Optimal motion control for energy-aware electric vehicles

Control Engineering Practice 38 (2015) 37–45

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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac

Optimal motion control for energy-aware electric vehicles$ Tao Wang, Christos G. Cassandras, Sepideh Pourazarm Division of Systems Engineering and Center for Information and Systems Engineering Boston University, United States

art ic l e i nf o

a b s t r a c t

Article history: Received 1 February 2014 Accepted 18 December 2014

We study two problems of optimally controlling how to accelerate and decelerate a non-ideal energyaware electric vehicle so as to (a) maximize its cruising range and (b) minimize the traveling time to a specified destination under a limited battery constraint. Modeling an electric vehicle as a dynamic system, we adopt an electric vehicle power consumption model (EVPCM) and formulate two respective optimal motion control problems. Although the full solutions can only be obtained numerically, we propose approximate controller structures such that the original optimal control problems are transformed into nonlinear parametric optimization problems, which are much easier to solve. Numerical examples illustrate the solution structures and support their accuracy. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Optimal control Electric vehicles Power consumption model Parametric optimization

1. Introduction The emergence of plug-in hybrid electric vehicles (HEV) and fully electric vehicles (EV) is motivated by the goals of reduced oil dependency and greenhouse gas emissions. However, both HEVs and EVs heavily rely on limited battery power, thus raising such issues as vehicle cruising ranges and accessibility to charging sources. In the HEV literature, work has been focused on designing optimal strategies for power distribution between the electric motor and the combustion engine in order to minimize fuel consumption (Sciarretta, Back, & Guzzella, 2004; Salmasi, 2007). Moreover, efforts have been dedicated to establishing controloriented models of the traction dynamics of the vehicles, which are the vital ingredients for active controllers to achieve desired accuracy and energy-efficiency. For example, a dynamic model of the power-train of HEVs is proposed in Powell, Bailey, and Cikanek (2002); Hori, Toyoda, and Tsuruoka (2002) address the traction control of an EV; and speed and acceleration controllers are studied for an energy-aware two-wheeled EV in Dardanelli et al. (2011). On the other hand, the expanding number of HEV and EV fleets brings out new research issues related to insufficient power supplied to vehicles, allocation of limited charging stations and power balance of electric grids. From the vehicle side, a circuitbased battery is commonly used to represent the electricity source

☆ The authors' work is supported in part by NSF under Grants CNS-1239021 and IIP-1430145, by AFOSR under Grant FA9550-12-1-0113, by ONR under Grant N00014-09-1-1051, and by ARO under Grant W911NF-11-1-0227. E-mail addresses: [email protected] (T. Wang), [email protected] (C.G. Cassandras), [email protected] (S. Pourazarm).

http://dx.doi.org/10.1016/j.conengprac.2014.12.013 0967-0661/& 2014 Elsevier Ltd. All rights reserved.

when considering the supervisory control of an HEV Guzzella and Sciarretta (2007). Sundstrom and Binding (2011) employed this model to optimally plan the charging behaviors of EV fleets in terms of grid power balancing. In order to minimize the waiting time for EV charging, a scheduling problem in a network of EVs and charging stations was studied in Qin and Zhang (2011). From the grid side, an optimization methodology of allocating the recharging infrastructure for EVs in an urban environment was developed in Gallego and Larrodeg (2011). More recently, a decentralized protocol for negotiating day-ahead charging schedules for EVs via pricing control was proposed in Gan, Topcu, and Low (2011) to fill the overnight electricity demand valley. However, despite the variety of research on HEV/EV energy-aware systems, there is little work investigating HEV/EV motion control from a power management perspective, which is mainly because the relationship between vehicle dynamics and power consumption is complicated. In Kuriyama, Yamamoto, and Miyatake (2010) an “Eco-Driving” technique based on dynamic programming is developed for EVs seeking an optimal speed profile to minimize the amount of total energy consumption under fixed origin and destination, running time and track condition. Assuming a complete knowledge of the upcoming traffic light timing via infrastructure to vehicle (I2V) communication, De Nunzio, Canudas de Wit, Moulin, and Domenico (2013) address the problem of energyoptimal velocity profiles in urban traffic network. Formulating this as an optimal control problem, an algorithm is proposed which provides a fast sub-optimal solution for the energy minimization problem. In Dib, Serroa, and Sciarretta (2011), a dynamic programming algorithm was developed to minimize energy consumption for an EV during a short trip by optimizing the speed trajectory. Two alternative formulations were proposed which are suitable for off-line and on-line implementation.

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T. Wang et al. / Control Engineering Practice 38 (2015) 37–45

