Aging-aware co-optimization of battery size, depth of discharge, and energy management for plug-in hybrid electric vehicles

Journal of Power Sources 450 (2020) 227638

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Aging-aware co-optimization of battery size, depth of discharge, and energy management for plug-in hybrid electric vehicles Shaobo Xie a, Xiaosong Hu b, **, Qiankun Zhang a, Xianke Lin c, *, Baomao Mu a, Huanshou Ji a a

School of Automotive Engineering, Chang’an University, Southern 2nd Road, Xi’an, 710064, China State Key Laboratory of Mechanical Transmissions, Department of Automotive Engineering, Chongqing University, Chongqing, 400044, China c Department of Automotive, Mechanical and Manufacturing Engineering, University of Ontario Institute of Technology, 2000 Simcoe St N, Oshawa, ON L1G 0C5, Canada b

H I G H L I G H T S

� Convex programming of battery size and energy management. � Battery degradation is considered. � The depth of battery discharge is optimized. � Comparison of convex programming and Pontryagin’s minimum principle. � Sensitivity analysis of the initial SOC and battery price is carried out. A R T I C L E I N F O

A B S T R A C T

Keywords: Plug-in hybrid electric bus Optimal depth of discharge Convex optimization Battery aging model Energy management

Plug-in hybrid electric vehicles (PHEVs) have a large battery pack, and the depth of discharge (DOD) signifi­ cantly affects the battery longevity. In this paper, the battery degradation is considered in the co-optimization of battery size and energy management for PHEVs using convex programming. The impact of DOD on battery degradation and energy management is also investigated. The cost function consists of fuel consumption, elec­ trical energy consumption, and equivalent battery life loss. A real-world speed profile collected from the urban city bus route up to about 70 km is used as an input to evaluate the proposed method. The results suggest that, for both cases with and without battery degradation, the total cost curve with respect to the preset final state of charge (SOC) is an upward parabola, where the optimal DOD can be identified, and the optimal battery size and energy management can be determined. The results also show that, with an initial SOC of 0.9, the proposed method can reduce the total cost by 3.6 CNY compared to other existing studies with the fixed final SOC. Moreover, a sensitivity analysis is conducted to explore the effect of battery price and initial SOC on the optimal DOD and total cost.

1. Introduction 1.1. Motivations In recent decades, the plug-in hybrid electric vehicle technologies have been improved significantly and have been deployed globally. The goals of these developments include easing the dependence on fossil fuel, improving range for long-distance travel, and enhancing vehicular performance, especially in terms of efficiency and cost-effectiveness [1]. Batteries are one of the main energy sources for PHEVs. In recent

years, battery technology has made significant progress in many areas, such as improved energy density and power density [2]. However, the cost of batteries still accounts for a large portion of the total cost of a PHEV. Therefore, optimizing battery capacity is one of the main goals of maximizing the vehicle’s overall lifetime value [3]. Meanwhile, for hybrid powertrains with two or more energy sources, such as a battery pack and an engine-generator-unit, the energy consumption in daily operations is another important factor affecting the fuel economy. Minimizing energy consumption is certainly beneficial for improving the whole-life value of PHEVs. Therefore, another important task of maxi­ mizing the whole-life value of a PHEV is to develop the energy

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (X. Hu), [email protected] (X. Lin). https://doi.org/10.1016/j.jpowsour.2019.227638 Received 2 October 2019; Received in revised form 2 December 2019; Accepted 17 December 2019 Available online 20 December 2019 0378-7753/© 2019 Elsevier B.V. All rights reserved.

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Journal of Power Sources 450 (2020) 227638

Nomenclature Pegu Tegu negu Pb mf be

ρf ηm

nm Tm i0

η1 δ m0 mb nb u ξ g Af CD r Pmr Paux

η2 η3

C Voc Voc Ib Imin Imax R E Qb

M c R1 T1 A Ea θ Atol Jb Δt cb ϕ1 cf ce SOC0 SOCf t tf min max G

EGU output power EGU output torque EGU rotational speed battery output power optimal fuel consumption optimal gas consumption natural gas density motor efficiency motor rotational speed motor torque gear ratio of the reducer mechanical efficiency of the reducer equivalent rotational inertia vehicle mass excluding battery mass battery module mass number of battery module vehicle speed rolling resistance coefficient gravitational acceleration front area of the vehicle air resistance coefficient wheel radius requested power of the motor power loss of the auxiliary components power efficiency from battery to terminal power efficiency from EGU to terminal battery equivalent capacity per voltage open-circuit voltage of each module rated battery voltage of each module electric current of the battery pack minimum current of the battery pack maximum current of the battery pack internal resistance of a battery module battery pack energy nominal capacity of the battery pack

pre-exponential coefficient charging and discharging rate ideal gas constant battery temperature throughput of the battery current active energy power exponent total Ah throughput total equivalent battery life loss cost time interval unit pirce of battery battery purchase cost price of fuel fossil price of electricity initial SOC final SOC time variable total time lower boundary value upper boundary value total cost of natural gas and electricity

