Adaptively coordinated optimization of battery aging and energy management in plug-in hybrid electric buses

Applied Energy 256 (2019) 113891

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

Adaptively coordinated optimization of battery aging and energy management in plug-in hybrid electric buses Shuo Zhanga,1, Xiaosong Hua,

⁎,1

T

⁎

, Shaobo Xieb, Ziyou Songc, , Lin Hud, Cong Houe

a

The State Key Laboratory of Mechanical Transmissions, Department of Automotive Engineering, Chongqing University, Chongqing 400044, China The School of Automotive Engineering, Chang’an University, Xi’an 710064, China c Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, MI 48109, USA d School of Automotive and Mechanical Engineering, Changsha University of Science and Technology, Changsha 410114, China e Powertrain R & D Institute, Chongqing Chang’an Automobile Company Ltd., Chongqing 400023, China b

H I GH L IG H T S

battery aging model is exploited in the optimization. • High-fidelity online coordinated optimization approach is developed. • An under three optimization scenarios are systematically compared. • Results economy of designed compromise scheme is comprehensively analyzed. • The • Optimization performance of the proposed control methodology is assessed.

A R T I C LE I N FO

A B S T R A C T

Keywords: Plug-in hybrid electric vehicle Energy management Pontryagin’s minimum principle Battery degradation Fuel economy Co-optimization

Plug-in hybrid electric buses with large battery packs exhibit salient advantages in increasing fuel economy and reducing toxic emissions. However, they may be subject to expensive battery replacement caused by battery aging. This paper designs an online, coordinated optimization approach, based on Pontryagin’s minimum principle, for a single-shaft parallel plug-in hybrid electric bus, aiming at minimizing the total cost of energy consumption and battery degradation. Specifically, three key contributions are delivered to complement the relevant literature. First, a capacity loss model for lithium ion batteries emulating dynamics of both cycle life and calendar life is exploited in the optimization framework, in order to highlight the importance of considering calendar life and its implication to overall energy management performance in real bus operations. Second, the online adaptive mechanism of the optimization method with respect to varying driving conditions is achieved by tracking two reference trajectories to adjust the state of charge and effective ampere-hour throughput of the battery. Finally, to verify the effectiveness of the proposed scheme, various comparative studies are carried out, accounting for different driving scenarios. Simulation results show that the maximum control errors between the proposed strategy and Pontryagin’s minimum principle are only 0.4% in the battery capacity loss and 2.7% in fuel economy under four random driving cycles, which indicates the prominent adaptability and optimization performance of the designed strategy.

1. Introduction Energy crisis, air pollution, and the staggering growth of mobility demands have been potently promoting the development of energysaving and environment-friendly automobiles [1,2]. As a thriving technology to counteract existing problems, electrified vehicles, such as hybrid electric vehicles (HEVs) and fuel cell electric vehicles (FCEVs),

are being intensively investigated [3]. Particularly, plug-in hybrid electric vehicles (PHEVs) recently have gained much attention in the public transit transport, because of their remarkable potential of improving fuel economy and reducing pollutants [4–7]. Compared with hybrid electric buses (HEBs), plug-in hybrid electric buses (PHEBs) with larger energy storage systems, e.g., lithium iron phosphate (LiFePO4) batteries, typically have a higher cost for maintaining and/or replacing

⁎

Corresponding authors at: No. 174, Shazhengjie, Shapingba District, Chongqing 400044, China (X. Hu). E-mail address: [email protected] (X. Hu). 1 Eqaully contributed to this research work. https://doi.org/10.1016/j.apenergy.2019.113891 Received 31 May 2019; Received in revised form 7 September 2019; Accepted 7 September 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

Pd mv g f θ A Cd ρair δ r va Teng Tem iT i0 ηT sign gav ṁ f Peng ωeng ηeng Qlhv Pem ωem ηem Pbat ̇ SOC Qbat Ibat Uoc Rint Qtot Qcyc Qcal Crate

gas constant Rgas Ea activation energy z power law factor σ severity factor cycling-induced capacity loss until EoL Qcyc, EoL effective Ah-throughput Aheff costf fuel cost coste electricity cost costb battery aging cost total operating cost costtot cf fuel price ce grid electricity price cbat price for replacing the battery ω weighting factor ωr reasonable weighting factor λ 0 (t ) co-state corresponding to SOC xP (t ) state variable uP (t ) control variable xṖ (t ) dynamics of the state SOCinitial initial SOC SOCtarget expected final SOC Ichg charging current Idis discharging current λ1 (t ) co-state corresponding to Aheff s1 (t ) , s2 (t ) adaptive factors of DA-ECMS kp1, kp2 proportional coefficients integral coefficients ki1, ki2 λ 0opt (0) optimal initial co-state of λ 0 (t ) λ1opt (0) optimal initial co-state of λ1 (t ) SOCref reference SOC reference Aheff Ahref total travel distance of the PHEB dn Dspl mileage of one driving mission opt Aheff optimal effective Ah-throughput c Aheff effective Ah-throughput for a single charge vn velocity vector of the random running cycle vspl velocity vector of the driving cycle sample pn noise intensity

power demand at wheels vehicle mass gravity acceleration rolling resistance coefficient slope of the road fronted area air drag coefficient air density mass factor wheel radius vehicular speed engine torque motor torque gear ratio of the AMT final drive ratio AMT efficiency sign function set of feasible gear positions instantaneous fuel consumption engine output power engine speed engine efficiency lower heating value of the fuel motor output power rotating speed of the motor motor efficiency battery terminal power dynamics of the battery SOC battery nominal capacity battery current open circuit voltage of battery internal resistance of battery total capacity loss capacity loss induced by cycling aging capacity loss induced by calendar aging current rate

