power split transmition

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 a b s t r a c t

Article history:Received 24 November 2010Received in revised form 6 May 2011Accepted 14 July 2011Available online 1 September 2011

The increasing attention to comfort, automation and drivability is pushing the drivelinetechnology to ever complex solutions, such as power-shift or continuously variable trans-missions. Between these, the hydro-mechanical solution seems promising for heavy dutyvehicle, due to the reliability and the capability of transferring high power. However, thedouble energy conversion occurring in the hydraulic branch of the transmission could lowerexcessively 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 optimizationproblem in which the objective function is the average efficiency of transmission, while thedesign variables are the displacements of the two hydraulic machines and gear ratios ofordinary and planetary gears. The optimization problem is solved by a “direct search” algorithmbased on the swarm method, which showed a good speed convergence and the ability toovercome local minima.The optimization design method will be applied to study the transmission of two vehicles: a62 kW compact loader and a high power agricultural tractor.

© 2011 Elsevier Ltd. All rights reserved.

Keywords:Hydro-mechanical transmissionPower split transmission

1. Introduction

The growing role of agricultural mechanization in worldwide food programs and the need to save fossil energy goes towardsreducing fuel consumption and emissions of agricultural machinery. This compels to reconsider the architecture of the vehicle'spropulsion system, addressing the designer toward more sophisticated and efficient solutions. The same trend holds for earth-moving and construction machinery.

In recent years, the evolution of the transmissions of agricultural and working machines was addressed to increasecomfort and drivability without penalty of efficiency and, possibly, costs. These objectives involve the use of continuoustransmissions that allow the elimination of the shifting gears, which is sometimes exhausting, and keep the fuel consumptionto 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 notcontrollable 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 doubleenergy conversion occurring in it.

The mechanical transmission, in its variants (metal chain, toroidal), is characterized by high efficiency, although sometimeswith 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 andstill high costs.

Mechanism and Machine Theory 46 (2011) 1901–1919

⁎ 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 transmissionwas designed, in which the power is transmitted partly via amechanical path, i.e. in anefficient path, and partly through a CVT. The two powers are then summed by means of a planetary gearing. In this way, thetransmission is still a continuously variable transmission but with efficiency higher than that of the CVT taken individually, sincethe transmission efficiency can be calculated as the weighted efficiency of the two paths. This type of transmission is called powersplit transmission.

The power-split configuration currently producing the best compromise between efficiency and cost seems to be the hydro-mechanical power split transmission, where the variable element is the hydrostatic transmission. It offers significant growthpotential 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 twoconfigurations 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 reachspeeds of 50 km/h. In 2000, the Steyr S-Matic was launched, followed by ZF (2001) and John Deere's “Auto Power”; all three haveinput 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], Kressstudies the performance of the four solutions in Figs. 1–4, assuming as constant, for simplicity, the efficiency of the hydrostaticsection. Kress does not suggest a best configuration among those examined, but highlights the potential of the power split solutionfor applications in traction. Resch and Renius [3] provide a comprehensive overview of the problem of hydro-mechanicaltransmissions. They define the minimum characteristics of a transmission technology for agricultural machines, in terms of speedand 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), andshows the advantages, while Kireijczyk [5] simulates the operation of a transmission of a similar configuration. Ross [6] outlinesthe 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 theircomplex versions (dual stage and compound, respectively); it follows that the complex solutions are suitable for large power andparticularly for agricultural tractors.

Krauss and Ivantysynova [8] observed that the purely hydrostatic solution with twin motors is still interesting for its simpleconstruction 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]: theypropose a dual stage drive with inertial energy storage for commercial vehicles and earth moving; the energy savings are up to22%.

The design of such a complex system requires the use of specific calculation codes in order to test different preliminary designsand find the most efficient solution. Two different approaches may be used in order to model the transmission system: the steadystate approach or the time dependent approach. Several commercial general purpose codes [11,12] refer to the latter, while manyacademic “ad hoc” codesmake use of the former [13,14]. Even though less detailed than dynamic analysis, the steady state analysisis generally faster and can still supply useful information on the transmission performance, as long as detailed loss functions forthe hydraulic machinesmodels are used, i.e. loss models taking into account the influence of the pressure, speed and displacement[13]. For example Ivantysynova andMikeska [13] proposed a modular software that enables the creation of various configurationsand their steady state performance simulation. The accuracy of results is guaranteed by loss models for volumetric machines of thepolynomial type; the models are derived from the measurement of a large number of machines. This same software tool is thenused by the research group in subsequent works [7,8,10]. Even Casoli et al. [14] proposed a procedure for the stationary simulationof an input coupled transmission of a commercial machine; similar to [13], the efficiencies of hydrostatic machines are polynomialexpressions derived from experimental data.

