Federal University of Tocantins - UFT, Palmas, Brazil. ABSTRACT: ... The Automotive Powertrain is the group of systems ... function of the design requirements.
Automotive powertrain optimization by genetic algorithm analysing transmission ratios G. B. Colherinhas & P. H. C. Dias & A. C. G. C. Diniz Department of Mechanical Engineering University of Brasilia - UnB, Brasilia, Brazil
A. P. S. P. Rodrigues Coordination of Computer Science Federal University of Tocantins - UFT, Palmas, Brazil
ABSTRACT: The automotive Powertrain refers to the group of components that generates power to move the vehicle, including wheels and tires, axles, transmission box and the engine. In general, a Powertrain is designed based on the designers knowledge with earlier experiences, taking into account distinct project requirements connected with performance and safety. It draws in complex analysis, hampering the process repeatability. This project purposes the optimization by genetic algorithm of the relation between acceleration, fuel economy and maximum velocity of an automotive vehicle, by selecting the best Powertrain configuration. To reach this goal, the performances of several Powertrain configurations were investigated given diverse driveline ratios for different gears, axles and wheel sizes using probabilistic methods. This work is applicable at the initial stage of automotive designs by the achievement of mathematical justification to select a particular Powertrain configuration and the optimization of the vehicle’s overall efficiency. This paper presents the tests used to assess the Powertrain performance relative to the target parameters of the optimization, enabling a satisfactory view of the methodology. The developed algorithm is used at a medium family vehicle equipped with manual transmission and the result is compared with a commercially available vehicle.
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INTRODUCTION
The Automotive Powertrain is the group of systems that apply power (mechanical, electric or thermal) to move the vehicle. In a general way, it consists of wheels (and tires), axles, gear box, engine, cooling, admission and exhaust systems. The operational parameters of the Powertrain affect the vehicle performance. Automotive designers must take into account several variables to determine these parameters and achieve all the project requirements as maximum acceleration, final velocity, fuel economy, minimum velocity at a particular climbing and others. These targets results in a conflicting solution that hampers the automotive design project. A Powertrain setting that increases the acceleration will naturally decreases the fuel economy. Nevertheless, better fulfillment in a simultaneous way of these two parameters is increasingly being required of the automotive industry by regulation laws. That’s why there is a huge room to apply an optimization. Nowadays, the Powertrain setting is done considering the designers background and also by complex analysis with
low repeatability. Accordingly, the computational optimization methods are a good way to find the best Powertrain configuration at automobiles. This paper proposes a powertrain optimization methodology to assist the automotive design project, enabling the selection of a configuration that targets an optimum relation between acceleration, maximum velocity and fuel economy. The variables parameters to be surveyed are: gear ratios, rear axle ratio and tire size. The gear ratio, rear axle ratio and tire geometry selection are made following some criteria (Hoff & Gregory 2003): vehicle purpose, required power, resistive forces (the ones the Powertrain must overcome), transmission type (manual, automatic, CVT), traction location (rear, front, all-wheel drive) and tire size. The automobile may be designed to be a family model, an economic model, a sport model or even a race model. According to the intent, the transmission ratios can be chosen to maximize acceleration, fuel economy, stability or other parameters. The goal of the vehicle will define the traction location as well. The engine power and fuel economy lines order the transmission ratios
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POWERTRAIN RATING - PERFORMANCE TESTS