Closely related work is presented in Petit and Sciarretta (2011) and Dib, Chasse, and Sciarretta (2012) which aim to minimize energy consumption for an EV over a time and distance horizon. Formulating this as an optimal control problem (OCP) and doing Hamiltonian analysis, a two point boundary value problem (TPBVP) was obtained in Petit and Sciarretta (2011). Then, using the inversion-based methodology, a closed-form solution for a simple case when the aerodynamic friction and the road slope are ignored was proposed. In Dib et al. (2012) a similar problem is revisited in an urban environment where a trip is broken into several segments and the optimal control is formulated for each segment. Similar to Petit and Sciarretta (2011), a TPBVP was obtained by Hamiltonian analysis and a closed form solution for the optimal trajectories was derived when the road slope is constant and the aerodynamic friction is ignored. In the current paper, we discuss two optimization problems for an EV traveling through a highway: (a) cruising range maximization and (b) traveling time minimization. In both, we start by formulating the problem as an OCP with a free terminal time. Note that for the cruising range maximization problem, which is closer to the problems discussed in Petit and Sciarretta (2011) and Dib et al. (2012), there is no terminal condition for amount to travel. Next, we solve them numerically and derive the structure of the optimal controllers based on which we propose simpler nonlinear parametric optimization problems giving approximate solutions. Using this method we can obtain approximate solutions for more general cases relative to Petit and Sciarretta (2011) and Dib et al. (2012) where we can include aerodynamic friction and road slope. An analytical power consumption model for an EV was proposed in Tanaka, Ashida, and Minami (2008), which presents a comprehensive relationship between velocity, acceleration and power consumption rate. Motivated by the model in Tanaka et al. (2008), in this paper we formulate two optimal motion control problems, i.e., a cruising range maximization problem and a vehicular traveling time minimization. Although the intricate state dynamics of the problems require resorting to numerical solutions, they do serve to formulate approximate solution structures so as to transform the original optimal control problems into simpler nonlinear parametric optimization problems. The approximate solutions possess a much simpler structure while preserving accuracy. This allows us to apply optimal motion planning to various interesting issues in EV-based systems, such as vehicular routing, charging station deployment and EV-to-smart-grid (V2G) charging scheduling, thus opening up a wide spectrum of research directions. In the area of optimal routing for electric vehicles, we have addressed several problems for a single EV with energy constraints as well as multi-vehicle routing problems considering traffic congestion effects on both traveling time and energy consumption on their paths (Wang, Cassandras, & Pourazarm, 2014; Pourazarm & Cassandras, 2014). Our results are illustrated by examples based on data available for certain EV classes. We point out, however, that with EV technology rapidly evolving, the model data used for specific examples are likely to become obsolete. Thus, the main contribution of the paper is the formulation of the optimal control problems and the derivation of controller structures which remain valid regardless of EV model used. The structure of the paper is as follows. In Section 2, we introduce the analytical power consumption model for EVs. In Section 3, we formulate an EV cruising range maximization problem based on a power consumption model with a prescribed initial energy. The explicit numerical solution, as well as an approximate one, is presented and the accuracy of the latter is verified. Based on the approximate solution structure, the optimal control problem is reduced to a nonlinear parametric optimization problem, which is easier to solve. Section 4 explores an EV

traveling time minimization problem with the same procedure as Section 3. Finally, conclusions and further research directions are described in Section 5.

2. Electric vehicle power consumption model (EVPCM) In Tanaka et al. (2008), an analytical electric vehicle power consumption model (EVPCM) was proposed, capturing the relationship between vehicular power consumption and motion metrics (velocity and acceleration). The accuracy of this model is verified based on tests involving two real EVs. The following table is the nomenclature used for the EVPCM: this model takes into account the power consumption on vehicle traction as well as heat loss of the motor. The former part can be expressed as P τ ðtÞ ¼ vðtÞFðtÞ ¼ vðtÞðmaðtÞ þ kv ðtÞ þ mg μÞ 2

ð1Þ

where F(t) represents the instantaneous motor traction force, incorporating the acceleration resistance, air resistance and rolling resistance (assuming that the vehicle travels on flat ground so that the inclination resistance is ignored). When a vehicle motor runs at high speed, the energy loss coming from the coil heating cannot be ignored. However, this is usually not considered when modeling the vehicle's engine-generator power consumption (Taha, Passarella, Rahim, & Sah, 2010). In Tanaka et al. (2008), the power loss due to the motor is modeled according to the relationship of the motor back-electromotive force E(t), the motor current I(t), the vehicle velocity v(t) and the motor traction force F(t), which are as follows: 8 < FðtÞ ¼ K a ϕN  IðtÞ ¼ KIðtÞ FðtÞ R ⟹IðtÞ ¼ : EðtÞ ¼ K a ϕN  vðtÞ ¼ KvðtÞ K R Along with the definition of F(t) in (1), the power loss due to the motor at time t becomes P m ðtÞ ¼ I 2 ðtÞr ¼

r K2

ðmaðtÞ þ kv ðtÞ þ mg μÞ2 2

Therefore, the total vehicle power consumption is P m ðtÞ þ P t ðtÞ. In other words, the instantaneous battery output power of an EV is P½aðtÞ; vðtÞ ¼

r K2

ðmaðtÞ þ kv ðtÞ þ mg μÞ2 2

þvðtÞðmaðtÞ þ kv ðtÞ þ mg μÞ 2

ð2Þ

Note that it is possible to have P½aðtÞ; vðtÞ o0 when aðtÞ o 0, which indicates that the EVPCM also incorporates the regenerative braking effect, i.e., the EV's battery recovering energy from braking. Remark 1. Note that adding road slope information does not change our analysis for the problems we will discuss in Sections 3 and 4. However, it will change the energy consumption and consequently cruising range of the EV. In this case, the instantaneous battery output power in (2) becomes P½aðtÞ; vðtÞ ¼

r K2

ðmaðtÞ þ kv ðtÞ þ mg μ þ mg sin θÞ2 2

þ vðtÞðmaðtÞ þ kv ðtÞ þ mg μ þ mg sin θÞ 2

where θ is the road slope. Therefore, if the vehicle has the sensing capability to measure the road slope (or road latitude) this information can be simply added to our formulation in the case of on-line implementation. The problem becomes more challenging when the road slope is a function of position, x(t), therefore, the instantaneous battery output power in (2) becomes a function of x(t). This is a topic of ongoing research and is not further addressed in this paper.