A tool like the one proposed in [13] is essentially a verification tool, which allows the performance evaluation of a fixedconfiguration whose design parameters (the displacement of machinery and gear ratios of the different gearing) have been

Fig. 1. Layout of the input coupled configuration.

1902 A. Macor, A. Rossetti / Mechanism and Machine Theory 46 (2011) 1901–1919

selected in advance by appropriate criteria. An optimization tool, on the contrary, could provide the optimal values of theseparameters allowing good preliminary design choices.

Since the problem is strongly nonlinear, direct search algorithms could be preferred to classical numerical methods, becausethey 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 highspeed 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 hydraulicmachine's efficiencies depending on the operating conditions, and a particle swarm method as optimizer; the average efficiencyover the entire vehicle speed range is used as an objective function, since it is related to the actual fuel economy of the vehicle. Theaim of the procedure is the sizing of the hydrostatic machines and the calculation of the transmission ratios of planetary andordinary 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 thecomponents of these vectors are the angular velocities and torques of all the transmission shafts. These vectors are related to thetransmission layout, by means of a nonlinear equation system:

Ω →ω;→m� ��

·→ω = bωM →ω;→m� �� ·→m = bm:

ð1Þ

The matrices Ω and M were obtained from kinematic and dynamic relations imposed by individual components andconnections 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. 2. Layout of the output coupled configuration.

Fig. 3. Layout of the hydraulic differential configuration.

1903A. Macor, A. Rossetti / Mechanism and Machine Theory 46 (2011) 1901–1919

image of Fig.�2

image of Fig.�3

of the machines, however, restrict this range, forcing the introduction of gears τI and τII in order to adjust the speed of eachhydrostatic unit inside an allowable range. The transmission ratios are defined as follows:

ω0I = τIωI

ω0II = τIIωII :

ð2Þ

The fundamental equations regulating the power transmission through the unit are listed below and are derived in idealconditions directly from the law of continuity and from the uniqueness of the pressure drop produced by the two units:

τIαIVI −τIIαIIVII½ � ωIωII

� �= 0 ð3Þ

1τIαIVI

− 1τIIαIIVII

� mImII

� �= 0 ð4Þ

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:

τIαIVIηkv I −τIIαIIVIIη

−kv II

h i ωIωII

� �= 0 ð5Þ

ηhymIηn Igear

�k

τIαIVI−

ηhymIIηn IIgear

�−k

τIIαIIVII

24

35 mI

mII

� �= 0 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 inthe opposite case; the efficiencies η v and ηhym are respectively the volumetric and hydro-mechanical efficiency of units I and II,

Fig. 4. Layout of the compound configuration.

Fig. 5. Hydrostatic CVT.


image of Fig.�4

image of Fig.�5

whereas the η gear takes into account themechanical losses in the ordinary gears. The exponents nI and nII represent the number ofgear pairs needed to achieve the gear ratio τI and τII; they are estimated on the basis of τmax, themaximumallowable value for eachgear pairs:

nI = int τI = τmaxð Þ ð7Þ

nII = int τII = τmaxð 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 assumedequal to 2.

In order to take into account the influence on operating conditions, i.e. influence of pressure, speed and displacement, thevolumetric and hydro-mechanical efficiencies of the hydraulic machines were expressed by the following relationship:

ηv = ηref ·χvω

ωωmax

� ·χv

α;p α;Δp

Δpmax

� ð9Þ

Fig. 6. Effect of speed on volumetric efficiency.