machine learning (ML) [16], have been also suggested. The DP approach can guarantee the globally optimal performance [17,18]. However, it is an optimization routine with a high computational cost, because of requiring handling grid selection and solution interpolations. The PMP, which permits redefining the global optimization problem in terms of local conditions, has a higher computational efficiency and is often regarded as an alternative to DP [19]. Given that both DP and PMP require prior information of driving cycle, which generally cannot be obtained in online applications [20], the solutions of these methods are usually used as offline benchmarks to measure other suboptimal yet causal strategies. With the predicted speed or power demand, DP or PMP can be applied in a moving horizon to build a model predictive controller for energy management purposes [21]. The uncertainty of prediction in such an MPC often contributes to a suboptimal solution compared with the global ones, and the length of the predicted horizon may result in an infeasible computational burden for onboard controllers. Among various instantaneous optimization methods, ECMS is more practical to be implemented in real-time, because of its lower computing cost [22,23]. Nonetheless the challenge of ECMS is to adaptively tuning its equivalent factor, which is highly related to driving environment. Fuel consumption reduction and battery life extension are always conflicting in the vehicle-level power-flow control. Therefore, some existing papers investigated techniques to simultaneously minimize battery aging and fuel consumption. For HEVs, a methodology base on

battery packs. Therefore, how to prolong PHEB battery life should be carefully considered in the design of supervisory energy management strategies for PHEBs. Energy management strategies for PHEVs that directly determine fuel economy and power performance of their propulsion systems, generally, can be divided into two categories: (i) the charge-depleting and charge-sustaining (CD-CS) strategy; (ii) the blended discharging strategy [8,9]. For the CD-CS strategy, PHEVs operate in an electriconly mode (i.e., CD mode) first, and then switch to the CS mode when the battery state of charge (SOC) drops to a pre-set threshold [10]. The CD-CS strategy can be easily implemented online, but it provably cannot achieve the optimal performance [11]. By contrast, the blended discharging strategy possesses an edge on improving fuel economy, despite it sometimes requires prior knowledge of driving cycle, such as driving speed and distance [12]. For PHEBs repeatedly operating under fixed routes, the roadway information can be potentially extracted from historical driving data, which offers opportunities for blended discharging controls. A wide variety of methods, including global optimization methods (e.g., dynamic programming (DP), convex programming, and Pontryagin’s minimum principle (PMP)) and instantaneous optimization methods (e.g., equivalent consumption minimization strategy (ECMS) and model predictive control (MPC)) have been reported to synthesize blended control strategies. In addition, data-driven methods, e.g., particle swarm optimization (PSO) [13], genetic algorithm (GA) [14], simulated annealing (SA) [15], and 2

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framework? (2) How will different penalties on battery aging affect the PHEB economy during its entire service life? (3) Is it possible to conceptualize effectively adaptive control architectures to optimize battery aging and power management online? How about the performance of such controllers in real time?

PMP is developed in [24] to include a capacity fading model in an optimal energy management problem. The study in [25] mathematically treats the battery state-of-health (SOH) as one of the powertrain model states and designs two separate controllers to regulate the SOH and SOC for realizing casual energy management. Since battery temperature is a prime factor affecting battery aging, Ref. [26] proposes a method to improve the SOH regulation by imposing soft constraints on battery internal temperature. Additionally, an analytical solution considering battery life, is presented for a hybrid electric truck in [27]. Moreover, ECMS with a constant equivalent factor is employed in [28] to optimize the energy management with the consideration of battery aging. For PHEVs, a battery-health-conscious energy management strategy is developed via stochastic dynamic programming (SDP) to explore interactions between fuel consumption and battery health [29]. Furthermore, an integrated control scheme concurrently considering battery charging, on-road energy management and battery degradation alleviation is constructed to minimize the daily operation cost of a PHEV via convex programming [30]. In Ref. [31], SDP and PSO are leveraged conjunctively to solve the multi-objective optimal control problem involving energy consumption and battery health degradation. Instead of considering battery aging in the objective function, the authors of Ref. [32] propose a novel MPC considering the optimal battery depth of discharge to minimize the total cost of fuel consumption and battery life loss. For a fuel cell/battery hybrid vehicle system, the SDP controller is designed to minimize the total operating cost, with both fuel consumption and fuel cell degradation considered simultaneously, by reducing the transient loading on the fuel cell stack [33]. Focusing on fuel cell and battery health, the study of [34] embeds the aging models of the both power sources into an energy management framework, where DP is used to solve the optimization problem. As a technology to enhance battery durability, hybrid energy storage system comprising batteries and ultracapacitors has been developed. For example, a neural network-based power management strategy is implemented to achieve real-time current allocation between both energy storage devices, with a reduction of SOH decay rate [35]. Achieving a reasonable trade-off between fuel economy and battery life in PHEVs is much more complicated than that in HEVs, due to an additional consideration of charging behavior and wider-range variations of SOC. Most aforementioned researchers merely analyzed interactions between battery cycle aging and fuel consumption. However, they ignored the impact of battery calendar aging on the optimization results and did not construct a sufficiently rational compromise scheme between battery aging and energy consumption from a perspective of the lifecycle cost. Meanwhile, multi-objective optimization problems also bring challenges to the online implementation of control strategies. The existing instantaneous optimization strategies mainly rely on two measures to obtain longer battery life: (1) imposing soft constraints on the SOH or capacity loss (direct control) [25,36]; (2) exerting management to the temperature or depth of discharge (indirect control) [26,32]. However, both control schemes exhibit limitations. For the former scheme, the timescales of SOH and capacity loss are typically several orders of magnitude larger than that of SOC, perhaps resulting in over-sensitivity of the control system to noise interference. Although manipulating the SOH at a substantially larger sampling interval can lessen this defect to some degree [25], it inevitably weakens the optimality of the results, because the SOH cannot be controlled synchronously with the SOC in real time. For the latter one, notwithstanding temperature control and optimum depth of discharge can slow down battery aging rate, the impact of high current rates on battery SOH cannot be completely avoided. In light of the above-mentioned downsides, the following interesting questions, pertaining to tradeoffs between battery aging and energy consumption, are far from resolved:

