Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV

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

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Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV Mitra Pourabdollah n, Bo Egardt, Nikolce Murgovski, Anders Grauers Chalmers University of Technology, Gothenburg, Sweden

art ic l e i nf o

a b s t r a c t

Article history: Received 12 April 2015 Received in revised form 7 February 2016 Accepted 9 February 2016

In this paper, the problem of optimal sizing of a series PHEV is studied by formulating a convex program that minimizes the sum of operational and component costs. The solution gives the optimal sizes of the main powertrain components, simultaneously with the vehicle's optimal energy management. Investigations are performed on driving cycles generated stochastically from real data using Markov chains, with different driving distance distributions and charging patterns. The results show that the optimal component sizing is affected more from the driving distances between charging opportunities, than the speed profile of the driving. With anticipated future battery and petroleum prices, larger battery sizes are obtained. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Hybrid electric vehicles Optimal sizing Driving cycles Convex optimization Markov models

1. Introduction Electrification of vehicles offers potential environmental benefits by reduction of emissions and fuel consumption. One of the most competitive concepts introduced to the market is the hybrid electric vehicle (HEV). HEVs can reduce fuel consumption and emissions due to the ability of recuperating braking energy, eliminating engine idling, and choosing more efficient operating points of the engine, without sacrificing performances. The next generation of HEVs are plug-in hybrid electric vehicles (PHEVs) that have all the merits of HEVs and, in addition, can partly run on energy charged from the electrical grid, thereby reducing the dependency on petroleum. However, the extent to which this can be achieved depends highly on the sizes of the main components, as well as the driving and charging behavior of the vehicle owner. The fuel efficiency, performance, and total cost of ownership can be improved if PHEVs are optimized to match the lifetime driving behaviors of drivers, including both the speed profile and the distance distribution of the trips (Hung & Wu, 2015). While the effect of single trips on fuel efficiency and performance has been studied in detail in many papers such as (Eren & Gorgun, 2015; Fontaras, Pistikopoulos, & Samaras, 2008; Gonder, Markel, Thornton, & Simpson, 2007; Kwon, Kim, Fallas, Pagerit, & n

Corresponding author. E-mail addresses: [email protected] (M. Pourabdollah), [email protected] (B. Egardt), [email protected] (N. Murgovski), [email protected] (A. Grauers).

Rousseau, 2008; Moawad, Singh, Hagspiel, Fellah, & Rousseau, 2009; Patil, Adornato, & Filipi, 2009; Rahman, Butler, & Ehsani, 1999), the effect of distance distribution of the trips has only been briefly mentioned in Markel et al. (2006), Frank (2007), Patil, Adornato, and Filipi (2010), Gao and Ehsani (2010), and Smith, Earleywine, Wood, Neubauer, and Pesaran (2012). However, in most of these studies, the distribution of driving distance has been estimated per day and over the population of many drivers instead of single drivers over their lifetime. Because of differences in movement patterns of different drivers, there will be a difference in the opportunity in utilizing a PHEV (Kullingsjö & Karlsson, 2012). The conclusion is that, in order to get the maximal benefits from a PHEV, the vehicle needs to be designed optimally with respect to the lifetime driving behavior. Another aspect of PHEVs that makes the optimization problem of PHEVs intriguing is that sizing is affected by the energy management, i.e., the control strategy that determines how the power/ energy should be split between battery and engine or enginegenerator unit at every time instant (Moura, Callawayb, Fathya, & Steina, 2010). Rule based energy management is often used with iterative methods to find the optimal component sizes as in Liu, Wu, and Duan (2007), Wu, Wang, Yuan, and Chen (2011), Wu, Cao, Li, Xu, and Ren (2011), Sorrentino, Pianese, and Maiorino (2013), and Lee, Kim, Jeong, Park, and Cha (2013). However, these algorithms do not guarantee that the solution is globally optimal. Moreover, the design results obtained from such optimization are biased toward the specific rules used in the algorithm, making it hard to compare different designs. To alleviate this problem,

http://dx.doi.org/10.1016/j.conengprac.2016.02.005 0967-0661/& 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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Nomenclature Af Ebat Ft Pg Pbat Pbat , int Pbat , loss PEGU PEM , loss Paux Pbrk PEM Pdem Pon Pf REM Rfg a cd cr d

the frontal area the stored energy of the battery the required traction force the grid power the battery terminal power the battery internal power the battery loss power the engine-generator unit power the EM power losses the electrical power used by auxiliary devices the power dissipated at the friction brakes the power of the electric motor the demanded power at the wheels the power threshold to turn the engine on fuel power of the EGU EM reduction gear ratio of the final gear the vehicle's acceleration the air drag coefficient the vehicle's expected age at the end of life the length of the cycle

