization is is that that integer integer variables, variables, set of of driving driving cycles. To To reflect reflect lifetime driving and charging charging engine on-off or gearoptimization selection, cannot included in the the drawback with convex is thatbe integer variables, set lifetime and behaviors of a cycles. driver, long driving cyclesdriving are needed to repre- e.g., set of driving cycles. To reflect lifetime driving and charging e.g., engine on-off or gear selection, cannot be included in the the behaviors of of aa driver, driver, long long driving driving cycles cycles are are needed needed to to reprerepre- problem e.g., engine on-off or gear selection, cannot be included in and therefore, should be given as an input a-priori to e.g., engine on-off or gear selection, cannot be included in the behaviors sent different driving situations. behaviors of a driver, long driving cycles are needed to repreproblem and therefore, should be given as an input a-priori to sent different different driving driving situations. situations. problem and therefore, should be given as an input a-priori to problem and therefore, should be given as an input a-priori to sent sent different driving situations. Copyright 2015 IFAC 16 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © 2015, IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 16 Copyright 2015 IFAC 16 Peer review© of International Federation of Automatic Copyright ©under 2015 responsibility IFAC 16 Control. 10.1016/j.ifacol.2015.10.003

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and EGU. The size of the electric motor is directly decided by the power demand from the driving cycle, and hence is not included in the problem. The control input variable u is defined as: u = [ud uc ], (2) where uc includes the continuous control inputs of length N , e.g., engine or motor torque, and ud includes the integer control inputs of size N , e.g., gear or engine on-off, and N is the number of time samples of the driving cycle. These variables are explained in more detail at the end of this section. The constraints in the problem are in the form of powertrain and component models, which are introduced in Section 2.2.

the optimization problem (Murgovski et al. [2011]). The value can be found either by heuristics e.g., as in Pourabdollah et al. [2013], or by iterative strategies as in [Murgovski et al., 2014], where the results are compared with DP. In this paper, convex optimization is used and compared with a particle swarm optimization for finding optimal design of a series PHEV. For the convex optimization method, engine onoff is found by two methods. First, the engine on-off decision for convex optimization is found based on a simple rule. As an alternative, convex optimization and dynamic programming are combined in an iterative manner to update the engine onoff decisions, and the iterations are performed until the cost converges. Particle swarm optimization (PSO) also finds the optimal design in a nested way with DP. This method, used by Ebbesen et al. [2012], Nuesch et al. [2012], Silvas et al. [2014], searches for the powertrain component sizes using PSO and for any fixed powertrain sizes, the optimal energy management is computed by DP.

2.1 Driving cycle and performance requirements In the optimization problem, we try to find the optimal component sizes and the energy management variables over a given driving cycle. Therefore, designing a vehicle requires knowledge about the lifetime driving of the vehicle. However, since it is impossible to predict the precise driving cycle and computational resources are limited, we use a long driving cycle that represents real-life driving [Kullingsj¨o et al., 2012].

The rest of this paper is organized as follows. An overall picture of the optimization problem, the driving cycle and the model of the powertrain and its components are presented in Section 2. A brief explanation of the optimization methods, convex optimization, dynamic programming, and particle swarm optimization are provided in Section 3. Illustrative results from the study are shown in Section 4. Finally conclusions are drawn in Section 5.

The long driving cycle used in the optimization can reflect real-life driving, but might not include extreme situations that require high performance. Acceleration requirement is considered as an important vehicle attribute by many drivers, and is hence added in the constraints. Acceleration as a function of speed on a flat road is used to make a so called performance cycle, which is then appended to the driving cycle. The performance cycle includes speeds from zero to maximum speed, increasing according to the desired accelerations as explained by Pourabdollah et al. [2013]. The driving cycle used in the simulations is shown in Section 4. We assume that the battery has the possibility to be charged with constant power from the grid at charging occasions overnight, when the car is parked for 8 hours.