T. Wang et al. / Control Engineering Practice 38 (2015) 37–45

3. Cruising range maximization problem

3.1. Problem statement

39

Then, the costate (Eular–Lagrange) equations λ_ ¼  ∂H=∂x give 8 > λ_ ðtÞ ¼ 0 > < 1 λ_ 2 ðtÞ ¼  λ1 ðtÞ þ λ3 ðtÞ∂Pða;vÞ ð12Þ ∂v > > : λ_ ðtÞ ¼ 0 3

EVs usually have a smaller maximum cruising range on a single charge than cars powered by fossil fuels. Therefore, the cruising range is critical for an EV to consider before it can reach its destination or a charging station. Motivated by this issue, we seek to control the acceleration process so as to maximize the EV's cruising distance with a given initial battery power, where the EV is modeled as an EVPCM. This problem can be formulated as a state constrained optimal control problem with an unspecified terminal time

∂Pða; vÞ r 2 ¼ 2 ð4k v3 ðtÞ þ 4mkvðtÞaðtÞ þ 4mg μkvðtÞÞ ∂v K þ maðtÞ þ 3kv ðtÞ þmg μ 2

ð13Þ

Moreover, due to (3) and (8), we must satisfy λðTÞ ¼ ∂ΦðxðTÞÞ=∂x, where ΦðxðTÞÞ ¼  xðTÞ þ ν1 vðTÞ þ ν2 eðTÞ and ν1 ; ν2 are unknown multipliers, so that

ð3Þ

λ1 ðTÞ ¼  1;

_ ¼ vðtÞ xðtÞ

ð4Þ

Since the terminal time is unspecified, the transversality condition is ðH þ ðdϕ=dtÞÞj t ¼ T ¼ 0 where ϕðxðTÞÞ ¼ xðTÞ, thus

_ ¼ aðtÞ vðtÞ

ð5Þ

min aðtÞ

 xðTÞ

In view of (2), we have

_ ¼  PðtÞ eðtÞ eð0Þ ¼ E0 ;

xð0Þ ¼ 0;

ð6Þ vð0Þ ¼ 0

ð7Þ

eðTÞ ¼ 0; vðTÞ ¼ 0

ð8Þ

eðtÞ Z0;

ð9Þ

0 rvðtÞ rvmax

amin r aðtÞ r amax

ð10Þ

where x(t) is the distance traveled by the EV by time t, e(t) is the instantaneous battery's residual energy at time t, v(t) and a(t) are defined in Table 1, and a(t) is the only control variable in this problem. The expression for P(t) is given in (2). The initial conditions in (7) initialize the battery energy, traveled distance and vehicle speed. The terminal time T is determined by (8), implying that the entire process ends when the battery energy reaches a given minimum value emin which, for simplicity, we take to be emin ¼ 0 without affecting the analysis. Finally, vmax bounds the top speed while amin and amax are, respectively, the maximum deceleration and acceleration, where amin o 0 and amax 4 0. 3.2. Hamiltonian analysis We begin by analyzing the unconstrained case in which the constraints in (9) are relaxed. In this case, the optimal state trajectory consists of an interior arc over the entire interval ½0; T. Let xðtÞ ¼ ðxðtÞ; vðtÞ; eðtÞÞT and λðtÞ ¼ ðλ1 ðtÞ; λ2 ðtÞ; λ3 ðtÞÞT denote the state and costate vector, respectively. The Hamiltonian for this problem is Hðx; λ; aÞ ¼ λ1 ðtÞvðtÞ þ λ2 ðtÞaðtÞ  λ3 ðtÞPða; vÞ

ð11Þ

Table 1 Nomenclature for the EVPCM. K ¼ K aRϕN

r m a(t) v(t) k μ g P(t)

K a : the DC motor innate armature constant; ϕ: the magnetic flux on the armature; R: the radius of the tire (m); N: the gear reduction ratio Coil resistance (Ω) The vehicle mass (kg) The acceleration at time t (m/s2) The vehicle velocity at time t (m/s) Air resistance constant (kg/m) Rolling resistance constant Gravity acceleration constant (m/s2) Instantaneous output power of the battery at t (J)

λ2 ðTÞ ¼ ν1 ;

λ3 ðTÞ ¼ ν2

ð14Þ

 vðTÞ  vðTÞ þ ν1 aðTÞ  ν2 P½aðTÞ; vðTÞ ¼ 0

ð15Þ

Note that from (14) and (12)

λ_ 2 ðtÞ ¼ 1 þ ν2

∂Pða; vÞ ∂v

with λ2 ðTÞ ¼ ν1

ð16Þ

where λ2 ðtÞ is a function of the control a(t) due to (13). Therefore, in view of (11), the Hamiltonian is not a linear function of a(t). As a result, we can use the optimality condition ∂H=∂a ¼ 0 so that

λ2 ðtÞ  λ3 ðtÞ

∂Pða; vÞ ¼0 ∂a

ð17Þ

where ∂Pða; vÞ rm 2 ¼ 2 ð2maðtÞ þ 2kv ðtÞ þ 2mg μÞ þ mvðtÞ ∂a K

ð18Þ

Now a two-point boundary value problem is fully specified using (12)–(18). However, owing to the complexity of ∂Pða; vÞ=∂v in (16), we are unable to analytically obtain λ2 ðtÞ. Therefore, we have to resort to numerical methods. 3.3. Numerical solution We solve this optimal control problem using bvp5c, a MATLAB solver for boundary value problems. To do so we convert our free terminal time problem into a fixed end time problem (Ascher & Russell, 1981). This is done by re-scaling time so that τ ¼ t=T, then τ A ½0; 1. Finally, we introduce final time as a dummy state r with the trivial dynamics r_ ¼ 0. Without loss of generality, we solve the problem for a class of EVs referred to as “neighborhood electric vehicles (NEV)”. This type of EVs is classified as “low speed vehicles” and is legally limited to roads with a speed limit as high as 45 mile/h (72 km/h). One example of such a vehicle is the Toyota COMS with a maximum speed of 60 km/h and a maximum range of 50 km. Typical parameter settings for such EVs, consistent with the literature (Shimizu, Harada, Bland, Kawakami, & Chan, 1997), are listed in Table 2. A numerical example of the optimal solution and optimal state trajectories is shown in Fig. 1. Table 2 Parameter settings for the EVPCM. K¼