This section presents the tests used to evaluate the fitness functions arguments for the Powertrain sets. Then, the function can be optimized. Each configuration created at the optimization process will undergo these tests, so the rating can be quantified as function of the design requirements. The acceleration test is regulated by Society of Automotive Engineers (SAE), named Vehicle Acceleration Measurement (SAEJ1491 2006), in which the maximum acceleration and the final velocity are determined. The second test is regulated by United States Environmental Protection Agency, named Federal Urban Dynamometer Driving Schedule (EPA 2013), taken to find the fuel economy performance in km/l. The tests were computer simulated at Matlab considering the equations, input and output data prescribed at the standards. 2.1 Acceleration Test Simulation The simulation is made under a vehicle moving on a straight road at wide open throttle and red line set at 6000 rpm. The test has two main targets to ascertain: i) acceleration time t0−100 from 0 to 100 km/h; ii) maximum final velocity vmf taken at 45 s test time (ti = 45s). Starting from null initial conditions, the velocity
Figure 1: Free Body Diagram for vehicle riding in inclined track (i)
vi and effective acceleration aef f are calculated for each time ti considering a constant increment inc. The velocities and positions are measured as an uniform movement with constant acceleration between each couple of instants ti and ti+1 . The effective acceleration for each moment is the minimum value between the acceleration provided by the Powertrain a(i) x , applying Newtons second law at the FBD (Figure 1), and the Maximum Possible Acceleration limited by friction amax . (i)
aef f = min(a(i) x , amax ),
(1)
2.2 Fuel Economy Test Simulation This test provides the fuel economy consumption of a vehicle moving in urban roads. The normalized velocity profile is presented by Figure 2. The fuel economy ec is determined as the ratio between the total distance Sf and the total fuel mass used mf , considering the fuel density ρf . The total distance depends only on the velocity profile, not changing with the Powertrain setting. At this case in Figure 2, the total distance at 1369 s is 11,987.68 m. The total fuel mass used is the sum of the fuel mass for each time increment: ec = Sf ρf (
1369 X
mf )−1 ,
(2)
i=0
The fuel mass used at each increment is the product (i) of the fuel mass flow m ˙ f by the time increment inc. 100 90 80 70 Velocity (km/h)
so it can operate at favorable conditions. The transmission type will set the number of gears. Typically, it is used 5 or 6 gears to manual and 4 gears to automatic transmission. In each case, the last gear is used to increase fuel economy. At long last, the tire size has a great importance at the vehicle acceleration and fuel economy performances. Two tests are going to be applied to inquire about the best Powertrain configuration. The first is the acceleration test regulated by the Society of Automotive Engineers (SAE), named Vehicle Acceleration Measurement (SAEJ1491 2006), in which the maximum acceleration and the final velocity are determined. The second is regulated by United States Environmental Protection Agency, named Federal Urban Dynamometer Driving Schedule (EPA 2013), taken to find the fuel economy performance in km/l. Through the result evaluation of each test, fitness is ascribed to each configuration and it is going to be inputted at a genetic algorithm to optimize the initial values. This optimization method was chosen due to its advantages at discontinuous functions with a high number of discrete variables. The following sections describe the evaluation tests for the Powertrain settings, the optimization method chosen (definitions of the chromosome, fitness function and convergence criteria) and the results.
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Figure 2: EPA Urban Dynamometer Driving Schedule (UDDS) Duration = 1369 seconds, Distance = 11.99 km, Average Speed = 31.53 km/h
This flow depends on the Engine Brake Power BP and the Brake Specific Fuel Consumption BSF C: (i)
(i)
mf = m ˙ f (ti − ti−1 ) = BP (i) BSF C (i) inc.
(3)
Brake Power and Brake Specific Fuel Consumption are calculated for each time increment as function of the engine rotation and the optimization target parameters: rear axle ratio, tire size, transmission ratios. 3
rr = 280, 283, 286, 287, 289, 292, 295, 296, 298, 299, 302, 305, 306, 308, 311, 312,
OPTIMIZATION
