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A new non-geometric transmission parameter optimization design method for HMCVT based on improved GA and maximum transmission efficiency
Computers and Electronics in Agriculture 167 (2019) 105034
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Computers and Electronics in Agriculture journal homepage: www.elsevier.com/locate/compag
A new non-geometric transmission parameter optimization design method for HMCVT based on improved GA and maximum transmission efficiency
T
Zhun Cheng, Zhixiong Lu , Jin Qian ⁎
College of Engineering, Nanjing Agricultural University, Nanjing 210031, China
ARTICLE INFO
ABSTRACT
Keywords: HMCVT Improved GA Transmission parameters Optimization design Work efficiency Transmission characteristic
The HMCVT (Hydro-Mechanical Continuous Variable Transmission) can realize continuously variable speed in a large range and transmit a large load torque, and thus it is very applicable to engineering vehicles, such as the tractor. This paper proposes a new transmission parameter optimization design method for HMCVT in order to enhance the practicability of the HMCVT, improve its transmission efficiency, realize the fast determination of HMCVT transmission parameters, and allow HMCVT to meet the work requirements of engineering vehicles like the tractor. Specifically, this paper first conducts a characteristic analysis of a certain model of HMCVT. Then, according to the operation requirements of the tractor and the driving characteristic continuity of the HMCVT, the improved design of the transmission scheme, the maximization of working efficiency, and the most proper positions of junction points of variable speed stages, this paper puts forward an HMCVT transmission parameter optimization design model. This paper uses the GA (genetic algorithm) to implement the transmission parameter optimization, and improves the GA to obtain a higher algorithm running speed and accuracy. The improvement in GA primarily includes avoiding the super individual in the initial population, the adaptive changes of the population size in the iteration process, and the adaptive changes of crossover probability and mutation probability. The research results demonstrate that the improved GA has a faster rate of convergence and better optimization accuracy. The new proposed optimization design method can design HMCVT transmission parameters flexibly, efficiently, and quickly, as expected. The HMCVT with optimized transmission parameters exhibits driving characteristics that meet the working requirements of the tractor, significantly improves working efficiency, and offers theoretical guidance and a basis of design for the determination of the proper positions of junction points of variable speed stages.
1. Introduction Engineering or agricultural vehicles, such as the tractor, face complicated working conditions. For example, the tractor faces both the medium/low-speed field operation requirements of farming and seeding, and the medium/high-speed transportation driving requirements in the field or on good roads. In addition, the tractor bears different loads in different working conditions (Fang et al., 2017; Raikwara et al., 2015; Liu et al., 2017). In this case, it is hard to ensure the economic and power performance of the tractor in operation. More gears can meet the economic and power requirements of the tractor for work. The HMCVT (Hydro-Mechanical Continuous Variable Transmission) can realize the continuously variable speed within a large range, and also inherits the high efficiency of a mechanical transmission and the high power of a hydraulic transmission (Zhang and Zhou, 2014; Zhu et al., 2016a,b). As an increasing number of high-power vehicles comes
⁎
into service, the research on HMCVT has great significance (Xue, 2017; Du et al., 2009; Cheng et al., 2006; Zhang et al., 2016). The transmission parameter design of the HMCVT is the primary and key content in the research on the HMCVT. Vehicle working or driving conditions have certain requirements for the speed changing range of the HMCVT (Gao et al., 2013). The multi-stage HMCVT requires a junction point between the front and back variable speed stages to ensure the continuity of the transmission ratio during stage shifting, which reduces the impact of stage shifting and improves ride comfort and the service life of parts. A good design of the HMCVT transmission parameter can meet these requirements. In addition, the transmission efficiency of the HMCVT is closely linked to its transmission parameter design. Currently, there is little research on HMCVT, especially its transmission parameter design. Xu et al. (2006) expounded the single-line planetary mechanism of the mechanical part and the selection method
Corresponding author at: College of Engineering, Nanjing Agricultural University, 40 Dianjiangtai Road, Nanjing 210031, China. E-mail address: [email protected] (Z. Lu).
https://doi.org/10.1016/j.compag.2019.105034 Received 29 March 2018; Received in revised form 8 July 2019; Accepted 30 September 2019 0168-1699/ © 2019 Elsevier B.V. All rights reserved.
Computers and Electronics in Agriculture 167 (2019) 105034
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Nomenclature
k1 k2 k3 i1 i2 i3 i4
iHM1 iHM2 iHM3 iHM4 P1
i1 i2
H P2
P2 P3
P3
P4
q1
q2 i3 i4
j1
of HM1 and HM2 discharge ratio value corresponding to the junction point of HM2 and HM3 j3 discharge ratio value corresponding to the junction point of HM3 and HM4 fun_1 sum of errors of transmission ratios of key positions in different stages fun_2 integral value of efficiency error i HMn1( = n2) transmission ratio value in speed change stage n1 when discharge ratio is n2 weight coefficient of fun_1 a2 weight coefficient of fun_2 b1 the proportion of time used by stage HM1 b2 the proportion of time used by stage HM2 b3 the proportion of time used by stage HM3 b4 the proportion of time used by stage HM4 a ratio coefficient of relative importance of fun_1 and fun_2 set with the empirical method best _fun _1 optimal solution value obtained when considering fun_1 alone as the objective function best _fun _2 optimal solution value obtained when considering fun_2 alone as the objective function Fi fitness of the ith individual c1 constant in calculation formula NIND the number of individuals in the population N1 the number of individuals with the poorest fitness N2 changing value of population size mean reducing rate of every 10 generations in the operF ating process of the GA favg average fitness of the population the density of population in the steady-state area N3 the number of individuals in the steady-state area px crossover probability pm1 probability of the first kind of mutation pm2 probability of the second kind of mutation pm overall probability of mutation
j2
characteristic parameter of planetary line P1 characteristic parameter of planetary line P2 characteristic parameter of planetary line P3 characteristic parameter of planetary line P4 gear pair transmission ratio of i1 in Fig. 1 gear pair transmission ratio of i2 in Fig. 1 gear pair transmission ratio of i3 in Fig. 1 gear pair transmission ratio of i 4 in Fig. 1 displacement ratio of variable-pump-constant-displacement-motor system transmission ratio of stage HM1 transmission ratio of stage HM2 transmission ratio of stage HM3 transmission ratio of stage HM4 transmission ratio of planetary line P1 loss coefficient gear pair transmission efficiency of i1 gear pair transmission efficiency of i2 transmission efficiency of pump-motor system transmission efficiency from planetary line P2’s gear ring to sun gear transmission efficiency from planetary line P2’s planet carrier to sun gear transmission efficiency from planetary line P3’s planet carrier to gear ring transmission efficiency from planetary line P3’s planet carrier to sun gear planetary line P4’s transmission efficiency proportion of the torque input to planetary line P2 in total input torque proportion of the torque obtained by planetary line P3’s planet carrier in total input torque transmission efficiency of gear pair corresponding to i3 transmission efficiency of gear pair corresponding to i 4 discharge ratio value corresponding to the junction point
of the gear pair transmission ratio in the axle, and analyzed the governor control characteristic of the HMCVT. Xu (2007) also made an optimization design of the transmission parameters of the HMCVT based on the standard GA, and his objective function of the optimization analysis took into account the varying range of speed and the dividing ratio of hydraulic power of variable speed stages. Liu et al. (2006) optimized the arithmetic transmission parameters of the HMCVT using the penalty function, and the objective function used three parameters to approximate and substitute the transmission efficiency: the rotating speed of the planetary gear, the relative power coefficient of the planetary line, and the hydraulic transmission ratio. Zhu et al. (2015) conducted a kinetic and kinematics analysis on highpower tractors, and designed a single planetary line confluence HMCVT to compute and determine the gear ratio of the single planetary gear and gearshift. Zhang et al. (2017) designed the HMCVT used in cotton pickers, inferred the conditions for the continuous changes of transmission ratio via the principles of the arithmetic transmission scheme, and determined the transmission ratio of each gear pair. Most researchers, such as Zhang (2017), Zhu (2016), Zhang (2016), Xiao (2016), first determine that the HMCVT adopts a geometric or an arithmetic transmission scheme, then determine the possible value range of the transmission ratio of each transmission part according to the operating requirements of agricultural machinery, and finally determine transmission parameters with an empirical method. It is evident that the research on the transmission parameter design of the HMCVT currently has significant shortcomings. On the one hand,
HMCVT transmission parameters generally need to be designed on the basis of adopting an arithmetic or a geometric transmission scheme, and then determined via theoretical calculation using an empirical method. On the other hand, only a few researchers have used modern optimization design methods to conduct the optimization design of HMCVT transmission parameters. In addition, the objective function of the optimization design is characterized by four problems. First, it uses the hydraulic power dividing ratio or other parameters to approximate and substitute the transmission efficiency. Second, it computes the transmission efficiency only in the working conditions of the maximum or minimum displacement ratio, and fails to consider the transmission efficiency of all transmission ratios during the work of the HMCVT. Third, it requires a large speed varying range (the larger the better), but fails to consider the driving or working requirements of vehicles. Finally, it only connects the ends of the transmission ratios of variable speed stages to ensure the continuous transmission, but fails to consider the proper positions of the junction points of the front and back variable speed stages or reserve the overlapping area of stage change. In addition, there are generally several transmission parameters and targets to be optimized, increasing the difficulty in the iterative optimization of the algorithm. There may be a phenomenon of “prematurity” in the iteration process, causing the failure in convergence to the optimal solution, and reducing the optimization accuracy of the algorithm. To solve these problems, this paper proposes a new HMCVT parameter optimization design method. The method considers the nongeometric transmission scheme with the optimization goals of matching 2
Computers and Electronics in Agriculture 167 (2019) 105034
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the HMCVT transmission characteristics and the working and driving demands, the continuity of the transmission characteristics, obtaining the highest weighted transmission efficiency during the work of the HMCVT, and determining the proper position setting of junction points of transmission stages. To build an HMCVT parameter optimization model, the transmission and efficiency characteristics of the HMCVT based on a certain model of the HMCVT are first researched. This paper then improves the GA in the following four aspects: avoiding the super individual in the initial population, allowing for the adaptive changes of population size in the iteration process, and allowing for the adaptive changes of crossover probability and mutation probability. An illustration of the advantages of the improved GA is also provided by obtaining the minimum of a function. The research results indicate that the proposed optimization method can result in a higher transmission efficiency of the HMCVT, as well as a better transmission characteristic and overlapping area of stage change. The proposed improved GA can effectively enhance the convergence rate and optimization accuracy, and is more applicable to multi-target and multi-parameter optimization problems.
