Theoretical Study on Eco-Driving Technique for an Electric Vehicle with Dynamic Programming
Motoi Kuriyama*, Sou Yamamoto** and Masafumi Miyatake***
Abstract – Eco-driving technique for electric vehicles (EVs) is investigated in this paper.
Many findings on EVs have been reported; however, they did not deal with eco-driving from the viewpoint of theoretical study. The authors have developed an energy-saving driving technique – the so-called “eco-driving” technique based on dynamic programming (DP).
Optimal speed profile of an EV, which minimizes the amount of total energy consumption, was determined under fixed origin and destination, running time, and track conditions. DP algorithm can deal with such complicated conditions and can also derive the optimal solution. Using the proposed method, simulations were run for some cases. In particular, the author ran simulations for the case of a gradient road with a traffic signal. The optimization model was solved with MATLAB.
Keywords: Dynamic Programming, Electric Vehicle, Eco-driving, Intelligent Transportation Systems
1. Introduction
In recent years, social demands for global environmental protection such as reduction of greenhouse gases have been growing. One of the most important goals in many kinds of environmental problems is reduction of total energy consumption. In the transportation sector, there is no doubt that electric vehicles (EVs) [1], with their high efficiency, should play a principal role in the near future [2]-[5].
But EVs have shortcomings. For example, mileage per charge of an EV is low. Although a high-performance battery can serve as a solution, the energy-saving driving technique called “eco-driving” is the method that will be used in this paper for solving this problem. When automatic operation is realized based on the intelligent transportation systems (ITS) technology [6], eco-driving will be implemented more easily and effectively.
Eco-driving technique for a conventional vehicle with an internal combustion engine (ICE) was studied in [7]. In that paper, a soft driving method, which especially restricted acceleration, was proposed. That eco-driving technique can reduce fuel consumption by 24%. Eco-driving assistance system for drivers has already been implemented in some commercial vehicles and reduces fuel consumption in real operation, although theoretical optimality is not always
achieved.
However, the eco-driving technique for the ICE vehicle cannot be applied to an EV because the characteristics of the power train are completely different and regenerative braking is available in the case of the ICE vehicle.
In this paper, the authors aim to clarify the eco-driving technique for an EV by theoretical study. Many previous works on optimal control problems adopt the numerical technique of calculus of variations. These methods often have difficulties accounting for actual vehicle running conditions, which are complicated. Bellman’s dynamic programming (DP) has a substantial advantage in this area, since it can deal directly with the difficult constraints of optimal control problems, except for the terminal boundary conditions. There are several papers that deal with the energy-saving operation of vehicles with a certain kinds of optimization techniques as seen in [8]-[10]. However, they consider only the control of power trains. The authors have modified and re-implemented these techniques for EVs. In particular, the goal is to optimize the speed profile of an EV to minimize the energy consumption under fixed running time and running distance between two points [9]. Having a fixed running time an important conditions because the sensitivity of running time to energy consumption is very high. The problem is formulated as an optimal control problem to find the speed profile that saves the most energy.
The authors demonstrate the effectiveness of the proposed method under some practical conditions such as the presence of gradients and traffic signals.
* Master’s Course of Electrical and Electronics Engineering, Sophia University, Japan.
** Dept. of Electrical and Electronics Engineering, Sophia University, Japan.
*** Dept. of Engineering and Applied Sciences, Sophia University, Japan.
(miyatake@sophia.ac.jp)
Received 17 June 2011 ; Accepted 24 November 2011
control input u Traction mode u = 1 Maximum acceleration 0 < u < 1 Acceleration
u = 0 Coasting
-1 < u < 0 Deceleration u = -1 Maximum deceleration
2. Mathematical formulation of the EV operation 2.1 Motion equation of the EV
Let us define EV motion by the following equations [11].