A Genetic Algorithm (GAs) optimization is proposed to find the best Powertrain setting. GAs are search techniques based on the processes of natural selection for survival through population genetics (Holland 1992). For the GAs to start evolving, we can use the following steps: Selection, Recombination (crossover), Mutation and Replacement, where the survival-of-the-fittest mechanism can be applied to the candidate solutions (Goldberg 1989 Haupt & Haupt 1998). In the initialization, the algorithm replies the evolutionary genetics and generates a random population by uniform distribution. The chromosomes represent the elements in the evolutionary algorithm space, where their features, called genes (quantified by values called alleles), are the problem inputs to be rated by the fitness function. The Selection allocates more copies of those solutions with higher fitness values and the roulette-wheel selection is the procedure selected to accomplish this idea. After that, recombination combines parts of two parental solutions to create new, possibly better solutions. The Mutation, on the other hand, randomly modifies the solutions’ allele. At the end a percentage of the best elements are just copied to the next generation using an elitism probability. This is a mechanism to keep the best solutions from the current generation (Goldberg 1989). The algorithm feeds the system back and evaluates each element from the population. Using cited evolutionary strategies, the fittest elements will have more probability to pass their features to the next generations. They are depicted by real base, where each gene matches to the hereditary characteristics to be combined and evaluated. At this problem, the inputs are: rr, rolling radius; Ra, rear axle ratio; Rtx , gear ratios to 5 gears, x = 1, 2, 3, 4, 5. The chromosome is describe by the following function: C = [rr; Ra; Rt1 ; Rt2 ; Rt3 ; Rt4 ; Rt5 ],
each gene represents one problem variable (Rodrigues 2007). Since the wheels are limited by the market; the Rolling Radius, rr, described as the mean effective radius, will be defined as a discrete variable. From a sample of medium size vehicles, this gene was restricted to:
(4)
The real base is useful when the parameters to be optimized are continuous variables (Rahmat-Samii & E. 1999). The computer uses flow point numbers to represent the chromosome and its size is equal to the vector that represents the problem solution; thereby
313, 315, 316, 317, 321, 322, 323, 324,
(5)
329, 330, 331, 345, 351, 356, 358, 359, 361, 365, 366, 369, 373, 376, 378, 379, 386, 392. Then rr can assume 42 values. The Rear Axle Ratio Ra is defined as one of several commercial values applicable to this study and they are located in between 2.70 and 4.30 (Hoff & Gregory 2003). 2.70 ≤ Ra ≤ 4.30,
(6)
The Gear Ratios (Rt1 to Rt5 ) were defined as usual values for a medium size family car. The data was taken from the manufacturers (Hoff & Gregory 2003). From a statistical analysis of this data, the lane of possible values was defined for each gear. Histograms were plotted representing the values and the functions were approximated by a normal distribution. Then the average and variance were calculated. Finally, these values were used to define the minimum and maximum for the lanes in a way that there was no overlap between them. Using two decimal digits of precision, the lanes are: 2.94 < Rt1 ≤ 4.36,
(7)
1.70 < Rt2 ≤ 2.94,
(8)
1.18 < Rt3 ≤ 1.70,
(9)
0.82 < Rt4 ≤ 1.18,
(10)
0.55 ≤ Rt5 ≤ 0.82,
(11)
The performance of each element of the population is measured by the fitness function. It rates the capability of the element (given a powertrain setting) to minimize fuel consumption and maximize final velocity and acceleration (less time from 0 to 100 km/h). The values of Fuel Economy ec, final velocity vm f and time from 0 to 100 km/h t0−100 are calculated using the two tests described on Section 2. The results are
combined on the fitness function (Equation 12), generating each elements fitness.
Optimized Vehicles
(12)
At every generation the code calculates the values of ec, vmf and t0−100 of all the elements using the process of Section 2, where wf and ws are the weight for the Family and the Sport car, respectively. These weights enable analysis from directed optimization for faster or economic configurations. After certain quantities of generations evaluated, the algorithm converges when the means of the fitness values of each generation floats about one specific value, presenting stability about this point (local maximum). To avoid a biased local maximum that have a small probability to correctly represent the best convergence curve, the algorithm parameters must be set. This is made adjusting the probability of crossover, mutation, elitism, population and generation sizes, through values found in specialized bibliographies as well as changing values found from many tests simulations.
Setting Fitness Fuel Cons. (km/l) 0 to 100 km/h (sec) rr (mm) Ra Rt1 Rt2 Rt3 Rt4 Rt5
Best 244.87 11.07 8.9 283 4.17 3.62 2.72 1.55 0.88 0.72
Sport 241.53 10.65 8.8 286 4.03 3.44 2.41 1.62 0.98 0.80
Family 232.58 13.84 10.9 289 3.40 3.51 2.84 1.26 1.09 0.66
Original 224.15 11.64 10.1 290 2.95 3.25 2.06 2.42 1.03 0.73
0.45 Best Fitness Original
0.4 0.35 0.3 Acelleration (g)
vmf ecwf f= , (t0−100 )ws
Table 1: Optimization between Best fitness, Sport and Family cars configurations in comparison to original vehicle.
