Speed Trajectory Optimisation for Electric Vehicles in Eco-approach and Departure using Linear Programming Shaofeng Lu1 , Fei Xue1 , Tiew On Ting1 and Yang Du1 Abstract— With the fast development of regenerative braking technologies in modern transportation systems, it has become popular to take into account the regenerated electric energy of electric vehicles for energy-saving purposes. In railway transportation, it was found that given the monotonicity of the vehicle speed during an acceleration or braking process, a partial speed optimisation model can be set up and solved by Mixed Integer Linear Programming. Taking into account the similarity between road traffic and rail transportation, this paper aims to build up a linear programming model to optimise the speed trajectory of an electric vehicle (EV) during eco-approach and departure (EAD) to achieve a minimum energy cost. Three cases have been studied. First, we consider an optimisation model when the preceding vehicle is at a full-stop status, for example, when it is at a road crossing. We set up a case scenario with a constant running distance but different running time when the following EV initiates the car-following process. We will further investigate if the following EV has to use up all available running time before it fully stops behind the preceding vehicle. Second, an optimisation model is proposed by predicting the movement of the preceding vehicle. In this way, we are considering an optimisation problem with varying distance and time for the target car. Third, we try to consider a case where the following EV tries to accelerate to the same speed of the preceding vehicle under the time and distance constraints. The motivation of this paper lies on the successful applications of linear programming for partial train speed trajectory optimisation, the capability of regenerative braking of plug-in all electric vehicles (PA-EV) and speed trajectory optimisation in the application EAD. The proposed model takes advantage of its robustness, computational efficiency and readiness of potential on-line energy-saving applications in intelligent and connected vehicle systems.
I. I NTRODUCTION With the development of battery storage technology and the charging facilities, electric vehicles (EVs) can further extend the driving range and contribute extensively to carbon and emission reduction in transportation sector [1], [2]. EVs can also be integrated into the power grid and act as energy storage system (ESSs) and to improve the power system stability and economic benefits based on V2G and demand response technology [3]. Further development of Vehicleto-Vehicle(V2V) and Vehicle-to-Infrastructure (V2I) provide This project is supported and sponsored in part by 2014 Jiangsu University Natural Science Research Programme, Project No. 14KJB580010 and in part by the Research Conference Fund at Xi’an Jiaotong-Liverpool University. 1 Shaofeng Lu, Fei Xue, Tiew On Ting and Yang Du are with the Department of Electrical and Electronic Engineering, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China
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information about other vehicles and the signal phase and timing (SPaT) information. It is more likely to develop a more comprehensive model to control the vehicle with higher efficiency, safety and comfort for vehicle driving [4]. Taking advantage of future route information, Vajedi et al. [5] proposed ecological adaptive cruise controller for plugin hybrid vehicles using nonlinear model predictive control. Validated by proportional-integral-derivative (PID) and linear Model Predictive Control (MPC), the proposed control method is demonstrated in simulations to achieve energy saving up to 19%. To develop the speed advisory system for drivers at signalized intersections, Xiang et al. [6] proposed a closed-loop speed advisory model which is able to adapt to driver’s behaviour for eco-driving. Fuel economy performance and driver’s behaviour adaptability have been studied in comparisons between three proposed models. From 20112015, Barth et al. have developed a number of eco-driving control approaches considering real-time traffic sensing and infrastructure information [7], preceding car information [8], and actuated signals [9] in the signalized corridor. Muñoz Organero et al. [10] tried to implement and validate an expert system based on which, optimal deceleration patterns can be identified to reduce fuel consumption for vehicles when approaching the traffic signals which may lead to a full stop of the vehicle. The above-mentioned papers on eco-driving of automobiles are based on control-oriented models or frameworks, such as model predictive control with a consideration of vehicle dynamics and future route information. The ecoapproach and departure application (EAD) in signalized intersections is one of the interesting topics in this area focusing on vehicles crossing the signalized intersections to reduce the fuel cost and emissions. The main