HMCVT input speed to the output speed when working at stage HM3. When clutch CL2 and clutch CL4 work, the HMCVT is in the working mode of stage HM4. We can infer the transmission characteristic of stage HM4 as follows:
iHM4 =
1
i1 i2
) (1 + k )(1 + k ) 1
2
(1)
where k1, k2, k3 , and are the characteristic parameters of planetary lines P1, P2, P3, and P4;i1, i2 , i3 , and i 4 are the transmission ratios of the gear pairs in corresponding positions in Figure 1; is the displacement ratio of the variable-pump-constant-displacement-motor system; finally, iHM1 is the transmission ratio in stage HM1, i.e., the ratio of the HMCVT input speed to the output speed when working at stage HM1. When clutch CL2 and brake B work, HMCVT is in the working mode of stage HM2. We can infer the transmission characteristic of stage HM2 as follows:
iHM2 =
(1 + k1)(1 + k 4 ) i3 i 4
(1 + k )(1 + k ) 1
2
(
k2 k1
i1 i2
)
i1 i2
)
(2)
where iHM2 is the transmission ratio in stage HM2, i.e., the ratio of the HMCVT input speed to the output speed when working at stage HM2. When clutch CL2 and clutch CL3 work, the HMCVT is in the working mode of stage HM3. We can infer the transmission characteristic of stage HM3 as follows:
iHM3 =
(1 + k1) i3 i 4 k1 i i 12
(4)
The efficiency characteristic is one of the important features of the HMCVT. The efficiency characteristic research of the HMCVT is the key to obtaining the optimal transmission ratio of variable speed, making variable speed control strategies, and assessing the transmission efficiency level of the designed HMCVT. A certain model of HMCVT is different from most single planetary line confluence mechanisms in the market. It generally requires the joint action of multiple planetary line mechanisms to work, so analyzing its transmission efficiency is a very complicated task. In addition, this paper aims to offer a theoretical basis for the optimization design of transmission parameters in the research, development, and design stages of HMCVT. Because the variable-pumpconstant-displacement-motor system is also a complicated system, this paper does not focus on it. In addition, the efficiency of the system generally changes in a limited interval. Therefore, this paper only approximately evaluates the transmission efficiency of a certain model of HMCVT. For transmission systems that are similar to a certain model of HMCVT that requires the joint action of multiple planetary lines for transmission, this paper proposes a tandem computation method for transmission efficiency. The method considers each single planetary line as the analysis object, and computes the transmission efficiency of the energy of each planetary line from the input end to the output end. It then computes the ratio of power transmitted by each planetary line to the input end power of the overall transmission system. Finally, it computes and infers the output end power of the overall transmission system according to the transmission efficiency of each planetary line, and the ratio of the power transmitted by each planetary line to the overall input power. The ratio of the input end power to the output end power is the total transmission efficiency. First, the transmission efficiency of planetary line P1 is calculated.
k3 (1 + k1)(1 + k 4 ) i3 i4
)(k
(
k2 k1
2.2. The efficiency characteristic analysis of a certain model of HMCVT
Fig. 1 illustrates the transmission principles of a certain model of HMCVT. The power output from the engine is input into the gear ring and sun gear of planetary line P1 from the mechanical route and the hydraulic route. The two power sources converge, and then are output by the planet carrier of planetary line P1. The HMCVT is mainly composed of the variable-pump-constant-displacement-motor system, 4 planetary lines (P1, P2, P3, and P4, the brake B, and the clutches (CL1, CL2, CL3, CL4, kV, and KR). It can realize the continuously variable speed of 4 working stages (HM1, HM2, HM3, and HM4 with the help of different clutches. Clutches kV and KR can allow the vehicle to realize four forward and four backward driving states, respectively, and the forward and backward speeds are same under the same conditions. When clutch CL1 and brake B work, the HMCVT is in the working mode of stage HM1. According to the computational method of planetary gear transmission (Xiao et al., 2018), we can infer the transmission characteristic of stage HM1 as follows:
(
2
where iHM4 is the transmission ratio in stage HM4, i.e., the ratio of the HMCVT input speed to the output speed when working at stage HM4. According to the original parameter data of a certain model of HMCVT, the values of k1, k2, k3 , and are 3, 2, 3, and 3, respectively, and the values of i1, i2, i3 , and i 4 are all 1. Fig. 2 shows the transmission characteristics obtained through computation.
2.1. The transmission characteristic analysis of a certain model of HMCVT
1 + k2 + k3
(1 + k )(1 + k ) 1
2. The characteristic analysis of a certain model of HMCVT
iHM1 =
(1 + k1) i3 i4
(3) Fig. 1. Transmission principles of a certain model of HMCVT.
where iHM3 is the transmission ratio in stage HM3, i.e., the ratio of the 3
Computers and Electronics in Agriculture 167 (2019) 105034
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Supposing the ratio of the torque obtained by the planet carrier of planetary line P3 to the total input torque is q2 , then, after computation, we obtain:
q2 =
q1 (1 + k3) k1 1 + k2
(7)
i1 i2
In summation, the approximate estimation results of the transmission efficiencies of variable speed stages of a certain model of HMCVT are as follows:
Fig. 2. Transmission characteristics of a certain model of HMCVT.
= {1 + |iIc|[|icIb
icIb icI |
+ |icIa | m]}
1
(5)
1 0 bI ; 1 < 0 bI bI = i1 H i2; is the loss coefficient, and its value is 0.039 in the paper; i1 and i2 are the transmission efficiencies of the corresponding gear pairs; finally, H is the transmission efficiency of the pump-motor system. The paper chooses 0.98 as the value of the gear transmission efficiency. As mentioned above, the value of the transmission efficiency of the pump-motor system is generally a constant value, 0.75, for computation. Next, the transmission efficiencies of planetary lines P2 and P3 are computed. The two planetary lines both have two input ends and one output end, so we use the Kpeиˇhec method (transmission ratio method) to compute their transmission efficiencies (Jiao et al., 2012). Computation results find that the transmission efficiency of power from the gear ring to the sun gear of planetary line P2 is P2 = 0.96 , and the transmission efficiency from the planet carrier to the sun gear of planetary line P2 is P2 = 0.9731; the transmission efficiency of power from the planet carrier to the gear ring of planetary line P3 is P3 = 0.9897 , and the transmission efficiency from the planet carrier to the sun gear of planetary line P3 is P3 = 0.96 . The gear ring of planetary line P4 will be locked by the brake during work, so we only consider the power input of the sun gear to the planet carrier as the output. Using the Kpeиˇhec method to compute the transmission efficiency, we obtain P4 = 0.97 . Finally, the ratio of the power transmitted by each planetary line to the total input power is computed. The total input power is input into planetary line P1 and planetary line P2, respectively. Supposing the ratio of the torque input to planetary line P2 to the total input torque is q1, then, according to the planetary line torque computation theory (Liu et al., 2019), we obtain: q1 =
(
(1 + k1) i1 i2
; icIb =
k1 ;i 1 + k1 cI
= icIa + icIb ;iIc =
1 ;m icI
HM2
=
HM3
=
HM4
=
P1 P3
+ (1 + q1
q2)
P1 P2 ] P4
1 + 2q1 [(1 + q1)
P1 P2
+ q1
P2 ] P4
1 + 2q1
+ q1
P2 P3 P4
i3 i 4
i3 i 4
P1 P2
1 + 2q1
+ q1
P2
i3 i 4
=
)(k + ) (1 + k )(1 + k ) 1
i1 i2
1
2
(11)
Considering the use requirements of vehicles, the transmission and efficiency characteristics of the HMCVT, and the overlapping area setting for stage change, this paper puts forward a transmission parameter optimization analysis model for the HMCVT. The analysis object is the high-power tractor. A tractor’s maximum required driving speed is 40 km/h, the wheel radius is 0.8 m, the engine speed range is 750–2200 r/min, and the transmission ratio of the other transmission mechanism is 26.866. After computation, the minimum transmission ratio of the HMCVT is determined to be 0.62. HM1 is set as a starting stage, and primarily serves as the starting stage of the tractor. HM2 is set as a transition stage or auxiliary stage, and mainly meets some of the low-speed operation requirements of the
(1 + k1)(1 + k2 )
1 + k2 + k3
(9) (10)
P1 i3 i 4
(1 + q1)
(8)
3. Build a transmission parameter optimization analysis model for HMCVT
1
where icIa =
=
where i3 and i 4 are the transmission efficiencies of gear pairs corresponding to i3 and i 4 , respectively. According to the original parameter data of a certain model of HMCVT, the values of k1, k2, k3 , and are 3, 2, 3, and 3, respectively, and the values of i1, i2, i3 , and i 4 are all 1. Fig. 3 shows the efficiency characteristics obtained through computation, and indicates that when the displacement ratio is 0, only the mechanical components transmit the power, and the HMCVT has the highest transmission efficiency. The highest efficiencies of stages HM1, HM2, HM3, and HM4 are 0.9234, 0.8940, 0.9511, and 0.9217, respectively.
