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Optimization of hydro-mechanical power split transmissions
Mechanism and Machine Theory 46 (2011) 1901–1919
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Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
Optimization of hydro-mechanical power split transmissions Alarico Macor, Antonio Rossetti ⁎ Department of Engineering and Management, Università di Padova, Stradella S. Nicola, 3-36100 Vicenza, Italy
a r t i c l e
i n f o
Article history: Received 24 November 2010 Received in revised form 6 May 2011 Accepted 14 July 2011 Available online 1 September 2011 Keywords: Hydro-mechanical transmission Power split transmission
a b s t r a c t The increasing attention to comfort, automation and drivability is pushing the driveline technology to ever complex solutions, such as power-shift or continuously variable transmissions. Between these, the hydro-mechanical solution seems promising for heavy duty vehicle, due to the reliability and the capability of transferring high power. However, the double energy conversion occurring in the hydraulic branch of the transmission could lower excessively the total efficiency, highlighting the needs for a careful design of the whole system. In this work, the design of a hydro-mechanical transmission is defined as an optimization problem in which the objective function is the average efficiency of transmission, while the design variables are the displacements of the two hydraulic machines and gear ratios of ordinary and planetary gears. The optimization problem is solved by a “direct search” algorithm based on the swarm method, which showed a good speed convergence and the ability to overcome local minima. The optimization design method will be applied to study the transmission of two vehicles: a 62 kW compact loader and a high power agricultural tractor. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The growing role of agricultural mechanization in worldwide food programs and the need to save fossil energy goes towards reducing fuel consumption and emissions of agricultural machinery. This compels to reconsider the architecture of the vehicle's propulsion system, addressing the designer toward more sophisticated and efficient solutions. The same trend holds for earthmoving and construction machinery. In recent years, the evolution of the transmissions of agricultural and working machines was addressed to increase comfort and drivability without penalty of efficiency and, possibly, costs. These objectives involve the use of continuous transmissions that allow the elimination of the shifting gears, which is sometimes exhausting, and keep the fuel consumption to a minimum. Current technology provides some solutions for continuous variable transmission (CVT): the hydrodynamic, the hydrostatic, the mechanical and the electrical. The hydrodynamic transmission (torque converter) is not suitable for these applications because its transmission ratio is not controllable by the user and it has a good efficiency only at high speeds. The hydrostatic transmission, on the contrary, can easily control the speed, but with a very low efficiency because of the double energy conversion occurring in it. The mechanical transmission, in its variants (metal chain, toroidal), is characterized by high efficiency, although sometimes with limits of power and speed range. This requires the use of a mechanical gearbox further downstream. Electric transmission, although particularly suitable for speed control, sometimes has unsatisfactory levels of performance and still high costs.
⁎ Corresponding author. Tel.: + 39 049 827 7474; fax: +39 049 827 6785. E-mail address: [email protected] (A. Rossetti). 0094-114X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2011.07.007
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From these considerations, it can be inferred that the efficiency requirements cannot be satisfied only by a directly coupled CVT. For this reason, a novel type of transmission was designed, in which the power is transmitted partly via a mechanical path, i.e. in an efficient path, and partly through a CVT. The two powers are then summed by means of a planetary gearing. In this way, the transmission is still a continuously variable transmission but with efficiency higher than that of the CVT taken individually, since the transmission efficiency can be calculated as the weighted efficiency of the two paths. This type of transmission is called power split transmission. The power-split configuration currently producing the best compromise between efficiency and cost seems to be the hydromechanical power split transmission, where the variable element is the hydrostatic transmission. It offers significant growth potential thanks to its many possible configurations. Kress [1] classifies four configurations: the so-called “input coupled” (Fig. 1), “output coupled” (Fig. 2), “hydraulic differential” (Fig. 3), and the 4 shafts solution, called “compound” (Fig. 4). Within the first two configurations there are also other possible variations, as shown by Jarkow [2]. The hydro-mechanical transmission, although known for decades, only recently has found applications in standard machines. The first was the Fendt Vario in 1996. It is an integrated output coupled by two traditional gears allowing the vehicle to reach speeds of 50 km/h. In 2000, the Steyr S-Matic was launched, followed by ZF (2001) and John Deere's “Auto Power”; all three have input coupled compound transmissions with 4 or 5 shafts. The literature on this subject is abundant, but here only the main work will be summarized. In the above cited work [1], Kress studies the performance of the four solutions in Figs. 1–4, assuming as constant, for simplicity, the efficiency of the hydrostatic section. Kress does not suggest a best configuration among those examined, but highlights the potential of the power split solution for applications in traction. Resch and Renius [3] provide a comprehensive overview of the problem of hydro-mechanical transmissions. They define the minimum characteristics of a transmission technology for agricultural machines, in terms of speed and load. Furthermore they analyze the operation and the efficiency of the two classical types: input and output coupled. Orshansky [4] analyzes the operation of the hydro-mechanical transmission with 3 planetary gearings (multi-range), and shows the advantages, while Kireijczyk [5] simulates the operation of a transmission of a similar configuration. Ross [6] outlines the design procedure of the Sundstrand Corporation Responder's transmission, an input coupled for heavy vehicles. A comparison of different solutions for a heavy vehicle is made by Blake et al. [7]: input coupled and output coupled and their complex versions (dual stage and compound, respectively); it follows that the complex solutions are suitable for large power and particularly for agricultural tractors. Krauss and Ivantysynova [8] observed that the purely hydrostatic solution with twin motors is still interesting for its simple construction compared with an output coupled solution. The power-split transmission is suitable also for the hybrid solutions, as suggested by Miller [9] and Kumar et al. [10]: they propose a dual stage drive with inertial energy storage for commercial vehicles and earth moving; the energy savings are up to 22%. The design of such a complex system requires the use of specific calculation codes in order to test different preliminary designs and find the most efficient solution. Two different approaches may be used in order to model the transmission system: the steady state approach or the time dependent approach. Several commercial general purpose codes [11,12] refer to the latter, while many academic “ad hoc” codes make use of the former [13,14]. Even though less detailed than dynamic analysis, the steady state analysis is generally faster and can still supply useful information on the transmission performance, as long as detailed loss functions for the hydraulic machines models are used, i.e. loss models taking into account the influence of the pressure, speed and displacement [13]. For example Ivantysynova and Mikeska [13] proposed a modular software that enables the creation of various configurations and their steady state performance simulation. The accuracy of results is guaranteed by loss models for volumetric machines of the polynomial type; the models are derived from the measurement of a large number of machines. This same software tool is then used by the research group in subsequent works [7,8,10]. Even Casoli et al. [14] proposed a procedure for the stationary simulation of an input coupled transmission of a commercial machine; similar to [13], the efficiencies of hydrostatic machines are polynomial expressions derived from experimental data. A tool like the one proposed in [13] is essentially a verification tool, which allows the performance evaluation of a fixed configuration whose design parameters (the displacement of machinery and gear ratios of the different gearing) have been
Fig. 1. Layout of the input coupled configuration.
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Fig. 2. Layout of the output coupled configuration.
selected in advance by appropriate criteria. An optimization tool, on the contrary, could provide the optimal values of these parameters allowing good preliminary design choices. Since the problem is strongly nonlinear, direct search algorithms could be preferred to classical numerical methods, because they do not require the calculation of the derivatives, but only that of the objective function. In particular, the method of swarms (Particle Swarm Optimization (PSO)) and its evolutions seems well suited for the present problem, because, in addition to high speed of convergence, it is able to “escape” the local extrema and to reach the global one [15]. The optimization procedure for power-split transmissions presented here is based on a stationary model, with hydraulic machine's efficiencies depending on the operating conditions, and a particle swarm method as optimizer; the average efficiency over the entire vehicle speed range is used as an objective function, since it is related to the actual fuel economy of the vehicle. The aim of the procedure is the sizing of the hydrostatic machines and the calculation of the transmission ratios of planetary and ordinary gearings, in order to lead to the maximum transmission efficiency. The procedure will be applied to the cases of a compact loader and a high power tractor. 2. Power-split model The instantaneous state of the transmission was described by means of the speed vector → ω and torque vector → m, where the components of these vectors are the angular velocities and torques of all the transmission shafts. These vectors are related to the transmission layout, by means of a nonlinear equation system: → → → Ω ; m · ω = bω ω → → M → ω; m · m = bm :
ð1Þ
The matrices Ω and M were obtained from kinematic and dynamic imposed by individual components and → →relations connections of the transmission and depend on the transmission state ω; m when the load effects of efficiency are considered. The terms bω and bm are the known speeds and torques. The model will be described in detail in the following sections, along with the optimization technique. 2.1. The hydrostatic variable unit The key element of the power split drives is the hydrostatic transmission (HST), which is sketched in Fig. 5. It is made up of two reversible variable displacement machines with maximum displacement equal to VI and VII. By varying the displacements it is possible to continuously change the speed in the ideal range [+ ∞, − ∞]. Energy losses and structural limitations
Fig. 3. Layout of the hydraulic differential configuration.
