Genetic algorithm based gear shift optimization for electric vehicles.
In this paper, an optimization method is proposed to improve the efficiency of a transmission equipped electric vehicle (EV) by optimizing gear shift strategy. The idea behind using a transmission for EV is to downsize the motor size and decrease overall energy consumption. The efficiency of an electric motor varies with its operating region (speed/torque) and this plays a crucial role in deciding overall energy consumption of EVs. A lot of work has been done to optimize gear shift strategy of internal combustion engines (ICE) based automatic transmission (AT), and automatic-manual transmissions (AMT), but for EVs this is still a new area. In case of EVs, we have an advantage of regeneration which makes it different from the ICE based vehicles. In order to maximize the efficiency, a heuristic search based algorithm - Genetic Algorithm (GA) is used. The problem is formulated as a multi-objective optimization problem (MOOP) where overall efficiency and acceleration performance are optimized. A mathematical formulation is provided to calculate the maximum possible efficiency for a given drive cycle. Non-dominated Sorting Genetic Algorithm (NSGA-II) is used to optimize the gear shift lines. A comparative study of fuel economy improvement is provided to validate the concept of using different downshift lines during regeneration.
CITATION: Saini, V, Singh, S., NV, S., and Jain, H., "Genetic Algorithm Based Gear Shift Optimization for Electric Vehicles," SAE Int. J. Alt. Power. 5(2):2016.
With advent of increasing pollution and rising demand and price for crude oil, there has been a constant push to explore alternative source of energy. The concerns for environmental change and to maintain sustainability of conventional fuels have inspired researchers around the world to develop technology required for adoption of renewable sources of energy. A similar trend is seen in the automobile industry which has been craving to develop more efficient and less polluting vehicles. Alternative sources (hydrogen ICE, fuel cell, solar etc.) are being explored and evaluated to power the vehicles.
Most of the cities around the world are facing increased levels of smog, largely attributed to tailpipe emissions from ICE based vehicles. With emission norms becoming stringent over years, have forced original equipment manufacturers (OEMs) to look beyond optimization of ICE powertrain and solve the problem by introducing a partial or complete electrical solution. There has been a lot of thrust through government by investing in EV-enabling infrastructure and providing free charging stations. Incentives to ease burdens in terms of purchase, lease and taxes on EVs are other ways taken by governments to make it attractive for consumers, fleet owners and OEMs.
Pro terra LLC's ProDrive[TM] have demonstrated operation of a permanent magnet motor to operate in high efficiency (92-95%) during traction and regenerative modes  using a three speed transmission. CodrinGruie et al.  have proposed a two AC traction motor electric axle configuration which successfully completed pilot run on a prototype truck. In , a fuzzy-logic based controller that optimizes power output of IC engine and motor is proposed. They had achieved close to 7% improvement in efficiency for urban cycles. Siemens AG also developed a direct drive configuration for a Bus Transit Line (BRT).
There have been an ample amount of research and studies to select right configuration, architecture and sizing of components for hybrid and EVs. However, the concept of using an automated transmission with an electric vehicle is relatively new and a limited amount of literature available for optimizing the shift control strategies for transmission based EVs. In this study, a slightly modified gear shift strategy is proposed. We have used Non-dominated Sorting Genetic Algorithm (NSGA-II ) to improve the efficiency of an EV by optimizing the shift strategy for both traction and regeneration mode. Gear shift strategy comprises of gear shift lines for upshift and downshift of transmission gear. Gear shift lines are optimized to operate the ICE/motor in the most efficient zone. In case of EV, we have an advantage of regeneration. To ensure maximum regeneration, we need to have different downshift lines from the usual ones, so that most of the energy can be regenerated. In the current study, normal upshift and downshift lines as well as downshift lines during regeneration are optimized. The problem formulated here is a nonlinear problem with multiple contradictory objectives (efficiency and performance) and multiple operating points to be optimized (18 points for a 4 speed transmission in this work). Genetic algorithms are widely used where the objective functions are non-linear, a large number of parameters are required to be optimized and have multiple local minima. Keeping that in mind, we have used NSGA-II to optimize the gear shift lines. The key contributions to this paper are 1) novel gear shift strategy for electric vehicle, and 2) mathematical formulation to calculate global minima.
