Flywheel energy storage for vehicle applications

Transcription

1 Scuola di Ingegneria Industriale e dell Informazione Laurea Magistrale in Ingegneria Meccanica Flywheel energy storage for ehicle applications Ettore Rasca Superisor: prof. Francesco Braghin Academic Year

2 Esprimo il mio ringraziamento a Stefano Sorti per tutto il supporto fornito.

3 Abstract 1 Abstract In recent years, a significant increase in the market share of electric ehicles was obsered. Most of these ehicles are meant for priate use and are equipped with chemical batteries. Despite the huge improements made on the capacity of the new generation lithium ion batteries, the long charging time remains a main drawback of this technology and opens the possibility for alternatie solutions. The present work describes a preliminary study aimed at inestigating the possibility to realize an electric ehicle relying on the flywheel energy storage technology as a primary energy source. First, a numerical and an analytical model of such a system are proposed and ealuated. Next, two sets of optimizations are performed on these models. Through the first optimization set, the optimal geometry for the rotors in the energy storage system is identified. This first process is repeated seeral times considering different alternaties for the rotors material, maximum rotational speed and basic geometry. Through the second optimization set, the ideal displacement and orientation of the rotors on the ehicle frame, as well as the total number of rotors, are inestigated. Finally, three multi-rotor configurations for the energy storage system are proposed and described. The data collected after performing simulations on the dynamics of these systems are then studied. In conclusion, after presenting obserations on the feasibility of such a technical solution, a set of future steps for the deelopment of the flywheel energy storage technology for ehicle applications are proposed.

4 2 Flywheel energy storage Contents Abstract... 2 Contents Introduction and aims of the work Energy storing in flywheel-based deices Historical oerlook High performance flywheels Characteristics of flywheel energy storage deices Kinetical energy storage systems applications Vehicle applications Kinetic energy recoery system Oerlikon Gyrobus Objecties of the study Analytical model of the rotor-frame system Introduction Degrees of freedom for the model Multibody system Inertial and non-inertial reference frames Cardan angles and their properties General procedure Motion equation for the two subsystems Preliminary obserations for subsystem coupling Analysis of the rotor motion equations Preparing the rotor motion equations for coupling Analysis of the frame motion equations Preparing the frame motion equations for coupling Motion equations coupling Rotor motion equation Rotor subsystem oerlook Lagrange equation components Soling the rotor motion equation... 36

5 Contents Simulations and results Preparing the rotor motion equation for coupling Variable change procedure Rotor subsystem boundary displacements Frame motion equation Frame subsystem oerlook Lagrange equation components Soling the frame motion equation Simulations and results Equations coupling Final obserations and SimMechanics models Single rotor optimization SimMechanics models Analysis of the problem General ariables of the optimization System models for the optimization System forcing Limits of the optimization field Additional constrains of the optimization Elementary cost functions Definition of the optimization procedure for the rotor geometry Preliminary ealuation Preliminary ealuation Preliminary ealuation Introduction to the rotor geometry optimization The selected optimization procedure The optimal rotor geometry and orientation First attempt optimal geometry Perfecting the results Orientation independent geometry General procedure Results aeraging Adanced single-rotor optimization Ealuation of the results of chapter

6 4 Flywheel energy storage 4.2 Improements on the optimization process Low density flywheel Low density flywheel at higher elocity Geometry improement Ealuation of the results Rotor Jtb Rotor Ltb Rotor Ntb Rotor Ptb Rotor Rtb Final rotor selection Multi-rotor optimization Rotor positioning Selection of the rotation direction Final solutions Solution 1: from rotor Ntb Solution 2: from rotor Ptb Solution 3: from rotor Rtb Conclusions Analysis of the results Mass and mass-based energy density Encumbrance and olume-based energy density Self-discharge Effects on the ehicle dynamics Final conclusions and future deelopments Bibliography Appendix Appendix A Appendix B Appendix C Appendix D Appendix E Appendix F

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9 1 Introduction and aims of the work In this first chapter a general introduction on the flywheel -based energy storage systems is proided. In the initial sections a historical oerlook on this technology is presented, with particular focus on how the energy storage systems of this nature hae always played a role in human technical deelopment. A discussion on the more modern use of these deices is then proided together with a description of their main characteristics and properties. It follows an analysis of some of the most interesting implementations related to the ehicle sector, and some significant examples are also presented. In the closing section of this chapter the aims of this work are detailed. Furthermore, an oerlook to the general procedure is proided with regard for the different areas of the study.

10 8 Flywheel energy storage 1.1 Energy storing in flywheel-based deices Historical oerlook The first flywheel-like deices where deeloped far before the kinetic energy conseration principle was clearly understood. In fact, the failure of ancient thought to recognize the basic principles of motion did not impede the deelopment of deices which exploit the inertia of bodies. It must be considered that in the ancient world, science and technology, were far more separated than in modern times. Beside this, deices exploiting inertial properties were deeloped and established thousands of years before the first attempts were made to explain the energy conseration of a rotating object. [1] The first eer use through history of tools that makes use of the principle of inertia, howeer, was not related with rotational motions. Linear motion, in fact, can be inestigated in a much easier way and with more basic tools. The utensils making use of this principle are all the hammer like deices. Howeer, hundreds of thousands of years were to elapse before rotary inertia could play a role in human life. The first tool using rotary motion was probably the drill, with the main application of lighting fires. But the hand drill, like its immediate successors, did not need the addition of a flywheel to work properly. [2] The first technically adanced application of the flywheel to appear in ery ancient times is the potter's wheel. With these deices some major difficulties are to be found both with the significant size and energy stored and the need for some sort of bearing. This last problematic is due to the fact that potters' wheels must rotate about a fixed axis. Since most of these machines where almost entirely built of wood, not many archaeological eidences of them are aailable. For this, it is not easy to ealuate the exact deelopment age of the first deices. Perhaps the potter s wheel preceded the wheeled ehicle. The first findings are from Sumer and Susiana, and date from between 3500 and 3000 BC. [1] [2] With these deices, the wheel needs to maintain its rotation for long enough to complete the operation of shaping the jar, whether in one or more stages. It is therefore a true flywheel energy storage system. It is assumed that the wheel should maintain its rotation for fie to seen minutes with an initial angular elocity of 100 re/min. Thousands of years later the first mechanical prime moer appeared. These were mainly the water mill, introduced in the 1st century AD and in widespread use by the 4th century, and the windmill. This last one appeared in Western Europe about a thousand years later. One of the first and most common use of mills was corn grinding. For this application the inertia of the water wheel and of the grinders, combined with the high torque proided by the prime moer, was sufficient for operating. Een for other more demanding applications an additional flywheel was not required and rarely introduced. This does not mean that a high inertia for the system is not required, on the contrary this keeps playing a fundamental role in proiding a smooth and affectie operation, but in this case, this is proided by the rotating elements composing the machine. [1] [3] During the middle ages gearwheels started to appear with increasing frequency. A high moment of inertia was often added to all the shafts because of the poor kinematic characteristics of the gear wheels of arious types then in use. An exception is for the shafts which carried the grinders and the water wheel. The high inertia was usually introduced by building heay gear wheels. Anyhow in some machines flywheels were purposely added, at least on some shafts. [2]

11 Introduction and aims of the work 9 In the 18th century, with the industrial reolution, the steam engine started its expansion as a new prime moer. Furthermore, the drop in the production costs of iron, and particularly of cast iron and later of steel, and the need for metal parts of the complex steam engines, resulted in the use of iron initially in some parts of the machines and later in the machines as a whole. Two important deelopments related to this age affected the eolution of flywheels technology. In first place flywheels where extensiely applied and studied for the realization of the new steam engines and, finally, the widespread use of metal in the construction of machines instead of wood allowed for the obtainment of iron flywheels. The first flywheels to be realized of a single iron piece could be built. Because of the greater density of iron with respect to wood these deices could embody a greater moment of inertia in the same space. Most of the practical applications of the age operated at low speed and so the use of high density and low resistance materials like cast iron was of no disadantage. [1] [2] In the years following the Industrial Reolution, the steam engine was deeloped to gie higher power and efficiency. Very large flywheels were built for the largest engines. From soon after the industrial reolution new engines operating at much higher rotational speed were deeloped. Among these new prime moers, the internal combustion engines, like the petrol and diesel engines, became widely used for both ground and naal operations. To cope with the higher centrifugal forces, new designs for the flywheel were deeloped. The solid disc-type flywheel started to become popular after it was first introduced in 1889 on the V-type twin cylinder power plant for automotie use, a small internal combustion engine capable of ery high rotational speed for the time. [1] [2] [4] Figure 1: V-type twin cylinder power plant for automotie use Nowadays flywheels are employed in a wide ariety of applications, ranging from slow to ery fast rotational speed operations. These deices are designed with ery different characteristics on the base of the application and of the amount of energy they are intended to store. Other than steel, new and more adanced materials are commonly applied, like compound materials for high rotational speed operations.

12 10 Flywheel energy storage High performance flywheels A notable deelopment in the flywheel technology occurred in two phases at the end of the 60s and at the beginning of the 70s. Two factors led to this breakthrough. Firstly, adances in the field of high-strength composite materials made it possible to build flywheels which could store much more energy for the same mass than conentional ones. Secondly, the growing concerns of ecological problems pushed the deelopment for energy storage deices for a wide range of applications. These clean and cheap energy storages were meant to be coupled with internal combustion engines mainly on road ehicles to reduce the oerall pollutants production. [1] Many important achieements in this field were obtained at the beginning of the 70s after many goernments proided substantial research funding. Many successful applications where realize employing a 'conentional' technology. The flywheels, in fact, were steel discs and were of 'conentional', although often ery good, design. The research was also dedicated to the deelopment of a new generation of flywheels, or the superflywheels. The main difference between these and conentional flywheels can be assessed in terms of the energy density accumulated, a parameter obtained as the ratio between energy stored and mass of the system. This is often achieed with an increase of the nominal rotational speed. A second peculiar feature of these new generation flywheels is that in case of anomaly the failure is much less destructie than with conentional ones. Despite all the efforts made in this direction, currently no adanced flywheel design which is successful enough to be mass produced is yet achieed Characteristics of flywheel energy storage deices A flywheel can be introduced in a mechanical system for two different purposes. The first reason is to proide more stability to an otherwise not steady state system by waring off oscillations in the angular elocity of the rotating machine. This result is achieed by introducing such a high inertial element on the fast-moing shaft. In this case the flywheel acts as a short-term energy storage system and so it is not necessary to hae much energy accumulated. On the other hand the second reason why a flywheel can be introduced in a mechanical system is to store kinetic energy and to slowly proiding it to the user load. In this second case the flywheel acts as a long-term energy storage system; it is now of greater importance that the deice is capable of storing significant amounts of energy. The latter case is the one that is analysed in detail in this work. In both cases the flywheel is a deice introduced in a mechanical system for its capability of storing and releasing energy. A parameter that ealuates the quality with which the energy is accumulated is the energy density, which is defined as the ratio between the useful energy stored oer the flywheel mass. With useful energy it is to be intended only the amount of energy that can actually be extracted from the system during nominal operation. Usually this parameter is related to the critical failure speed, the rotational speed at which the centrifugal forces are sufficiently high to cause a main failure in the deice. In this case the following expression for the energy density is defined. ( e m ) u.f. = K σ u ρ (1) Where e m is the flywheel energy density ealuated with respect to the ultimate failure stress, σ u is the ultimate tensile stress, ρ is the flywheel density and K is a parameter called shape factor. In case the rotor is composed of an isotropic material, the shape factor is function of the flywheel geometry only.

13 Introduction and aims of the work 11 Of course, during nominal operation, the flywheel speed does not work between null and critical failure speed and this must be considered while ealuating the energy density. In particular, three factors are introduced: - Safety factor α. This factor represents the ratio between the energy stored in the flywheel at operational conditions and the energy stored ad failure. - Depth of discharge α. This is the factor that considers the ratio between the stored energy and the useful one and it can be calculated in the following way. α = (1 ω 2 min ω2 ) (2) max An excessie reduction in the rotor rotational speed leads to technological issues and so, for this reason, it is often imposed that the minimum rotational speed is equal to half of the maximum. - Mass factor α. This factor is defined as the ratio between the flywheel mass and the total mass of the energy storage deice. With these parameters the energy density of the flywheel during nominal operation can be stated. This is defined as the oerall energy density. ( e m ) = α α α ( e o.o. m ) u.f. = α α α K σ u ρ (3) Flywheels can be classified in three categories on the base of the energy density. - Low energy density: when the energy stored is lower than 36 KJ/Kg. - Medium energy density: when the energy stored is between 36 and 90 KJ/Kg. - High energy density: when the energy stored is higher than 90 KJ/Kg. The high energy density is one of the greatest adantages of kinetic energy storage deices oer other energy accumulators. Moreoer, the speed up phase in a flywheel energy storage can be performed in a limited amount of time and so the loading phases of such deices is far faster than the loading of chemical batteries. Finally, when energy is stored in the form of kinetical energy this can be extracted at ery fast rates proiding a high-power output to the user load. [5] [6] The storage efficiency of flywheel-based energy storage systems or the capability of presering the energy stored for long periods with little losses is typically ery high, usually een higher than the efficiency of chemical based battery. Howeer, this parameter generally decreases when the accumulation time gets longer. High performances can be achieed with magnetically suspended rotors een if, for this kind of applications, still more deelopment is needed before the magnetic bearing technology can be wildly used. It has also to be considered that the transmission system has a crucial releance on the oerall efficiency of the system, especially because a continuously ariable ratio transmission is often needed to couple the rotor to the generator/motor. [1] The main disadantages of kinetic energy storage systems are related with the intrinsically problematic nature of a big and heay object rotating at fast angular speed. First, it must be considered that a catastrophic failure may occur and so the rotor design must be deeloped in order to reduce all the possible risks to a minimum. Besides also other complications related to ibrations, noise, wear and fatigue may arise. Howeer, the fatigue problem is usually far more releant when considering chemical-based batteries.

14 12 Flywheel energy storage Finally, the design of a continuously ariable transmission usually causes a heay penalty on the oerall system efficiency, to the point that constant speed and ariable inertia flywheels designs hae recently been considered. [1] [7] Kinetical energy storage systems applications On the base of the application the flywheel energy storage systems can be categorized in two groups: the storage systems operating on stationary machines and the storage systems operating on ehicles. In this work the second class is studied in detail and, since a closer look on this is proided in the following section, a brief introduction on the stationary applications is now presented. Most of the stationary application of the flywheel energy storage systems are related to low power applications. These are usually emergency deices mainly used to proide load leelling for uninterruptible power supply like, for example, for data centres, as they sae a considerable amount of space compared to battery systems. [8] Flywheels are sometimes used as short term spinning resere for momentary grid frequency regulation and balancing sudden changes between supply and consumption. No carbon emissions, faster response times and ability to buy power at off-peak hours are among the adantages of using flywheels instead of traditional sources of energy like natural gas turbines. Operation is ery similar to batteries in the same application, their differences are primarily economic. [9] Flywheel based high power energy storages are technically possible but currently these are not economically conenient, unless for the case in which other traditional energy storage systems are not feasible. [1] 1.2 Vehicle applications The idea of introducing a flywheel energy storage system on a ehicle is not new and many different applications hae already been inestigated. From an energetic point of iew two main different configurations can be considered. Firstly, the flywheel can be coupled with a prime moer like, for example, an internal combustion engine to recoer part of the energy that would be lost during the deceleration of the ehicle. This energy is then stored and used to reduce the prime moer energy needed during acceleration. Secondly, the flywheel energy storage system can be employed as the main energy source of the ehicle. Anyhow the use of a fast-rotating element on a moing machine could lead to gyroscopic effects that must be carefully ealuated. In the following paragraphs an oerlook is gien on these two main applications with respect to the case of road ehicles. It is estimated that on a road ehicle almost 60% of the energy is used for the acceleration, and later lost during breaking. This is a particularly critical problem for the mobility of priate ehicles in cities and for public transportation, where a high number of acceleration-deceleration cycles are usually needed. Many different options hae been considered to recoer the lost energy. Nowadays hybrid ehicles with internal combustion and electrical engines are one of the most popular alternaties to address this problem. Flywheel energy storage systems could also be a alid alternatie in that a high energy density can be achieed. More importantly this kind of energy storage allows for fast charging and fast discharging, yielding high-power input and output energy fluxes. Technical problems, mainly related to the design of a continuously arying transmission held back the spreading of such a deice. Also other applications, like the use of a flywheel to help with restarting an internal combustion engine, are currently under deelopment. [1]

15 Introduction and aims of the work 13 Nowadays electrical ehicles are rapidly getting more popular. Despite the many adantages of an electric ehicle, some main disadantages mostly related to the battery technology affect them. A battery storage system, in fact, needs long time to be recharged and consists of a heay and bulky deice. Finally, it has also to be considered that with current technology the lifespan of chemical energy storage systems is rather short. All three of these problems could be addressed by using a flywheel-based energy storage system. Some prototypes and working models of ehicles with this kind of energy storage hae already been deeloped, howeer no one successful enough to be mass produced. In the following subsections one significant example for each of the two applications now introduced is presented. Figure 2: flywheel and motor-generator of the Gyrobus Kinetic energy recoery system A kinetic energy recoery system (KERS) is an example of ehicle application for flywheel energy storage systems, designed to recoer the energy lost during breaking. As it was preiously stated not all the kinetic energy recoery systems are flywheel based: some of them use a high oltage battery instead. One of the most interesting and adance applications of KERSs is with Formula One race cars. In 2009 FIA allowed the use of a KERS in the regulations for the 2009 Formula One season to push for the deelopment of a responsible solutions to the world enironmental challenges. This application is particularly fascinating because the regulation allows for the use of either a mechanical energy storage ia flywheel or an electrical storage ia battery or supercapacitors. These three technologies hae thus been deeloped and are currently competing to proide the best possible solution. As typically happens for the use of a flywheel energy storage system on ehicles, one of the biggest challenges related to the deelopment of a flywheel based KERS for Formula One applications was related to the design of a continuous ariable transmission. The kinetic energy recoery system allowed from the FIA regulation was a 60 KW deice in 2009 and the power was later increased to 120 KW. According to rules, the stored power can be used for not more than 6.6 seconds and the total energy that can be accumulated is KJ. Finally, the angular speed of flywheels based KERSs is of rpm. [10]