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Fig. 4. Layout of the compound configuration.
of the machines, however, restrict this range, forcing the introduction of gears τI and τII in order to adjust the speed of each hydrostatic unit inside an allowable range. The transmission ratios are defined as follows: 0
ωI = τI ωI
ð2Þ
0
ωII = τII ωII : The fundamental equations regulating the power transmission through the unit are listed below and are derived in ideal conditions directly from the law of continuity and from the uniqueness of the pressure drop produced by the two units: ½ τI αI VI
−τII αII VII
1 τI αI VI
−
1 τII αII VII
ωI ωII
mI mII
ð3Þ
=0
ð4Þ
=0
where αI and αII are the actual to maximum displacement ratio of the two units. In actual conditions, Eqs. 3 and 4 are modified as follows: h
k
τI αI VI ηv I
2
k nI 4 ηhym I ηgear τI αI VI
−k
−τII αII VII ηv II
i
ωI ωII
ð5Þ
=0
−k 3 II ηhym II ηngear 5 mI = 0 − mII τII αII VII
ð6Þ
where the exponent k is a function of the power flow direction. In particular, k = 1 if the power goes from I unit to II unit; k = − 1 in the opposite case; the efficiencies η v and η hym are respectively the volumetric and hydro-mechanical efficiency of units I and II,
Fig. 5. Hydrostatic CVT.
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Fig. 6. Effect of speed on volumetric efficiency.
whereas the η gear takes into account the mechanical losses in the ordinary gears. The exponents n I and n II represent the number of gear pairs needed to achieve the gear ratio τI and τII; they are estimated on the basis of τmax, the maximum allowable value for each gear pairs: nI = int ðτI = τmax Þ
ð7Þ
nII = int ðτII = τmax Þ:
ð8Þ
The efficiency of each gear pair ηgear was assumed to be 0.98, while the maximum gear ratio for each pair τmax was assumed equal to 2. In order to take into account the influence on operating conditions, i.e. influence of pressure, speed and displacement, the volumetric and hydro-mechanical efficiencies of the hydraulic machines were expressed by the following relationship: v
ηv = ηref ·χω
ω Δp v ·χα;p α; ωmax Δpmax
Fig. 7. Effect of pressure and actual to maximum displacement ratio on volumetric efficiency.
ð9Þ
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Fig. 8. Effect of speed on hydro-mechanical efficiency.
hy
ηhy = ηref ·χω
ω Δp hy ·χα;p α; : ωmax Δpmax
ð10Þ
ηref is the reference efficiency value, assumed to be 0.96 for both efficiencies, reduced by coefficients χω and χα, p. The former coefficient depends on the speed, and the latter depends on the pressure and displacement, as shown in Figs. 6–9. The laws of χω and χα, p were derived from Wilson-type [16] loss models, whose coefficients were calibrated according to a set of experimental data [17]; the laws are also in agreement with the trends shown by Casoli et al. [14].
2.2. Mechanical elements The main mechanical elements of the power split transmission, i.e. the planetary gearing and the three shaft ordinary gearing, are showed in Fig. 10.
Fig. 9. Effect of pressure and actual to maximum displacement ratio on hydro-mechanical efficiency.
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Fig. 10. Scheme of a three shaft ordinary gear (a), and of a simple planetary gear (b).
The kinematic equations used for the three shaft gear are the following:
1 0 1 1
½1
−1
9 8 < ω1 = −1 =0 ω 0 : 2; ω3
ð11Þ
9 8 < m1 = 1 m2 = 0 ; : m3
ð12Þ
in which a unit transmission ratio has been used. Any reduction or multiplication ratios, as well as the gears efficiency, can be considered incorporated in the gears modeled by τI and τII, as part of the CVT element. The speed and torque equations of a simple planetary gearing are:
½ ð1−T Þ
−1
8 9 < ωc = + T ωr = 0 : ; ωs
ð13Þ
Fig. 11. Scheme of a possible power split configuration.
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Fig. 12. Flow chart of the iterative solution of the system.
2 4
1
1−Tη−t 0
0
Tη−t 0
9 38 0 < mc = 5 mr = 0 ; : ms 1
ð14Þ
where T is the standing transmission ratio and ηo is the efficiency of the planetary gearing, assumed equal to 0.985. The exponent t depends on the power flow inside the gear: for a three shaft planetary gearing 12 different operating conditions can be identified [11]. The parameter t can take ±1 values and can be synthetically expressed as a function of the incoming power from the solar, measured in relative reference to the carrier: t = sign([ms(ωs − ωc)]). 2.3. Definition and resolution of the system Eqs. 5, 6 and 11–14 allow the definition of a system of four equations and eight unknowns; three more equations that take into account the links between elements are then added; for the case of Fig. 11 (input coupled) they are: 8 < ω3 −ωs = 0 ð15Þ ω −ωI = 0 : 2 ωr −ωII = 0 and similarly for the torques: 8 < m3 + ms = 0 m + mI = 0 : : 2 mr + mII = 0
ð16Þ
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The last constraint necessary to make the system determined is the speed and the torque of one of the two extreme shafts. For an input coupled drive (Fig. 11), where the engine is connected to shaft 1, the equations are: ω1 = ωengine and m1 = mengine. In this way, a system of 8 equations and 8 unknowns is obtained. The model can then be summarized in its matrix form: Ω·→ ω = bω M·→ m = bm :
ð17Þ
The variable efficiencies of the hydraulic machines don't allow the direct solution of Eq. (17). An iterative solution, which assumes unitary efficiencies at the first iteration, has been set up, with convergence tolerance equal to 0.1%. The flow chart of the iterative process is shown in Fig. 12. 3. System optimization The simulation model was coupled with an evolutionary optimization system, which is based on the swarm method, in order to identify the transmission with the best performance among those satisfying the constraints required by the application. The optimization process can be generally formulated as follows: Find x = ½ x1
x2
T
:: xn minimizing f ðxÞ; subject to the constraints gj ðxÞ ≤ 0 j = 1::m and lj ðxÞ = 0 j = 1::p:
where: • • • •
x1 x2 :: xn are the free optimization variables (degree of freedom); the equality constraints lj(x) = 0 j = 1..p are the design parameters; the inequality constraints gj(x) ≤ 0 j = 1..m are the design constraints; f(x) is the objective function.
In the following sections, the problem's elements will be presented according to the previous sequence; finally the optimization technique will be discussed. 3.1. Degrees of freedom The variables subject to the optimization are: • • • •
the displacements VI and VII; the transmission ratios τI and τII; The standing transmission ratio T of the planetary gearing; The transmission ratio τOut of the gearings (both ordinary and planetary) between the output shaft of the power split transmission and the wheel axis.
Fig. 13. Overall dimension of the test case vehicle [mm].
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Table 1 Design data of the test vehicles.
I.C.E. power I.C.E. design speed Wheel radius Maximum wheel pulling force Maximum vehicle speed
Compact loader
Tractor
62 kW 2500 rpm 0.42 m 28 kN 16 km h− 1
180 kW 2200 rpm 0.965 m 120 kN 40 km h− 1
3.2. Design parameters The design parameters supplied as input to the optimization procedure are: • • • • •
internal combustion engine (ICE) design power (generally close to the best efficiency power); design speed of ICE; maximum speed of the vehicle; maximum wheel pulling force; wheel radius.