List of abbreviations HEV hybrid electric vehicle PHEV plug-in hybrid electric vehicle DOD depth of discharge DP dynamic programming CP convex programming PMP Pontryagin’s minimum principle SOC state of charge EGU engine-generator-unit EM electric machine BSFC brake specific fuel consumption EBLLC equivalent battery life loss cost TC total cost FCC fuel consumption cost ECC electricity consumption cost

management strategy, i.e., the power split ratio between different en­ ergy sources based on the battery capacity. However, these two sub-problems, i.e., sizing the battery capacity and developing the energy management strategy, are coupled [4], for two major reasons. First, it is preferred that a PHEV depletes the battery until the allowable minimum state of charge (SOC) level at the end of the trip [5] in order to maximize the consumption of electricity rather than fossil fuel because electricity is usually cheaper. Also, when more elec­ trical energy is stored in the battery, the discharge rate can be higher, which affects energy management. Therefore, it is necessary to simul­ taneously optimize the battery size and energy management to achieve the overall economy of a PHEV.

teaching-learning based optimization [10], the non-dominated-sorting-genetic algorithm-II (NSGA-II) [11], and convex programming (CP) [12], or even a combination of dynamic program­ ming (DP) and CP [13]. In a CP optimization problem where any local solution is also a global solution, CP exhibits a remarkable advantage in terms of computational efficiency and has been extensively used for co-optimization of energy source sizing and energy management. For example, convex problems were formulated to solve this problem in hybrid electric vehicles (HEV) [14], plug-in hybrid electric vehicles [15], fuel cell hybrid powertrains [16,17], or even powertrains with a hybrid energy storage system consisting of battery and super-capacitor [11,18]. Particularly, Sorrentino et al. [17] proposed a flexible pro­ cedure of co-optimizing the design and energy management of fuel cell hybrid vehicles. Moreover, the battery state of health (SOH) was considered in the co-optimization problem for PHEVs [19]. Unlike HEVs where the battery acts as a temporary power buffer [20], there is more electrical energy stored in a PHEV battery, and overuse can accelerate battery aging [21]. Therefore, many approaches have been proposed to address battery degradation in co-optimization, where the cost caused by battery aging is included in the cost function [21,22]. Since charging behavior also affects battery life, an untied framework including charging, power management, and battery degradation was proposed for PHEVs [23]. Multiple-objective optimization problems can be formulated to optimize the component size, minimize the total energy

1.2. Literature review Many methods have been proposed for simultaneously optimizing battery size and energy management. For example, a dual-loop opti­ mization structure can be implemented by first determining the battery capacity in the external loop and then determining the power split ratio in the internal loop. Many optimization approaches can be used to solve this problem, including particle swarm optimization [6], genetic algo­ rithms [7], Pontryagin’s minimum principle (PMP) [3,8] and model predictive control [9]. The coupled problem (battery sizing and energy management) can also be solved by other approaches, such as 2

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Table 1 Initial and final SOC values used in existing PHEV battery sizing optimization studies. Reference No.

Initial SOC

Final SOC

[4] [6] [8] [10] [12] [15] [23] [26] [27] [28] [29]

0.60 0.90 0.80 1.0 0.60 0.75 0.70 0.90 0.70 0.75 0.75

0.40 0.20 0.30 0.02 0.20 0.20 0.30 0.20 0.20 0.25 0.25

consumption, and maximize the battery longevity [12,24,25]. Although the co-optimization of battery size and energy manage­ ment, including battery aging, has received a lot of research attention, especially for PHEVs, almost all optimization schemes using CP, DP, PMP, as well as other methods, assumed fixed, preset initial and final SOC values. That is, the co-optimization was solved with a predefined battery depth of discharge (DOD). Table 1 summarizes the preset SOCs for PHEVs in the literature. Therefore, the essential relationship between energy consumption, battery longevity, and DOD cannot be used to optimize battery capacity and the resulting overall cost. The DOD is also associated with battery aging in PHEVs. A higher DOD means more use of battery energy, which is relatively cheaper for propulsion compared with fossil fuel and would seemingly reduce the overall energy cost. However, the excessive use of battery energy will inevitably accelerate battery aging, increase the equivalent battery life loss cost (EBLLC) [21]. Therefore, in the daily operation of a PHEV, using a preset final SOC level (or given battery DOD) does not guarantee a minimal sum of en­ ergy consumption cost and EBLLC. To achieve the overall economy for a PHEV, finding the proper DOD should be part of an optimal solution for battery size and energy management.

Fig. 1. (a) The prototype of the PHEV [30]; (b) The powertrain of the PHEV [30]. Table 2 Specifications of the PHEV used in this study [30].