The gradient is considered the running resistance R. The control input u is defined in Table 1.
t v dx =
d (1)
) , ( ) , d (
d F u v R xv
t
m v= - (2)
) 0
, (u v uf
F = (0£v<v0) (3)
0 0
) ,
( f
v v uv u
F = (v³v0) (4)
)) ( 2 sin(
) , (
2
x CSv mg
mg v x
R =m +r + q (5)
The variables are defined as follows:
x EV position m total mass [kg]
v EV velocity [m/sec]
t trip time [s]
F force of acceleration/deceleration [N]
u control input
R running resistance including gradient [N]
f maximum force of acceleration [N] 0
m kinetic coefficient of friction r air density [kg/m3]
S frontal projected area of the EV [m2] C drag coefficient
q angle of the gradient g gravity acceleration; 9.8m/s
The equation for the force of acceleration/deceleration changes depending on the EV speed, as seen in (3) and (4).
The speed at which the upper bound of the constant acceleration/deceleration force area in Fig.1 is assumed to be v0. The value differs whether the EV is in accelerate or decelerate due to variation of inverter input voltage. The value of v0 is defined as v0+ and v0- for acceleration and deceleration, respectively, as shown in (6).
ïî ïí ì
=
= =
- +
on) decelerati (in
[m/s]
3 50
on) accelerati (in
[m/s]
2 25
0 0
0 v
v
v (6)
Fig. 1. Force of acceleration/deceleration characteristics and running resistance
2.2 Objective function
The objective function to minimize gives the total energy consumption and is given in (7).
ïî ïí ì
<
= ³
®
=
ò
) 0 ( ) , (
) 0 ( ) , 1 ( ) , (
min d ) , ( ]
[ 0
u v v u F G
u v v u M F v
u p
t v u p u
J
e e T
(7)
where
J[u] energy consumption T total time Me motor efficiency Ge generator efficiency
2.3 Transformation into multistage decision process Since the original problem in (7) is theoretically unsolvable, conversion of the problem is inevitable in order to implement the optimization as a computer program.
Generally, this conversion is accomplished by linearization and time-uniform discretization. Quantization of x-v state space is also indispensable, namely the state space is finely divided and many lattice points are dotted in the state space.
In addition, DP needs to transform terminal boundary conditions to specify the goal into a penalty function since it cannot be considered explicitly. When using penalty
coefficients c1 and c2, the evaluation value J[{uk}] is given by (8).
å
-= + + D
+
=
1 0
1)}
, ( ) , ( {
) , ( }]
~[{
N k
k k k k N N k
t v u p v u p
v x u
J f
(8)
where
Fig. 2. Searching method in state space
2 2 1( )2
) ,
(xN vN =c xN -L +cvN f
ïî ïí ì
<
= ³
) 0 ( ) , (
) 0 ( ) , 1 ( ) , (
u v v u F G
u v v u M F v u p
k k k e
k k k k e
k
N and Δt are number of divisions and sampling period, respectively, thus the relation of (N-1)Δt=T is satisfied. L is the position of the goal. If there is no calculation error,
xN should be L. The penalty term in (8) gives the penalty value considering error between the terminal boundary condition and calculated result. The coefficients c1 and c2
should be adjusted by evaluating the results so as to obtain the best solution that best satisfies the terminal conditions and provides low energy consumption.
Next, we can obtain discretized linearized state equations as (9) using first-order Taylor expansion
) , , ( ) d (
) ( d
0 0 0 u x v Bf
t t A
t = y +
y (9)
where
÷÷ø çç ö è
=æ úû ê ù ë
=é úû ê ù ë
=é
1 , 0 1 , 0
) (
) ) (
( A B
t v
t t x
b
y a ,
{
( , ) ( , )}
, ) , ( ) , ( ) , ( ) , , (0 0 0
0 0
0 0 0 0 0 0 0
0 0
v x f v x r v
v x r x v x r v u f v x u f
v v
v
- +
+ -
=
).
, (
), , ( ) , (
0 0
0 0 0
0
v x r
v x r v x f
x
v v
-
=
-
= b a
Finally we can obtain (10) using the trapezoidal rule for numerical approximation of integrals by defining
) ( tk
k =y D
y .