dynamic programming (DP) has been used to find the optimal energy management. To find the optimal component sizes, DP is used either in iterations as in Sundström, Guzzella, and Soltic (2008), or together with an evolutionary algorithm as in Filipi et al. (2004), Ravey, Roche, Blunier, and Miraoui (2012), Zou, Li, and Hu (2012), Zou, Sun, Hu, Guzzella, and Peng (2012), and Masih-Tehrani, Hairi-Yazdi, Esfahanian, and Safaei (2013). The main problem with DP is that its computation time grows exponentially with number of states and design variables. Even though the problem is solved off-line on very fast processors, time is still an issue since the problem of sizing needs to be solved for several long driving cycles, assuming different future prices of components and energy, and different components and configurations. An alternative way to optimize both component sizing and energy management is to formulate and solve a convex optimization problem. The problem was first introduced in Murgovski, Johannesson, Sjöberg, and Egardt (2012) and was later extended in e.g. Pourabdollah, Murgovski, Grauers, and Egardt (2013), Egardt, Murgovski, Pourabdollah, and Johannesson (2013), Hu, Murgovski, Johannesson, and Egardt (2013), and Hu, Johannesson, Murgovski, and Egardt (2015). This method finds the optimization variables, i.e. optimal sizes and energy management, simultaneously based on a convex optimization formulation. Convexity has two important implications: (1) the problem has a unique optimal solution, and (2) fast and reliable solvers are available to solve the problem. This paper extends the results presented in Pourabdollah, Grauers, and Egardt (2013) by focusing on the effect of driving behaviors and pricing scenarios on the optimal component sizes of a series PHEV. The cost function being minimized in this study comprises two parts. The first part reflects the cost of the key components of the vehicle, which for a series PHEV are battery, electric motor (EM), and engine-generator unit (EGU). The second part of the cost function is the operational cost of fuel and electricity. The constraints in the convex optimization problem are given by equations governing the power flow in the components and system, and the maximum component ratings. The decision variables are the component sizes (battery, electric motor, and engine-generator unit) and the optimal energy management at

dy eon g h mtot pc pc rw sbat sEM sEGU v w yv costcomp costbat costEM costEGU

ηg λ ρf ρel ρLHV

the lifetime driving distance of the vehicle the engine on-off signal the gravitational acceleration the time varying sampling interval the vehicle's total mass the yearly interest rate the air density wheel radius the scaling factor of the battery the scaling factor of the electric motor the scaling factor of the engine-generator unit the vehicle's speed the optimization weighting factor the vehicle's expected age at the end of life the total cost of components the cost of the battery the cost of the electric motor the cost of the engine-generator unit the efficiency of the grid the scale parameters for a Weibull curve energy price for diesel energy price for electricity lower heating value of fuel

every time instant. The optimization is performed over pre-specified driving cycles made of several stochastic trips. A trip is defined by the driving interval between two consecutive charging events at which the driver has the opportunity to charge the battery. A novel method is used to generate trips representing different driving distance distributions: a Markov process, trained from real driving data, is used to generate velocity trajectories, and driving distances between charging events are drawn from a Weibull distribution. The corresponding optimal sizes of the components for different driving cycles are found for different battery and fuel prices. The paper is organized as follows. In Section 2, an overall picture of the optimization problem, the driving cycles, and the performance requirements used in the problem are given. The powertrain and component models are introduced in Section 3. Then the mathematical formulation of the convex optimization problem is given in more detail. Illustrative results from the study are shown in Section 4. Finally conclusions are drawn.

2. Problem formulation The studied PHEV, depicted in Fig. 1, is a powertrain with series topology, where only the electric motor (EM) is mechanically linked to the drive-train and propels the wheels. The enginePg -

+ Battery Pbat

Engine Pf

Generator

PEGU PEM,el

Pdem Electric Motor

PEM Pbrk

Fig. 1. Series PHEV configuration (solid lines are mechanical link and dashed lines are electrical links).

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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lifetime driving of the vehicle is required (Kwon et al., 2008). However, not only is it impossible to predict the precise lifetime driving of a vehicle, but there is also the problem with limited computational resources. Therefore, shorter driving cycles are needed, still reflecting the lifetime behavior of a driver. As mentioned earlier, the design of a PHEV is influenced by both the driving distance distributions and the speed profile of the trips. In the next sections, the method used to generate driving cycles is presented considering these characteristics.

Fig. 2. Framework of the optimal PHEV design procedure.

generator unit (EGU) can directly feed the electric motor or charge the battery. The optimization problem will be described briefly in this section, however, the mathematical details are deferred to Section 3, after having introduced the component models. The framework of the optimal PHEV design procedure is shown in Fig. 2. For a vehicle model and a given driving cycle the engine on– off variables are decided based on some heuristics. The decisions together with the cost models, driving cycle and performance requirements are given as an input to solve a convex problem which finds the optimal component sizes and the energy management. The objective function which is minimized in the convex optimization problem is a weighted sum of the operational and component costs as:

cost = costop + w · costcomp.

