SYNTHESIS OF ENERGY- EFFICIENT ACCELERATION CONTROL LAW OF AUTOMOBILE

Energy efficiency is the property that ensures high dynamics of a vehicle with the lowest energy consumption. The improvement of energy efficiency is required due to the growing intensity of road traffic and the requirements to reduce the harmful effect of automobile transport on the environment. Solving a given task is positively influenced by the emergence of electric cars and hybrid vehicles with a combined electromechanical drive of drive wheels. The existence of electrical part in the drive of drive wheels makes it possible to rationally control the acceleration of a vehicle, in other words, to implement such a law of change in the acceleration of a car depending on speed that would ensure the lowest consumption of energy (power consumption) by the engine. SYNTHESIS OF ENERGYEFFICIENT ACCELERATION CONTROL LAW OF AUTOMOBILE


Introduction
Energy efficiency is the property that ensures high dynamics of a vehicle with the lowest energy consumption.The improvement of energy efficiency is required due to the growing intensity of road traffic and the requirements to reduce the harmful effect of automobile transport on the environment.Solving a given task is positively influenced by the emergence of electric cars and hybrid vehicles with a combined electromechanical drive of drive wheels.The existence of electrical part in the drive of drive wheels makes it possible to rationally control the acceleration of a vehicle, in other words, to implement such a law of change in the acceleration of a car depending on speed that would ensure the lowest consumption of energy (power consumption) by the engine.
A vehicle controlled by the driver is a movable physical (energy, thermal-mechanical) system for producing mechanical energy and its immediate utilization for the implementation of a transportation process.Therefore, energy issues are particularly important for the automobile.High indicators of vehicle dynamics should be attained at minimal energy consumption by the engine.Under acceleration mode, it is expedient to control a vehicle motion making use of electric part of the drive.
Energy efficiency is one of the most important operational properties of the automobile.Maximal fuel consumption is associated with the unstable mode of operation of vehicle's internal combustion engine.The appearance of cars with a hybrid electromechanical drive of drive wheels makes it possible to reduce the additional consumption of energy under unstable mode of ICE operation.