This article aims to address the above questions by making three distinct contributions. First, a high-fidelity capacity loss model emulating both calendar aging and cyclic aging dynamics for LiFePO4 batteries, for the first time, is integrated in an optimization framework, to explore the interplay between battery aging and energy consumption, with emphasis on the significance of keeping a good account of calendar life. Second, based on the PMP, the adaptively coordinated optimization framework is implemented by separately controlling the SOC and effective Ah-throughput to realize online, near-optimal battery usage and power distribution. Ultimately, numerous results under three different optimization scenarios (i.e., the energy-consumption-only case, the coordinated case, and the over-weighted battery-aging case) are systematically compared. Moreover, the effectiveness and adaptability of the proposed online optimization method are substantiated under varying driving conditions. The rest of the paper is organized as follows. Section 2 describes the parallel hybrid powertrain modeling, and Section 3 presents the battery aging model. The online coordinated optimization scheme is elucidated in Section 4. The results are shown in Section 5, with main conclusions summarized in Section 6. 2. Powertrain modeling The investigated PHEB has a single-shaft parallel powertrain equipped with an AMT, as shown in Fig. 1. The powertrain has two energy conversion units: a diesel engine and a permanent magnet synchronous machine. The shift of AMT can be dynamically controlled to improve the powertrain efficiency. Table 1 shows the main parameters of the PHEB. In the following subsections, a quasi-static modeling approach is employed to describe the system dynamics [37]. 2.1. Longitudinal dynamic model According to the vehicle longitudinal dynamics, the power demand Pd at wheels can be computed as

1 dv Pd = ⎡m v gf cosθ + ACd ρair va2 + m v g sinθ + m v δ a ⎤ va 2 dt ⎦ ⎣

(1)

where m v is the vehicle mass, g is the gravity acceleration, f is the rolling resistance coefficient, and θ is the slope of the road. Moreover, A is the fronted area, Cd is the air drag coefficient, ρair is the air density, δ is the mass factor caused by the rotating inertia of wheels and powertrain rotating components, r is the wheel radius, and va is the vehicular

(1) Can battery calendar life be neglected in the combined optimization of battery degradation and energy consumption? If not, how should it be incorporated, alongside cycle life, into the optimization

Fig. 1. Powertrain schematic of the plug-in hybrid electric bus. 3

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engine regions, Tem, min (ω (i)) and ωem, max are the minimum torque and maximum rotating speed of the motor, respectively. Additionally, card (gav ) is the number of elements contained in the set of feasible gear positions gav , and ηem is the vector of motor efficiency.

Table 1 Main parameters of PHEB [38]. Item

Description

Vehicle Mass Engine Motor AMT

14,500 kg Diesel engine, normal power: 130 kW Permanent magnet synchronous machine, normal power: 110 kW 12-speed, ratios: 1/1.2/1.5/2.1/2.6/3.36/4.4/5.5/7/9/12/15, Efficiencies: 0.94/0.94/0.94/0.94/0.94/0.96/0.96/0.96/0.96/ 0.96/0.96/0.98 Ratio: 4.7 Lithium iron phosphate, capacity: 69 Ah, total voltage: 511.5 V

Final Drive Battery pack

2.2. Engine model The engine efficiency is determined by its torque and speed, and its instantaneous fuel consumption is calculated as

ṁ f =

speed. Meanwhile, the engine and electrical motor can supply the demanded torque together, and the power balance can be expressed as

Pd = (Teng + Tem ) iT i 0

va ηT (iT ) sign (Tem ) r

Peng ηeng (ωeng , Teng ) Qlhv

Peng = Teng ωeng

(3) (4)

where Peng is the engine output power, the engine efficiency ηeng is expressed as a function of the engine speed ωeng and torque Teng , and Qlhv is the lower heating value of the fuel.

(2) 2.3. Motor model

where Teng and Tem are the engine torque and motor torque, respectively, iT is the gear ratio of the AMT, i 0 is the final drive ratio, ηT is Tem > 0 1 the AMT efficiency, and sign (Tem) = ⎧ is a sign function. ⎨− 1 Tem < 0 ⎩ Although the gear-shifting operation can be optimized together with the power-split decision, it complicates the optimization process and therefore significantly increases the computational burden. In this paper, a rule-based gear-shifting strategy is proposed to ensure that the engine can run in high-efficiency regions, and the motor can recover as much braking energy as possible. The shift-decision logic is illustrated in the Fig. 2, where k is the time index in discrete time, ω thr , min and ωthr , max are the preset speed thresholds corresponding to high-efficiency

The motor can operate in either a driving or a generating mode to support driving or regenerative braking torque. The motor efficiency is a function of its torque and rotational speed (see Eq. (5)), and its output torque Tem and power Pem are given by Eqs. (6) and (7), respectively.

ηem = f (ωem , Tem)

(5)

min(Tem, req, Tem, max (ωem )) if Tem, req > 0 Tem = ⎧ ⎨ ⎩ max(Tem, req, Tem, min (ωem )) if Tem, req < 0

(6)

Pem = Tem ωem

(7)

Fig. 2. Flow chart of gear-decision strategy. 4

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to a realistic battery operating condition. And the values of Ahnom and Ahcyc can be calculated by

where ωem is the rotating speed of the motor, Tem, req is the motor torque request, Tem, max (ωem) and Tem, min (ωem) are the torque bounds in the traction and braking modes, respectively. 2.4. Battery circuit model

Ahnom

The battery pack is represented by an equivalent circuit model considering the open circuit voltage Uoc and the internal resistance Rint , which is depicted as

Ibat =

Uoc −

Uoc2

−sign (Tem) Pbat = Pem ηem

̇ =− SOC

Ibat Qbat − Qtot

(

(8)

(10)

Aheff (t ) =

ΔAh =

)

⎤ ⎥ ⎥ ⎥ ⎦

(15)

1 3600

∫0

t

σ (τ )|Ibat (τ )| dτ

(16)

1 3600

∫t

tk + 1

k

−Ea + η ·Crate ⎞ z−1 ⎟ Ahk Rgas ·TK ⎠

|Ibat (t )| dt

(17) (18)

Ahk + 1 = Ahk + ΔAh

(19)

where Qcyc, k and Qcyc, k + 1 denote the capacity loss at the time instants tk + 1 and tk , respectively; ΔAh is the Ah-throughput from tk to tk + 1; Ahk and Ahk + 1 are the accumulated Ah-throughput until the time instants tk + 1 and tk , respectively.

