POLITECNICO DI BARI
DOTTORATO DI RICERCA
IN INGEGNERIA DELLE MACCHINE
XX Ciclo
Curriculum:Macchine a fluido (SSD ING-IND/08)
Sede di Bari
M
ODELLING,
O
PTIMIZATION AND
VERIFICATION OF POWER SPLIT
INFINITELY VARIABLE
TRANSMISSIONS
Salvatore Schembri Volpe
Relatori:
Dott. Ing. Giuseppe Carbone
Prof. Ing. Michele Napolitano
Dott. Ing. Enrico Sedoni
Controrelatori:
Prof. Ing. Massimo Borghi
Coordinatore:
Prof. Ing. Michele Napolitano
One must still have chaos in oneself
to be able to give birth to a dancing star.
F. Nietzche
on Earth, I will keep on running
on Air, I will keep on flying
on Water, I will keep on rowing
and the Fire inside me,
will always keep on burning
SSV
to Lucia
and to all of my Family
Abstract
The author presents an optimization procedure to design infinitely variable
transmission architectures which allows them to achieve a significant reduction
of power recirculation and, hence, an increase in mechanical efficiency. The
focus of this thesis is on infinitely variable transmissions used in off-highway
vehicles and in particular on input coupled and output coupled architectures.
The optimized solutions have been analyzed in depth, with particular attention
to the power flowing through the infinitely variable unit, which strongly
influences the overall efficiency of the transmission. The major result of this
study is that also the so far neglected output coupled solution, if properly
optimized, guarantees very good performance over the entire range of vehicle
speed. The analysis then shows that the particular choice of either input or
output coupled architecture by itself, or of a mixed solution, strictly depends on
the specific application under consideration and that none of them should be
discarded a priori.
Robust control systems playing a crucial role in order to guarantee human
operator safety and overall vehicle performance in different working conditions,
a virtual verification process is described focusing on the Model Based
Engineering, which allows to reduce the number of prototypes and, hence, lower
costs and development time.
Preface
This work has been carried out as part of a collaboration program between the
Politecnico di Bari and Case New Holland - Product Development. This being
so, the academic research has been applied to the industrial automotive context,
with particular focus on the Global Product Development process of power split
infinitely variable transmissions for off-highway, agricultural and construction
equipment vehicles.
The thesis aims to show an original and effective approach, based upon a design
optimization procedure and a virtual verification methodology, which turned out
to significantly improve the transmission performance, reducing the time to
market and the costs related to the product development.
Therefore, this study not only provided an original contribution for the
conceptual and performance analysis of infinitely variable transmissions, but
also constitutes a valid and helpful tool for the design engineers in order to
improve product quality and reliability, with a significant impact on the time to
market and on the product cost.
Chapter 1 provides a general introduction on the thesis work.
Chapter 2 provides an overview on agricultural and construction equipment
vehicles, highlighting the main technological features such as the Power Take
Off, the hydraulic and electronic systems, with a general description of the
general requirements during working conditions. Particular attention has also
been devoted to the
Global Product Development
process.
Chapter 3 provides a general survey on continuously and infinitely variable
transmissions, describing the state-of-the-art technological solutions available
and their general advantages and drawbacks. Specific attention is also paid on
the suitability of each solution to the off-highway vehicles.
Chapter 4 focuses on power split infinitely variable transmissions. A thorough
analysis of power and torque flows is provided, with particular attention to the
problem of power recirculation. Both the input and output coupled architectures
have been analyzed, highlighting the potential benefits of mixed solution.
An effective approach to the design phase is then presented by means of an
optimization procedure to minimize power recirculation through the variable
speed unit. This process allows to achieve a significant reduction of power
The analysis described in this chapter has been accepted for publication on the
ASME Journal of Mechanical Design.
Chapter 5 provides a general overview of a virtual verification process based
upon Model Based Engineering. Following the optimization procedure
described in Chapter 4, the optimized product specifications are used to develop
a dynamic model of the physical system which allows to perform
closed-loop
simulations.
Contents
1 Introduction 3
2 Fundamentals of agricultural machines 7
2.1 Transaxle and Power Take Off . . . 7
2.2 Hydraulic systems . . . 8
2.3 Electronic systems . . . 10
2.4 Performance and typical working operations . . . 11
2.5 Global Product Development for agricultural and construction equipment machines . . . 13
3 Introduction to Continuously and Infinitely Variable Transmissions 16 3.1 CVU types and principles . . . 17
3.1.1 Hydrodynamic torque converter . . . 17
3.1.2 Mechanical CVUs . . . 18
3.1.3 Hydrostatic CVUs . . . 21
3.1.4 Electric CVUs . . . 24
4 Design optimization of power split IVTs 27 4.1 Principles of power split IVTs . . . 27
4.2 Kinematic analysis of a power split IVT . . . 30
4.2.1 Input coupled equations . . . 32
4.3 Powerflow analysis . . . 36
4.3.1 Input coupled power flows . . . 37
4.3.2 Output coupled power flows . . . 41
4.4 Design optimization of power split IVTs . . . 45
4.4.1 Optimization problem formulation . . . 46
4.4.2 Numerical implementation of the optimization process . . . 48
4.4.3 The optimization algorithms . . . 49
4.4.4 Simulated Annealing . . . 51
4.4.5 Results . . . 53
4.5 Optimization conclusions . . . 56
5 Virtual verification of power split IVTs 61 5.1 Current scenarios for off-highway vehicles . . . 62
5.2 Model-Based Design . . . 64
5.3 Plant Model . . . 65
5.3.1 The actuation systems . . . 67
5.3.2 Driveline model . . . 70
5.4 Model and Software in the Loop . . . 71
5.5 Hardware in the Loop . . . 72
5.5.1 HIL simulations and automatic test sequences . . . 75
6 Conclusions 81
7 Acknowledgements 83
Chapter 1
Introduction
Automotive manufacturers are facing significant challenges arising from the continuous evolution of the market demand, lawmakers decisions in terms of polluting emissions and new compelling technologies. In particular, the product development process has to deal with tighter cost and time targets in order to increase profitability, gain and maintain a sustainable competitive advantage versus competitors: customer needs have to be properly understood and translated into effective and efficient technical solutions, matching cost targets and minimizing the time to market.
Agricultural and construction equipment machines are of utmost importance for the world’s nutrition and housing needs, with a technology content evolving to high sophis-ticated mechatronic systems in the developed countries. The tractor and the harvester remain still the most important machines and their transmission system is a key compo-nent representing about 35-40% of the total tractor first cost.
In the last few decades, a growing attention has been devoted to the environmental issue. Governments are continuously setting tighter limitations for polluting emissions to reduce green-house gases. In order to fulfil these requirements, automotive manufacturers are obliged to dramatically reduce fuel emissions while increasing vehicle performance and comfort.
au-tomotive industry, in particular in the off-highway market, thanks to their many advan-tages in terms of fuel economy, reduced emissions and human operator comfort. About 100 years ago, battery-driven electrical drives allowed already a continuously variable speed control and low noise levels, though the main problem was the poor capacity for stored energy. Early developments of hydrostatic drives have been introduced in mid 60s, at the same time agricultural engineers invented a friction drive CVT for a self-propelled plough in which the speed control was obtained by the radii of friction contacts.
Recently, off-highway vehicle manufacturers have introduced hi-tech CVTs on almost all the lines of product, with a small exception for the low power machines in which the cost of such a complex transmission cannot be justified.