Planetary line P1 couples the two parts of the power output by the engine with the hydraulic and mechanical parts, and outputs the power from its planet carrier so that it involves the variable-pump-constantdisplacement-motor system and the gear transmission pairs corresponding to i1 and i2 . For this kind of structure, this paper computes the transmission efficiency of planetary line P1 using the engaging power method (Li et al., 2017). The following is the computation formula: P1
[q2
HM1
Fig. 3. Efficiency characteristics of a certain model of HMCVT.
(6) 4
Computers and Electronics in Agriculture 167 (2019) 105034
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tractor, such as rotary tillage or sowing. HM3 is set to be a work stage to meet the medium/high-speed operation requirements of the tractor, such as ploughing, intertilling, and field transportation. Finally, HM4 is set to be a transportation stage, and allows the tractor to run at a high velocity on the road. Researchers often use the arithmetic or geometric transmission scheme to determine the HMCVT transmission parameter. The geometric transmission scheme can better meet the requirements of highpower tractors, and is characterized by its rotating speed, which increases slowly at the low-speed stage and then quickly at the high-speed stage. However, to avoid frequent stage changes, improve transmission efficiency, and set stages more properly, this paper improves the geometric transmission scheme and proposes a non-geometric transmission scheme. Stage HM3 is a work stage. Fig. 3 shows that stage HM3 has an interval with a low transmission efficiency, but also has the interval with the highest transmission efficiency. Therefore, properly expanding the variable speed range of stage HM3 and using the interval with the highest transmission efficiency can effectively meet the working requirements of the tractor, avoid frequent stage changes, and improve transmission efficiency. Table 1 presents the transmission ratios of the stages of HMCVT that were set after comprehensive consideration. From Table 1, it is evident that all stages satisfy the geometric transmission scheme, except for stage HM3. Only stage HM3 properly increases its variable speed range. It is necessary to introduce the transmission efficiency of the HMCVT into the optimization objective function. To more accurately and practically evaluate the effect of the transmission parameter optimization design on the transmission efficiency of the HMCVT, this paper uses the integral method for the transmission efficiencies of the stages, and considers the integral value of transmission efficiency as the basis for parameter optimization. In addition, this paper determines the weight of the transmission efficiency integral value of each stage according to the expected service time of the stage. Suppose the service time ratios (the weight of the transmission efficiency integral value of each stage) of stages HM1, HM2, HM3, and HM4 are 10%, 30%, 40%, and 20%, respectively. There should also be junction points of the stages to ensure the overlapping area for stage change, thus allowing the continuous changes between stages, the reasonable use of high-transmission-efficiency intervals in the stages, and the development of stage changing strategies in the subsequent study. Given the above, the objective function in the paper is
min
fun _2(X ) = b1
d + b3
1 + iHM2( iHM1( = 1)
= j1 )
= j2 )
3.09 + iHM3( = j2 )
3.09 + iHM3(
+ iHM4(
= j3 )
1.23 + iHM3(
0.62
= 1)
= j3 )
HM3
j2 = j1
d + b4
100% 1 = j3
HM2
100%
HM4
d
(15)
where a is the relative importance ratio coefficient of fun_1 and fun_2 set with the empirical method, of which the value is 0.5 in this paper, meaning that fun_1 and fun_2 are equal in importance;
a2 = 1
a
best _fun _1 best _fun _2
(16)
where best _fun _1 is the optimal solution value obtained when considering only fun_1 as the objective function, and best _fun _2 is the optimal solution value obtained when considering only fun_2 as the objective function. 4. Improvements of GA The GA is a heuristic intelligent optimization algorithm that is extensively applied. It uses the natural selection and DNA genetic mechanisms of the biological field, and is applicable to complicated, nonlinear, and multi-objective optimization problems (Kr. Jha and Eyong, 2018; Rakesh et al., 2018; Korchuganov et al., 2018; Sun et al., 2018). The HMCVT transmission parameter optimization analysis in this paper is complicated, and there are as many as 11 variables to be optimized. Therefore, to improve the speed, accuracy, and practicability of the GA, we make the following improvements: (1) Screening the super individual. “Prematurity” is a common problem in the operation process of the GA. In essence, the problem is that some super individuals appear in the initial iteration of the GA, and their fitness values are much larger than the mean individual fitness value in the population. Therefore, the optimization process of the algorithm is trapped in the local optimal solution. To solve the “prematurity” problem, this paper proposes a new, improved method. According to the previous analysis on the essence of “prematurity,” this paper tries to modify the generation process of the initial population from the source while guaranteeing population diversity. The specific modification method for the generation of the initial population is as follows: Step 1. To guarantee the diversity of population, the initial
6.17
+ iHM2(
100%
d + b2
a1 = a
fun _1(X ) 6.17 +
j2 = j3
HM1
In Eq. (14), the values of b1, b2 , b3, and b4 are equal to the time ratios of stages HM1, HM2, HM3, and HM4, respectively, as mentioned previously. In this paper, the values of b1, b2 , b3, and b4 are 10%, 30%, 40%, and 20%, respectively. This paper uses the minimum weighted transmission efficiency loss integral value to represent the maximum total transmission efficiency of the HMCVT. The weight coefficients and a2 are set to ensure the consistent order of magnitudes of fun_1 and fun_2 while meeting the relative importance requirement of fun_1 and fun_2 , respectively. We propose the following method to determine the weight coefficients and a2 :
where X is the decision variable and X = [k1, k2, k3, k 4, i1, i2 , i3, i4 , j1 , j2 , j3 ], in which j1 , j2 , and j3 are the values of the displacement ratios corresponding to the junction points between stage HM1 and stage HM2, between stage HM2 and stage HM3, and between stage HM3 and stage HM4, respectively; fun_1 is the sum of errors of the transmission ratios of key positions in the stages, as shown in Eq. (13); fun_2 is the integral value of the efficiency error; finally, and a2 are weight coefficients.
= iHM1( = j1 )
100%
(14)
(12)
f (X ) = a1 fun _1(X ) + a2 fun _2 (X )
1 = j1
Table 1 Setting of the transmission ratios of the stages of the HMCVT.
1.23
(13)
In Eq. (13), the value of the constant comes from Table 1, and i HMn1( = n2) is the value of the transmission ratio when the displacement ratio is n2 in the n1 th variable speed stage. 5
Stage
Vehicle Speed (km/h)
Transmission Ratio
HM1 HM2 HM3 HM4
0–4 4–8 8–20 20–40
6.17–+ 3.09–6.17 1.23–3.09 0.62–1.23
Computers and Electronics in Agriculture 167 (2019) 105034
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population is still generated randomly. Step 2. Judge whether there is any super individual in the initial population based on the criterion:
Fi
c1
NIND j=1
=
(17)
where Fi is the fitness value of the ith individual; c1 is a constant coefficient, of which the value in this paper is 0.05; NIND is the number of individuals in the population. Step 3. If the judgment result in Step 2 is that there are super individuals, replace the N1 individuals with the poorest fitness value. To guarantee the diversity of the initial population in the replacement, the replacing individuals should still follow the principles of random generation. The following is the computation formula of N1:
c1 N1 =
1 NIND 2
c1
px = 0.2037exp( ) + 0.3963
(21)
pm1 = 0.0058exp( ) + 0.0042
(22)
The two above equations meet the requirement that the bigger is, the faster the crossover or mutation probabilities increase. As the number of iterations increases, it is necessary to properly increase the mutation probability to improve the local search ability, and get out of the local optimal solution. In this case, the computation formula of mutation probability pm2 is as follows:
Fi NIND
NIND j = 1 Fj
(18)
pm2 = Step 4. Repeat Step 2 and Step 3 until there is no super individual. (2) Adaptive changes of the population size. The GA and other intelligent optimization algorithms are not sensitive to the original parameter values, but the population size (i.e., the number of individuals) plays a very important role as an initial parameter for the iteration of the algorithm. If the population size is large, super individuals will appear prematurely in the algorithm, resulting in the slow execution speed of each generation of the algorithm; this is contrary to the essence of evolution of the GA. If the population size is small, the algorithm may require many generations to be convergent to the optimal solution, thus increasing the iterative time and opposing the convergence to the optimal solution. To solve this problem, this paper proposes a population size adaptive change method in which the population size is changed quantitatively when the algorithm completes ten iterations. Supposing the value changing is N2 and the mean reducing rate of ten generations is F , the computation formula is then as follows: 1 NIND 2
N2 =
1 NIND 2
1
1 F 3
F
1 3F
NIND
1 3
f x , y = 0.5 +
(23)
sin2 x 2 + y 2
0.5
[1 + 0.001(x 2 + y 2 )]2
(24)
Schaffer F6 has only the global minimum f (0, 0) = 0 in the definitional domain (x [ 100, 100], y [ 100, 100]) . Fig. 5 shows the computation results when the initial population is set as 200. Fig. 5 shows that the standard GA easily exhibits the prematurity phenomenon and cannot be convergent to the optimal solution when the initial population size is small. The computation results of the standard GA are x = 1.94 and y = 2.47 , and the optimal solution is 0.0097. In contrast, the improved GA obtains the optimal solution in the 49th generation.