(11)

where Qcyc represents the capacity loss induced by cycling aging, Qcal represents the capacity loss caused by calendar aging. Generally, 20% capacity loss corresponds to End of Life (EoL) in automotive battery applications.

3.2. Calendar aging model For the calendar aging model, two main factors (i.e., the battery SOC and temperature) are taken into account. To balance the accuracy and simplicity, a semi-empirical calendar aging model is adopted [46]:

3.1. Cycle aging model Numerous cycle aging tests were performed for the LiFePO4 battery cells to develop control-oriented aging models [42–44]. Wang et al. [43] employed an Arrhenius law to establish a semi-empirical cycle model considering stress factors of the ambient temperature and current rate (C-rate). Furthermore, Suri et al. [44] calibrated a degradation model according to experimental data of HEV LiFePO4 battery. The cycle aging model [44] is adopted in this paper as follows:

Qcal = exp(k1·SOC + k2)·exp(k3/ T + k 4 )·t 0.5

(20)

where k1, k2 , k3 and k 4 are coefficients that can be calibrated using a least-squares method to fit the experimental data in [46], as shown in Fig. 3. Thereby the final calendar aging model used in this paper is given as

Qcal = exp(0.82·SOC + 9.02)·exp( −4427.08/ T + 3.85)·t 0.5

(21)

Note that the battery aging model is based on the following assumptions often used for vehicle-level energy management:

(12)

where α and β are fitting coefficients, Ea is the activation energy, η is the compensation factor of Crate , Rgas is the gas constant, and TK is the ambient temperature in [K] when testing batteries. Further, Ah is the Ah-throughput, and z is the power law factor. The parameters of the model are listed in Table 2. To quantify the aging effect in realistic cycles, in light of Ref. [45], a severity factor σ is defined as

Ahnom (SOCnom, Crate, nom, TK , nom) Ahcyc (SOC , Crate, TK )

(14)

1 z

Qcyc, k + 1 − Qcyc, k = ΔAh·z (αSOC + β )·exp ⎜⎛ ⎝

The primary degradation mechanisms inducing battery aging include: (i) loss of recyclable lithium; (ii) loss of active materials; and (iii) impedance growth [39]. In this work, an ANR26650 LiFePO4 battery pack, comprising cylindrical cells manufactured by A123 systems, is considered. It has been proven that the SEI growth is the dominant aging mechanism for this battery, and a key manifestation of its aging is capacity loss [40,41], as indicated by

σ (t ) =

)

which is used to characterize the effective cycle-life depletion owing to charge exchange within the battery [24]. In this paper, Qcyc is optimized by minimizing Aheff . Then, discrete-time equations of dynamic cycle aging are given by

3. Battery aging model

−E + η ·Crate ⎞ z Qcyc = (αSOC + β )·exp ⎜⎛ a ⎟ Ah Rgas ·TK ⎝ ⎠

1 z

⎤ ⎥ ⎥ ⎥ ⎦

where Qcyc, EoL is the cycling-induced capacity loss until EoL. Note that Qcyc, EoL needs to be estimated based on engineering experience, since it is closely related to calendar aging and cannot be obtained beforehand. Thus, the estimated value of Qcyc, EoL is set at 15%. Based on the concept of the severity factor, the effective Ahthroughput is defined as [45]

(9)

where Ibat is the battery current (positive for discharge and negative for charge), Qbat is the battery nominal capacity, Qtot is the total capacity loss induced by battery aging (see Eq. (11) in Section 3), Pbat is the ̇ is the dynamics of the battery SOC. battery terminal power, and SOC The sign function works in the same way as that in Eq. (2). For simplicity, vehicular auxiliary power consumption is neglected herein.

Qtot = Qcyc + Qcal

(

⎡ Qcyc, EoL Ahcyc = ⎢ ⎢ −Ea + η·Crate ⎢ (αSOC + β )·exp Rgas·TK ⎣

− 4Rint Pbat

2Rint

⎡ Qcyc, EoL =⎢ ⎢ −Ea + η·Crate, nom ⎢ (αSOCnom + β )·exp Rgas·TK , nom ⎣

Table 2 The parameters of the cycle aging model [44]. Battery Parameter

Value

Fitting coefficient α

⎧ 2896.6, ⎨ ⎩ 2694.5, ⎧ 7411.2, ⎨ 6022.2, ⎩

Fitting coefficient β

(13)

where Ahnom is the total Ah-throughput until EoL, with respect to a nominal test condition defined as SOCnom = 0.35, Crate, nom = 2.5C , and TK , nom = 298.15 K [24]; Ahcyc is the total Ah-throughput corresponding 5

Compensation factor η Activation energy Ea (J/mol) Gas constant Rgas [J/(mol·K)]

152.5 31,500 8.314

Power law factor z

0.57

SOC ≤ 0.45 SOC > 0.45 SOC ≤ 0.45 SOC > 0.45

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with t

f ⎧ costf = ∫t0 cf ṁ f dt ⎪ tf Pbat ⎪ coste = ∫t ce 3600 dt 0 tf ⎨ σ | Ibat | ⎪ costb = ∫t0 cbat 3600Ahnom dt ⎪ costtot = costf + coste + costb ⎩

where cf and ce are the prices of fuel and electricity, respectively; the price for replacing the battery is represented by cbat ; ω is a weighting factor to address the battery aging; costtot is the total operating cost. The overall framework is shown in Fig. 4. In the offline control frame, the PMP method is used to solve the bus operational costs with respect to different weighting factors. And the reasonable weighting factor ωr is determined by trial-and-error to extend the battery life to meet the bus life. Meanwhile, the PMP's solution corresponding to ωr is regarded as an ideal compromise scheme (ICS) between energy consumption and battery aging. In the online control frame, an adaptively coordinated optimization control strategy inspired by the ECMS is proposed. Like a single-state adaptive ECMS (SA-ECMS), the state constraints can be satisfied by tracking the reference SOC trajectory based on the SOC feedback. However, for a problem where the SOC and capacity loss of the battery are required to achieve the desired goals concurrently, the single-state adaptive ECMS is not competent. Specifically, based on the ICS obtained by previous offline control, a dual-state adaptive ECMS (DAECMS) is developed.