The scientific literature offers a high number of contributions focusing on CVTs. Carbone, Mangialardi and Mantriota have analyzed the dynamic performance of metal V-belt CVTs and toroidal traction drives [1, 2], in particular for road vehicles [3, 4]. Renius [5] has performed a very useful and detailed analysis of the market demand evolution for agriculture machines, focusing on several state-of-the-art CVT technologies.
The CVU can generally be realized using different technological solutions: hydro-static transmissions with variable displacement units, belt or chain drive, toroidal trac-tion drives, electric groups (generator-inverter-motor). Several technological solutrac-tions for CVUs offer the possibility to obtain a zero output speed with a non zero input speed even without any PGT connection: in this case, the CVU can actually be thought of as an Infinitely Variable Unit (IVU). Examples of IVU can be hydrostatic transmissions with at least one variable displacement unit, or hybrid electric transmissions with generator, inverters and motors.
Generally, both CVUs and IVUs present a lower efficiency compared to that of a fixed ratio mechanical transmission, therefore a direct CVT transmission, i.e., a transmission in which all the input power flows through the CVU, presents a poor overall efficiency and a significant heat dissipation. Furthermore, direct CVTs adopting CVU do not allow to obtain a zero output speed with a non-zero input one (the so-called zeroactive speed).
Power split IVTs are a particular CVT typology that offers the possibility to obtain a zero active speed: they can generally be obtained by coupling either a CVU or a IVU, a Planetary Gear Train (PGT) and a fixed ratio gear. The total power is split into two parts, one flowing through a constant ratio mechanical path and one through the vari-able speed unit. Therefore, adopting power split IVT architectures, the negative effects in terms of power dissipation will influence a reduced amount of total power; nonetheless a continuously variable output speed can be obtained over a wider speed range, in partic-ular with multiple-range architectures. As a main consequence, power split IVTs present an overall efficiency higher than that of a direct CVT, due to the higher efficiency of the mechanical path.
This thesis focuses on applications of virtual modelling and numerical simulations to the entire Global Product Development (GPD) process of power split infinitely variable transmissions for agricultural and construction equipment machines, introducing a thor-ough conceptual analysis of different architectures. The power and torqueflows through the variable speed units have been deeply investigated, since they are strictly related to the overall transmission efficiency. Particular attention has been devoted to the effects of the transmission gear ratios, especially for complex, multiple ranges architectures, for they strongly influence the amount of power recirculation through either the CVU or IVU.
A novel, effective approach for the design optimization process of power split IVTs for agricultural and construction machines has been developed by the means of different state-of-the-art optimization algorithms, which turns out to potentially increase the transmission efficiency, significantly improving the overall vehicle performance.
Robust and effective control system playing a crucial role, the Model Based Design ap-proach will befinally described, in order to implement a virtual verification process before production. The number of prototypes required to develop and test the control strategies can be reduced adopting numerical models which are used to performin-the-loop simula-tions with the control system. As resulting benefits, the overall vehicle performance can
be tested and improved earlier within the GPD process; a very high number of working scenarios can be safely simulated, including failure dangerous conditions; a significant reduction of costs and time can be achieved.
Chapter 2
Fundamentals of agricultural
machines
2.1
Transaxle and Power Take O
ff
The complete tractor transmission, also referred to as transaxle, can be schematically represented as in Fig. 2-1, [6], showing a combination of the vehicle speed change gearbox, the rear axle with brakes, the Power Take Off(PTO) and, if required, arrangements for the front axle drive and for the drive of auxiliary units.
The PTO has a crucial importance for agricultural machines for it is the only way to make the tractor to be a mobile power supply [21]. It is represented by a mechanical shaft generally located on the rear of the tractor which is frequently adopted to connect the machine with several kind of devices, such as trailers, ploughers, haying tools, by means of a mechanical joint. In some cases, a PTO is available also on the front of the vehicle. In general, the PTO has to be able to drive hydraulic pumps and any other auxiliary component, also with non-zero vehicle speed, Fig. 2-2.
In standard applications, the PTO speed is completely independent from the driveline output speed for it takes its motion directly from the primary shaft, typically using an engagement clutch followed by spur gears for different output PTO speeds. Generally,
Figure 2-1: Side view of a tractor transmission
the PTO presents two standard output speeds, namely 540 and 1000 rpm at 2000 rpm of engine speed. In some cases, a so called economic ratio can be found, with the aforementioned output speeds with the engine rotating at about 65% of its maximum power speed.
Modern tractors also offer asynchronized PTO which takes its motion from the output driven shaft of the transmission, therefore the angular speed is synchronous with the tractor wheels speed. This kind of application is especially useful for those kind of trailers that need to work as drivers.
2.2
Hydraulic systems
Following the historic development of the mechanical system, the adoption of hydraulic systems of growing complexity is one of the key feature of the modern agricultural and construction machines, starting in 70s and 80s, making a heavy use of hydraulic compo-nents in order to activate and control main compocompo-nents, such as the loader, the steer-ing system, the four-wheel drive mode, the power take-off, the range shift system, the clutches and the brakes. Fig. 2-3 shows a sample scheme of the hydraulic circuit of a
Figure 2-2: Example of a mechanical PTO, CNH all rights reserved.
light tractor. In recent applications, hydraulic circuits are also employed in combination with other mechanical devices to realize continuously variable transmissions. Combine and forage harvesters typically adopt hydro-mechanical transmissions characterized by multiple hydrostatic units and very prolonged piping systems. Construction equipment machines can operate only thanks to hydraulic systems able to actuate telescopic arms and loaders.
There are two principal typologies of hydraulic circuits: Open Center and Closed Center. Open center circuits, normally adopt such a pump so as to get a constant
flow rate and a variable pressure. The pump is directly actuated by the engine shaft and continuously pressurizes the oil, regardless from the components activated. Closed center circuits, on the other end, normally adopt pumps which provide a constant pressure and variable flow rate. In both cases, the hydraulic circuit only absorbs engine power when some hydraulic components is actuated, in particular for auxiliary ones.
Figure 2-3: Hydraulic circuit of a 100hp tractor. CNH all rights reserved.
2.3
Electronic systems
Starting from the mid 80s, electronic systems showed a strongly growing presence on off -highway vehicles. A high number of Electronic Control Units (ECUs) has been adopted on board for different purposes: (i) to properly measure and process physical signals, (ii) to better detect and respond in different working conditions, (iii) to automatically control the engagement and disengagement of mechanical components. ECUs allow to control in closed loop the proper sequences of transmission ranges, especially in loaded conditions, according to the engine speed and to the output vehicle speed requested by the human operator, as it is in the case of infinitely variable transmissions.
In general, electric systems are adopted to actuate any hydraulic component related to the driveline ranges, i.e. multi-disc clutches, synchronizers, brakes, infinitely variable units, engine speed request. The transmission can indeed be automatically controlled by taking into account the engine speed, the load torque acting on the wheels, the traction load, the vehicle speed and tire slipping.
Figure 2-4: Forces decomposition during towing operations
2.4
Performance and typical working operations
Agricultural and construction machines are mainly meant to provide the following func-tions:
• Provide a traction towing force and guarantee loading operations;
• Supply mechanical power by the means of the PTO;
• Supply hydraulic power thanks to the auxiliary components and to the loader. In particular the towing force is one of the crucial aspects since it allows the opera-tor to work in different scenarios, on different ground conditions and with different working devices, Fig.2-4.One of the key requirements for a tractor is indeed repre-sented by the traction load curve at fixed maximum engine power. The maximum traction load to be guaranteed is generally associated to the Gross Vehicle Weight (GVW) of the ballasted tractor, so as to obtain a behavior as the normalized one shown in Fig. 2-5, representing the so called Nominal Working Cycle, where the
maximum traction load can be saturated either by the tires adherence limits or by any other maximum sustainable load within the transaxle.