F <1
iter 0.02 I
where iter is the current number of iterations, and I is the total number of iterations. In conclusion, the mutation probability pm is the sum of pm1 and pm2 . Besides the gene variation of the individual, the mutation phenomenon also directly manifests as individual mutation. This paper introduces the concept of the direct mutation of an individual, and randomly regenerates the 10% lower-ranking individuals in terms of fitness in each generation with a probability of 0.05. Fig. 4 presents the improved GA flow chart. To verify the improved results of the GA, this paper uses the most representative Schaffer F6 test function for the test.
1 1 3
(20)
where N3 is the number of individuals in the stable-state area. In this paper, when varies in the range of 0–1, the variation ranges of the corresponding crossover probability px and mutation probability pm1 are 0.6–0.95 and 0.01–0.03, respectively. The respective computation formulas of crossover probability px and mutation probability pm1 are as follows:
Fj
NIND
N3 NIND
(19)
The population size is changed every ten generations. When N2 is negative, the evolution rate is too fast, in which case the population size should be reduced according to the evolution rate. Remove the |N2| individuals with the poorest fitness. When N2 is positive, the evolution rate is slow, in which case the population size should be enlarged according to the evolution rate. To guarantee the diversity of the population, we randomly generate N2 individuals and screen them for super individuals as mentioned previously. (3) Adaptive changes of the crossover and mutation probabilities. The stable-state area refers to the area near the mean fitness value. Empirically, if the offspring’s fitness values are not obviously distributed near the stable-state area, it is necessary to reduce the crossover and mutation probabilities. If the offspring’s fitness values are obviously distributed near the stable-state area, it is necessary to increase the crossover and mutation probabilities.
5. HMCVT transmission parameter optimization results and analysis According to the proposed HMCVT transmission parameter optimization method, this section conducts an analysis with a certain model of HMCVT as the optimization object. As mentioned previously, the optimization is comprised of four aspects: the continuous change of the transmission ratio, the transmission range matching the driving and working requirements of the tractor, the maximization of transmission efficiency, and the proper position setting of the junction points of the stages. This paper uses the improved GA for the optimization process, and sets the initial population size as 1000 and the maximum number of iterations as 100. To better demonstrate the effect of the improved GA, we also make an optimization analysis with the standard GA under the same conditions. Fig. 6 displays the iteration curves. Fig. 6 shows that the proposed improved GA has a higher optimization accuracy and can better avoid “prematurity.” Compared with the standard GA, the improved GA has found an optimal solution in the initial iteration period, and its descent rate in the optimal solution
Given the above, this paper defines the stable-state area to be within ± 30% of favg , the mean fitness value of the population. Supposing is the population density in the stable-state area, then its computation formula is as follows: 6
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Fig. 5. The iterative evolution curves when the initial population is 200.
Fig. 6. A certain model of the transmission parameter optimization iteration curves of the HMCVT. Table 2 Transmission parameter optimization results of a certain model of HMCVT. Parameter
Value
k1 k2 k3 k4 i1 i2 i3 i4
3.12 1.72 3.99 3.57 0.63 1.10 0.69 1.95
improved GA can get out of the local optimal solution, even in the 60th and 80th generations, and continuously improve the optimization result. Table 2 lists the transmission parameter optimization results of a certain model of HMCVT. Fig. 7 shows the transmission characteristics of the HMCVT after the optimization design. As shown in Fig. 7, the junction point of stages HM1 and HM2 is in the position where the displacement ratio is −0.69. The junction point
Fig. 4. The flow chart of the proposed improved GA.
research process is faster. In addition, at the end of iteration, the algorithm can still continuously improve the optimization result because of the change of population size, the adaptive change of gene mutation probability, and individual mutation. The partially enlarged detail in Fig. 5 shows that the standard GA is convergent to the local optimal solution and stops the evolution process. However, the proposed 7
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Before optimization, the weighted integral value of the transmission efficiency of the HMCVT is 0.3226, and it changes to 0.2596 after optimization, exhibiting an improvement of 19.53%. 6. Conclusions (1) According to the complex confluence mechanism running jointly with 4 planetary lines, this paper proposes a rapid estimation method for the transmission efficiency of the HMCVT. The method, using a tandem computation technique, combines the engaging power method and the Kpeиˇhec method. The calculation model requires only the transmission parameters of the HMCVT and the transmission efficiency of the pump-motor system to obtain the total transmission efficiency of the HMCVT. The results show that when the displacement ratio is 0, the HMCVT has the highest transmission efficiency, and the highest efficiencies of stages HM1, HM2, HM3, and HM4 are 0.9234, 0.8940, 0.9511, and 0.9217, respectively. (2) This paper proposes a new multi-objective, non-geometric HMCVT transmission parameter optimization design method by taking into account the transmission characteristic continuity, the working requirements of vehicles, transmission efficiency maximization, and the proper positions of the junction points of stages. The new objective function combines the HMCVT requirements of transmission characteristic continuity, speed range, and transmission efficiency maximization. The results show that after optimization, the transmission characteristics of the HMCVT change continuously with no discontinuity point, and the average error between the actual value and the theoretical design value of the transmission ratio change (the range of the transmission ratio change is 0.62̃6.17) is just 0.044. The weighted integral value of the transmission efficiency of the HMCVT improves by 19.53% after optimization. The total transmission efficiency improves significantly. (3) To solve the defects of the GA, this paper proposes a new, improved GA. The improvements are in the following four aspects: super individual screening, population size adaptability, and the adaptability of crossover and mutation probabilities. The research results show that for test function Schaffer F6, the improved GA has converged to the globally optimal solution in the 49th generation, while the standard GA with a prematurity phenomenon fails to converge to the globally optimal solution. The improved GA with a higher convergence rate and optimization accuracy improves the parameter optimization result of the HMCVT by 35.74% compared with the standard GA.
Fig. 7. Transmission characteristics of the HMCVT after optimization.
Fig. 8. A comparison of transmission efficiency before and after optimization.
Acknowledgement
Table 3 Comparison of integral values of transmission efficiency losses of the stages of HMVT before and after optimization.
HM1 HM2 HM3 HM4
Integral value of transmission efficiency loss before optimization
Integral value of transmission efficiency loss after optimization
Improvement after optimization (%)
0.2894 0.3357 0.3400 0.2849
0.2370 0.2754 0.2592 0.2480
18.11% 17.96% 23.76% 12.95%
This research was supported by the National Key Research and Development Plan (2016YFD0701103), Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_0647). References Cheng, G.W., Zhou, Z.L., Men, Q.Y., Deng, C.N., 2006. The Study on the measurement & testing technology of the HMCVT hydraulic pressure based on the data fusion technology. J. Phys. Conf. Ser. 48 (1), 1289–1294. Du, J.Y., Yuan, S.H., Wei, C., Zou, Y.F., 2009. Performance analysis of dual mode hydromechanical transmission. T. Chinese Soc. Agr. En. 25 (4), 86–90. Fang, S.S., Lu, Z.X., Wang, Z.C., Yong, D.X., Lu, Y., Gong, J.H., Zhu, C.Y., 2017. Design and prototype performance experiments of steering-by-wire hydraulic pressure system of tractor. T. Chinese Soc. Agr. En. 33 (10), 86–93. Gao, X., Zhu, Z., Zhu, Y., 2013. Analysis of transmission characteristics of HMCVT.J. Chongqing Jiao Ton. 32 (4), 715–716 724. Jiao, W.M., Ma, F., Yang, Y.D., Zhao, X.X., 2012. Calculation of transmission ratio and efficiency of planetary gear box. J. Mech. Transm. 36 (12), 114–116. Korchuganov, V.N., Smygacheva, A.S., Fomin, Ye.A., 2018. Multiobjective optimization of the synchrotron radiation source Siberia-2 lattice using a genetic algorithm. Phys. Part. Nucleilet. 15 (1), 92–96. Kr. Jha, S., Eyong, E.M., 2018. An energy optimization in wireless sensor networks by
