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Co-optimization of speed trajectory and power management for a fuel-cell/ battery electric vehicle

T

Youngki Kima, , Miriam Figueroa-Santosb, Niket Prakashb, Stanley Baekc, Jason B. Siegelb, Denise M. Rizzod ⁎

a

Mechanical Engineering, University of Michigan-Dearborn, Dearborn 48128, USA Mechanical Engineering, University of Michigan, Ann Arbor 48109, USA c Electrical & Computer Engineering, United States Air Force Academy, CO 80840, USA d U.S. Army CCDC Ground Vehicle Systems Center (GVSC), Warren, MI 48397, USA b

HIGHLIGHTS

GRAPHICAL ABSTRACT

of speed trajectory • Co-optimization and power management is investigated.

Minimum Principle is • Pontryagin’s used to reduce the search space for control.

performance of co- and sequential • The optimization is comprehensively analyzed.

ARTICLE INFO

ABSTRACT

Keywords: Eco-driving Co-optimization Speed profile optimization Power management Electric vehicles Optimal control

With the advent of vehicle connectivity via intelligent transportation systems, the simultaneous optimization of powertrain operation and vehicle dynamics has become an important research problem to improve energy efficiency. However, this problem is still very challenging due to computational loads in terms of time and memory usage, and hence very little effort has been spent to reveal the full potential of co-optimization. Studies in the literature on eco-driving or energy-efficient operation typically preferred using a sequential optimization or eliminating some constraints for fast computation. This paper extends the authors’ previous work of determining energy-efficient speed profiles focused on vehicle dynamics by including additional state and control variables due to vehicle hybridization in application to a fuel-cell/battery electric vehicle. The results from Pontriagyn’s Minimum Principle analysis reveal that the co-optimization can be formulated with one discrete variable describing vehicle operation and another continuous variable for power distribution to reduce computation in implementing Dynamic Programming. Then, the performance of co-optimization and sequential optimization for energy-efficient driving is comprehensively analyzed and compared in terms of energy consumption by each powertrain component, operating modes, and computational cost. The total energy savings by co-optimization range from 5.3% to 24.2% for aggressive driving on a hilly terrain. Meanwhile, on a relatively flat road, the benefit of co-optimization is less significant. Nonetheless, co-optimization achieved 0.5%–5.3% reduction in total energy consumption as compared to sequential optimization.

⁎

Corresponding author. E-mail address: [email protected] (Y. Kim).

https://doi.org/10.1016/j.apenergy.2019.114254 Received 17 May 2019; Received in revised form 23 November 2019; Accepted 23 November 2019 0306-2619/ © 2019 Elsevier Ltd. All rights reserved.

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Y. Kim, et al.

Nomenclature fc m

fb fe fr fm f mprop f mreg g Ib M p Pb Pdmd

Pfcin Pfcout Pmelec Pmmech Qb Rb s t u v Voc

Fuel-cell efficiency Motor efficiency Grade angle Distance domain states Friction braking force Mode-dependent motor force Resistive force Mechanical motor force Propulsion motor force Regenerative motor force Gravitational acceleration Battery current Vehicle mass Time domain Co-state Battery power Vehicle power demand

x DP PMP SOC

1. Introduction

Hydrogen fuel power Fuel-cell power Electrical motor power Mechanical motor power Battery capacity Battery internal resistance Distance Time Control variables Velocity Battery open-circuit voltage Battery terminal voltage Time domain states Dynamic programming Pontryagin’s Minimum Principle Battery state-of-charge

and control variables with hybridization can contribute to improved fuel efficiency, but it increases the complexity of the optimal control problems, e.g., battery state-of-charge (SOC) and power-split ratio among two (or more) energy sources. Therefore, the approaches or algorithms studied for vehicles with a single energy source (e.g, a conventional vehicle with an internal combustion engine or a batterypowered electric vehicle) may not lead to the global solution for ecodriving of hybridized vehicles and hence much effort has been spent to address the co-optimization problem for hybrid electric vehicles. Typically, these problems have been solved in two ways, either with sequential optimization that minimizes the energy demand at the wheel, or with co-optimization that takes into account the energy generation within the propulsion systems. A bi-level optimization or a sequential optimization has been applied based on the idea of decoupling the optimization of the speed profile (first level) and the power management (second level) to reduce computation cost in terms of time and memory usage. In the first level, the speed profile is optimized by minimizing (1) the energy consumption by a single power source such as the engine and the motor [15–19], or (2) the deviation of acceleration [20]. In the second level, minimum energy consumption is achieved by (1) optimal gear shifting, (2) optimal power-split between the engine and the motor, or (3) both gearshifting and power-split depending on vehicle architecture. For example, in [20], the authors presented the potential for fuel efficiency improvement by the sequential smoothing of a velocity profile given traffic constraints and the optimization of its charge depletion strategy in a plug-in hybrid electric vehicle application. In [17,19], the authors investigated the effectiveness of hierarchical approach to addressing vehicle dynamics and energy management of hybrid electric vehicles using model predictive control and adaptive equivalent consumption minimization based strategies, respectively. Approaches to co-optimization have also been discussed in several papers. In [21], a co-optimization was carried out by allowing for small deviations in velocity and controlling the power split for a hybrid electric vehicle. This work was further improved in [22], where first the long-horizon SOC trajectory was optimized and then within a short horizon, the co-optimization was implemented. The authors in [23–25] developed algorithms that combine Dynamic Programming with the Energy Management System design for a hybrid electric vehicle to calculate eco-driving cycles. Notably, the performance of co-optimization was presented in comparison with sequential optimization approaches for a parallel hybrid electric vehicle in [18] where four methods were used to solve the optimal control problem and compared in terms of fuel consumption saving, state trajectories, computation time and memory to find the trade off between optimality and