A mathematical approach is proposed to calculate optimal theoretical value of energy consumed per unit distance (EPD), which is used to check if the objective function is approaching global minima. Furthermore, a comparative study is done to validate the concept of using different downshift lines during regeneration.
This paper is organized in four sections. In the first part, a brief introduction of electric vehicle and gear shift strategy are provided. Next section talks about Non-dominated Sorting Genetic Algorithm, where the flow chart of the algorithm is presented. After that, an approach to calculate efficiency of an electric vehicle is proposed to obtain global minima for a given transmission, motor configuration. In the end, a detailed analysis of NSGA-II algorithm and modeling and simulation of electric vehicle are discussed.
BATTERY ELECTRIC VEHICLE (BEV)
Designing powertrain for an electric vehicle is relatively easier as the torque offered by a motor over lower range speed is high. With sophisticated motor control and a wide range of torque availability, eliminate the need for a clutch. For a small passenger car, motor is directly connected to the differential through a single gear reduction. There are similar direct drive solutions for other larger commercial vehicles like bus/trucks as well. However, challenge faced by the OEM's in electric buses with direct drive configuration is the startability and gradeability. Using a powerful and bigger motor is not the best solution as it not only makes the powertrain bulky but also the motor capability becomes redundant while cruising and normal city driving.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
Here, F(t) is the traction force available on wheels after accounting friction, D(t) includes all aero drag forces, M is mass of the vehicle. The major challenge, however, still lies in extending the range of any electric vehicle. The available battery technology and its cost restrict the storage available for use in vehicles. Stringent goals have been set to reduce the cost of the batteries. 'Range anxiety' is one of the major barriers to large scale production of the electric vehicles. This involves providing charging solutions at various geographic locations and technology to quickly charge the battery.
Researchers are trying to develop strategies to improve utilization of available strong on-board batteries. There is always a trade-off between performance and range and it becomes more pronounced in case of a direct drive configuration. Also, an attempt made to increase motor size/power to meet vehicle performance adds to the cost of the design. A simple solution to dampen this trade-off is to go for a powertrain with an Automated Manual Transmission (AMT). The two major advantages are that it can lead to reduction in motor size without affecting performance and other being flexibility to allow motor to run in its most efficient zone. The total package size of the AMT with downsized motor is about 5-10% more than a direct drive configuration. The additional cost of the transmission can be realized in terms of savings due to improved efficiency and reduced cost of downsized motor.
Good hardware design without proper control may not produce intended results. Maximum benefits of adding an automated transmission can be harvested using an optimized shift strategy. Designing a shift control strategy has been an area of interest in the past for automated manual and automatic transmissions. Even as Hybrid Electric Vehicles (HEVs) gained popularity in market development of energy management system attracted a lot of attention . Efforts to reduce energy consumption for EV bus and to improve comfort and shift quality have been made by Hongbuet et al. . They have made an attempt to achieve this through efficient shifting control as well as some changes in the transmission synchronizer design. In another study by Jun-Qiang et al. , have shown that using a multi-speed transmission with proper control strategy for shifting enhanced efficiency by 9% with reduction in acceleration time by 18%. They had used vehicle speed and throttle input as the two main parameters that decide shifting thresholds.
In the current study, we have attempted to optimize the overall efficiency of a vehicle while satisfying minimum requirements for acceleration and gradeability. The powertrain, in an EV, has a different structure and dynamics than a conventional ICE powered vehicle. The formulation of a shift strategy involves allowing motor to operate in its most efficient zone during powering and regeneration mode. In figure-1, a motor map is presented, where the most efficient zone is marked by dark red color. An alternate strategy for downshifting, taken at the time of regeneration (brake pedal pressed), ensures maximum amount of kinetic energy from the vehicle is absorbed into the battery rather than using the normal downshifting lines. This is one of the key differentiator in this study.