of stages HM2 and HM3 in the position where the displacement ratio is 0.90. The junction point of stages HM3 and HM4 is in the position where the displacement ratio is −0.52. The transmission characteristics meet the requirement designed in Table 1, and the mean error is 0.044. Fig. 8 provides a comparison of transmission efficiencies of all working stages of the HMCVT of a model before and after optimization. Table 3 presents the integral values of transmission efficiency losses of the stages of the HMVT before and after optimization. 8
Computers and Electronics in Agriculture 167 (2019) 105034
Z. Cheng, et al. using genetic algorithm. Telecommun. Syst. 67 (1), 113–121. Li, J.L., Liu, L.T., Xiao, M.H., Wang, T.T., Wang, X., Zhang, H.J., 2017. Research on dynamic characteristics of hydro-mechanical continuously variable transmission. J. Mech. Strength 39 (1), 14–19. Liu, B., Yuan, S.H., Guo, X.L., 2006. Parametric optimization based on the study of hydromechanical continuously variable transmission of arithmetic type. J. Mech. Transm. 30 (6), 47–48 69. Liu, M.N., Zhou, Z.L., Xu, L.Y., Zhao, J.H., Yan, X.H., 2017. Multi-objective optimization and design of tractor trailer systems. T. Chinese Soc. Agr. En. 33 (8), 62–68. Liu, F., Wu, W., Hu, J., Yuan, S., 2019. Design of multi-range hydro-mechanical transmission using modular method. Mech. Syst. Signal. Pr. 126, 1–20. Raikwara, S., Tewaria, V.K., Mukhopadhyayb, S., Verma, C.R.B., Rao, M.S., 2015. Simulation of components of a power shuttle transmission system for an agricultural tractor. Comput. Electron. Agr. 114, 114–124. Rakesh, V., Sharma, U., Murugan, S., Venugopal, S., Asokan, T., 2018. Optimizing force closure grasps on 3D objects using a modified genetic algorithm. Soft Comput. 22 (3), 759–772. Sun, Y.J., Zhang, X.F., Jian, M., Wang, S.K., Wu, Z.J., Su, Q.T., 2018. Beijing Chen. An improved genetic algorithm for three-dimensional reconstruction from a single uniform texture image. Soft Comput. 22 (2), 477–486. Xiao, D., 2016. Design the Hydro-mechanical Continuously Variable Transmission of Three-stage Tractors Based on High-power. Nanjing Agriculture University, Nanjing, China. Xiao, M., Zhao, J., Wang, Y., Zhang, H., Lu, Z., Wei, W., 2018. Fuel economy of multiple conditions self-adaptive tractors with hydro-mechanical CVT. Int. J. Agric. Biol. Eng. 11 (3), 102–109. Xu, L.Y., 2007. Study on Characteristics of Hydro-mechanical Continuously Variable
Transmission of Tractor. Xi’an University of Technology, Xi’an, China. Xue, Z.F., 2017. Modeling and simulation of hydro-mechanical continuously variable transmission based on simscape. Tractor Farm Transpo. 44 (2), 12–16. Xu, L.Y., Zhou, Z.L., Zhang, M.Z., Li, Y., 2006. Design of hydro-mechanical continuously variable transmission of tractor. T. Chinese Soc. Agr. Ma. 37 (7), 5–8. Zhang, S., 2016. Study on the Large Tractor HMCVT Design and Finite Element Analysis. Nanjing Agriculture University, Nanjing, China. Zhang, P., 2017. Design and Characteristic Analysis of Hydro-mechanical Continuous Variable Transmission. Shihezi University, Shihezi, China. Zhang, M.Z., Zhou, Z.L., 2014. Modeling and control simulation for the multi-range hydro-mechanical CVT. Key. Eng. Mat. 621, 462–469. Zhang, M.Z., Wang, Q.S., Bai, D.Y., Xin, Y.Y., Hao, X.Y., 2016. Speed changing law of hydro-mechanical CVT based on maximum efficiency of tractors. T. Chinese Soc. Agr. En. 32 (21), 74–78. Zhang, P.C., Ni, X.D., Mei, W.J., Peng, X.R., 2017. Design and characteristic analysis of hydro-mechanical continuous variable transmission of cotton picker. Machinery Design Man. 10, 64–66 70. Zhu, Z., 2016. Optimization Research on the Performance of Hydro-mechanical Continuously Variable Transmission. Jiangsu University, Zhenjiang, China. Zhu, Z., Gao, X., Pan, D.Y., Zhu, Y., 2015. Design and test analysis of HMCVT. Mach. Des. Res. 31. pp. 53–56 61. Zhu, Z., Gao, X., Cao, L.L., Cai, Y.M., Pan, D.Y., 2016a. Research on the shift strategy of HMCVT based on the physical parameters and shift time. Appl. Math. Model. 40 (15–16), 6889–6907. Zhu, Z., Gao, X., Pan, D.Y., Zhu, Y., Cao, L.L., 2016b. Study on the control strategy of shifting time involving multigroup clutches. Math. Probl. Eng. 2016, 17 Article ID 9523251.
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Computers and Electronics in Agriculture journal homepage: www.elsevier.com/locate/compag
A new non-geometric transmission parameter optimization design method for HMCVT based on improved GA and maximum transmission efficiency
T
Zhun Cheng, Zhixiong Lu , Jin Qian ⁎
College of Engineering, Nanjing Agricultural University, Nanjing 210031, China
ARTICLE INFO
ABSTRACT
Keywords: HMCVT Improved GA Transmission parameters Optimization design Work efficiency Transmission characteristic
The HMCVT (Hydro-Mechanical Continuous Variable Transmission) can realize continuously variable speed in a large range and transmit a large load torque, and thus it is very applicable to engineering vehicles, such as the tractor. This paper proposes a new transmission parameter optimization design method for HMCVT in order to enhance the practicability of the HMCVT, improve its transmission efficiency, realize the fast determination of HMCVT transmission parameters, and allow HMCVT to meet the work requirements of engineering vehicles like the tractor. Specifically, this paper first conducts a characteristic analysis of a certain model of HMCVT. Then, according to the operation requirements of the tractor and the driving characteristic continuity of the HMCVT, the improved design of the transmission scheme, the maximization of working efficiency, and the most proper positions of junction points of variable speed stages, this paper puts forward an HMCVT transmission parameter optimization design model. This paper uses the GA (genetic algorithm) to implement the transmission parameter optimization, and improves the GA to obtain a higher algorithm running speed and accuracy. The improvement in GA primarily includes avoiding the super individual in the initial population, the adaptive changes of the population size in the iteration process, and the adaptive changes of crossover probability and mutation probability. The research results demonstrate that the improved GA has a faster rate of convergence and better optimization accuracy. The new proposed optimization design method can design HMCVT transmission parameters flexibly, efficiently, and quickly, as expected. The HMCVT with optimized transmission parameters exhibits driving characteristics that meet the working requirements of the tractor, significantly improves working efficiency, and offers theoretical guidance and a basis of design for the determination of the proper positions of junction points of variable speed stages.
1. Introduction Engineering or agricultural vehicles, such as the tractor, face complicated working conditions. For example, the tractor faces both the medium/low-speed field operation requirements of farming and seeding, and the medium/high-speed transportation driving requirements in the field or on good roads. In addition, the tractor bears different loads in different working conditions (Fang et al., 2017; Raikwara et al., 2015; Liu et al., 2017). In this case, it is hard to ensure the economic and power performance of the tractor in operation. More gears can meet the economic and power requirements of the tractor for work. The HMCVT (Hydro-Mechanical Continuous Variable Transmission) can realize the continuously variable speed within a large range, and also inherits the high efficiency of a mechanical transmission and the high power of a hydraulic transmission (Zhang and Zhou, 2014; Zhu et al., 2016a,b). As an increasing number of high-power vehicles comes
⁎
into service, the research on HMCVT has great significance (Xue, 2017; Du et al., 2009; Cheng et al., 2006; Zhang et al., 2016). The transmission parameter design of the HMCVT is the primary and key content in the research on the HMCVT. Vehicle working or driving conditions have certain requirements for the speed changing range of the HMCVT (Gao et al., 2013). The multi-stage HMCVT requires a junction point between the front and back variable speed stages to ensure the continuity of the transmission ratio during stage shifting, which reduces the impact of stage shifting and improves ride comfort and the service life of parts. A good design of the HMCVT transmission parameter can meet these requirements. In addition, the transmission efficiency of the HMCVT is closely linked to its transmission parameter design. Currently, there is little research on HMCVT, especially its transmission parameter design. Xu et al. (2006) expounded the single-line planetary mechanism of the mechanical part and the selection method
Corresponding author at: College of Engineering, Nanjing Agricultural University, 40 Dianjiangtai Road, Nanjing 210031, China. E-mail address: [email protected] (Z. Lu).
https://doi.org/10.1016/j.compag.2019.105034 Received 29 March 2018; Received in revised form 8 July 2019; Accepted 30 September 2019 0168-1699/ © 2019 Elsevier B.V. All rights reserved.