Fig. 3. Fitting curves of the calendar aging model (data are from [46]).

• The cells in the battery pack have perfect consistency, and each cell can be used to represent the condition of the battery pack; • The battery thermal management system can ensure that the battery works at an average temperature of 30 ℃.

4. Coordinated optimization approach The cost function of the studied optimal control problem includes the fuel cost costf , electricity cost coste , and concomitant battery aging cost costb :

J=

∫t

tf

0

⎛cf ṁ f + ce Pbat + ωcbat σ |Ibat | ⎞ dt 3600 3600Ahnom ⎠ ⎝

⎜

(23)

4.1. PMP framework According to the PMP method, the Hamiltonian function of Eq. (22) can be described as

⎟

(22)

Fig. 4. The overall control strategy framework. 6

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H = cf ṁ f + ce

Pbat σ |Ibat | ̇ + ωcbat + λ 0 (t ) SOC 3600 3600Ahnom

consumption and battery aging, respectively when proper coefficients λ 0 and λ1 or factors s1 and s2 are chosen. Alternatively, the instantaneous optimal cost function can be written as

(24)

where λ 0 (t ) is the co-state variable. The state variable xP (t ) and the control variable uP (t ) are represented by

xP (t ) = SOC (t ),

G = cf ṁ f + s1 (t ) ce

(25)

uP (t ) = Pem (t )

Then the dynamics of the state and co-state can be expressed as

xṖ (t ) =

(26)

t

⎧ s1 (t ) = s10 + kp1 (SOCref (t ) − SOC (t )) + ki1 ∫0 (SOCref (t ) − SOC (t )) dt

∂H ∂SOC ̇ ∂ṁ f ce ∂Pbat ωcbat σ |Ibat | ∂SOC = − cf − − − λ 0 (t ) ∂SOC 3600 ∂SOC 3600Ahnom ∂SOC ∂SOC

λ 0̇ (t ) = −

⎨ s2 (t ) = s20 + kp2 (Ahref (t ) − Aheff (t )) + ki2 ∫t (Ahref (t ) − Aheff (t )) dt 0 ⎩ (34) (27)

where

To guarantee global optimality, the following necessary condition (Eq. (28)) and boundary conditions (Eq. (29)) should be satisfied:

H (x p∗ (t ), uP∗ (t ), λ 0∗ (t ), t ) ≤ H (x p∗ (t ), uP (t ), λ 0∗ (t ), t )

⎧ s10 = 1 − ⎪ ⎨ ⎪ s20 = 1 + ⎩

(28)

∗ ⎧ SOC (t0) = SOCinitial SOC ∗ (t f ) = SOCtarget ⎨ ⎩

where the superscript ∗ is the identifier of optimal solution; SOCinitial is the initial SOC, and SOCtarget is the expected final SOC, both of which are preset values. Meanwhile, the physical constraints for states and control inputs are

(36) where Rd (·) is a round-down function for returning the integer part of the value; dn is the total travel distance; Dspl is the mileage of one driving mission between two adjacent charging events (see red points in Fig. 5 in Section 5). Given that the effective Ah-throughput is cumulative along the driving distance, an approximate linear relationship between effective Ah-throughput and driving distance is adopted, Table 3 Pseudocode of the PMP Algorithm.

For the studied PHEB commuting on a fixed route, the variations in the speed profile are caused by varying traffic flow and different driver’s driving styles. To track optimal battery capacity degradation of the ICS, an additional state variable Aheff is introduced here. According to Eq. (22), the Hamiltonian function can be transformed as follows:

Algorithm: PMP Algorithm 1: λ 0 ∈ [a, b]; λ 0opt (0) = (a + b)/2 ; 2:while 1 3:for k = 1:N 4: Pem (k ) = Pem, min (k ): ΔPem: Pem, max (k ) ;

Pbat σ |Ibat | ̇ + λ1 (t ) Aḣ eff + ωr cbat + λ 0 (t ) SOC 3600 3600Ahnom

5:

(31)

8:Calculate severity factor σ (k )← Eq. (13); 9:

∗ [Pem (k ), SOC ∗ (k )] = argumin(t ) H (k ) ; P

10:end 11:if |SOC ∗ (N ) − SOCtarget |≤ ε

1000λ 0 (t ) ⎞ Pbat 3600λ1 (t ) Ahnom ⎞ H = cf ṁ f + ⎛1 − ce + ⎛1 + ωr cbat Ebat ce ⎠ 3600 ωr cbat ⎝ ⎝ ⎠ s⏟ s⏟ 1 (t ) 2 (t ) σ |Ibat | 3600Ahnom (32) ⎜

Peng (k ) = Pd (k )/ ηT (iT ) sign (Pd (k )) − Pem (k ) ;

̇ (k )← Eqs. (3), (9), (10); 6:Calculate ṁ f (k ), Pbat (k ), SOC ̇ (k ) ; SOC (k + 1) = SOC (k ) + SOC 7:

̇ = − Ibat ; Pbat = Ibat Uoc ; Aḣ eff = σ | Ibat | ; λ1 (t ) is the co-state where SOC 3600 Qbat corresponding to Aheff . Then the Hamiltonian is rewritten as ⎟

(35)

ωr cbat

d (t ) ⎞ ⎛ d (t ) SOCref (t ) = SOCinitial − ⎜ n − Rd ⎛⎜ n ⎞⎟ ⎟ (SOCinitial − SOCtarget ) Dspl Dspl ⎠ ⎝ ⎝ ⎠

(30)

4.2. Dual-State Adaptive ECMS (DA-ECMS)

⎜

Ebat ce λ1opt (0) Ahnom

i = eng , em

where Ichg and Idis are the charging and discharging current, respectively; the subscripts max and min denote the maximum and minimum values, respectively. The optimization problem shown in Eq. (24) is a typical two-point boundary value problem that the shooting method can be leveraged to solve [47]. For a given stochastic initial co-state value, an optimal initial co-state value λ 0opt (0) can be captured by the shooting method, and to accelerate the convergence, the dichotomy is introduced, and the pseudocode of this algorithm is described in Table 3.