Literature review and problem statement
Papers [1,2] consider choosing rational motion speed under condition of high dynamic indicators of vehicle.Authors of [3] established the relationship between the power of the engine used to accelerate the vehicle, motion speed, and recommended acceleration where m a is the mass of the vehicle; V a , • a V are the linear speed and vehicle acceleration; ΔN e is the additional engine power used for the acceleration of the vehicle; η tr is the transmission performance efficiency; δ rm is the accounting factor for the rotating masses of engine and transmission; U gr is the gear ratio of the gearbox; τ 1 ; τ 2 are the coefficients that take into account the effect of rotating masses, which are connected to the drive wheels through the constant and variable transfer ratios, τ 1 =0.03-0.05;τ 2 =0.04-0.06.Acceleration of the vehicle from speed V a to speed V a +ΔV a will last before a new equilibrium is reached (the traction balance [3]).In papers [1,2], authors derived an equation that determines the condition for establishing the following equilibrium at a higher speed of the vehicle equal to V a +ΔV a ( ) where g is the free fall acceleration, g=9.81 m/s 2 ; ψ is the total coefficient of road resistance; k is the coefficient of air resistance; F is the frontal drag area (midsection) of the vehicle; V a1 is the speed at which the vehicle reaches the new equilibrium,
Authors of articles [1,2] derived from equation (3), with respect to relations (1) and ( 4), a cubic algebraic equation relative to parameter ΔV a (5) The task solved in papers [1,2] implied determining a function ΔV a (V a ) to be followed by obtaining rational speed (V a ), which ensures (ΔV a ) max .The authors of scientific studies [1,2], rather than solving equation ( 5) using the Cardano's expression, applied approximate and numerical solutions.Fig. 1 shows dependences ΔV a (V a ), constructed in papers [1,2] employing the approximate and numerical methods without an accurate analytical solution to equation (5).
Curve 3 in Fig. 1 is built based on the numerical solution to equation (5).Curve 1 in Fig. 1 -for the case when terms (ΔV a ) 3 and (ΔV a ) 2 are removed from equation ( 5); and curve 2 -at (ΔV a ) 3 =0.The authors of papers [1,2] were interested in the points of maximum for curves 1, 2, 3.An analysis of these curves allowed them to conclude that the points of maximum of the indicated curves coincide.They are determined from dependence However, papers [1,2], to determine (ΔV a ) max , proposed an approximate analytical expression since solving a cubic algebraic equation using the Cardano's expression was time consuming.In order to obtain the exact solution, it is necessary to solve equation (5).kF/m a =0.0003 m -1 ; δ rm =1.176 [1,2] Acceleration time to the assigned speed is the indicator of traction-speed properties of the vehicle.That is why it is required to determine the acceleration time of a vehicle at In addition, it is necessary to compare the functions of change in the acceleration time of a vehicle on speed, implemented at the existing rational techniques of control over acceleration and at Targeted measures that affect technologies of vehicles increasingly become a tool for the developers of energy policy in the EU as a means of achieving the objectives in the field of energy efficiency, renewable energy, climate change, and ensuring energy security.There are known papers whose authors explore energy costs of vehicles from ( ) ( ) certain fleets and predict their growth: article [4] proposed a model that estimates current energy consumption by the automobile fleet in Andorra, [5] in the Republic of Ireland, and paper [6] -the city of Chang-Zhu-Tan, People's Republic of China.Authors of article [7] suggest a method for calculating energy efficiency of car fleet on the roads in terms of "traffic factor".However, these studies do not solve the problem on determining the relationship between the maximum increase in speed and the minimal increase in the vehicle engine power.Energy efficiency of the vehicle is addressed in papers [8,9].To model the external speed characteristic of the internal combustion engine, the following empirical dependence is applied where λ is the ratio of the current angular velocity ω e to the angular velocity ω N at maximum power, a, b, c are the empirical coefficients of equation ( 7) (Table 1); N emax is the maximum engine power.Papers [10,11] report the study into stability of the vehicle during acceleration.However, their authors failed to assess the effective work performed by ICE when accelerating a crankshaft from angular velocity ω min (minimally stable rotation speed) to maximal angular velocity ω max .
Thus, the analysis of known studies revealed the following: -there is no analysis of the engine energy used for the acceleration of the vehicle when shifting different gears; -there is no analytical solution to equation ( 5): this makes it impossible to state the law on the rational control over acceleration when accelerating a vehicle.

The aim and objectives of the study
The aim of present study is to determine the relationship between the maximum increase in speed and minimal increase in engine power, which would produce a rational law to control vehicle acceleration.
To accomplish the aim, the following tasks have been set: -to build a rational dynamic characteristic of a vehicle; -to estimate acceleration time of the vehicle to the assigned speed under various laws of control over acceleration; -to evaluate effectiveness of ICE performance when accelerating the vehicle at different gears.