1.3. Contributions To bridge the research gaps identified above, this paper develops an optimization framework for battery size, energy management, and battery degradation using convex programming, with the DOD as an optimization variable. The main contributions are as follows:

Parameter

Value

Mass (kg) Two-gear ratio Tire radius (m) Rolling resistant coefficient

14500 13.9 0.57 0.0076 þ 0.000056u

Front area (m2) Aerodynamic drag coefficient Equivalent rotational inertia

(1) A CP framework is proposed to co-optimize battery size and en­ ergy management, which considers energy consumption, battery aging, and optimal DOD, and aims to minimize the total cost of fuel consumption cost, electricity consumption cost, and equiv­ alent battery life loss cost. (2) Cases with and without a battery aging model included in the cost function are both discussed, and the results reveal that whether battery aging is considered or not, an optimal DOD can be iden­ tified which corresponds to a minimal sum of energy consump­ tion cost and EBLLC, with a given initial SOC. That is, an optimal solution for battery capacity and power split can be determined based on the optimal DOD, which provides a useful guideline for co-optimizing the size of the energy storage system and the power split ratio of the powertrain for PHEVs. (3) A real-world speed profile over roughly 70 km with battery SOC measurements was collected from an urban bus route in Xi’an, China, and used as an input to evaluate the proposed method, whereas, in other studies, only short-distance driving cycles are used to evaluate the optimization problem. (4) A sensitivity analysis is conducted to explore the effect of varia­ tion in battery price and initial battery SOC level on the optimal DOD and total cost.

8 0.65 1.07

Note: u is the vehicular speed.

1.4. Organization of this paper The remainder of this paper is organized as follows. Section 2 de­ scribes the powertrain model, and Section 3 establishes the battery dy­ namics. The convex co-optimization problem is formulated in Section 4, results are discussed in Section 5, and conclusions are given in Section 6. 2. Powertrain modeling 2.1. Physical description of the powertrain The prototype of the investigated PHEV, which serves as a public bus in Xi’an, China, is shown in Fig. 1(a). Fig. 1(b) shows a schematic of its powertrain. A natural gas engine is mechanically coupled to an integrated-starter-generator (ISG), to form an engine-generator-unit (EGU), which can supply electrical energy for propulsion or charge the battery. Electric motors on each side are connected to the wheels 3

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Journal of Power Sources 450 (2020) 227638

Fig. 2. (a) Efficiency map of the engine; (b) Efficiency map of the generator; (c) Optimal gas consumption rate versus EGU output power; (d) Efficiency map of the electric motor; (a),(b) and (d) © [2018] IEEE. Reprinted, with permission, from Ref. [31].

through a two-gear reducer. The other energy source is a lithium iron phosphate battery which can also be charged through the electrical grid. The specifications of this PHEV are summarized in Table 2.

� � m_ f Pegu ¼ maxðf1 ; f2 Þ ¼ max a1 Pegu ; a2 Pegu þ a3

where the fitting coefficients a1 ​ a2 , and a3 are 0.0000865, 0.0001685 and 0.0068406, respectively. This function is labeled as the “fitted curve” in Fig. 2(c). Moreover, the relationship between the optimal fuel consumption rate and EGU power is made convex through Equation (2). The motor efficiency ηm , as shown in Fig. 2(d) is considered to be a function of its rotational speed nm and torque Tm :

2.2. Model of the powertrain In this section, the mathematical model of the powertrain is described, where several expressions are mathematically relaxed to formulate the convex problem.

ηm ¼ ηm ðnm ; Tm Þ

2.2.1. EGU and electric motor model Fig. 2(a) and (b) show brake specific fuel consumption (BSFC) map of the natural gas engine and efficiency map of the generator, respectively. By combining the efficiency maps of the two, the EGU efficiency can be determined. The optimal gas consumption rate versus output power operating curve for the EGU can then be calculated, as shown in Fig. 2 (c). The relationship between the optimal fuel consumption rate (m_ f ) by the EGU and its output power (Pegu ) can be described by, m_ f ¼

� be Pegu ¼ ζ Pegu 3600ρf

(2)

(3)

where nm is the motor rotational speed, rpm, and Tm is the motor output torque, N.m. 2.2.2. Longitudinal dynamics In the independent two-motor system, the torque output from the electric machine (EM) is calculated as follows, � � r du CD Af u2 Tm ¼ 0:5 δðm0 þ nb mb Þ þ ðm0 þ nb mb Þgξ þ (4) i0 ηm η1 dt 21:15

(1)

where i0 is the gear ratio of the two-gear reducer, η1 is the mechanical efficiency of the reducer, r is the wheel radius, m, δ is the equivalent rotational inertia, m0 is the vehicle mass excluding the battery mass, kg, mb is the battery module mass, kg, u is the vehicle speed, km/h, ξ is the rolling resistance coefficient, CD is the air resistance coefficient, and Af is the front area of the vehicle, m2. The rotational speed of the EM, nm , rpm, is

where mf is fuel consumption, m3, Pegu is the EGU output power, kW, be is the optimal gas consumption for an output power Pegu , g/kWh, and ρf is the density of natural gas, kg/m3. ζ denotes the functional relationship between m_ f and Pegu , which is defined by the efficiency map of the EGU. Moreover, optimal gas consumption rate with respect to EGU output power can be fitted using a piecewise function, with one part (f1 ) in the low power region, and another part (f2 ) in the high power region,

nm ¼ 4

ui0 0:377r

(5)

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Journal of Power Sources 450 (2020) 227638

Fig. 3. Convex fitting of (a) open-circuit voltage and (b) internal resistance as a function of SOC.