(1)

The operational cost is determined by the consumed fossil fuel and electrical energy over the driving cycle. The component cost is the sum of the costs of battery, electric motor, and engine-generator unit. The remaining costs of the vehicle are assumed to be independent of sizing and are therefore excluded from the problem. The coefficient, w, is used to calculate the depreciation of the component cost over the investigated driving cycle and is given by the ratio between the length of the cycle, d, and the lifetime driving distance of the vehicle. If payment is equally divided in vehicle's lifetime with yearly interest rate of pc = 5% , the weighting factor w is calculated as

w=

y + d ⎛ ⎜ 1 + pc v dy yv ⎝ 2

1⎞ ⎟, ⎠

2.1.1. Trips and driving distance distribution One important factor that significantly influences the optimal design of a PHEV is the distance driven between charging opportunities, usually shown by the driving distance distribution of the driver. Data from over 200 privately driven cars provided by Test Site Sweden (Kullingsjö & Karlsson, 2012) was used to study how people drive in real life. The data includes the time series of position, velocity, and ID of the used satellite. In the data base, trips are assumed to be finished every time the engine is turned off for at least 10 s. The time during which the engine is off is assigned as the parking time between two trips. For PHEVs, however, trips are separated by charging events instead of parking events. Since the data is from conventional vehicles, information about parking time is used to separate the trips in this study. It takes about 4 h to fully charge the battery of a PHEV with a fast charger, and about 8 h with a conventional plug used at home during night. These assumptions are used to calculate the cumulative distribution function (CDF) of the trip distances. The CDF of single trips in the database, as well as aggregated trips with minimum four and 8 h (daily trips) of charging time in between, are depicted in Fig. 3 for the 200 cars. The monthly driving distances of the drivers are illustrated by the color of the curves, i.e., the darker the color is, the longer is the driving distance. It can be seen from the figure that charging time assumptions influence the driving distance distribution noticeably. This is discussed in more detail at the end of this section.

(2)

where yv is the vehicle's expected age at the end of life and dy is the average traveled distance of the vehicle in one year (Al-Alawi & Bradley, 2013). The decision variables of the optimization problem are the component scaling parameters, used to size the components, and vector decision variables related to the energy management, which are determined for every time instant of the driving cycle. The optimization variables are marked in bold in the paper. The constraints in the problem are caused by powertrain and component limitations and are introduced in Section 3. 2.1. Driving cycle The optimization finds the optimal component sizes and the energy management parameters over a given driving cycle. Therefore, to optimally design a vehicle, knowledge about the

Fig. 3. The cumulative probability densities of the single trips, trips with 4 h of charging in between, and daily trips. The monthly driving distance of the drivers are illustrated by the color of the curves, i.e., the darker the color is, the longer is the driving distance. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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average driving cycle, which is in turn proportional to λ. This case is studied in Section 4.1. On the other hand, the availability of charging stations also affects the driving distance distribution. For example, if charging is possible at work or during shopping using a fast charger, a CDF with smaller λ will be more representative. Therefore, in the second scenario the lifetime driving distance is kept constant, but the distance distribution varies between drivers (different λs) depending on the availability of charging stations. This case is studied in Section 4.2.

Fig. 4. The cumulative probability functions of trip distances for over 200 drivers (considering daily charging). Weibull cumulative distribution function for k ¼1.5 and different values of λ = [15, 30, 50, 80, 100, 120, 140] are shown with black curves, and the trip lengths in the synthesized driving cycle are shown with circles.

The distance distributions depicted in Fig. 3 are described analytically, in order to use them in a systematic way in the optimization problem. It was found that the Weibull distribution, widely used in data analysis, can be used to approximately represent the daily driving distance data, see Fig. 4. The CDF of a Weibull distribution is given by x k

F (x; λ , k ) = 1 − e−( λ ) ,

x≥0

2.1.2. Speed profile A trip includes the vehicle's velocity, v(k), possibly the road inclination, β (k ), and charging times after each trip. To include different driving styles and reduce the dependency of the sizing on one specific given driving cycle, stochastic trips are generated from a discrete time, discrete state, time invariant Markov process (Gong, Midlam-Mohler, Marano, & Rizzon, 2011; Lee, 2011; Lee & Filipi, 2010, 2011; Lee, Bareket, Gordon, & Filipi, 2012; Muratori, Moran, Serra, & Rizzoni, 2013). The states of this Markov process are speed, V, and acceleration A. The initial process states are zero; the process then moves from one state to another according to specified probabilities in a so-called transition matrix. To construct the transition matrix, the real data provided by Test Site Sweden is used. The transition matrix describes the probabilities of transitions from each possible combination of present states (vk, ak) to possible next states (vk + 1, ak + 1) in the database as

Pk (Vk + 1 = vk + 1, Ak + 1 = ak + 1|vk , ak ).

(4)

Since the style of driving can be different for different trip distances, the data can be sorted into different classes (Lee & Filipi, 2011). Here, the driving cycles are divided based on the trip distances into three classes, city (less than 5 km), mixed (5–30 km) and highway (more than 30 km). Each class results in one transition matrix. The properties of these cycles are given in Table 1. To generate a stochastic trip with a desired driving distance, the corresponding transition matrix is used.