K a ϕN (V s/m) R

3.5 g (m/s2) 9.8

r (Ω)

k (kg/m)

amax (m/s2)

amin (m/s2)

1 μ 0.025

0.3 m (kg) 650

4 E0 (J)

4 Vmax (m/s) 17

2:5  107

40

T. Wang et al. / Control Engineering Practice 38 (2015) 37–45

4

5

0

Instantaneous cruising distance(m)

x 10

0

1000

2000

3000

4000

5000

2

Instantaneous vehicle acceleration (m/s ) 0.5 0 −0.5

0

1000

2000

3000

4000

5000

4000

5000

Instantaneous vehicle speed (m/s) 20 10 0

1000

2000

3000

Instantaneous battery residual energy (J)

x 10

2 0

Note that since T ¼ t 1 þt 2 , then v~ n ðTÞ ¼ 0 by (20), which satisfies the terminal condition (8). However, we still have a constraint on the terminal energy value, i.e., eðTÞ ¼ 0. Therefore, substituting n a~ ðtÞ and v~ n ðtÞ into (2) and integrating (6) from 0 to T, we can establish a new equality constraint gðA; t 1 ; t 2 Þ ¼ 0 defined by Z T Z T eðTÞ  eð0Þ ¼  PðtÞ dt⟹gðA; t 1 ; t 2 Þ ¼  PðtÞ dt þ E0 ¼ 0 0

0 7

4

Moreover, in light of (4) and (7), we can write the objective function (3) as Z T A x~ n ðTÞ ¼ xð0Þ þ v~ n ðtÞ dt ¼ ð3t 1 t 2 þ t 21 Þ ð21Þ 6 0

0

1000

2000

3000

4000

5000

time (sec)

We use the software package Mathematica (Wolfram Research Inc) to complete the integration, the result of which is presented in the Appendix. Now along with (21), we can transform the original optimal control problem into a nonlinear parametric optimization problem A  x~ n ðTÞ ¼  ð3t 1 t 2 þ t 21 Þ 6

min

A;t 1 ;t 2

Fig. 1. Optimal solution of the cruising range maximization problem.

s:t: In Fig. 1, the numerical solution displays a clear structure for the optimal acceleration: it starts with a small value (0.2528 m=s2 in the solution), slowly decreases to 0 at some time, gradually increasing the speed up to a value (11:21 m=s equivalently 40.3 km/h), and then remains at 0 for a while, and in the end slowly decreases to a negative value ð  0:2528 m=s2 Þ, bringing the speed down to 0. The accelerating process in the beginning and the decelerating process in the end are symmetric in terms of the absolute values of acceleration and velocity. Under the control process, the energy trajectory e(t) and the cruising distance x(t) seem to be respectively linearly decreasing and increasing throughout ½0; T. In this example, T n ¼ 4512 s and xn ðTÞ ¼ 49:66 km.

3.4. Approximate parametric optimization problem Motivated by the numerical solution and its clear structure, we n propose an approximate optimal control a~ ðtÞ as follows: 8 A > > ðt 1  tÞ > > > < t1 n a~ ðtÞ ¼ 0 > > > A > > :  t ðt  t 2 Þ 1

gðA; t 1 ; t 2 Þ ¼ 0

0 o A ramax ;

0 o t 1 rt 2

ð22Þ

Using the same parameter settings as in Table 2, we solve this problem using the optimization solver Opti (MATLAB toolbox for optimization). In the numerical solution, An ¼ 0:205, t n1 ¼ 109:0944 and t n2 ¼ 4403:9 such that x~ n ðTÞ ¼ 49:64 km, which is a very accurate approximation of the optimal objective xn ðTÞ ¼ 49:66 km for the optimal control problem. Fig. 2 shows the optimal solution and optimal state trajectories. To further justify the effectiveness of our approximate solution, we compare the objective values in the solutions of the two problems based on different values of E0 in Table 3. The comparison shows that the solution to the parametric optimization problem is almost indistinguishable from the one of the optimal control problems. The numerical solutions indicate that the critical cruising speed vcr is 11.18 m/s (around 40 km/h) which is a reasonable speed for an NEV. We observe that this speed is robust with respect to the initial battery energy. The reason is that the optimal velocity profile for the maximum range problem is equivalent to the one leading to the most efficient energy consumption. Table 4 shows the critical speed for different values

t A ½0; t 1  4

t A ðt 1 ; t 2 

ð19Þ

5

Instantaneous cruising distance (m)

x 10

t A ðt 2 ; T 0

where T ¼ t 1 þ t 2 and A; t 1 and t2 are the unknown parameters to be determined, also satisfying 0 o A r amax and 0 o t 1 r t 2 . This structure approximates the numerically obtained optimal control an ðtÞ by linearizing the gradually changing curves at the beginning and ending parts shown in Fig. 1. Accordingly, since the acceleration structure has been determined, the velocity structure can be obtained through (5)

v~ n ðtÞ ¼

0

8 A 2 > > t At  > > 2t > 1 > > < At 1 2 > >   > > > A t 21  t 22 t 2 > >  þ t2 t : t1 2 2

t A ½0; t 1  t A ðt 1 ; t 2 

ð20Þ

t A ðt 2 ; T

We let vcr ¼ At 1 =2, which is the critical cruising speed of the vehicle over ðt 1 ; t 2 .