Then, the requested power of the EM, Pmr , kW, is calculated as follows, Pmr ¼

2Tm nm

rffiffiffiffiffiffiffiffiffiffi 2Enb Pb ¼ I b C

(6)

ηm

Then, for convex programming, Pb can be relaxed as [19]. rffiffiffiffiffiffiffiffiffiffi 2Enb Pb � Imin nb RI 2min C

So, during driving or braking, the power balance equation of the PHEV can be defined as: � � Pb Pmr þ Paux ¼ min Pb η2 ; (7) þ Pegu η3 where Paux is the power loss of the auxiliary components including the steering and braking systems, kW, η2 and η3 are the power transfer ef­ ficiencies from the battery and EGU to the terminals, respectively, and Pb is the output power of the battery, kW. To make it convex, Equation (7) can be relaxed to Ref. [19]: � � Pb Pmr þ Paux � min Pb η2 ; (8) þ Pegu η3

Pb ðtÞ �

3. Battery model

Then, � 1 Voc 2R

3.1. Battery electric model The battery pack is formed by connecting a string of modules in se­ ries. For each module, the nominal capacity is 120 Ah, and voltage is 3.2 V, which is equivalent to a string of A123 cells in parallel [32]. For the battery pack, the output power can be expressed as follows, � Pb ¼ Voc Ib I 2b R nb (9)

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 4RPb � Imax V 2oc nb

(15)

(16)

(17)

or, E

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi E2 2RCPb � RImax 2CEnb

(18)

Fig. 3 shows plots of the open-circuit voltage Voc and internal resis­ tance R of a battery module with respect to SOC based on experimental data. An approximately linear relationship can be established between Voc and SOC,

(10)

Voc ¼ p1 SOC þ p2

where C is the battery equivalent capacity per voltage, C, and can be represented as: 2Qb Voc

nb V 2oc 4R

By substituting Equation (10) into (16), we obtain, �rffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � 1 2E 2E 4RPb � Imax 2R Cnb Cnb nb

where Voc is the open-circuit voltage of a module, V, R is the internal resistance for a module, ohm, Ib is the electric current flowing through the battery pack, A, and nb is the total number of modules. The battery pack energy can be expressed as follows,

C¼

(13)

where Imax is the maximum battery current, A. As a necessary condition of Equation (14), the following constraint should be imposed,

η2

CV 2oc nb 2

(12)

where Imin is the minimum current of the battery pack, A. Equation (9) can then be rearranged to express the battery current as follows, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � � 1 4RPb Ib ¼ Voc 2 ½Imin ​ Imax � V 2oc (14) 2R nb

η2

E¼

nb RI 2b

(19)

where the fitting coefficients p1 and p2 are 0.1205 and 3.1778, respec­ tively. Internal resistance R, is assumed to be constant within the scope of SOC used in this study:

(11)

where Qb is the nominal capacity of the battery pack, Ah, and Voc is the rated battery voltage of each module, V. By combining Equations (9) and (10), Pb can be rewritten as follows,

R ¼ R0

(20)

where R0 is the value of the internal resistance of the module. The derivative of the battery SOC with respect to time is defined as, 5

Journal of Power Sources 450 (2020) 227638

S. Xie et al.

to the general case. Electricity and natural gas prices are based on actual market prices in the local market in Xi’an, China. Natural gas density is from the Mechanical Design Handbook (Fifth edition, ISBN: 9787122014092, 2008). The mass of the battery module is based on the actual product (product model: SE130AHA) of China Lithium Battery Technology Co., Ltd. The gas constant and pre-exponential coefficient are from Refs. [22,32]. The model capacity and internal resistance are derived based on the actual product.

Table 3 Parameters of the powertrain and battery system [22,32]. Parameter

Value

Parameter

Value

Power transfer efficiency η1 , η2 and η3

0.97, 0.94 and 0.94

Module mass mb0 (kg)

4.5

Electricity price ce (CNY/ kWh)

0.995

Module capacity C (Ah)

120

Natural gas price cf (CNY/m3)

4.5

Module internal resistance R0 (ohm)

6.5 � 10 4

Natural gas density ρf

0.717

Gas constant R1 (J/mol. K)

8.31

Battery price cb (CNY/ kWh)

900

Pre-exponential coefficient M

31630

(kg/m3)

dSOC ¼ dt

Ib Qb

4. Convex problem formulation In this section, the cost function is first defined, then a convex pro­ gramming problem is formulated. 4.1. Cost function

(21)

Without considering battery degradation, a cost function consists of the total cost of the consumed natural gas and electricity: Z tf � cf m_ f dt þ ce SOC0 SOCf Qb Voc nb (29) min ​ G ¼

3.2. Battery aging model To quantify the degradation of battery capacity, a widely used exponential model is adopted [32], � � Ea ðcÞ ΔQb ¼ MðcÞexp (22) AðcÞθ R1 T1