Fig. 7. Effect of pressure and actual to maximum displacement ratio on volumetric efficiency.


image of Fig.�6

image of Fig.�7

ηhy = ηref ·χhyω

ωωmax

� ·χhy

α;p α;Δp

Δpmax

� : ð10Þ

ηref is the reference efficiency value, assumed to be 0.96 for both efficiencies, reduced by coefficients χω and χα,p. The formercoefficient 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 fromWilson-type [16] loss models, whose coefficients were calibrated according to a setof 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. 8. Effect of speed on hydro-mechanical efficiency.

Fig. 9. Effect of pressure and actual to maximum displacement ratio on hydro-mechanical efficiency.


image of Fig.�8

image of Fig.�9

The kinematic equations used for the three shaft gear are the following:

1 0 −11 1 0

� ω1ω2ω3

8<:

9=; = 0 ð11Þ

1 −1 1½ �m1m2m3

8<:

9=; = 0 ð12Þ

in which a unit transmission ratio has been used. Any reduction or multiplication ratios, as well as the gears efficiency, can beconsidered 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:

1−Tð Þ −1 + T½ �ωcωrωs

8<:

9=; = 0 ð13Þ

Fig. 10. Scheme of a three shaft ordinary gear (a), and of a simple planetary gear (b).

Fig. 11. Scheme of a possible power split configuration.


image of Fig.�10

image of Fig.�11

1 1−Tη−t0

�0

0 Tη−t0 1

24

35 mc

mrms

8<:

9=; = 0 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 powerfrom 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 intoaccount the links between elements are then added; for the case of Fig. 11 (input coupled) they are:

ω3−ωs = 0ω2−ωI = 0ωr−ωII = 0

8<: ð15Þ

and similarly for the torques:

m3 + ms = 0m2 + mI = 0mr + mII = 0

:

8<: ð16Þ

Fig. 12. Flow chart of the iterative solution of the system.


image of Fig.�12

The last constraint necessary to make the system determined is the speed and the torque of one of the two extreme shafts. Foran input coupled drive (Fig. 11), where the engine is connected to shaft 1, the equations are: ω1=ωengine andm1=mengine. In thisway, a system of 8 equations and 8 unknowns is obtained.

The model can then be summarized in its matrix form:

Ω·→ω = bωM·→m = bm:

ð17Þ

The variable efficiencies of the hydraulic machines don't allow the direct solution of Eq. (17). An iterative solution, whichassumes unitary efficiencies at the first iteration, has been set up, with convergence tolerance equal to 0.1%. The flow chart of theiterative process is shown in Fig. 12.

3. System optimization

The simulationmodel was coupledwith an evolutionary optimization system, which is based on the swarmmethod, in order toidentify 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 = x1 x2 :: xn½ �T minimizing f xð Þ; subject to the constraints gj xð Þ ≤ 0 j = 1::m and lj xð Þ = 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 theoptimization 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 splittransmission and the wheel axis.

Fig. 13. Overall dimension of the test case vehicle [mm].


image of Fig.�13

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: ω 'I and ω 'I ≤ β(α) ωmax(V).

The maximum speed of the hydraulic units was considered to be a function of the nominal displacement, (ωmax (V)). Becausethe maximum speed of the units refers to the full displacement condition, an overspeed factor β (α) was added to model theincrease 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 distancefrom the allowed domain.

3.4. Objective function

The purpose of the optimization was the minimization of fuel consumption caused by transmission losses. Therefore, theobjective function is the integral average loss calculated between the zero speed andmaximum design speed of the vehicle:

e xð Þ =∫

vmax

01−ηDrivelineð 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 evolutionaryalgorithms, which are able to reduce the importance of the initial research point and the trapping in local minima far from absoluteminimum.

Table 1Design data of the test vehicles.

Compact loader Tractor

I.C.E. power 62 kW 180 kWI.C.E. design speed 2500 rpm 2200 rpmWheel radius 0.42 m 0.965 mMaximum wheel pulling force 28 kN 120 kNMaximum vehicle speed 16 km h−1 40 km h−1

Table 2Optimization constraints.