16 14 Flywheel energy storage Een if the use of this deice in the Formula One competitions had ups and downs in the last years, this led to important deelopments in the technology. Many other automotie competitions are also currently pushing for the deelopment of such a knowledge and ultimately contribute to the spread of such systems in priate ehicles, and in the transfer of this technology to road cars. [11] Oerlikon Gyrobus An interesting ehicle application of flywheels as prime moer are gyrobuses, electrical buses that uses flywheel-based energy storages. With comparison to trolleybuses, gyrobuses don t need to be continuously connected to the grid by wires. While there are no gyrobuses currently in use commercially, deelopment in this area continues. [1] Figure 3: Maschinenfabrik Oerlikon Gyrobus, 1953 The concept of a flywheel-powered bus was first deeloped and realized during the 40s by the swiss company Oerlikon, with the aim of creating a feasible alternatie to battery energy storage in road ehicles, that didn t inole any oerhead-wire electrification. The flywheel designed to store energy was a large metal disc spun at up to rpm. During the charging phase, this was accelerated by an electrical motor powered by means of three booms mounted on the ehicle roof, which contacted charging points located as required or where appropriate. Natural charging position were at passenger stops and at terminals. During moement, the energy to power the ehicle electric motor was obtained by using the flywheel charging motor as a generator and gradually slowing down the metal disk. In this way the kinetic energy of the flywheel was once more conerted into electricity. Finally, since the bus breaking was electric, part of the breaking energy was recoered and used to speed up the flywheel and increase the oerall range. [12] [13] A gyrobus could typically trael as far as 6 km on a leel route at speeds of up to 50 to 60 km/h, depending on ehicle load. Charging a flywheel took between 30 seconds and 3 minutes and, in order to reduce this

17 Introduction and aims of the work 15 time, the option of increasing the charging oltage was considered. Gien the relatiely restricted energy stored, issues related to the ehicle range could arise in case of dense urban traffic. [14] Some of the greatest adantages of this kind of ehicle are that it is quiet, pollution-free (or at least pollution confined to generators on electric power grid) and that it can operate flexibly at arying distances. The main disadantages are howeer related to the great weight introduced by the flywheel, the risks related to haing a heay fast-moing object on board, the short range that can be achieed and the additional complexities in driing a ehicle affected by strong gyroscopic effects. Howeer, it must be considered that with the use of modern technology most of these drawbacks could be soled or at least reduced. 1.3 Objecties of the study In this work a preliminary study to the design of an electric ehicle based on the flywheel energy storage technology is performed. With this, a feasible alternatie to the more popular chemical battery energy storage for road ehicles is inestigated. A comparison between these two alternaties is made in terms of both the encumbrance and weight of the system, of the storable energy and of the achieable range in practical use. This study is limited to a configuration in which the rotor support system is obtained by mechanical bearings. Adantages and disadantages of the two energy storage systems are analysed, and particular attention is put on the effects that can arise from the accelerating and decelerating dynamics of a big and heay rotating object mounted onto a moing ehicle. This study is split in four main areas that are now introduced. The first step is the deelopment of a working model for the rotor-frame system. This model is initially deeloped in an analytical form with Matlab and later ealuated using a numerical model obtained with SimMechanics. Comparable results are expected. This model is first introduced for a single rotor configuration but the possibility of later considering other flywheels on the same frame must be left open. All the following optimization processes are performed on these models. An optimization on the geometry of the rotor for a single flywheel configuration is performed. In this phase the cost functions to be minimized are carefully selected together with the ariables to be optimized and the boundary to be imposed. The possibility of optimizing the rotor geometry independently from its displacement and orientation on the frame is inestigated and outcomes for both cases are proided. The obtained results are ealuated and discussed. Since the goal of this work is to perform a feasibility analysis, the structural assessment of the solutions obtained is not analyzed in detail. Howeer, ealuations on the iability of these results are performed. The identified optimal geometry for the single rotor is later used as the starting point in that of the geometry for the rotors in the ehicle energy storage system for the multirotor optimization. After the single rotor geometry optimization, a new optimization on a multirotor system is performed. The optimal multi-rotor configuration is found imposing the constraint that this solution must be capable of storing a sufficient amount of energy for the ehicle on road use. The olume occupied by the rotors and the mass of the whole system are carefully ealuated and compared to those that would be found using chemical battery for storing the same amount of energy. A detailed study of the final configuration obtained by performing the two optimization processes is carried out. General conclusions are drawn on the possibility of realizing a road ehicle, for priate or public use, that uses a flywheel based energy storage system with mechanical support system for the rotors.

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19 2 Analytical model of the rotor-frame system In this chapter the procedure to deelop an analytical model for the rotor-frame system is detailed. This mathematical work frame is deeloped using Matlab, and later compared with a numerical model gained with SimMechanics. Equal results are expected. The model is initially build for a single rotor configuration since this is needed for the rotor geometry optimization described in chapter 3. Subsequently, a multi-rotor model is built to perform the multi-rotor optimization of chapter 5. This latter is deeloped on the single rotor model, which is designed to be easily modified to account for more than one flywheel. The general procedure is here introduced together with a description of the complications that were encountered during this deelopment. Crucial information on how the Matlab function s are written are also presented. Finally, the SimMechanics model is explained and adantages and disadantages related to the use of this latter option are discussed.

20 18 Flywheel energy storage 2.1 Introduction Degrees of freedom for the model As preiously stated, the analytical model for the rotor-frame system is deeloped for the purpose of performing an optimization process, aimed at finding the best rotor geometry. In a second moment in chapter 5 this same model is also used to identify the optimal multi-rotor configuration. Finally, this same mathematical framework allows to simulate the behaiour of the system with respect to time in any gien configuration and for any releant external excitation. For this reason, it is with this model that the system dynamics is simulated. In the model deelopment, to completely describe the system, the appropriate number of degrees of freedom must be selected. The set of degrees of freedom must include enough coordinates to express the system dynamics of interest, plus seeral others to be used for introducing the external excitations. On the base of the selected degrees of freedom the system model is represented, and the equation of motion deeloped. For the sake of simplicity, the system is split into two subsystems which are analysed separately. These are the rotor subsystem and the frame subsystem. Figure 4: the rotor subsystem (11 degrees of freedom) When it comes to the rotor, it is clear that a set of six degrees of freedom is needed to describe its dynamics. Three of them represent the rotor displacement on the frame, while the other three the rotor orientation. Moreoer, fie additional degrees are introduced for considering the boundary imposed displacements, and this is done in accordance with the modelling choices detailed in the following sections. Now that the subsystem degrees are selected, its model can be established. A representation of the rotor subsystem model is proided in Figure 4. For what concernes the frame, instead, the choice on the degrees of freedom number is not that obious, and many different alternaties can be ealuated. On the base of how many degrees of freedom are considered for the frame subsystem, different models can be defined for it. Among all these possibilities, three are selected and used to define the frame subsystem models, and these are represented from Figure 5 to Figure 7.

21 Analytical model of the rotor-frame system 19 Figure 5: Option 1 for the frame subsystem (7 degrees of freedom) Figure 6: Option 2 for the frame subsystem (11 degrees of freedom) An increasing complexity characterizes the three models. Further details on these are proided in the following subsections. All three of these system representations hae been deeloped, each into a different analytical model composed of a set of differential equations. For the sake of simplicity, in this chapter we focus on the deelopment of the motion equations for the system composed of the rotor subsystem combined with the first option for the frame subsystem. Anyway, the method changes only slightly when it comes to obtaining the motion equations for a system defined with one of the other two alternaties for the frame representation.

22 20 Flywheel energy storage Multibody system Figure 7: Option 3 for the frame subsystem (15degrees of freedom) As it can be denoted from the representations in the preceding subsection the system objectie of the study is a multibody system characterized by multiple degrees of freedom. In this subsection the main elements appearing in the models are described. The focus is put on both the rigid bodies and on the stiffness and dumping elements. Some of these elements are common to all three of the models, other are present only when the second or the third alternaties for the frame subsystem are considered. In the description of the model elements, the parameters are presented without detailing any numerical alue. Reason of this stays in the fact that the model is deeloped in a completely analytical way, and no numerical alues are introduced. In this phase of the study, in fact, it is not considered which of these parameters are to be imposed as constant and which are the ariable of the optimization process Rotor Quantity Unit Symbol Mass Kg m rot Moment of inertia (x-axis) Kg m 2 xx J rot Moment of inertia (y-axis) Kg m 2 J rot Moment of inertia (z-axis) Kg m 2 zz J rot Rotor density Kg/m 3 ρ dens,rot Rotor eccentricity m x O Table 1: rotor characteristics The rotor is a rigid body composed of three cylindrical elements. The rotation axis is the symmetry axis common to the three elements. The rotor mass centre is placed outside from the rotation axis. The distance between this axis and the mass centre is denoted as x O. Table 1 is proided to summarize the main geometrical and physical characteristics of this rigid body. yy

23 Analytical model of the rotor-frame system Frame The frame represents the main ehicle body. It is modelled by a single homogenous brick. Table 2 summarizes the frame characteristics Rotor support system Quantity Unit Symbol Mass Kg m tel Moment of inertia (x-axis) Kg m 2 xx J tel Moment of inertia (y-axis) Kg m 2 J tel Moment of inertia (z-axis) Kg m 2 zz J tel Rotor density Kg/m 3 ρ dens,tel Table 2: frame characteristics The rotor support system is the physical linking element between rotor and frame. It is composed of a pair of roller bearings and one axial bearing. The roller bearings are placed at the ery end of the rotor shaft. The axial bearing is placed in such a way that the whole axial load acts on the central element of the rotor. For what concerns the model of this latter element, we can consider as its load was directly applied to the central point of the symmetry axis of the rotor central element. In the next chapter also a two-roller bearing and a three-roller bearing per side options are inestigated. In these cases, the load of any additional bearing is applied to the shaft at a certain distance from the shaft end. This distance is equal to the clearance needed to place the more external bearing on the same part of the shaft. Eery roller bearing is represented in the model as two spring-dumper subsystems placed in the two directions the bearing can handle the load. The axial bearing is represented as a spring-damper subsystem. The subsystems are composed of a spring in parallel with a dumper. The ariables of the rotor suspension system are listed in table Vehicle support system Quantity Unit Symbol(s) Roller bearing stiffness N/m K A,x K B,x K A,y K B,y Roller bearing dumping (N s)/m r A,x r B,x r A,y r B,y Axial bearing stiffness N/m K O,z Axial bearing dumping (N s)/m r O,z Table 3: rotor support system characteristics The ehicle support system is composed of the ehicle suspension system and of the ehicle wheels. It is, therefore, the mechanical subsystem linking the ehicle to the ground. With the three alternaties for the frame subsystem introduced in subsection 2.1.1, three different representations of the ehicle support are presented. It can be noted that the three options hae an increasing leel of detail. It follows a short description of these alternaties. With the first option, a ery simple modeling of the ehicle support system is proided. In this case, the whole system is represented with four spring-dumper subsystems placed ertically on the ehicle corners. The spring-damper couples should hae such characteristics to represent the assembly composed of both wheel and suspension. yy

24 22 Flywheel energy storage The second alternatie is a further deelopment of the first option. At each corner of the ehicle two springdumper subsystems are placed. The wheel is modelled as a suspended mass in between these two. The spring-damper connecting the suspended mass to the ground represents the wheel elastic and dumping characteristics, while the spring-damper connecting the suspended mass to the frame models the suspension properties. The additional inertia that the suspended mass proides accounts for the effects that the wheel introduces on the system dynamics. With the third model, also the wheel transersal and longitudinal stiffness and dumping are considered. With this option it is also possible to simulate the behaiour of the ehicle during acceleration and deceleration phases. Moreoer, with this model, also the dynamics of the ehicle during cures can be ealuated. It is important to notice that with the first and second alternaties the oerall system is not bounded in the longitudinal and transersal directions, and according to rotation along the ertical axis. To preent this problem, during the deelopment of the motion equation, some additional constraints are introduced along these directions. Finally, to gain consistency whit the selected model, the equations for the unconstrained degrees of freedom are simply neglected. The main characteristics of the ehicle support system are listed in table 4. Quantity Unit Symbol(s) Suspension stiffness N/m K a,sx K a,dx K p,sx K p,dx Suspension damping (N s)/m r a,sx r a,dx r p,sx r p,dx Wheel radial stiffness N/m K w,r Wheel radial damping (N s)/m r w,r Wheel transersal stiffness N/m K w,t Wheel transersal damping (N s)/m r w,t Wheel longitudinal stiffness N/m K w,l Wheel longitudinal damping (N s)/m r w,l Table 4: frame support system characteristics Inertial and non-inertial reference frames To deelop the motion equations for the oerall system some reference frames should be introduced. It is reminded that the equations of motion must be defined in an inertial reference system. With this in mind, the reference system selected for defining the motion equations is a reference placed on the ground. Since the oerall rotor-frame assembly is rather complicated, some additional reference systems are defined to keep all the procedure steps as most clear and simple as possible. Before introducing the references that is used, a conention to identify position and orientation of a body in these references must be defined. First it is reminded that at least six coordinates are needed to fully describe the position and the orientation of a rigid body in any gien 3D reference. The set of coordinates that was chosen is composed of three displacements and three rotations. The displacements are defined as cartesian coordinates while for the rotations cardan angles are used. This choice is common to eery reference system adopted. The details of the chosen conention are now proided; it is shown how the six coordinates are used to identify the body position into space. The body is initially considered to be placed in the origin of the reference system with its director axis lying on the z direction. The following transformations are applied to moe from such position to the one identified by the set of coordinates:

25 Analytical model of the rotor-frame system 23 - Three cardan rotations are applied in sequence. The magnitude of these rotations is expressed by the three angles in the coordinates ector. - Only after the final orientation for the body is identified, it is displaced in space according to the three translations in the coordinates ector. From here to the end of this subsection the reference systems are introduced and briefly described. More details on these frames and their use are gien as the motion equation deelopment method is described in detail Local reference frame {x L, y L, z L } This reference system is fixed on the rotor. Its origin is placed on the rotation axis of the rotor at the minimum distance point from the rotor centre of mass. The z L axis lies on the rotor rotation axis and the x L axis is placed in such a way that the rotor centre of mass is a point on it. This reference system is introduced to define the moments of inertia of the rotor along the three axes in the simplest way. Reason of this is that olume integrals defining the moments and products of inertia are independent from time when they are ealuated in the local reference. Moreoer, in this reference system the characteristic of eccentricity of the rotor is introduced. The local reference is denoted with letter L in Figure Global reference frame {x O G, y O G, z O G } This reference system is fixed to the ehicle frame. When the rotor is in its neutral position the origin of the global reference and of the local reference, together with the z L and the z G axis, coincide. Moreoer when the rotor lies in its initial position the two reference frames oerlap. The global reference system is introduced to study the dynamics of the rotor around its equilibrium position. In fact, in this system six degrees of freedom can be introduced to study the rotor behaior. The motion equations for the rotor are first written in this reference, gien the many adantages that this proides for the rotor dynamics analysis. The global reference is denoted with letter G in Figure Absolute reference frame {x O A, y O A, z O A } As for the global reference frame, also the absolute frame is fixed to the ehicle chassis. Howeer, the origin and the orientation of these two reference systems are generally not the same. The origin of this latter one is placed in the centre point of the lower face of the parallelepiped representing the frame. The z A axis is placed along the longitudinal direction while the x A axis is in the ertical one. This frame is introduced to create a reference on the chassis that is oriented with the ehicle direction, and it is also particularly useful when studying a multi-rotor configuration. The absolute reference is denoted with letter A in Figure Ground reference frame {x T, y T, z T } The ground reference is the only inertial reference system introduced. It is the reference system the motion equations are to be written in. The ground reference is fixed to the ground and its origin is placed ertically under the absolute reference system origin. The x terr axis is the ertical one while the z terr axis is in the longitudinal ehicle direction.

26 24 Flywheel energy storage The ground reference is denoted with letter T in Figure 8. Figure 8: the reference frames The motion equations for the frame can be simply written in this reference system, while the motion equation for the rotor can be easily ealuated in the global reference. A procedure must be deeloped in order to couple these two set of motion equations, and this is detailed within this chapter Cardan angles and their properties In this subsection some important properties of the reference systems are described. These properties are later used to define the system motion equations. First, it is reminded that rotations in space are not commutatie. This makes it much harder to describe the dynamics of a system in space with respect to the dynamics on a flat surface. For this reason, a conention for defining the orientation of a body in a threedimensional space must be introduced. As already stated in the preceding section, the chosen conention is the cardan angle conention. Let s consider the example in which the cardan angles σ, β and ρ are used to define the local reference system with respect to the global one. In this case the cardan angles consist of three consecutie rotations that are applied in a determined sequence to transform the directions of the global reference axes into the local reference axes ones. The three rotations are: 1. A first rotation of magnitude σ performed around the x G axis, of the global reference. This transforms the global reference in the axes of an intermediate reference {x I, y I, z I }. 2. A rotation of magnitude β performed around the y I axis. This transforms the axes of the intermediate reference {x I, y I, z I } in the axes of a second intermediate reference {x II, y II, z II }. 3. A final rotation of magnitude ρ performed around the z II axis. This transforms the axes of the intermediate reference {x II, y II, z II } in the axes of the local reference {x L, y L, z L }. Since the rotations in space are not commutatie the order in which they are performed is releant. Different application orders proide different results.