Rational dynamic characteristic of the vehicle
In a classic statement, first it is required to solve equation ( 5) relative to ΔV a , and then proceed to search for the maximum employing known methods [1,2] ( ) First, we propose to differentiate equation ( 5) for V a .In this case, the specified equation will take the form A maximum of function (that is, the extremum shown in Fig. 1) ΔV a (V a ) is reached at dΔV a /dV a =0.At dΔV a /dV a =0, expression (10) takes the following form (11) or The solution to equation ( 12) with respect to the root that has a physical meaning will be obtained in the form (13) Expression (13) connects the maximum increase in the motion speed (ΔV a ) max and rational speed (V a ) rat , at which it is possible to implement the specified increase.
It is proposed to solve equation ( 5) relative to V a .In this case, the indicated equation transforms to the form Solution to quadratic solution (14) Dependence chart of V a (V a ) will represent a chart turned at 90° shown in Fig. 1.The chart shown in Fig. 2 demonstrates that (V a ) max is implemented at V a =(V a ) rat and at V a1 =V a2 .Condition V a1 =V a2 is satisfied when a discriminant of the quadratic equation equals zero, that is, at We shall derive from equation ( 16) Thus, considering expression (15) By comparing expressions ( 6) and (18), we can conclude that the analytical solution to equation ( 5) allowed us to refine the approximated result that was obtained earlier (6).Correction factor Expressions ( 17) and (18) allow us not only to determine parameters of ((ΔV a ) max and (V a ) rat , but also to obtain a rational dynamic characteristic of the vehicle.
Equation (18) will be solved relative to acceleration • .a V We shall obtain as a result An analysis of dependence (20) shows that when equality ( 6) is fulfilled, the magnitude 20) describes the law of acceleration control, which ensures acceleration dynamic of the vehicle at the lowest energy consumption.We shall obtain a differential equation of vehicle acceleration from expression (20) 21) is a differential equation with variables that are divided As a result of solving differential equation (22), we find acceleration time t p from speed V a to speed V a1 =V a +(ΔV a ) max at rational control over vehicle acceleration An analysis of equation ( 23) shows that a decrease in δ rm leads to a decrease in the acceleration time t p of vehicle from speed V a to speed V a1 =V a +ΔV a .The magnitude of accounting factor for the rotating masses of transmission and engine is close to unity if a crankshaft is under constant speed mode with the vehicle acceleration driven by electric motors.
By multiplying the left and right sides of equation (20) by the magnitude m a V a δ rm /η tr , we shall obtain a rational law of engine power control at vehicle acceleration ( ) where η tr is the performance efficiency of vehicle transmission, it is possible to accept that η tr ≈0.8.
Fig. 3 shows dependence charts of ( ) • , a a V V built at different values of ψ (curves 2, 3).The same figure shows a chart (curve 1) of dependence of boundary acceleration on speed, which corresponds to the implementation of boundary forces for adhesion on the drive wheels of the vehicle.For an all-wheel drive vehicle, at the implementation of boundary forces for adhesion on the drive wheels where φ x is the longitudinal coefficient of drive wheels adhesion to the road, we accept φ x =0.8.Therefore, the obtained analytical expressions make it possible to implement a rational dynamic characteristic of a vehicle.It was determined for the example under consideration (Fig. 3) that the rational dynamic characteristic can be implemented over the entire range of possible speeds from zero to V a =70 m/s (252 km/h).In the interval of speeds from 10 m/s up to 30 m/s maximal vehicle acceleration is limited by the boundary adhesive capacity of drive wheels to the road.At the point that corresponds to the fulfilment of condition (6), there is a rupture of function ( )

Estimation of vehicle acceleration time to the assigned speed under various acceleration control laws
A perfect dynamic characteristic of the vehicle (an acceleration characteristic) is implemented at equality of tangential reactions on wheels to the boundary forces for ad- and upon solving differential equation (25) with the variables that are divided, we shall obtain, at vehicle acceleration from speed V a to speed V a1 , Fig. 4 shows dependence charts t p =t p (V a ), built at different values of φ x .A decrease in adhesion coefficient φ x from 0.8 to 0.2 increases the minimum possible acceleration time (for example, from 0 to V a1 =30 m/s) by about four times (Fig. 4).