With the advent of connected and autonomous vehicles, optimal energy management has become an important part of vehicle control strategies. The feature of connectivity and autonomy allows the vehicle controllers to directly manipulate the speed of an autonomous vehicle in such a way that it minimizes fuel or energy consumption by utilizing the numerous pieces of information available via vehicle-to-vehicle, vehicle-to-infrastructure, and vehicle-to-cloud about its driving environment. Connected and autonomous vehicle technologies and their impacts on energy saving potentials are comprehensively discussed in [1]. The so-called eco-driving problems of determining the speed trajectory that minimizes a vehicle’s energy consumption for various powertrain architectures such as conventional vehicles and electrified vehicles have been extensively explored in the literature. For conventional vehicles with internal combustion engines, the authors in [2,3] have utilized Pontryagin’s Minimum Principle (PMP) and a polynomial fueling rate function to address highway cruising scenarios. In [4], the effectiveness of the Pulse-and-Gliding operation over maintaining vehicle speed is analytically shown for gasoline engine vehicles in fuel efficiency perspective. The authors in [5] proposed a fuel-efficient eco-drive algorithm using the Relaxed Pontryagin’s Minimum Principle for driving on hilly terrains, and its effectiveness was experimentally shown in [6]. The optimization for real-world urban driving scenarios with consideration of Signal Phasing and Timing signals has been studied in [7–9]. In [10], the authors have extended the idea of the velocity optimization profile presented in [11] to co-optimize the velocity as well as the gear-shift strategy. In both cases, Dynamic Programming (DP) was used to solve the optimal control problems. A more approximate but faster optimization technique utilizing a pseudospectral method was applied to solve the optimal speed profile through traffic lights in [7]. For electric vehicles, the authors in [12] proposed an energy management and driving strategy based on information from the road terrain profile and the preceding vehicle to minimize the battery power. The optimization problem was carried out with DP as well as Model Predictive Control for online implementation. The authors in [13,14] synthesized the results of PMP analysis for all electric vehicles into DP formulation to reduce computational loads. Further, in [15], an efficient approach using a sequential quadratic programming method was investigated to determine a globally optimal solution. In [4], energy efficient driving of electric vehicles is analytically studied based on the assumption that energy losses are a quadratic function of traction force. On the other hand, vehicle hybridization introduces new challenges to the energy management problem. The increase in the number of state 2

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optimization and the sequential optimization is comprehensively analyzed and compared in terms of energy consumption by each powertrain component, operating modes, and computational cost for the case of fuel-cell/battery electric vehicles. This paper is organized as follows: Section 2 details models to capture the longitudinal dynamics and powertrain behavior of a fuelcell/battery powered electric vehicle. Section 3 addresses the problem formulation for co-optimization of a speed profile and power management and its PMP analysis. Approaches to co-optimization and sequential optimization are discussed as well. Then, the performance of the two approaches in terms of energy consumption and driving time is investigated through a case study in Section 4. Finally, conclusions and directions for future work are presented in Section 5.

complexity. From the four methods, the two step sequential optimization of the control variables achieves near optimal fuel savings while maintaining a reasonable computation time. The performance of cooptimization was also presented in comparison with sequential optimization approaches for a series hybrid electric vehicle in [26]. The cooptimization consisted on finding the optimal speed profile and the power split simultaneously, which performed better in terms of fuel economy compared to the sequential optimization approach. For vehicles utilizing a wide range of battery SOC such as plug-in hybrid electric vehicles and electric vehicles with multiple energy sources, the trajectory of battery SOC depletion has significant impact on fuel or energy consumption [27,20]. Therefore, in this study, the basic idea of PMP analysis is extended to address the problem of a simultaneous optimization of a speed profile and a power-split strategy for hybrid vehicles with charge depletion which fully utilizes the battery. Dynamic Programming, implemented in MATLAB environment [28], is used to solve the problems of interest because of its capability of finding a global solution to the optimal control problem, despite the curse of dimensionality [18,14,29]. This paper focuses on electric vehicles with a hybrid energy storage system consisting of a hydrogen fuel-cell and a battery. Fuel-cell hybrid energy storage systems in applications to ground vehicles have been intensively studied in literature for their potential to improve energy efficiency and emissions [30,31]. In [30] a multi-input multi-output controller was developed that minimizes the primary energy consumption and startup time while balancing energy generation among a fuel cell, thermoelectric device, and battery. In [31] a finite state machine strategy was proposed for energy management for a hybrid vehicle that consisted of fuel cell/battery/supercapacitor system. Energy management techniques for fuel-cell hybrid electric vehicles and trams have also been investigated and are well discussed in [32,33], respectively. There are additional benefits for military applications because of their potential for increased mobility, ability to export electrical power, silent watch, and low-temperature operation (i.e., no heat signature or noise). Especially, the Proton Exchange Membrane Fuel Cell has high efficiency (around 50% of peak efficiency) and low operating temperature (around 70 °C). It should be noted that although the considered vehicle is an electric vehicle, the materials presented in this paper can be applied to plug-in hybrid electric vehicles as well, for (1) the vehicle hybridization and (2) the level of battery utilization. The main contribution of this paper is twofold. First, PMP analysis is applied to the co-optimization, including the speed trajectory and power management, of a fuel-cell/battery electric vehicle. The results from the PMP analysis reveal that the co-optimization can be formulated with two control inputs to reduce computational cost: one discrete variable describing vehicle operation and the other continuous variable for the fuel-cell power. Second, the performance of the co-

2. Vehicle and powertrain model The vehicle of interest in this study is a lightweight military ground robot electrically hybridized by a battery pack and a fuel cell. This hybridized architecture allows it to smooth out transient loads and to provide auxiliary power during startup and shutdown of the fuel cells. Especially, Proton Exchange Membrane Fuel Cells are of great interest to the army because of their advantages such as silent mobility and low heat signature ( 70 °C). The topology of the fuel-cell/battery powertrain considered in this study is illustrated in Fig. 1. The fuel-cell is coupled to the battery by a DC/DC converter that regulates the power flow from the fuel-cell to account for the change in cell terminal voltage. The decrease in cell voltage with increasing current is indicated by the polarization curve shown in Fig. A.1. The main focus of this work is on optimization of vehicle speed and power management. In general, a DC/DC converter has its maximum efficiency around 95% and has a similar shape to the efficiency curve of the Proton Exchange Membrane Fuel-cell system which typically has poor efficiency at low power, reaches a peak between 20 and 50% of the rated power, and then tapers off slightly at high power levels. Therefore, detailed models for power electronics and efficiency are not considered, that is, power losses by the DC/DC converter are not included in this study. 2.1. Point-mass longitudinal dynamics The longitudinal dynamics of the vehicle to determine propulsion and braking power are described by: (1)

s = v, Mv = fm + fb

fr ,

(2)

where M is the vehicle mass; s and v denote the distance and velocity of the vehicle, respectively; fm is the mechanical force provided by the electric motor; and fb is the force by the friction brake. The resistive

Fig. 1. A simple schematic of a fuel-cell/battery powered electric vehicle. 3

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force fr is computed based on the grade given by,

and velocity of a vehicle as

fr = A + Bv + Cv 2 + Mgsin ,

where Q b is the capacity of the battery. When it comes to power management, the battery output power P b is generally used as an input to the battery system and hence the following equation is used to convert the battery power output to the corresponding current:

(3)

where coefficients A, B , and C are used to determine resistance forces by rolling and aerodynamic drag, and the grade angle is a function of distance, i.e. = (s ) .