2. PROBLEM FORMULATION AND MODELING In this section, the problem formulation and modeling details are introduced. The studied PHEV, depicted in Fig. 1, is a series powertrain, where only the electric motor (EM) is mechanically linked to the drivetrain and can propel the wheels. Pg -

+ Battery Pbat

Pf

Engine

Generator

PEGU PEM,el

Pdem Electric Motor

17

2.2 Modeling

PEM

In this section the models of the powertrain and its components are presented. The same models are used both for DP and convex optimization. A part of the problem in (1) fulfills convexity requirements, as it will be shown in the following. A convex function satisfies f (λx + (1 − λ)y) ≤ λf (x) + (1 − λ)f (y) for all x, y ∈ Rn with 0 ≤ λ ≤ 1 [Boyd and Vandenberghe, 2009]. Powertrain components that are being described by quasi-static functions are approximated with nonlinear convex functions, together with some necessary variable changes. The accuracy of these approximations is acceptable and discussed in detail by Murgovski et al. [2011].

Pbrk

Fig. 1. Series PHEV powertrain configuration (solid lines: mechanical link, and dashed lines: electrical links). The problem is formulated to find design and control variables that minimize an objective function under the presence of constraints. Here, the objective function is a weighted sum of operational costs over the driving cycle, Jop , and component costs, Jcomp . The operational cost includes the consumed fossil fuel and electrical energy, and the components cost is the sum of the costs of battery and engine-generator unit (EGU). The problem can be stated as: (1) min J = Jop (u, s) + Jcomp (s)

Powertrain: Starting from the driving cycle that is fully described by the velocity v(t) and acceleration a(t) (and, in this particular case, a zero road slope) at discrete time instants, the required traction force, Ft , and power demand, Pdem , are calculated from the driving cycle as

u,s

subject to: x(t + 1) = f (x(t), u(t), s) x ∈ X, u ∈ U, s ∈ S, where x is the state variable vector, e.g., the battery state of charge; and s is the scaling factor for components, i.e., battery

2

cd Af ρv(t) + mtot gcr + mtot a(t), 2 Pdem (t) = Ft (t)v(t),

Ft (t) =

17

(3)

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power losses are approximated by a second-order polynomial in torque as PEM,loss (t) = a1 (t)PEM (t)2 + a2 (t)PEM (t) + a3 (t), (9) PEM,el (t) = PEM (t) + PEM,loss (t), (10) 2 = a1 (t)PEM (t) + a2 (t)PEM (t) + a3 (t), (11) where the coefficients a1 ≥ 0, a2 = a2 + 1 and a3 are functions of EM speed, ωEM , and hence time. The values are calculated from data using least squares method for a number of grid points of ωEM . For the speed values not belonging to the grid nodes, the coefficients are obtained by linear interpolation (Murgovski et al. [2011]).

Table 1. Vehicle parameters parameter

value

Baseline mass Glider mass Frontal area Rolling resistance Aerodynamic drag coefficient Air density Wheel radius Ratio of the final gear EM reduction gear

m=1600 kg mg =1280 kg Af =2.37 m2 cr =0.009 cd =0.33 ρ= 1.293 kg/m3 rw =0.3 m rf g =4.2 rEM =2

where cd , Af , mtot , ρ, g, cr , and a are air drag coefficient, frontal area, total vehicle mass, air density, gravitational acceleration, rolling resistance coefficient, and acceleration, respectively, with values given in Table 1. The total vehicle mass, mtot , is the sum of the masses of the glider, battery, and EGU.