0

where G is the total cost of the consumed natural gas and electricity over the entire trip, CNY, tf is the total time of the trip, s, cf and ce are the prices of natural gas and grid electricity, respectively, CNY/m3 and CNY/kWh, and SOC0 and SOCf are the initial and preset final SOCs, respectively. After including battery aging, the EBLLC is added into the cost function as follows, Z tf � � min ​ G ¼ cf m_ f þ Jb dt þ ce SOC0 SOCf Qb Voc nb (30)

where ΔQb is the capacity loss expressed as a percentage, %, MðcÞ is a pre-exponential coefficient, which is a function of the C-rate, and R1 is the ideal gas constant, J/mol.K. Here, the battery temperature T1 is assumed to be constant at 25 � C. AðcÞ is the throughput of the battery current, Ah, which is also the function of the C-rate. The active energy Ea ðcÞ, J/mol, and power exponent θ are defined as [32]: � Ea ðcÞ ¼ 31700 163:3c ​ (23) θ ¼ 0:57

0

4.2. Convex programming

In the automobile industry, a 20% loss in battery capacity is gener­ ally accepted as the end-of-life [32]. Using this number, the total Ah throughput Atol in the battery’s life span can be written as, 2

The general formulation of convex programming can be described as follows [33], minimize

31θ

6 20 � Atol ðcÞ ¼ 6 4 MðcÞexp

7 �7 5

Ea ðcÞ R1 T1

subject ​ to ​

(24)

hj ðxÞ ¼ 0

It is assumed that the total Ah throughput over the entire lifespan is constant [32]. The EBLLC can be calculated as follows, Jb ¼ ϕ1

jIb j Δt 3600Atol ðcÞ

(25)

dE ¼ dt

2nb p1

can be expressed as follows,

Ib 3600

(27)

By substituting Equations (24) and (27) into (25), the instantaneous EBLLC can be transformed to, Jb ¼

� �� � �� � � �1 � �dE� ϕ1 MðcÞ z 31700 ��dE�� 587880 � � exp exp zR1 T1 � dt � 2nb p1 zR1 T1 Qb � dt � 2nb p1 20

(33)

(1) Start with the measured speed profile, and known parameters of the powertrain and battery system. (2) Discretize the final SOC SOCf with its available range. (3) For the time step k ¼ 1; 2; 3⋯N (where N is the total number of steps in the trip), perform the optimization below with each final SOC: Variables: the sequence of Pb and Pegu , E, nb ; N P Cost function: min ðcf m_ f ðkÞ þ Jb ðkÞÞ þ ce ðSOC0 SOCf ÞQb Voc nb ;

According to Equations (10), (19) and (21), the derivation of the dEðtÞ dt

j ¼ 1; 2; 3⋯p

(32)

Table 4 The framework of the convex programming with uncertain DOD.

(26)

battery energy

fi ðxÞ � 0 ​ ​ ​ ​ ​ i ¼ 1; 2; 3⋯m

where the expressions fi ðxÞ ​ ði ¼ 0; 1; 2; 3⋯mÞ are convex, and hj ðxÞ are affine. When the final SOC is not fixed, the optimization variables for such a convex programming scheme include ​ Pb ðtÞ, Pegu ðtÞ ​ EðtÞ, ​ SOCf and nb . In Section 2, several equations are relaxed to formulate the convex problem. Finally, the constraints imposed are given as follows:

where Jb is the EBLLC, CNY, Δt is the time interval, s, and ϕ1 is the battery purchase cost, CNY, which is obtained by the product of its price cb , CNY/kWh and the battery energy E, kWh: ϕ1 ¼ cb E

(31)

f0 ðxÞ

k¼1

Constraints: (34)–(43). (4) Identify the optimal final SOC and obtain the optimal solution, including the optimal number of battery modules and power split ratios between the battery and EGU.

(28)

The parameters of the powertrain and battery system are given in Table 3. The power transfer efficiencies (η1 , η2 and η3 ) are set according 6

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Journal of Power Sources 450 (2020) 227638

EðtÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E2 ðtÞ 2RCEðtÞPb ðtÞ � Imax R 2CEðtÞnb

dEðtÞ � dt

p1 � EðtÞ RQb

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi � E2 ðtÞ 2RCEðtÞPb ðtÞ

nb C 2 nb C 2 V ðSOCmin Þ � EðtÞ � V ðSOCmax Þ 2 oc 2 oc Eðt0 Þ ¼

nb C 2 V ðSOC0 Þ 2 oc

125 � nb � 280 Fig. 4. Measured speed profile along an urban bus route in Xi’an.