27 Analytical model of the rotor-frame system The rotation matrix Figure 9: cardan angles application The example in which the cardan angles σ, β and ρ are used to define the local reference system with respect to the global one is analyzed. In this case, the rotation matrix is a matrix that allows to transform a ector a G, defined in the global reference, into the corresponding ector a L in the local reference. This matrix is indicated as follows: a L = [Λ LG ]a G (4) This matrix depends on the orientation of the local reference with respect to the global one and therefore, ultimately, from the alue of the Cardan angles. To define the expression of the rotation matrix as function of the Cardan angles, we proceed according to the three successie rotations defined aboe. Therefore, the matrixes that define the three intermediate transformations are introduced. This are planar rotation matrixes [Λ I G ] = [ 0 cos(σ) sin (σ)] (5) 0 sin (σ) cos(σ) [Λ I G ] = [ 0 cos(σ) sin (σ)] (6) 0 sin (σ) cos(σ) cos(ρ) sin(ρ) 0 [Λ L II ] = [ sin(ρ) cos(ρ) 0] (7) Finally, the rotation matrix is obtained as the product of these three matrixes. [Λ LG ] = [Λ L II ][Λ II I ][Λ I G ] (8) It is to be noted that the planar rotation matrices are orthogonal, so that their inerse matrix coincides with the transpose. Since a matrix obtained as a product of orthogonal matrices is orthogonal, it follows that the rotation matrix is also orthogonal. Where: [Λ GL ] = [Λ LG ] 1 = [Λ LG ] T (9) a G = [Λ GL ]a L (10)

28 26 Flywheel energy storage Rigid body angular speed with cardan angles The angular speed ector of a body in space is defined as the ratio between the infinitesimal rotation oer the infinitesimal time in which the rotation takes place. Since the cardan angles are used to define the rotation of the body, the angular speed ector is obtained as the sum of three ectors: 1. A first ector with modulus σ pointing along the x G axis. 2. A second ector with modulus β pointing along the y I axis of the first intermediate reference. 3. A final ector with modulus ρ pointing along the z II axis of the second intermediate reference. The angular speed ector can now be defined in the global or in the local references. 1. In the global reference the speed ector is defined as follows. σ 0 0 ω G = { 0} + [Λ G I ] ({ β } + [Λ I II ] { 0}) (11) 0 0 ρ The expressions for the matrixes, reported in equations from (5) to (7), are substituted, and the equation (11) is rearranged as follows. 1 0 sin(β) σ ω G = [ 0 cos(σ) sin(σ) cos(β) ] { β } = [A G ]{θ } (12) 0 sin(σ) cos(σ) cos (β) ρ 2. In the local reference the speed ector is defined as follows. 0 0 σ ω L = { 0} + [Λ L II ] ({ β } + [Λ II I ] { 0}) (13) ρ 0 0 The expressions for the matrixes, reported in equations from (5) to (7), are substituted, and the equation (13) is rearranged as follows. cos(ρ) cos (β) sin(ρ) 0 σ ω L = [ sin(ρ) cos(β) cos(ρ) 0] { β } = [A L ]{θ } (14) sin(β) 0 1 ρ 2.2 General procedure In this subsection the oerall approach that has been used to deelop the equations of motion of the whole system is described in detail. The one now introduced is just one out of many different aailable alternaties. As briefly stated in the last section, the basic concept of this whole process is to deelop the equations of motion for the frame separately from the equations of motion for the rotor. In fact, in a first moment the motion equations for the rotor in the global reference frame and the motion equation for the frame in the ground reference frame are ealuated. Subsequently the two sets of equations must be coupled. Before coupling, the rotor equations must be defined in the ground reference frame. This means that a ariable changing procedure needs to be defined. In the following subsections details on this procedure are proided. To make the explanation easier to understand, a simplified ersion of the model is used through the whole section for the purpose of illustration. This model is based on the coupling of the rotor subsystem together with the first alternatie for

29 Analytical model of the rotor-frame system 27 the frame subsystem. The simplified model used for illustration differs from the real one because of its twodimensional nature. Of course, it does not mean that this procedure applies to a 2D model only, but this is just a way of making the problem easier to describe and to understand. It follows a representation of the simplified model. Figure 10: The simplified rotor-frame model It is to be noted that in the figure the points O O, A A and B B do not coincide. This is always the case if the rotor is not in its initial position. Howeer, the represented configuration is the initial or equilibrium one, and the points are not illustrated as oerlapped only in the interest of clarity. Figure 11: The rotor subsystem As briefly introduced in the preceding sections, this system can be split in to two subsystems: the frame subsystem and the rotor subsystem. In Figure 11 and Figure 12 these two are shown. From these, the selected

30 28 Flywheel energy storage boundaries between the two subsystems can be obsered. Obiously, this is not the only possible choice for tracing the boundary, and other options can also be considered. It is already possible to note that the two subsystems exchanges forces in the points A, B, and O. Moreoer, it is clear that, once the spatial position of the frame is known, the boundary moement acting on the rotor can be calculated. From now on, with motion equation the motion equations of the whole system are intended. That is the motion equations of the frame combined to the motion equations of the rotor. Figure 12: The frame subsystem Motion equation for the two subsystems In this step the deelopment of the motion equation for the two separate subsystems is considered. No particular challenge is found in this phase, if the rotor motion equations are deeloped in the global reference and the frame equations in the ground one. Since with this section only an oerlook on the general procedure is proided, the details of this phase and all the others are presented in the following sections. Results of this initial stage are the motion equations for the rotor in the global reference and the equations for the frame in the ground one Preliminary obserations for subsystem coupling Let s now suppose that the motion equations for the rotor in the global reference and the motion equations for the frame in the ground reference are already aailable. To couple the motion equations of the two subsystems, two fundamental steps are required: Step 1: Write both sets of equations in the same reference. This means that in both sets of equations the same ariable should be used to express quantities that are not independent. Step 2: Identify and consider all the effects (in terms of displacements and forces) that the two systems mutually exchange. This means that, for example, the boundary moement acting on the rotor subsystem must be expressed as a function of the frame degrees of freedom.

31 Analytical model of the rotor-frame system 29 Summarizing, to couple the two sets of equations these must be expressed as functions of the least number of ariables. No ariable that is a combination of others should be present. Moreoer, all the exchanged forces should be considered Analysis of the rotor motion equations As preiously stated the rotor motion equation are obtained according to the global reference frame. This reference is fixed with the frame. This means that it is not an inertial reference. The rotor motion equations in this form are a function of the rotor degrees of freedom in the global reference frame and of the imposed boundary moement. In this case the boundary moement is represented by the motion of the spring-damper anchor point on the frame (points O, A and B on the picture). If we now consider the 2D model of the system that is used as an example, the motion equations for the rotor are in the following form: (motion eq. ) global rotor = f(x G, z G, β G, x A, x B, z O ) (15) 2D model: 6 equations and 6 degrees of freedom. 3D model: 11 equations end 11 degrees of freedom. It is once more reminded that the 3-dimensional model is the one which was object of the motion equation inestigation, while the 2- dimensional one is introduced only for the sake of making the explanation more understandable. In the preceding expression, the rotor degrees of freedom in the global reference frame are indicated in blue. These are two displacements and one rotation. In red are represented the boundary imposed motions, each for one of the spring-damper systems. Of course, if we now consider the 3-dimensional model, the number of degrees of freedom for the rotor in the global reference would be six (three displacements and three rotations). Moreoer, we would also hae fie imposed boundary moements (x and y displacements for points A and B and z displacement for point O ) Preparing the rotor motion equations for coupling The motion equations for the rotor in the current form are correct and sufficient for the study of the isolated rotor dynamics, and the same can be said for the frame equations. Howeer, since the final goal is to define the motion equation for the whole system, coupling of these two sets of equations must be performed. To prepare these sets for coupling, the operations summarized in the preiously introduced steps step 1 and step 2 should be done. In this subsection, a detailed description of these steps for the rotor motion equations is proided. With step 1, the coordinates representing the rotor displacement and orientation in the global reference frame (indicated in red in expression 15) are substituted with a new set of coordinates expressing the rotor displacement and orientation, but this time defined in a reference system consistent with the one used to describe the chassis equations. As stated before, this last reference is the ground reference, and this is an inertial reference system. This ariable change is performed into two different sub-steps that are now presented. Since the goal of this section is to proide an oerall presentation to the motion equation deelopment method, the details of the ariable change procedure are oerlooked. These are considered in the following section.

32 30 Flywheel energy storage In the first sub-step a ariable change between global coordinates and absolute ones is performed. 1 Once this first sub-step is performed the motion equations for the rotor in the absolute reference are obtained. These are represented in equation 16. (motion eq. ) absolute rotor = f(x A, z A, β A, x A, x B, z O ) (16) 2D model: 6 equations and 6 degrees of freedom. 3D model: 11 equations end 11 degrees of freedom. The second sub-step inoles a further ariable change. In this case the ariables in the absolute reference are substituted with those in the ground one. This new ariable change differs from the preceding in that the two reference systems inoled are not fixed together. This means that some terms, keeping into account that the absolute reference system is dragged by the frame motion, are to be introduced. These are the frame and so the absolute reference system displacements and rotations ealuated with respect to the ground reference. These new coordinates are denoted in green and are the frame degrees of freedom in the ground reference. As for the preiews ariable change the details are shown in the following chapter. The new rotor motion equations are the one represented in equation 17. (motion eq. ) ground rotor = f(x T, z T, β T, x tel, z tel, β tel, x A, x B, z O ) (17) 2D model: 6 equations and 9 degrees of freedom. 3D model: 11 equations end 17 degrees of freedom. It is interesting to note that, with this new expression for the rotor motion equation, also the frame degrees of freedom appear. This underlines the physical connection between rotor and frame. The ariable change of step 1 is now completed, and the rotor motion equations are expressed in the ground reference frame. In accordance with Figure 10, that is used to remind which are the boundaries traced between the two subsystems, step 2 is performed. Whit this step, forces and displacements acting on these boundaries are considered. The forces that are applied by the frame to the rotor are exerted on points O, A and B. These forces hae already been considered in the Lagrange component while writing the rotor motion equation, howeer the irtual displacements associated to these forces must be expressed in accordance to the ariables used to describe the frame dynamics. In fact, it can be noted that the terms x A, x B and z O (represented in red in equation 17) can be expressed as a function of the frame degrees of freedom. Howeer, it is of crucial importance to understand that the displacements of points A, B and O that appears in the rotor motion 1 It is to be noted that both the absolute and the global reference frame are fixed with the chassis. This means that they are also fixed together. With this, the two references differ of a quantity that is constant with time. The reasons why the absolute reference is introduced, and this first sub-step is performed are now listed. - With this it is easier to consider that the resting position for the rotor has a certain displacement and orientation on the frame. - Since more rotors are later considered on the same frame, it is possible to define different placements and different orientations for them. Both of this reference systems are non-inertial.

33 Analytical model of the rotor-frame system 31 equations is not a generic moement of those points. Instead only the components of this displacements in the correct directions should be considered. The directions are defined according to the rotor point of iew, and so according to the global reference frame. Summarizing it is possible to define a relation in the form of equation 18. Like for the preceding ariable change all the details are shown in the following section. x A = f(x tel, z tel, β tel ) (18) Now this substitution is applied to rotor motion equation in the ground reference frame. (motion eq. ) ground rotor = f(x T, z T, β T, x tel, z tel, β tel ) (19) 2D model: 6 equations and 6 degrees of freedom. 3D model: 11 equations end 11 degrees of freedom Analysis of the frame motion equations As preiously stated the frame equations are written in the ground reference. These hae the following form: ( motion eq. ) ground fraim = f(x tel, y tel, β tel, x ant, x post ) (20) 2D model: 4 equations and 4 degrees of freedom. 3D model: 7 equations end 7 degrees of freedom. The terms in orange represent the constraint imposed motion. These are the degrees of freedom of the points where the spring-dampers subsystems, representing the frame support system, are linked to the ground. In the final model these are to be the inputs that the ground proides to the system Preparing the frame motion equations for coupling In this subsection, the procedure related to the implementation of steps 1 and 2 are presented for the frame motion equations. It can be noted that the frame motion equations are written in the ground reference system. The objecties of step 1 are therefore already achieed. For what concerns the first step, no changes are performed on the equations. When considering step 2, howeer, it can be noted that some forces are exchanged between rotor and frame. More in detail these forces are exerted on points O, A and B. These forces should appear into the Lagrange component of the frame motion equations. The irtual displacements that regard these forces must be expressed as a function of the frame degrees of freedom. No further ariable is introduced Motion equations coupling Let s consider what was achieed; - Both the rotor and the frame motion equations are written within the same reference. - The used reference system is inertial. - All the effect mutually exchanged between the two systems hae been considered. It is now possible to couple the two set of equation to obtain the dynamics of the whole system.

34 32 Flywheel energy storage { (motion eq. ) ground rotor = f(x T, z T, β T, x tel, y tel, β tel ) (motion eq. ) ground frame = f(x tel, y tel, β tel, x ant, x post ) (21) 2.3 Rotor motion equation Deelopment in the global reference system There are many different options to obtain the rotor motion equation. It has been decided to use the Lagrange equation in the form reported in equation 22. Where: d T dt x i - T stands for the system kinetic energy. - D stands for the system dissipatie term. - V stands for the system potential energy. - Q is the Lagrange component of external forces. In the following subsections all these terms are analyzed and computed Rotor subsystem oerlook T + D + V x i x i x = Q i (22) The rotor subsystem is composed of one rigid body the rotor and the spring-damper subsystems representing the rotor support system. Figure 13: the rotor subsystem The rotor itself is characterized by six degrees of freedom. This are three displacements and three rotations. As preiously stated we consider these degrees of freedom in the global reference during the motion equation deelopment phase. The boundary displacement proides other fie degrees of freedom. Table 5 summarizes the subsystem degrees of freedom. Degree of freedom Rotor displacement Rotor rotation Displacement of the boundary in O Symbol x O G, y O O G, z G σ, β, ρ z O

35 Analytical model of the rotor-frame system 33 Displacement of the boundary in A x A, y A Displacement of the boundary in B x B, y B Total number of degrees of freedom 11 Table 5: the rotor subsystem degrees of freedom These eleen degrees of freedom are arranged into a ector. This is referred to as the degrees of freedom ector for the rotor in the global reference frame. The same order for the degrees of freedom has been adopted in the writing of the Lagrange equation. It follows, in equation 23, the degrees of freedom ector. X G = O x G { } y G O z G O σ { β } ρ x A y A { x } B { y B } X GO = { θ } (23) In the subsystem, the rotor is the only element with no neglectable mass. The graitational effects on the potential term appearing in the Lagrange equations are ignored, gien their minor influence on the system dynamics. The fie spring-damper subsystems are responsible for the whole subsystem stiffness and dumping. Finally, fie external forces act on the subsystem according to the boundary displacements Lagrange equation components In the following subsections the components appearing in the Lagrange equation are ealuated Rotor kinetic energy T In this subsection, the analytical expression for the rotor subsystem kinetic energy is obtained. This expression must be deried as a function of the system independent coordinates. The kinetic energy expression is later introduced in the Lagrange equation, allowing to obtain the inertial term in the body motion equations. The formula for the kinetical energy of a rigid body moing in a tridimensional space is reported in equation (24). X b T = 1 2 ( G c.o.m. ) T [m] G G translational (ω L) T [J L ]ω L rotational (24) Where: - Matrix [m] is a diagonal matrix where eery diagonal element is equal to the rotor mass m rot. - Matrix [J L ] is the rotor tensor of inertia defined in the local reference frame and ealuated on the rotor center of mass. For the translational component, the elocity of the center of mass (c.o.m.) defined in the global reference is used. For the rotational component, the angular speed must be defined in the local reference frame. Only in this way, in fact, it is possible to use the rotor tensor of inertia ealuated in the local reference, which elements are constant with time, and depend only on the rotor mass distribution.

36 34 Flywheel energy storage Expression (24) can be rewritten as follows. T = 1 2 { G c.o.m. T } ω L [m] [0] [ [0] [J L ] ] { G c.o.m. ω L } (25) Since the origin of the local reference frame O does not coincide with the rotor center of mass, the following relation can be considered. c.o.m. O O c.o.m. O O c.o.m. G = G + [ΛGL ] [x L ] ωl = x G + [ΛGL ] [x L ] [AL ]{θ } (26) Now, the relation between the kinematic quantities and the rotor independent coordinates can be written. { G c.o.m. } = [ [I] [Λ O c.o.m. GL] [x L ] [AL ] ] { X GO } (27) ω L [0] [A L ] θ Expression (27) is now substituted into equation (25). By rearranging, the following formulation for the rotor kinetic energy is obtained. T = 1 T 2 {X GO } θ [m] [ O c.o.m. ([m][λ GL ] [x L ] [AL ]) T O c.o.m. [m][λ GL ] [x L ] [AL ] [A L ] T O c.o.m. T ([x L ] [m] [x L O c.o.m. ] + [JL ]) [A L ] ] {X GO θ } (28) It is reminded that the degrees of freedom ector for the rotor subsystem does not include only the rotor independent coordinates. In fact, also the boundary displacements should be considered. Howeer, these hae no effects on the subsystem kinetic energy. For this reason, a null term is placed in correspondence with them in the system mass matrix. T = 1 O X G 2 { θ } X b T [ [m] O c.o.m. ([m][λ GL ] [x L ] [AL ]) T O c.o.m. m][λ GL ] [x L ] [AL ] [0] [A L ] T O c.o.m. T O c.o.m. ([x L ] [m] [x L ] + [JL ]) [A L ] [0] [0] [0] [0]] X GO { θ X b } (29) Finally, it can be noted that the term appearing in the 3 3 central block of the mass matrix can be interpreted as the tensor of inertia of the body ealuated with respect to the local reference system origin O. centerd in O O c.o.m. T O c.o.m. [J L ] = [x L ] [m] [x L ] + [JL ] (30) Figure 14: rotor kinematic analysis Rotor potential energy V The subsystem potential energy term is related to the fie spring elements supporting the rotor. The spring position is ealuated by defining the kinematic linkages. These can be obtained using the graphical representation of Figure 14.

37 Analytical model of the rotor-frame system 35 From simple trigonometrical obseration the following relations can be defined. l A,x = x G O d 1 sin β x A l A,y = y G O + d 1 sin σ cos β y A l B,x = x G O + d 2 sin β x B (31) l B,y = y G O d 2 sin σ cos β y B Where d 1 and d 2 are defined as follows. l O,z = z G O z O d 1 = l 1 + l 2 2 d 2 = l l 3 Now the mathematical relations between spring length and degrees of freedom are known. Using these relations, the potential energy is written according to equation (33). V = 1 2 k A,x( l A,x ) k A,y( l A,y ) k B,x( l B,x ) k B,y( l B,y ) k O,z( l O,z ) 2 (33) (32) Rotor dissipatie energy D The elements responsible for the energy dissipation in the system are the fie dampers. It can be noted that these elements are placed in parallel to the springs. Therefore, the damper motion speed can be calculated deriing equations (31) with respect to time. Therefore, the following mathematical relations between dampers speed and degrees of freedom are obtained. l A,x = x G O ω G,2 d 1 cos β x A l A,y = y G O + d 1 ( ω G,2 sin β sin σ + ω G,1 cos β cos σ) y A l B,x = x G O + ω G,2 d 2 cos β x B (34) l B,y = y G O d 2 ( ω G,2 sin β sin σ + ω G,1 cos β cos σ) y B l O,z = z GO z O Where ω G,1 and ω G,2 are defined as in equation (35), where the matrix [A G ] is obtained according to equation (12). ω G,1 σ { ω G,2 } = [A G ] { β } (35) ω G,3 ρ Now, by substituting equations (34) into equation (36), the dissipation term in the Lagrange equation can be written.