(3)

where k and λ are shape and scale parameters respectively (Miller & Childers, 2004). In Fig. 4, trip distance distributions for over 200 drivers are shown along with driving distance distributions for seven fictitious drivers with Weibull distributions shown in black. With the shape parameter used here (k ¼1.5), a useful interpretation of λ is obtained from the formula for the median, x0.5 = λ (log 2)1/ k ≈ 0.8λ . A Weibull CDF with λ = 15 thus has a median of 12 km, indicating that the driver drives less than 12 km for half of the trips. Similarly, a Weibull CDF with λ = 140 has a median of 110 km. These numbers can be verified in Fig. 4. In the numerical examples that follow, the Weibull distributed driving distance is represented by sampling the distribution function uniformly along the y-axis, and forming a driving cycle based on the corresponding trips with different distances. The samples, usually 10, are shown by dots in Fig. 4. Another factor that directly relates driving behavior of a driver to component sizing is the lifetime driving distance. It can be seen from (2) that for a car with a shorter lifetime driving distance, the component costs have higher weight in the optimization objective, whereas for a car with longer lifetime driving distance, it is the fuel and electricity consumptions that have larger influence. To properly investigate the influence of both lifetime driving distance and daily driving cycle distance distribution, two different scenarios will be investigated. In the first scenario, it is assumed that the number of driving cycles per year is equal for all drivers. This means that the total driving distance of a driver is proportional to the distance of an

2.2. Performance requirements A long driving cycle can reflect real-life driving, but might not include extreme situations that require high performance. However, high performance is still considered as an important vehicle attribute by many drivers when buying a car. Very high performance requirements, such as long all electric range, high acceleration and charge sustaining top speed, can decide the components size directly, but increases the costs significantly (Pourabdollah et al., 2013). Therefore, a medium level of acceleration requirements is included in this problem. The acceleration requirements can be fulfilled by different combinations of battery and EGU maximum power. Therefore, the constraint is indirectly imposed by including extreme driving conditions into the driving Table 1 Parameters of the real-life trips in the data base used for generating Markov transition matrices. Parameter

City

Mixed

Highway

Number of trips Total distance Total time Max. speed Average speed Max. acceleration Avg. positive acceleration Max. deceleration

943 5765 km 194 h 120 km/h 30 km/h 4.8 m/s2 0.4 m/s2  5.6 m/s2

481 8293 km 164 h 123 km/h 50 km/h 4.8 m/s2 0.3 m/s2  4.9 m/s2

208 7269 km 97 h 123 km/h 74 km/h 4.7 m/s2 0.2 m/s2  4.9 m/s2

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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cycle. A performance cycle is introduced as just another trip appended to the end of the driving cycle, artificially designed to include speeds with a required acceleration.

3. Modeling Modeling of the powertrain and its components is a crucial step in formulating and solving the optimization problem. Since the problem is solved using convex optimization, convexity has to be guaranteed when modeling the powertrain. This section first presents the powertrain and the component models, in addition with the steps needed to reformulate them as convex. The power losses of EM and fuel power of EGU are described by convex functions, including the scaling of size. In the last part, the cost model of the components and the complete detailed optimization problem are presented. 3.1. Powertrain model Having the velocity and the acceleration at discrete time instants, k, from the driving cycle, the required traction force, Ft(k), and the demanded power at the wheels, Pdem, for a vehicle on a flat road are calculated as

Ft (k ) =

Af cd ρv (k )2 + mtot (gcr + a (k )), 2

Pdem (k ) = Ft (k ) v (k ),

(5)

where Af, cd, ρ, g, and cr are frontal area, air drag coefficient, air density, gravitational acceleration, and rolling resistance coefficient, respectively. The total vehicle mass, mtot, is the sum of the masses of the glider, battery, electric motor, and engine-generator unit. The mass of the components is modelled as linear functions of sizes as mj = mj,0 sj , ∀ j ∈ {bat , EM , EGU}. The vehicle's parameters are given in Table 2. The powertrain model is described through mechanical and electrical power balance equations; the former is given by

Pdem (k ) + Pbrk (k ) = P EM (k ),

(6)

where Pdem(k) is calculated from (5), P brk is the power dissipated at the friction brakes when the regenerative braking is limited, and PEM is the EM mechanical power. The electrical power balance equations during driving, 2 drive , is given by

P bat, int (k ) − Pbat, loss (k ) + P EGU (k ) = P EM (k ) + PEM, loss (k ) + Paux.

(7)

During braking times Pdem in Eq. (6) is negative, and hence PEM becomes negative. Then in Eq. (7), the battery is charged from the power of the electric motor, deducting the power losses in the battery and the auxiliary power. During charging, 2 charge , the electrical power balance equations is given by Table 2 Vehicle's parameters.