0

1000

2000

3000

4000

5000

Instantaneous vehicle acceleration (m/s2)

0.5 0 −0.5

15 10 5 0

0

1000

3000

4000

5000

4000

5000

Instantaneous vehicle speed (m/s)

0

1000 7

4

2000

x 10

2000

3000

Instantaneous battery residual energy (j)

2 0

0

1000

2000 3000 time (sec)

4000

Fig. 2. Optimal solution of the parametric optimization problem.

5000

T. Wang et al. / Control Engineering Practice 38 (2015) 37–45

Table 3 Comparison of solutions to the optimal control problem and the parametric optimization problem. 0.6  107 12.07 12.05

E0 (J) xn ðTÞ (km) x~ n ðTÞ (km)

1.2  107 23.62 23.61

2.5  107 49.66 49.64

5  107 99.72 99.71

Table 4 Robustness in the critical speed with respect to initial battery energy E0 in the solutions to the optimal control problem and the parametric optimization problem. 0.6  107 11.21 11.08

E0 (J) vncr (m/s) v~ ncr (m/s)

1.2  107 11.21 11.15

2.5  107 11.21 11.18

5  107 11.21 11.2

of initial energy by solving both the optimal control and the parametric optimization problems. We should mention that solving the original optimal control problem is a challenging task and extremely dependent on finding a good initial guess as well as available computer memory. Also, it may take more than 30 min to reach an answer in each try. On the other hand, based on our simulation results, the parametric optimization problem is a precise approximation of the original problem which can provide the optimal control policy in less than 2 s using the same desktop computer with 1.86 GHz Intel Core 2 CPU and 4 GB RAM. To sum up, by solving the parametric problem (22) to determine the control policy (19), we obtain an approximate solution to the EV cruising range maximization problem and observe that this policy has a simple, easy to implement structure dependent only on two critical times and a fixed cruising speed maintained between these times.

41

4.2. Hamiltonian analysis We begin again by relaxing the state constraints (29) and analyzing the unconstrained case, where the optimal state trajectory consists of an interior arc throughout the entire process. As before, let xðtÞ ¼ ðxðtÞ; vðtÞ; eðtÞÞT and λðtÞ ¼ ðλ1 ðtÞ; λ2 ðtÞ; λ3 ðtÞÞT denote the state and costate vector, respectively. The Hamiltonian for this problem is Hðx; λ; aÞ ¼ 1 þ λ1 ðtÞvðtÞ þ λ2 ðtÞaðtÞ  λ3 ðtÞPða; vÞ

ð31Þ

and the costate equations λ_ ¼  ∂H=∂x are the same as (12) 8_ λ ðtÞ ¼ 0 > > > 1 < ∂Pða; vÞ λ_ 2 ðtÞ ¼  λ1 ðtÞ þ λ3 ðtÞ ∂v > > > :_ λ 3 ðtÞ ¼ 0

where ∂Pða; vÞ=∂v is the same as (13). Moreover, due to (28), we must satisfy λðTÞ ¼ ∂ΦðxðTÞÞ=∂x, where ΦðxðTÞÞ ¼ νðxðTÞ  SÞ and ν are an unknown multiplier, so that

λ1 ðTÞ ¼ ν;

λ2 ðTÞ ¼ 0;

λ3 ðTÞ ¼ 0

ð33Þ

Solving (32) with the boundary conditions (33), we get 8 > < λ1 ðtÞ ¼ ν λ2 ðtÞ ¼  νðt TÞ > : λ ðtÞ ¼ 0

4.1. Problem statement

Still owing to the unspecified terminal time, the transversality condition ðH þðdϕ=dtÞÞj t ¼ T ¼ 0 (ϕðxðTÞ ¼ 0 here) requires that 1 þ νvðTÞ ¼ 0

ð35Þ

Thus, with the terminal costate condition (33), we have

ν¼ 

1 vðTÞ

Hðxn ; λ ; an Þ ¼ min Hðx; λ; aÞ n

ð36Þ

aðtÞ

We are now interested in how fast the EV can cover a desired traveling distance with a given initial battery load. This is a traveling time minimization problem. We still adopt the EVPCM to model the EV and formulate an optimal control problem as follows: Z T min dt ð23Þ 0

_ ¼ vðtÞ xðtÞ

ð24Þ

_ ¼ aðtÞ vðtÞ

ð25Þ

_ ¼  PðtÞ eðtÞ

ð26Þ

eð0Þ ¼ E0 ;

xð0Þ ¼ 0;

vð0Þ ¼ 0

xðTÞ ¼ S eðtÞ Z 0;

ð27Þ ð28Þ

0 r vðtÞ r vmax

amin r aðtÞ r amax

ð34Þ

3

It follows that ν o 0 with vðTÞ Z 0, which makes λ2 ðtÞ o 0 over ½0; TÞ. We can now apply the Pontryagin minimum principle using (31)

4. Traveling time minimization problem

aðtÞ

ð32Þ

ð29Þ ð30Þ

The problem formulation is almost the same as the one of the cruising range maximization problems except for the objective function (23) and the terminal condition (28), where S is the desired traveling distance, assumed to be less than the maximum cruising range achievable given the same E0 (otherwise, there exists no feasible solution.) We also no longer require vðTÞ ¼ 0 and eðTÞ ¼ emin .

where an ðtÞ, t A ½0; TÞ, denotes the optimal control. We can then see that since λ2 ðtÞ o 0 and λ3 ðtÞ ¼ 0 over ½0; TÞ an ðtÞ ¼ amax ;

t A ½0; T

ð37Þ

which is the optimal control policy for the unconstrained case. Accordingly, the optimal velocity trajectory is achieved by (25) vn ðtÞ ¼ amax t;

t A ½0; T

and the optimal objective value is attained by (28), which gives sffiffiffiffiffiffiffiffiffiffi 2S Tn ¼ amax