3.3. Design constraints The optimization procedure must be forced to satisfy some design and structural constraints. In particular: • • • •
maximum pressure of hydraulic machines: Δp ≤ Δpmax; minimum and maximum transmission ratio of the planetary gear: Tmin ≤ T ≤ Tmax; maximum and minimum volumes allowed for the units displacement: Vmin ≤ VI and VII ≤ Vmax; maximum speed of hydraulic units as a function of displacement: ωI' and ωI' ≤ β(α) ωmax(V).
The maximum speed of the hydraulic units was considered to be a function of the nominal displacement, (ωmax (V)). Because the maximum speed of the units refers to the full displacement condition, an overspeed factor β (α) was added to model the increase of the maximum speed in the partial displacement conditions (α b 1). The constraints were implemented according to penalty functions, which add a quantity proportional to the exceeded distance from the allowed domain. 3.4. Objective function The purpose of the optimization was the minimization of fuel consumption caused by transmission losses. Therefore, the objective function is the integral average loss calculated between the zero speed and maximum design speed of the vehicle:
eðxÞ =
∫
vmax
0
ð1−ηDriveline Þ dv vmax
:
ð18Þ
3.5. Optimization algorithm Since the object function doesn't have an analytical formulation, direct-search algorithms seem to be suitable for this problem. Furthermore, the particular form of the objective function, probably with many local minima, suggests the use of evolutionary algorithms, which are able to reduce the importance of the initial research point and the trapping in local minima far from absolute minimum. Table 2 Optimization constraints.
Hydrostatic system Unit displacement Maximum pressure Maximum unit speed Planetary gearing Standing gear ratio
Symbol
Min
Max
V Δpmax
28 cm3 – 0
250 cm3 400 bar Figs. 14 and 15
T
− 1/2
− 1/7
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Fig. 14. Maximum speed at full displacement as a function of the unit maximum displacement [17].
The search algorithms based on swarm (PSO Particle Swarm Optimizer) are stochastic search algorithms based on social behavior demonstrated by flocks of birds or swarms of insects, in which the interaction between the individual information and the information sharing with other individuals leads to many evolutionary advantages, such as greater efficiency in finding food. The most important feature of the particle swarm algorithm is the process that moves the particle of the swarm (initialized in random positions in the first iteration), between two subsequent iterations. The position of the i-th particle in the space of the allowable solutions at the n + 1 iteration is obtained by summing a displacement d to the position x of the same particle at the n-th iteration as follows: i
i
i
xn +1 = xn + dn+1 :
ð19Þ
The displacement is computed according to the following definition:
i i i i i dn+1 = wdn + c1 rand yn + xn + c2 rand yn + xn :
Fig. 15. Over speed factor as a function of the actual to maximum displacement [19].
ð20Þ
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Fig. 16. Swarm dispersion as function of the optimizer iterations; (a) compact loader, (b) agricultural tractor.
The displacement includes three terms, which model the prime basis of the social behaviors: • the first term models the resistance to changes: an individual tends to repeat its behavior and maintain its opinions even if new environmental changes suggest to modify them. This term is ruled by the inertial coefficient w, which multiplies the displacement at the previous iteration; • the second one represents the tendency of any singular individual to move toward the position of best fitness, according to its own experience. The position yni is, in fact, defined as the best position discovered by the i-th particle from the beginning of the optimization to the current iteration. This component is weighted by the so-called “self confidence” coefficient c1 and the uniformly distributed random variable rand that can take any value between 0 and 1; • the third one models the interaction between individuals such as knowledge sharing and emulations. The goal of this part of the displacement is to move toward the position yn, which is the position of best fitness ever discovered by the swarm. This component is weighted by the so called “swarm confidence” coefficient c2 and the random variable rand. In the current optimization values of 0.6, 1.5 and 1.5 respectively for the inertial coefficient w, c1 and c2. The convergence criterion used was based on the mean dispersion of the swarm, assuming the criterion was fulfilled when at least 80% of the particles were collected inside 0.05% of the design space centered around the best known solution yn. The optimizer used in this work, originally presented by Shu-Kai S. Fan et al. [15], combines the classical formulation of PSO with the Nelder–Mead algorithm [18]. This formulation improves the efficiency of local convergence, keeping the research capabilities of the algorithm unchanged.
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Fig. 17. Best fitness as function of the optimizer iterations; (a) compact loader, (b) agricultural tractor.
4. Application of the optimization procedure In order to test the optimization procedure, two different work vehicles were considered: a compact loader and a high power tractor. The dimensions and the design data of the two vehicles are shown respectively in Fig. 13 and Table 1. In Table 2 the design constraints are summarized. The maximum unit speed is calculated (section 3.3) as the product of the maximum speed and the over speed factor, respectively represented in Figs. 14 and 15. The maximum speed at full displacement was defined as a function of the unit size and could be well interpolated as a linear trend in double logarithmic axes, with smaller units characterized by higher velocities (Fig. 14). The over speed factor is instead described as a function of the actual to maximum displacement and
Table 3 Optimization results for the compact loader test case. T VI VII τI τII τout ηmean
−0.44 30 cm3 83 cm3 1.66 2.35 17.15 0.785
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Fig. 18. Overall efficiency and hydrostatic path efficiency for the compact loader.
expresses the capability of the unit to withstand higher velocity at partial displacement (Fig. 15). The used data were derived from online catalogs for axial piston machines [17,19]. The different characteristics of the vehicles suggested two different transmission configurations. The small size and power of the compact loader indicated an input coupled solution, which allows to achieve better performance with smaller hydrostatic units than with the output coupled configuration [7]. For the agricultural tractor, instead, an output coupled transmission was considered, which leads to higher performance, even with larger size of the hydrostatic units. However, to contain the value of the displacements, given the high value of the maximum force exerted by the tractor, a two-speed gearbox with a transmission ratio equal to 4 was placed downstream from the CVT transmission. The speed change point was chosen where the transmission efficiency line in the first gear crossed the transmission efficiency line in the second gear.
1st gear 2nd gear ηDriveLine vchange = ηDriveLine vchange :
ð21Þ
The objective function was then rewritten for this case as:
eðxÞ =
gear gear max dv + ∫vvchange dv ∫ vchange 1−η1st 1−η2nd Driveline Driveline 0
vmax
:
Fig. 19. Unit's speed and speed limits for the compact loader.
ð22Þ
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Fig. 20. Unit's displacement as a function of the vehicle ground speed for the compact loader.
4.1. Results 4.1.1. Optimizator performance The optimization procedure took 6.8 10 3 s and 1.1 10 4 s for the compact loader and the agricultural tractor respectively using parallel computing on a 2 processor dual core AMD Opteron 265 computer. The convergence history was monitored by considering both the objective function value and the mean dispersion of the swarm on the design space. As previously introduced, the convergence of the algorithm was defined on the basis of the mean normalized distance from the current best position of the nearest 80% of the swarm. Fig. 16 describes the swarm dispersion by means of three different measures: • the average distance from the best position, calculated on the overall swarm (r100%); • the average distance from the best position considering only the nearest 80% of the swarm particles, i.e. the convergence control variable (r80%); • the average distance from the best position considering only the nearest 50% of the swarm particles (r50%). In both the cases, the three lines move parallel to each other on the logarithmic scale. This proves the capability of the algorithm to balance between the refinement of the search near the current actual optimum and the scouting of the design space in order to avoid early convergence to a local optimum. Half of the swarm, in fact, searches in the 10% of the design space near the current
Fig. 21. HST pressure as a function of the vehicle ground speed for the compact loader.