Engine-generator unit: The fuel power, Pf,base , of a baseline EGU is a function of the generator power and is approximated with a second-order polynomial in PEGU as 2 + b2 PEGU,base + b3 (12) Pf,base (PEGU,base ) = b1 PEGU,base The powertrain model is described by mechanical power bal- where bj ≥ 0; j ∈ {0, 1, 2} [Murgovski et al., 2011]. The model of the scaled EGU is obtained by applying linear relation ance equation, given as to the fuel and electric power of the baseline EGU, i.e. Pf = (4) sEGU Pf,base and PEGU = sEGU PEGU,base . The fuel power Pdem (t) + Pbrk (t) = PEM (t), of the scaled EGU then becomes where Pbrk and PEM are the power dissipated at friction P 2 (t) Pf (t) = b1 EGU + b2 PEGU (t) + eon (t)b3 sEGU , (13) brakes and EM mechanical power. The electrical power balance sEGU equations during driving times, Tdrive , and charging times, where eon is a binary signal that is introduced to remove the Tcharge , are given as idling losses b3 sEGU , when the EGU is off. To preserve the , PEM,el (t) + Paux (t) = PEGU (t) + Pb,int (t) − Pb,loss (t), t ∈ Tdrive problem convexity, eon is decided prior to the optimization. (5) Pb,int (t) − Pb,loss (t) = −Pg (t)ηg , k ∈ Tcharge , The EGU mass scales linearly with the mass of the baseline where PEM,el , PEGU , Pb,int , Pb,loss , Pg , and ηg are the EM EGU with 100 kW power , i.e. m EGU = mEGU,1 sEGU , where electrical power, the EGU power, the battery internal power m EGU,1 is a linear weight coefficient. (before the losses), the battery loss power, grid power, and grid efficiency. The electrical power used by auxiliary devices, 2.3 Problem Formulation Paux , is estimated by a constant value of 500 W. For simplicity, the losses in the power electronics are included in the EM losses The problem introduced briefly in (1) is now given in more and the rotational inertia (including the inertia of the wheels, the detail. The operational cost Jop includes the consumed fossil differential, the EM and the EGU) is neglected in the models. fuel and electrical energy as N  Battery: The battery consists of sb identical cells, each modwf Pf (t)h(t) + we Pg (t)h(t), (14) Jop = eled as a constant open circuit voltage Voc in series with a t=1 constant internal resistance, R. The power losses and the stored ρf ρel where wf = ρLHV , we = 1000·3600 , and ρLHV is the lower energy of the battery, Eb , are calculated as heating value of gasoline. The fuel power, Pf , and charger RPb,int (t)2 power, Pg , are converted to an equivalent cost in EUR usPb,loss (t) = , (6) 2 sb Voc ing energy prices ρf for gasoline and ρel for electricity. The Eb (t + 1) = Eb (t) − h(t) Pb,int (t). (7) sampling interval h(t) is equal to 1 s at the driving instances The battery internal power is positive when discharging. Dur- and at charging instances is equal to the entire charging period ing the available parking periods the vehicle is charged with [Pourabdollah et al., 2013]. constant current and power. Then, without loss of generality, The component cost, Jcomp (s), is the sum of the costs of battery the charging energy can be modeled as if entering the battery and EGU. The remaining cost of the vehicle is independent in one extra long sample at the parking occasions. The battery of sizing and can be therefore excluded from the problem. cells have capacity of Q = 159W h, power of 880W , and mass The components cost is calculated as the depreciation over the of mb,1 , giving the total pack mass as mb = mb,1 sb . driving cycle, i.e., the proportion of the components cost given by the ratio between the length of the cycle, d, and the lifetime Electric Machine: The losses of the EM and power elec- driving distance of the vehicle. If payment is equally divided tronics are gathered in a power loss map, PEM,loss , where the in vehicle lifetime with yearly interest rate of p = 5%, the c losses are measured at steady-state for different torque-speed components cost is given by   combinations. At each time instant on the driving cycle, the EM yv + 1 d angular speed, ωEM , is calculated in advance as Jcomp = (costb + costEGU ), (15) 1 + pc d y yv 2 rf g ωEM (t) = rEM v(t) , (8) where yv is the vehicle lifetime, and dy is the average traveled rw distance of the vehicle in one year. For each component, the where rw , rf g , and rEM are the wheel radius, ratio of the final cost model is a linear function costb = costj,0 + costj,1 sj ∀ j ∈ {b, EGU }, gear (differential), and EM reduction gear respectively. The (16) 18