The total energy consumed per unit distance (EPD) by an electric vehicle is given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where P is the required power at time instant t
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[[omega].sub.Motor] > [[tau].sub.Motor] and [[eta].sub.Motor] represent motor speed, torque and efficiency, respectively. In case of regeneration [[tau].sub.Motor] is going to be negative. Overall efficiency of an electric vehicle is inversely proportional to total energy consumed per unit distances.
In the current study, optimization problem is formulated to minimize EPD and drive cycle track error. Drive cycle track error is defined by subtracting the desired vehicle speed from the actual vehicle speed. The objectives are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
To optimize [f.sub.1] and [f.sub.2], a vector of 18 decision variables X is considered, subject to their upper and lower limits. These variables define upshift lines, downshift lines and downshift lines during regeneration, tabulated in Table-1. The upper and lower bound of each variable are defined based on the motor map and the step size of gears, explained in results section. In order to reduce the number of decision variables, the same shift line is considered for all the gears.
A typical shift pattern is shown in figure-2 for ICE based vehicles.
NON-DOMINATED SORTING GENETIC ALGORITHM-II
Genetic algorithms have been used to optimize nonlinear problems in various domains. It is a heuristic search based algorithm - a subset of evolutionary algorithms. Evolutionary algorithms are a process of modeling biological evolution. There are various types of genetic algorithms, mainly classified in two categories: binary-coded and real-parameter genetic algorithm. In the current study, NSGA-II, a real-parameter genetic algorithm is used for optimization. In , NSGA-II is claimed to have better performance compared with other constrained multi objective optimization algorithms.
Multi-objective optimization problem, in the case of conflicting objectives, has a number of possible solutions. To sort out these solutions, the concept of dominations is used where each solution is compared with other solutions. The solutions are categorized into two categories: (1) non-dominating solutions, and (2) dominating solutions. An exact definition of dominating and non-dominating solutions is given in , where a solution is said to be dominating if it outperforms other solutions in all objectives. The set of all non-dominated solutions is called the Pareto-optimal set and the curve formed by joining all non-dominated solutions is called Pareto-optimal front. In figure-3, solution spaces for two objective optimization problems are plotted by the gray area and Paretooptimal fronts are plotted by solid dark lines.
A flow diagram of NSGA-II is shown in figure-4, where all the major steps of NSGA are shown. A detailed analysis of the algorithm can be found in .
ELECTRIC VEHICLE EFFICIENCY
As pointed out earlier, GA is a heuristic search based algorithm. It doesn't provide any information about the global minima. To have a mathematical proof of convergence of an optimization method, globally optimal value should be known. In order to do that, a mathematical model is proposed to find out the global minima. It calculates the total energy consumption per unit distance of an electric vehicle on a route for a certain transmission and motor configuration assuming that the drive cycle is achievable for the given vehicle configuration.
Initially, total force required at the wheels to drive the vehicle is calculated based on the drive cycle.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
Where [F.sub.Wheel] is force required at the wheel, [M.sub.Veh] is vehicle mass, [M.sub.inertia] is mass due to inertia, V is vehicle speed and [dV/dt] is vehicle acceleration. The drag force would be given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Where [C.sub.d] is a drag coefficient, p is the density of the air, and A is the frontal area of the vehicle. The friction force would be given by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
where [mu] is road rolling coefficient, g is gravitational coefficient, and 8 is gradient of the road (slope). Force due to gradient would be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
Mass due to inertia can be calculated from the below equation:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
where [I.sub.Engine] = Engine Inertia(kg - [meter.sup.2]),
[I.sub.Gear] = Gear Inertia(kg - [meter.sup.2]),
[I.sub.FD] = Final Drive Inertia (kg - [meter.sup.2]),
[I.sub.wheel] = Wheel Inertia (kg - [meter.sup.2]),
GR = Gear Ratio of Selected Gear,
[GR.sub.FD] = Final Drive Gear Ratio.