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Nomenclature
k1 k2 k3 i1 i2 i3 i4
iHM1 iHM2 iHM3 iHM4 P1
i1 i2
H P2
P2 P3
P3
P4
q1
q2 i3 i4
j1
of HM1 and HM2 discharge ratio value corresponding to the junction point of HM2 and HM3 j3 discharge ratio value corresponding to the junction point of HM3 and HM4 fun_1 sum of errors of transmission ratios of key positions in different stages fun_2 integral value of efficiency error i HMn1( = n2) transmission ratio value in speed change stage n1 when discharge ratio is n2 weight coefficient of fun_1 a2 weight coefficient of fun_2 b1 the proportion of time used by stage HM1 b2 the proportion of time used by stage HM2 b3 the proportion of time used by stage HM3 b4 the proportion of time used by stage HM4 a ratio coefficient of relative importance of fun_1 and fun_2 set with the empirical method best _fun _1 optimal solution value obtained when considering fun_1 alone as the objective function best _fun _2 optimal solution value obtained when considering fun_2 alone as the objective function Fi fitness of the ith individual c1 constant in calculation formula NIND the number of individuals in the population N1 the number of individuals with the poorest fitness N2 changing value of population size mean reducing rate of every 10 generations in the operF ating process of the GA favg average fitness of the population the density of population in the steady-state area N3 the number of individuals in the steady-state area px crossover probability pm1 probability of the first kind of mutation pm2 probability of the second kind of mutation pm overall probability of mutation
j2
characteristic parameter of planetary line P1 characteristic parameter of planetary line P2 characteristic parameter of planetary line P3 characteristic parameter of planetary line P4 gear pair transmission ratio of i1 in Fig. 1 gear pair transmission ratio of i2 in Fig. 1 gear pair transmission ratio of i3 in Fig. 1 gear pair transmission ratio of i 4 in Fig. 1 displacement ratio of variable-pump-constant-displacement-motor system transmission ratio of stage HM1 transmission ratio of stage HM2 transmission ratio of stage HM3 transmission ratio of stage HM4 transmission ratio of planetary line P1 loss coefficient gear pair transmission efficiency of i1 gear pair transmission efficiency of i2 transmission efficiency of pump-motor system transmission efficiency from planetary line P2’s gear ring to sun gear transmission efficiency from planetary line P2’s planet carrier to sun gear transmission efficiency from planetary line P3’s planet carrier to gear ring transmission efficiency from planetary line P3’s planet carrier to sun gear planetary line P4’s transmission efficiency proportion of the torque input to planetary line P2 in total input torque proportion of the torque obtained by planetary line P3’s planet carrier in total input torque transmission efficiency of gear pair corresponding to i3 transmission efficiency of gear pair corresponding to i 4 discharge ratio value corresponding to the junction point
of the gear pair transmission ratio in the axle, and analyzed the governor control characteristic of the HMCVT. Xu (2007) also made an optimization design of the transmission parameters of the HMCVT based on the standard GA, and his objective function of the optimization analysis took into account the varying range of speed and the dividing ratio of hydraulic power of variable speed stages. Liu et al. (2006) optimized the arithmetic transmission parameters of the HMCVT using the penalty function, and the objective function used three parameters to approximate and substitute the transmission efficiency: the rotating speed of the planetary gear, the relative power coefficient of the planetary line, and the hydraulic transmission ratio. Zhu et al. (2015) conducted a kinetic and kinematics analysis on highpower tractors, and designed a single planetary line confluence HMCVT to compute and determine the gear ratio of the single planetary gear and gearshift. Zhang et al. (2017) designed the HMCVT used in cotton pickers, inferred the conditions for the continuous changes of transmission ratio via the principles of the arithmetic transmission scheme, and determined the transmission ratio of each gear pair. Most researchers, such as Zhang (2017), Zhu (2016), Zhang (2016), Xiao (2016), first determine that the HMCVT adopts a geometric or an arithmetic transmission scheme, then determine the possible value range of the transmission ratio of each transmission part according to the operating requirements of agricultural machinery, and finally determine transmission parameters with an empirical method. It is evident that the research on the transmission parameter design of the HMCVT currently has significant shortcomings. On the one hand,
HMCVT transmission parameters generally need to be designed on the basis of adopting an arithmetic or a geometric transmission scheme, and then determined via theoretical calculation using an empirical method. On the other hand, only a few researchers have used modern optimization design methods to conduct the optimization design of HMCVT transmission parameters. In addition, the objective function of the optimization design is characterized by four problems. First, it uses the hydraulic power dividing ratio or other parameters to approximate and substitute the transmission efficiency. Second, it computes the transmission efficiency only in the working conditions of the maximum or minimum displacement ratio, and fails to consider the transmission efficiency of all transmission ratios during the work of the HMCVT. Third, it requires a large speed varying range (the larger the better), but fails to consider the driving or working requirements of vehicles. Finally, it only connects the ends of the transmission ratios of variable speed stages to ensure the continuous transmission, but fails to consider the proper positions of the junction points of the front and back variable speed stages or reserve the overlapping area of stage change. In addition, there are generally several transmission parameters and targets to be optimized, increasing the difficulty in the iterative optimization of the algorithm. There may be a phenomenon of “prematurity” in the iteration process, causing the failure in convergence to the optimal solution, and reducing the optimization accuracy of the algorithm. To solve these problems, this paper proposes a new HMCVT parameter optimization design method. The method considers the nongeometric transmission scheme with the optimization goals of matching 2
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the HMCVT transmission characteristics and the working and driving demands, the continuity of the transmission characteristics, obtaining the highest weighted transmission efficiency during the work of the HMCVT, and determining the proper position setting of junction points of transmission stages. To build an HMCVT parameter optimization model, the transmission and efficiency characteristics of the HMCVT based on a certain model of the HMCVT are first researched. This paper then improves the GA in the following four aspects: avoiding the super individual in the initial population, allowing for the adaptive changes of population size in the iteration process, and allowing for the adaptive changes of crossover probability and mutation probability. An illustration of the advantages of the improved GA is also provided by obtaining the minimum of a function. The research results indicate that the proposed optimization method can result in a higher transmission efficiency of the HMCVT, as well as a better transmission characteristic and overlapping area of stage change. The proposed improved GA can effectively enhance the convergence rate and optimization accuracy, and is more applicable to multi-target and multi-parameter optimization problems.
HMCVT input speed to the output speed when working at stage HM3. When clutch CL2 and clutch CL4 work, the HMCVT is in the working mode of stage HM4. We can infer the transmission characteristic of stage HM4 as follows:
iHM4 =
1
i1 i2
) (1 + k )(1 + k ) 1
2
(1)
where k1, k2, k3 , and are the characteristic parameters of planetary lines P1, P2, P3, and P4;i1, i2 , i3 , and i 4 are the transmission ratios of the gear pairs in corresponding positions in Figure 1; is the displacement ratio of the variable-pump-constant-displacement-motor system; finally, iHM1 is the transmission ratio in stage HM1, i.e., the ratio of the HMCVT input speed to the output speed when working at stage HM1. When clutch CL2 and brake B work, HMCVT is in the working mode of stage HM2. We can infer the transmission characteristic of stage HM2 as follows:
iHM2 =
(1 + k1)(1 + k 4 ) i3 i 4
(1 + k )(1 + k ) 1
2
(
k2 k1
i1 i2
)
i1 i2
)
(2)
where iHM2 is the transmission ratio in stage HM2, i.e., the ratio of the HMCVT input speed to the output speed when working at stage HM2. When clutch CL2 and clutch CL3 work, the HMCVT is in the working mode of stage HM3. We can infer the transmission characteristic of stage HM3 as follows:
iHM3 =
(1 + k1) i3 i 4 k1 i i 12
(4)
The efficiency characteristic is one of the important features of the HMCVT. The efficiency characteristic research of the HMCVT is the key to obtaining the optimal transmission ratio of variable speed, making variable speed control strategies, and assessing the transmission efficiency level of the designed HMCVT. A certain model of HMCVT is different from most single planetary line confluence mechanisms in the market. It generally requires the joint action of multiple planetary line mechanisms to work, so analyzing its transmission efficiency is a very complicated task. In addition, this paper aims to offer a theoretical basis for the optimization design of transmission parameters in the research, development, and design stages of HMCVT. Because the variable-pumpconstant-displacement-motor system is also a complicated system, this paper does not focus on it. In addition, the efficiency of the system generally changes in a limited interval. Therefore, this paper only approximately evaluates the transmission efficiency of a certain model of HMCVT. For transmission systems that are similar to a certain model of HMCVT that requires the joint action of multiple planetary lines for transmission, this paper proposes a tandem computation method for transmission efficiency. The method considers each single planetary line as the analysis object, and computes the transmission efficiency of the energy of each planetary line from the input end to the output end. It then computes the ratio of power transmitted by each planetary line to the input end power of the overall transmission system. Finally, it computes and infers the output end power of the overall transmission system according to the transmission efficiency of each planetary line, and the ratio of the power transmitted by each planetary line to the overall input power. The ratio of the input end power to the output end power is the total transmission efficiency. First, the transmission efficiency of planetary line P1 is calculated.
k3 (1 + k1)(1 + k 4 ) i3 i4
)(k
(
k2 k1
2.2. The efficiency characteristic analysis of a certain model of HMCVT
Fig. 1 illustrates the transmission principles of a certain model of HMCVT. The power output from the engine is input into the gear ring and sun gear of planetary line P1 from the mechanical route and the hydraulic route. The two power sources converge, and then are output by the planet carrier of planetary line P1. The HMCVT is mainly composed of the variable-pump-constant-displacement-motor system, 4 planetary lines (P1, P2, P3, and P4, the brake B, and the clutches (CL1, CL2, CL3, CL4, kV, and KR). It can realize the continuously variable speed of 4 working stages (HM1, HM2, HM3, and HM4 with the help of different clutches. Clutches kV and KR can allow the vehicle to realize four forward and four backward driving states, respectively, and the forward and backward speeds are same under the same conditions. When clutch CL1 and brake B work, the HMCVT is in the working mode of stage HM1. According to the computational method of planetary gear transmission (Xiao et al., 2018), we can infer the transmission characteristic of stage HM1 as follows:
(
2
where iHM4 is the transmission ratio in stage HM4, i.e., the ratio of the HMCVT input speed to the output speed when working at stage HM4. According to the original parameter data of a certain model of HMCVT, the values of k1, k2, k3 , and are 3, 2, 3, and 3, respectively, and the values of i1, i2, i3 , and i 4 are all 1. Fig. 2 shows the transmission characteristics obtained through computation.
2.1. The transmission characteristic analysis of a certain model of HMCVT
1 + k2 + k3
(1 + k )(1 + k ) 1
2. The characteristic analysis of a certain model of HMCVT
iHM1 =
(1 + k1) i3 i4
(3) Fig. 1. Transmission principles of a certain model of HMCVT.
where iHM3 is the transmission ratio in stage HM3, i.e., the ratio of the 3
Computers and Electronics in Agriculture 167 (2019) 105034
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Supposing the ratio of the torque obtained by the planet carrier of planetary line P3 to the total input torque is q2 , then, after computation, we obtain:
q2 =
q1 (1 + k3) k1 1 + k2
(7)
i1 i2
In summation, the approximate estimation results of the transmission efficiencies of variable speed stages of a certain model of HMCVT are as follows:
Fig. 2. Transmission characteristics of a certain model of HMCVT.