Fig. 3. Fundamental optimizing algorithm
1 1
2 ) ( 2 ) ( subject to
}]
~[{
min
0 1 1
} {
£
£ -
þý ü îí
ì D + D
D + -
= - -
k
k k
u k
u
t Bf tA
I tA I
u J
k
y
y (10)
3. Dynamic programming-based optimization 3.1 Introduction of dynamic programming
Bellman’s DP was employed for solving optimal control problems in [8]-[10]. This DP was proposed by Richard Bellman and is used as a method for solving optimal control problems. It was previously used for the eco-driving analysis of a rail vehicle [12]. It can obtain the global optimal control input and it can be easily reconfigured against disturbances. The DP is based on the following optimality principle:
An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state from the first decision [13].
acceleration/deceleration reference, used as a control input, that is optimized.
New paragraph: DP process can be implemented with a digital computer as follows:
(a) Set k=N
(b) Determine optimal policy on every lattice point of the state space.
Fig.2 shows the searching method. From one of the lattice points, the trajectory moves to the different place in the state space at the next time stage. By using the information of optimal policy at the next time stage already calculated, the optimal policy of the lattice point can be found.
(c) Reduce k by 1. If k¹1, return to (b).
Fig. 4. Modeling of a traffic signal
Table 2. Specific parameters of EV
Total weight [kg] 1500
Maximum speed [km/h] 62
Efficiency of motor/generator 0.90 Maximum acceleration [m/s2] 2.67 Maximum deceleration [m/s2] 3.00 m : kinetic coefficient of friction 0.011
r : air density [kg/m3] 1.2
S: frontal projected area [m2] 2.5
C: drag coefficient 0.35
Table 3. Specific operation parameters
Total length [m] 500
Total time [sec] 50
(d) Search the optimal trajectory from the origin (k=0) to the destination (k=N) along the optimal policies already stored at every lattice points in the processes (a)-(c).
Fig.3 shows the searching method. The whole trajectory can be easily generated just like connecting the optimal policies.
In general, the conversion from the optimal control problem to the multistage decision process, which can be achieved by the DP algorithm, will increase the amount of
amount of calculation time related to the spacing of admissible state spaces. Therefore, it is important to set appropriate number of lattice pointsin order to avoid long calculation time and large error.
3.2 Consideration of speed limitations and traffic signals In practical situations, when analyzing the driving of automobiles, speed limitations should be considered. DP can easily include the effects of speed limitation by eliminating the lattice points that do not satisfy the speed limitations. This can be formulated as in (11).
) ,
max(xt v
vk £ (11) The limitations are normally given as time-invariant conditions, vmax(x,t)=vmax(x). However, they can also be given as time-variant conditions. The technique of using the time-variant conditions enables us to model the behavior of traffic signals. As shown in Fig. 4, when the traffic signal is red, the speed around the intersection should be infinitely small.
Table 4. Correspondence of line color of graphs to simulation conditions
sinq Line color of the graph
0.010 Black
0.025 Orange
0.050 Green
0.100 Red
0.150 Blue
Fig. 5. Track profile from up-gradient to down-gradient
Fig. 6. Track profile from down-gradient to up-gradient
Presently, when driving, a driver does not know the timing of the red lights. However, in the near future, a car will automatically acquire the information of the status of the signal lights by using the ITS technology. In this paper, the timing of the red light is regarded as known.
4. Numerical study with simulations
The authors developed the optimization program with MATLAB. Examples of the optimization under several conditions, including gradients and traffic lights, are demonstrated using this program. Specific parameters are tabulated in Tables 2 and 3.
4.1 Simulation with gradients
Several vertical profiles with different gradients, as tabulated in Table 4, were assumed in this simulation. The track profiles are shown in Figs. 5 and 6. Figs. 7 and 8 show the results for the case in which the EV runs from up- gradient to down-gradient. On the other hand, Figs. 9 and 10 show the results for the case in which the EV runs from down-gradient to up-gradient. The values of energy consumption are shown in Tables 5 and 6.