[R.sub.wheel] = Radius of Wheel (meter)
The mass due to inertia depends upon GR. In order to calculate GR, we need to know the selected gear (unknown). To overcome this problem, selected gear is assumed based on the desired vehicle speed and the maximum vehicle speed attained in the gears. For example, if the desired vehicle speed is less than
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] it ig assumed to be in the first gear. In the same manner, vehicle speed limit is calculated for other gears and the selected gear is assumed.
The required torque at the output shaft to drive the vehicle will be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
and the output shaft speed would be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
For each gear based on its gear ratio, motor speed and required torque is calculated. The required torque at the required speed for the selected gear is mapped in the motor efficiency map. Gears, in which the required torque can't be provided at the required speed, have been discarded. A gear which has maximum efficiency is selected. The same procedure is followed for the complete drive cycle. Based on the selected gear, corresponding motor speed and torque is calculate. The basic architecture of the above process is shown in figure-4 and figure-5.
With the help of above formulation and Eq. (3) and Eq. (4), global minima of EPD can be calculated which is used to validate the convergence of NSGA-II.
RESULTS AND DISCUSSION
A point mass model of an electric vehicle is simulated in MathWorks, Inc.'s MATLAB[R] and Simulink[R] environment. This analysis is done for medium duty (MD) electric bus for HDDS (Urban Dynamometer Driving Schedule) drive cycle. The vehicle configuration is tabulated in Table-2.
The Simulink vehicle model consists of three major blocks:
3. Plant and Controls
The purpose of Environment block is to provide inputs to the model which includes drive cycle with grade information. This information is given to the operator block which is like a virtual driver. This simulates the driver behavior and generates accelerator and brake commands in response to the grade data and the drive cycle. The actual vehicle model with the controller sits in the plant and controls block.
Plant and Controls
This subsystem takes inputs from the operator output and revert the vehicle speed to the operator subsystem. Various components in this subsystem are modeled based on physics and have different levels of fidelity based on requirement. Major building blocks of the system are explained in this paper:
Motor/Generator System Block
Motor provides driving torque and regeneration capability to the complete vehicle powertrain. Motor block is simulated to get the motor/generator torque and corresponding efficiency. Steady state efficiency maps corresponding to the motor used in actual hardware is used to obtain the efficiency. The motor map for maximum torque available both during traction and regeneration is given as an input to the model.
This subsystem model is a look up table based model. Battery model calculates the State of Charge (SoC) of the battery based on the power drawn from it. It also limits the maximum amount of charge/discharge current. The voltage from the battery is decided based on the SoC.
Transmission block is physics based model which calculates the torque output based on the input speed and torque obtained from motor. Vehicle System
The output torque of the transmission is imported to the vehicle block as a force. The drag, friction, and grade forces on a vehicle are subtracted from input differential force to get the net force acting on the vehicle. This force is then divided by the equivalent mass (driveline inertia and vehicle mass) to get vehicle acceleration with the help of Eq. (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
Vehicle speed is obtained by integrating the acceleration:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
[v.sub.0] is vehicle's initial speed.
The overall vehicle system is modeled based on the following approach: Driving torque propagates from engine to the wheels in forward direction and vehicle speed propagates from wheels to engine in reverse direction. Every subsystem takes driving torque of previous system as an input and feedbacks its current speed as an input to it.
Electrical Vehicle Efficiency
For UDDS drive cycle and the given electric vehicle configuration, maximum possible vehicle efficiency is calculated with the help of mathematical formulation described earlier. The overall efficiency or energy consumption per unit distance (EPD) is 0.6793 kWatt hr/km. In figure-6, efficiency of the electric motor in all possible gear is plotted at every point of drive cycle. The most optimal gear at every point is shown, in the next subplot.