= {1 + |iIc|[|icIb
icIb icI |
+ |icIa | m]}
1
(5)
1 0 bI ; 1 < 0 bI bI = i1 H i2; is the loss coefficient, and its value is 0.039 in the paper; i1 and i2 are the transmission efficiencies of the corresponding gear pairs; finally, H is the transmission efficiency of the pump-motor system. The paper chooses 0.98 as the value of the gear transmission efficiency. As mentioned above, the value of the transmission efficiency of the pump-motor system is generally a constant value, 0.75, for computation. Next, the transmission efficiencies of planetary lines P2 and P3 are computed. The two planetary lines both have two input ends and one output end, so we use the Kpeиˇhec method (transmission ratio method) to compute their transmission efficiencies (Jiao et al., 2012). Computation results find that the transmission efficiency of power from the gear ring to the sun gear of planetary line P2 is P2 = 0.96 , and the transmission efficiency from the planet carrier to the sun gear of planetary line P2 is P2 = 0.9731; the transmission efficiency of power from the planet carrier to the gear ring of planetary line P3 is P3 = 0.9897 , and the transmission efficiency from the planet carrier to the sun gear of planetary line P3 is P3 = 0.96 . The gear ring of planetary line P4 will be locked by the brake during work, so we only consider the power input of the sun gear to the planet carrier as the output. Using the Kpeиˇhec method to compute the transmission efficiency, we obtain P4 = 0.97 . Finally, the ratio of the power transmitted by each planetary line to the total input power is computed. The total input power is input into planetary line P1 and planetary line P2, respectively. Supposing the ratio of the torque input to planetary line P2 to the total input torque is q1, then, according to the planetary line torque computation theory (Liu et al., 2019), we obtain: q1 =
(
(1 + k1) i1 i2
; icIb =
k1 ;i 1 + k1 cI
= icIa + icIb ;iIc =
1 ;m icI
HM2
=
HM3
=
HM4
=
P1 P3
+ (1 + q1
q2)
P1 P2 ] P4
1 + 2q1 [(1 + q1)
P1 P2
+ q1
P2 ] P4
1 + 2q1
+ q1
P2 P3 P4
i3 i 4
i3 i 4
P1 P2
1 + 2q1
+ q1
P2
i3 i 4
=
)(k + ) (1 + k )(1 + k ) 1
i1 i2
1
2
(11)
Considering the use requirements of vehicles, the transmission and efficiency characteristics of the HMCVT, and the overlapping area setting for stage change, this paper puts forward a transmission parameter optimization analysis model for the HMCVT. The analysis object is the high-power tractor. A tractor’s maximum required driving speed is 40 km/h, the wheel radius is 0.8 m, the engine speed range is 750–2200 r/min, and the transmission ratio of the other transmission mechanism is 26.866. After computation, the minimum transmission ratio of the HMCVT is determined to be 0.62. HM1 is set as a starting stage, and primarily serves as the starting stage of the tractor. HM2 is set as a transition stage or auxiliary stage, and mainly meets some of the low-speed operation requirements of the
(1 + k1)(1 + k2 )
1 + k2 + k3
(9) (10)
P1 i3 i 4
(1 + q1)
(8)
3. Build a transmission parameter optimization analysis model for HMCVT
1
where icIa =
=
where i3 and i 4 are the transmission efficiencies of gear pairs corresponding to i3 and i 4 , respectively. According to the original parameter data of a certain model of HMCVT, the values of k1, k2, k3 , and are 3, 2, 3, and 3, respectively, and the values of i1, i2, i3 , and i 4 are all 1. Fig. 3 shows the efficiency characteristics obtained through computation, and indicates that when the displacement ratio is 0, only the mechanical components transmit the power, and the HMCVT has the highest transmission efficiency. The highest efficiencies of stages HM1, HM2, HM3, and HM4 are 0.9234, 0.8940, 0.9511, and 0.9217, respectively.
Planetary line P1 couples the two parts of the power output by the engine with the hydraulic and mechanical parts, and outputs the power from its planet carrier so that it involves the variable-pump-constantdisplacement-motor system and the gear transmission pairs corresponding to i1 and i2 . For this kind of structure, this paper computes the transmission efficiency of planetary line P1 using the engaging power method (Li et al., 2017). The following is the computation formula: P1
[q2
HM1
Fig. 3. Efficiency characteristics of a certain model of HMCVT.
(6) 4
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tractor, such as rotary tillage or sowing. HM3 is set to be a work stage to meet the medium/high-speed operation requirements of the tractor, such as ploughing, intertilling, and field transportation. Finally, HM4 is set to be a transportation stage, and allows the tractor to run at a high velocity on the road. Researchers often use the arithmetic or geometric transmission scheme to determine the HMCVT transmission parameter. The geometric transmission scheme can better meet the requirements of highpower tractors, and is characterized by its rotating speed, which increases slowly at the low-speed stage and then quickly at the high-speed stage. However, to avoid frequent stage changes, improve transmission efficiency, and set stages more properly, this paper improves the geometric transmission scheme and proposes a non-geometric transmission scheme. Stage HM3 is a work stage. Fig. 3 shows that stage HM3 has an interval with a low transmission efficiency, but also has the interval with the highest transmission efficiency. Therefore, properly expanding the variable speed range of stage HM3 and using the interval with the highest transmission efficiency can effectively meet the working requirements of the tractor, avoid frequent stage changes, and improve transmission efficiency. Table 1 presents the transmission ratios of the stages of HMCVT that were set after comprehensive consideration. From Table 1, it is evident that all stages satisfy the geometric transmission scheme, except for stage HM3. Only stage HM3 properly increases its variable speed range. It is necessary to introduce the transmission efficiency of the HMCVT into the optimization objective function. To more accurately and practically evaluate the effect of the transmission parameter optimization design on the transmission efficiency of the HMCVT, this paper uses the integral method for the transmission efficiencies of the stages, and considers the integral value of transmission efficiency as the basis for parameter optimization. In addition, this paper determines the weight of the transmission efficiency integral value of each stage according to the expected service time of the stage. Suppose the service time ratios (the weight of the transmission efficiency integral value of each stage) of stages HM1, HM2, HM3, and HM4 are 10%, 30%, 40%, and 20%, respectively. There should also be junction points of the stages to ensure the overlapping area for stage change, thus allowing the continuous changes between stages, the reasonable use of high-transmission-efficiency intervals in the stages, and the development of stage changing strategies in the subsequent study. Given the above, the objective function in the paper is
min
fun _2(X ) = b1
d + b3
1 + iHM2( iHM1( = 1)
= j1 )
= j2 )
3.09 + iHM3( = j2 )
3.09 + iHM3(
+ iHM4(
= j3 )
1.23 + iHM3(
0.62
= 1)
= j3 )
HM3
j2 = j1
d + b4
100% 1 = j3
HM2
100%
HM4
d
(15)
where a is the relative importance ratio coefficient of fun_1 and fun_2 set with the empirical method, of which the value is 0.5 in this paper, meaning that fun_1 and fun_2 are equal in importance;
a2 = 1
a
best _fun _1 best _fun _2
(16)
where best _fun _1 is the optimal solution value obtained when considering only fun_1 as the objective function, and best _fun _2 is the optimal solution value obtained when considering only fun_2 as the objective function. 4. Improvements of GA The GA is a heuristic intelligent optimization algorithm that is extensively applied. It uses the natural selection and DNA genetic mechanisms of the biological field, and is applicable to complicated, nonlinear, and multi-objective optimization problems (Kr. Jha and Eyong, 2018; Rakesh et al., 2018; Korchuganov et al., 2018; Sun et al., 2018). The HMCVT transmission parameter optimization analysis in this paper is complicated, and there are as many as 11 variables to be optimized. Therefore, to improve the speed, accuracy, and practicability of the GA, we make the following improvements: (1) Screening the super individual. “Prematurity” is a common problem in the operation process of the GA. In essence, the problem is that some super individuals appear in the initial iteration of the GA, and their fitness values are much larger than the mean individual fitness value in the population. Therefore, the optimization process of the algorithm is trapped in the local optimal solution. To solve the “prematurity” problem, this paper proposes a new, improved method. According to the previous analysis on the essence of “prematurity,” this paper tries to modify the generation process of the initial population from the source while guaranteeing population diversity. The specific modification method for the generation of the initial population is as follows: Step 1. To guarantee the diversity of population, the initial
6.17
+ iHM2(
100%
d + b2
a1 = a
fun _1(X ) 6.17 +
j2 = j3
HM1
In Eq. (14), the values of b1, b2 , b3, and b4 are equal to the time ratios of stages HM1, HM2, HM3, and HM4, respectively, as mentioned previously. In this paper, the values of b1, b2 , b3, and b4 are 10%, 30%, 40%, and 20%, respectively. This paper uses the minimum weighted transmission efficiency loss integral value to represent the maximum total transmission efficiency of the HMCVT. The weight coefficients and a2 are set to ensure the consistent order of magnitudes of fun_1 and fun_2 while meeting the relative importance requirement of fun_1 and fun_2 , respectively. We propose the following method to determine the weight coefficients and a2 :
where X is the decision variable and X = [k1, k2, k3, k 4, i1, i2 , i3, i4 , j1 , j2 , j3 ], in which j1 , j2 , and j3 are the values of the displacement ratios corresponding to the junction points between stage HM1 and stage HM2, between stage HM2 and stage HM3, and between stage HM3 and stage HM4, respectively; fun_1 is the sum of errors of the transmission ratios of key positions in the stages, as shown in Eq. (13); fun_2 is the integral value of the efficiency error; finally, and a2 are weight coefficients.