0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 Ground Speed (a ) T ra ct io n L o ad 0 0.2 0.4 0.6 0.8 1 Ground Speed (b ) E n g in e P o w er
Figure 2-5: Normalized traction load (a) at maximum rated normalized engine power (b) Table 2.1 shows other typical agricultural applications with their relative vehicle work-ing speed.
Ground speed Application
0,2−2km/h Digging
2−5 km/h Rotary harrows, hoeing
5−10km/h Plowing, packing
10−50 km/h Mowing, sprinkling, transportation
Table 2.1: Ground speed as function of agricultural application
Focusing on the rated engine power, typically four main tractor families can be dis-tinguished which differ in terms of diesel engine type, number of vehicle missions to be accomplished, and comfort level required for the human operator, see Table 2-6.
Figure 2-6: Major classification of tractors
2.5
Global Product Development for agricultural and
construction equipment machines
Global Product Development (GPD) process represents the key strategy at the basis of all Research & Development (R&D) efforts in developing new concepts and technologies to match customer needs and improve product quality and vehicle performance main-taining profitability.
The starting point is represented by the voice of the Customer, who has to actively in-teract with marketing department, namely the Brand, and specific product platforms to express needs and requirements: these information have to be thoroughly understood and translated into effective design concept and technical specifications [20].
Generally, the GPD process for automotive industry can be divided in five main steps, as shown in Fig. 2-7.
Once the customer needs have been properly collected and analyzed, the first step is represented by the Program Planning. At this stage, the product platforms and the marketing department share the potential programs to match the targets that have been set by the Customers, defining the key strategies, the overall program duration - which constitutes the Time To Market (TTM) - and the program milestones. The engineering departments have to be involved as well since this phase is strictly related to theConcept
Figure 2-7: Main steps of the Global Product Development process for automotive in-dustry.
Development which represents the early design process either of a new product or a core feature. In this phase, all the ideas and concepts addressing the targets are taken into consideration so as to create a set of potential technical solutions.
Next step is represented by the Feasibility Analysis, during which engineers and plat-forms thoroughly evaluate advantages and drawbacks related to any potential solution arisen from the concept development. Different considerations and constraints are taken into account, namely costs, legislation, potential markets, competitors offer, available and patented technologies, manufacturing and logistic constraints. The output from this phase can be thought of as the Company’s answer to the Customer in terms of feasible and appropriate product design concepts. If this output is considered appropriate in terms of quality, feasibility and profitability, the program can be considered approved (Program Approval) and the GPD process can move ahead.
The Optimization phase consists in all of those actions and analyses meant to properly define product specifications and features so as to match customer expectations and to maximize the added value. For the specific applications of agricultural and construction
equipment machines, the value added is certainly represented by reduced fuel consump-tion, low emissions, improved vehicle performance, maximum human operator comfort. In this step, very important is the What-If analysis, which is basically meant to thor-oughly investigate as many design solutions as possible, in order to guarantee the optimal product performance given the related design constraints.
Product design specifications have then to be validated and tested in order to ensure prod-uct quality and safety prior to manufacturing: this happens in the Verification phase. Physical and virtual prototypes are verified and tested; also, the control logics and the relative control software are developed and validated.
Last step is the Implementation, in which the product design is frozen and released to manufacturing plants that are now allowed to produce the product (OK to Build) and then forward it to the various Dealers (OK to Ship).
Virtual analysis and numerical simulations turn out to be of utmost importance dur-ing the overall aforementioned process. Product and feature models allow to speed up the early GPD phases so as to be able to detect any potential failure in significant advantage with respect to the latest steps; the optimization phase can be managed in a rigorous and effective way implementing Design of Exploration activities so as to deeply investigate and maximize the product performance by varying the design parameters over a wide range of acceptable values; dangerous and unsafe working conditions can be tested and analyzed minimizing the risks for employees and for the Customer itself.
From an economic point of view, the cost associated to a design problem heavily in-creases in the latest step of the GPD: too late problems can determine the need of extra prototypes, significantly increasing the time and the costs.
In this work, applications of numerical simulations will be shown following the base-line of the GPD process, starting with the model and the analysis of a new product, then showing the benefits obtained from an effective optimization procedure in terms of ve-hicle performance, and finally illustrating astate-of-the-art approach for control system development and test, which significantly supports product robustness and reliability.
Chapter 3
Introduction to Continuously and
In
fi
nitely Variable Transmissions
The transmissions adopted in the automotive industry can generally be split into two main families: stepped and stepless transmissions. The former category represents the most popular solution, especially for European car market; the latter is based on Con-tinuously Variable Transmissions (CVTs) and recently has been widely introduced in the automotive industry, in particular in the off-highway market, thanks to their many advantages in terms of fuel economy, reduced emissions and human operator comfort. Fo-cusing on agricultural machines, CVTs with automatic controls have been introduced in Europe in 1996 for standard tractors, opening a new era of power train design principles. A CVT is a power transmission device which allows to continuously vary the speed ratio between two finite extremes thanks to the adoption of a Continuously Variable Unit (CVU) connected between two mechanical shafts. When the speed ration can be continuously varied between two values including the zero output speed with a non-zero input one, the CVU can be referred to as an Infinitely Variable Unit (IVU).
Infinitely Variable Transmissions (IVTs) are particular kind of CVT that offer the possibility to continuously vary the output velocity including the possibility to obtain a zero output speed wit a non-zero input speed. Therefore, an IVU can be thought of as
Figure 3-1: Main typologies of continuously variable units. an IVT when adopted by itself.
This chapter provides a general survey of the known available technologies in terms of continuously and infinitely variable units, describing the working principles and the main advantages and drawbacks, with specific reference to off-highway applications.
3.1
CVU types and principles
In general, four main CVU typologies can be considered as fundamentals, differing for the physical principle adopted, ratio control system and field of application: hydrodynamic torque converter; mechanical; hydrostatic; electric, see Fig. 3-1, [5].
3.1.1
Hydrodynamic torque converter
The hydrodynamic torque converter has recently achieved the highest production volumes for cars and construction machinery, offering the lowest production cost. On the other hand, this technology presents two major weak points for off-highway applications: (i)
the maximum efficiency is not generally poor, but is available only within a very limited range of transmission ratios; (ii) the speed ratio cannot be controlled in Closed Loop as it is automatically related to the load. A solution to diminish point (i) is to add a free wheeling element or even a clutch blocking the unit, but this would reduce the system effectiveness in terms of continuous output velocity thus requiring a higher number of additional conventional ranges. Weak point (ii) cannot be significantly improved . These are the main reasons why the torque converter have not been successfully adopted, in particular on agricultural machines.
3.1.2
Mechanical CVUs
Mechanical CVUs allow the closed loop ratio control by the means of variable effective radii, whereas the torque is transmitted by the mean of the traction force between the friction contacts. Furthermore, the efficiency is significantly high with respect to the other CVT types. Therefore, mechanical CVUs can be successfully employed in the off -highway markets, though generally limited to the low to medium power applications. An example of mechanical CVT transmission for agricultural machines is shown in Fig. 3-2. In general, two main traction types CVUs are available : V-Belt CVUs and toroidal traction drives.