Symbol Min Max

Hydrostatic systemUnit displacement V 28 cm3 250 cm3

Maximum pressure Δpmax – 400 barMaximum unit speed 0 Figs. 14 and 15

Planetary gearingStanding gear ratio T −1/2 −1/7


The search algorithms based on swarm (PSO Particle Swarm Optimizer) are stochastic search algorithms based on socialbehavior demonstrated by flocks of birds or swarms of insects, in which the interaction between the individual information andthe 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 inrandom positions in the first iteration), between two subsequent iterations. The position of the i-th particle in the space of theallowable solutions at the n+1 iteration is obtained by summing a displacement d to the position x of the same particle at the n-thiteration as follows:

xin+1 = xin + din+1: ð19Þ

The displacement is computed according to the following definition:

din+1 = wdi

n + c1rand yin + xi

n

�+ c2rand yn + xi

n

�: ð20Þ

Fig. 14. Maximum speed at full displacement as a function of the unit maximum displacement [17].

Fig. 15. Over speed factor as a function of the actual to maximum displacement [19].


image of Fig.�14

image of Fig.�15

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 newenvironmental changes suggest to modify them. This term is ruled by the inertial coefficient w, which multiplies thedisplacement at the previous iteration;

• the second one represents the tendency of any singular individual to move toward the position of best fitness, according to itsown experience. The position yn

i is, in fact, defined as the best position discovered by the i-th particle from the beginning of theoptimization to the current iteration. This component is weighted by the so-called “self confidence” coefficient c1 and theuniformly 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 thedisplacement is to move toward the position yn, which is the position of best fitness ever discovered by the swarm. Thiscomponent 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 convergencecriterion used was based on the mean dispersion of the swarm, assuming the criterion was fulfilled when at least 80% of theparticles 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 PSOwith the Nelder–Mead algorithm [18]. This formulation improves the efficiency of local convergence, keeping the researchcapabilities of the algorithm unchanged.

Fig. 16. Swarm dispersion as function of the optimizer iterations; (a) compact loader, (b) agricultural tractor.


image of Fig.�16

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 powertractor. The dimensions and the design data of the two vehicles are shown respectively in Fig. 13 and Table 1. In Table 2 the designconstraints are summarized. The maximum unit speed is calculated (section 3.3) as the product of the maximum speed and theover speed factor, respectively represented in Figs. 14 and 15. The maximum speed at full displacement was defined as a functionof the unit size and could be well interpolated as a linear trend in double logarithmic axes, with smaller units characterized byhigher velocities (Fig. 14). The over speed factor is instead described as a function of the actual to maximum displacement and

Fig. 17. Best fitness as function of the optimizer iterations; (a) compact loader, (b) agricultural tractor.

Table 3Optimization results for the compact loadertest case.

T −0.44VI 30 cm3

VII 83 cm3

τI 1.66τII 2.35τout 17.15ηmean 0.785


image of Fig.�17

expresses the capability of the unit to withstand higher velocity at partial displacement (Fig. 15). The used data were derived fromonline 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 thanwith the output coupled configuration [7]. For the agricultural tractor, instead, anoutput 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 themaximum force exerted by the tractor, a two-speedgearbox with a transmission ratio equal to 4 was placed downstream from the CVT transmission. The speed change point waschosen where the transmission efficiency line in the first gear crossed the transmission efficiency line in the second gear.

η1st gearDriveLine vchange

�= η2nd gear

DriveLine vchange �

: ð21Þ

The objective function was then rewritten for this case as:

e xð Þ =∫ vchange0

1−η1st gearDriveline

�dv + ∫ vmax

vchange1−η2nd gear

Driveline

�dv

vmax: ð22Þ

Fig. 18. Overall efficiency and hydrostatic path efficiency for the compact loader.

Fig. 19. Unit's speed and speed limits for the compact loader.


image of Fig.�18

image of Fig.�19

4.1. Results

4.1.1. Optimizator performanceThe optimization procedure took 6.8103 s and 1.1104 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 byconsidering both the objective function value and themean 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 thenearest 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 controlvariable (r80%);

• the average distance from the best position considering only the nearest 50% of the swarm particles (r50%).

In both the cases, the three linesmove parallel to each other on the logarithmic scale. This proves the capability of the algorithmto balance between the refinement of the search near the current actual optimum and the scouting of the design space in order toavoid early convergence to a local optimum. Half of the swarm, in fact, searches in the 10% of the design space near the current

Fig. 20. Unit's displacement as a function of the vehicle ground speed for the compact loader.