= iHM1( = j1 )
100%
(14)
(12)
f (X ) = a1 fun _1(X ) + a2 fun _2 (X )
1 = j1
Table 1 Setting of the transmission ratios of the stages of the HMCVT.
1.23
(13)
In Eq. (13), the value of the constant comes from Table 1, and i HMn1( = n2) is the value of the transmission ratio when the displacement ratio is n2 in the n1 th variable speed stage. 5
Stage
Vehicle Speed (km/h)
Transmission Ratio
HM1 HM2 HM3 HM4
0–4 4–8 8–20 20–40
6.17–+ 3.09–6.17 1.23–3.09 0.62–1.23
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population is still generated randomly. Step 2. Judge whether there is any super individual in the initial population based on the criterion:
Fi
c1
NIND j=1
=
(17)
where Fi is the fitness value of the ith individual; c1 is a constant coefficient, of which the value in this paper is 0.05; NIND is the number of individuals in the population. Step 3. If the judgment result in Step 2 is that there are super individuals, replace the N1 individuals with the poorest fitness value. To guarantee the diversity of the initial population in the replacement, the replacing individuals should still follow the principles of random generation. The following is the computation formula of N1:
c1 N1 =
1 NIND 2
c1
px = 0.2037exp( ) + 0.3963
(21)
pm1 = 0.0058exp( ) + 0.0042
(22)
The two above equations meet the requirement that the bigger is, the faster the crossover or mutation probabilities increase. As the number of iterations increases, it is necessary to properly increase the mutation probability to improve the local search ability, and get out of the local optimal solution. In this case, the computation formula of mutation probability pm2 is as follows:
Fi NIND
NIND j = 1 Fj
(18)
pm2 = Step 4. Repeat Step 2 and Step 3 until there is no super individual. (2) Adaptive changes of the population size. The GA and other intelligent optimization algorithms are not sensitive to the original parameter values, but the population size (i.e., the number of individuals) plays a very important role as an initial parameter for the iteration of the algorithm. If the population size is large, super individuals will appear prematurely in the algorithm, resulting in the slow execution speed of each generation of the algorithm; this is contrary to the essence of evolution of the GA. If the population size is small, the algorithm may require many generations to be convergent to the optimal solution, thus increasing the iterative time and opposing the convergence to the optimal solution. To solve this problem, this paper proposes a population size adaptive change method in which the population size is changed quantitatively when the algorithm completes ten iterations. Supposing the value changing is N2 and the mean reducing rate of ten generations is F , the computation formula is then as follows: 1 NIND 2
N2 =
1 NIND 2
1
1 F 3
F
1 3F
NIND
1 3
f x , y = 0.5 +
(23)
sin2 x 2 + y 2
0.5
[1 + 0.001(x 2 + y 2 )]2
(24)
Schaffer F6 has only the global minimum f (0, 0) = 0 in the definitional domain (x [ 100, 100], y [ 100, 100]) . Fig. 5 shows the computation results when the initial population is set as 200. Fig. 5 shows that the standard GA easily exhibits the prematurity phenomenon and cannot be convergent to the optimal solution when the initial population size is small. The computation results of the standard GA are x = 1.94 and y = 2.47 , and the optimal solution is 0.0097. In contrast, the improved GA obtains the optimal solution in the 49th generation.
F <1
iter 0.02 I
where iter is the current number of iterations, and I is the total number of iterations. In conclusion, the mutation probability pm is the sum of pm1 and pm2 . Besides the gene variation of the individual, the mutation phenomenon also directly manifests as individual mutation. This paper introduces the concept of the direct mutation of an individual, and randomly regenerates the 10% lower-ranking individuals in terms of fitness in each generation with a probability of 0.05. Fig. 4 presents the improved GA flow chart. To verify the improved results of the GA, this paper uses the most representative Schaffer F6 test function for the test.
1 1 3
(20)
where N3 is the number of individuals in the stable-state area. In this paper, when varies in the range of 0–1, the variation ranges of the corresponding crossover probability px and mutation probability pm1 are 0.6–0.95 and 0.01–0.03, respectively. The respective computation formulas of crossover probability px and mutation probability pm1 are as follows:
Fj
NIND
N3 NIND
(19)
The population size is changed every ten generations. When N2 is negative, the evolution rate is too fast, in which case the population size should be reduced according to the evolution rate. Remove the |N2| individuals with the poorest fitness. When N2 is positive, the evolution rate is slow, in which case the population size should be enlarged according to the evolution rate. To guarantee the diversity of the population, we randomly generate N2 individuals and screen them for super individuals as mentioned previously. (3) Adaptive changes of the crossover and mutation probabilities. The stable-state area refers to the area near the mean fitness value. Empirically, if the offspring’s fitness values are not obviously distributed near the stable-state area, it is necessary to reduce the crossover and mutation probabilities. If the offspring’s fitness values are obviously distributed near the stable-state area, it is necessary to increase the crossover and mutation probabilities.
5. HMCVT transmission parameter optimization results and analysis According to the proposed HMCVT transmission parameter optimization method, this section conducts an analysis with a certain model of HMCVT as the optimization object. As mentioned previously, the optimization is comprised of four aspects: the continuous change of the transmission ratio, the transmission range matching the driving and working requirements of the tractor, the maximization of transmission efficiency, and the proper position setting of the junction points of the stages. This paper uses the improved GA for the optimization process, and sets the initial population size as 1000 and the maximum number of iterations as 100. To better demonstrate the effect of the improved GA, we also make an optimization analysis with the standard GA under the same conditions. Fig. 6 displays the iteration curves. Fig. 6 shows that the proposed improved GA has a higher optimization accuracy and can better avoid “prematurity.” Compared with the standard GA, the improved GA has found an optimal solution in the initial iteration period, and its descent rate in the optimal solution
Given the above, this paper defines the stable-state area to be within ± 30% of favg , the mean fitness value of the population. Supposing is the population density in the stable-state area, then its computation formula is as follows: 6
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Fig. 5. The iterative evolution curves when the initial population is 200.
Fig. 6. A certain model of the transmission parameter optimization iteration curves of the HMCVT. Table 2 Transmission parameter optimization results of a certain model of HMCVT. Parameter
Value
k1 k2 k3 k4 i1 i2 i3 i4
3.12 1.72 3.99 3.57 0.63 1.10 0.69 1.95
improved GA can get out of the local optimal solution, even in the 60th and 80th generations, and continuously improve the optimization result. Table 2 lists the transmission parameter optimization results of a certain model of HMCVT. Fig. 7 shows the transmission characteristics of the HMCVT after the optimization design. As shown in Fig. 7, the junction point of stages HM1 and HM2 is in the position where the displacement ratio is −0.69. The junction point
Fig. 4. The flow chart of the proposed improved GA.
research process is faster. In addition, at the end of iteration, the algorithm can still continuously improve the optimization result because of the change of population size, the adaptive change of gene mutation probability, and individual mutation. The partially enlarged detail in Fig. 5 shows that the standard GA is convergent to the local optimal solution and stops the evolution process. However, the proposed 7
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Before optimization, the weighted integral value of the transmission efficiency of the HMCVT is 0.3226, and it changes to 0.2596 after optimization, exhibiting an improvement of 19.53%. 6. Conclusions (1) According to the complex confluence mechanism running jointly with 4 planetary lines, this paper proposes a rapid estimation method for the transmission efficiency of the HMCVT. The method, using a tandem computation technique, combines the engaging power method and the Kpeиˇhec method. The calculation model requires only the transmission parameters of the HMCVT and the transmission efficiency of the pump-motor system to obtain the total transmission efficiency of the HMCVT. The results show that when the displacement ratio is 0, the HMCVT has the highest transmission efficiency, and the highest efficiencies of stages HM1, HM2, HM3, and HM4 are 0.9234, 0.8940, 0.9511, and 0.9217, respectively. (2) This paper proposes a new multi-objective, non-geometric HMCVT transmission parameter optimization design method by taking into account the transmission characteristic continuity, the working requirements of vehicles, transmission efficiency maximization, and the proper positions of the junction points of stages. The new objective function combines the HMCVT requirements of transmission characteristic continuity, speed range, and transmission efficiency maximization. The results show that after optimization, the transmission characteristics of the HMCVT change continuously with no discontinuity point, and the average error between the actual value and the theoretical design value of the transmission ratio change (the range of the transmission ratio change is 0.62̃6.17) is just 0.044. The weighted integral value of the transmission efficiency of the HMCVT improves by 19.53% after optimization. The total transmission efficiency improves significantly. (3) To solve the defects of the GA, this paper proposes a new, improved GA. The improvements are in the following four aspects: super individual screening, population size adaptability, and the adaptability of crossover and mutation probabilities. The research results show that for test function Schaffer F6, the improved GA has converged to the globally optimal solution in the 49th generation, while the standard GA with a prematurity phenomenon fails to converge to the globally optimal solution. The improved GA with a higher convergence rate and optimization accuracy improves the parameter optimization result of the HMCVT by 35.74% compared with the standard GA.
Fig. 7. Transmission characteristics of the HMCVT after optimization.
Fig. 8. A comparison of transmission efficiency before and after optimization.
Acknowledgement
Table 3 Comparison of integral values of transmission efficiency losses of the stages of HMVT before and after optimization.
HM1 HM2 HM3 HM4
Integral value of transmission efficiency loss before optimization
Integral value of transmission efficiency loss after optimization
Improvement after optimization (%)
0.2894 0.3357 0.3400 0.2849
0.2370 0.2754 0.2592 0.2480
18.11% 17.96% 23.76% 12.95%
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