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19

introduced in Section 2.2. The complete problem formulation is given as

where costj,0 is the initial cost and costj,1 is the linear cost coefficients. The state variable, x, in (1) is the energy in the battery given in (7). The decision variables firstly include, s, the dimensionless component sizes for battery and EGU. The continuous input variables, uc , which are related to the energy management, are determined for every time instant. These variables are EGU power, PEGU (t), battery internal power, Pb,int (t), battery state of energy, Eb (t), grid power, Pg (t), and braking power, Pbrk (t). The integer input variable, ud , consists of the binary engine on-off variable, eon (t).

min wf

N  t=1

Pf (t)h(t) + we

N 

Pg (t)h(t) + Jcomp

(18a)

t=1

variables: PEGU , Pb,int , Eb , Pg , Pbrk , sb , sEGU subject to: (18b) Pdem (t) + Pbrk (t) = PEM (t) PEM,el (t) + Paux (t) = PEGU (t) + Pb,int (t) − Pb,loss (t) (18c) Pg (t)ηg = −Pb,int (t) + Pb,loss (t), t ∈ Tcharge (18d) RPb,int (t)2 (18e) Pb,loss (t) ≥ 2 sb Voc PEM,el (t) ≥ a1 (t)PEM (t)2 + a2 (t)PEM (t) + a3 (t) (18f) (18g) Eb (t + 1) = Eb (t) − h(t)Pb,int (t) PEGU (t)2 + b2 PEGU (t) + eon (t)b3 sEGU (18h) Pf (t) = b1 sEGU Eb (t) ∈ sb [Eb,min , Eb,max ] Pb,int (t) ∈ sb [Pb,int,min , Pb,int,max ] Pg (t) ∈ [0, Pg,max (t)] PEGU (t) ∈ sEGU [0, PEGU,max,base ]

As mentioned earlier, the total vehicle mass in (3) is computed as the sum of the masses of the glider, m0 , battery, and EGU, mtot = m0 + mb,1 sb + mEGU,1 sEGU . (17) Here, we distinguish the concept of a baseline vehicle that has predefined values for the scaling factors sb and sEGU . The demanded power of the baseline vehicle is hereafter denoted by Pdem,base . 3. OPTIMIZATION METHODS The optimization methods used in this paper, namely dynamic programming, convex optimization and particle swarm optimization are explained in this section.

for t ∈ Tdrive unless is defined otherwise.

In order to guarantee the problem convexity, the equality signs in (18e) and (18f) are relaxed to inequalities. The optimal result will satisfy equality because otherwise energy is wasted which is not optimal. Moreover, the number of cells, sb , is relaxed to a real value. The relaxation will introduce a rounding error that has a small influence on the optimal result [Egardt et al., 2014, Murgovski et al., 2011]. To solve the problem, a tool, CVX [Grant and Boyd, 2010], is used to automatically translate it to a second order cone form, required by a publicly available solver, e.g. Sedumi [Strum, 2011].

3.1 Dynamic programing Dynamic programming is a method to solve optimal control problems based on the Bellman’s principle of optimality (Bellman [1957]). The dynamic programming algorithm proceeds backward in time. The problem starts with a final time cost, which is assumed to be zero for PHEVs, because, unlike HEVs, there is no constraint on the final battery state of charge. At each time instant, DP finds the optimal energy management and engine on-off that minimizes a total cost. DP has been widely used in automotive applications to find the optimal energy management which minimizes fuel consumption, since it can handle nonlinear constraints and integer variables (Lin et al. [2003], Hofman et al. [2012]). The main drawback of DP is its computational time which increases exponentially with the number of states and number of components sizes. Therefore, in order to find the optimal design of vehicles, DP is often used with other optimization methods. 3.2 Convex optimization

Integer variables in convex optimization problem As mentioned earlier, integer variables cannot be included in the convex optimization problem. Therefore the decision on engine on-off is found a-priori to the problem. The problem has been studied by different researchers. Elbert et al. [2014] derived the globally optimal engine on-off analytically, whereas Murgovski et al. [2014] used a heuristic method based on pontryagin’s maximum principle. However, these approaches do not include sizing of the engine.