Ib =

2 Voc

Voc

4Rb P b

2Rb

.

(9)

2.2. Fuel-cell model 2.4. Motor model

In this study, a liquid-cooled Proton Exchange Membrane Fuel-cell system is considered based on the model of a 15 kW Hydrogenics HD15 stack. To include a parasitic load for cooling and air supply, a static power consumption is considered, and hence the fuel-cell system is modeled with a simple static function of the fuel cell system efficiency, in fc ; hence, the hydrogen fuel power Pfc can be expressed as follows:

Pfcin =

Pfcout

,

The electrical power for propulsion and regenerative braking by the motor Pmelec is determined from the mechanical power and the efficiency of the motor, Pmmech and m , respectively. The electrical power Pmelec should be provided by the fuel cell and the battery and hence the following equation is obtained:

Pmelec =

(4)

fc

m,1

where is the fuel-cell net power. In general, the fuel cell system efficiency fc is influenced by the operating temperature of the fuel-cell system. Since the fuel cell generates a significant amount of heat, auxiliary power is required for regulating temperature. In this study, it is assumed that the temperature of the fuel-cell system is perfectly regulated at 60 °C such that the maximum efficiency can be achieved for various power levels. A more detailed description of the fuel-cell model is provided in Appendix A. Under this assumption, fc can be expressed in terms of fuel cell’s net system power (Pfcout ):

Pfcout

fc

=

(Pfcout ).

(5)

Pfcout /

(Pfcout )

2.3. Battery model The terminal voltage Vt of the battery system is captured by an equivalent circuit model comprising the open-circuit voltage Voc , the internal resistance Rb , and the current Ib as following:

Vt = Voc

Ib Rb

(7)

Typically, Voc and Rb are dependent on the state of charge (SOC) of the battery, as shown in Fig. 3; and the SOC dynamics are represented by

SOCb =

Ib , Qb

= P b + Pfcout

DC

(10)

In this section, two approaches to optimization of the speed profile and power management for the all-electric vehicle considered in this study are presented: co-optimization and sequential optimization. For simplicity, the assumptions in [14] are adopted in this study: (1) the efficiency of the motor is invariant, that is, the motoring and generating efficiency of the motor is fixed; (2) the vehicle moves forward only; and (3) road grade and velocity limit are distance-dependent and known. In general, the efficiency of an electric motor is nonlinear [12]. However, the efficiency is relatively constant above certain force or torque levels covering the nominal operating range, and hence the assumption of invariant efficiency is considered reasonable.

= where which is shown in Fig. 2. Here two stacks are required to achieve the power requirement of the vehicle. (Pfcout )

m,2

3. Optimal control problem formulation (6)

(Pfcout ),

+ f mreg v

where f mprop and f mreg represent the positive force for propulsion and the negative force for regeneration by the motor, respectively; DC denotes the efficiency of the DC/DC converter; and m,1 and m,2 denote motoring and generating efficiency of the motor, respectively. Note that the main focus of this work is on optimization of vehicle speed and power management and hence is concerned on power flows among powertrain components, i.e., the fuel-cell, the battery, and the motor; therefore, detailed models for power electronics are not considered, that is, power losses by the DC/DC converter are not included in this study and hence DC = 1 is assumed.

From Eqs. (4) and (5), the hydrogen fuel power is given by:

Pfcin =

f mprop v

3.1. System dynamics For PMP analysis, the battery dynamics are simplified via linearization as given by:

(8)

Fig. 2. The static efficiency of the fuel-cell system; the power for the blower and cooling system load to regulate the temperature of the fuel-cell is included in this efficiency. 4

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Fig. 3. The open-circuit voltage and internal resistance of the battery as a function of battery SOC. 2 Voc

Voc +

SOC

4Pb0 Rb

2Q b Rb

+ P 0=0 b

1 4Rb P b 2 2Q b Rb 2 Voc 4Pb0 Rb

= P b0 = 0

computational burden due to the curse of dimensionality. The primary goal of the optimization is to determine a speed profile and a power split trajectory that minimize the total energy consumption over a given route. Thus, the cost function to be minimized is the total hydrogen fuel energy, defined by:

1 P b. Q b Voc

(11) It should be noted that similar approaches to approximating the battery SOC dynamics can be found in the literature [34,35]. Finally, from Eqs. (1), (2), (10), and (11), the following system dynamics can be obtained:

x2 = u1 + u2 + u3 x3 =

(13)

(x1, x2),

(14)

(x2 , u1, u2 , u4 ),

where

x= [x1, x2 , x3] = [s, v, SOC] , u= [u1, u2, u3, u4] =

prop fm

M

,

reg fm

M

,

fb M

Pfcin (Pfcout ) dt =

0

(u4 ) dt

(16)

Since the given optimal control problem suffers from the curse of dimensionality when it is solved by Dynamic Programming, simplified control actions are much preferred. To that end, Pontryagin’s Minimum Principle is considered, similarly to the approach in [14]: the solution to the optimal control problem leads to the minimization of the Hamiltonian comprised of the cost function subject to constraints via the system dynamics. The Hamiltonian is defined as:

(12)

x1 = x2 ,

tf

=

, Pfcout ,

=

(u4 ) + p1 x2 + p2 (u1 + u2 + u3

(x1, x2))

Mp3

x2 u1

Q b Voc (x3 )

m,1

+ x2 u2

and

= =

m,2

u4 M

(17) fr (x 1, x2) M

where the first term corresponds to the cost function in (16) and the other terms are related to the system dynamics described by (12)–(14) associated with the adjoint variables p1 , p2 , and p3 . The dynamics of the adjoint variables are given by:

,

1 Q b Voc (x3)

Mx2 u1 m,1

+ Mx2 u2

m,2

u4 .