P bat, int (k ) − Pbat, loss (k ) = − Pg (k ),

Value

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

m¼ 1600 kg mg ¼ 1280 kg Af ¼ 2.37 m2 cr ¼ 0.009 cd ¼0.33 ρ¼ 1.293 kg/m3 rw ¼ 0.3 m Rfg ¼ 4.2 REM ¼ 2 Paux ¼ 500 W

(8)

where Pbat , loss , PEM , loss , PEGU , Paux, P bat , int , and Pg are battery loss power, EM power losses, EGU electrical power, electrical power used by auxiliary devices, which is assumed to be constant, battery internal power, and charger power. The battery internal power is chosen to be positive when discharging. For simplicity, the losses in the power electronics are included in the EM losses, and the rotational inertia (including the inertia of the wheels, the differential, and the electric motor) are included as a small equivalent mass added to the vehicle mass in the models. The inertia of the engine-generator unit is neglected. 3.2. Battery The battery consists of sbat = ns np identical cells, where ns is the number of cells in a string connected in series and np is the number of strings connected in parallel. The cells arrangement is not affecting the sizing, as long as the optimal battery size is big enough to provide the voltage with one string of cells. The final series/parallel configuration can be decided after the battery sizing, such that a desired terminal voltage is achieved. Each battery cell is modeled as an open circuit voltage, Voc, in series with a constant internal resistance, R. The open circuit voltage over each cell, Voc, is a function of the battery state of charge; however in this study Voc is approximated to be constant. This assumption can be justified when operating in limited SoC ranges (Murgovski et al., 2012). The battery terminal power, Pbat, the battery loss power, Pbat , loss , and the stored energy of the battery, Ebat, are calculated as

Pbat (k ) = P bat, int (k ) − Pbat, loss (k ),

(9)

RP bat, int (k )2 , 2 sbat Voc

(10)

Ebat (k + 1) = Ebat (k ) − h (k ) P bat, int (k ),

(11)

Pbat, loss (k ) =

where the time varying sampling interval h(k) is equal to 1 s when driving, while greater values are assigned for charging events. During the available charging periods, it is assumed, without loss of generality, that the vehicle is charged with constant current and power as

Pbat, int (k ) − Pbat, loss (k ) = Pg (k ) ηg .

(12)

3.3. Electric motor The electric motor and power electronics losses are measured at steady-state and given in a map, PEM , loss, base . The power losses for each gridded EM speed can be approximated by a second-order polynomial in the power as 2 PEM , loss, base (PEM , base, ωEM ) = a1 (ωEM ) P EM + , base

Parameter

5

a2 (ωEM ) PEM , base + a3 (ωEM ),

(13)

where the coefficients a1 ≥ 0, a2 and a3 are functions of ωEM and hence time. These coefficients are calculated using least squares method for grids over ωEM to fit the second-order polynomial to the measured data (Murgovski et al., 2012). At each time instant on the driving cycle, the angular speed of the electric motor, ωEM (t ), is calculated in advance as

ωEM (k ) = REM Rfg

v (k ) , rw

(14)

where REM, Rfg, and rw are the EM reduction gear, ratio of the final

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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gear (differential), and wheel radius, respectively. To vary the size of the electric motor, the maximum and minimum powers, PEM ,max, base (ωEM ) and PEM ,min, base (ωEM ), are scaled linearly by the scaling factor sEM (Filipi et al., 2004; Wu et al., 2011). The losses are also assumed to change linearly with sEM . Hence, given a baseline EM described by power PEM , base and losses PEM , loss, base , the losses of the scaled EM are calculated at each time instant as

PEM, loss (k ) = a1 (k )

2 (k ) P EM + a2 (k ) P EM (k ) + a3 (k ) sEM . sEM

(15)

This nonlinear model is convex in PEM and sEM > 0. These steps are explained in more detail in Pourabdollah (2015). 3.4. Engine-generator unit The fuel power of a baseline EGU, Pf , base , is modeled as a second-order polynomial in PEGU , base as

Pf , base (k ) = b1PEGU, base (k )2 + b2 PEGU, base (k ) + b3,

(16)

where bj ≥ 0; j ∈ 0, 1, 2, are calculated in a similar way as a1, a2, and a3 for the electric motor (Murgovski et al., 2012). The model of the scaled EGU is obtained by applying linear relation to the fuel and electric power of the baseline EGU, i.e. Pf = sEGU Pf , base and PEGU = sEGU PEGU , base . The fuel power of the scaled EGU then becomes

Pf (k ) = b1

2 (k ) P EGU + b2 P EGU (k ) + eon (k ) b3 sEGU . sEGU

costj = costj,0 + costj,1 sj , for j ∈ {bat , EM , EGU} . (17)

The binary signal eon is introduced to remove the idling loss, b3 sEGU , when the engine is off. The engine on/off state, eon, is decided prior to the optimization to preserve the convexity of the problem and it is decided based on the baseline power demand required by the vehicle (Pourabdollah et al., 2013). 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. This is shown as

⎧ 1 if Pdem, base (k ) > Pon eon (k ) = ⎨ ⎩ 0 otherwise.

Fig. 5. Predicted future prices of diesel and battery.

(18)

The optimization is iterated over several values of Pon, as shown in Fig. 2, to find the best result.