The maximum efficiency (inverse of EPD) can be achieved if gear shifting happens as shown in figure-6. But in real scenario, there are constrains which make above shift pattern not feasible. For example, frequent shifts have adverse effect on the reliability of shift actuation system and drive quality experienced by driver. When the transmission shifts gear frequently, there is a loss of traction at the wheels (torque hole) which affects the performance and drivability. Another important aspect is to have a buffer of power available if the driver needs it. In simple terms, it is not desirable to be in the most efficient gear if that requires motor to deliver its maximum torque thereby leaving no room for further increase without making a shift. In the current vehicle model, these contains are modeled in following ways:
1. Gear Hunt Prevention: This algorithm is used in all automatic transmission controls to prevent occurrence of frequent shifts. This algorithm temporarily modifies the shift points to an extent that there is still room for shift to happen in case of sudden change in demand.
2. Drive cycle track error: This value is indicative of loss in performance of the vehicle even though all constrains are met.
Iterative process of tuning these shift lines brings the efficiency closer to the above theoretical limit, but achieving this minimum is practically not possible. As a standard process, shift points are tuned and tested on these vehicle models to quantify improvements in efficiency while meeting all the constraints. Once this process is complete, those shift calibrations are flashed on to a field vehicle to validate the simulation results. This whole process consumes lot of time and computation resource (still does not guarantees a minima) in the absence of a methodology which can optimize the shift points. For these reasons, the problem becomes a perfect candidate for an optimization problem.
As explained in the previous section, 18 decision variables are considered to optimize the overall vehicle efficiency. These decision variables are related to gear shift points (motor speed) at various throttle position. First 6 variables dictate upshift points. The upper bound for upshift points are defined by the point where maximum power is available from the motor, shown in figure-8. The lower bound for upshift points are considered to be the point where motor is less efficient comparatively, as shown in figure-9. The next 6 variables define downshift points and the last 6 points define downshift points during regeneration. The upper bound for downshift points is a multiplication of step size of gear ratio (max(2.6/4.83,l .69/2.6,1/1.69)) and the upper bound of upshift points. The lower bound for downshift points is assumed to be zero.
By defining the upper and lower bounds of decision variables, MOOP problem is formulated to minimize EPD and drive cycle tracking error. The population size of 100 is simulated for 100 generation. The crossover probability of [p.sub.c] = 0.9, a mutation probability of [p.sub.m] = 1/n (n is number of decision variables) and the crossover, mutation operator for NSGA-II [n.sub.c] =15, [n.sub.m] =20 are used for simulating NSGA-II code .
In figure-10. a convergence plot of Pareto-fronts of NSGA-II is shown for 100 generations. For the given configuration, EPD should be 0.6793 kWatt hr/km and the tracking error should be zero to reach global minima.
From the figure-10, it is clearly visible that NSGA is approaching towards global minima, though it doesn't reach to theoretical global minima due to several reasons. A few of them are (1) gear shift strategy is based on gear shift lines (2) upshift and downshift lines are same for all the gears, (3) shift delay, and transmission/ axle losses are not considered while calculating global minima.
As discussed earlier, Pareto-optimal front is a set of non-dominating solutions, as shown in figure-11. Since both the objectives are contradictory to each other, a mid-point of the optimal front is taken which has EPD of 0.7716 kWhr/km and track error of 14.96% to run the vehicle simulation for the corresponding gear shift lines.
The gear shift lines for the selected optimized solution are shown in figure-12. These lines are defined by decision variables optimized through NSGA. It can be seen during regeneration downshift lines are different from the normal downshift lines. In the current simulation, grade is not considered. Because of this, the vehicle would never come to a state where upshift is required at lower throttle pedal and downshift is required at higher throttle pedal. That is why some of the decision variables won't be optimized for the current simulation. The decision variables are represented by solid circles in figure-12. The upshift line has higher motor speed than the downshift line which is expected. These shift lines are dependent on the drive cycle, for different drive cycle different shift points will be obtained.