Fig. 4. Dependence of vehicle acceleration time from zero
to the speed of V a1 when implementing a perfect dynamic characteristic kF/m a =0.0003 m -1 : 1 - The existing technique to control vehicle acceleration implies combined control over the engine and the gearbox.Engine acceleration is associated with significant additional energy expenditures, since the moment of inertia of its moving parts, applied to the drive wheels, reaches considerable magnitudes.
To model the impact of an ICE speed characteristic, we employ dependences [4] ( ) where λ is the ratio of the current angular velocity ω e to the angular velocity ω N of maximum power N emax , a, b, c are the empirical coefficients of equation ( 27) [8] (Table 1); N e max is the maximum engine power.
To model partial speed characteristics of ICE, we shall use the following dependence: where K N is the coefficient of engine power utilization, equal to the ratio of maximum power at a speed characteristic that is considered relative to the maximum power of the engine The assumption, made in [3] for expression (29), implies that ω N =const does not change when shifting from one speed characteristic to the other.
Effective torque at the engine shaft can be determined from: With respect to ratio (28) where P k is the traction force on the drive wheels (34) U 0 , U gr are the transfer ratios of the main transmission and the gearbox at the i-th gear; r d is the dynamic radius of the drive wheels; η tr is the transmission performance efficiency.
We shall define linear acceleration of the vehicle from expression (33) with respect to (34) and (32) where A, B, C are the coefficient corresponding to the i-th gear, The solution to the differential equation with variables that are divided determines vehicle acceleration time at the i-th gear when accelerating a crankshaft from ω min to ω max Vehicle acceleration time from V a to V a1 =V a max that employs all gears is determined as follows where n is the quantity of gears used in the gearbox; t sh is the time of gear shifting.When a vehicle reaches maximum speed value V a max on top gear, the angular velocity of crankshaft ω e may be less than ω max .Therefore, when determining vehicle acceleration time on top gear t рt , the upper bound of integration ω t must be taken equal not to ω max , but to the magnitude, which is determined from expression where U gbt is the gear ratio of the gearbox on top gear.
In paper [3], authors obtained the rational dynamic characteristic of a vehicle (dependence of acceleration on car speed).Fig. 5 shows a chart of change in the rational value of At the third section, we shall also employ expression (20), but with the limits of change in speed V a .In this case, Depending on the section where speed V a1 is reached, a general acceleration time is determined by summing up all components of t p1 , t p2 , t p3 , or only parts of them.
To determine the boundaries of speed intervals * а V and * * , а V it is required to solve a system of equations ( 20) and (6).A solution to the given system of equations can be obtained upon solving an algebraic equation of sixth degree Two of the six roots of equation (48) will have a physical sense.