R Tn 0

ð38Þ n

v ðtÞ dt ¼ S in

ð39Þ

When it comes to the constrained case with only 0 r vðtÞ r vmax incorporated, we can directly check whether the constraint is active or not by comparing amax T n with vmax: if amax T n r vmax , then the constraint is not active, therefore (37) is still the optimal control policy; otherwise, the constraint vðtÞ r vmax is active at t ¼ vmax =amax and the optimal control (37) cannot apply anymore. In this case, we can still carry out the Hamiltonian analysis. We can immediately exclude vðTÞ o vmax since, if this holds, then no change would occur from (31) to (35), hence the optimal control would still be (37). Now with vðTÞ ¼ vmax , we have a different ΦðxðTÞÞ ¼ ν1 ðxðTÞ  SÞ þ ν2 ðvðTÞ  vmax Þ (νi are the unknown multipliers), which implies that

λ1 ðTÞ ¼ ν1 ;

λ2 ðTÞ ¼ ν2 ;

λ3 ðTÞ ¼ 0

ð40Þ

42

T. Wang et al. / Control Engineering Practice 38 (2015) 37–45

In view of (32), as long as λ3 ðtÞ ¼ 0, ∂PðtÞ=∂v will not be involved in λ_ ðtÞ, which will make λ ðtÞ independent of the control a(t). 2

2

Therefore, we can proceed with a similar derivation to (34)–(37) and determine the optimal control as   8 vmax > > > amax ; t A 0; < amax   ð41Þ an ðtÞ ¼ v > max n > > 0; t A ; T : amax where Tn can be determined by (24)–(25) as Tn ¼

S vmax þ vmax 2amax

ð42Þ

Lastly, let us consider the constrained case with both 0 r vðtÞ r vmax and eðtÞ Z0 incorporated. Regarding the optimal controls (37) and (41), an ðtÞ Z 0 throughout ½0; T n . Then, according to (26) and (2), P n ðtÞ 4 0 over ½0; T n  and en ðtÞ is monotonically decreasing. Thus, we only need to check the value of en ðTÞ for the two possible optimal controls (37) and (41). If en ðTÞ Z 0 under the optimal control, then the constraint eðtÞ Z0 is not involved during t A ½0; TÞ. Otherwise, we have to revise the Hamiltonian analysis letting ΦðxðTÞÞ ¼ ν1 ðxðTÞ  SÞ þ ν2 ðvðTÞ  vmax Þ þ ν3 eðTÞ (νi are the unknown multipliers), which will make λ3 ðTÞ ¼ ν3 . As a result, from (32), λ2 ðtÞ becomes a function of a(t). Similar to the case of the cruising range maximization problem, this analysis cannot generally yield an explicit analytical solution for the optimal control problem, so that we once again proceed with numerical solutions. 4.3. Numerical solution Since we are to address the case where the optimal controls (37) and (41) cannot apply, we assume that eðTÞ ¼ 0. At first, we relax the state constraint vðtÞ r vmax . As in Section 3, by converting the problem to a boundary value problem with a fixed final time, we can obtain a numerical solution to this optimal control problem. A numerical example of the optimal solution and optimal state trajectories is shown in Fig. 3. The parameter settings are the same as in Table 2 except that the required traveling distance S is set as 10,000 m and the value of E0 is 0:5  108 , which is designed to achieve a higher top speed so as to incorporate state constraint vðtÞ r vmax in what follows. Also, we still employ the commonly used vmax value, which is 17 m/s. Instantaneous cruising distance (m) 10000 5000 0

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From Fig. 3, it is clear that in the numerical solution, the whole acceleration process consists of four parts: (1) starting with full accelerating at amax ð4 m=s2 Þ; (2) gradually descending to 0; (3) maintaining aðTÞ ¼ 0; (4) gradually descending to a decelerating value. Accordingly, along with the acceleration process the speed increases until it reaches part (3), and then remains at a fixed value until some time after which the vehicle is required to decelerate for the remaining traveling time. Under this optimal control, we obtain T n ¼ 161:5 s. Moreover, in this numerical example, we notice that the maximum velocity value is 65 m/s, which is unrealistically high in practice for this class of EV, i.e., vmax o 65 m=s. By numerically perturbing the value of vmax we find that there exist two additional types of solutions. The first one can be expressed as   8 vmax > > a ; t A 0; max > > amax > <   vmax ð43Þ an1 ðtÞ ¼ 0; t A ; t > s > > amax > > : amin t A ðt s ; T in which case eðTÞ ¼ 0 and the value of ts depends on vmax and E0. The second one arises when vmax and E0 are such that t s ¼ T in (43) but eðTÞ 4 0, in which case the solution is the same as (41). This case corresponds to a scenario where the vehicle has so much initial energy that it can complete the entire process as fast as possible with a positive residual energy. 4.4. Approximate parametric optimization problem Motivated by the Hamiltonian analysis and the results obtained in the last section, we again propose an approximate optimal n control a~ ðtÞ as follows: 8 > < amax t A ½0; t 1  t A ðt 1 ; t 2  ~a n ðtÞ ¼ 0 ð44Þ > :a t A ðt 2 ; T min where t 1 ; t 2 and T are unknown parameters to be determined, also satisfying 0 o t 1 r t 2 rT and t 1 r vmax =amax . This control structure contains all three types of optimal controls obtained through numerical solutions. Accordingly, the velocity structure can be expressed using (5) 8 t A ½0; t 1  > < amax t t A ðt 1 ; t 2  v~ n ðtÞ ¼ amax t 1 ð45Þ > : a t þ a ðt t Þ t A ðt ; T max 1 2 2 min Therefore, in light of (24) and (27), we can represent the terminal condition (28) by a new equality constraint   t1 a ðT t 2 Þ2 þ min S ¼ 0 ð46Þ g 1 ðt 1 ; t 2 ; TÞ ¼ amax t 1 T  2 2 Moreover, as pointed out earlier, the condition en ðTÞ 4 0 could only occur in the optimal solution (41). Thus, we can simply check the feasibility of the solution (41): if, under (41), eðTÞ 4 0, then the optimal solution is (41); otherwise, we can add the additional terminal condition eðTÞ ¼ 0 without affecting the solution. n Thus, let us assume eðTÞ ¼ 0 in what follows. Substituting a~ ðtÞ n ~ and v ðtÞ into (2) and integrating (6) from 0 to T, we can replace the condition eðTÞ ¼ 0 with an equality constraint g 2 ðt 1 ; t 2 ; TÞ ¼ 0, where gðt 1 ; t 2 ; TÞ is Z T Z T eðTÞ  eð0Þ ¼  PðtÞ dt⟹g 2 ðt 1 ; t 2 ; TÞ ¼  PðtÞ dt þ E0 ¼ 0 0