� � Pb ðtÞ Pmr ðtÞ þ Paux ðtÞ � min Pb ðtÞη1 ; þ Pegu ðtÞη2

η1

Pb ðtÞ �

EðtÞ 2RC

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2EðtÞnb Pb ðtÞ � Imin C

(38) (39) (40) (41)

where the boundary values of the optimal number of battery modules are set to 125 and 280, respectively, to represent a very low and large battery capacity. Also, it is noteworthy that nb is taken as a continuous variable in the optimization, and the solution is rounded to the nearest integer. Also, the minimum SOC is not a fixed value, but is set to a range:

(34) (35)

0:2 � SOCf � 0:8 RI 2min nb

(37)

Additionally, physical limitations should also be satisfied:

(36)

Fig. 5. The solution of the CP for the PHEV in real-world speed profile of Xi’an. 7

(42)

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Journal of Power Sources 450 (2020) 227638

where Tegu and negu are the torque and rotational speed of the EGU, and the indexes min and max are the lower and upper boundary values, respectively. Using the following definitions, 8 > > > z ¼ Jb > > > > �1 � � � > > Qb Voc cb MðcÞ z 31700 > > > c ¼ exp > 1 > 20 zR1 T1 2000p1 > > > > < 587880 ​ ​ (44) c2 ¼ > > 2p > 1 zR1 T1 Qb > > > > > dE > > x¼ > > dt > > > > > > y ¼ nb :

Table 5 The optimal solution generated by convex programming. Final SOC

TC (CNY)

FCC (CNY)

ECC (CNY)

EBLLC (CNY)

Optimal number of battery modules

Optimal battery energy (kWh)

0.369 0.20 0.60

78.10 81.80 79.26

6.28 22.33 32.14

45.85 35.83 27.86

25.98 23.64 19.76

226 134 239

86.78 51.46 91.78

Note: TC: total cost, FCC: fuel consumption cost, ECC: electricity consumption cost, EBLLC: equivalent battery life loss cost.

Equation (28) can be rewritten as follows, (45)

jxj

z ¼ c1 jxjec2 y

If x � 0, the Hessian Matrix of 2 2 3 2 ∂ z ∂2 z 6 2 7 ∂x∂y 7 c1 c2 c2 x 6 6 ∂x 6 7¼ H¼6 e y6 7 6 2 4 y 4 ∂z ∂2 z 5 2 ∂y∂x ∂y

z, denoted as H, can be computed as x c2 þ 2 y 2x y

c2

x2 y2

2x y

3 x2 y2 7 7 7 x3 5

c2

2x2 þ c2 3 y2 y

(46)

2

In Matrix (46), ∂∂x2z > 0 and the determinant jHj ¼ 0, so H, the Hessian matrix of Jb , is positive semi-definite, that is, Equation (45) is a convex function of x and y when x � 0. Also, if x � 0, Jb is a monotonically increasing function of x. Moreover, since Jb is an even function of ​ x, it is a monotonically decreasing function of x when x � 0. Therefore, it can

Fig. 6. Total cost versus final SOC. Case 1: battery aging is included in the objective function. Case 2: battery aging is not included in the objective function.

8 < Tegu;min � Tegu ðtÞ � Tegu;max Pegu;min � Pegu ðtÞ � Pegu;max : negu;min � negu ðtÞ � negu;max

. be concluded that Jb is a convex function of nb and dEðtÞ dt Based on the above analysis, it can be concluded that Jb is convex. Also, it can be easily shown that the term ce ðSOC0 SOCf ÞQb Voc nb in the cost function is also convex with respect to nb . The convexity of the term m_ f was ensured by using a piecewise linear function to represent it, as described in Section 2. As a result, both cost functions (29) and (30) are convex, and any local optimal solution will also be the global solution.

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Table 6 Optimal solutions for cases with and without battery degradation in the convex programming objective function. EBLLC is included in the total cost for both. Battery aging included or not Included (case 1) Not (case 2)

Optimal solution Final SOC

TC (CNY)

FCC (CNY)

ECC (CNY)

EBLLC (CNY)

Number of battery modules

Battery energy (kWh)

0.369 0.434

78.11 79.62

6.28 2.60

45.85 49.17

25.98 27.54

226 276

86.78 105.98

Fig. 7. (a) SOC versus route time for routes optimized using CP and PMP; (b) Battery output power versus route time for routes optimized using CP and PMP. 8

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Journal of Power Sources 450 (2020) 227638

Table 7 Comparison of optimized costs using CP and PMP. Method

TC (CNY)

FCC (CNY)

ECC (CNY)

EBLLC (CNY)