Convex optimization is also used to solve the problem of finding the optimal design and energy management [Egardt et al., 2014, Murgovski et al., 2011, Pourabdollah et al., 2013]. The powertrain and component models, in addition to the cost and weight models, are formulated as convex to solve a convex problem. Modeling of the powertrain and its components to guarantee the convexity is the main step of the optimization method. Once the problem is defined as a convex optimization problem, it can be solved in a relatively short time, using efficient solvers. Integer variables such as engine on-off cannot be included in the problem and hence need to be decided a-priori to the convex optimization to preserve convexity. The constraints in the convex problem include the physical limits of the components in addition to the powertrain and component models,

Heuristics have been used also by Pourabdollah et al. [2013] to find the integer variables in the problem of component sizing. The engine is turned on if at a certain time instant the power demand is higher than a power threshold, Pon , and is turned off otherwise. The optimization is then repeated over different values of Pon to find the threshold that minimizes the cost. An alternative way to find engine on-off decision is by using convex optimization and DP methods alternately. In order to do this, the optimization starts with DP with initial component sizes. DP finds the optimal energy management including engine on-off for the given component sizes. The engine on-off decision is then given to the convex optimization to find the new component sizes. This iteration is continued until the cost and component sizes converge. The procedure is illustrated in the right part of in Fig. 2. 19

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Sizing Optimization

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based on the baseline power demand required by the vehicle when following the driving cycle, Pdem,base (t). At every time instant, if the power demand is higher than a power threshold, Pon , the engine is turned on and is turned off otherwise. The convex optimization problem is iterated over several values of Pon to find the best result. The result of battery size and total cost for different values of Pon are shown in Fig. 3. The optimal power threshold to turn the engine on is equal to 16kW which gives total cost as e 20.81, battery size equal to 8.07 kWh (sb = 50.64) and EGU size of 66.87 kW (sEGU = 0.69).

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Fig. 2. Bi-level optimization frameworks compared for component sizing and control: (Method 1/right) the proposed combined convex optimization and DP, and (Method 2/left) the combination of Particle Swarm Optimization and DP.

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3.3 Particle swarm optimization Particle swarm optimization (PSO) is a stochastic search method inspired by the coordinated motion of animals living in groups, originally developed by Kennedy and Eberhart [1995]. Using a population of candidate solutions, called particles, in the design space, called swarm, PSO optimizes the problem by improving the candidate solution with regard to a given quality measure. To begin with, the method initializes the swarm randomly in the design space, with each particle having assigned a position, χ, and a velocity, υ. Then, the objective function is evaluated for each particle and a global best is determined, G. Iteratively the velocity, υi , and position, χi , of each particle are changing towards their own best, Bi , and globally best location by

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Fig. 3. Number of battery cells and the total cost obtained by convex optimization using different power threshold to turn engine on. The convex optimization is then combined with DP in an iterative manner to optimize the engine on-off as shown in Fig.2. As mentioned, the iteration starts with DP, with initial component sizes. These initial inputs can have some impact on the final results ending in a local optimum. For example, if the initial battery size is much larger than the optimal value, in the first iteration of DP, the vehicle is propelled most of the time by the cheap energy from the battery and the engine is turned off in order to reduce the losses. Giving this engine onoff to the convex optimization, the battery may stay oversized since the convex optimization is not able to alter the engine onoff decision. In this way, the iterative optimization never gets a chance to converge to a smaller battery size. To avoid this problem, a small battery with sb = 13, or size equal to 2 kWh, is chosen as the initial value.

υik+1 = φk υik + α1 [β1,i (Bi − χki )] + α2 [β2,i (G − χki )], (19) = χki + υik+1 , χk+1 i with χki and υik the current position and velocity of particle i in generation number k, and φ the particle inertia which causes a certain momentum of the particles. Parameters β1,2 ∈ [0, 1] are uniformly distributed random values and α1,2 are acceleration constants. In nested combinations with other algorithms, PSO has proven as a good candidate in design of HEVs, as shown by Williamson et al. [2005], Gao et al. [2007], Sundstr¨om [2009], Nuesch et al. [2012], Silvas et al. [2014]. Motivated by these studies, we will combine PSO with DP, as depicted in Fig. 2 and use this bi-level design method as a benchmark comparison.