38 36 Flywheel energy storage D = 1 2 r A,x( l A,x) r A,y( la,y) r B,x( lb,x) r B,y( lb,y) r O,z( l O,z) 2 (36) Lagrange component As preiously stated, the forces acting on the subsystems are the one exchanged between the rotor bearing and the frame. Since these fie forces act on fie degrees of freedom, it is easy to define the Lagrange component ector. Q = { F A,x F A,y F B,x F B,y } T (37) This completes what is required for the subsystem motion equation Soling the rotor motion equation By substituting the components introduced in subsection into the Lagrange equation, the rotor motion equations in the global reference are computed. These are a set of eleen second-order differential equation in eleen time-ariating ariables. These equations can be used to simulate the dynamic behaior of the isolated rotor. To do so, these must be rearranged. First, it must be noted that the first six equations are related to the rotor dynamics while the last fie equations refer to the force equilibrium on the boundaries. Now, the first six equations must be rearranged in order to extract the second-order deriatie of the rotor degrees of freedom. Moreoer, the last fie equations can be rearranged to obtain an expression for the force acting on the boundary of the system Simulations and results In this subsection two ealuations are performed on the rotor model. With the first one the system poles are calculated end plotted in a real-imaginary plane. With the second one some simulations of the rotor dynamics are performed. Both the pole plotting, and the dynamics simulation are computed using a set of first attempt data for the system physical properties. Reason of these ealuations is to gie a rough estimation of the correctness of the model. Figure 15: the rotor subsystem poles 2 To perform these rearrangements in the motion equations the Matlab command sole is used.

39 Analytical model of the rotor-frame system 37 The rotor subsystem poles are represented in Figure 15. From this representation some obserations can be made: - There are no poles with positie real part. This means that the model is not unstable. - There is one pole with real part equal to zero. This means that the system presents lability. These two characteristics are consistent with what expected for the rotor subsystem. The lability is related to the possibility for the rotor to rotate without friction or other stopping forces. The dynamic simulation of the model is performed ia a Simulink model. A representation of this is reported in Figure 16. The simulation is based on the time eolution of a state ector. This is composed of the eleen degrees of freedom and their first-order deriatie, as shown in equation (38). X state = {x G O O x G y G O O O y G z GO z G σ σ β β ρ ρ} (38) In the scheme, the two blocks Rotor dynamics and Force ealuation take as input the state ector and the ector of the boundary displacements, and proides as output the time deriatie of the state ector. This last is referred to the following simulation step. These two blocks are defined as a Simulink user-defined functions. The six expressions that were obtained in subsection 2.3.3, soling the first six motion equations, are introduced in the Rotor dynamics block. In the Force ealuation block, instead, the fie expressions that were obtained soling the last fie motion equations are introduced. A set of fie simulations are performed to assess the correctness of the model. These are defined in such a way that the results can be predicted and ealuated. Whit such a procedure, conclusions on the correctness of the model can be drawn. Figure 16: Simulink model for the rotor subsystem simulation In particular, the fie simulations ealuate the response of the subsystem to an initial displacement of the rotor off its equilibrium point. For these simulations a rotor with null eccentricity is considered, together with an initial rotational speed of 6000 rpm. The off equilibrium initial displacements for the rotor are presented in table 6.

40 38 Flywheel energy storage Simulation x G [m] y G [m] z G [m] σ G [rad] β G [rad] ρ G [rad] 1 0, , , , ,01 0 Table 6: initial configurations for the rotor subsystem simulations For these fie cases, the expected results are those of a damped second order system. This system must be characterized by a mass, a stiffness and a damping that can be easily ealuated on the base of the data introduced for the rotor subsystem. More details on these ealuations are presented in appendix A. The results obtained are in good accordance with the expectations. The correctness of the rotor subsystem motion equations is therefore assessed. 2.4 Preparing the rotor motion equation for coupling In this section the procedure introduced in the subsection is detailed. It is to be reminded that the procedure inoles two distinct tasks. The first task deals with changing the reference frame for the motion equation: from the global reference to the absolute one, first, and from the absolute reference to the ground one, later. The second task aims at defining and substituting into the motion equation an expression for the boundary displacements. This expression should relate the boundary displacements to the frame degrees of freedom Variable change procedure In this subsection the ariable change procedure is detailed. This procedure was deeloped into a Matlab function and applied for both the global-to-absolute and the absolute-to-ground ariable changes. Before getting into details the rotor degrees of freedom in the three releant reference systems are listed in table 7. Reference frame Degrees of freedom Global reference frame O O O x G y G z G σ β ρ Absolute reference frame O O O x A y A z A σ A β A ρ A Ground reference frame x T y T z T σ T β T ρ T Table 7: the rotor degrees of freedom Moreoer, the position and orientation of the global reference with respect to the absolute one and the position and orientation of the absolute reference with respect to the ground one are listed in the following table. These also represent how the rotor was placed on the frame and how the frame is located into the ground reference. Item Positioning Global reference with respect to absolute reference x rot y rot z rot σ rot β rot ρ rot Absolute reference with respect to ground reference x tel y tel z tel σ tel β tel ρ tel Table 8: reciprocal displacement of reference systems These six coordinates ectors are defined according to the conention introduced in section for identifying the position and orientation of a rigid body in space.

41 Analytical model of the rotor-frame system 39 As preiously stated, the absolute reference is fixed with the global one, and so the coordinates identified with subscript rot are constant with time. On the other hand, the absolute reference is not fixed to the ground one, and so the coordinates with subscript tel are ariables. The task that the Matlab function must perform is to express the rotor degrees of freedom in the global reference as a function of the rotor degrees of freedom in the absolute one, and express the rotor degrees of freedom in the absolute reference as a function of the rotor degrees of freedom in the ground one. These two tasks are performed into two different stages each. In the first stage a relation between the displacement quantities is found. In the second stage the relation between the orientation angles is inestigated. The procedure is presented for the global-to-absolute ariable change, but it does apply also to the absolute-tolocal ariable change Displacement ariable change Figure 17: displacement, from absolute to global reference To perform this ariable change some initial obserations must be done. It can be noted from Figure 17 that the ector identifying a generic position of point O in the absolute reference is composed by the sum of two different ectors. These are the ector finding the global reference origin and the ector that locates point O starting from the global reference origin. These two ectors are defined as follows. x rot { y rot } { z rot G x O G y O G z O } (39) Howeer, the first ector is defined in the absolute reference while the second one in the global reference. Before summing them they must be referred to the same reference. This is done by using the transformation matrix preiously introduced. By doing this we gain the mathematical relation between displacement degrees of freedom in the absolute and in the global reference. x O G x O A G { y O } = [Λ AG ] 1 ({ G z O A { y O } = [Λ AG ] { A z O x O A y O A z O A x rot } { y rot x O G y O G z O G x rot } + { y rot } (40) z rot z rot }) = [Λ GA ] ({ A x O A y O A z O } { x rot y rot z rot }) (41)

42 40 Flywheel energy storage Note that for the transformation from global to absolute reference the rot-ector, representing the position of the global reference with respect to the absolute reference, is constant. The rot-ector is substituted with the tel-ector in case the transformation from the absolute to the ground reference is considered. In this second case, this ector is not constant Orientation ariable change The orientation ariable change is less intuitie. The procedure that now is introduced is one out of the many deeloped for soling this task. This is the most reliable and the easiest to understand. The crucial idea of this procedure is to conert the cardan angles into quaternions to make the ariable change easier. Finally, the quaternions are transformed back to cardan angles in the new reference system. To make the explanation more understandable the representation in Figure 18 is presented. In this representation with the ector R is illustrated the rotor orientation. The ector R represents the neutral orientation for the rotor in the global reference while the ector R represents the neutral orientation in the absolute reference. Figure 18: orientation ariable change By looking at the picture the following obserations can be made: - The orientation of the rotor in the global reference system is defined with the three cardan angles that represent the transformation from R to R. This transformation is specified as T R R. - The orientation of the rotor in the absolute reference system is define with the three cardan angles that represent the transformation from R to R. This transformation is specified as T R R. - The orientation of the global reference in the absolute frame is define with the three cardan angles that represent the transformation from R to R. This transformation is specified as T R R. - Finally, the transformation T R R can be expressed as the sum of transformation T R R and a transformation T R R. It follows that: T R R = T R R + T R R (42) T R R = T R R ext T R R int (43) To make the notation easier to understand the two transformations on the right-hand side of the preious expression are indicated as external (ext) and internal (int).

43 Analytical model of the rotor-frame system 41 This means that the transformation that identifies the rotor orientation in the global reference can be obtained as the difference between the transformation that identifies the rotor orientation in the absolute frame and the transformation that identifies the global reference orientation in the absolute frame. These obserations are deeloped into an algorithm first, and into a Matlab function later, according to the following eight steps. 1. The cardan angles representing the T R R transformation are conerted into quaternions. This operation is performed as expressed in equation (44). Since the cardan angles of the T R R transformation where defined in the absolute reference, also the quaternion now obtained is defined in the absolute reference. ext q1 A ext q2 A ext = q3 A ext { q4 A } sin ( 1 2 σ A) sin ( 1 2 β A) sin ( 1 2 ρ A) + cos ( 1 2 σ A) cos ( 1 2 β A) cos ( 1 2 ρ A) +sin ( 1 2 σ A) cos ( 1 2 β A) cos ( 1 2 ρ A) + sin ( 1 2 β A) sin ( 1 2 ρ A) cos ( 1 2 σ A) sin ( 1 2 σ A) sin ( 1 2 ρ A) cos ( 1 2 β A) + sin ( 1 2 β A) cos ( 1 2 σ A) cos ( 1 2 ρ A) { +sin ( 1 2 σ A) sin ( 1 2 β A) cos ( 1 2 ρ A) + sin ( 1 2 ρ A) cos ( 1 2 σ A) cos ( 1 2 β A) } (44) 2. The cardan angles representing the T R R transformation are conerted into quaternions. This operation is performed as follows. Since also the cardan angles of the T R R transformation where defined in the absolute reference, also the quaternion now obtained is defined in the absolute reference. q1 A int q2 A int q3 A int { q4 int A } = sin ( 1 2 σ rot) sin ( 1 2 β rot) sin ( 1 2 ρ rot) + cos ( 1 2 σ rot) cos ( 1 2 β rot) cos ( 1 2 ρ rot) +sin ( 1 2 σ rot) cos ( 1 2 β rot) cos ( 1 2 ρ rot) + sin ( 1 2 β rot) sin ( 1 2 ρ rot) cos ( 1 2 σ rot) sin ( 1 2 σ rot) sin ( 1 2 ρ rot) cos ( 1 2 β rot) + sin ( 1 2 β rot) cos ( 1 2 σ rot) cos ( 1 2 ρ rot) { +sin ( 1 2 σ rot) sin ( 1 2 β rot) cos ( 1 2 ρ rot) + sin ( 1 2 ρ rot) cos ( 1 2 σ rot) cos ( 1 2 β rot) } (45) 3. Now that the transformations T R R and T R R hae been defined with their quaternion representation, these can be subtracted according to the expression in equation (43). The result is expressed in the absolute reference. The algebraic sum between quaternion is performed as reported in the following equation. tot q1 A q1 int A q1 ext A q2 int A q2 ext A q3 int A q3 ext A q4 int ext A q4 A tot q2 A q1 int tot = A q2 ext A + q2 int A q1 ext A + q3 int A q4 ext A q4 int ext A q3 A q3 A q1 int A q3 ext A + q3 int A q1 ext A + q4 int A q2 ext A q2 int ext A q4 A tot { q4 A } { q1 int A q4 ext A + q4 int A q1 ext A + q2 int A q3 ext A q3 int ext A q2 A } (46) 4. The quaternion that was obtained in step 3 represents the transformation T R R. This is the transformation connected to the rotor orientation in the global reference. Howeer, the transformation as obtained in step 3 is defined in the absolute reference. To adapt this to the global reference the ector defining this transformation is introduced. To understand the transformation ector physical meaning it must be reminded that the transformation is a rigid rotation. The ector has a direction lying on the rotation axis and a modulus equal to the rotation magnitude. Once the transformation ector is defined, this can easily be obtained also in a new reference system. The transformation ector in the absolute reference is calculated as follows.

44 42 Flywheel energy storage Modulus: ω A tot = 2 acos(q1 A tot ) (47) Direction ector: { tot ax A tot ay A tot az A } = { q2 A tot q3 A tot q4 A tot sin ( ω A tot 2 ) sin ( ω A tot 2 ) sin ( ω A tot 2 ) } (48) 5. The transformation ector is now conerted from the absolute reference to the global one. This task is performed using the transformation matrix already introduced. { tot ax G tot ay G tot az G } = [Λ GA ] { tot ax A tot ay A tot az A } (49) 6. From this ector the quaternion for the same transformation in the global reference frame is obtained. tot q1 G tot q2 G tot = q3 G tot { q4 G } cos ( ω G tot 2 ) ax tot G sin ( ω G tot 2 ) ay tot G sin ( ω G tot 2 ) az tot G sin ( ω G tot { 2 ) } (50) 7. To go back to the cardan angles representation of this transformation, first the transformation matrix must be calculated. This is done according to the following formulation. tot (q1 G ) 2 tot + (q2 G ) 2 tot (q3 G ) 2 tot (q4 G ) 2 2(q2 tot G q3 tot G q1 tot G q4 tot G ) 2(q2 tot G q4 tot G + q1 tot G q3 tot G ) [M] = [ 2(q2 tot G q3 tot G + q1 tot G q4 tot tot G ) (q1 G ) 2 tot (q2 G ) 2 tot + (q3 G ) 2 tot (q4 G ) 2 2(q3 tot G q4 tot G q1 tot G q2 tot G ) ] (51) 2(q2 tot G q4 tot G q1 tot G q3 tot G ) 2(q3 tot G q4 tot G + q1 tot G q2 tot tot G ) (q1 G ) 2 tot (q2 G ) 2 tot (q3 G ) 2 tot + (q4 G ) 2 8. From the transformation matrix, it is possible to define the rotor orientation, expressed as cardan angles in the global reference system. In this way a relation between cardan angles in the global reference and cardan angles in the absolute one is obtained. σ G = atan ( M(2,3) M(3,3) ) ; M(1,3) β G = atan ( sqrt (1 (M(1,3)) 2 ) ; ) ρ G = atan ( M(1,2) M(1,1) ) ; (52)

45 Analytical model of the rotor-frame system Rotor subsystem boundary displacements As it was preiously introduced, the boundary displacement of the rotor subsystem can be related to the frame degrees of freedom. In this subsection the mathematical relation between these two quantities is inestigated. As for the ariable change, a procedure is deeloped and implemented into a Matlab function. It is reminded that the boundary displacements are not any generic moements of the points O, A and B. Instead they are only the displacements of these points in the specific directions defined in the global reference frame. To make the procedure description easier to read, the following 2D scheme of the system is presented. Here the frame is represented with the three points linking it to the rotor. Figure 19: imposed boundary displacements on the rotor The procedure to define the mathematical relation between boundary displacements and frame degrees of freedom is explained in the following steps. 1. The three displacements for the three points are defined. These are ealuated as the difference between the final and initial positions of these points. The mathematical deelopment for A is hereafter illustrated. This same procedure applies also to points O and B. Position of point A in the absolute reference: { A x A A y A A z A Final position of point A in the ground reference: { T x A T y A T z A x rot 0 } = [Λ AG ] { 0 } + { y rot } (53) d 1 z rot } = [Λ TA ] { Initial position of point A in the ground reference: { T x A T y A T z A } t=0 = [Λ TA ] σt =0 { β T =0 ρ T =0 Displacement for point A in the ground reference: A x A A y A A z A A x A A y A A z A x tel } + { y tel } (54) z tel x tel } + { y tel z tel } t=0 = { A x A A y A A z A } (55)

46 44 Flywheel energy storage disp (x T A ) T ) { disp (y A } = [Λ TA ] { T ) disp (z A A x A A y A A z A x tel } + { y tel z tel } { A x A A y A A z A } (56) 2. These displacements are brake down into the three components along the global reference. This is performed with the following mathematical procedure. As for the first step, the procedure is reported for point A, but this can also be applied to points O and B. disp (x A ) disp (z A ) disp (x T A ) { disp (y A )} = [Λ GA ][Λ AT ] { disp (y A } (57) T ) T ) disp (z A 3. The six desired displacements are selected between the ones calculated. These are the x and y displacements for points A and B and z for point O. 2.5 Frame motion equation Deelopment in the ground reference There are many different options to obtain the frame motion equation. It has been decided to use the Lagrange equation in the following form. Where d T dt x i - T stands for the system kinetic energy. - D stands for the system dissipatie term. - V stands for the system potential energy. - Q is the Lagrange component of external forces. T + D + V x i x i x = Q i (58) The motion equations for the frame are directly obtained in the ground reference frame. Other for this the oerall procedure is similar to the one already described for the rotor. In the following subsection all these terms are analyzed and computed. Fewer details are proided gien the similarities between this method and the one already described for the rotor Frame subsystem oerlook The frame subsystem is composed of one rigid body the frame itself and the frame support system. As preiously introduced three different alternaties for the model of the support system are considered. For this description of the deelopment of the analytical model the first and simplest model is studied. Only slight changes must be introduced to the method if one of the other two models is analyzed. Seen degrees of freedom characterize the first alternatie of the frame subsystem. Three degrees of freedom are associated to the frame 3. This can moe ertically and rotate along the longitudinal and 3 It is to be noted that, een if in the first model the frame degrees of freedom are only three, no boundaries are put on this while creating the analytical model. Doing this a system with a six degrees of freedom frame is obtained. The equations related to the

47 Analytical model of the rotor-frame system 45 transersal axis. The boundary displacement proides other four degrees of freedom to the frame subsystem. In this case, the boundary displacement represents the effects of the road roughness on the ehicle. The subsystem degrees of freedom are listed in the following table. Degree of freedom Symbol z tel Frame displacement Frame rotation β rot, ρ rot Displacement of the boundaries x a,sx, x a,dx, x p,sx, x p,dx Total number of degrees of freedom 7 Table 9: the frame subsystem degrees of freedom The seen degrees of freedom, plus the three that are later ignored, are arranged into a ector. This is referred to as the degrees of freedom ector for the frame in the ground reference system. It is introduced now to underline in which order the ten degrees of freedom of the system are considered in the Lagrange equation. X tel = {x tel y tel z tel σ tel β tel ρ tel }T (59) Figure 20: the frame subsystem Lagrange equation components The terms appearing in the Lagrange equation are now computed Frame kinetic energy T The only element with non-neglectable mass in this subsystem is the frame. The mass matrix appearing in the Lagrange equation is computed with the same procedure already described for the rotor. In this case the eccentricity of the frame is linked to the fact that it was chosen to put the absolute reference frame origin on the bottom of the chassis. [m tel ] [M] = [ ([m tel ][Λ TA ][x A O c.o.m. ][A A ]) T O c.o.m. [m tel ][Λ TA ][x A ][A A ] [A A ] T O c.o.m. ([x A ] T [m tel ][x A O c.o.m.] + [J A ]) [A A ] ] (60) undesired degrees of freedom are simply ignored to gain consistency with the frame subsystem first model. This can be done because no cross-effects are produced by longitudinal and transersal displacements of the frame and by the rotation along the ertical axis.