PMP CP

77.94 78.10

6.17 6.28

45.98 45.85

25.78 25.98

4.3. Numerical solution In this paper, the formulated CP is solved using the MATLAB CVX toolbox, with the solver Sedumi [34]. The framework for solving the CP is given in Table 4. 5. Results and discussion A measured real-world speed profile from a round-trip bus route in Xi’an, China, as shown in Fig. 4, was used as the input to assess the proposed method. The speed profile lasts about 5.2 h, covering a total distance of about 70 km. The initial SOC is set to 0.9. Compared to conventional hybrid electric vehicles, plug-in hybrid electric vehicles are usually equipped with a larger battery pack to provide more power at a lower cost by the electricity in the battery pack and overcome the long-range anxiety. If the distance of the testing driving cycle is short, for example, a New European Driving Cycle (NEDC, 1180s, and 11.007 km), then pure electric mode will be the best solution for this task (because electricity provides power at a lower cost than fossil fuels), and the hybrid mode cannot be activated. Therefore, longer driving cycle is needed to trigger the hybrid mode and optimizing the power split ratios between different energy sources is necessary for the hybrid mode. The total cost and the optimal number of battery modules or total capacity corresponding to different final SOCs obtained by solving the CP are shown in Fig. 5(a). For each preset final SOC, there is an optimal number of battery modules/capacity. The total cost with respect to the preset final SOC for all cases shows an approximate upward parabola, and the turning point of the curve, which reaches the minimal total cost, can be identified. In fact, the existence of such a minimum in the total cost curve is guaranteed by the convexity of the cost function in Equa­ tion (30). The optimal solution has a total cost of 78.1 CNY, with the final SOC set to 0.369, and an optimal number of 226 battery modules. In contrast, if the final SOC is set to a different value, for example, 0.20, which represents a common DOD. This means supplying more (inexpensive) energy from the battery, and the total cost increases to 81.80 CNY, which is 3.7 CNY greater than that of the optimal value when the final SOC is a variable. This can be explained as follows. With the final SOC set to 0.20, the optimal number of battery modules ac­ cording to the CP drops to 134, from the 226 determined at the global optimum, greatly reducing the total capacity of the battery. Meanwhile, the cost of the consumed natural gas (fuel consumption cost or FCC) sharply rises to 22.33 CNY from 6.28 CNY, with a slight drop in EBLLC from 25.98 to 23.64 CNY, as summarized in Table 5. The combined effect of the three individual costs (electricity consumption cost or ECC, FCC and EBLLC) contributes to the increase in the total cost. If the preset final SOC is increased, which could be done as a con­ servative measure for battery longevity, the total cost also increases. For example, increasing the final SOC to 0.60 from the optimal point of

Fig. 8. Total cost (top) and the optimal number of battery modules (bottom) versus final SOC for three initial SOC values.

0.369, leads to a total cost of 79.26 CNY, an increase of 1.16 CNY. The optimal number of battery modules slightly increases, from 226 to 239, and the battery capacity increases from 86.78 to 91.78 accordingly. At the same time, the electricity consumption cost decreases significantly from 45.85 CNY to 27.86 CNY at the minimum. The EBLLC also de­ creases, from 25.98 to 19.76 CNY and the fuel consumption cost is increased significantly, from 6.28 to 32.14 CNY, as shown in Table 5. From Fig. 5(a), it can be found that the total cost increases faster as the final SOC decreases from the optimal value. This is due to the change in optimal battery module number or total capacity. As shown in Fig. 5 (b), the optimal number of battery modules increases slightly from 226 at optimum to a peak value of 241 (at a final SOC between 0.5 and 0.6.), then decreases to 214 as the final SOC is further increased. Relatively, the optimal number of battery modules changes little on the right branch of the total cost vs. final SOC curve. As the final SOC increases, the available electrical energy from the battery decreases, which reduces the ECC and EBLLC. On the left branch, as the final SOC decreases, the optimal battery capacity falls significantly, but the available battery energy increases and consequently, the EBLLC is roughly held constant while the ECC decreases less and FCC increases less compared to the right branch. This combination of effects leads to a higher rate of in­ crease in the total cost on the left branch than the right. ECC and EBLLC usually move together, first, increase, then decrease as final SOC varies from 0.2 to 0.8. FCC moves in the opposite direction, due to the trade-off between energy from the battery and the EGU. The minimum total cost occurs at the maximum of the ECC and the minimum of the FCC. In order to clearly demonstrate the advantage of the proposed method in reducing total cost over the counterpart as well as the con­ ventional method which usually neglects the battery aging cost, a comparison between the model that includes battery degradation in the objective function (case 1, same as above) and the model that does not (case 2) is made in Fig. 6 and Table 6. The three curves plotted in Fig. 6 represent the total cost for case 1, the total cost for case 2, and the total cost for case 2 excluding EBLLC, respectively. As expected, the total cost for case 2 significantly exceeds case 1 for all final SOCs. Moreover, the

Table 8 Optimal solutions with different initial SOCs. Initial SOC 0.80 0.90 0.95

Optimal solution Final SOC

TC (CNY)

FCC (CNY)

ECC (CNY)

EBLLC (CNY)

Number of battery modules

Battery energy (kWh)

0.265 0.369 0.417

78.29 78.10 78.01

6.59 6.28 6.43

45.78 45.85 45.63

25.93 25.98 25.96

224 226 224

86.02 86.78 86.02

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Journal of Power Sources 450 (2020) 227638

Table 9 The sensitivity of the optimal solution to battery price. Battery price (CNY/kWh) 900 1050 1200

Optimal solution Final SOC

TC (CNY)

FCC (CNY)

ECC (CNY)

EBLLC (CNY)

Number of battery modules

Battery energy (kWh)