characteristic of this problem is that the vehicle is required to conduct acceleration and deceleration operations. In EAD, speed trajectory optimisation can be conducted to provide the driver with recommended speed curves based on SPaT [9]. In 2016, Lu et al. [11] proposed a Mixed Integer Linear Programming model to study the partial train speed trajectory optimisation problem in acceleration and deceleration phases during train departure and braking operations. It was concluded in the paper that the model can be solved in a short computational time and become particularly interesting for online cases where a train is altering the speed in a fixed distance and time. Considering the similarity between the train and connected vehicle systems, the total travel time and distance available for the host vehicle could be available in both systems. In addition, the traction characteristic could
II. M ODELLING
10
Tractive or braking effort and resistance (kN)
be modelled in a similar way for electric vehicles (EVs) and electric rail vehicles. Different from [11], the main contribution of this paper lies in the study of the traction characteristic of EV and the trajectory optimisation problem for EV during EAD. This paper is motivated by the successful application of linear programming for partial train speed trajectory optimisation [11], the capability of regenerative braking of plug-in all electric vehicles (PA-EV) and speed trajectory optimisation in the application of EAD. Without the need for a detailed vehicle dynamics and control model, this paper proposes a linear programming model to optimise the speed trajectory for EVs during their acceleration and deceleration phases across the signalized intersections to achieve an operation of EAD. The proposed optimisation model takes into account the SPaT and distance information from the preceding vehicles and thus applicable in carfollowing operations.
Resistance Tractive or Braking Effort
9
8
7
6
5
4
3
2
1
0
0
50
100
150
200
250
Speed (km/h)
Fig. 1. Motor traction and resistance characteristic for a typical electric vehicle. TABLE I M ODELING PARAMETERS FOR A TYPICAL ELECTRIC VEHICLE
A. Model of a typical EV traction system The energy consumption of an EV is mainly related to the tractive effort and the resistance forces including the aerodynamic drag forces and rolling resistance versus the vehicle. In this paper, vehicle characteristic have been simply modelled using tractive effort and resistance versus the vehicle speed while vehicle dynamics have been greatly simplified as presented in Fig. 1. Such a model, as has been adopted in a number of references on rail vehicles [12], will enable us to focus on the energy consumption without facing the complex dynamics of the vehicle. It is found that practical motor traction and resistance characteristic are readily applicable for the proposed model. In engineer applications, EV will be able to apply the electric traction and braking effort according to driver’s commands which should be under the characteristic line. This characteristic line is usually different for traction and regenerative operations. In this paper, for the sake of simplicity and generality, it is assumed that the traction and regenerative braking have the same characteristic curve and the power efficiency of electric motors is 100%. It is very easy to incorporate different characteristic of the motor during regenerative braking and traction by changing the parameter setup in Eg. (9). Further study has found that conditional controls on the optimisation model can be imposed to tackle the practical efficiency of motors at different operation modes. The other parameters of EV are summarized in Table I. These parameters are re-engineered based on available specification and performance data of Tesla Model S (70D) [13]. As listed in Table I, M is the mass of EV; M0 is the mass considering rotary effect; Vmax is the maximum speed, A, B and C are the coefficients to model the resistance. B. Model of speed trajectory optimisation of EV during EAD As presented in [11], it is assumed that there are a series of monotonously changing speeds along the trajectory during EAD, i.e. v1 , v2 , . . . , vN and v1 < v2 < ... < vN . This is
M(t) 2.09
M0 (t) 2.26
Vmax (m/s) 63.9
A(kN) 0.30723
kN B( m/s ) 0
kN C( (m/s) 2) 0.0003545