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−0.29 156 cm3 244 cm3 3.82 3.97
optimum for most of the convergence history (r50% b 0.1), while the significant difference between the r80% and r100% demonstrates that the farthest 20% of the particles are spread on the design space even in the late part of the convergence. Fig. 17 reports the trend of the best efficiency as a function of the iteration number for the two test cases. The highly discontinuous trend shows that most of the improvements are due to a radical change in the transmission configuration instead of through the local adjustment of the current optimum. This could be related to the positive contribution of the diverging particles. The discontinuous movements of the optimum configuration as the iteration increases could be clearly seen comparing Figs. 16 and 17: the steep increase of the best fitness is related to a sudden expansion of the average swarm radius, proving the repositioning of the optimum configuration used as the origin of the radial measurements. The optimizer was capable of avoiding the low-fitness local optimum and was able to converge toward an acceptable highfitness optimum, for both cases. 4.1.2. Test case 1: compact loader The optimum design of the compact loader is reported in Table 3, while Figs. 16 to 19 report the behavior of the most important variables of the power split transmission. In Fig. 18, the overall efficiency of the transmission and the efficiency of HST are reported. The former shows the typical trend for input coupled transmissions: the recirculation of power between mechanical and hydraulic paths leads to a sudden decrease of performance for speeds lower than the full mechanical point; at speeds higher than the full mechanical point the overall efficiency is close to the CVT hydraulic efficiency as the power transferred by the hydraulic path increases. The mean overall efficiency of the optimum transmission in the range 0–16 km/h is 0.785, which could be considered a good result for such a small and compact solution. Fig. 19 shows the rotational speed of the two hydraulic units compared to the maximum allowable speed for the considered displacement. The displacement of the hydrostatic units is instead reported in Fig. 20, while the differential pressure inside the hydrostatic system is reported in Fig. 21. 4.1.3. Test case 2: Agricultural tractor The optimum design parameters are presented in Table 4. Fig. 22 shows the efficiency of the overall transmission and of the hydrostatic CVT element for the design velocity range. The initial choice of a gear ratio of 4 seems to be appropriate in terms of efficiency because the transmission presents the highest
Fig. 22. Overall efficiency and hydrostatic path efficiency for the agricultural tractor.
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Fig. 23. Unit's speed and speed limits for the agricultural tractor.
efficiency in the velocity ranges typical of the most power expensive condition: in field work and street transport. In fact, the transmission shows efficiency higher than 85% in the low velocity range (4–10 km/h), which corresponds to the typical range of field working velocity, while the maximum efficiency of the high gear is instead compatible with the street transport condition. The mean overall efficiency of the transmission is 84%, which is notably higher than the standard proposed by Renius for CVT hydraulic transmission [3]. As for the previous test case, Figs. 23–25 complete the description of the system behavior, giving the unit's speed, displacement and the differential pressure of the HST system.
5. Conclusions The advantage of the continuous speed variation of the power-split drives is counterbalanced by a reduced efficiency, caused by the double energy conversion taking place in the hydrostatic transmission. Therefore, the design of the power split drive must be carefully studied. In this paper, the design is treated as an optimization problem, where the objective function to be minimized is the total loss of the transmission, and the free variables are the displacements of the hydraulic machines and the gear ratios of the ordinary and planetary gearings.
Fig. 24. Unit's displacement as a function of the vehicle ground speed for the agricultural tractor.
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Fig. 25. HST pressure as a function of the vehicle ground speed for the agricultural tractor.
An automatic procedure, made up by a steady state simulator of the driveline and an evolutionary optimizer, was set up to solve this problem. The steady simulator takes into account the losses of the hydraulic units, which heavily influence the driveline efficiency, and the losses of the other mechanical components. A “direct search” algorithm based on the swarm method was used as the optimizer, which showed a good speed convergence and the ability to overcome local minima. The proposed approach offers significant advantages over traditional design methods: the procedure does not depend on experience and previous knowledge because no assumption had to be made on the component's sizing; the optimality of the output is based on the implemented search algorithm while the quality of the classical designs depends strongly on the designer's experience. Furthermore the procedure can handle more complex constraint formulation than traditional design methods and verify them for all the working conditions. The procedure was then tested on two heavy duty vehicles, which were chosen to be very different in terms of power, size and velocity range: a compact loader with a nominal power of 62 kW, and an agricultural tractor of 180 kW. The procedure demonstrated to be capable of handling the optimized design of both test cases, fulfilling all the design constraints and obtaining high efficiency drivelines. The procedure could also be further developed in order to be applied in more complex power split configurations, such as the “dual stage” and the compound solutions.
Nomenclature m torque [Nm] n iteration [–] p pressure [Pa] v velocity [m s − 1] z gear tooth number [–] M transmission torque matrix [–] P power [W] T = − zs/zr standing gear ratio of the planetary gear [–] V maximum unit displacement [m 3]
Greek letters α actual to maximum displacement ratio [–] ηv volumetric efficiency [–] hydro mechanical efficiency [–] ηhym τ gear ratio [–] gear ratio of gearbox between power split transmission and axle [–] τout ω rotational speed [rad s − 1]
A. Macor, A. Rossetti / Mechanism and Machine Theory 46 (2011) 1901–1919
χ Ω
non dimensional penalty coefficient for unit efficiency calculation [–] transmission velocity matrix [–]
Subscript I II rif c s r ′
and Superscripts hydrostatic transmission element inlet shaft hydrostatic transmission element outlet shaft condition of unit maximum efficiency planetary gear, carrier shaft planetary gear, sun shaft planetary gear, ring shaft unit shaft
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Acknowledgment The results presented in the paper were obtained as part of the research program “Studio di trasmissioni power-split per trattrici agricole” of the PRIN 2007 (Programmi di Ricerca di Interesse Nazionale — Research Programs of National Interest 2007) funded by Ministero dell'Istruzione, dell'Università e della Ricerca (Ministry of Education, University and Research) of Italian Republic. References [1] J.H. Kress, Hydrostatic power splitting transmissions for wheeled vehicles — classification and theory of operation. SAE paper 680549, Society of Automotive Engineers, Warrendale, PA, 1968. [2] F. Jarchow, Leistungsverzweigte Getriebe (Power split transmissions), VDI-Z. 106 (6) (1964) 196–205. [3] K.T. Renius, R. Resch, Continuously Variable Tractor Transmissions, ASAE Distinguished Lecture Series, Tractor Design No. 29, 2005. [4] E. Orshansky, W.E. Weseloh, Characteristics of Multiple Range Hydromechanical Transmissions. SAE Paper 720724, Society of Automotive Engineers, Warrendale, PA, 1972. [5] J. Kirejczyk, Continuously Variable Hydromechanical Transmission for Commercial Vehicle By Simulation Studies. SAE paper 845095, Society of Automotive Engineers, Warrendale, PA, 1984. [6] W.A. Ross, Designing a Hydromechanical Transmissions for Heavy Duty Trucks. SAE paper 720725, Society of Automotive Engineers, Warrendale, PA, 1972. [7] C. Blake, M. Ivantysynova, K. Williams, Comparison of operational characteristics in power split continuously variable transmissions, SAE 2006 Commercial Vehicle Engineering Congress & Exhibition, SAE Technical Paper Series 2006-01-3468, October 2006. [8] A. Krauss, M. Ivantysynova, Power split transmissions versus hydrostatic multiple motor concepts — a comparative analysis, SAE 2004 commercial vehicle engineering congress & exhibition, SAE Technical Paper Series 2004-01-2676, October 2004. [9] J. Miller, Comparative assessment of hybrid vehicle power split transmissions, 4th VI Winter Workshop Series, 2005. [10] R. Kumar, M. Ivantysynova, K. Williams, Study of energetic characteristics in power split drives for on highway trucks and wheel loaders, SAE 2007 commercial vehicle engineering congress & exhibition, SAE Technical Paper Series 2007-01-4193, October 2004. [11] Matlab 2008, copyright 1984–2008, The MathWorks Inc. [12] AMESim 2008, copyright 1996–2008, LMS imagine SA. [13] D. Mikeska, M. Ivantysynova, Virtual prototyping of power split drives, Proc. Bath Workshop on Power Transmission and Motion Control PTMC2002, Bath, UK, 2002, pp. 95–111. [14] P. Casoli, A. Vacca, G.L. Berta, S. Meletti, A numerical model for the simulation of diesel/CVT power split transmission, ICE2007 8th International Conference on Engines for Automobile; Capri, Naples, September 16–20, 2007. [15] Shu-Kai S. Fan, Liang Yun-Chia, Zahara Erwie, A genetic algorithm and a particle swarm optimizer hybridized with Nelder–Mead simplex search, Comput. Ind. Eng. 50 (2006) 401–425. [16] W.E. Wilson, Positive-Displacement Pumps and Fluid Motors, Sir Isaac Pitman and Sons, LTD, London, 1950. [17] http://www.boschrexroth.com/mobile-hydraulics-catalog/Vornavigation/VorNavi.cfm?Language=EN&VHist=g54076&PageID=g5406917. [18] J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 7 (1965) 308–313. [19] http://www.samhydraulik.it/public/samhydraulik/it/prodotti/AxialPistonMotors.html.