(21)

The cost and mass functions are calculated from a baseline EM with 100 kW power, and a baseline EGU with power equal to 100 kW. The battery is energy optimized with cell capacity of 44.3 Wh and power of 881 W. To evaluate the cost in (19) and (20), the prices for electricity, fuel, battery cells, electric motor and engine-generator unit are needed. To find the optimal design of a vehicle, the current and predicted future prices are used in the optimization problem. There is an expected rapid change in the predicted future price of batteries and a slower change in price of fuel until 2050, as shown in Fig. 5 (A portfolio of power). The prices for electric motor, engine-generator unit and electricity are assumed not to change in the problem and are given in Table 3. 3.6. Optimization problem

3.5. Cost model As mentioned briefly in Section 2, the objective function to be minimized in the convex optimization problem is a weighted sum of the operational and component costs as given in (1). The operational cost includes the consumed fossil fuel and electrical energy as N

N

∑ Pf (k) h (k) + we ∑ Pg (k ) h (k),

costop = wf

k=1

where wf =

ρf ρLHV

k=1

and we =

ρel 1000·3600

(19)

are weighting coefficients, and N

is the length of the driving cycle. The fuel power, Pf, and the grid power, Pg , are converted to equivalent costs in EUR using the energy prices ρf for diesel, ρel for electricity, and the lower heating value of diesel, ρLHV. The component costs ar the sum of the costs of battery, electric motor, and engine-generator unit as

costcomp = costbat + costEM + costEGU .

(20)

The cost model for each component is a linear function of the scaling factor (Wu et al., 2011), expressed as

In this section, the complete optimization problem is presented. However, there are some steps that need to be taken in order to write the problem as a second-order cone program (SOCP). The equalities in (10), (15), and (17) in the problem are relaxed to inequalities to keep the convexity. It is easy to see that at optimum solution, the constraints will be satisfied with equality since otherwise energy is wasted unnecessarily (Egardt et al., 2013). Moreover, to keep the problem convex, sbat is relaxed to a real value. The relaxation will introduce a rounding error that has a small influence on the optimal result. This is because either the Table 3 Parameters for calculating costs. Price of electricity Lower heating value of diesel Yearly interest rate Vehicle lifetime Average yearly traveled distance Cost of EM

ρe ¼ 0.15 EUR/kWh ρLHV ¼ 38.6 MJ/kg pc ¼ 5% yv ¼ 15 years dy ¼ 12,000 km costEM,1 = 20 EUR/kW

Cost of EGU

costEGU,1 = 30 EUR/kW

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

M. Pourabdollah et al. / Control Engineering Practice ∎ (∎∎∎∎) ∎∎∎–∎∎∎

N

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2 (k ) P EGU + b2 P EGU (k ) + eon (k ) b3 sEGU sEGU

Ebat (k + 1) = Ebat (k ) − h (k ) P bat, int (k ) Pg (k ) ∈ [0, Pg,max (k )] P bat, int (k ) ∈ sbat [Pbat, int, min, Pbat, int, max ] Ebat (k ) ∈ sbat [Ebat, min, Ebat, max ] P EM (k ) ∈ sEM [PEM, min, base (k ), PEM, max, base (k )] P EGU (k ) ∈ sEGU [0, PEGU, max, base ]

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cell capacity can be considered very small to give large number of cells, or the result can be interpreted as an indication of the optimal pack capacity (Murgovski et al., 2012). The complete problem given below is automatically translated by CVX (Grant, 2010) to a standard SOCP form, and solved by a publicly available solver, Sedumi (Sturm, 1999):

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The decision variables, x, to be found by the optimization solver N N N +1 are sbat , sEM , and sEGU , which are scalars and Pbat , int , Ebat , PEM , N N , P gNc , and Pbrk , which are vectors. The superscripts show the PEGU length of the vectors, where N is the number of time samples of the driving cycle and Nc is the number of charging occasions. In our problem, there are around 60,000–400,000 variables, which are found by the solver in one round of optimization in about 200– 2000 s on a standard PC with 4 GB RAM and 2.66 GHz.

4. Results In this section, the results of the simultaneous optimization for different driving, charging, and pricing scenarios are presented. In the first part, the optimal component sizing for driving cycles with different lifetime driving distances are presented. Similar study is done for driving cycles with different charging behaviors in the second part. For both cases, the effect of the predicted future fuel and battery prices on the optimal cost and component sizes is studied and results are given. 4.1. Effect of different lifetime driving distances on sizing The driving cycles used in the optimization problem in this section have different driving distance distributions as explained in Section 2.1.1. The driving distance distributions also affect the lifetime driving distance of vehicles. Each driving cycle is a combination of 10 single trips with lengths chosen to represent a specific driving distance distribution. To help understanding the results better, first the result of the optimization over single trips with different driving distances is investigated. These results are then used to explain the effect of the driving cycles on optimal sizing.