48 46 Flywheel energy storage Frame potential energy V Like for the rotor, the potential term connected to the graitational field is neglected. In fact, the static spring preload counterbalances this. Only the four spring-damper subsystems, responsible of supporting the frame, account for the potential energy term. To ealuate these the new kinematic linkages are obtained. The following illustration helps with this task. The kinematic linkages are now reported. Figure 21: frame kinematics analysis l a,sx = x tel + l y 2 sin(ρ tel)cos(β tel ) + l z 2 sin(β tel) x a,sx l a,dx = x tel l y 2 sin(ρ tel)cos(β tel ) + l z 2 sin(β tel) x a,sx l p,sx = x tel + l y 2 sin(ρ tel)cos(β tel ) l z 2 sin(β tel) x a,sx (61) l p,dx = x tel l y 2 sin(ρ tel)cos(β tel ) l z 2 sin(β tel) x a,sx Now the mathematical relations between spring length and degrees of freedom are known. Substituting these relations into equation (62), the potential energy is written. V = 1 2 k a,sx( l a,sx ) k a,dx( l a,dx ) k p,sx( l p,sx ) k p,dx( l p,dx ) 2 (62) Frame dissipatie energy D The elements responsible for the energy dissipation in the system are the fie dampers. It can be noted that the dampers are placed in parallel to the springs. Therefore, the damper motion speed can be calculated deriing the fie expressions obtained for the springs position with respect to time. Therefore, the following mathematical relations between dampers speed and degrees of freedom are obtained. l a,sx = x tel + l y 2 (ω tel,3 cos(ρ tel ) cos(β tel ) ω tel,2 sin(β tel ) sin(ρ tel )) + l z 2 ω tel,2 cos(β tel ) x a,sx (63)

49 Analytical model of the rotor-frame system 47 l a,dx = x tel l y 2 (ω tel,3 cos(ρ tel ) cos(β tel ) ω tel,2 sin(β tel ) sin(ρ tel )) + l z 2 ω tel,2 cos(β tel ) x a,sx l p,sx = x tel + l y 2 (ω tel,3 cos(ρ tel ) cos(β tel ) ω tel,2 sin(β tel ) sin(ρ tel )) l z 2 ω tel,2 cos(β tel ) x a,sx l p,dx = x tel l y 2 (ω tel,3 cos(ρ tel ) cos(β tel ) ω tel,2 sin(β tel ) sin(ρ tel )) l z 2 ω tel,2 cos(β tel ) x a,sx Where ω tel,2 and ω tel,3 are defined as preiously introduced. ω tel,1 { ω tel,2 } = [A T ] { ω tel,3 σ tel β tel ρ tel Now the dissipation term in the Lagrange equation can be written. } (64) D = 1 2 r a,sx( l a,sx) r a,dx( la,dx) r p,sx( lp,sx) r p,dx( l p,dx) 2 (65) Lagrange component The Lagrange component in the frame motion equation can be split into two main elements. The first one is the one proided by the boundary imposed displacements. It is referred to this as the Lagrange component for the external forces. A second component is related to the effects of the forces exerted by the rotor onto the frame. This is the internal forces component. These two elements are deeloped separately. The external forces component is obtained in a similar way to that it is done for the rotor Lagrange component. This is as follows 4. Q tel Fext = { F a,sx F a,dx F p,sx F p,dx } T (66) The internal forces component is deeloped considering the mathematical relation between rotor degrees of freedom and displacement of point A, B and O. This relation was already deeloped. The procedure consists in calculating the irtual work for the forces exchanged between frame and rotor first, and the Lagrange component later. The irtual work of these forces is computed as follows, where the displacements are calculated in accordance with equation (57). Virtual work = F A,x disp (x A ) + F A,y disp (y A ) + F B,x disp (x B ) + F B,y disp (y B ) + F O,z disp (z O ) (67) Finally, the Lagrange component of the internal forces is calculated. 4 Note how in this formulation six degrees of freedom for the rotor are considered. Reason of this is what was stated in section

50 48 Flywheel energy storage Q tel Fint = (Virtual work) (68) Where with is intended the Jacobean, with respect to the frame subsystem degrees of freedom ector Soling the frame motion equation By substituting the components introduced in the last subsections into the Lagrange equation, the motion equations for the frame in the ground reference are computed. These are a set of ten second-order differential equation in ten time-ariating ariables. These equations can be used to simulate the dynamic behaior of the isolated frame. To do this they must be rearranged. First, it must be noted that the first six equations are related to the frame dynamics while the last four equations refer to the force equilibrium on the boundaries. To gain consistence with the frame model represented in Figure 20, among the first six equations, the three representing the dynamics of the constrained degrees of freedom are neglected. Now, the remaining equations must be rearranged in order to extract the second-order deriatie of the rotor degrees of freedom. Moreoer, the last four equations can be rearranged to obtain an expression for the force acting on the boundary of the system. To perform these rearrangements in the motion equations the Matlab command sole is used Simulations and results In this subsection a set of simulations on the isolated frame model dynamics are performed. These are done using the first attempt data preiously introduced. Reason of this ealuations is to gie a rough estimation of the correctness of the model. The dynamic simulation is performed ia the Simulink model reported in Figure 22. As it was done for the rotor, the simulation is based on the time eolution of a state ector. This is composed of the eleen degrees of freedom and their first-order deriatie as shown in equation (69). X state = {x tel x tel y tel y tel z tel z tel σ tel σ tel β tel β tel ρ tel ρ tel } (69) The system dynamics and the force measurement blocks are designed, as it was done for the rotor Simulink model, by introducing the correct set of equations ealuated in section Now the set of simulations, performed with the aim of ealuating the motion equations correctness, are defined with the same basic concepts that were used for assessing the rotor motion equations. These, in fact, are defined in such a way that the results can be predicted. The predicted results are introduced together with the actual results from the system simulation. It is obsered that the results are in good accordance with what was expected. More information on these ealuations are proided in appendix B. 2.6 Equations coupling In this chapter the motion equations for the two subsystems were achieed. A series of steps were performed to write both the two sets of equations in the same reference, and considering all the mutually exchanged effects. Now the two sets can be coupled without performing any further operation. The motion equation for the whole system are therefore achieed. Also the two Simulink models for the two subsystems can be coupled to obtain a model capable of simulating the whole system dynamics. Instead of creating a whole new Simulink model, where in a single block all the equations are introduced, the two models are linked together. It is belieed, in fact, that with this procedure

51 Analytical model of the rotor-frame system 49 Figure 22: Simulink model for the frame motion equation ealuation Figure 23: Simulink model for the rotor-frame motion equation ealuation

52 50 Flywheel energy storage a more intuitie representation on how the two subsystems exchanges forces and displacements is proided. The model is represented in Figure 23. Finally, with a similar procedure to the one here presented, the motion equations for the whole system, composed of rotor subsystem and the two other alternaties for the frame subsystem, are obtained. 2.7 Final obserations and SimMechanics models To ealuate the correctness of the motion equations for the whole system, three SimMechanics models are deeloped, each for one of the different alternaties for the frame subsystem. For the sake of breity, the procedure to obtain the SimMechanics models is oerlooked. A representation of the three models block schemes is presented in figures from 24 to 27. A set of simulations, with random initial conditions, are performed on both the analytical and the SimMechanics models. In both cases, results are expected to be equal. It is to be noted that the execution time and the amount of Ram used for the obtainment of the analytical model exceeded the aailable resources. For this reason, in order to compare the analytical results to the numerical ones, and in the attempt of soling these problems, it was decided to introduce numerically some quantities before the execution of the script. These parameters are the geometrical and physical characteristics of the system, including the rotor positioning and orientation. Furthermore, some additional simplifications must be introduced to achiee the motion equation for the whole system. It is, in fact, necessary to impose the rotor eccentricity equal to zero, and a two-step process is applied to assess the correctness of the equations for any gien eccentricity. In a first phase, the analytical equation for the whole system characterized by a rotor of null eccentricity is compared with the numerical equation of the same system. In the second phase, the analytical equation for the isolated rotor with non-null eccentricity is ealuated. Under this hypothesis good accordance was obsered between the results of the analytical and numerical models. The long execution time and the heay use of resources associated to the execution of the script deeloping the analytical motion equation do not come unexpected. The main reason for these problems, in fact, is related to the use of the cardan angles, especially when it comes to the ariable changes associated to the two reference systems changes performed. Indeed, the summation of two sets of cardan angles is known to be a particularly challenging task from the analytical point of iew. For this reason, in the reference changes, eery degree of freedom representing the rotor orientation, and its deriaties, are substituted with a complicated expression, leading to the mentioned computational problems. These problematics are better analysed in section and are known in literature as reported in [15]. Figure 24: SimMechanics graphical representation of the system

53 Analytical model of the rotor-frame system 51 Figure 25: SimMechanics model first alternatie for the frame subsystem

54 52 Flywheel energy storage Figure 26: SimMechanics model second alternatie for the frame subsystem

55 Analytical model of the rotor-frame system 53 Figure 27: SimMechanics model third alternatie for the frame subsystem

56

57 3 Single rotor optimization In this chapter the rotor geometry is defined through an optimization process. The research of the optimal geometry is performed on the model of a system composed of the frame and a single flywheel. The flywheel is positioned in the frame origin and different alternaties for its orientation are considered. An important preliminary study is performed to address which of the many problem ariables are to be imposed as constants, which other to ary and which to be selected as objects of the optimization. Moreoer, an analysis on which boundary to impose to the optimization problem must be considered. Finally, the cost functions to be minimized must be carefully selected according to the aims of the optimization. The possibility of optimizing the rotor geometry independently from its displacement and orientation on the frame is inestigated. Outcomes for this option, and the one of optimizing together both the rotor geometry and orientation, are presented. A conclusion is drawn on whether an optimal geometry for eery gien rotor orientation exists. The rotor geometry that minimizes the preiously defined cost functions is obtained; this result is ealuated and discussed. Finally, it must be reminded that the optimal geometry for the single rotor identified in this chapter is later used as the starting point in that as the geometry for eery rotor in the ehicle energy storage system for the multirotor optimizations described in chapter 5.

58 56 Flywheel energy storage 3.1 SimMechanics models In chapter 2 both an analytical and a SimMechanics numerical model where deeloped. This is done for the three systems corresponding to the three alternaties for the frame subsystem. Finally, the results from both alternaties were ealuated and compared. Since these are in good agreement, during the optimization phases of this chapter and the two following ones, it can be decided to work with the numerical or the analytical model. It was decided to use the numerical model for the following reasons: - The running times for the script deeloping the analytical model are extremely long. For this reason, in the attempt of reducing the needed resources and the execution times, it was decided to modify the script, introducing the rotor positioning and orientation numerically before the execution of the program. In this way, the produced system motion equations regard a specific displacement an orientation of the rotor. Een if this solution has drastically reduced the needed running times, still these are considerably long and, moreoer, now it is necessary to run the script for eery new rotor placement. Finally, as stated in section 2.7, additional simplifications must be introduced to achiee the motion equations in an analytical form with the aailable resources. All these makes the analytical model inconenient for optimization purposes. - With the SimMechanics numerical model it is ery fast and easy to introduce little changes in the system. Moreoer, the simulation execution times are short enough to perform an optimization process. Finally, with the SimMechanics model comes a graphical representation of the system that helps understanding the simulated dynamics. 3.2 Analysis of the problem Gien the complexity of the problem, before starting with the optimization process, a study to categorize the wide number of ariables is performed. The goal is to identify which ariables can be independently selected, which can be measured and which are the objects of the optimization process. Moreoer, by creating a table with all the ariable figuring in the system, it is easier to deelop an effectie procedure to identify the optimal rotor geometry and not getting lost in a huge choice of possible tests and optimizations. In the following subsections the list of the ariables featuring the system is proided. This are grouped according on the effects that they are performing on the model. Not all the ariables are listed but only those which hae a crucial influence on the setting of the optimization process. Finally, fie cost functions are presented, each representing one of the main negatie aspects related to the flywheel dynamics that must be minimized General ariables of the optimization The ariables now listed are the main parameters of the optimization process. A short explanation is proided together with the broad use that is for them intended. A numerical alue is gien for the ones that are not going to change in eery optimization. Variable 1 (1) Variable representing the rotor geometry. This ariable is composed of the six geometrical dimensions of the solid rotor, and so of three diameters and three lengths. Since the final goal of the process is finding the optimal rotor geometry, this ariable is left free to change during the optimization.

59 Single rotor optimization 57 Variable 2 (2) Variable 3 (3) Variable 4 (4) Initial angular speed of the rotor. In a first phase the rotor angular elocity is imposed equal to rpm. This is the maximum nominal speed of a KERS operating on Formula One cars, which is used as a first attempt guideline. In a second phase it is also possible to consider different alues of this parameter. Kinetic energy initially stored in the flywheel. In a first phase the rotor initial kinetic energy is imposed equal to J, this is the energy stored in a Formula One KERS. In a successie phase it is also possible to consider different alues of this parameter. Rotor geometry before optimization. This ariable represents the initial geometry of the rotor before the optimization process is performed. Reason for this parameter to be introduced is to ealuate the conergence of the solution on the same results, independently of the initial conditions. Three initial configurations are defined and these are reported in the following table. Configuration l 1 l 2 l 3 D 1 D 2 D 3 α 1 0,01 0,05 0,03 0,07 0,15 0,07 α 2 0,03 0,03 0,03 0,08 0,08 0,08 α 3 0,1 0 0,30 0,10 0,04 0,40 0,04 Table 10: initial rotor geometry [m] Variable 5 (5) Variable 6 (6) Variable 7 (7) Variable 8 (8) Variable 9 (9) Rotor density. This is a parameter that represents the rotor material. An initial alue of kg/m 3 is considerd. In a successie phase it is also possible to consider different alues of this parameter. Percental eccentricity of the rotor. This parameter expresses as a percentage based on the rotor aerage diameter the distance between the rotor rotation axis and the rotor centre of mass. The aerage diameter is calculated as a weighted aerage on the three diameters of the solids composing the rotor. A realistic initial alue of 0,2% is considered. Rotor rotational axis. This ariable is defined by two angles representing the orientation of the rotor with respect to the frame. In some first ealuations this parameter is imposed, while, later, it is considered as one of the optimization ariables of the problem. Number of bearings. This ariable represents the number of bearings per side of the rotor rotation axis. By considering more than one bearing per side it is possible to increase the rotor support system stiffness beyond the limits imposed by mechanical bearings. Vehicle elocity. This parameter represents the speed at which the ehicle goes on a rough terrain. A first attempt alue of 20 m/s is considered System models for the optimization In this subsection the models employed during the optimization process are illustrated. Out of the three that were deeloped, only the following two are actually used for this task. Since these models differs by the frame representation only, the rotor is not considered in the illustrations and in the count of the degrees of freedom.