The control inputs are bounded by their limits such that u1 [0, u1max ], u2 [u 2min , 0], u3 [u3min , 0], and u4 [0, u4max ]. It is noted that the power losses by the DC/DC converter are not considered 1, u4 = Pfcout · DC needs to and hence DC = 1 is assumed. When DC be used instead. The use of four control inputs is deliberate for Pontryagin’s Minimum Principle analysis, which will be discussed in the following section, similar to the approach in [14]. Initial and final conditions of the states are:

x1 (0) = 0, x2 (0) = v0, x3 (0) = SOC 0 , x1 (t f ) = sf , x2 (t f ) = vf , x3 (t f ) = SOCf ,

p1 =

x1

p2 =

x2

p3 =

x3

= p2

=

x 1,

(18)

p1 + p2

x2 u1

=

x2

+ p3

+ x2 u2

m,1

m,2

u1

+ u2

m,1

m,2

,

(19)

u4 p , M 3x3

(20)

where

(15)

p3 =

where t f and sf are operational time and distance traveled, and these are bounded, meaning that the vehicle stops at time t f after traveling a given distance, sf .

Mp3 Q b Voc (x3)

,

(·)

and (·) = . The Hamiltonian function is further simplified by factoring out the control input u in order to determine optimal control inputs. Then Eq. (17) becomes:

3.2. Simplification of control modes In this subsection, the co-optimization problem to minimize hydrogen fuel consumption is presented and analyzed to determine a small subset of control actions via Pontryagin’s Minimum Principle (PMP) analysis. In the authors’ previous work [14], a finite number of operational modes were analytically found for minimum electrical energy consumption. Since this study considers additional state and control variables, the battery SOC and power split ratio, respectively, a similar analysis has been conducted so that the results can be used to effectively implement Dynamic Programming to mitigate the

= p2

p3 x2 m,1

u1 + (p2

p3 x2

m,2 ) u2

+ p2 u3 +

(u 4 ) +

p3 u4 M

+ p1 x2

p2 (x1, x2).

(21) As can be seen from Eq. (21), the control inputs u1, u2 , and u3 appear as linear, meaning that, based on the values of the switching functions p3 x2 p2 , p2 p3 x2 m,2 , and p2 , the control inputs u1, u2 , and u3 to m,1

minimize given by: 5

can be found. On the other hand, the control input u4 is

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From Eq. (3), it can be found that the resistance force fr is the sum of two terms, and each term is described by one state only, leading to x1 x 2 = 0 ; moreover, the resistance force is a monotonic function with

0, if u 0 if 0 < u < u4max u4 = u , u4max , o. w.

> 0 and x2 > 0 , resulting in candidate solution is obtained as x1,

where u4 is determined from the stationary condition u = 0 . 4 Since each switching function for u1, u2 , and u3 has three options, of being greater than, less than, or equal to 0, 27 different modes are possible. However, not all of the modes are feasible due to the assumptions considered for the problem. For instance, the conditions p3 x2 p2 > 0, p2 p3 x2 m,2 = 0 , and p2 0 cannot occur at the same m,1

time. From the first two conditions, p3 x2

m,2

>

p3 x2 m,1

p3 x2 m,1

< 0, p2

p3 x2

m,2

p3 x2 m,1

p3 x2 m,1

p3 x2

p2

m,1

p3 x2

p2

m,1

p3 x2

p2

m,1

p3 x2

p2

m,1

p3 x2

p2

m,1

< 0, p2

p3 x2

> 0, p2

p3 x2

m,2

< 0, p2 < 0

> 0, p2

p3 x2

m,2

> 0, p2 < 0

> 0, p2

p3 x2

m,2

> 0, p2 > 0

= 0, p2

p3 x2

> 0, p2

p3 x2

m,2

m,2

dt i

= 0, i = 0, 1, …..,2l

d2l dt2l

(22c) (22d)

min

d

( ) d2l

(22f)

1

= p2

m,1

¨1 = p2

(

2 x2 x2

+

x2

+

x2 x2

p3

)

m,1

(u1

x 1 x2 x2

2

=

(u 4)

u1

x2 x2

0,

ds

2

fe s, 2, uc + f f (uc )

( )

,

M2

=

Voc

1 (0)

= 0,

3 (0)

2 Voc

4Rb (Pdmd

2 2 Q b Rb 1 (sf )

u 4)

,

= tf ,

2 (sf ) = vf , 3 (sf )

= SOC0 ,

= SOCf ,

{uc,1, uc,2, uc ,3, uc ,4 , uc,5}, [0, u4max ],

where

= [ 1,

Pdmd =

m (fe )· fe (s ,

T

2,

3]

(26)

= [t , v, SOC ]T ; since the equations of motion d

are expressed in terms of s, = ds is used. The mode-dependent force by the motor fe (s, 2, uc ) is detailed in Appendix B. The power demand Pdmd is given by: 2,

(27)

uc ) 2.

Table 1 Possible control actions.

= 0 . Then, its

Control u

Description

[u1max , 0, 0] × u4 [0, 0, 0] × u4

Full Propulsion

[0, u 2min , 0] × u 4

[0, u 2min , u3min] × u 4 [ (x1, x2), 0, 0] × u 4 [0, (x1, x2), 0] × u 4

= 0, ) + p2

1

uc u4

first and second time derivatives are expressed as follows:

p1 + p2

sf

2 (0) = v 0,

(23) p3 x2

+

(25)

0

= 1/ 2,

3

1,

where l is an integer variable. In case (22e), u1 is undefined. Consider

1=

=

(22e)

0

dt 2l

1

s.t.