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time (sec) Fig. 3. Optimal solution to the traveling time minimization problem.

0

ð47Þ We use Mathematica to do the integration, the result of which is shown in the Appendix.

T. Wang et al. / Control Engineering Practice 38 (2015) 37–45

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g 1 ðt 1 ; t 2 ; TÞ ¼ 0

g 2 ðt 1 ; t 2 ; TÞ ¼ 0 0 o t 1 rt 2 r T;

t1 r

vmax amax

ð48Þ

Note that the constraint t 1 r vmax =amax is equivalent to v r vmax . Using the same parameter settings as in the optimal control problem, we use Opti to solve this problem. In order to verify the accuracy of our approximate solution, we first relax the constraint t 1 r vmax =amax and compare the result with the numerical solution in Fig. 3. In the solution to the parametric problem (48), t n1 ¼ 15:74 s and T n ¼ t n2 ¼ 166:8 s, which can be seen to be

close to the numerical optimal objective T n ¼ 161:5 s for the optimal control problem. Fig. 4 shows the solution and corresponding state trajectories. If we incorporate the constraint t 1 r vmax =amax with vmax ¼ 17 m=s (the typical vmax values for NEVs), then we apply the control policy (41) and test the corresponding e(T), which turns out to be 4:163  107 J, much greater than eðTÞ ¼ 0 in the case without the constraint t 1 rvmax =amax (Fig. 4). Consequently, (41) is the optimal control in this case, under which T n ¼ 590:4 s. The solution, including the control and state trajectories, is shown in Fig. 5. If we reduce the initial energy E0, then the optimal top speed will correspondingly decrease. Thus, when E0 is small enough, the optimal top speed does not exceed the speed limit vmax. In this scenario, the optimal controller behaves as (44) and the solution is shown in Fig. 6, where the optimal cruising speed is 11.95 m/s (about 43 km/h). We conclude that the value of an optimal control approach manifests itself when E0 is small, which is precisely when a driver, faced with limited remaining energy, needs assistance to ensure the best EV performance possible.

5. Conclusions and future work We have used an electric vehicle power consumption model (EVPCM) to study two problems of optimally controlling the acceleration (and deceleration) of a non-ideal energy-aware electric vehicle so as to (a) maximize the cruising range and (b) minimize the traveling time to a prescribed destination with limited battery power. In the cruising range maximization problem, due to the complicated relationship between power consumption and vehicle dynamics, the solution can only be attained numerically. However, based on the numerical solution, an approximate solution structure is proposed such that the original optimal control problem can be transformed into a nonlinear parametric optimization problem, which is easier to solve. The accuracy of the approximate solution is also verified. For the traveling time minimization problem, using Hamiltonian analysis, we can first analytically determine the solution when the constraint eðtÞ Z0 is not active, which can be shown to imply en ðTÞ 4 0. If en ðTÞ ¼ 0, which can be tested by checking whether eðTÞ 4 0 under solutions (37) and (41), the solution requires resorting to the numerical method. Then, a similar process to the

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T. Wang et al. / Control Engineering Practice 38 (2015) 37–45

cruising range maximization problem applies. Assuming a typical highway trip or one with few or no mandatory stops, an on-board implementation of these controllers is possible by embedding the necessary software to compute the values of t1, t2 and A in (20) or t1, t2 and T in (45), which determines the optimal speed controller v~ n ðtÞ, and providing it to the driver through standard visualization tools. Alternatively, a simpler approach is to perform these computations off line for a range of reasonable values of the situation-dependent parameters in (49), (46), or (50), such as S or E0, and store the results in on-board memory. The driver may then simply select desired parameter values and obtain the best approximation available for v~ n ðtÞ through a simple table lookup operation. This line of research opens up a wide spectrum of extensions. For instance, as mentioned in Remark 1, when the road slope is a function of the EV position, the Hamiltonian analysis is substantially modified. In a different direction, with the approximate simple solution structure for both problems we can integrate the EVPCM into a number of problems including vehicular routing, charging station deployment and EV to smart grid (V2G) charging scheduling, where either EV traveling time or distance are metrics to optimize over.