based on the observation that EV is kept on monotonous speed trajectory during acceleration or deceleration in practice. It is undesirable to speed up a car during braking if the car is able to arrive at the full stop within the demanded time. It has been proved in [12], that if only negative braking efforts are imposed, not only the speed trajectory should be monotonous, but also the optimal operations of the EV can be concluded. However, as an important direction of the future work, it needs to be proved from the optimal control viewpoint if the monotonicity will always be able to achieve the optimal solutions or solutions which are close to the optimal ones. As shown in Figs. 2 and 3, each speed candidate represented by a black circle has a corresponding vehicle position. The initial and final speed will be very much depending on the initial states of the following host EV and the preceding vehicle. Different scenarios could be modelled using different requirement on v1 , vN and the total journey time and distance for the trajectory. For example, in a deceleration case, if v1 being the current speed of the preceding vehicle is set as zero, and vN the initial speed of the following host EV, we could model a case scenario where the EV is approaching a signalized intersection with the preceding car waiting before the signal light. The optimisation is to find out what is the trajectory the EV should have to brake down to full stop within the required distance and time. The journey time will largely depend on the signalling information. On the contrary, in the departure or acceleration cases, the EV is required to speed up from v1 to vN . In such a case, the initial speed is v1 and the final speed is vN . While similarity exists in the modellling, some difference should be taken in to account
Speed
Following Host EV
Travelling Direction
𝑣𝑁−1
The maximum electric braking and traction effort between vi and vi+1 is denoted by Fi,em between speeds vi and vi+1 respectively, can be obtained by linearly interpolating the electric traction characteristic shown in Fig. 1. The distance and time between vi and vi+1 is denoted by ∆di and ∆ti , where i = 1, 2, 3, ..., N − 1. ∆di is the determining variable of the model. The linear relationship between ∆di and ∆ti is denoted by (4).
𝑣𝑁
𝑣𝑁−2
𝑣2
Preceding Vehicle
∆ti = ∆di /vi,avg
𝑣1
Distance
Fig. 2. case
Modelling of speed trajectory of EV during EAD: a decleration
Speed Travelling Direction
𝑣𝑁
Preceding Vehicle
Once ∆di is determined, the entire solution for optimal braking trajectory is obtained. The sum of each distance interval should be equal to the total distance D as shown in (5). D=
𝑣𝑁−1
N −1 X
∆di = dN −1
(5)
i=1
The total time constraint is presented by (6). The total allowed time window as determined by two optimisation parameters, i.e. Tmin and Tmax is proposed. If there is a fixed time for the entire journey, Tmin and Tmax will be equal to a constant.
𝑣𝑁−2
Following Host EV
(4)
𝑣2
Tmin ≤
𝑣1
N −1 X
∆ti ≤ Tmax
(6)
i=1
Distance Fig. 3. case
Modelling of speed trajectory of EV during EAD: an acceleration
between the approaching and departure cases which will be covered in detail later in this section. Something in common between deceleration and acceleration cases will be introduced first. With regard to each candidate speed in the trajectory, the distance and elapsed time between vi and vi+1 are corresponding to the distance between the positions where the EV is at the speed of vi and vi+1 and the time used when the EV is travelling between vi to vi+1 . The total number of the speed candidate N is set sufficiently large to ensure that the average speed between two adjacent speeds can precisely reflect the current vehicle speed and characteristic in relation to the EV speed. If N is set, the entire speed series can be determined by (1). vN − v1 i = 1, 2, ..., N (1) N −1 The average speed vi,avg between vi and vi+1 is calculated by (2): vi = v1 + (i − 1)
vi,avg = (vi + vi+1 )/2
i = 1, 2, ...N − 1
(2)
The resistance between vi and vi+1 can be obtained by (3). 2 Fi,r = A + Bvi,avg + Cvi,avg
i = 1, 2, ..., N − 1
(3)
In the following, different modelling constraints should be imposed for different operation modes. Deceleration cases. Since the braking effort is not only from the electric motor, but also from the mechanical braking force, a maximum braking rate should be imposed. The deceleration rate between vi and vi+1 should not exceed the maximum braking rate defined by (7). abr i =
2 vi+1 − vi2 ≤ Abrm 2∆di
(7)
where Abrm is a positively constant 4 m/s2 in this paper and can be adjusted for different cases. Hence, it yields the lower bound of di as shown in (8). ∆di ≥
2 vi+1 − vi2 2Abrm
(8)
Two factors limit the regenerative braking effort in the deceleration case: the electric motor characteristic and the maximum allowed total braking efforts determined by the actual braking rate. As a result, two more constraints should be imposed on the model as shown in (10) and (9), where Eiem is the regenerated braking energy or consumed electric energy in the distance interval ∆di . Note that Eiem is positive for braking operation and negative for traction operation in each distance interval ∆di , indicating that the braking effort is regarded positive, in the same direction of the resistance forces and the traction effort is regarded negative. Such a virtually backward calculation will not change the optimisation result but will enhance the
Speed
modelling efficiency when switching between deceleration and acceleration cases. Note that in some cases where the journey time is small, traction operations are needed to keep a close-to-constant speed and overcome the resistance.