V-Belt CVUs
The gear ratio variation is obtained by two fixed sheaves with opposing two movable sheaves so that their relative movement allows to change the belt pitch radius at the input and output shafts, Fig. 3-3. The torque is generally a function of the normal force, the friction coefficient and the radius, namely T = μFNr. The normal force FN
is generally obtained using a hydraulic actuation driven with electro valves. In terms of friction, these CVUs are normally lubricated by oil, therefore the maximum usable friction coefficient (steel/steel) ranges between 0.06 and 0.12 depending on the type of
Figure 3-2: Continuously variable transmission realized with a mechanical CVU, Munich Research Tractor 1988.
obtained for instance with rape seed oils.
In general, V-Belt CVUs present the best potential efficiency, likely the highest effi -ciency among the different CVU concepts. If the clamping forces are properly controlled and adjusted to the actual torque load, mechanical full load efficiencies up to 95% can be achieved over a wide range of speed ratio. Actual values are generally smaller due to the losses in the hydraulic actuation and control system. If a simple variable displace-ment pump is used, the efficiency can be reduced down to 90%, whereas a significant improvement to 92.5% can be obtained using a variable displacement pump [5].
In terms of drawbacks, these CVUs present two significant limitations: (i) the CVU transmission range is limited between twofinite values, thus it is not possible to obtain a zero output CVU velocity with a nonzero one; (ii) the transmission output speed range is generally limited, especially with respect to the combination between forward and reverse ranges; (iii) it is not possible to obtain an effective active zero output speed (power-zero) condition.
Figure 3-3: V-Belt chain CVT, concept of PIV
As will be shown in the next chapter, the main consequence of weakness (i) is that V-Belt CVUs cannot be used for output coupled power split architectures.
Toroidal traction drives
Toroidal traction drives make use of power rollers, whose rotating axis is able to modify its position and to change the input and the output contact points. Two main classifications are available: "Full" and "Half" toroidal CVUs, Fig. 3-5.A typical advantage of toroidal CVUs is the potential for high torque capacity with compact design due to the parallel power flow. The slip is generally higher than in the case of chain drive CVUs, and the same applies for drilling friction.
In general, half toroidal present a higher efficiency with respect to full-toroidal case, although in both cases the efficiency drops at high loads and at high speed reductions. Also, like in the case of V-Belt CVUs, the transmission ratio is limited between twofinite
Figure 3-4: Measured mechanical efficiency of a V-belt CVU with a variable displacement pump for the actuation system. Courtesy of P.I.V.
values. These are the most significant limitations for off-highway vehicles, in particular for power split IVTs.
3.1.3
Hydrostatic CVUs
Hydrostatic CVUs are formed by the combination of at least one hydrostatic pump and at least one hydrostatic motor, [5]. Moreover, at least one unit must have a continuously variable displacement. Since these circuits allow to obtain a zero output speed with a non-zero input one, they can be referred as IVUs. A simple scheme of a hydrostatic cir-cuit is shown in Fig. 3-6, [6].In general, these circir-cuits work with a fully reversible variable displacement pump (1) that is connected to the input shaft, and a fully reversible motor (2) that can have constant of variable displacement according to the design architecture adopted and is connected to the output shaft. The charge pump (3), with the safety valve (4), always feeds the low pressure pipe passing through the filter (5) and one of the check valves (6). The surplus oil leaves the low pressure pipe automatically thanks to the flush valve (7) arriving at the tank through the pressure relief valve (8) and the cooler (9).
Figure 3-5: a) Half toroidal and b) Full toroidal CVU geometry
The charging system is often adopted in order to replace oil leakage maintaining a min-imum pressure in the low pressure pipe (namely 20bar), to control oil temperature, to control fluid contamination, to serve as an auxiliary power and to enable high pump speeds.
The pressure relief valves (10) are safety elements. If the charging system fails, an emer-gency re-filling of the circuit is operated by the suction check valves (11), which however are not used in general. If a blow out takes place for a long period of time, thefluid tem-perature can rise to very high values exceeding the IVU limits, determining dangerous failures. That’s one of the main reasons why modern systems adopt pressure limitation without blow out: the pump will decrease its displacement automatically once a pressure signal exceeds a given limit.
For these systems, the torque transmitted is directly related to the circuit pressure and to the displacement of the units, therefore, if the pressure in the circuit is saturated by either the relief valves or the pump itself, also the maximum transmittable torque will be saturated. Therefore, for off-highway vehicles, pressure relief valves can constitute the limit to the maximum towing force defined in Sec. 2.4.
The transmission ratio can be effectively controlled in closed loop by varying the variable displacement of either unit according to the input speed, the desired vehicle ground speed and the load conditions. Different strategies can be adopted in order to optimize vehicle performance and guarantee safe working conditions, Fig. 3-7, shows a standard concept of pump and motor displacement as a function of the ground speed, [5].
Hydrostatic IVUs present a lower efficiency if compared to mechanical CVUs and gener-ally turn out in heavier and more voluminous solutions: these are the main reasons why these solution are not frequently adopted for passenger cars. However, they are success-fully employed for off-highway vehicles and mobile machinery, especially in the medium to high power ranges, where the overall vehicle weight is not a crucial design constraint. Moreover, this kind of transmissions allow the possibility to obtain the complete set of power split Infinitely Variable Transmission (IVT) possible architectures, which provide
Figure 3-7: Sample pump and motor displacement variation as a function of the ground speed.
higher performance, as will be described in the following sections.
Fig. 3-8, shows two typical connections between hydrostatic IVUs and the gearbox for a tractor and a wheel loader transmission, respectively.In order to cover the entire speed range, the IVU is connected to a stepped gearbox with generally 2, 3 or 4 forward ranges and no more than 2 reverse ones. Dual clutch systems and synchronizers can be adopted in order to automatically control the range shift together with the variation of the overall IVU ratio.
3.1.4
Electric CVUs
Electric CVUs constitute an upcoming technology that present significant advantages in terms of reduced noise level, reduced maintenance costs, high effectiveness of control systems with low energy required for the ratio control, environmental sustainability. A significant set of critical points still limit the adoption of such systems on a wide basis: safety aspects related to the high voltage required; high costs of the required components; limited efficiency related to the high number of energy conversions; high volumes and weights associated to the batteries. Toyota Prius, Fig. 3-9, constitutes one of the best
Figure 3-8: Typical connections between a hydrostatic IVU and stepped gearboxes: a) tractor, b) wheel loader machines.
Figure 3-9: Toyota Prius transmission: first commercial hybrid car CVT working with an electric IVU, 1997.
Chapter 4
Design optimization of power split
IVTs
This chapter focuses on the analysis of power split infinitely variable transmissions, fo-cusing on the main concepts and applications. Particular attention will be devoted to the problem of power recirculation, providing a thorough conceptual analysis of power and torque flows, in particular through the variable speed unit, for different possible trans-mission architectures. Then, a novel and effective approach to optimize transmission performance is presented, showing the benefits obtained in terms of overall transmission performance.