Appendix Derivation of equality constraint in nonlinear parametric optimization problem (22) Z T gðA; t 1 ; t 2 Þ ¼ PðtÞ dt E0 ¼ 0 0

  mA A 2 2 ðt  tÞ þ k At  t 1 2 t1 2t 1 0 K     2 A 2 mA A 2 2 t ðt 1 tÞ þ k At  t þ mg μ þ At  2t 1 t1 2t 1 !2  2   Z t2  r At 1 At 1 At 1 2 k k þ mg μ þ þ mg μ dt þ 2 2 2 2 t1 K  Z t1 þ t2   r mA A 2 2  ðt  t 2 Þ þ k ðt  t Þ þ mg μ dt þ t1 2t 1 1 2 K2 t2 !2   2  2 2 A t A 2 2 A t  þ t2t þ mg μ þ ðt 1  t 2 Þ þ  þ t1 2t 1 t1 2 2   2 2 mA A 2 2 A t þ t 2 tÞÞ  ðt t 2 Þ þk ðt  t Þ þ  þ t2 t t1 2t 1 1 2 t 1 2  þ mg μ dt  E0 ¼ 0    r 1 ⟹gðA; t 1 ; t 2 Þ ¼ 2 m2 g 2 μ2 þ m2 Ag μ þ m2 A2 t 3 K !   mg μA mA2 2 r 1 4 2 þ mA3 k þ mg μkA t 3 þ t þ 2 3 15 8 K 12 #t1 " 2 2 2 3 8rk A4 t 5  r k A4 4 kmg μA2 2 t1 þ t1 þ kA t 4 þ  þ 2 2  35 16 2 315K K 0 # t 2   kA3 3 mg μA  t1 t t1 þ þ m2 g 2 μ2 þ  2 8 t1 " r 2 2 2 2 þ ðA ðt 1  t 2 Þ k þ 4at 1 t 2 m þ 4t 21 μgmÞ2 t 16K 2 t 41 ⟹gðA; t 1 ; t 2 Þ ¼

Z

t1

r

þ 4AðAt 2 ðt 21  t 22 Þk  t 1 mÞðA2 ðt 21  t 22 Þ2 k þ 4At 1 t 2 m 4 2 þ 4t 21 μgmÞt 2  A2 ðA2 ðt 21  7t 22 Þðt 21  t 22 Þ2 k þ 4At 1 t 2 ð3t 21 3  5t 22 Þkm þ4t 21 ðt 21 μgk  3t 22 μgk mÞmÞt 3  2A2 kðA2 t 2 ð3t 41  10t 21 t 22 þ 7t 42 Þk  2At 1 ðt 21  5t 22 Þm þ4t 21 t 2 μgmÞt 4

2 þ A2 kðA2 ð3t 41  30t 21 t 22 þ 35t 42 Þk þ 20At 1 t 2 m þ 4t 21 μgmÞt 5 5 4 4 2  A3 kð  3At 21 t 2 k þ 7At 32 k þ t 1 mÞt 6 A4 ðt 21  7t 22 Þk t 7 3 7   1 A 4 2 2  A4 t 2 k t 8 þ A4 k t 9 þ 3  t 21 μgmtð  3t 21 þ 3t 22 9 3 8t 1  3t 2 t þ t 2 Þ þ At 1 mtð  2t 2 þ tÞð 2t 21 þ 2t 22  2t 2 t þ t 2 Þ A2 kt ð35t 61  35t 41 ð3t 22 3t 2 t þ t 2 Þ þ 21t 21 ð5t 42  10t 32 t 35 þ 10t 22 t 2 5t 2 t 3 þt 4 Þ 5ð7t 62  21t 52 t þ 35t 42 t 2  35t 32 t 3 t1 þ t 2  þ 21t 22 t 4 7t 2 t 5 þt 6 ÞÞ  E0 ¼ 0 þ

ð49Þ

t2

Derivation of equality constraint g 2 ðt 1 ; t 2 ; TÞ in nonlinear parametric optimization problem (48) Z T g 2 ðt 1 ; t 2 ; TÞ ¼ PðtÞ dt  E0 ¼ 0⟹g 2 ðt 1 ; t 2 ; TÞ 0 Z t1 r 2 ¼ ðmamax þ kamax t 2 þ mg μÞ2 2 0 K Z t2 r 2 2 þamax tðmamax þ kamax t 2 þ mg μÞ dt þ kamax t 21 2 K t1 2 2 þmg μ þamax t 1 ðkamax t 21 þ mg μÞ dt Z T r þ ðmamin þ kðamax t 1 þ amin ðt  t 2 ÞÞ2 þ mg μÞ2 2 t2 K ðamax t 1 þamin ðt  t 2 ÞÞðmamin þkðamax t 1 þamin ðt t 2 ÞÞ2 þ mg μÞ dt  E0 ⟹g 2 ðt 1 ; t 2 ; TÞ   2 2 r

¼ 2 mamax þ mg μ t þ ðmamax 3 K ! 2 4  2 k amax 5 1 t þ ðma2max þmg μ kamax t 3 þ 2 5 #t 1  3  ka r  2 þmg μamax Þt 2 þ max t 4  þ 2 ðkamax t 21 þ mg μÞ2  4 K 0 i t 2  r

 2 2 þamax t 1 ðkamax t 1 þ mg μÞ t  þ 2 mg μ þ a2max t 21 k t1 K 2 2 2amin amax t 2 k þ amin ðamin t 2 k þ mÞ t 2amin ð  amax t 1 þ amin t 2 Þkðmg μ þ a2max t 21 k 2amin amax t 1 t 2 k 2 þamin ðamin t 22 k þ mÞÞt 2 þ a2min kðmg μ þ 3a2max t 21 k 3 6amin amax t 1 t 2 k þ amin ð3amin t 22 k þ mÞÞt 3  1 2 2 a2min ð  amax t 1 þ amin t 2 Þk t 4 þ a4min k t 5 5 1 þ tð2amax t 1 þ amin ð  2t 2 þ tÞÞð2mg μ þ 2a2max t 21 k 4 þ2amin amax t 1 kð 2t 2 þ tÞ T ð50Þ þamin ð2m þ amin kð2t 22  2t 2 t þ t 2 ÞÞÞ t  E0 ¼ 0 2

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