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Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
Optimization of hydro-mechanical power split transmissions Alarico Macor, Antonio Rossetti ⁎ Department of Engineering and Management, Università di Padova, Stradella S. Nicola, 3-36100 Vicenza, Italy
a r t i c l e
i n f o
Article history: Received 24 November 2010 Received in revised form 6 May 2011 Accepted 14 July 2011 Available online 1 September 2011 Keywords: Hydro-mechanical transmission Power split transmission
a b s t r a c t The increasing attention to comfort, automation and drivability is pushing the driveline technology to ever complex solutions, such as power-shift or continuously variable transmissions. Between these, the hydro-mechanical solution seems promising for heavy duty vehicle, due to the reliability and the capability of transferring high power. However, the double energy conversion occurring in the hydraulic branch of the transmission could lower excessively the total efficiency, highlighting the needs for a careful design of the whole system. In this work, the design of a hydro-mechanical transmission is defined as an optimization problem in which the objective function is the average efficiency of transmission, while the design variables are the displacements of the two hydraulic machines and gear ratios of ordinary and planetary gears. The optimization problem is solved by a “direct search” algorithm based on the swarm method, which showed a good speed convergence and the ability to overcome local minima. The optimization design method will be applied to study the transmission of two vehicles: a 62 kW compact loader and a high power agricultural tractor. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The growing role of agricultural mechanization in worldwide food programs and the need to save fossil energy goes towards reducing fuel consumption and emissions of agricultural machinery. This compels to reconsider the architecture of the vehicle's propulsion system, addressing the designer toward more sophisticated and efficient solutions. The same trend holds for earthmoving and construction machinery. In recent years, the evolution of the transmissions of agricultural and working machines was addressed to increase comfort and drivability without penalty of efficiency and, possibly, costs. These objectives involve the use of continuous transmissions that allow the elimination of the shifting gears, which is sometimes exhausting, and keep the fuel consumption to a minimum. Current technology provides some solutions for continuous variable transmission (CVT): the hydrodynamic, the hydrostatic, the mechanical and the electrical. The hydrodynamic transmission (torque converter) is not suitable for these applications because its transmission ratio is not controllable by the user and it has a good efficiency only at high speeds. The hydrostatic transmission, on the contrary, can easily control the speed, but with a very low efficiency because of the double energy conversion occurring in it. The mechanical transmission, in its variants (metal chain, toroidal), is characterized by high efficiency, although sometimes with limits of power and speed range. This requires the use of a mechanical gearbox further downstream. Electric transmission, although particularly suitable for speed control, sometimes has unsatisfactory levels of performance and still high costs.
⁎ Corresponding author. Tel.: + 39 049 827 7474; fax: +39 049 827 6785. E-mail address: [email protected] (A. Rossetti). 0094-114X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2011.07.007
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From these considerations, it can be inferred that the efficiency requirements cannot be satisfied only by a directly coupled CVT. For this reason, a novel type of transmission was designed, in which the power is transmitted partly via a mechanical path, i.e. in an efficient path, and partly through a CVT. The two powers are then summed by means of a planetary gearing. In this way, the transmission is still a continuously variable transmission but with efficiency higher than that of the CVT taken individually, since the transmission efficiency can be calculated as the weighted efficiency of the two paths. This type of transmission is called power split transmission. The power-split configuration currently producing the best compromise between efficiency and cost seems to be the hydromechanical power split transmission, where the variable element is the hydrostatic transmission. It offers significant growth potential thanks to its many possible configurations. Kress [1] classifies four configurations: the so-called “input coupled” (Fig. 1), “output coupled” (Fig. 2), “hydraulic differential” (Fig. 3), and the 4 shafts solution, called “compound” (Fig. 4). Within the first two configurations there are also other possible variations, as shown by Jarkow [2]. The hydro-mechanical transmission, although known for decades, only recently has found applications in standard machines. The first was the Fendt Vario in 1996. It is an integrated output coupled by two traditional gears allowing the vehicle to reach speeds of 50 km/h. In 2000, the Steyr S-Matic was launched, followed by ZF (2001) and John Deere's “Auto Power”; all three have input coupled compound transmissions with 4 or 5 shafts. The literature on this subject is abundant, but here only the main work will be summarized. In the above cited work [1], Kress studies the performance of the four solutions in Figs. 1–4, assuming as constant, for simplicity, the efficiency of the hydrostatic section. Kress does not suggest a best configuration among those examined, but highlights the potential of the power split solution for applications in traction. Resch and Renius [3] provide a comprehensive overview of the problem of hydro-mechanical transmissions. They define the minimum characteristics of a transmission technology for agricultural machines, in terms of speed and load. Furthermore they analyze the operation and the efficiency of the two classical types: input and output coupled. Orshansky [4] analyzes the operation of the hydro-mechanical transmission with 3 planetary gearings (multi-range), and shows the advantages, while Kireijczyk [5] simulates the operation of a transmission of a similar configuration. Ross [6] outlines the design procedure of the Sundstrand Corporation Responder's transmission, an input coupled for heavy vehicles. A comparison of different solutions for a heavy vehicle is made by Blake et al. [7]: input coupled and output coupled and their complex versions (dual stage and compound, respectively); it follows that the complex solutions are suitable for large power and particularly for agricultural tractors. Krauss and Ivantysynova [8] observed that the purely hydrostatic solution with twin motors is still interesting for its simple construction compared with an output coupled solution. The power-split transmission is suitable also for the hybrid solutions, as suggested by Miller [9] and Kumar et al. [10]: they propose a dual stage drive with inertial energy storage for commercial vehicles and earth moving; the energy savings are up to 22%. The design of such a complex system requires the use of specific calculation codes in order to test different preliminary designs and find the most efficient solution. Two different approaches may be used in order to model the transmission system: the steady state approach or the time dependent approach. Several commercial general purpose codes [11,12] refer to the latter, while many academic “ad hoc” codes make use of the former [13,14]. Even though less detailed than dynamic analysis, the steady state analysis is generally faster and can still supply useful information on the transmission performance, as long as detailed loss functions for the hydraulic machines models are used, i.e. loss models taking into account the influence of the pressure, speed and displacement [13]. For example Ivantysynova and Mikeska [13] proposed a modular software that enables the creation of various configurations and their steady state performance simulation. The accuracy of results is guaranteed by loss models for volumetric machines of the polynomial type; the models are derived from the measurement of a large number of machines. This same software tool is then used by the research group in subsequent works [7,8,10]. Even Casoli et al. [14] proposed a procedure for the stationary simulation of an input coupled transmission of a commercial machine; similar to [13], the efficiencies of hydrostatic machines are polynomial expressions derived from experimental data. A tool like the one proposed in [13] is essentially a verification tool, which allows the performance evaluation of a fixed configuration whose design parameters (the displacement of machinery and gear ratios of the different gearing) have been
Fig. 1. Layout of the input coupled configuration.
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Fig. 2. Layout of the output coupled configuration.
selected in advance by appropriate criteria. An optimization tool, on the contrary, could provide the optimal values of these parameters allowing good preliminary design choices. Since the problem is strongly nonlinear, direct search algorithms could be preferred to classical numerical methods, because they do not require the calculation of the derivatives, but only that of the objective function. In particular, the method of swarms (Particle Swarm Optimization (PSO)) and its evolutions seems well suited for the present problem, because, in addition to high speed of convergence, it is able to “escape” the local extrema and to reach the global one [15]. The optimization procedure for power-split transmissions presented here is based on a stationary model, with hydraulic machine's efficiencies depending on the operating conditions, and a particle swarm method as optimizer; the average efficiency over the entire vehicle speed range is used as an objective function, since it is related to the actual fuel economy of the vehicle. The aim of the procedure is the sizing of the hydrostatic machines and the calculation of the transmission ratios of planetary and ordinary gearings, in order to lead to the maximum transmission efficiency. The procedure will be applied to the cases of a compact loader and a high power tractor. 2. Power-split model The instantaneous state of the transmission was described by means of the speed vector → ω and torque vector → m, where the components of these vectors are the angular velocities and torques of all the transmission shafts. These vectors are related to the transmission layout, by means of a nonlinear equation system: → → → Ω ; m · ω = bω ω → → M → ω; m · m = bm :
ð1Þ
The matrices Ω and M were obtained from kinematic and dynamic imposed by individual components and → →relations connections of the transmission and depend on the transmission state ω; m when the load effects of efficiency are considered. The terms bω and bm are the known speeds and torques. The model will be described in detail in the following sections, along with the optimization technique. 2.1. The hydrostatic variable unit The key element of the power split drives is the hydrostatic transmission (HST), which is sketched in Fig. 5. It is made up of two reversible variable displacement machines with maximum displacement equal to VI and VII. By varying the displacements it is possible to continuously change the speed in the ideal range [+ ∞, − ∞]. Energy losses and structural limitations
Fig. 3. Layout of the hydraulic differential configuration.