H = cf ṁ f + ce

1000λ 0opt (0)

where kp1 and kp2 are the proportional coefficients; ki1 and ki2 are the integral coefficients; SOCref and Ahref are the reference SOC and Aheff , respectively; λ1opt (0) is the optimal initial co-state of λ1 (t ) . It should be emphasized that the offline calculation of the PMP only has one state variable, i.e., the battery SOC, so according to the ICS, it can be determined that λ1opt (0) = 0 . As the final SOC is expected to reach the targeted value, a reference SOC trajectory is planned as [23]

(29)

⎧Ti, min (ωi ) ≤ Ti (t ) ≤ Ti, max (ωi ) ⎪ ωi, min ≤ ωi (t ) ≤ ωi, max ⎨ Pi, min (ωi ) ≤ Pi (t ) ≤ Pi, max (ωi ) ⎪I ≤ Ibat (t ) ≤ Idis, max ⎩ chg, max

(33)

where s1 (t ) and s2 (t ) are the adaptive factors of DA-ECMS. The dualstate adaptive control for the DA-ECMS is performed by two proportional-integral (PI) controllers:

Uoc2

− 4Rint Pbat Uoc − ∂H =− ∂λ 0 (t ) 2Rint Qbat

Pbat σ |Ibat | + s2 (t ) ωr cbat 3600 3600Ahnom

⎟

break;

12:elseif SOC ∗ (N ) − SOCtarget < 0 13:

b = λ 0opt (0) ; λ 0opt (0) = (a + b)/2 ;

14:elseif SOC ∗ (N ) − SOCtarget > 0 15: 16:end 17:end

where Ebat = Uoc Qbat is the battery energy [Wh]. The Hamiltonian in Eq. (32) contains three terms, where the second and third terms essentially reveal the optimally instantaneous costs related to electrical

a = λ 0opt (0) ; λ 0opt (0) = (a + b)/2 ;

18:return λ 0opt (0)

7

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Fig. 6. Costs with respect to different values of the weighting factor.

are listed in Table 4.

5.1. Analysis of operating costs under different penalties for battery aging For one driving mission, the operating costs with respect to different weighting factors are depicted in Fig. 6, and the optimal initial values of co-state are shown in Fig. 7. It can be seen that the relationship between cost and weighting factor can be divided into three regions. In the first stage ([0, 2.6]), the fuel consumption cost and the battery aging cost are almost constant because the battery aging accounts for a minor proportion in the total cost, and the energy consumption still dominates the cost function. However, increased penalties for battery aging will modify the SOC trajectory by limiting the electricity consumption. To ensure that the final SOC reaches its target value, the optimal initial costate should be increased to make the SOC decline fast. In the second stage ([2.6, 4.8]), the total cost increases significantly with the growing weighting factor, because the fuel consumption cost increases faster than the battery aging cost declines. The electricity consumption cost even slightly decreases since the increasing weighting factor related to the battery aging cost reduces the battery utilization. At the same time, the optimal initial value of co-state changes slowly, which can be regarded as a neutralizing effect of the energy recovery decrease and the fuel consumption increase. For the third stage (i.e., [4.8, 6.0]), each cost curve tends to be stable again, which indicates that reducing the battery utilization is unallowable when the weight of battery aging is large.

Fig. 5. Gothenburg, Sweden, driving cycle [38,48] and the daily PHEB operation.

d (t ) ⎞ opt d (t ) ⎛ d (t ) opt c Ahref (t ) = ⎜ n − Rd ⎛⎜ n ⎞⎟ ⎟ Aheff + Rd ⎛⎜ n ⎞⎟ (Aheff + Aheff ) Dspl D spl ⎝ ⎠⎠ ⎝ Dspl ⎠ ⎝ (37) opt Aheff

where is the optimal effective Ah-throughput obtained by offline c is the effective Ah-throughput for a PMP for one driving mission; Aheff single charge corresponding to the SOC rising from SOCtarget to SOCinitial . 5. Results and discussion A bus route in Gothenburg, Sweden, is used as the driving scenario, as shown in Fig. 5(a), and the daily operation can be divided into three sections, which are driving, charging, and parking, as shown in Fig. 5(b). The driving process starts at 6AM, 12PM, and 6PM, respectively in each day. And each driving mission consists of four consecutive driving cycles. The battery pack is charged after every driving mission, and the charging time lasts for about half hour by using a fast charging mode. It is assumed that the PHEB travels 365 days a year and reaches its end-of-life after 8 years. Detailed parameters for simulations Table 4 Detailed parameters for simulation [38]. Vehicle Parameter Frontal area A (m2 ) Aerodynamic drag coefficient Cd Rolling resistance coefficient f Wheel radius r (m) Air density ρair (kg/m3 ) [a] Diesel price cf (EUR/L) [a]

Grid electricity price ce (EUR/kWh) [b] Battery pack price cb (EUR/kWh) Fuel lower heating value Qlhv (MJ/kg)

Value

Battery cell Parameter

Value

7.54

Battery cell mass mb (kg)

0.07

0.7 0.007 0.509 1.184

Battery cell resistance Rint (Ω ) Nominal battery capacity Qbat (As) Nominal battery voltage Uoc (V) Average Inverter efficiency

0.01 8280 3.3 0.96

1.46

Maximum discharge current (A)

35

0.17 900 42.426

Maximum charge current (A) Initial SOC SOCinitial Final SOC SOCtarget

−35 0.8 0.3

Note: the battery pack price is an estimate of the producer price for the battery pack formed by ANR26650 LiFePO4 cells, including manufacturing cost and average profit, which may vary by region. [a] Adopted from [49]. [b] Adopted from [30,50]. 8

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Fig. 10. Calendar aging and cycle aging of the ICS.