4.1
Principles of power split IVTs
Power split IVTs are a particular CVT typology that offers the possibility to obtain a zero output speed with a non-zero input one; IVTs can generally be obtained by coupling a CVU, a Planetary Gear Train (PGT) and a fixed ratio gear. The total power is split into two parts, one flowing through a constant ratio mechanical path and one through either the CVU or the IVU. In general CVUs and IVUs have a lower efficiency compared to the constant ratio path, thus, adopting power split IVT architectures, their negative
effects in terms of power dissipation will influence a reduced amount of total power; nonetheless a continuously variable output speed can be obtained over a wider speed range, in particular with multiple-range architectures. As a main consequence, power split IVTs present an overall efficiency higher than that of a "direct" CVT, due to the higher efficiency of the mechanical path.Fig. 4-1 shows one possible basic configuration
Figure 4-1: Principles of a power split IVT.
for a power split IVT, [5]. In general, the IVU output speed can be reversed according to the working conditions, this meaning that the power flow through the IVU can be reversedflowing from right to left. In this case, the power is superimposed to the system input power and therefore must be transferred through the mechanical path again to the right of the system. This condition is typically referred to as power recirculation, which can strongly affect the overall IVT efficiency. If the value of the power recirculating through the IVU is low compared to the input power, the total system efficiency is still higher than the one of a direct CVT, otherwise, significant power and heat dissipation will occur, deteriorating the overall system performance.
the driven/output shaft, Input Coupled (IC) or Output Coupled (OC) architectures can be obtained, Fig. 4-2.The IC and OC architectures can be thought of as mirrored one to
Figure 4-2: Basic concept of Input (A) and Output (B) coupled concept for power split IVTs, [5].
each other, although their operational behavior is completely different in terms of power recirculation and system efficiency. Renius [5] has performed a qualitative steady state analysis showing the IVT efficiency for the two cases assuming a mechanical efficiency of 97% for each gear meshing and a constant IVU efficiency of 85%, Figs. 4-3 and 4-4.It can be noticed that the best values efficiency are obtained when all the power is transmitted through the mechanical path: this condition is usually referred to as the lockup point. Also, the power split region without power recirculation is characterized by the higher efficiency and limited power and heat dissipation.
In the following sections, a thorough kinematic analysis of input and output coupled architectures is provided with a deep focus on the power and torque flows through the IVU. An effective optimization procedure is the presented to determine the optimal set of transmission gear ratios that minimize the power recirculation according to the specific vehicle nominal working cycle and in compliance with the major design constraints. This analysis constitutes a very helpful tool for the designer for it provides a detailed benchmarking between the IC and OC concepts and avoids the traditionaltrial and error
approach to determine the gear ratios, in particular for multiple ranges architecture with a high number of degrees of freedom, that minimize power recirculation and improve
Figure 4-3: Power and efficiency charateristics for an input coupled power split IVT. vehicle performance.
4.2
Kinematic analysis of a power split IVT
A general useful scheme of input and output coupled power split IVTs can be obtained using the schematic diagram of Fig. 4-5.
Yan and Hsiech [8] have performed a preliminary analysis of both concepts with particu-lar reference to a Differential Transmission (DT); they state: "if an output-coupled DT is used as an IVT and the output of the DT has zero speed, we have a case that the member of the PGT adjacent to the CVU also has zero speed. This is physically impossible when the input member of the DT is in motion". Thus, Yan and Hsiech concluded that only the input coupled architecture can be applied to an IVT. This conclusion is only partially
Figure 4-4: Power and efficiency characteristics of an output coupled power split IVT. correct as it holds true solely for the particular case of CVUs that do not allow a zero output speed with a non-zero input speed, as in the case of expandable-pulley CVUs. Therefore, observing that IVUs allow one to obtain the so calledlockup point, occurring when the IVU output speed is zero with a non-zero input speed, the conclusion given by Yan and Hsiesh does not apply when a IVU (e.g., a hydrostatic transmission) is adopted as
a CVU.
Also notice that IVTs present the Power-Zero condition when the output speed is zero with a non-zero input speed. It follows that if a IVU is used as an IVT without any PGT, the lockup and power-zero conditions will coincide.
The consequence is that the output coupled concept can be reassessed for IVT applica-tions. Examples of IVUs are hydrostatic transmissions with one or two variable units (pump/motor); hybrid architectures composed by an electric circuit with a generator, a
Figure 4-5: Circuit schematic diagrams of input coupled (a) and output coupled (b) architectures
motor and an inverter. Therefore, both the IC and OC architectures are analyzed in this work, which provides a novel contribution on the efficient implementation of an OC IVT. In order to cover the entire vehicle speed range, the great majority of IVT transmissions presents multiple range architectures. In this paper, the authors will concentrate on a dual range, fully synchronized IVT transmission for agricultural machines [10], which is frequently adopted for off-highway transportation. This choice will not limit the gener-ality of the analysis and methodology adopted for both the IC and OC architectures.
4.2.1
Input coupled equations
The conceptual transmission stick diagram of Fig. 4-6 shows a dual range IC IVT archi-tecture, in which the outlet power is always transmitted by the Carrier (C), infirst range, and by the Sun (S2), in second range. By expressing the speed ratio of the PGT in a frame of referencefixed to the carrier, two Willis [5] transmission ratios can be evaluated in terms of the various angular speeds, as:
τwA =
ω5−ωC
ω1−ωC
(4.1) and
Figure 4-6: Input coupled power-split architecture. CNH, all rights reserved.
τwB =
ω5−ωC
ωS2−ωC
, (4.2)
whereωi represents the angular speed of the ith gear. The angular speed of the ring 5 is
related to the input speed through the IVU, so that:
ω5 =τF RG τ∗IV U ω1 =τIV U ω1, (4.3)
τF RG being the fixed ratio of the gears before and after the IVU and τ∗IV U the IVU
transmission ratio. From Eqs. (4.1-4.3), the IVT transmission ratio in first and second ranges are obtained as:
τIIV T = ωC ω1 = τwA−τIV U τwA−1 =τ1τIV T (4.4) and
τIIIV T = ωS2 ω1 = [τwA(τwB−1) + (τwA−τwB)τIV U] τwB(τwA−1) =τ2τIV T, (4.5)
respectively. In Eqs. (4.4) and (4.5) thefirst and second range gear ratios are defined as
τ1 = ωC/ωI and τ2 = ωS2/ωII, respectively ωI and ωII are the corresponding output
angular speeds, and τIV T the global transmission ratio,τIV T =ωout/ωIN.
The specific IVU under evaluation is a hydrostatic transmission made up by reversible variable displacement pump and a reversible bidirectionalfixed displacement motor. Fur-thermore, two independent clutches respectively connect the carrier and the sun to the transmission final reduction in order to obtain a synchronized range shift without any power or speed discontinuity. In standard applications, during the vehicle’s acceleration the variable unit displacement is varied from negative to positive values in first range and from positive to negative values in second range. Therefore the following design constraint results for the transmission ratios:
τwA−τwB <0. (4.6)
Also, defining τIV T_s = ωout_s/ωin the transmission ratio at the range shift point, the
synchronization condition requires the output speeds in first and second ranges to be equal, with the same IVU ratio, i.e.,
¡ τIV U_s ¢I =¡τIV U_s ¢II . (4.7) Since¡τIV T_s ¢I =¡τIV T_s ¢II
, using Eqs. (4.4,4.5) and (4.7), the following constraint for
τwB is obtained as: τwB = τwA ¡ τ1τIV T_s−1 ¢ (τ1−τ2)τIV T_s . (4.8)
Figure 4-7: Output coupled power-split architecture.
4.2.2
Output coupled equations
The conceptual stick diagram for the OC architecture is shown in Fig. 4-7, where the IVU is linked to thefinal gears through the carrier and the sun, infirst and second ranges, respectively.Following the same approach as in Sec. 4.2.1, the ring angular speed will now be defined as ω5 = τIV UωC, and ω5 = τIV UωS2, in first and second range respectively .
The corresponding IVT transmission ratios are:
τIIV T = ωC ω1 = τwA τIV U +τwA−1 =τ1τIV T, (4.9) and τIIIV T = ωS2 ω1 = τwA(τwB −1) τwB(τwA−1) + (τwB−τwA) τIV U =τ2τIV T; (4.10)
τwB =
τ1−τ2 τwA+τ1 τ2 τIV T_s(τwA−1)
(τ1−τ2)
. (4.11)
4.3
Power
fl
ow analysis
Power split IVTs basically present three possible power flows, as analyzed by Yan and Hsiech [8] and Mantriota [11, 12, 13], for the IC case, Fig. 4-8. Here, a thorough power
flow analysis is provided also for the OC architecture Fig. 4-9.