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Fig. 4. Layout of the compound configuration.
of the machines, however, restrict this range, forcing the introduction of gears τI and τII in order to adjust the speed of each hydrostatic unit inside an allowable range. The transmission ratios are defined as follows: 0
ωI = τI ωI
ð2Þ
0
ωII = τII ωII : The fundamental equations regulating the power transmission through the unit are listed below and are derived in ideal conditions directly from the law of continuity and from the uniqueness of the pressure drop produced by the two units: ½ τI αI VI
−τII αII VII
1 τI αI VI
−
1 τII αII VII
ωI ωII
mI mII
ð3Þ
=0
ð4Þ
=0
where αI and αII are the actual to maximum displacement ratio of the two units. In actual conditions, Eqs. 3 and 4 are modified as follows: h
k
τI αI VI ηv I
2
k nI 4 ηhym I ηgear τI αI VI
−k
−τII αII VII ηv II
i
ωI ωII
ð5Þ
=0
−k 3 II ηhym II ηngear 5 mI = 0 − mII τII αII VII
ð6Þ
where the exponent k is a function of the power flow direction. In particular, k = 1 if the power goes from I unit to II unit; k = − 1 in the opposite case; the efficiencies η v and η hym are respectively the volumetric and hydro-mechanical efficiency of units I and II,
Fig. 5. Hydrostatic CVT.
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Fig. 6. Effect of speed on volumetric efficiency.
whereas the η gear takes into account the mechanical losses in the ordinary gears. The exponents n I and n II represent the number of gear pairs needed to achieve the gear ratio τI and τII; they are estimated on the basis of τmax, the maximum allowable value for each gear pairs: nI = int ðτI = τmax Þ
ð7Þ
nII = int ðτII = τmax Þ:
ð8Þ
The efficiency of each gear pair ηgear was assumed to be 0.98, while the maximum gear ratio for each pair τmax was assumed equal to 2. In order to take into account the influence on operating conditions, i.e. influence of pressure, speed and displacement, the volumetric and hydro-mechanical efficiencies of the hydraulic machines were expressed by the following relationship: v
ηv = ηref ·χω
ω Δp v ·χα;p α; ωmax Δpmax
Fig. 7. Effect of pressure and actual to maximum displacement ratio on volumetric efficiency.
ð9Þ
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Fig. 8. Effect of speed on hydro-mechanical efficiency.
hy
ηhy = ηref ·χω
ω Δp hy ·χα;p α; : ωmax Δpmax
ð10Þ
ηref is the reference efficiency value, assumed to be 0.96 for both efficiencies, reduced by coefficients χω and χα, p. The former coefficient depends on the speed, and the latter depends on the pressure and displacement, as shown in Figs. 6–9. The laws of χω and χα, p were derived from Wilson-type [16] loss models, whose coefficients were calibrated according to a set of experimental data [17]; the laws are also in agreement with the trends shown by Casoli et al. [14].
2.2. Mechanical elements The main mechanical elements of the power split transmission, i.e. the planetary gearing and the three shaft ordinary gearing, are showed in Fig. 10.
Fig. 9. Effect of pressure and actual to maximum displacement ratio on hydro-mechanical efficiency.
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Fig. 10. Scheme of a three shaft ordinary gear (a), and of a simple planetary gear (b).
The kinematic equations used for the three shaft gear are the following:
1 0 1 1
½1
−1
9 8 < ω1 = −1 =0 ω 0 : 2; ω3
ð11Þ
9 8 < m1 = 1 m2 = 0 ; : m3
ð12Þ
in which a unit transmission ratio has been used. Any reduction or multiplication ratios, as well as the gears efficiency, can be considered incorporated in the gears modeled by τI and τII, as part of the CVT element. The speed and torque equations of a simple planetary gearing are:
½ ð1−T Þ
−1
8 9 < ωc = + T ωr = 0 : ; ωs
ð13Þ
Fig. 11. Scheme of a possible power split configuration.
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Fig. 12. Flow chart of the iterative solution of the system.
2 4
1
1−Tη−t 0
0
Tη−t 0
9 38 0 < mc = 5 mr = 0 ; : ms 1
ð14Þ
where T is the standing transmission ratio and ηo is the efficiency of the planetary gearing, assumed equal to 0.985. The exponent t depends on the power flow inside the gear: for a three shaft planetary gearing 12 different operating conditions can be identified [11]. The parameter t can take ±1 values and can be synthetically expressed as a function of the incoming power from the solar, measured in relative reference to the carrier: t = sign([ms(ωs − ωc)]). 2.3. Definition and resolution of the system Eqs. 5, 6 and 11–14 allow the definition of a system of four equations and eight unknowns; three more equations that take into account the links between elements are then added; for the case of Fig. 11 (input coupled) they are: 8 < ω3 −ωs = 0 ð15Þ ω −ωI = 0 : 2 ωr −ωII = 0 and similarly for the torques: 8 < m3 + ms = 0 m + mI = 0 : : 2 mr + mII = 0
ð16Þ
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The last constraint necessary to make the system determined is the speed and the torque of one of the two extreme shafts. For an input coupled drive (Fig. 11), where the engine is connected to shaft 1, the equations are: ω1 = ωengine and m1 = mengine. In this way, a system of 8 equations and 8 unknowns is obtained. The model can then be summarized in its matrix form: Ω·→ ω = bω M·→ m = bm :
ð17Þ
The variable efficiencies of the hydraulic machines don't allow the direct solution of Eq. (17). An iterative solution, which assumes unitary efficiencies at the first iteration, has been set up, with convergence tolerance equal to 0.1%. The flow chart of the iterative process is shown in Fig. 12. 3. System optimization The simulation model was coupled with an evolutionary optimization system, which is based on the swarm method, in order to identify the transmission with the best performance among those satisfying the constraints required by the application. The optimization process can be generally formulated as follows: Find x = ½ x1
x2
T
:: xn minimizing f ðxÞ; subject to the constraints gj ðxÞ ≤ 0 j = 1::m and lj ðxÞ = 0 j = 1::p:
where: • • • •
x1 x2 :: xn are the free optimization variables (degree of freedom); the equality constraints lj(x) = 0 j = 1..p are the design parameters; the inequality constraints gj(x) ≤ 0 j = 1..m are the design constraints; f(x) is the objective function.
In the following sections, the problem's elements will be presented according to the previous sequence; finally the optimization technique will be discussed. 3.1. Degrees of freedom The variables subject to the optimization are: • • • •
the displacements VI and VII; the transmission ratios τI and τII; The standing transmission ratio T of the planetary gearing; The transmission ratio τOut of the gearings (both ordinary and planetary) between the output shaft of the power split transmission and the wheel axis.
Fig. 13. Overall dimension of the test case vehicle [mm].
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Table 1 Design data of the test vehicles.
I.C.E. power I.C.E. design speed Wheel radius Maximum wheel pulling force Maximum vehicle speed
Compact loader
Tractor
62 kW 2500 rpm 0.42 m 28 kN 16 km h− 1
180 kW 2200 rpm 0.965 m 120 kN 40 km h− 1
3.2. Design parameters The design parameters supplied as input to the optimization procedure are: • • • • •
internal combustion engine (ICE) design power (generally close to the best efficiency power); design speed of ICE; maximum speed of the vehicle; maximum wheel pulling force; wheel radius.
3.3. Design constraints The optimization procedure must be forced to satisfy some design and structural constraints. In particular: • • • •
maximum pressure of hydraulic machines: Δp ≤ Δpmax; minimum and maximum transmission ratio of the planetary gear: Tmin ≤ T ≤ Tmax; maximum and minimum volumes allowed for the units displacement: Vmin ≤ VI and VII ≤ Vmax; maximum speed of hydraulic units as a function of displacement: ωI' and ωI' ≤ β(α) ωmax(V).