= h1 (x , p) + h2 (x , p) u,

( 1)l + 1 du

x2

x2

In general, the problem of optimizing a speed profile is formulated in the distance domain, since it is easy to handle road data; e.g., terrain type, grade, and speed restrictions are distance-dependent [12,21,22,14]. Thus, the optimal control problem discussed in Section 3 can be reformulated as follows:

The optimal control inputs u1, u2 , and u3 for the first four cases can be easily determined to minimize the Hamiltonian, resulting in full propulsion, coasting, full regenerative braking, and full braking, respectively. On the other hand, for the last two cases, u1 and u2 are undefined because the optimal control actions cannot be determined uniquely; in this case, Kelley’s condition [36] can be used to determine the optimal control inputs: for a single control problem, the control u for a singulararc can be found by differentiating = u with respect to time t until the control input u appears explicitly, and then Kelley’s condition must be satisfied. That is: di

2

= p2

3.3. Co-optimization

(22b)

= 0, p2 < 0

1

Clearly, both cases (22e) and (22f) describe a cruising operation of the vehicle, as the same amount of input is applied to the system to maintain the speed. Thus, five possible control actions are determined for u1, u2 , and u3 , and they are summarized in Table 1. It should be highlighted that the vehicle dynamics are controlled by the five control modes only, regardless of the fuel-cell power u4 . This result allows a substantial decrease in computational cost in solving the co-optimization problem with Dynamic Programming. In the following subsections, Dynamic Programming formulation for co-optimization and sequential optimization is explained in detail.

(22a)

< 0, p2 < 0

(24)

u2 = (x1, x2).

0 cannot occur

m,1

< 0, p2 < 0

> 0 . Therefore, the

2 x

< 0 and p2 > 0 can be eliminated as

m,2

x2 x2

because x 2 + x2 x2 > 0 and p2 < 0 . 2 Similarly, in the case (22f), the solution can be found to be:

both propulsion force and friction braking force are used, which should be avoided to minimize the energy consumption. The following six cases are feasible in vehicle operation:

p2

d2 u1 dt 2

( 1)2

simultaneously. It is also obvious that the motor cannot provide both propulsion and regeneration forces simultaneously, which allows for p3 x2 < 0 and p2 p3 x2 m,2 > 0 . ruling out the conditions p2 Moreover, the conditions p2

+

and this solution satisfies the Kelley’s condition:

; because of the fact

> 0 , and P2

2 x2 x2

u1 = (x1, x2 ),

that efficiency of the motor is positive and less than 1 (i.e., 0 < m < 1) and the assumption that the electric vehicle moves forward only (i.e., x2 > 0 ), p3 has to be negative, and p2 has to be negative. Similarly, the

conditions p2

x2 x2

= 0. 6

Coasting Full Regenerative Braking Full Braking Cruising

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Y. Kim, et al.

where 1 m (fe )

=

m,1

, fe

m,2 ,

Table 2 Vehicle parameters and constraints.

0,

fe < 0.

(28)

In order to further reduce the computational cost, the optimal control problem (26) is modified such that a weighted penalty on a travel time is included as follows:

min s.t.

1

sf

=

2

=

3

=

0

(

(u 4 )

+

2

1 2

fe s, 2, uc + f f (uc )

) ds ( )

,

M2 2 Voc

Voc

4Rb (Pdmd

u 4)

2 2 Q b Rb

2 (0)

= v 0,

3 (0)

= SOC0 ,

uc u4

2 (sf )

,

= SOCf ,

{uc,1, uc,2, uc ,3, uc ,4 , uc,5}, [0, u4max ],

s.t.

sf 0

1

= 1/ 2,

2

=

(

m (fe )· fe (s ,

fe s, 2, uc + f f (uc )

2 (0)

uc

= 0, = v 0,

1 (sf )

2,

s.t.

3

=

=

3 (0)

u4

sf

(u 4)

0

2

Voc

2 Voc

= SOC0 ,

4Rb (Pdmd

[0, u4max ]

Ns Np

SOCmax SOCmin out Pfc,max out Pfc,min

Nfc Ncell

−6

kW

1 0.3 32

– – kW

0

kW

2 65

– –

To solve the optimal control problems using DP, the dynamic equations with state and control variables in a time domain need to be discretized. The level of discretization affects the accuracy of a solution and the computation time: coarse discretization results in fast computation but large numerical errors. Therefore, step sizes for the discretization should be carefully chosen. The level of discretization for time, velocity, battery SOC, and distance for this study is provided in Table 3. Since this study finds an optimal speed profile, the initial and terminal speed constraints are set to be the same as the original driving cycles as given by:

(30)

• Churchville-B Driving: (0) = (s ) = 12, • Convoy Driving: (0) = (s ) = 0, 2

2

u 4)

2

2

f

f

The initial and terminal constraints of the battery SOC are set as follows:

ds

3 (sf )

– –

P bmin

4.1. Optimization set-up

,

= vf ,

2 2 Q b Rb

14 1

m,max

Pm,max Vm P bmax

A light-weight military ground vehicle hybridized with a fuel cell and a battery is considered as a target vehicle. The parameters and operational constraints are summarized in Table 2. The drive cycles for the development of ground vehicles in the Army Ground Vehicle Programs typically focus on operating at a constant speed over varied terrain for practical reasons [37]. Thus, two different drive conditions as shown in Fig. 4 are studied: 1) the Convoy Cycle - long distance driving on a relatively flat road (small variation in grade), and 2) the Curchville-B Cycle - constant speed driving on a hilly road (more discernible and aggressive grade change).

uc )) ds

{uc,1, uc,2, uc ,3, uc ,4 , uc,5}.

2,2

kg N Ns/m Ns2/m2 m – m/s m/s2 m/s2 RPM kW V kW

4. Case study - lightweight military ground robot

Note that the cost function 2,1 is the total energy consumption by the motor for propulsion and regenerative braking. It is possible to further reduce the computational cost by following the approach used in (29), i.e., adding a penalty on a travel time and eliminating one state 1. However, for the purpose of performance comparison, travel times obtained from solutions to the optimal control problem (29) are used as a terminal constraint in this optimization problem. Then, at stage II, the problem of optimal power management over the speed profile obtained from Stage I is solved with the following formulation:

min

453.6 0.17 0.06804 13.608 0.2794 7.54 23 2 −2 6500 30 50 6

where Pdmd is the power demand to traverse the optimized drive cycle obtained at stage I; that is, Pdmd is known. In the following section, a case study to compare the performance of two optimization approaches will be presented.