60 58 Flywheel energy storage Model 1 (4) 15 degrees of freedom model. This model allows for better results and doesn t add as much extra numerical stiffness to the system as the 11 degrees of freedom one does, howeer the simulation times are longer. Finally, with this model it is also possible to simulate the ehicle dynamics during turning. Figure 28: 11 degrees of freedom model for the frame Model 2 (5) 11 degrees of freedom model. The results obtained with this model are expected to be rougher, while the running times shorter. In any case the exact amount in which the results from the two models differ can only be assessed by running a multiple number of simulations. Figure 29: 15 degrees of freedom model for the frame

61 Single rotor optimization System forcing The factors proiding the excitation to the frame, and responsible for the dynamics of the system, are now listed. A short explanation is proided. Moreoer, information related on which model must be used to introduce these excitations is specified. Excitation 1 (1) Excitation 2 (2) Excitation 3 (3) Excitation proided by the rotor dynamics on the frame. Gien the nature and the goals of the optimization process, this excitation is considered in eery test and optimization. All three of the models preiously introduced can consider this kind of forcing. Excitation proided by the road roughness and acting through the ehicle tires and suspensions. In general, this excitation is always considered unless some reason for neglecting is found. The road roughness can be considered in all three of the models preiously introduced. Excitation proided by the ehicle dynamics during turning or in acceleration and deceleration phases. This forcing can be introduced only with the 15 degrees of freedom model, because of its additional degrees of freedom Road roughness from legislation The road roughness is produced by means of a random based process, according to the specifications proided by legislation. For the sake of breity, the details related to the legislation and to the procedure that was implied in deeloping the road roughness profile are oerlooked. Howeer, a graphical representation of a road profile obtained with this method is presented in Figure 30. Moreoer, since the legislation imposes constraints on this profile power spectral density, also a representation of this quantity is presented with Figure 31. Figure 30: road roughness profile Limits of the optimization field In the optimization process, a set of constraints must be considered on the optimization ariables. This is particularly true for the rotor geometrical characteristics that, as preiously stated, are always introduced in the list of the optimization ariables. The main reasons why these constrains are considered are now listed:

62 60 Flywheel energy storage - Degenerated configurations, characterized for example by geometrical structures with negatie dimensions, can be aoided. - Unlikely structures are excluded from the results of the optimization process. These are those structures characterized by extremely thin elements, and that would not be able to bear loads or be manufactured. Reasonable boundary must be selected. It is desired that the final results of the optimization process to not lie on the boundary for any optimization ariable. If this is the case, a careful ealuation of the results must be performed. Figure 31: road profile power spectral density Additional constrains of the optimization Additional constrains are imposed to the system to grant a faster conergence to the solution. These constraints are selected with the intent of not influencing the soler results, but only of making it easier to get to them. These are described in the following table. Extra 1 (6) Extra 2 (7) Imposed rotor symmetry. Gien the symmetrical nature of the system it is reasonable to expect that also the optimal solution would hae some kind of symmetry. A series of preliminary optimizations confirms this hypothesis. By the imposition of a symmetry constraint, the dimension ariables characterizing the rotor size are reduced from six to four. Imposition of a correlation between diameter and length for the two external cylindrical elements, composing the rotor body. With this constraint it is considered that these elements should hae the right size to fit a bearing haing the inner diameter equal to the cylinder one 5. The relation between diameter a length of these elements is obtained by extracting data from the bearing manufacturers catalogues. The points obtained are fitted by an order three polynomial. With this it is possible to further reduce the number of ariables characterizing the rotor geometry from four to three. 5 It was decided to consider a not ery tight fit for the bearing in the axial direction. Reason of this is that in the final assembly also some washers wold be probably introduced.

63 Single rotor optimization 61 Figure 32: relation between the element diameter and length to proide the bearing fit Elementary cost functions In this subsection the elementary cost functions for the system optimization are introduced. Each of these functions represents one of the main negatie aspects related to the flywheel dynamics, and these must be minimized. The exact procedure that is used to minimize these cost functions is analysed only in the following section. In any case, these elementary cost function are not minimized indiidually, during the optimization process, for ealuating the final rotor geometry. In fact, the function to be minimized is a composed function that is introduced in the following sections. Cost C1 (1) Cost function that represents the speed of decay of the kinetic energy stored in the rotor. This function is obtained by drawing the time eolution of the kinetic energy, calculating the angular coefficient of the tangential line to this cure at a gien time instance and, finally, calculating the square of this quantity. This cost function is of great alue to define the performances of an energy storage system. A representation of the kinetic energy eolution and its tangent line are represented in Figure 33. C 1 = q 2 (70) Cost C2 (2) This second elementary cost function is equal to the rotor mass. C 2 = m rot (71) Cost C3 (3) Cost function representing the rotor ibration. It is defined by calculating the mean alue of the rotor acceleration in the three x G, y G and z G directions, obtaining the square of these three alues and finally by summing them together. An example of the rotor ibration during time is represented in Figure 34. C 3 = (mean(x rot(t))) 2 + (mean(y rot(t))) 2 + (mean(z rot (t))) 2 (72) Cost C4 (4) Cost function representing the frame ibration. It is defined by calculating the mean alue of the frame acceleration in the three x T, y T and z T directions, obtaining the square of these three alues and finally by summing them together. An example of the frame ibration during time is represented in Figure 34.

64 62 Flywheel energy storage C 4 = (mean(x tel(t))) 2 + (mean(y tel(t))) 2 + (mean(z tel (t))) 2 (73) Cost C5 (5) Cost function that represents the encumbrance of the rotor. This function is identified by the olume of the smallest parallelepiped that can contain the rotor. The olume of this can be ealuated as the sum of the lengths of the three cylindrical elements composing the rotor, multiplied by the square of the largest diameter. C 5 = (l 1 + l 2 + l 3 )D 2 2 (74) A fast-transitory phase can be obsered at the beginning of eery simulation. For this reason, it is decided to start to collect the data to ealuate the fie cost functions only after this first phase wears off. This usually happens in a fraction of a second. It is belieed that this transitory phase has a numerical cause. Reason of this is that eery spring element has been introduced in the model with an initial position equal to the equilibrium one. Figure 33:example of the kinetic energy time eolution 3.3 Definition of the optimization procedure for the rotor geometry In this section a series of preliminary ealuations are performed before starting with the optimization process. Goal of these ealuations is to gather enough information to set the oerall optimization process by defining a roadmap of the procedure. In other words, the problem that is to be addressed in this section is the one of creating a clear step-by-step technique to identify the rotor optimal geometry, and so to define how to proceed in the following sections. Some of the problems that are faced are concerned with defining the sequence of optimizations between the multitude of possible alternaties. Moreoer, for each of them, which parameters to fix, which to change and on which parameter to perform the optimization on. In the following subsections some initial ealuations are performed and, finally, the last subsection presents the oerall procedure deeloped.

65 Single rotor optimization 63 Figure 34: example of the rotor ibration Figure 35: example of the frame ibration Preliminary ealuation 1 Before getting into the optimization process, an ealuation on how the elementary cost functions C 1, C 3 and C 4 changes is performed. During this process all the ariables are kept constant with except for the rotor support bearing stiffness and the percent eccentricity of the rotor. The elementary cost functions C 2 and C 5 are not considered in this phase since they represent the rotor mass and the rotor encumbrance, and these are independent from the parameters that ary during the ealuation. For this initial optimization the rotor geometry to be used is the one identified as α 1.

66 64 Flywheel energy storage In this phase only one bearing for rotor axis side is considered. For what concerns the stiffness of these elements, a range of alues that goes beyond what is normally achieable with mechanical bearings is considered. Reason for this is that the goal is to obtain an oerall trend for the elementary cost functions as the rotor support stiffness and the eccentricity ary, and this without considering the boundary imposed by manufacturing problems. In the graphs from Figure 36 to Figure 38 the results obtained using the 11 degrees of freedom model, and for the rotor with the rotation axis in the ehicle progression directions, are reported. On the ertical axis, the logarithm of the elementary cost functions is presented. Figure 36: cost function C 1, 11 degrees of freedom model, α 1 rotor Figure 37: cost function C 3, 11 degrees of freedom model, α 1 rotor Figure 38: cost function C 4, 11 degrees of freedom model, α 1 rotor

67 Single rotor optimization 65 It can be noted from these results that an increase of the bearing stiffness has a positie effect on all three of the considered elementary cost functions. For this reason, ery high stiffness bearings are selected among those aailable on the market. Moreoer, to achiee higher stiffness, the opportunity of building a rotor with more than one bearing for side of the rotation axis is inestigated. It can also be noted that the effects of the percent eccentricity on the cost functions are rather small with respect to those of the bearing stiffness. In any case a reduction in the eccentricity leads to better results. It has also to be considered that, while it is possible to select the desired stiffness for the rotor support system, it is not easy to reduce the eccentricity under a certain amount. This means that while the stiffness is a parameter that can be freely selected, the eccentricity is a consequence of the system imperfections. Additional ealuations are reported in appendix C Preliminary ealuation 2 Gien the results of the preliminary ealuation 1, some more assessments on the possibility of introducing more than one bearing per side of the rotor axis are conducted. Getting into detail, since from the preious ealuation it was clear that the stiffer the rotor support system the better, the stiffest mechanical bearings are selected. These are roller bearings. The trends for the elementary cost functions C 1, C 3 and C 4 at the change of the eccentricity are obtained for the configuration haing one, two or three bearing per part, and these are reported in graphs from Figure 39 to Figure 41. Figure 39: cost function C 1, 11 degrees of freedom model, α 1 rotor Figure 40: cost function C 3, 11 degrees of freedom model, α 1 rotor

68 66 Flywheel energy storage Figure 41: cost function C 4, 11 degrees of freedom model, α 1 rotor As for the preious case, this ealuation is conducted on the 11 degrees of freedom model, with the rotor haing the rotation axis along the ehicle progression directions. The rotor configuration used to obtain these results is α 1. It can be stated that the options with two and three bearings per side gie better results with respect to the one with a single bearing per side. These alternaties are considered while the optimization process is conducted. Additional ealuations are reported in appendix C Preliminary ealuation 3 The effects that the rotor orientation has on the elementary cost functions C 1, C 3 and C 4 are now inestigated. As it was for the two preious cases the cost function C 2 and C 5 are not considered in this analysis since they represent the mass and the bulkiness of the rotor, which are not influenced by the rotor orientation. Goal of this analysis is to define if and in which measure the cost functions are related to the orientation of the rotor axis. In case a strong correlation is obsered, the option of considering also the angles expressing the rotor orientation as optimization ariables must be considered. In this case finding an optimal geometry for the rotor independently from its orientation won t be an easy task. On the contrary, in case no correlation or weak correlation between cost functions and rotor orientation is found, the optimal geometry found is probably always be the same, independently from which orientation is imposed. Figure 42: cost function C 1, 11 degrees of freedom model, α 1 rotor

69 Single rotor optimization 67 The angles defining the rotor orientation inside the frame are σ rot and β rot. The elementary cost functions are ealuated for both the α 1 and α 2 configurations, arying these orientation angles between 0 and π/2. The results are obtained using the 11 degrees of freedom model and the cost functions are represented on a linear scale. The results for α 1 configuration are reported in graphs from Figure 42 to Figure 45. The only reported result for the system with α 2 rotor is presented in Figure 46Figure 46: cost function C 1, 11 degrees of freedom model, α 2 rotor; this is for the elementary cost function C 1. The reason is that this is the only cost function with notable differences from the case with α 1 rotor. From the representations it can be noted that all three of the elementary cost functions are significantly influenced by the rotor orientation. From this it can be concluded that, in order to optimize the rotor geometry, also the rotor orientation must be kept into account. Moreoer, the possibility of obtaining a geometry for the rotor that is optimal independently from the rotor orientation must be carefully inestigated. This possibility, in fact, must not be considered as granted since different optima rotor geometries could exist for different rotor orientations. A particular in-depth analysis must be done on the trend of the C 4 cost function, representing the oerall ibrations of the frame in the z T, y T and z T directions. It is reminded that the model used for this analysis is the 11 degrees of freedom one, in which the frame is bounded to moe and ibrate only on the ertical axis x T. Keeping this in mind it is clear why an alignment of the rotor axis whit the z T one produces a minimum almost equal to zero in the C 4 cost function. In this case, in fact, the rotor dynamics produces an excitation in the z T - y T plane, that are constrained degrees of freedom for the frame. Therefore, it might be think that the C 4 cost function trend is completely wrong and caused only by the extra numerical stiffness that the model has in the z T and y T directions. Howeer, een if the 11 degrees of freedom model introduces such a numerical stiffness to the system, also with a model that allows the frame motion on the z T - y T plane, these directions would result with a much higher stiffness, with respect to the x T direction. This means that a similar trend for the C 4 function is expected also with the 15 degrees of freedom model, een if the minimum is probably much higher. This trend is calculated and displayed in the following picture for the α 1 geometry. Figure 43: cost function C 4, 15 degrees of freedom model, α 1 rotor

70 68 Flywheel energy storage Figure 44: cost function C 3, 11 degrees of freedom model, α 1 rotor Figure 45: cost function C 4, 11 degrees of freedom model, α 1 rotor Figure 46: cost function C 1, 11 degrees of freedom model, α 2 rotor Een if the trend is consistent with the one expected, it is belieed that, after a first optimization performed with the 11 degrees of freedom model, a second optimization must be performed. This is conducted with the 15 degrees of freedom model, to perfect the results and assess if the extra numerical stiffness introduced by the 11 degrees of freedom model is not affecting the correctness the results.

71 Single rotor optimization Introduction to the rotor geometry optimization The first step to perform, in order to set an optimization process, is to define an oerall cost function. This cost function is now introduced as the algebraic sum of the fie elementary cost functions, presented in the preceding section. Cost = P 1 C 1 + P 2 C 2 + P 3 C 3 + P 4 C 4 + P 5 C 5 (75) Where C i are the elementary cost functions, while P i are the weight that each of the elementary cost functions hae on the oerall cost. One of the most crucial steps in the whole optimization process consists in defining reasonable alues of the weights. In the following subsections the selected procedure is described and analysed in detail. A first optimization process is then presented together with the results achieed Rotor geometry optimization with rotation axis along z T direction Before going into details with the research of the rotor optimal geometry, a set of optimizations with rotor axis imposed in the z T directions are performed. It is reminded that the optimal rotor geometry depends on the orientation of the rotor itself, so the results that are now achieed are related to a rotor with axis along the z T direction only. Anyway, with this first optimization, some interesting results are expected. Firstly, the possibility of introducing more than one bearing per side of the rotor is analysed, secondly, an oerlook at the method applicability is made. In table 11 a schematic representation of the optimizations performed, and of the parameters for them introduced is proided. It can be noted that for the optimizations made, the weights are selected all to be null whit except for one. Finally, in the last four rows of the table the results are presented, consisting of both the optimal geometrical characteristics and of the optimal cost. From an analysis of these data the following obserations are drown: 1. The alidity of the constructie alternaties characterized by more than one bearing for side of the rotor axis are confirmed. For the cost functions C 1, C 3 and C 4 an improement can be noted with increasing the number of bearings, while, for the cost functions C 2 and C 5 representing weight and bulkiness of the rotor the better results are obtained with fewer bearings. 2. It can be noted that for what concerns the solutions obtained with the cost functions C 1 4, these are obtained whit a rotor constituted by a disk of the biggest diameter possible and as thin as possible, whit the smallest supports. A different result is only achieed by minimizing the elementary cost function C Finally, it can be noted that the same results are obtained in terms of geometry and similar in terms of costs by starting the optimization from two different first-attempt solutions. This can be noted by the pairs of optimizations [A1a, A1d], [A1b, A1e], [A1c, A1f] and others not reported. For this reason, from now on the same first-attempt solution geometry α 1 is used for all the tests The selected optimization procedure From the experience made with the preliminary ealuations and with the first attempt optimization two procedure are deeloped. The first one aims at optimizing the rotor geometry together with the rotor orientation to find the best single rotor configuration. Whit the second one the opportunity of defining a rotor geometry that is optimal, no matter for the orientation, is inestigated. These procedures are now discussed and in the following sections are performed.

72 A1a A1b A1c A1d A1e A1f A2a A2b A2c A3a A3b A3c A4a A4b A4c A5a A5b A5c 1 kinetic energy decade rate rotor mass rotor ibration frame ibration rotor encumbrance rotor geometry ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry a1 a1 a1 b1 b1 b1 a1 a1 a1 a1 a1 a1 a1 a1 a1 a1 a1 a1 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 7 rotation axis z z z z z z z z z z z z z z z z z z 8 number of bearings (per side) ehicle speed [m/s] forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x x x x x x x x x x x x x x x x 4 11 dof model 5 15 dof model x x x x x x x x x x x x x x x x x x 6 imposed rotor symmetry 7 simplified rotor geometry 1 length l2 [cm] 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5: Flywheel energy storage 1 total cost function 7,82E+05 1,06E+05 4,22E+04 1,01E+05 1,38E+04 5,45E+03 4,32E+00 4,42E+00 4,52E+00 1,72E+07 3,11E+06 1,28E+06 1,74E+03 1,28E+03 1,21E+03 2,22E-03 3,10E-03 3,86E-03 2 optimal length l2 [m] 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 6,295E-02 9,567E-02 8,312E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,502E-02 2,500E-02 2,501E-02 2,528E-02 5,272E-02 5,923E-02 6,413E-02 4 optimal diameter D2 [m] 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,467E-01 1,313E-01 1,346E-01 Optimization code Cost function weights General parameters Forcing and soler Boundaries of the field Results Table 11: first attempts in optimizing the rotor geometry

73 Single rotor optimization 71 Let s start with the geometry and orientation optimization. First a rough optimization is performed by considering the 11 degrees of freedom model and by optimizing the three ariables representing the rotor geometry and the two indicating the rotor orientation. When the optimal solution is identified, the results are ealuated with the 15 degrees of freedom model. The goal in this phase is to assess whether the minimum in the cost function that was found with the 11 degrees of freedom model exists also with the 15 degrees of freedom one, for a similar configuration. Main reason of these concern is what was already discussed for the effects of rotor orientation on the 11 and on the 15 degrees of freedom models when considering the elementary cos function C 4. If in this phase a preferential direction for the rotation axis is obsered, this is imposed in the following optimization stage. Finally, a fine optimization using the 15 degrees of freedom model is performed to perfect the results obtained in the preious phase. In this last optimization, in case a preferential rotation axis was identified, the optimization ariables wold be only the three quantities representing the rotor geometry. On the contrary, in case no preferential direction axis was identify, the optimization is once more performed on the fie ariables representing geometry and orientation. Figure 47: optimization processes The second optimization process aims at finding the optimal solution for the rotor geometry, independently from the rotor orientation. This possibility must be carefully ealuated and must not be gien for granted. The procedure that has been selected consists in defining a set of optimizations using the solution identified in the preious case as first attempt solution. This time, different orientations are imposed and the results are ealuated. If a diergence in the rotor geometry is obsered it is concluded that no configuration that is optimal no matter the rotor orientation can be found. On the other hand, in case only slight changes are obsered, the optimal solution can be identified with a weighted aerage between these results. Of course,