Fig. 7. Optimal initial co-state with respect to different values of the weighting factor.

Fig. 8. Daily temperature of Gothenburg, Sweden in 2016 [51].

Fig. 11. Optimized motor power profiles for the three optimization scenarios.

Fig. 9. Capacity loss with different weighting factors.

5.2. Comparative analysis of three optimization scenarios The daily temperature of Gothenburg, Sweden in 2016 [51], is collected to predict the battery calendar life accurately, as shown Fig. 8. According to PMP results and the principle of ICS development, the ideal value of the weighting factor is determined to be 3.0, i.e., ωr = 3.0, which is an important datum reference for online coordinated optimization. Fig. 9 shows the battery capacity loss curves with three typical optimization scenarios, namely:

Fig. 12. Battery current distributions for one driving mission.

(1) The case that only energy consumption is considered (OEC, ω = 0 ); (2) The balanced case of the ideal compromise scheme (ICS, ω = 3.0 ); (3) The case that battery aging is over weighted (BOW, ω = 3.4 ). The result demonstrates that the battery can only be used for 9

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Fig. 13. The battery SOC, effective Ah-throughput growth, and capacity loss curves under different scenarios during one day.

shown in Fig. 13(a)–(c), respectively. As compared with OEC, the battery SOC changes more gently under ICS and BOW; for the ICS, the effective Ah-throughput decreases by 9.1%, and the battery capacity loss decreases by 3.5%; for BOW, the effective Ah-throughput and battery capacity loss decrease by 23.1% and 10.6%, respectively. Fig. 13(c) shows that when the PHEB is in a parking state, the battery capacity loss still happens due to the calendar aging effect. In the charging process, the SOC rises from the termination value to 80% within half an hour, resulting in a sharp increase in effective Ahthroughput and capacity loss. However, the increments during the charging operation are not as significant as the case one during the running operation. As depicted in Fig. 6, when battery aging is considered, the fuel consumption cost increases dramatically, so it is necessary to analyze the ICS economy. Fig. 14 illustrates the cost of each optimization scenario, including the fuel consumption cost, electricity consumption cost, and the battery replacement cost at the end of the bus lifetime. The OEC achieves the best fuel economy, but has an additional cost for replacing a battery pack. For BOW, the battery aging cost is further

7.3 years without considering battery aging, while over-weighted battery aging creates a surplus in battery capacity at the end of the bus life. For the ICS, the calendar aging accounts for 19.2% of the total capacity loss, as shown in Fig. 10. Compared with the case unaware of battery aging, ICS and BOW limit the motor peak power and the amount of regenerative energy, as shown in Fig. 11. This is beneficial to prolonging the battery life at a sacrifice of fuel economy to some degree. Fig. 12 shows frequency distributions of the battery current amplitude for the three optimization scenarios. It can be observed that the battery charging/discharging current is below 0.5C most of the time. The time of a high current operation (> 2C) decreases in turn under OEC, ICS, and BOW. The comparison of optimization results under the three scenarios reveals the underlying reasons for the prolongation of battery life: the weighting factor reduces the battery current by adjusting the peak C-rate of battery during discharge and limiting the current impact of the motor on the battery during regenerative braking. The battery SOC, effective Ah-throughput growth, and capacity loss curves for different scenarios (OEC, ICS, and BOW) during one day are 10

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Fig. 14. Accumulated costs of each scenario over the bus lifetime.

Fig. 15. Effect of battery pack pricing on the accumulated total cost.

Fig. 16. Comparisons of the DA-ECMS and PMP methods.

reduced at the expense of more fuel consumption. But the battery pack has to retire when the bus is scrapped, and the cost of battery aging is the same as that of ICS. The total cost of ICS reduces by 15,925 euros and 14,937 euros when compared with OEC and BOW. Note that ICS is always better than BOW, but not OEC. The advantages of ICS over OEC depend on the battery pack pricing, as shown in Fig. 15. When the battery pack price is 600 EUR/kWh, ICS saves 5337 euros compared with OEC. However, when the battery pack price is 449 EUR/kWh, the total costs of ICS and OEC are the same. The ICS is therefore not recommended when the battery pack price is less than 449 EUR/kWh.

effective Ah-throughput is approximately linear with driving distance. As shown in Fig. 16(c), the capacity loss profiles obtained by the PMP and DA-ECSM match well, demonstrating that it is feasible to regulate battery capacity loss with the effective Ah-throughput. To further illustrate the effectiveness of the proposed strategy, a mathematical method for imitating a real-life driving scenario is presented. First, white noise of different intensities is added to the driving cycle sample to simulate sudden changes in road conditions:

vn = vspl + wgn (card (vspl ), 1, pn ),

n = 1, 2, 3, 4

(38)

where vn is the velocity vector of the random running cycle generated by the driving cycle sample with added noise; wgn (·) is a function producing Gaussian white noise; card (·) represents the number of elements in a set; pn is the noise intensity in [dBW], which represents the complexity of traffic conditions. The added white noise may cause velocity oscillation, which is inconsistent with the actual driving condition, so the locally weighted scatterplot smoothing (LOWESS) method is employed to process speed data:

5.3. Optimization performance of adaptively coordinated optimization Fig. 16 shows the optimization results of the proposed DA-ECMS strategy and the PMP strategy at the end of a driving mission. It is clear that the SOC profile corresponding to the DA-ECMS has a similar downward trend to that yielded by the PMP strategy and fluctuates around the reference trajectory. Moreover, it is validated that the DAECMS can realize the SOC planning appropriately before the starting of each driving mission. As a feedback controller of the proposed method, the reference SOC essentially provides SOC constraints to generate a control input. In Fig. 16(b), it can be observed that the effective Ahthroughput traces of both methods are in accordance with the reference curve. This observation testifies the previous hypothesis that the

vn' = smooth (vn, m , 'lowess')

(39)

where m is the window width of the smooth function in MATLAB, which affects the smoothness of the speed profile and thereby different driving styles. On the premise that the driving distance and average speed of 11

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Fig. 17. Four driving cycles with random white noise.