0.369 0.435 0.630

78.10 82.68 86.24

6.28 17.68 37.90

45.85 37.34 23.03

25.98 27.67 25.32

226 210 223

86.78 80.64 85.63

total cost for case 2 that does not include EBLLC represents the nominal total cost of a conventional method, which ignores battery degradation. The results illustrate that the total cost of case 2 (denoted as the red dashed curve) actually exceeds case 1, which is a result of the increased EBLLC due to the battery degradation being ignored in case 2. Also, in both cases, a global optimum is determined. When battery degradation is excluded, the optimal final SOC is 0.434, which is higher than case 1. The results in Table 6 demonstrate that the optimal total cost can be reduced by considering battery aging. Excluding battery aging leads to the consumption of more energy from the battery, leading to a higher EBLLC. Note that, in case 2, the optimal number of battery modules has increased from 226 in case 1 to 276, with the optimal battery energy increasing from 86.78 to 105.98 kWh. In addition, it has been demonstrated that despite a global optimi­ zation approach, dynamic programming has a low calculation efficiency [35] compared to the Pontryagin’s minimum principle (PMP). So, the PMP herein is employed to evaluate the performance of the power split of the CP by using the original nonlinear model of the powertrain. The SOC and battery output power versus time along the route are plotted in Fig. 7(a) and (b), for both methods. The results show essentially over­ lapping SOC and battery power profiles. Table 7 also shows similar FCC, ECC, EBLLC and total cost.

implementing the convex programming. 6. Conclusions This paper proposes a method to optimize battery size and energy management for a PHEV simultaneously. Unlike the existing studies which use a predefined final SOCs (essentially, a fixed DOD when the initial SOC is fixed), the battery depth-of-discharge (DOD) is treated as a variable in this study, and battery degradation is also included in the cost function. The proposed method is implemented using convex pro­ gramming and tested using a real-world city bus speed profile with a length of about 70 km. The conclusions of this study are: (1) There is an optimal DOD to achieve a minimal sum of FCC, ECC, and EBLLC, for the simultaneous optimization of battery sizing and energy management, whether or not battery aging is considered. The numerical results of the CP show that the total cost with respect to final SOC is an upward parabolic curve, where a solution corresponding to an optimal final SOC, battery size, and power distribution resulting in minimal total cost can be determined for a given initial SOC. (2) With an initial SOC of 0.9, when the final SOC is reduced from the optimal level of 0.369 to 0.2, the total cost grows rapidly. When the final SOC value is increased from 0.369, the total cost in­ creases more slowly. (3) A sensitivity analysis suggests that changing the initial SOC be­ tween 0.8 and 0.95 did not have a significant effect on optimal DOD or total cost. In a separate analysis, it was found that increasing battery price lowered the optimal DOD, shifted more of the energy from the EGU, and increased the total cost.

5.1. Sensitivity analysis 5.1.1. Sensitivity to initial SOC Three different initial SOCs (0.80, 0.90 and 0.95) are used to assess the effect of variable battery charge levels. The results in Table 8 show that the optimal final SOC increases with initial SOC. That is, the optimal DOD remains roughly constant (0.535, 0.531, and 0.533 for the three cases, respectively). The optimal number of battery modules or capacity, as well as the total cost, FCC, ECC, and EBLLC are very close among these three cases. Fig. 8 shows the optimized total cost and number of battery modules versus the final SOC in the three cases. The results show that when the final SOC is set to a high level, for example, between 0.45 and 0.8, the optimal number of modules drops slightly with increasing final SOC. When the final SOC is smaller, for example, between 0.2 and 0.3, the optimal number of battery modules and total cost are more sensitive to changes in the initial SOC.

Future research will focus on the co-optimization of battery size, charging, power split, and battery longevity, with consideration of the depth of discharge. Acknowledgments This part was supported by the Shaanxi Province Natural Science Foundation (Grant No. 2019JQ-439), Fundamental Research Funds for the Central Universities (Grant No. 300102319307), National Natural Science Foundation of China (Grant no. 51875054), Chongqing Natural Science Foundation for Distinguished Young Scholars (Grant No. cstc2019jcyjjq0010), and Natural Sciences and Engineering Research Council of Canada (NSERC) through the Discovery Grant Program (RGPIN-2018-05471).

5.1.2. Sensitivity to battery price To explore the effect of battery price on the optimal solution, two additional cases, with battery prices of 1050 CNY/kWh and 1200 CNY/ kWh, are considered, using an initial SOC of 0.9 (note that the baseline cases above use a battery price of 900 CNY/kWh). The results in Table 9 indicate that the optimal final SOC and the total cost increase signifi­ cantly with the increasing battery price, and the DOD falls to 0.465 and 0.270 from the baseline value of 0.531. Conversely, fuel consumption is increased, which increases the total cost. EBLLC is relatively unchanged as the battery price increases. This is because, despite the battery price increase, the optimal final SOC is also increased, holding the EBLLC at a stable level. Additionally, it should be noted that DOD and the current flowing into and out of the battery (which contribute to battery degradation) basically include information on SOC and discharge capacity when

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