Figure 4-8: Powerflow types for the IC architecture
Figure 4-9: Power flow types for the OC architecture
The type III power flow is the only one which guarantees that the power crossing the IVU is equal to or smaller than the input power [11]. Since IVUs have a lower
efficiency with respect to PGTs, higher efficiencies are obtained with lower power fractions
flowing through the variable path, the optimum being obtained at the lockup point [5]. Thus, type III regions enjoy a high transmission efficiency and a low heat dissipation. Moreover the size of IVU components is strictly related to maximum power and torque to be guaranteed during working conditions, so that type IIIflows generally lead to more compact technological solutions.
The IVT efficiency, ηIV T, is a weighted function of the mechanical efficiency, ηM, and of the IVU efficiency,ηIV U, the weights being the power fractionsflowing through the IVU
and the PGT,PIV U/PIN and1−PIV U/PIN,see Table 4.1.
Type I ηIV T =ηM ³ 1− PIV U PIN ´ +η−IV U1 PIV U PIN Type II ηIV T =η−M1 ³ 1− PIV U PIN ´ +ηIV UPPIV UIN Type III ηIV T =ηM ³ 1− PIV U PIN ´ +ηIV U PIV U PIN
Table 4.1: Expressions of ηIV T as function of possible types of power flows
It is noteworthy that type I, II and III flows occur forPIV U/PIN <0, PIV U/PIN >1
and0< PIV U/PIN <1,respectively. Fig. 4-10 shows the IVT efficiency versusPIV U/PIN
for different values of ηIV U, and fixed ηM = 0.95. Maximum possible efficiency is thus obtained for lockup conditions where PIV U/PIN = 0, ηIV T = ηM.The main focus of
this work is therefore on power recirculation, which is strictly related to the overall transmission efficiency.
4.3.1
Input coupled power
fl
ows
Considering the IC architecture and defining Ti,j the torque exerted from theith element
to the jth one, the following equations can be written for the control volume 2 (CV2)
shown in Fig. 4-11, in first range:
-20 -1.5 -1 -0.5 0 0.5 1 1.5 2 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 P IVU/PIN η IV T η IVU = 0.8 η IVU = 0.75 η IVU = 0.7 η M
Figure 4-10: ηIV T as a function of the ratio PIV U/PIN; ηM = 0,95.
and
T12,1ω1+T6,5ω5+TCωC = 0. (4.13)
In order to have type III powerflow, the following conditions must be satisfied simul-taneously:
T12,1 ω1 >0; T6,5 ω5 >0; TCωC <0. (4.14)
In Eqs. (4.12) and (4.13), TC is the load torque acting on the carrier shaft and ωC
the corresponding angular speed. Rearranging Eqs. (4.12) and (4.13) and using the kinematic conditions (4.1-4.6) evaluated in Sec. 4.2.1, one has:
T6,5 = µ 1−τIV T τIV U −1 ¶ TC (4.15) and T12,1 = µ τIV T −τIV U τIV U −1 ¶ TC. (4.16)
Considering the inequalities in Eqs. (4.14), rearranging Eqs. (4.15) and (4.16) and substituting the expression of the IVT ratio from Eq. (4.4), type III power flow is obtained in first range for:
τIV U >0. (4.17)
Likewise, in second range, Eqs.(4.12) and (4.13) can be rewritten as:
T12,1+T6,5+TS2,4 = 0 (4.18)
and
respectively. Therefore, using the expression of the IVT ratio (4.5), the condition for type III power flow in second range is:
τIV U <0. (4.20)
Considering the control volume CV1 in Fig. 4-11 containing the IVU, and being T5,6 =
−T6,5 and T1,12 =−T12,1, the ratio between the powerflowing through the IVU and the
input power can be derived as:
PIV U
PIN
= τIV U (τIV T −1)
τIV T (1−τIV U)
. (4.21)
Rearranging Eqs. (4.4) and (4.21), eliminating τwB using the constraint Eq. (4.8), the
power ratios in first and second ranges can be written as:
PIV U PIN I = τwA+ (1−τwA)τ1 τIV T (τwA−1)τ1 τIV T (4.22) and PIV U PIN II = τ2 £ τIV T_s−τIV T ¡ τ1τIV T_s−1 ¢ (τwA−1) ¤ +τ1τIV T_s(τwA−1)−τwA τ2τIV T ¡ τ1τIV T_s−1 ¢ (τwA−1) , (4.23) respectively. From Eq. (4.22), it can be observed that for an IC architecture, the power ratio is theoretically infinite at zero output power, meaning that a significant heat dissipa-tion occurs at nearly zero speed. The analysis is completed studying the torque behavior at the input and output shafts of the IVU as a function of the output load torqueTout and
of the IVT gear ratioτIV T. Considering that power is transmitted through the carrier in
first range and through the sun in second range, power conservation can be respectively applied for both ranges asTC,3ωC =Toutωout, TS2,4ωS2 =Toutωout,ωout being the output
TC,3 =Tout ωout/ωC =Tout/τ1 (4.24)
and
TS2,4 =Tout ωout/ωS2 =Tout/τ2. (4.25)
Using Eqs. (4.24) and (4.25) in the torque and power-balance equations, the torques at the output shaft of the IVU infirst and second ranges, can be written as:
TIV U,outI =T5,6 =Tout 1 τ1 µ 1 τwA−1 ¶ (4.26) and TIV U,outII =T5,6 =Tout 1 τ2 " −1 +τ2τIV T_s ¡ τ1τIV T_s−1 ¢ (τwA−1) # , (4.27)
respectively. Finally, by applying the power conservation between the input and output shafts of the IVU, the corresponding torques at the IVU input shaft can be obtained as:
TIV U,inI =T12,13 =Tout 1 τ1 ∙τ wA+ (1−τwA)τ1τIV T_s τwA−1 ¸ (4.28) and TIV U,inII =T12,13= Tout τ2 ¡ τ1τIV T_s−1 ¢ ∙µ1 −τ2 τIV T_s 1−τwA ¶ −1 +τ1τIV T_s+τ2τIV T ¡ 1 +τ1τIV T_s ¢¸ , (4.29) respectively.
4.3.2
Output coupled power
fl
ows
For the output coupled architecture shown in Fig. 4-12, using the same approach as in Sec. 4.3.1, the torque and power balance equations in first range and steady state conditions can be written applying the power conservation theorem to CV 1, to give:
TIN +T6,5+T10,C = 0 (4.30)
and
TIN ω1+T6,5 ω5+T10,C ωC = 0, (4.31)
respectively. The conditions for type III powerflow can be written as:
TINω1 >0; T6,5ω5 <0; T10,CωC <0, (4.32)
T10,C being the load torque acting on the carrier. Note that in second range, Eqs.
(4.30-4.32) are still valid, provided thatT10,C andωC are replaced byT8,S2andωS2 respectively.