= tf ,

2 (sf )

M A B C r ia v lim amax amin

Number of Fuel-cell Stacks Number of Fuel-cell Cells

( )

M2

1 (0)

Mass A B C Tire radius Final drive ratio Speed limit Max. Acc. Min. Acc. Max. Motor Speed Max. Motor Power Motor Operation Voltage Max. Battery Power

Min. FC Power

Although a significant amount of reduction in computation can be achieved by using the finite operational modes of the vehicle dynamics, co-optimization is still computationally costly. Therefore, the co-optimization problem can be sequentially solved, which has been typically used in the literature [16–18,15,20]. In sequential optimization, an optimal speed profile is determined at the first stage, and then a power management strategy is optimized over the speed profile at the second stage. Obviously, the number of state and control variables considered at each stage is small compared to the co-optimization problem, allowing for faster computation and less memory requirement. At stage I, the problem of optimizing a speed profile is formulated as:

=

Unit

Max. SOC Min. SOC Max. FC Power

(29)

3.4. Sequential optimization

2,1

Value

Number of Battery Cells in Series Number of Battery Cells in Parallel

where is a weighting factor to penalize a travel time. Apparently, this formulation can effectively eliminate one state variable 1 from the cooptimization problem. It should be noted that a desired travel time can be achieved by adjust the weighting factor .

min

Symbol

Min. Battery Power

= vf ,

3 (sf )

Parameter

,

SOC0 = 0.9, SOCf

= SOCf ,

[SOCmin , SOCmax ].

Note that terminal SOCs from the co-optimization and the sequential optimization can be different, since no specific value is assigned for the

(31) 7

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Fig. 4. Elevation and grade of the considered driving environment: (a) Churchville-B Cycle and (b) Convoy Cycle. Table 3 Discretization of optimization variables in dynamic programming. State Velocity Time Distance SOC

Table 4 Average computation time.

Symbol

Value

Unit

v t s SOC

0.1 1 60 0.007

m/s s m –

Cycle

Churchville-B Convoy

Co-optimization

5 min 33.1 s 19 min 49.9 s

Sequential Optimization Speed Opt

Power Opt

4.83 s 15.60 s

0.34 s 1.18 s

percentage reduction by co-optimization: negative values indicate that co-optimization leads to less energy consumption than sequential optimization. It can be observed from Fig. 5(a) that the sequential optimization leads to lower energy consumption by the motor than the cooptimization. This makes sense since the energy consumption by the motor is the cost function in the sequential optimization. On the other hand, as shown in Fig. 5(b), more hydrogen energy is consumed in the sequential optimization than in the co-optimization, which is not surprising because the co-optimization finds a solution that minimizes the hydrogen energy consumption. The battery energy consumption to complement the fuel cell for vehicle operation is presented in Fig. 5(c). Since the solutions obtained by the co-optimization prefer not using the fuel cell, a large amount of energy needs to be provided by the battery. The total energy consumed for each optimization approach is compared in Fig. 5(d). Overall, co-optimization can achieve lower total energy consumption than sequential optimization. Particularly, the total energy savings by co-optimization range from 5.3% to 24.2%, which highlights the benefit of co-optimization against sequential optimization. However, for driving cycles with a longer time traveled, the energy savings achieved by co-optimization are not as significant compared to the shorter cases. For better understanding of the energy savings by co-optimization, the trajectories of the vehicle speed and the battery SOC are presented

terminal SOC; this will be shown in the following subsection. A solution to the DP problem is numerically obtained by using the dpm function implemented in the MATLAB environment [28]. To run DP over the considered driving environment, the following hardware is used: Windows 10 with Intel® Core™ i7-7700HQ CPU @ 2.80 GHz and 32.0 GB RAM. 4.2. Optimization results The computation time to run the co-optimization and sequential optimization are provided in Table 4. The co-optimization is about 64–71 times slower than the sequential optimization. It should be noted that the co-optimization is performed with limited control inputs uc and without including time as a state variable, meaning that the original problem with discretized continuous control inputs will lead to a significant increase of the computation time. 4.2.1. Churchville-B driving Fig. 5(a)–(d) show comparisons of energy consumption by the motor, the fuel cell, the battery, and both the fuel-cell and the battery, respectively, over the driving cycles obtained by the co-optimization and the sequential optimization for different driving times. Particularly, the numbers indicate the difference in energy consumption and the

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Fig. 5. Comparison of energy consumption obtained by co-optimization and sequential optimization for various times traveled on Churchville-B road: (a) energy consumption by the motor, (b) energy consumption by the fuel cell, (c) energy consumption by the battery, and (d) the total energy consumption by the fuel cell and the battery.

in Fig. 6(a) and (b) for the shortest cycle, and in Fig. 6(c) and (d) for the longest cycle, respectively. As can be seen from Fig. 6(a), for the shortest drive cycle, the speed profile obtained by co-optimization has less deviation than that obtained by sequential optimization. It can also be observed that the vehicle speeds selected by the two approaches are different. Particularly, the solutions by co-optimization prefer coasting during downhills, whereas those by sequential optimization prefer regenerative braking and cruising during downhills. The use of coasting is beneficial as it can minimize unnecessary energy conversion; i.e., recuperated braking energy used later for vehicle propulsion. For instance, as shown in Fig. 6(b), the battery SOC increases around 100 s in sequential optimization. However, no change is observed in co-optimization during the same time period. Eventually, the terminal SOC in sequential optimization becomes significantly higher than that in co-

optimization. Fig. 6(c) also shows that the deviation of vehicle speed in co-optimization is less than that in sequential optimization for the longest cycle. Since solutions by both optimization approaches prefer using cruising from 450 s until the end of the trip, the difference in terminal SOC of two solutions is less than that observed in the shortest drive cycle. In sequential optimization, the total energy consumed by the motor is minimized and the efficiency of the powertrain system including the fuel cell and the battery is not included. Therefore, high-power operation is preferred as long as the total motor energy is minimized and the use of coasting operation can be maximized as seen from Fig. 7(a) and (d) at Stage I. Since the battery power is insufficient to provide such high motor power, the fuel-cell needs to be used as well despite the fact that the fuel-cell system has lower efficiency than the battery, as shown