4.1.1. Results for single trips In this case, single trips with 60 different lengths uniformly spaced from 1 to 178 km are generated using Markov chain. For each length, 10 random trips are generated in order to remove strong dependency on a particular speed profile. The mean and the standard deviation of optimal sizes of battery, electric motor, and engine-generator unit for each trip length are shown in Fig. 6, using cost model from year 2020. The battery size is shown by its power, but there is a linear function between the power and the capacity of the battery used in this study, with a battery of capacity 1 kWh having a power of 20 kW. Assuming that the number of trips during vehicle's lifetime is constant, the lifetime driving distance used in each optimization is linearly dependent on the driving cycle's distance. In this case, the weighting factor, w in (2), becomes a constant value. Hence, the component cost per kilometer becomes equal for different driving distances, whereas the operational cost is lower for shorter cycles and higher for longer cycles. In other words, in the optimization over shorter cycles, the component cost has higher weight than in the optimization over longer cycles. This also results in higher total cost per kilometer for shorter trips. As shown in Fig. 6, the EM size does not vary much for cycles with different lengths. This is because the size is decided by the maximum power demand from the driving cycle, which happens during the common performance trip. The small difference, however, is the effect of the component weights. For the vehicles with a big EGU, the optimal EM size is also bigger, since the vehicle has greater total mass and therefore the EM has to deliver slightly higher power demand. The sum of the battery and EGU power should be equal or greater than the EM maximum power to provide the power. We have to bear in mind that this is also affected by the choice of battery, for example if the battery technology supports high power to energy ratio, then the sum of battery and EGU power might be much higher than the EM power, since the battery is sized by the energy demand. For trips up to 100 km, the battery power increases from 15 kW to 110 kW and EGU size decreases from 110 kW to zero. As mentioned, for short trips it is the component cost that has higher weight in the multi-objective optimization. Therefore, since battery is a more expensive way to meet the power demand compared to EGU, the optimal size of the battery is smaller. By increasing the trip length, the operational cost gets larger weight in the optimization and becomes more influential. The increase in battery size as shown in Fig. 6 results in reduction in the

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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operational cost, as shown in Fig. 7. For trips with 100 km length, the entire trip is driven electrically and the fuel cost is close to zero. For trips longer than 100 km, there is a shift in the trend. Since the EGU size is already zero, increasing the battery size cannot result in smaller EGU and any gain in cost. Moreover, to meet the energy demand by electrical energy, the battery size has to increase linearly with the trip length, which increases the total mass and hence operational costs. Therefore, the battery size starts decreasing and the EGU size starts rising. Here, it is the energy demand that is influencing the sizing, rather than the power demand. For very long trips, the size of EGU converges to the average power demand of the trips, which is around 15 kW. One important observation to notice is that for 10 trips with different speed profiles at each distance, the optimal component sizes have low variance. This is because for a series PHEV, the power demand directly affects the EM size, whereas the size of EGU and battery are affected by the power and energy demand. If performance cycle is included in the driving cycle, the optimal sizes of EM is not changed significantly, and therefore the sum of EGU and battery are also insensitive to different speed profiles. It is important to mention that the optimization studies in this paper only include the hybrid vehicles, but not conventional vehicles (without batteries) or electric vehicles (without enginegenerator units). This is because the convex cost functions of these components are linear with an offset, namely the packaging cost. With this cost model, even if the sizes are zero, the cost still includes the packaging cost. Therefore, to exclude the total cost of battery or engine-generator unit, the optimization has to be run

Fig. 7. Optimal cost/km of fuel, and electricity and weighted cost of battery, EM, and EGU, for single trips with different lengths, when vehicle's lifetime number of trips are kept constant.

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Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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Fig. 9. Optimal cost/km of fuel, and electricity and weighted cost of battery, EM, and EGU for different driving cycles (mean and variance), for driving cycles with different driving distance distributions (different values of λ), during 2020–2050.

separately but with zero cost for these components. 4.1.2. Results for driving cycles In this section the optimization is done for seven driving cycles, representing different driving distance distributions. To study the sensitivity of the results to the speed profile, five different driving cycles are stochastically generated and used in the optimization for each λ. Similar to the results from single trips, the lifetime driving distance changes with respect to the driving distance distribution. This means that the drivers who often drive short distances, represented by lower values of λ, have shorter lifetime driving distances compared to drivers who drive often longer distances, represented by higher values of λ. Including the total driving distance in the cost function, the former drivers get higher total cost, and the latter drivers get a lower total cost. The optimal sizes of the battery, electric motor, and enginegenerator unit for different projected battery and fuel prices in four different years are shown in Fig. 8. As shown earlier in Fig. 5, the price of battery is assumed to be the most expensive at 2020 and drop continuously till 2050, whereas the price of fuel is expected to increase slightly during these years. Therefore, in 2020, the optimal battery size for a driver who drives very short distances most of the times (the driving cycle distribution represented by λ = 15) is quite small. This is because batteries are relatively expensive for this driver and the component cost dominates the operational costs. As the driver drives longer distances (the driving cycle corresponding to the CDF with λ = 30 ), the component costs become less dominant and therefore the optimal battery size increases, see Fig. 9. The EGU size, however, is around 20 kW to provide the average energy on very long trips.