74 72 Flywheel energy storage the newly identified optimal configuration is not as good as the one achieed by the first optimization process, but it is independent from orientation. In Figure 47 the two optimization processes are summarized. 3.4 The optimal rotor geometry and orientation First attempt optimal geometry Solution for the configuration haing two bearings per side of the rotor axis In this section the procedure used to define the rotor optimal geometry is presented and the obtained results are reported and discussed. As indicated, the two bearings per side solution has been selected. Reason on this stays in the fact that this option proides better results than the single bearing one and is responsible for less encumbrance with respect to the three bearings case. In this section the optimizations are performed on the 11 degrees of freedom model. Finally, because of the correlation between cost functions and rotor orientation, the ariables of the optimization process are both the rotor geometry and the rotor orientation. A set of different solutions is reported. For these a dynamic simulation of the system behaiour is performed and analysed The minimum costs In order to identify the weights to introduce in the oerall cost function, a set of optimizations with elementary cost functions is performed. The aim of these optimisations is that of ealuating the minimum cost that can be achieed for each of the elementary cost functions. As preiously stated, the number of optimization ariables are now fie, the three parameters defining the rotor dimensions and the two indicating the rotor axis orientation. Gien the newly introduced degrees of freedom for the optimizer, it is reasonable to expect that the new results hae a lower or equal cost respect to what was obtained in the preious section, where the rotor axis was imposed in the z T direction. In table 12 the data related to the optimizations performed are presented together with the results obtained. As it can be noted the forecasted decrease in the costs with respect to the corresponding cases in the preious section is confirmed. The minimum obtainable alues of the elementary cost functions are obtained Weights and rough optimization The weights in the oerall cost functions are now defined as the products of two factors according to the following expression. P i = (F scale ) i Q i (76) Where F scale is a scaling factor for the costs. This is introduced to make comparable costs that differ for orders of magnitude. It is defined as one oer the minimum alue that was obtained of each elementary cost function. (F scale ) i = 1 (C min ) i (77)

75 Single rotor optimization 73 Optimization code B1b B2b B3b B4b B5b 1 Cost function weights kinetic energy decade rate rotor mass rotor ibration frame ibration rotor encumbrance General parameters rotor geometry ariable ariable ariable ariable ariable 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry a1 a1 a1 a1 a1 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 7 rotation axis ariable ariable ariable ariable ariable 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x x x 4 11 dof model 5 15 dof model x x x x x 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:100 2,0:100 2,0:100 2,0:100 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 8,99E+04 4,42E+00 3,13E+06 3,85E-02 3,10E-03 2 optimal length l2 [m] 2,000E-02 2,000E-02 2,000E-02 2,005E-02 9,567E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,505E-02 5,923E-02 4 optimal diameter D2 [m] 1,959E-01 1,959E-01 1,959E-01 1,958E-01 1,313E-01 5 σ_rot [rad] 1,360E ,761E-05 1,396E β_rot [rad] 2,995E ,477E-01 1,571E Table 12: minimum cost ealuation The factor Q i, on the other hand, is a parameter arbitrary selected. It is introduced to specify the emphasis gien to each of the elementary cost functions. In the first case all of these factors are set equal to one, in a latter moment also different alues are considered. Now that the weighs are defined, a series of optimizations on the rotor geometry and on the rotor axis orientation are set and performed. For these optimizations the scale factors are calculated with the newly introduced formula together with the results achieed in the preious section, while the selected factors Q i are reported in table 13. In the same table are also presented all the other ariables to set the optimization problem, together with the results obtained. A few words are now spent to explain the choice made for the factors Q i. It can be noted that three alternaties are presented. In the first one all fie of these parameters are set equal to one. In this the same importance is gien to the fie elementary cost functions, and this can be considered as the base case. In the second alternatie it was decided to gie particular emphasis to the C 5 cost function. This can be easily justified from the fact that one of the main characteristics that an energy storage deice for on ehicle use

76 74 Flywheel energy storage should hae is that of high olumetric energy density. It can be noted how the results obtained in this second case differ from those obtained in the first one. Finally, a third scenario is introduced in which high emphasis is put on both the C 1 and C 5 cost functions. This is because a second really important factor for an energy storage deise for use on ehicles is that of storing energy without losses for the longest time possible. In this last scenario the results are once more different from the preious cases. Optimization code Cost function weights C Ctb-1 F_scala C Ctb-2 F_scala C Ctb-3 F_scala 1 kinetic energy decade rate 1 1,11E ,11E ,11E-05 2 rotor mass 1 5,00E ,00E ,00E+01 3 rotor ibration 1 4,00E ,00E ,00E+01 4 frame ibration 1 5,10E ,10E ,10E+00 5 rotor encumbrance 1 7,35E ,35E ,35E-01 1 General parameters rotor geometry ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry a1 a1 a1 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 7 rotation axis ariabile ariabile ariabile 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x 4 11 dof model 5 15 dof model x x x 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 Results 1 total cost function 5,55E+00 1,40E+01 8,22E+00 2 optimal length l2 [m] 2,002E-02 2,158E-02 2,000E-02 3 optimal diameter D1 [m] 2,502E-02 3,825E-02 2,521E-02 4 optimal diameter D2 [m] 1,959E-01 1,920E-01 1,959E-01 5 σ_rot [rad] 1,546E+00 8,751E-01 1,571E+00 6 β_rot [rad] 1,570E+00 1,571E+00 1,571E+00 Table 13: final rotor geometry optimization, 11 degrees of freedom model It is interesting to note that the solutions obtained are characterized by a rotor haing the rotation axis in the ertical x T direction. Since in this section only the 11 degrees of freedom model is used, these results can be justified from the obserations made in subsection As preiously stated, while the 15 degrees of freedom model is used to perfect these results, it must be inestigated whether the ertical orientation of the rotation axis remains an optimal solution. In appendix D a fast analysis of the obtained results is performed. A more in-depth study of the results is reported for the configurations obtained for the optimizations on the 15 degrees of freedom model.

77 Single rotor optimization Perfecting the results Figure 48: graphical representation of the optimal configuration Ctb-1 The 15 degrees of freedom model is now considered with the aim of refining the results for the rotor geometry. These new optimizations are set considering a first attempt geometry equal to the one that was obtained with the optimizations on the 11 degrees of freedom model, and considering a range for the optimization ariables around this result. As it was already pointed out in the preious subsection, all the solutions obtained till now hae a rotation axis almost ertical (x T direction). Gien the adantages related to designing a ehicle haing a perfectly ertical rotor axis, both for the constructie ease and for problems related to the rotor encumbrance, this solution would be desirable. Moreoer, by imposing the rotor axis to be along the x T direction, there would be also adantage in the optimization process. In this case, in fact, the ariables to be optimized would be only the three related to the rotor dimensions while the two ariables representing the rotor orientation wold simply be imposed. Before imposing this condition on the rotor orientation, it must be assured that also with the 15 degrees of freedom model the optimal rotor solution is with rotation axis in the x T direction. This ealuation is performed in the following subsection Ealuations of the rotor orientation axis The obserations performed in subsection 3.3.3, Preliminary ealuation 3, are reminded. With these it was stated that an obious optimal solution was to be found when using the 11 degrees of freedom model by placing the rotor whit rotation axis in the ertical direction. This is related to the fact that whit this orientation the dynamic of the rotor excites two constrained frame degrees of freedom the longitudinal and the transersal one and so the cost function C 4 tends to zero. With the 15 degrees of freedom model, howeer, these degrees of freedom are not constrained, een if these are characterized by a higher stiffness with respect to the ertical direction. In this section it is ealuated if also with the 15 degrees of freedom model the optimal solution has ertical rotor axis. In the representations from Figure 49 to Figure 51 the trends for the C 1, C 3 and C 4 elementary cost functions are represented. These trends are obtained with the 15 degrees of freedom model and for the α 1 rotor configuration. Now, a set of optimizations on the rotor geometry and on the rotor orientation is performed to address this doubt. These optimizations are done on the 15 degrees of freedom model considering the elementary cost

78 76 Flywheel energy storage functions. If from these optimizations it is found that the ertical rotation axis is still the optimal solution, then this is imposed in the refined optimizations performed with the 15 degrees of freedom model. Results of this assessment are presented in table 14. From these optimizations it is clear that also with the 15 degrees of freedom model the ertical orientation of the rotation axis is the optimal solution that minimizes the elementary cost functions C 1, C 3 and C 4. Howeer, it must be pointed out that the effects of the rotor orientation, when using the 15 degrees of freedom model, are not as seere on the cost functions as when the 11 degrees of freedom model is used. This can be noted from the graphs from Figure 49 to Figure 51, representing the trend of the three cost functions with the change of the rotor orientation. This is a consequence of the fact that the 11 degrees of freedom model was oerestimating the system stiffness in the longitudinal and transersal directions. In a latter study, the fact that the rotor orientation does not affect as much the cost function as it was preiously estimated, could make it easier to consider an optimal geometry for the rotor independently from its orientation. Other data from these optimizations are neglected since the minimal costs and the rotor optimal geometry are ealuated imposing the rotation axis in the ertical direction. Figure 49: cost function C 1, 15 degrees of freedom model, α 1 rotor Figure 50: cost function C 3, 15 degrees of freedom model, α 1 rotor

79 Single rotor optimization 77 Figure 51: cost function C 4, 15 degrees of freedom model, α 1 rotor Optimization code D1b- D2b- D3b- D4b- D5b- 1 Cost function weights kinetic energy decade rate rotor mass rotor ibration frame ibration rotor encumbrance General parameters rotor geometry ariabili ariabili ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry a1 a1 a1 a1 a1 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 7 rotation axis ariabili ariabili ariabili ariabili ariabili 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x x x 4 11 dof model x x x x x 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] ±10% ±10% ±10% ±10% ±10% 2 diameter D1 [cm] ±10% ±10% ±10% ±10% ±10% 3 diameter D2 [cm] ±10% ±10% ±10% ±10% ±10% Results 1 total cost function 1,06E+05 4,42E+00 2,76E+07 4,46E+03 3,10E-03 2 optimal length l2 [m] 2,000E-02 2,000E-02 4,400E-02 3,770E-02 9,567E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 9,980E-02 2,540E-02 5,923E-02 4 optimal diameter D2 [m] 1,959E-01 1,959E-01 1,452E-01 1,672E-01 1,313E-01 5 σ_rot [rad] 1,571E ,571E+00 1,571E β_rot [rad] 1,571E ,571E+00 1,571E Table 14: rotor ertical axis ealuation with the 15 degrees of freedom model

80 78 Flywheel energy storage The new optimal costs The new minimum costs for the elementary cost functions are now ealuated. In this case the 15 degrees of freedom model is used, and the rotation axis is imposed in the ertical direction. The procedure is the same that was preiously adopted for the first research of the minimum costs. In this case, howeer, the first attempt rotor geometry is imposed equal to the optimal geometry obtain from the preious research of the minimum costs, moreoer, the interal where the optimization ariables can ary is set as a ±10% ariation on these alues. - Optimization code D1b D2b D3b D4b D5b Cost function weights 1 kinetic energy decade rate rotor mass rotor ibration frame ibration rotor encumbrance General parameters 1 rotor geometry ariabili ariabili ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry ott ott ott ott ott 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler 1 forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x x x 4 11 dof model x x x x x 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] ±10% ±10% ±10% ±10% ±10% 2 diameter D1 [cm] ±10% ±10% ±10% ±10% ±10% 3 diameter D2 [cm] ±10% ±10% ±10% ±10% ±10% Results 1 total cost function 1,05E+05 4,42E+00 3,03E+06 2,17E+03 3,10E-03 2 optimal length l2 [m] 2,000E-02 2,000E-02 2,000E-02 2,000E-02 9,567E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,500E-02 5,923E-02 4 optimal diameter D2 [m] 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,313E-01 - Table 15: research of the minimum costs, 15 degrees of freedom model The tests now performed are described in table 15, together with their results. In the table, the rotor geometrical configurations indicated with ott are defined as followings: - For the simulation D1b, as the results of optimization B1b. - For the simulation D2b, as the results of optimization B2b. - For the simulation D3b, as the results of optimization B3b. - For the simulation D4b, as the results of optimization B4b. - For the simulation D5b, as the results of optimization B5b.

81 Single rotor optimization 79 It can be noted that the results achieed with the 15 degrees of freedom model are only slightly different from those obtained with the 11 degrees of freedom one. For this reason, it can be stated that an 11 degrees of freedom model might be sufficient to define the rotor optimal geometry. In any case, it is decided to proceed with the last phase of the analysis with also the 15 degrees of freedom model, een if only slight changes to the already obtained results are expected Weight definition and precise optimization The weights are defined with the same method that was introduced for the 11 degrees of freedom model. P i = (F scale ) i Q i (78) These new weights, howeer, are obtained on the base of the newly calculated minimum costs. The new optimizations are performed using the same Q i factors that were earlier introduced. The first attempt rotor geometry is set equal to the optimal geometry obtained with the 11 degrees of freedom system. A range of the optimizations ariables of ±10% around these alues is considered. The optimizations performed are presented in the table 16. Optimization code Etb-1 Etb-2 Etb-3 Cost function weights C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 9,57E ,57E ,57E-06 2 rotor mass 1 2,26E ,26E ,26E-01 3 rotor ibration 1 3,30E ,30E ,30E-07 4 frame ibration 1 4,61E ,61E ,61E-04 5 rotor encumbrance 1 3,23E ,23E ,23E+02 General parameters 1 rotor geometry 2 initial rotor elocity [rpm] 3 stored energy [J] 4 initial rotor geometry 5 rotor density [kg/m^3] 6 perceptual eccentricity 7 rotation axis 8 number of bearings (per side) 9 ehicle speed [m/s] Forcing and soler 1 forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration 4 11 dof model 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] 2 diameter D1 [cm] 3 diameter D2 [cm] Results 1 total cost function 2 optimal length l2 [m] 3 optimal diameter D1 [m] 4 optimal diameter D2 [m] ariabili ariabili ariabili ott ott ott ,5 0,5 0,5 x x x x x x x x x 2,0:100 2,5:10 2,0:101 2,5:10 2,0:102 2,5:10 2,5:100 4,98E+00 2,5:100 5,69E+00 2,5:100 6,46E+00 2,000E-02 2,074E-02 2,122E-02 2,503E-02 2,859E-02 2,751E-02 1,959E-01 1,941E-01 1,930E-01 Table 16: final rotor geometry optimization, 15 degrees of freedom model

82 80 Flywheel energy storage As it was predicted, only a slight change from the preious results can be noted. In the following sections some data on the optimal configurations are presented. In detail, these are information related to the physical properties of the optimal rotors, to the dynamics of a single rotor configuration and to the basic physical characteristics of a multi-rotor configuration based on these optimal solutions. For the multirotor configuration, the constraint of storing 20 KWh is set. Additional data on these configurations is proided in appendix E.

83 Single rotor optimization Optimal solution Etb-1 General information is here listed for the optimal rotor configuration Etb-1. Figure 52: graphical representation of optimal rotor Etb-1 Quantity Unit Value Energy stored per rotor J Number of rotors (multirotor) Kinetical energy decay rate J/s -247,74 Kinetical energy decay rate %/s -0,0619% Single rotor mass Kg 4,3241 System mass (multirotor) Kg 778,3302 Single rotor olume m 3 0,0019 System olume (multirotor) m 3 0,3425 Table 17: optimal rotor Etb-1 characteristics Figure 53: time eolution of the kinetical energy stored in the rotor, configuration Etb-1

84 82 Flywheel energy storage Optimal solution Etb-2 General information is here listed for the optimal rotor configuration Etb-2. Figure 54: graphical representation of optimal rotor Etb-2 Quantity Unit Value Energy stored per rotor J Number of rotors (multirotor) Kinetical energy decay rate J/s -268,67 Kinetical energy decay rate %/s -0,0672% Single rotor mass Kg 4,6664 System mass (multirotor) Kg 839,9594 Single rotor olume m 3 0,0021 System olume (multirotor) m 3 0,3757 Table 18: optimal rotor Etb-2 characteristics Figure 55: time eolution of the kinetical energy stored in the rotor, configuration Etb-2

85 Single rotor optimization Optimal solution Etb-3 General information is here listed for the optimal rotor configuration Etb-3. Figure 56: graphical representation of optimal rotor Etb-3 Quantity Unit Value Energy stored per rotor J Number of rotors (multirotor) Kinetical energy decay rate J/s -247,89 Kinetical energy decay rate %/s -0,0620% Single rotor mass Kg 4,3250 System mass (multirotor) Kg 778,4970 Single rotor olume m 3 0,0019 System olume (multirotor) m 3 0,3433 Table 19: optimal rotor Etb-3 characteristics Figure 57: time eolution of the kinetical energy stored in the rotor, configuration Etb-3

86 0,0E+00 1,0E-02 2,0E-02 3,0E-02 4,0E-02 5,0E-02 6,0E-02 7,0E-02 8,0E-02 9,0E-02 1,0E-01 1,1E-01 1,2E-01 1,3E-01 1,4E-01 1,5E-01 1,6E-01 1,7E-01 1,8E-01 1,9E-01 2,0E Flywheel energy storage 3.5 Orientation independent geometry General procedure The solution that is chose to perform this second optimization process is the one identified with the name Etb-3. This solution is characterized by a rotor with rotation axis in the ertical x T direction. In this section the opportunity of obtaining starting from this result a solution that is optimal independently from the rotor orientation is inestigated. To do this the Etb-3 solution is used as the first attempt solution of a new optimization process, where the rotor axis is imposed in different directions. If the results of these new optimizations are only slightly different, then an aerage solution between these can be considered. Before proceeding with the newly mentioned optimizations, a new ealuation of the scale factors must be performed. This is conducted considering a first attempt geometry equal to the optimal geometry obtained from the preious research of the scale factors. On the contrary the same alues for the Q i factors that were used in the Etb-3 optimization are now considered. The axis on which to perform this optimization are the x T -axis for which the results are already aailable the y T and z T axis and other two axes randomly selected. These are defined as in the following table. Axis σ rot β rot x T 0 π/2 y T π/2 0 z T 0 0 R 1 0,1547 0,014 R 2 1, Table 20a: imposed rotor orientation axis In table 20 the settings and the results of the series of optimizations, performed imposing the rotor axes in the x T, y T and z T directions, are presented. For each of these axes, the procedure is diided into two steps: a weight ealuation phase and a final optimization phase. The geometrical characteristics of the optimal rotors obtained with these settings are presented. It can be noted that these results differ significantly depending on the rotor axis orientation. These differences can be easily noted with the graph in Figure 58Figure 58: optimal configuration comparison. Gtb Ftb Atb-r optimal diameter D2 [m] optimal diameter D1 [m] optimal length l2 [m] Figure 58: optimal configuration comparison