Considering the kinematic conditions (4.9,4.10) evaluated in Sec. 4.2.2, one has:
T6,5 = µ 1−τIV T τIV U ·τIV T −1 ¶ T10,C (4.33) and TIN = µ 1−τIV U τIV U ·τIV T −1 ¶ τIV TT10,C. (4.34)
Thus, using Eqs. (4.32-4.34), with the definition of speed ratios provided in Eqs. (4.1) and (4.2), and applying the design constraint, Eq. (4.6), the condition for type III powerflow infirst range reduces to:
τIV U <0. (4.35)
Likewise, substitutingT10,C with T8,S2 in second range, one has:
τIV U >0. (4.36)
can be written as: PIV U PIN I = τwA−(τwA−1)τ1τIV T τwA (4.37) and PIV U PIN II =−1 +τ2τIV T + τ1τIV T ¡ 1−τ2τIV T_s ¢ ¡ τ1τIV T_s−1 ¢ τwA , (4.38) respectively.
Furthermore, the corresponding torques at the output and input shafts of the IVU can be written as: TIV U,outI =T5,6 =Tout 1 τ1 µ τ1τIV T τwA ¶ , (4.39) TIV U,outII =T5,6 =Tout 1 τ2 " τ1τIV T ¡ 1−τ2τIV T_s ¢ τwA ¡ τ1τIV T_s−1 ¢ # , (4.40) TIV U,inI =T8,9 =Tout 1 τ1 ∙ τ1τIV T +τwA(1−τ1τIV T) τwA ¸ , (4.41) and TIV U,inII =T10,11 = Tout τ2 " τ1τIV T ¡ 1−τ2τIV T_s ¢ + (1−τ2τIV T) ¡ τwA−τwAτ1τIV T_s ¢ τwA ¡ τ1τIV T_s−1 ¢ # , (4.42) respectively.
Unlike the IC case, the torque at the output shaft of the IVU depends also on the IVT transmission ratio. Nonetheless, from Eq. (4.37) it can be observed that the power ratio at zero speed has a unit value, this meaning that the OC architecture behaves like a pure-hydraulic direct transmission near power-zero condition, with no power recirculation.
both the IC and OC architectures.
1st Range 2nd Range IC τIV U >0 τIV U < 0
OC τIV U <0 τIV U > 0
Table 4.2: Conditions to have type III power flow for the IC and OC architectures.
4.4
Design optimization of power split IVTs
The power flow analysis carried out in Sec. 4.3 has shown different possible working scenarios pointing out recirculation conditions through the IVU and their negative effects in terms of overall transmission efficiency, fuel consumption and heat dissipation.
Standard designs of mechanical transmissions typically rely on atrial and error approach. The main inputs to this process are the vehicle nominal working cycle, the number of ranges, the engine power and torque curves. The outputs are then the corresponding transmission ratios: following this methodology, given a nominal engine speed, each gear ratio corresponds to a specific output speed and thus can be determined uniquely, according also to pre-existing components. A verification process with design and layout constraints will follow next.
For an IVT transmission, it is clear that each range has an infinite number of output speeds, depending on the IVU ratio; therefore the aforementioned iterative approach results in a heavy computational load also because the number of degrees of freedom to be evaluated is much higher than for a standard mechanical transmission.
Here, an effective approach to transmission design has been implemented by means of optimization algorithms, which allow one to explore a wide range of designs and to optimize the overall transmission efficiency. As will be shown in the next sections, the design of complex IVTs, such as the one under evaluation, can improve significantly thanks to a more effective way of evaluating design data and parameters with respect to
the standard approach. As a practical application, the case of an agricultural vehicle has been considered in the present work. Given specific design requirements and constraints, an optimization process has been implemented to evaluate the optimal set of transmis-sion parameters, namely τ1,τ2 and τwA. A nominal working cycle in maximum power
conditions has been considered as shown in Fig. 4-13.
0 0.5 1 1.5 2 2.5 0 0.5 1 τIVT (a ) PIN /PM A X 0 0.5 1 1.5 2 2.5 0 0.5 1 τIVT (b ) TIN /T M A X
Figure 4-13: Nominal working cycle: (a) normalized torque; (b) normalized power
PIN/Pmax
4.4.1
Optimization problem formulation
An analytical form for the optimization process can be obtained by choosing the objective function to be minimized as:
Ψ(τwA,τ1,τ2) =
Z
I
where I = £τIV T_M in;τIV T_M ax
¤
, f = PIV U/PIN_M ax and ρ is the Probability Density
Function (PDF) of τIV T. All design requirements have led to the formulation of
opti-mization constraints. The maximum allowable speed and torque values, in particular at the input and output shafts of the IVU, have to be guaranteed at each point within the integration domain I; thus, the resulting constraints for the input and output coupled cases can be written for a fixed value of the synchronization ratioτIV T_s as:
¯ ¯TIV U_in(τwA,τIV T, τ1,τ2) ¯ ¯≤¡TIV U_in ¢ M AX (4.44) and ¯ ¯TIV U_out(τwA,τIV T, τ1,τ2) ¯ ¯≤¡TIV U_out ¢ M AX. (4.45)
In terms of the maximum allowable speed, for the generic ith driveline shaft, the
following constraints have also to be satisfied
|ωi|≤ωM AX. (4.46)
Therefore, for each value of τIV T ∈I, a domain DτIV T can be defined as:
DτIV T ={(τwA,τ1,τ2)| ¡ TIV U_in ¢ M AX ≥ ¯ ¯TIV U_in ¯ ¯; (4.47) ¡ TIV U_out ¢ M AX ≥ ¯ ¯TIV U_out ¯ ¯; |ωi|≤ωM AX}.
Since the inequalities in Eqs. (4.44), (4.45) and (4.46) have to be verified simultaneously within the entire interval I, the final optimization constraints can be derived imposing the condition that:
(τwA,τ1,τ2)∈D=
\
τIV T ∈I
4.4.2
Numerical implementation of the optimization process
In order to numerically implement the optimization problem defined in Sec. 4.4.1, the integral in Eq. (4.43) has been discretized as:
ΨD = N
X
i=1
fi2(τwA,τIV T, τ1,τ2) wi, (4.49)
where the integration intervalI has been uniformly divided intoN sub-intervals of equal amplitude∆τIV T, and wirepresents a weighting function related to the machine working
cycle, defined as:
wi =ρi∆τIV T = ti ttot , (4.50) 0 0,5 1 1,5 2 2,5 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 τ IVT wi τIVTs
Figure 4-14: Weighting function wi versus speed distribution
where ti is the amount of time in which the vehicle speed lies in the interval Ii =
Fig. 4-14).
Three state of the art global optimization algorithms have been used to solve the present design optimization for both IC and OC architectures, namely Differential Evolution (DE) [14, 15], Simulated Annealing (SA) [16] and Nelder & Mead (NM) [17]. All these algorithms belong to the category of Direct Search methods, since they do not make use of derivative information.
4.4.3
The optimization algorithms
Numerical algorithms for constrained nonlinear optimization can be broadly categorized into gradient-based methods and direct search methods. Gradient-based methods use
first derivatives (Gradients) or second derivatives (Hessians) of the objective function. Examples are the sequential quadratic programming (SQP) method, the augmented La-grangian method, and the (nonlinear) interior point method. Direct search methods do not use derivative information. Examples are Nelder & Mead, Genetic algorithm and Differential Evolution, and Simulated Annealing. Direct search methods tend to con-verge more slowly, but can be more tolerant to the presence of noise in the function and constraints.
Therefore, when the objective function is nonlinear and non-differentiable, as it is in the problem under evaluation, direct search methods are the methods of choice. For all of these methods, the key feature is the strategy that generates variations of the parameter vectors. Once a variation is generated, a decision must be made whether or not to accept the newly derived parameters. All standard direct search methods, such as NM, use the greedy criterion to make this decision. Under the greedy criterion, a new parameter vector is accepted if and only if it reduces the value of the objective function. Although the greedy decision process converges fairly fast, it runs the risk of becoming trapped in a local minimum. Parallel search techniques, in turns, like genetic algorithms and evolution strategies (DE) have some built-in safeguards to forestall misconvergence: by running several vectors simultaneously, superior parameter configurations can help other
vectors escape local minima.