In 2030, a similar result is obtained, with the difference that due to the cheaper price of battery, the optimal size of the battery is in general larger. From 2040, the batteries are expected to become so cheap that the optimal vehicle is similar to an electric vehicle. For drivers with long driving distances, a small EGU is needed to provide part of the energy on the long trips. In the cases with large battery sizes, however, the optimal cost should be compared with the cost of electrical vehicles, excluding the initial cost and weight of the EGU. This can be solved by the optimization, but has not been done in this study, since the result are then more sensitive to the packaging and base cost, and there is a need for better cost models. 4.2. Effect of different charging behaviors on sizing In this section, similar optimization as in Section 4.1 is done to investigate the effect of different charging behaviors on the optimal sizing of series PHEVs. Mathematically, the main difference from the previous study is that in this case, the lifetime driving distances for all driving cycles are constant. The frequency of charging only affects the trip lengths and hence the driving distance distribution, but not the lifetime driving distance. In other words, assuming that the lifetime driving distance is constant means that the driver had more chances to charge the vehicle. This in return makes the trips shorter and hence reduces the λ that represents the driver. The optimal component sizes for driving cycles represented by different values of λ and at different years are shown in Fig. 10. In 2020, batteries are expected to be too expensive to make electric vehicles cost efficient. The drivers with short driving distances

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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need less electrical energy on the short trips, therefore, the expensive batteries can be sized small. With decreasing charging opportunities, for driving cycles up to λ = 80 , both the fuel consumption and optimal battery size increase, since the longer trips demand more energy. However, if the driver has even less access to charging, for driving cycles with λ > 80, the optimal battery size gets smaller and the EGU size increases, resulting in a vehicle similar to an HEV. Starting in 2030, the optimal vehicle is an electric vehicle or a series range extender with a small EGU. This is due to the expected decrease in battery price, similar to the previous case. However, compared to the case in Section 4.1.2, driving cycles represented by low values of λ have longer yearly driving distance, hence the component cost has lower weight in the optimization, which results in larger size of battery. As can be seen in Fig. 11, having less access to charging from the grid (higher λ) results in increase in total cost of the vehicle. On the other hand, having more charging opportunities enables the driver to use the cheaper energy from the grid, and also the battery can be sized smaller since it does not need to carry as much energy, which results in decrease in total cost.

5. Conclusion In this paper, convex optimization techniques are used to solve the problem of sizing the major components of a series plug-in hybrid electric vehicle, i.e. the battery, electric motor, and enginegenerator unit. To define the problem as a convex optimization problem, the model of the vehicle and components is approximated using convex functions. The main focus of the paper is on

the effect of driving, charging, and pricing scenarios on component sizing. The driving cycles used in the optimization are generated randomly by Markov chains, to represent behavior of drivers with different driving distance distributions and charging behaviors. The optimization results show that the size of the electric motor for a series PHEV is decided mainly by performance requirements. Moreover, to provide enough power to the electric motor, the sum of EGU and battery power has to be equal or higher than the power of the electric motor. With the current price of battery, HEVs are the optimal vehicles. From 2030, assuming that the battery price decreases, electric vehicles will be competitive options for many drivers. At 2020, the size of the engine-generator unit and battery are primarily affected by the distribution of trip lengths. For instance, for a driver that often drives short distances, a hybrid vehicle with a small battery is optimal. If the driver drives longer distances, the optimal battery size increases, while the EGU size decreases. However, as the trips get even longer, the optimal EGU size increases slightly, to provide the energy for the very long driving cycles. Moreover, charging opportunities can also affect the optimal cost and sizing. For a driver who seldom has access to charging, the optimal battery size increases, resulting in higher cost of ownership. If the driver can charge the battery often, then the optimal battery can be reduced, and more electricity can be used instead of fuel, which decreases the total cost. To include the effect of different speed profiles, several driving cycles are generated randomly from real data. However, it is shown that the optimal component sizes are more sensitive to driving distance distribution and charging behavior than the speed profile.

Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i

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Fig. 11. Optimal cost of cost/km of fuel, and electricity and weighted cost of battery, EM, and EGU for different driving cycles with different charging behaviors (different values of λ), during 2020–2050.

It should be mentioned that the study did not include conventional or electric vehicles. Future work can include comparison of hybrid vehicles with large batteries or large EGU with electric vehicles and conventional vehicles. Moreover, only one battery type is used in the study. It is relevant to investigate the results using both power optimized and energy optimized batteries. The problem should also be extended to include a SoC dependant battery voltage model and the battery health model.

Acknowledgment The simulations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) (Grant no. 30734-2) at C3SE. This project is financially supported by Energy Agency and the Swedish Hybrid Center.

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Please cite this article as: Pourabdollah, M., et al. Effect of driving, charging, and pricing scenarios on optimal component sizing of a PHEV. Control Engineering Practice (2016), http://dx.doi.org/10.1016/j.conengprac.2016.02.005i