Moreover, the sensitivity of the economy of the designed compromise scheme to price changes of battery pack is analyzed. Finally, the optimization performance of the proposed dual-state adaptive ECMS strategy is assessed under four random driving cycles. Several important findings are summarized below:

Table 5 The parameters for random running cycles. Driving cycle

pn [dBW]

m

1 2 3 4

0.1 5 10 15

9 13 19 19

(1) Besides the battery cycle aging model, it is useful to consider the calendar aging for a coordinated optimization of battery life and energy management. And the analysis indicates that the capacity loss caused by calendar aging can account for up to 19.2% of the total battery capacity loss, which has a critical impact on the operating cost of the bus throughout its entire service life. (2) For the investigated plug-in hybrid propulsion system, the increased weighting factor related to battery aging in the cost function prolongs the battery life by adjusting the peak C-rate of battery during discharge and limiting the current impact of the motor on the battery during regenerative braking, which also leads to the increase of fuel consumption cost. (3) Compared with the energy-consumption-only case and the overweighted battery-aging case, the designed compromise scheme achieves the best performance in the current research scenario, with total cost reductions of 15,925 euros and 14,937 euros, respectively. While the option of replacing the battery pack may become an alternative in the future, as the battery price is continuously decreasing. (4) The random driving cycles are generated and used to confirm that the proposed strategy can realize the adaptively bi-objective, coordinated optimization. Compared with the optimization results of PMP, the maximum control errors of the dual-state adaptive ECMS are only 0.4% in the battery capacity loss and 2.7% in fuel economy under four random driving cycles, indicating the good adaptability and optimization performance of the proposed method.

driving cycle sample remain unchanged, four random running cycles shown in Fig. 17 are generated by adopting parameters in Table 5. The optimized SOC and effective Ah-throughput trajectories under the random driving cycles are compared in Fig. 18. It is apparent that both the SOC and effective Ah-throughput curves match well with the reference trajectories, indicating the adaptability and robustness of the proposed DA-ECMS. The quantitative results are given in Table 6 where differences from results with PMP are accounted for. It is evident that the proposed method achieves similar operating costs in all four cases as those of the PMP method. Compared with the optimization results of PMP, the maximum control errors of the DA-ECMS are only 0.4% in the battery capacity loss and 2.7% in fuel economy, which occurs under random driving cycles 1. The outcomes reveal that the DA-ECMS can realize the bi-objective coordinated optimization well, leading to a close performance to that of the PMP method. 6. Conclusions This paper proposes a novel real-time coordinated optimization approach based on the PMP, which can simultaneously improve fuel economy and extend battery lifetime for plug-in hybrid electric vehicles. The proposed scheme analyzes the interplay between energy consumption and battery aging, where a more realistic control-oriented battery aging model is embodied into the optimization framework. 12

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Fig. 18. Comparisons of the optimized trajectories under 4 random driving cycles: (a) SOC trajectories; (b) effective Ah-throughput trajectories. Table 6 Comparison of the DA-ECMS and PMP under 4 random driving cycles. Driving cycle

Strategy

F.C. [L]

E.C. [kWh]

Aheff [Ah]

costtot [EUR]

Final SOC

Capacity loss Qtot [%]

1

DA-ECMS PMP

15.86 16.30

62.66 61.34

775.3 771.7

45.89 46.25

0.286 0.298

0.2228 0.2219

2

DA-ECMS PMP

17.17 17.10

61.64 61.77

776.4 780.2

47.64 47.68

0.296 0.295

0.2226 0.2226

3

DA-ECMS PMP

15.57 15.76

61.60 61.62

775.7 777.7

45.30 45.61

0.297 0.297

0.2221 0.2212

4

DA-ECMS PMP

16.66 17.10

62.34 60.36

777.0 775.6

47.03 47.32

0.289 0.308

0.2222 0.2217

Note: F.C. and E.C. are abbreviations of fuel consumption and electricity consumption, respectively. 13

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Our future research could focus on the applicability of the proposed optimization strategy in the field test. Furthermore, with the rapid development of vehicle-to-vehicle and vehicle-to-infrastructure communication technologies, the appropriate equivalent factors can be calculated in advance by accurate prediction of vehicle loading regimes, which facilitates better optimization results.

In order to ensure the effectiveness of the proposed control scheme, the following implementation issues should be considered in practical applications:

• For on-board batteries different from the type discussed in this • •

paper, aging tests should be performed to re-identify the parameters of battery aging model; The technical level of the battery thermal management system and ambient weather in specific applications should be considered when screening weighting factors; When designing the supervisory controller, the parameters of the proportional-integral controllers and the weighting factor should be adjusted appropriately according to the selected bus route.

Acknowledgements This work was supported in part by National Natural Science Foundation of China (Project No. 51875054).

Appendix A Acronyms

FCEV HEV PHEV AMT HEB PHEB LiFePO4 CD-CS SOC DP PMP ECMS MPC PSO GA SA ML Ah-throughput SOH EoL C-rate ICS SA-ECMS DA-ECMS OEC BOW LOWESS

Fuel cell electric vehicle Hybrid electric vehicle Plug-in hybrid electric vehicle Automatic mechanical transmission Hybrid electric bus Plug-in hybrid electric bus Lithium iron phosphate Charge-depleting and charge-sustaining State of charge Dynamic programming Pontryagin’s minimum principle Equivalent consumption minimization Model predictive control Particle swarm optimization Genetic algorithm Simulated annealing Machine learning Ampere-hour throughput State-of-health End of Life Current rate Ideal compromise scheme Single-state adaptive ECMS Dual-state adaptive ECMS The case that only energy consumption is considered The case that battery aging is over weighted Locally weighted scatterplot smoothing

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