Another method which can extricate a parameter vector from a local minimum is the Simulated Annealing (SA) which relaxes the greedy criterion by occasionally permitting an uphill move. Such moves potentially allow a parameter vector to climb out of a local minimum. As the number of iterations increases, the probability of accepting an uphill move decreases. In the long run, this leads to the greedy criterion.
In general, an effective optimization technique should fulfill two basic requirements: (i) the method should converge to a global minimum regardless of the initial condition; (ii) the convergence should be fast.
Differential Evolution
Differential Evolution is a parallel, direct search minimizer of multidimensional func-tions. The method presents globally and locally correlated step sizes, which self-adapt over time in relation to the location of the population of individuals in the search space. The method uses m parameter vectors uG = {x1, x2, ..., xm}, as a population for each
generation G.The initial population is generally assumed randomly by adding normally distributed random deviation to the initial solution. Then, the key feature of the DE method is to generate trial parameter vectors: DE generates new parameter vectors by adding a weighted difference vector between two population members to a third mem-ber. In fact, unlike stochastic techniques such as Genethic Algorithms and Evolutionary Strategies, where the perturbation occurs in accordance with a random quantity, DE uses weighted differences between decision space vectors to perturb the population.
A sample algorithm can be the following:
• Step 1 i= 1;
• Step 2 Randomly select r1, r2, r3 ∈{1,2, ..., m} such that x1 6=x2 6=x3 6=i where
i is the index of the currently selected individual in the population;
the current individualxi,G. Namelyui,G+1 =xi,G+K(xr3,G −xi,G)+F (xr1G−xr2,G),
K andF being two control parameters.
The coefficientK is responsible for the level of combination that occurs betweenxr3,G
and the current individualxi,G, whereasF is responsible for scaling the step size resulting
from the vector subtractionxr1G−xr2,G.
Typically, in single objective problems, if the new individual ui,G+1 evaluates better
than the current individual xi,G, than the current individual is replaced by the new one.
However, in multi-objective problems, individuals cannot directly replace the parents without without either a dominance comparison with the current parent, or a sort of all the offspring with all the parents, with respect to their dominance level. This feature prevents the method to converge to local minima.
4.4.4
Simulated Annealing
In this algorithm, each pointx of the search space is analogous to thestate of a physical system, whereas the function f(x) to be minimized is analogous to the internal energy of the system in the statex. Therefore, the goal is to bring the system from an arbitrary initial state to a state with them minimum possible internal energy.
At each step, the SA algorithm considers some neighbors x0 of the current state x, and
decides between moving to state x0 or staying in x. The probabilities are chosen so that the system ultimately tends to move to the states of lower energy. The process is then iterated until the system energy falls below a given threshold or the computational budget has been exhausted.
The probability of making the transition from the current state xtox0 is specified by anacceptance probability function P(e, e0, T),that depends on the energiese=f(x)and
e0 =f(x0)of the two states, and on a global time-varying parameterT calledtemperature.
One essential requirement for the probability functionP is that it must be nonzero when
e0 > e, meaning that the system can move to the new state even if it presents a higher
from becoming stuck into a local minimum.
On the other end, when the time-varying temperature T approaches to zero, the prob-ability function P must tend to zero as if e0 > e, and to a positive value if e0 < e. In
particular, when T goes to zero, the method reduces to the greedy algorithm.
Given these properties, the evolution of the solutionx crucially depends on the tem-perature T since it is sensitive to coarser energy evolution when T is high, whereas to
finer variations whenT is low, where T is gradually decreased during the simulation.
Nelder & Mead
Differently from DE and SA, the Nelder & Mead method belongs to the class of direct-search methods. For a function of n variables, the algorithm maintains a set of n+ 1
points forming the vertices of a polytope in n−dimensional space.
At each iteration, n+ 1 points x1, x2, ..., xn+1 form a polytope. The points are ordered
so that f(x1) ≤ f(x2) ≤ ... ≤ f(xn+1). A new point is then generated to replace the
worst pointxn+1.
Let c be the centroid of the polytope consisting of the best n points, c= n1 Pni=1xi;
a trial point xt, is generated by reflecting the worst point through the centroid, xt =
c+α(c−xn+1), being α a positive parameter.
If the new pointxtis worse than the second worst point,f(xt)≥f(xn), the polytope
is assumed to bee too large and needs to be contracted. Thus a new trial point is defined as:
xc = c+γ(xn+1−c), if f(xt)≥f(xn+1) ;
xc = c+γ(xt−c), if f(xt)< f(xn+1).
where 0 < γ <1. If f(xc)< M in[f(xn+1), f(xt)] the contraction is successful and xc replacesxn+1, otherwise another contraction is carried out.
In general, NM is not a true global optimization algorithm. However, in practice it tend to work reasonably well for problems that do not present many local minima and strong nonlinearities.
4.4.5
Results
The results obtained from the optimization process are compared versus those resulting from the standard design approach in order to highlight the benefits obtained in terms of power recirculation and machine size reduction. The torque and power results are normalized with respect to the maximum recirculating value provided by the standard design approach, unless otherwise specified.
For the IC architectures the three optimization algorithms have provided almost iden-tical results. Fig. 4-15 shows the effects of the optimization process on power recirculation compared to the standard approach: a considerable reduction can be observed during the entire vehicle working cycle, resulting in a lower amount of heat generation, in particular near power-zero conditions, and thus improved transmission efficiency. Figs. 4-16 and 4-17 show the normalized torque behavior respectively at the input and output shafts of the IVU. An important overall decrease of the torque at the IVU input shaft is observed whereas the torque at the output shaft shows a significant improvement only in second range. For the sake of completeness, the relationship between the IVU ratio, τIV U, and
the overall transmission ratio, τIV T, has been analyzed and compared versus that
pro-vided by standard gears for both architectures, in order to evaluate the type of power
flow obtained from the optimization process along the entire working cycle. Fig. 4-18 shows how all optimization algorithms provided lockup points infirst and second ranges for lower output speeds with respect to the standard ratios, leading to a wider operating region with type III flow.
Fig. 4-19 shows how the OC optimal solution presents a significantly lower amount of power flowing through the IVU, mainly in second range. Important benefits obtained from the optimization process are also evident in Figs. 4-20 and 4-21 in terms of the
0 0,5 1 1,5 2 2,5 -1 -0.5 0 0.5 τIVT N o rm al iz ed P IV U Standard approach Differential Evolution Simulated Annealing Nelder & Mead
τIVTs
Range II Range I
Figure 4-15: NormalizedPIV U for the IC architecture
torques at the input and output shafts of the IVU. As a matter of fact, the DE and SA algorithms turned out to outperform the NM one, not only in terms of the optimal solu-tion provided, but also for the reduced level of computasolu-tional complexity; furthermore, the NM algorithm presented poor convergence properties for the OC problem, due to its nonlinearity, see [18, 19]. Standard DE SA NM IC (time) 52.9 % 69.7 % 69.7 % 69.7 % IC (vel.) 46.7 % 60.0 % 60.0% 60.0 % OC (time) 38 % 40.6 % 35.7 % 23 % OC (vel.) 33 % 50 % 40 % 10 %
Table 4.3: Time and speed percentages with type III flow for IC and OC
Table 4.3 shows the percentages of time and speed with type III power flow over the entire vehicle worki