In this section, the results of the simultaneous optimization of energy management and component sizing are given. The optimization is performed over near 3 hours/176 km long real life driving cycle, followed by a performance cycle. There are also 4 occasions where the car has the possibility to charge the battery with constant grid power for 8 hours. As mentioned, the size of the EM is decided by the maximum power demand of the driving cycle, which is equal to 108.5 kW.

The procedure of using convex optimization and DP is continued until the cost and component sizes converge. In Fig. 4 the results of 12 iterations are given. As we can see, the optimal sizes of the engine-generator unit and the battery converge after the first iteration. The cost however, decreases in the next iteration, where DP finds the optimal energy management for the given sizes. The total cost decreases from e 22.93 in the first iteration to e 20.54 in the last. The optimal battery size is equal to 8.2 kWh (sb = 51.42) and EGU size is equal to 66.17 kW (sEGU = 0.66). The driving cycle used in the optimization together with the result of the optimal state of charge from the last iteration are shown in Fig. 5.

In the first method, convex optimization is used to find the optimal design. A simple rule is used to find the engine on-off

In the second method, and for comparison, we solve the same sizing and control problem using PSO for finding component

4. RESULTS

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Speed profile

In this paper, we address the problem of finding the optimal powertrain component sizing and control algorithm of a PHEV, with respect to hybridization and fuel costs, using a real-life measured driving cycle. To solve this problem, we use convex optimization together with heuristics or dynamic programming to find the engine on-off decisions. The results are compared with another existing method, i.e., PSO/DP. As shown by results, the convex optimization based method proves to be faster alternative to current methods that can find the global unique solution. Using simple rules to find the engine on-off gives results close to optimal, whereas DP can improve the result with a very high accuracy, with a relatively short time compared to an evolutionary based algorithm, PSO. Moreover, extra states, such as battery state of health or engine thermal state, or design variables, such as electric motor scaling, can simply be added to convex optimization problem without much increase in computational burden. However, the computational time of DP or evolutionary methods explodes exponentially by increasing the number of states or design parameters.

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Fig. 5. The optimal SoC (top) and the speed profile of the driving cycle (bottom). The charging occasions are denoted by black dots. sizing and DP for finding the optimal control as shown in Fig.2. The PSO algorithm is simulated for different particles number, p ∈ [5, 40], and generations number, g ∈ [5, 10]. It is required to have at least 30 particles for the algorithm to reach the global optimum (see Fig. 6), or 10 particles to reach a ’close to’ optimum value (within 0.5% of the global optimum), while 8 generations are required in average for the algorithm to converge to a solution. The optimal battery size is equal to 8.2 kWh (sb = 51.44), EGU size is equal to 66.17 kW (sEGU = 0.66) and the total cost is 20.54 euros.

In this work we have investigated a series configuration, on one driving cycle. Future work can include more studies, including different configurations, larger set of driving cycles, different battery technologies, and pricing scenarios. Although both methods used in this study have converged to the same results, there is no general proof that the global minimum is reached. Further studies is needed for a theoretical basis to proof the convergence of the algorithms used.

4.1 Computational Efficiency As shown in the results, both convex optimization/DP and PSO/DP methods find the same optimized solution. However, the computational time of the two methods differ. For the 167 minutes long driving cycle, the evaluation of each DP simulation takes 250 seconds and each convex optimization

Simulations for method 1 were performed on a PC with Intel core 2 processor, at 2.67 GHz and 8 GB memory, and for method 2 on a similar PC, with Intel i7 processor at 2.2 GHz and 8 GB of memory. 1

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6. ACKNOWLEDGMENT This project is financially supported by the Swedish Hybrid Center.

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