Fig. 21. HST pressure as a function of the vehicle ground speed for the compact loader.


image of Fig.�20

image of Fig.�21

optimum for most of the convergence history (r50%b0.1), while the significant difference between the r80% and r100% demonstratesthat 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 highlydiscontinuous trend shows that most of the improvements are due to a radical change in the transmission configuration instead ofthrough 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. 16and 17: the steep increase of the best fitness is related to a sudden expansion of the average swarm radius, proving therepositioning 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 high-fitness optimum, for both cases.

4.1.2. Test case 1: compact loaderThe optimum design of the compact loader is reported in Table 3, while Figs. 16 to 19 report the behavior of themost 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 betweenmechanical and hydraulic paths leads to a sudden decrease ofperformance for speeds lower than the full mechanical point; at speeds higher than the full mechanical point the overall efficiencyis close to the CVT hydraulic efficiency as the power transferred by the hydraulic path increases. The mean overall efficiency of theoptimum transmission in the range 0–16 km/h is 0.785, which could be considered a good result for such a small and compactsolution.

Fig. 19 shows the rotational speed of the two hydraulic units compared to the maximum allowable speed for the considereddisplacement. The displacement of the hydrostatic units is instead reported in Fig. 20, while the differential pressure inside thehydrostatic system is reported in Fig. 21.

4.1.3. Test case 2: Agricultural tractorThe 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

Table 4Optimization results for the agriculturaltractor.

T −0.29VI 156 cm3

VII 244 cm3

τI 3.82τII 3.97

Fig. 22. Overall efficiency and hydrostatic path efficiency for the agricultural tractor.


image of Fig.�22

efficiency in the velocity ranges typical of the most power expensive condition: in field work and street transport. In fact, thetransmission shows efficiency higher than 85% in the low velocity range (4–10 km/h), which corresponds to the typical range offield 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 CVThydraulic transmission [3].

As for the previous test case, Figs. 23–25 complete the description of the system behavior, giving the unit's speed, displacementand 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 bythe double energy conversion taking place in the hydrostatic transmission. Therefore, the design of the power split drive must becarefully studied.

In this paper, the design is treated as an optimization problem, where the objective function to be minimized is the total loss ofthe transmission, and the free variables are the displacements of the hydraulic machines and the gear ratios of the ordinary andplanetary gearings.

Fig. 23. Unit's speed and speed limits for the agricultural tractor.

Fig. 24. Unit's displacement as a function of the vehicle ground speed for the agricultural tractor.


image of Fig.�23

image of Fig.�24

An automatic procedure, made up by a steady state simulator of the driveline and an evolutionary optimizer, was set up to solvethis problem. The steady simulator takes into account the losses of the hydraulic units, which heavily influence the drivelineefficiency, 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 convergenceand the ability to overcome local minima.

The proposed approach offers significant advantages over traditional design methods: the procedure does not depend onexperience and previous knowledge because no assumption had to be made on the component's sizing; the optimality of theoutput is based on the implemented search algorithmwhile the quality of the classical designs depends strongly on the designer'sexperience. Furthermore the procedure can handle more complex constraint formulation than traditional design methods andverify 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 andvelocity range: a compact loader with a nominal power of 62 kW, and an agricultural tractor of 180 kW. The proceduredemonstrated to be capable of handling the optimized design of both test cases, fulfilling all the design constraints and obtaininghigh 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.

Nomenclaturem 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/zrstanding gear ratio of the planetary gear [–]V maximum unit displacement [m3]

Greek lettersα actual to maximum displacement ratio [–]ηv volumetric efficiency [–]ηhym hydro mechanical efficiency [–]τ gear ratio [–]τout gear ratio of gearbox between power split transmission and axle [–]ω rotational speed [rad s−1]

Fig. 25. HST pressure as a function of the vehicle ground speed for the agricultural tractor.


image of Fig.�25

χ non dimensional penalty coefficient for unit efficiency calculation [–]Ω transmission velocity matrix [–]

Subscript and SuperscriptsI hydrostatic transmission element inlet shaftII hydrostatic transmission element outlet shaftrif condition of unit maximum efficiencyc planetary gear, carrier shafts planetary gear, sun shaftr planetary gear, ring shaft′ unit shaft

Acknowledgment

The results presented in the paper were obtained as part of the research program “Studio di trasmissioni power-split pertrattrici 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 ItalianRepublic.

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