Estimation of the ICE operation efficiency when accelerating a vehicle at different gears
In order to model partial speed characteristics of ICE, we shall apply the following dependence: where K N is the coefficient of engine power utilization, equal to the ratio of maximum power for a speed characteristic, which is considered relative to the maximum power of the engine The assumption, made for expression (49), implies that ω N =const does not change in the transition from one speed characteristic to the other.
For a more general representation of dependence (49), it is expedient to transform it into the following form where N rel is a relative indicator of ICE effective power; Dependence chart of N rel is shown in Fig. 6 (with respect to the accepted assumptions).
Note that engine acceleration cannot happen in full by the external speed characteristic because a given process takes place at a gradual increase in fuel feed to the cylinders.
In this case, the engine gradually passes from one partial speed characteristic to another, until, at a certain angular speed ω p of crankshaft, it switches to the external ICE speed characteristic (Fig. 7).
Accelerating characteristics of automobile ICE are given in papers [1,3].The authors obtained boundary ICE accelerating characteristics for adhesion of the vehicle drive wheels to the road.This allowed them to estimate boundary parameters for dynamic properties of a vehicle and to determine the boundary engine power.We shall denote section 2 of the engine accelerating characteristic as the characteristic's regulatory section; section 1 -the section of external speed characteristic (Fig. 7).
Determine coordinates of point P (Fig. 2) by solving the system of equations (53) (54) where K p is the angular coefficient of straight line 2 in Fig. 7. Fig. 7. ICE accelerating characteristic: 1 -working section of the external speed characteristic; 2 -section of accelerating characteristic at growing fuel feed to cylinders of the engine Solution to the system of equations ( 53) and (54), with respect to the root that has a physical sense, will take the following form: We shall derive from expression (54) and we shall determine from expression (53) Thus, the effective work of ICE when accelerating the crankshaft from ω min to ω max can be determined from where λ min , λ max are the minimum and maximum value for parameter λ, corresponding to ω min and ω max .
Determine dλ/dt.To this end, we shall apply equation of the vehicle power balance where С x is the coefficient of frontal drag; ρ is the air density; F is the area of the frontal drag (a midsection) of the vehicle; δ rm is the accounting factor for the rotating masses of the transmission and the engine, where U gb is the gear ratio of the gearbox.Linear speed and acceleration of the vehicle Equation ( 60) with respect to (62) and (63) takes the following form: We derive from equation (64) Considering equation (58), we shall obtain upon transforms ; Expression (66) with respect to (67)-(69) will take the form Equation ( 59) considering (70) will be transformed in the following way: Fig. 8 shows dependence of vehicle's ICE effective work (using KrAZ-5233 as an example) on the gear of its GB, obtained using dependence (71).The effective work of ICE of the truck KrAZ-5233, as shown in Fig. 8, decreases when the vehicle shifts to a higher gear. .
An analysis of the results obtained reveals that a transition from lower to higher gears leads to a significant reduction in the engine energy consumption required to accelerate the vehicle.When accelerating, in the same range of speeds, the internal combustion engine (from ω min to ω max ) of the vehicle KrAZ-5233, energy expenditure at gear I is A e =16.0 MJ, and at gear VIII -A e =2.0 MJ; in other words, eight times less.This allows us to recommend that hybrid vehicles should accelerate at lower gears employing an electric drive.It is appropriate to implement a rational law for vehicle acceleration control at all gears.

Discussion of results of research into a vehicle acceleration control law
The obtained rational law of acceleration control makes it possible to reach a maximum increase in the vehicle speed at minimum consumption of the engine energy.If, at each minimal interval of speed change, one would change, in a stepwise manner, engine power, then the result of acceleration to the maximal speed will be a minimal expenditure of engine energy.
The synthesis of the rational law for vehicle acceleration control became possible through the transformation of equation for vehicle acceleration dynamics when considering it over a small range of change in speed ΔV.
Similar research aimed at synthesizing rational laws for vehicle acceleration control implemented tangential reactions of the road, boundary for adhesion, on drive wheels.However, known laws demanded a multiple increase in the engine power and considerable energy consumption to control a motor-transmission unit.
The results proposed could be further developed for electric cars and hybrid vehicles, in which the implementation of the resulting rational law for acceleration control might be achieved by the simplest means with the lowest consumption of energy.

Conclusions
1.The analytical expressions obtained make it possible to implement such a change in the vehicle acceleration depending on its speed that ensures maximum dynamics at minimum engine power consumption, taking into consideration a nonlinear change in external resistance.The maximum acceleration, which is possible to implement using the rational dynamic characteristic, can amount to 2. Based on the dependences obtained, it is possible to determine effective work of ICE required to accelerate a vehicle at different gears.An analysis of calculation results revealed that the transition from lower to higher gears is accompanied by a sharp decrease in engine energy expenditure required to accelerate the vehicle (from A e =16.0 MJ at gear I of the truck KrAZ-5233 to A e =2.0 MJ at gear VIII).
3. In the case of hybrid vehicles, acceleration using the electric drive, rather than accelerating at lower gears of the mechanical drive, makes it possible to reduce energy losses by 20 % (for a four-cylinder internal combustion engine).Energy preservation is accomplished by reducing the fluctuation of traction force, as well as the possibility of a step-free change in motion speed.

Fig. 6 .
Fig. 6.Dependence of N rel (λ, K N ): 1 -external speed characteristic; 2 -partial speed characteristics left and right sides of equation (58) by N e max , we shall obtain