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RESULTS
After the algorithm of optimization was finished, many tests were made to determine the best probabilities of crossover, mutation and elitism, as well as the population and generation sizes. The parameters were adjusted with: 7 % probability of mutation; 70 % crossover; 15 % elitism; 250 chromosomes per generation. After 150 generation, the Figure 3 was plotted containing the curves of the best general fitness for each generation with their mean values. Also the fitness for a Family and Sport optimization and the Original Saturn SL2 vehicles were plotted to be compared. The optimization shows that the best vehicle has a rating 9.2 % higher than the original Saturn SL2, 7.8 % for the Sport car and 3.8 % for the Family car. Table 1 shows the results for the vehicles with maximum fitness value at each case (General, Family and Sport car). The original Saturn SL2 values can also be found. It can be seen that the performance parameter values for the Best Fitness and the Original vehicles are located in between the ones found for the Sport and the Family cars. This was expected, because these two last ones represent extreme intents. The Family vehicle should be the most economic and The Sport vehicle the faster. It is confirmed, comparing to the original model, The Family vehicle is 18.9 % more economic and The Sport vehicle is 14.8 % faster with a maximum final velocity 2.6 % higher. Seen from an isolated angle, the optimization target parameter values have a random distribution. The standardless results show that the optimization
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Figure 4: Acceleration Test for Best fitness and Original Vehicle (Saturn SL2)
method using genetic algorithms was really useful to solve this particular problem. The acceleration test is plotted at Figure 4 and Figure 5. Each step represents a gear been shifted. All the curves go from 1st gear in time 0 to 5th gear in the end of the test by ascending one by one. Note that the maximum overall acceleration is performed by The Best Fitness vehicle, 0.43 g, and not by The Sport configuration. However this one has the highest area under the curve, therefore it can perform the test distance in less time. Figure 6 and Figure 7 shows the power provided at the wheels at the test. As the acceleration plot, the 0.45 Family Sport
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Figure 5: Acceleration Test for Sport and Family Vehicles
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Fitness
235 230 225 220 Best Fitness Mean (Best) Original Family Sport
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Figure 3: Optimization Results by fitness per generations for Best fitness, max. fit Family and Sport, and Original vehicles
fuel economy, however in this case a different solution was found.
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Power (KW)
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Figure 6: Power provided by Best fitness and Original Vehicle (Saturn SL2)
steps represents the gears been shifted. P rl is the road load curve, it is the power that the vehicle needs to fulfill to start acceleration and their interception sets the final maximum velocity. The difference between the power provided and the road load power is the acceleration power. The Family vehicle keeps a long time at 3rd gear, while the Sport vehicle is the first to reach the 5th gear. It goes away from what traditionally Power Train designers do. Usually the 5th gear is used to 90 80 70
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Figure 7: Power provided by Sport and Family Vehicles
CONCLUSIONS
The optimization solution was very effective. The best fit vehicle overcame the original model in 9.2 %. Also the Sport model has acceleration 12.8 % faster and the Family model has fuel consumption 18.9 % higher. It highlights that this tool can be very helpful at the initial stages of Powertrain designing. The fitness function rates the different settings according to its performance, however several other parameters could be taken into account when choosing it as user friendliness, manufacturing technics and initial conditions. This study can be improved by refining the calculation and adding other target parameters. REFERENCES EPA (2013). Urban Dynamometer Driving Schedule (UDDS). United States Environmental Protection Agency. Goldberg, D. E. (1989). Genetic Algorithms in Search, Optimization and Machine Learning. The University of Alabama: Society of Automotive Engineers. Haupt, R. L. & S. E. Haupt (1998). Pratical Genetic Algorithm. John Wiley G. Sons Inc. Hoff, C. J. & D. W. Gregory (2003). Introduction to Automotive Powertrains. Kettering University. Holland, J. H. (1992). Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control and Artificial Intelligence. Cambridge, MA, USA: MIT Press. Rahmat-Samii, Y. & M. E. (1999). Electromagnetic Optimization by Genetic Algorithms. John Wiley & Sons. Rodrigues, A. P. S. P. (2007). Parametrization and Numerical Simulation of the Hidrokinetic Turbine Otimization Using Genetics Algorithm, Dissertation in Mechanical Engineering. University of Bras´ılia - UnB. SAEJ1491 (2006). Vehicle Acceleration Measurement. Society of Automotive Engineers.