The maximum speed of the hydraulic units was considered to be a function of the nominal displacement, (ωmax (V)). Because the maximum speed of the units refers to the full displacement condition, an overspeed factor β (α) was added to model the increase of the maximum speed in the partial displacement conditions (α b 1). The constraints were implemented according to penalty functions, which add a quantity proportional to the exceeded distance from the allowed domain. 3.4. Objective function The purpose of the optimization was the minimization of fuel consumption caused by transmission losses. Therefore, the objective function is the integral average loss calculated between the zero speed and maximum design speed of the vehicle:
eðxÞ =
∫
vmax
0
ð1−ηDriveline Þ dv vmax
:
ð18Þ
3.5. Optimization algorithm Since the object function doesn't have an analytical formulation, direct-search algorithms seem to be suitable for this problem. Furthermore, the particular form of the objective function, probably with many local minima, suggests the use of evolutionary algorithms, which are able to reduce the importance of the initial research point and the trapping in local minima far from absolute minimum. Table 2 Optimization constraints.
Hydrostatic system Unit displacement Maximum pressure Maximum unit speed Planetary gearing Standing gear ratio
Symbol
Min
Max
V Δpmax
28 cm3 – 0
250 cm3 400 bar Figs. 14 and 15
T
− 1/2
− 1/7
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Fig. 14. Maximum speed at full displacement as a function of the unit maximum displacement [17].
The search algorithms based on swarm (PSO Particle Swarm Optimizer) are stochastic search algorithms based on social behavior demonstrated by flocks of birds or swarms of insects, in which the interaction between the individual information and the information sharing with other individuals leads to many evolutionary advantages, such as greater efficiency in finding food. The most important feature of the particle swarm algorithm is the process that moves the particle of the swarm (initialized in random positions in the first iteration), between two subsequent iterations. The position of the i-th particle in the space of the allowable solutions at the n + 1 iteration is obtained by summing a displacement d to the position x of the same particle at the n-th iteration as follows: i
i
i
xn +1 = xn + dn+1 :
ð19Þ
The displacement is computed according to the following definition:
i i i i i dn+1 = wdn + c1 rand yn + xn + c2 rand yn + xn :
Fig. 15. Over speed factor as a function of the actual to maximum displacement [19].
ð20Þ
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Fig. 16. Swarm dispersion as function of the optimizer iterations; (a) compact loader, (b) agricultural tractor.
The displacement includes three terms, which model the prime basis of the social behaviors: • the first term models the resistance to changes: an individual tends to repeat its behavior and maintain its opinions even if new environmental changes suggest to modify them. This term is ruled by the inertial coefficient w, which multiplies the displacement at the previous iteration; • the second one represents the tendency of any singular individual to move toward the position of best fitness, according to its own experience. The position yni is, in fact, defined as the best position discovered by the i-th particle from the beginning of the optimization to the current iteration. This component is weighted by the so-called “self confidence” coefficient c1 and the uniformly distributed random variable rand that can take any value between 0 and 1; • the third one models the interaction between individuals such as knowledge sharing and emulations. The goal of this part of the displacement is to move toward the position yn, which is the position of best fitness ever discovered by the swarm. This component is weighted by the so called “swarm confidence” coefficient c2 and the random variable rand. In the current optimization values of 0.6, 1.5 and 1.5 respectively for the inertial coefficient w, c1 and c2. The convergence criterion used was based on the mean dispersion of the swarm, assuming the criterion was fulfilled when at least 80% of the particles were collected inside 0.05% of the design space centered around the best known solution yn. The optimizer used in this work, originally presented by Shu-Kai S. Fan et al. [15], combines the classical formulation of PSO with the Nelder–Mead algorithm [18]. This formulation improves the efficiency of local convergence, keeping the research capabilities of the algorithm unchanged.
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Fig. 17. Best fitness as function of the optimizer iterations; (a) compact loader, (b) agricultural tractor.
4. Application of the optimization procedure In order to test the optimization procedure, two different work vehicles were considered: a compact loader and a high power tractor. The dimensions and the design data of the two vehicles are shown respectively in Fig. 13 and Table 1. In Table 2 the design constraints are summarized. The maximum unit speed is calculated (section 3.3) as the product of the maximum speed and the over speed factor, respectively represented in Figs. 14 and 15. The maximum speed at full displacement was defined as a function of the unit size and could be well interpolated as a linear trend in double logarithmic axes, with smaller units characterized by higher velocities (Fig. 14). The over speed factor is instead described as a function of the actual to maximum displacement and
Table 3 Optimization results for the compact loader test case. T VI VII τI τII τout ηmean
−0.44 30 cm3 83 cm3 1.66 2.35 17.15 0.785
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Fig. 18. Overall efficiency and hydrostatic path efficiency for the compact loader.
expresses the capability of the unit to withstand higher velocity at partial displacement (Fig. 15). The used data were derived from online catalogs for axial piston machines [17,19]. The different characteristics of the vehicles suggested two different transmission configurations. The small size and power of the compact loader indicated an input coupled solution, which allows to achieve better performance with smaller hydrostatic units than with the output coupled configuration [7]. For the agricultural tractor, instead, an output coupled transmission was considered, which leads to higher performance, even with larger size of the hydrostatic units. However, to contain the value of the displacements, given the high value of the maximum force exerted by the tractor, a two-speed gearbox with a transmission ratio equal to 4 was placed downstream from the CVT transmission. The speed change point was chosen where the transmission efficiency line in the first gear crossed the transmission efficiency line in the second gear.
1st gear 2nd gear ηDriveLine vchange = ηDriveLine vchange :
ð21Þ
The objective function was then rewritten for this case as:
eðxÞ =
gear gear max dv + ∫vvchange dv ∫ vchange 1−η1st 1−η2nd Driveline Driveline 0
vmax
:
Fig. 19. Unit's speed and speed limits for the compact loader.
ð22Þ
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Fig. 20. Unit's displacement as a function of the vehicle ground speed for the compact loader.
4.1. Results 4.1.1. Optimizator performance The optimization procedure took 6.8 10 3 s and 1.1 10 4 s for the compact loader and the agricultural tractor respectively using parallel computing on a 2 processor dual core AMD Opteron 265 computer. The convergence history was monitored by considering both the objective function value and the mean dispersion of the swarm on the design space. As previously introduced, the convergence of the algorithm was defined on the basis of the mean normalized distance from the current best position of the nearest 80% of the swarm. Fig. 16 describes the swarm dispersion by means of three different measures: • the average distance from the best position, calculated on the overall swarm (r100%); • the average distance from the best position considering only the nearest 80% of the swarm particles, i.e. the convergence control variable (r80%); • the average distance from the best position considering only the nearest 50% of the swarm particles (r50%). In both the cases, the three lines move parallel to each other on the logarithmic scale. This proves the capability of the algorithm to balance between the refinement of the search near the current actual optimum and the scouting of the design space in order to avoid early convergence to a local optimum. Half of the swarm, in fact, searches in the 10% of the design space near the current
Fig. 21. HST pressure as a function of the vehicle ground speed for the compact loader.