In figure-13, results are plotted by simulating the vehicle model for the optimized gear shift points. On the left hand side, results are shown for the complete drive cycle whereas right hand side represents a zoomed version of the same plots. In the figure, vehicle speed and the commanded vehicle speed are plotted. As one can see, with the optimized gear shift points the vehicle is able to follow the drive cycle. In the next plot, motor torque is plotted; during regeneration (when the vehicle is slowing down - brake pedal applied) motor torque is negative. In the next plots, motor speed and transmission current gear are displayed. While upshifting motor speed drops down, whereas while downshifting the motor speed shoots up due to change in gear ratios. In figure-14, motor operating points are displayed on the motor map. The motor is being operated in its most efficient zone as much as possible. As it is quite clear from the figure that most of the points are in the region where motor efficiency is more than 90 %.
A comparitive study was done to see if we use the normal downshift lines while regeneration, which is generally done, instead of using different downshift lines how it is going to impact the overall vehicle efficiency. The energy consumption per unit distance for the same configuration is 0.8072 kWhr/km when the same downshift lines were considered which is higher than the case when we use different downshift lines during regenration as shown in Table-3. Lastly, also shown in the table is EPD obtained from manual tuning of shift points. This process involves selecting base shift points and making iterative improvements in those points by visulazing the operating points on the motor points.
A genetic algorithm based gear shift optimization strategy for electric vehicle is proposed in the paper. The optimization problem is formulated to maximize overall vehicle efficiency and minimize drive cycle track error. NSGA-II (non-sorting genetic algorithm) is used to get a Pareto optimal front which represents all possible optimized values without assigning any weightage to any of the objectives. A mathematical formulation is proposed to calculate theoretical global minima of energy consumption per unit distance (EPD) for a drive cycle and engine, transmission combination. The simulations validates that the Pareto optimal front approaches toward global minima with the help of NSGA-II. This optimization yields 1.21% higher efficiency than manual tuning of shift points. An alternate strategy for downshifting is proposed during regeneration, which results in net 5.02 % improvement in the electrical vehicle efficiency for the simulated scenarios.
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EV - Electric Vehicle
AMT - Automatic Manual Transmission
GA - Genetic Algorithm
AT - Automatic Transmission
NSGA - Non-dominated Sorting Genetic Algorithm
ICE - Internal Combustion Engine
MOOP - Multi Objective Optimization Problems
OEM - Original Equipment Manufacturer
DoE - Department of Energy
PEV - Plug-in Electric Vehicles
EPD - Energy Consumption Per Unit Distance
BEV - Battery Electric Vehicle
SoC-State of Charge
MD - Medium Duty
HDDS - Urban Dynamometer Driving Schedule
Vinod Saini, Sanchit Singh, and Shivaram NV
Indian Institute of Technology Delhi
Table 1. Decision Variables for NSGA-II Throttle Pedal Upshift (RPM) Downshift (RPM) Downshift during (Percentage) Regeneration (RPM) 0 X(1) X(7) X(13) 20 X(2) X(8) X(14) 30 X(3) X(9) X(15) 70 X(4) X(10) X(16) 80 X(5) X(ll) X(17) 100 X(6) X(12) X(18) Table 2. Vehicle Configuration  Gross Vehicle Weight 18000 Kg Axle Ratio 5.74 Tire Diameter 0.46 m Aerodynamic Drag Coefficient 0.7 Frontal Area 6.6 [m.sup.2] Rolling Resistance Coefficient 0.0075 Transmission Gear Ratio [4.83,2.6, 1.69, 1.0] Tractive Motor Power 130 KW Tractive Motor Torque 800 Nm Table 3. A Comparison of Improvement in EV Efficiency (lower EPD is better) EPD with Manual EPD for Normal EPD for Different tuning ([10.sup.h] Downshift Lines Downshift Lines iteration) 0.8171 0.8072 0.7716 Improvement 1 21% 5 02% (%)
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|Author:||Saini, Vinod; Singh, Sanchit; NV, Shivaram; Jain, Himanshu|
|Publication:||SAE International Journal of Alternative Powertrains|
|Date:||Jul 1, 2016|
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