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Fig. 6. Comparison of vehicle and powertrain operation obtained by co-optimization and sequential optimization over Churchville-B road for travel times of 329 s and 795 s: (a) vehicle speed, (b) electrical power demand by the motor, (c) power consumed by the fuel cell, (d) power provided by the battery, and (e) battery SOC.

in Fig. 7(b) and (e). The battery can be used in a wide range within its limits to minimize fuel consumption at Stage II, compared to the fuelcell (see Fig. 7(c) and (f)). On the other hand, high-power operation is not preferred in cooptimization due to low efficiency of the fuel cell as seen from Fig. 7(a) and (d). Therefore, the battery can be used as a primary power source and the use of the fuel-cell can consequently be reduced as shown in Fig. 7(b) and (d). More specifically, for the shortest cycle, the co-optimization hardly uses the fuel-cell at higher power, i.e., Pfcin > 10 kW, which can be explained by the efficiency of the fuel-cell (see Fig. 2).

Consequently, lower terminal SOCs are obtained in co-optimization than those in sequential optimization (see Fig. 6(b) and (d)). It is found that the longest cycle can be driven without using the fuel-cell. 4.2.2. Convoy driving Comparisons of energy consumption by the motor, the fuel cell, the battery, and both the fuel-cell and the battery obtained by the co-optimization and the sequential optimization during the Convoy driving for different driving times are presented in Fig. 8(a)–(d). The results show similar trend as observed for the Churchville-B driving shown in

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Fig. 7. Histograms of normalized occurrence of powertrain operation obtained by co-optimization and sequential optimization on Churchville-B road for two drive cycles: (a) motor power demand, (b) fuel-cell input power, and (c) battery power with t f = 329 s ; (d) motor power demand, (e) fuel-cell input power, and (f) battery power with t f = 795 s .

Fig. 5. However, the energy savings by co-optimization in the Convoy driving are much smaller than those in the Churchvill-B driving; specifically, the maximum energy saving via co-optimization is found to be 5.3% only. This small benefit in energy saving by co-optimization can be explained from the trajectories of the vehicle speed and the battery SOC for two driving times presented in Fig. 9, in which Fig. 9(a) and (b) show the results for the shortest cycle while Fig. 9(c) and (d) show those for the longest cycle. As seen from Fig. 9(a), for the shortest cycle, both optimization approaches use the so-called Pulse-and-Glide operation in a very similar way. Note that this Pulse-and-Glide operation is one of the most typical energy-efficient driving strategies for proper roads in the plains or with a slope less than 2% [38,4]. In this operation, the vehicle accelerates quickly to a higher speed and then keeps coasting or gliding as a result of the vehicle’s kinetic energy stored in the period of acceleration. Because of this similar driving pattern, the resulting SOC

trajectories obtained from both optimization approaches become similar as well (see Fig. 9(b)). For the longest cycle, very similar trajectories in vehicle speed and battery SOC are observed as shown in Fig. 9(c) and (d), respectively. It can be also found that Pulse-and-Glide operation is not preferred in relatively low speed operation for both cooptimization and sequential optimization. Since both co-optimization and sequential optimization find similar solutions, the normalized occurrences of powertrain look also similar for both shortest and longest cycles. Especially, for the shortest cycle, the Pulse-and-Glide strategy dominates vehicle operation. To provide the high propulsion power, both fuel-cell and battery need to be used as shown in Fig. 10(a)–(c). On the other hand, for the longest Convoy cycle the fuel-cell is hardly used for both co-optimization and sequential optimization since power demand is sufficiently low such that the battery can only provide propulsion and braking power, as seen from Fig. 10(d)–(f).

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Fig. 8. Comparison of energy consumption obtained by co-optimization and sequential optimization for various times traveled on Convoy road: (a) energy consumption by the motor, (b) energy consumption by the fuel cell, (c) energy consumption by the battery, and (d) the total energy consumption by the fuel cell and the battery.

5. Conclusion

instead. Then, two specific driving scenarios for a military ground vehicle over the Churchville-B Cycle and Convoy Cycle are simulated to compare the two approaches in terms of energy consumption. The total energy savings by co-optimization range from 5.3% to 24.2% for aggressive driving on a hilly terrain. Meanwhile, on a relatively flat road, the benefit of co-optimization is less significant. Nonetheless, co-optimization achieved 0.5%-5.3% reduction in total energy consumption as compared to sequential optimization. Future work will seek ways to properly enforce a terminal SOC constraint for sequential optimization. Considering the tradeoff between performance and computational cost (time and memory requirements), sequential optimization is a good option for real-time implementation. As provided in this study, the co-optimization approach takes about 5 min to optimize a driving cycle over which a

This study investigates the co-optimization and sequential optimization approaches for determining the speed profile and power-split of fuel-cell/battery hybrid electric systems based on Dynamic Programming. In order to reduce computational cost for Dynamic Programming, the optimal control problem is formulated and analyzed by applying Pontryagin’s Minimum Principle to reduce the number of control inputs. It is found that the simplification of the battery SOC dynamics leads to the use of discrete vehicle control modes, similarly to [14]. This result allows for a substantial decrease in computational cost of Dynamic Programming. To further reduce computational load, the original co-optimization problem is modified by eliminating the state variable corresponding to time and adding a penalty on operation time

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Fig. 9. Comparison of vehicle and powertrain operation obtained by co-optimization and sequential optimization over the Convoy cycle for travel times of 1259 s and 2579 s: (a) vehicle speed, (b) electrical power demand by the motor, (c) power consumed by the fuel cell, (d) power provided by the battery, and (e) battery SOC.

vehicle operates around 5–8 minutes as compared to the sequential approach which takes about 5 s. Therefore, it will be important to modify the sequential optimization cost function such that it can approximate the co-optimization result. Moreover, investigation of the influence of thermal system and power electronics system including a DC/DC converter on the performance of the fuel-cell/battery electric vehicle could be another direction of future work. The effects of battery degradation and fuel-cell durability were studied in [39,40], and should

be further investigated in co-optimization problem framework. Particularly, a machine learning techniques (e.g., Reinforcement Learning or Approximate Dynamic Programming used in [41]) can be applied to deal with a system with many states and uncertainties in co-optimization problem.