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A. Macor, A. Rossetti / Mechanism and Machine Theory 46 (2011) 1901–1919 Table 4 Optimization results for the agricultural tractor. T VI VII τI τII
−0.29 156 cm3 244 cm3 3.82 3.97
optimum for most of the convergence history (r50% b 0.1), while the significant difference between the r80% and r100% demonstrates that the farthest 20% of the particles are spread on the design space even in the late part of the convergence. Fig. 17 reports the trend of the best efficiency as a function of the iteration number for the two test cases. The highly discontinuous trend shows that most of the improvements are due to a radical change in the transmission configuration instead of through the local adjustment of the current optimum. This could be related to the positive contribution of the diverging particles. The discontinuous movements of the optimum configuration as the iteration increases could be clearly seen comparing Figs. 16 and 17: the steep increase of the best fitness is related to a sudden expansion of the average swarm radius, proving the repositioning of the optimum configuration used as the origin of the radial measurements. The optimizer was capable of avoiding the low-fitness local optimum and was able to converge toward an acceptable highfitness optimum, for both cases. 4.1.2. Test case 1: compact loader The optimum design of the compact loader is reported in Table 3, while Figs. 16 to 19 report the behavior of the most important variables of the power split transmission. In Fig. 18, the overall efficiency of the transmission and the efficiency of HST are reported. The former shows the typical trend for input coupled transmissions: the recirculation of power between mechanical and hydraulic paths leads to a sudden decrease of performance for speeds lower than the full mechanical point; at speeds higher than the full mechanical point the overall efficiency is close to the CVT hydraulic efficiency as the power transferred by the hydraulic path increases. The mean overall efficiency of the optimum transmission in the range 0–16 km/h is 0.785, which could be considered a good result for such a small and compact solution. Fig. 19 shows the rotational speed of the two hydraulic units compared to the maximum allowable speed for the considered displacement. The displacement of the hydrostatic units is instead reported in Fig. 20, while the differential pressure inside the hydrostatic system is reported in Fig. 21. 4.1.3. Test case 2: Agricultural tractor The optimum design parameters are presented in Table 4. Fig. 22 shows the efficiency of the overall transmission and of the hydrostatic CVT element for the design velocity range. The initial choice of a gear ratio of 4 seems to be appropriate in terms of efficiency because the transmission presents the highest
Fig. 22. Overall efficiency and hydrostatic path efficiency for the agricultural tractor.
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Fig. 23. Unit's speed and speed limits for the agricultural tractor.
efficiency in the velocity ranges typical of the most power expensive condition: in field work and street transport. In fact, the transmission shows efficiency higher than 85% in the low velocity range (4–10 km/h), which corresponds to the typical range of field working velocity, while the maximum efficiency of the high gear is instead compatible with the street transport condition. The mean overall efficiency of the transmission is 84%, which is notably higher than the standard proposed by Renius for CVT hydraulic transmission [3]. As for the previous test case, Figs. 23–25 complete the description of the system behavior, giving the unit's speed, displacement and the differential pressure of the HST system.
5. Conclusions The advantage of the continuous speed variation of the power-split drives is counterbalanced by a reduced efficiency, caused by the double energy conversion taking place in the hydrostatic transmission. Therefore, the design of the power split drive must be carefully studied. In this paper, the design is treated as an optimization problem, where the objective function to be minimized is the total loss of the transmission, and the free variables are the displacements of the hydraulic machines and the gear ratios of the ordinary and planetary gearings.
Fig. 24. Unit's displacement as a function of the vehicle ground speed for the agricultural tractor.
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Fig. 25. HST pressure as a function of the vehicle ground speed for the agricultural tractor.
An automatic procedure, made up by a steady state simulator of the driveline and an evolutionary optimizer, was set up to solve this problem. The steady simulator takes into account the losses of the hydraulic units, which heavily influence the driveline efficiency, and the losses of the other mechanical components. A “direct search” algorithm based on the swarm method was used as the optimizer, which showed a good speed convergence and the ability to overcome local minima. The proposed approach offers significant advantages over traditional design methods: the procedure does not depend on experience and previous knowledge because no assumption had to be made on the component's sizing; the optimality of the output is based on the implemented search algorithm while the quality of the classical designs depends strongly on the designer's experience. Furthermore the procedure can handle more complex constraint formulation than traditional design methods and verify them for all the working conditions. The procedure was then tested on two heavy duty vehicles, which were chosen to be very different in terms of power, size and velocity range: a compact loader with a nominal power of 62 kW, and an agricultural tractor of 180 kW. The procedure demonstrated to be capable of handling the optimized design of both test cases, fulfilling all the design constraints and obtaining high efficiency drivelines. The procedure could also be further developed in order to be applied in more complex power split configurations, such as the “dual stage” and the compound solutions.
Nomenclature m torque [Nm] n iteration [–] p pressure [Pa] v velocity [m s − 1] z gear tooth number [–] M transmission torque matrix [–] P power [W] T = − zs/zr standing gear ratio of the planetary gear [–] V maximum unit displacement [m 3]
Greek letters α actual to maximum displacement ratio [–] ηv volumetric efficiency [–] hydro mechanical efficiency [–] ηhym τ gear ratio [–] gear ratio of gearbox between power split transmission and axle [–] τout ω rotational speed [rad s − 1]
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χ Ω
non dimensional penalty coefficient for unit efficiency calculation [–] transmission velocity matrix [–]
Subscript I II rif c s r ′
and Superscripts hydrostatic transmission element inlet shaft hydrostatic transmission element outlet shaft condition of unit maximum efficiency planetary gear, carrier shaft planetary gear, sun shaft planetary gear, ring shaft unit shaft
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Acknowledgment The results presented in the paper were obtained as part of the research program “Studio di trasmissioni power-split per trattrici agricole” of the PRIN 2007 (Programmi di Ricerca di Interesse Nazionale — Research Programs of National Interest 2007) funded by Ministero dell'Istruzione, dell'Università e della Ricerca (Ministry of Education, University and Research) of Italian Republic. References [1] J.H. Kress, Hydrostatic power splitting transmissions for wheeled vehicles — classification and theory of operation. SAE paper 680549, Society of Automotive Engineers, Warrendale, PA, 1968. [2] F. Jarchow, Leistungsverzweigte Getriebe (Power split transmissions), VDI-Z. 106 (6) (1964) 196–205. [3] K.T. Renius, R. Resch, Continuously Variable Tractor Transmissions, ASAE Distinguished Lecture Series, Tractor Design No. 29, 2005. [4] E. Orshansky, W.E. Weseloh, Characteristics of Multiple Range Hydromechanical Transmissions. SAE Paper 720724, Society of Automotive Engineers, Warrendale, PA, 1972. [5] J. Kirejczyk, Continuously Variable Hydromechanical Transmission for Commercial Vehicle By Simulation Studies. SAE paper 845095, Society of Automotive Engineers, Warrendale, PA, 1984. [6] W.A. Ross, Designing a Hydromechanical Transmissions for Heavy Duty Trucks. SAE paper 720725, Society of Automotive Engineers, Warrendale, PA, 1972. [7] C. Blake, M. Ivantysynova, K. Williams, Comparison of operational characteristics in power split continuously variable transmissions, SAE 2006 Commercial Vehicle Engineering Congress & Exhibition, SAE Technical Paper Series 2006-01-3468, October 2006. [8] A. Krauss, M. Ivantysynova, Power split transmissions versus hydrostatic multiple motor concepts — a comparative analysis, SAE 2004 commercial vehicle engineering congress & exhibition, SAE Technical Paper Series 2004-01-2676, October 2004. [9] J. Miller, Comparative assessment of hybrid vehicle power split transmissions, 4th VI Winter Workshop Series, 2005. [10] R. Kumar, M. Ivantysynova, K. Williams, Study of energetic characteristics in power split drives for on highway trucks and wheel loaders, SAE 2007 commercial vehicle engineering congress & exhibition, SAE Technical Paper Series 2007-01-4193, October 2004. [11] Matlab 2008, copyright 1984–2008, The MathWorks Inc. [12] AMESim 2008, copyright 1996–2008, LMS imagine SA. [13] D. Mikeska, M. Ivantysynova, Virtual prototyping of power split drives, Proc. Bath Workshop on Power Transmission and Motion Control PTMC2002, Bath, UK, 2002, pp. 95–111. [14] P. Casoli, A. Vacca, G.L. Berta, S. Meletti, A numerical model for the simulation of diesel/CVT power split transmission, ICE2007 8th International Conference on Engines for Automobile; Capri, Naples, September 16–20, 2007. [15] Shu-Kai S. Fan, Liang Yun-Chia, Zahara Erwie, A genetic algorithm and a particle swarm optimizer hybridized with Nelder–Mead simplex search, Comput. Ind. Eng. 50 (2006) 401–425. [16] W.E. Wilson, Positive-Displacement Pumps and Fluid Motors, Sir Isaac Pitman and Sons, LTD, London, 1950. [17] http://www.boschrexroth.com/mobile-hydraulics-catalog/Vornavigation/VorNavi.cfm?Language=EN&VHist=g54076&PageID=g5406917. [18] J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 7 (1965) 308–313. [19] http://www.samhydraulik.it/public/samhydraulik/it/prodotti/AxialPistonMotors.html.