Travelling Direction 𝑣𝑁
Following Host EV
𝑣𝑁−1 𝑣𝑁−2
− Fi,em ∆di ≤ Eiem ≤ Fi,em ∆di
(9) 𝑣2
0
Eiem ≤ (M abr i − Fi,r )∆di
(10)
The objective function of the optimisation model for deceleration cases is shown in Eq.(11). max
N −1 X
∆di
Eiem (∆di )
(11)
(1) − (10)
Acceleration cases. As discussed above, in acceleration cases, the EV needs to impose a positive traction effort all the time to overcome the resistance and there is no longer constant maximum acceleration rate imposed as this will be constrained by the varying traction effort determined by vehicle operations. A new constraint on the consumed electric energy shown in (12) is updated from (10). Meanwhile, the objective function will need to minimize the total electric traction energy, but not to maximize it as in (13). 0
Eiem = (M atr i + Fi,r )∆di
(12)
Finally, the objective function shown in (11) will be changed into (13) subject to constraints.
Subject to:
Distance Fig. 4.
Case 1 illustration: deceleration case with a full stop at the end
A. Case 1: deceleration with a full stop at the end
∆di ∈ R
∆di
Initial Speed: 𝑣𝑁 = 60 m/s Final Speed: 𝑣1 = 0 m/s Maximum Braking Time: 20s − 60s Total Braking Distance: 500 m
i=1
Subject to:
min
𝑣1
Preceding Vehicle
N −1 X
Eiem (∆di )
(13)
i=1
(1) − (9) and (12)
∆di ∈ R As illustrated and demonstrated in [11], gradients and speed limits can also be modelled using Mixed-integer Linear Programming. Given that similar information is not as always available in road traffic as in rail transportation, this paper adopts level gradient and no speed limit. III. R ESULTS AND D ISCUSSION In this section, the optimisation model will be manipulated to suit different case scenarios. With the preceding vehicle’s instant distance and speed information available, the optimisation model can be modified to search for optimal speed trajectory of EV with the objective function to maximise the regenerative braking energy or minimise the electric traction energy.
In Case 1, the preceding vehicle is considered to stop before the signal section and the total waiting time time is known. An EV is supposed to approach the preceding vehicle with a final full stop within the waiting time and the total braking distance equals to the distance between the host EV and the preceding vehicle located at the signalised cross section. The model will be built to maximize the regenerative braking energy during the entire braking procedure. As shown in Fig. 4, the host EV will brake from 60 m/s to 0 m/s. Referring to (6), Tmax may range from 20 s to 60 s and Tmin = 0 with a fixed total braking distance equivalent to 500 m. Based on the optimisation results for different cases with the maximum allowed braking time ranging from 20 s to 60 s, it is found that there is only one optimal solution found when the total braking time equals to 16.22 s. The solution generates a total regenerated braking energy of 0.8248 kWh taking 82.48% of the initial kinetic energy of EV. The optimal trajectory and braking effort distribution are shown in Fig. 5. For the cases with Tmax