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Fig. 10. Histograms of normalized occurrence of powertrain operation obtained by co-optimization and sequential optimization on the Convoy cycle for two driving times: (a) motor power demand, (b) fuel-cell input power, and (c) battery power with t f = 1259 s ; (d) motor power demand, (e) fuel-cell input power, and (f) battery power with t f = 2579 s .

CRediT authorship contribution statement

interests or personal relationships that could have appeared to influence the work reported in this paper.

Youngki Kim: Conceptualization, Methodology, Software, Data curation, Visualization, Writing - original draft. Miriam FigueroaSantos: Software, Data curation, Writing - original draft. Niket Prakash: Software. Stanley Baek: Methodology. Jason B.Siegel: Conceptualization, Supervision, Funding acquisition, Writing - review & editing. Denise M. Rizzo: Conceptualization, Supervision, Project administration.

Acknowledgment The authors would like to acknowledge the technical and financial support of the Automotive Research Center (ARC) in accordance with Cooperative Agreement W56HZV-19-2-0001 U.S. Army CCDC Ground Vehicle Systems Center (GVSC) Warren, MI. DISTRIBUTION A. Approved for public release; distribution unlimited. OPSEC#1991.

Declaration of Competing Interest The authors declare that they have no known competing financial Appendix A. Details of the fuel-cell model

This section provides detailed equations to model the 15 kW hydrogenics liquid-cooled fuel-cell system (consisting of a graphite bipolar plate). It is assumed that the temperature is regulated by the cathode airflow. The fuel-cell model is based on the polarization curve as given by the following equation:

Vcell = E (Tfc, p)

RTfc sinh F

1

i + iloss 2i oc

iRmb + Bc log 1

i , i max

where Vcell is the voltage of the fuel-cell, Tfc is the fuel cell temperature, and i is the current density determined by the fuel-cell current Ifc and the active area Afc = 523cm2 , i.e., i = Ifc/ Afc . The stack voltage can be found by multiplying the cell voltage by the number of cells in each stack, 14

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Ncell = 65. The limiting current density, the exchange current density and the loss current density are denoted by i max , i oc , and iloss , respectively; the membrane resistance and the mass transport loss coefficient are represented by Rmb and Bc , respectively. The fuel-cell potential E is given by the Nernst equation as follows: E Tfc, p =

G (Tfc ) RTfc log + nF nF

p H2

pO 2

p0

p0

pV

,

psat (Tfc)

G (T )

fc where p0 is the reference pressure, psat (T ) is the saturation pressure, and is the temperature-dependent Gibbs free energy. The oxygen, nF hydrogen, and water vapor pressures are denoted as p H2 , pO2 , and pV , respectively. The polarization curve parameters are obtained by using experimental data at 60 ° C as shown in Fig. A.1. The oxygen, hydrogen, and water vapor pressures values used to obtain the polarization curve parameters where pO2 = 0.21·(pca pv ) bar, p H2 = 120 kPa, and pV = psat (Tfc ) bar where the cathode pressure varies between pca = 103 to 120 kPa depending on the airflow rate and current. The relative humidity in the cathode inlet is assumed to be 1. The saturation pressure is given in kilopascals (kPa) by:

psat (Tfc ) = 0.61121exp

18.678

Tfc 234.5

Tfc 257.14 + Tfc

.

The parameters of the polarization model are provided in Table A.1. The fuel cell system efficiency ( fc

fc )

and is calculated as:

Pfc Paux P out Pfcout I ·N N ·V Paux = fcin = = = fc fc cell cell HLHV , H2 WH2 Eth Ifc Nfc Ncell Eth Ifc Nfc Ncell Pfc

where Pfc = Ifc ·Nfc Ncell·Vcell is the fuel-cell gross power, Pfcout = Pfc Paux is the fuel-cell’s net power, Paux is the parasitic load that comes from the radiator pump, cooling fan and cathode air supply blower, HLHV , H2 is the lower heating value and WH2 is the hydrogen flow rate. In the efficiency calculation, the hydrogen flow rate is proportional to the current, since the anode is dead ended and stoichiometry 1 operation is assumed. The theoretical potential Eth is based on the lower heating value, and Ifc is the fuel cell’s current, and Nfc is the number of stacks and Ncell is the number of cells per stack. Therefore, the fuel power is equivalent to Eth Ifc Nfc Ncell .

Fig. A.1. Fuel cell model polarization curve and comparison with experimental data at 60° C. Table A.1 Parameters for the fuel cell polarization curve obtained from experimental data. Parameters

Values

iloss i oc Bc imax Rmb

8.72 mA/cm2 0.17 µ A/cm2 0.1028 V 1.4508 A/cm2 0.1548 cm2

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Appendix B. Detailed computation for DP implementation For each of the five control actions in OCPs (26) and (30), the following equations are used to compute the corresponding acceleration, velocity, and force for a given distance of s : Mode 1, Full Propulsion

{

a=

min a , 2 2, k

2, k + 1=

fe = ff =

u1max

(s, 2) M

},

+ 2a s ,

Ma + (s, 0,

2 ),

Mode 2, Coasting fr

a=

fg

M 2 2, k

2, k + 1=

fe = ff =

,

+ 2a s ,

0, 0,

Mode 3, Regenerative braking

{

a=

max a, 2 2, k

2, k + 1=

fmin

fr

fg

M

},

+ 2a s ,

fe =

Ma + fr + fg ,

ff =

0,

Mode 4, Full braking

a=

a, 2 2, k

2, k + 1=

+ 2a s ,

fe =

max{fmin , Ma + fr + fg },

ff =

min{Ma + fr + fg

fe , 0},

Mode 5, Cruising

a= 2, k + 1=

0, 2, k ,

fe =

fr + fg ,

ff =

0,

In the equations above, the larger magnitude of the resisting force, fr , or grade force, fg , in addition to whether the grade is positive or negative, determines whether the force applied to the vehicle is a propulsion force or braking force. Thus, the force in cruising mode can be either positive or negative depending on the grade and its dominance over the other resisting forces.

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