Fig. 7. Control input for the optimization result for the track profile of Fig.5
Fig. 8. Velocity of the optimization result for the track profile of Fig.5
Table 5. The value of energy consumption for the case of going from up-gradient to down-gradient
sinq Energy consumption [Wh]
0.010 37.1
0.025 37.6
0.050 38.0
0.100 38.7
0.150 43.1
Table 6. The value of energy consumption for the case of going from down-gradient to up-gradient
sinq Energy consumption [Wh]
0.010 37.7
0.025 37.4
0.050 36.4
0.100 36.3
0.150 34.8
The correspondence of the line colors of the graphs to the simulation conditions is shown in Table 4.
From the results, the optimal speed profile of the vehicle can be solved with the DP-based method. In the case of going from up-gradient to down-gradient, the control input becomes more severe as the gradient becomes steeper. By comparing Figs. 7 and 9, it can be seen that the input signal of Fig. 7 is more intense near 150m. In particular, the energy consumption is intense in the up-gradient.
Fig. 9. Control input for the optimization result for the track profile of Fig. 6
Fig. 10. Velocity of the optimization result for the track profile of Fig. 6
presence of a traffic signal
Case case 1 case 2 case 3 case 4 Timing of red light [s] 0-10 5-15 10-20 20-25 Energy consumption [Wh] 38.3 38.4 41.0 38.3
Increased Energy [%] 0.0 0.2 7.0 0.0 Note: Cases 1 and 4 are used as the reference point for
‘Increased Energy’.
In the case of going from down-gradient to up-gradient, the energy consumption decreases as sinq is larger as tabulated in Table 6. As seen in Fig. 10, the EV accelerated in the down-gradient section as well as starting phase in order to shorten the acceleration time, whereas it coasted before reaching the down-gradient section.
4.2 Simulation with traffic signals
The simulation in which a traffic signal was considered is also demonstrated. An intersection is assumed to be located between 115£ x£135[m]. While the red light was present for 10 seconds in each case, the timing of the red light was shifted in the four cases.
The optimized speed profile is plotted in Fig. 11. Figs. 12–
15 show the optimal distance curves that are used for analyzing how to get through the red light. The position and times of the red light are shown as rectangles in these figures. In all cases, the EV was driven so as to avoid stopping by controlling the speed before the intersection.
Fig. 11. Velocity of the optimization result for the case of the presence of a traffic signal
Fig. 12. Distance curve of the optimization result for case 1
Fig. 13. Distance curve of the optimization result for case 2 The energy consumption is tabulated in Table 7. The minimum energy consumption was accomplished in cases 1 and 4. The speed profiles were almost ideal in these cases, since the timing of the traffic signal did not affect the motion of the EV. However, in cases 2 and 3, energy consumption was increased by 0.2-7.0% because of the control required for avoiding the red light.
5. Conclusion
In this paper, the optimal control problem for minimizing energy consumption by an EV was numerically solved with the proposed algorithm based on Bellman's dynamic programming. The substantial advantage of the proposed method was demonstrated by simulating the operation of an EV under the conditions of the presence of gradients and traffic signals.
Fig. 14. Distance curve of the optimization result for case 3
Fig. 15. Distance curve of the optimization result for case 4
In future work, the authors will continue to analyze the behavior of EVs under more practical operation conditions, such as the presence of two or more signals, the interaction between cars, and so on.
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Motoi Kuriyama received M.S. degree in electrical and electronics engineering from Sophia University. His research interests are energy management systems for electric vehicles.
Sou Yamamoto received B.S degree in electrical and electronics engineering from Sophia University. His research interests are energy management systems for electric vehicles.
Masafumi Miyatake received the Ph.D.
degree in information and communi cation engineering in 1999 from the University of Tokyo. In 2000, he joined Sophia University. From 2004, he has been an Associate Professor at Sophia University. His research interests include energy management control and their applications to transportation systems.