87 Single rotor optimization 85 Optimization code A1b-r A2b-r A3b-r A4b-r A5b-r Atb-r F1b F2b F3b F4b F5b Ftb G1b G2b G3b G4b G5b Gtb Cost function weights C F_scala C F_scala C F_scala 1 kinetic energy decade rate ,11E ,81E ,84E-05 2 rotor mass ,00E ,26E ,26E-01 3 rotor ibration ,00E ,21E ,31E-07 4 frame ibration ,10E , , rotor encumbrance ,35E ,23E ,23E+02 General parameters 1 rotor geometry ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry ott ott ott ott ott ott ott ott ott ott ott ott ott ott ott ott ott ott 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 0,5 7 rotation axis z z z z z z y y y y y y z z z z z z 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler 1 forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x x x x x x x x x x x x x x x x 4 11 dof model x x x x x x x x x x x x 5 15 dof model x x x x x x 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] ±10% ±10% ±10% ±10% ±10% 2,0:102 ±10% ±10% ±10% ±10% ±10% 2,0:102 ±10% ±10% ±10% ±10% ±10% 2,0:102 2 diameter D1 [cm] ±10% ±10% ±10% ±10% ±10% 2,5:10 ±10% ±10% ±10% ±10% ±10% 2,5:10 ±10% ±10% ±10% ±10% ±10% 2,5:10 3 diameter D2 [cm] ±10% ±10% ±10% ±10% ±10% 2,5:100 ±10% ±10% ±10% ±10% ±10% 2,5:100 ±10% ±10% ±10% ±10% ±10% 2,5:100 Results 1 total cost function 1,06E+05 4,42E+00 3,11E+06 1,28E+03 3,10E-03 8,22E+00 1,13E+05 4,42E+00 3,12E+06 2,84E+02 3,10E-03 1,01E+01 2,61E+04 4,42E+00 3,02E+06 2,36E+03 3,10E-03 7,39E+00 2 optimal length l2 [m] 2,000E-02 2,000E-02 2,000E-02 2,000E-02 9,567E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 9,567E-02 2,081E-02 2,000E-02 2,000E-02 2,000E-02 2,000E-02 9,567E-02 2,046E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,501E-02 5,923E-02 2,521E-02 2,910E-02 2,500E-02 2,500E-02 2,500E-02 5,923E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 5,923E-02 3,008E-02 4 optimal diameter D2 [m] 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,313E-01 1,959E-01 1,958E-01 1,959E-01 1,959E-01 1,959E-01 1,313E-01 1,940E-01 1,959E-01 1,959E-01 1,959E-01 1,959E-01 1,313E-01 1,947E-01 Table 20b: research for the optimal geometry, independently from the rotor orientation

88 86 Flywheel energy storage Results aeraging From the optimizations performed in the preceding subsection it can be noted that the optimal geometry for the rotor differs significantly, depending on the rotor orientation. The ariances are too important to consider an optimal configuration defined as a weighted aerage between the results for the different orientations. Howeer, a procedure for obtaining an optimal solution no matter the rotor orientation could still be considered. The base idea of this procedure is to implement an optimization on the rotor geometry where a new multidirectional cost function is adopted. This new cost is defined as the sum of the cost functions of the same nature of the one introduced in the preceding chapter for a set of different rotor orientations. The result obtained would be the rotor geometry that statistically minimizes the costs if the rotor orientation is not known in adance. In any case this result would hae higher costs if compared with the result obtained optimizing the rotor on the specific direction of interest. Furthermore, an een better configuration is represented by the rotor haing the rotation axis in the ertical direction. Since, as stated in this chapter, the best rotor configuration is the one with ertical rotation axis, and since this configuration is particularly conenient when designing a flywheel energy storage system for on ehicle use, this solution is adopted for the multi-rotor optimizations that are performed in the next chapter.

89

90

91 4 Adanced single-rotor optimization In this chapter the results and the procedure deeloped in chapter 3 a re analyzed to ealuate the possibility of obtaining an energy storage unit for on ehicle applications. In particular, the opportunity of realizing a light and compact system that can act as primary energy storage on road ehicles is inestigated. The focus is put on not exceeding what are the natural boundaries of weight and bulkiness for such a system with, at the same time, keeping good results for the cost functions that were preiously introduced. It must be underlined, though, that in this chapter it is not performed a multirotor optimization, but simple quantities like the oerall system mass and encumbrance are considered. The study is conducted according to the following steps. In a first section the optimal rotor configuration defined in the preious chapter is analyzed with the aim of using this solution as a rotor in the energy storage system. Pros and cons are listed and analyzed and the possibility of improing these results with changes in the rotor material or rotor basic geometry are inestigated. In the following sections, if an improement is considered possible with such changes, a new series of optimizations is performed with the same procedure that was used for the already obtained results. Finally, a selection of possible rotors for the energy storage system is presented and studied.

92 90 Flywheel energy storage 4.1 Ealuation of the results of chapter 3 In this first section the rotor characterized by the geometry obtained with the optimization process of chapter 3 is analyzed. As preiously stated, the focus of this study is to define weather an energy storage system based on such a flywheel would hae a reasonable weight and encumbrance, making it suitable for the use on a road ehicle. If from this study it results that said flywheel is not compatible with this use, additional optimization processes are defined to sole these problems. In particular, in the following sections, new possibilities to address the problematics related to the fitting of a multirotor energy storage system on ehicle mast be considered. Optimization code Etb-1 Etb-2 Etb-3 Cost function weights C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 9,57E ,57E ,57E-06 2 rotor mass 1 2,26E ,26E ,26E-01 3 rotor ibration 1 3,30E ,30E ,30E-07 4 frame ibration 1 4,61E ,61E ,61E-04 5 rotor encumbrance 1 3,23E ,23E ,23E+02 General parameters 1 rotor geometry 2 initial rotor elocity [rpm] 3 stored energy [J] 4 initial rotor geometry 5 rotor density [kg/m^3] 6 perceptual eccentricity 7 rotation axis 8 number of bearings (per side) 9 ehicle speed [m/s] Forcing and soler 1 forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration 4 11 dof model 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] 2 diameter D1 [cm] 3 diameter D2 [cm] Results 1 total cost function 2 optimal length l2 [m] 3 optimal diameter D1 [m] 4 optimal diameter D2 [m] Additional results 1 olume [m^3] 2 mass [Kg] 3 rotor number [---] 4 total olume[m^3] (20 kwh) 5 total mass [kg] (20 kwh) ariabili ariabili ariabili ott ott ott ,5 0,5 0,5 x x x x x x x x x 2,0:100 2,0:101 2,0:102 2,5:10 2,5:10 2,5:10 2,5:100 2,5:100 2,5:100 4,98E+00 5,69E+00 6,46E+00 2,000E-02 2,074E-02 2,122E-02 2,503E-02 2,859E-02 2,751E-02 1,959E-01 1,941E-01 1,930E-01 3,04E-03 3,13E-03 3,07E-03 4,424E+00 4,575E+00 4,601E ,547 0,563 0, ,3 823,5 828,2 6 maximum stress [MPa] 1,094E+03 1,074E+03 1,062E+03 Table 21: final rotor geometry optimization with additional results, 15 degrees of freedom model With a dot it is specified how the results compare with the others, obtain across the whole work: green dot stands for good results, orange for aerage and red for poor results

93 Adanced single-rotor optimization 91 To define and ealuate a flywheel based energy storage system a target energy capacity should be selected. A alue sufficient for short-middle range ehicle operation is chosen. If it is found that this energy storage capacity is easily obtained with such a system, also improed capacity energy storage systems are considered. E tot = 20 kwh = J (79) Now that a target energy to store is defined, the number of rotors necessary in the energy storage system can be calculated and, therefore, also the total mass related to the rotor and the total olume occupied. These mass and encumbrance parameters are introduced in table 21. This is an updated ersion of the already presented table 16. From this, it can be noted that, een if the olume required for the system is not much, it is easy to understand that its weight is excessie, especially considering the limited energy that the system is capable of storing. Een if these solutions are to be discarded, it is interesting to ealuate the maximum stresses the rotating disk is subject to during rotation. It is now reminded that it is not the goal of this work to proide a final solution for the geometry of a flywheel based energy storage system, but only to perform a feasibility analysis. Consequently, the structural assessment of the solutions proided is not analyzed in detail. Howeer, it is important to ealuate whether the proided results are feasible. With this in mind, the maximum tensile stresses the rotating disk is subject to for centrifugal force, are ealuated with the following simplified formula. σ r = σ θ = 3 + ν 8 ρω2 2 r ext (80) Where ν and ρ are respectiely the material Poisson s coefficient and the material density, while ω is the disk rotational speed and r ext is the radius of the disk. With this formula it is easy to conclude that the loads produced exceed what most steel alloys are capable to bare. The alue calculated for the maximum stresses is reported in table 21. For these two reasons (oerall system weight and maximum material stresses) additional optimizations are performed in the following section, considering lighter weight materials. 4.2 Improements on the optimization process Gien the drawbacks identified for the rotor solution of the optimization process performed in chapter 3, a new set of optimizations is performed. Goal of these is to define the ideal geometry and characteristics for a flywheel to be used for an on-ehicle energy storage system. This implies to identify a solution characterized by low weight and encumbrance, and that produces reasonable stresses on the rotor disk, but that at the same time proides good results in terms of the elementary cost functions. In particular, it is desired that the solution is capable of presering the energy stored for a reasonably long time. These new optimization processes are performed by changing the alues assigned to some of the ariables set as constants in the optimization process detailed in chapter 3. These changes are related to the rotor material characteristics, to the flywheel initial speed and oerall geometry and to the amount of energy stored in each single rotor. A comparison between the results is then proided.

94 92 Flywheel energy storage Optimization code Jtb-1 Jtb-2 Jtb-3 Cost function weights C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 1,86E ,86E ,86E-05 2 rotor mass 1 3,85E ,85E ,85E-01 3 rotor ibration 1 4,92E ,92E ,92E-07 4 frame ibration 1 6,08E ,08E ,08E-04 5 rotor encumbrance 1 1,80E ,80E ,80E+02 1 General parameters rotor geometry ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry ott no dic 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 7 rotation axis x x x 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x 4 11 dof model x x x 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 Results 1 total cost function 6,78E+00 8,25E+00 2,20E+01 2 optimal length l2 [m] 2,016E-02 6,494E-02 1,004E-01 3 optimal diameter D1 [m] 9,518E-02 2,524E-02 2,500E-02 4 optimal diameter D2 [m] 2,465E-01 1,888E-01 1,693E-01 Additional results 1 olume [m^3] 7,22E-03 4,43E-03 4,57E-03 2 mass [Kg] 4,160E+00 4,620E+00 5,724E+00 3 rotor number [---] total olume[m^3] (20 kwh) 1,300 0,798 0,823 5 total mass [kg] (20 kwh) 748,8 831,5 1030,4 6 maximum stress [MPa] 5,923E+02 3,474E+02 2,794E+02 Table 22: final rotor geometry optimization, low weight material, 15 degrees of freedom model Low density flywheel The most natural change in the system that is to be ealuated is that of considering a low-density flywheel. Firstly, a low-density flywheel could proide better results in terms of the oerall weight and encumbrance for the energy storage system, secondly, it has also to be considered that all flywheel based system with such a high rotational speed are realized in aluminum and composed materials such as carbon fiber. The reason of this stays in the fact that, with high density, huge stresses arise from the centrifugal load the system is subject to. As preiously stated, since this aims to be a preliminary study, a detailed stress analysis on the rotor is neglected. Anyway, since realistic results are to be obtained, the iron flywheel is not considered as a iable possibility.

95 Adanced single-rotor optimization 93 The new flywheel density is set to 2500 Kg/m 3. The optimization procedure that is detailed in chapter 3 is implemented. For breity only the most significant results are presented in table 23, and for these the oerall system weight and encumbrance are ealuated. Optimization code Ltb-1 Ltb-2 Ltb-3 Ltb-4 Ltb-5 Ltb-5 Cost function weights C F_scala C F_scala C F_scala C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 1,86E ,86E ,18E ,86E ,86E ,86E-05 2 rotor mass 1 3,85E ,85E , ,85E ,85E ,85E-01 3 rotor ibration 1 4,92E ,92E ,61E ,92E ,92E ,92E-07 4 frame ibration 1 6,08E ,08E , ,08E ,08E ,08E-04 5 rotor encumbrance 1 1,80E ,80E , ,80E ,80E ,80E+02 1 General parameters rotor geometry ariabili ariabili ariabili ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry ott ott ott ott ott ott 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x x 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x x x x 4 11 dof model x x x x x x 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry Boundaries of the field 1 length l2 [cm] 2,0:100 2,0:101 2,0:102 2,0:101 2,0:101 2,0:101 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 2,50E+01 2,95E+01 8,09E+01 8,83E+01 9,55E+01 1,42E+01 2 optimal length l2 [m] 6,131E-02 6,15E-02 8,169E-02 6,14E-02 6,49E-02 2,00E-02 3 optimal diameter D1 [m] 2,500E-02 3,760E-02 2,500E-02 6,512E-02 6,236E-02 2,001E-02 4 optimal diameter D2 [m] 1,659E-01 1,656E-01 1,544E-01 1,644E-01 1,624E-01 2,192E-01 Additional results 1 olume [m^3] 3,31E-03 3,59E-03 3,36E-03 4,01E-03 3,97E-03 4,40E-03 2 mass [Kg] 3,385E+00 3,507E+00 3,896E+00 3,984E+00 4,013E+00 2,118E+00 3 rotor number [---] total olume[m^3] (20 kwh) 0,597 0,647 0,604 0,722 0,714 0,792 5 total mass [kg] (20 kwh) 609,2 631,3 701,2 717,2 722,4 381,2 6 maximum stress [MPa] 4,767E+02 4,752E+02 4,130E+02 4,683E+02 4,571E+02 8,329E+02 Table 23: final rotor geometry optimization, low weight material and faster rotation speed, 15 degrees of freedom model It can be noted that, een if the rotor density is reduced, the oerall energy storage system mass increases. Moreoer, the lower rotor density affects negatiely also the oerall olume needed. In any case, because of the much better results when it comes to maximum stresses, a low-density flywheel is the only option considered. It is now clear that some alternatie methods for reducing the energy system weight and bulkiness must be considered.

96 94 Flywheel energy storage Low density flywheel at higher elocity An option to increase the energy storage density is that of increasing the maximum or initial angular speed of the rotor. An optimization process is performed keeping the rotor density at 2500 Kg/m 3 and increasing the initial rotor elocity to rpm. The results are presented in table 23. A considerable reduction in the oerall system mass is obtained together with positie effects also on the system encumbrance. Een if this seems like a promising method, the technical difficulties of realizing such a fast spinning deice are of fundamental importance. For this reason, starting from the next optimization, the initial rotor angular speed is set once more to rpm. The quality of the just obtained results is considered as a target to seek with the reduced rotor angular speed, and by considering alternatie methods. It is to be noted that, despite the high rotational elocity of the rotor in these configurations, the maximum stresses on the rotor axis are neer exceeding the limits for a carbon fiber flywheel Geometry improement In this subsection an improement on the low-density flywheel geometry is introduced. The rotor geometry, in fact, is not identified by three cylindrical elements any more, but now also a forth solid part is introduced. This element is a cylindrical ring, with inner diameter equal to the diameter D 2. The difference between inner and outer radius is indicated by s 1, and the cylinder height by l 4. The element is positioned symmetrically with respect to the disk of diameter D 2, and its symmetry axis lies on the rotor rotation axis. Now the optimization process is performed on fie degrees of freedom instead of three. This proides some additional difficulties to achiee a good conergence of the results on the best possible solution. The problem is addressed by repeating eery optimization many times, setting for each of them a different first attempt solution. Finally, the best result is selected as solution of the optimization process. The final results are presented in table 24. The results of the optimization process performed improing the rotor geometry are noticeably better that those preiously obtained, both in terms of the oerall bulkiness and oerall mass. In fact, the adantages related to considering such a geometry oercame those of considering a higher rotational speed. In the following sections an in-depth analysis is performed to assess whether such an improed geometry proides also good results with regard to rotor and frame ibrations and to kinetic energy decay rate. With this new geometry it is much more complicated to ealuate the maximum stress the rotor is prone to, and the simplified formula (80) introduced in the preceding section is no longer iable. In this case, in fact, to obtain the maximum stresses the rotor is subject to a finite elements method must be implemented. Reasons for this are the complicated geometry of the rotor and the raised notch effects. Anyway, gien to low stress alues that where obtained in the preious cases, it can be stated with high confidence that a rotor with such a geometry can be realized. The only cases that are to be carefully ealuated are those where a thin disk is coupled with a big cylindrical ring. Gien the good quality of the results obtained with the improed rotor geometry, some additional cases are ealuated. With these new optimizations the same improed geometry is considered together with a higher alue for the energy stored in the single rotor. The results are reported in the following tables 25 and 26.

97 Adanced single-rotor optimization 95 C F_scala C F_scala C F_scala C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 2,36E ,36E ,36E ,36E ,36E ,36E-05 2 rotor mass 1 3,85E ,85E ,85E ,85E ,85E ,85E-01 3 rotor ibration 1 6,03E ,03E ,03E ,03E ,03E ,03E-07 4 frame ibration 1 0, , , , , , rotor encumbrance 1 3,10E ,10E ,10E ,10E ,10E ,10E+02 1 rotor geometry 2 initial rotor elocity [rpm] 3 stored energy [J] 4 initial rotor geometry 5 rotor density [kg/m^3] 6 perceptual eccentricity 7 rotation axis 8 number of bearings (per side) 9 ehicle speed [m/s] 1 forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration 4 11 dof model 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry 1 length l2 [cm] 2 diameter D1 [cm] 3 diameter D2 [cm] 1 total cost function 2 optimal length l2 [m] 3 optimal diameter D1 [m] 4 optimal diameter D2 [m] 5 optimal length l4 [m] 6 optimal length s1 [m] 1 olume [m^3] 2 mass [Kg] Optimization code Cost function weights General parameters Forcing and soler Boundaries of the field Results Additional results 3 rotor number [---] 4 total olume[m^3] (20 kwh) 5 total mass [kg] (20 kwh) Ntb-1 Ntb-2 Ntb-3 Ntb-4 ariabili ariabili ariabili ariabili ott ott ott ott ,5 0,5 0,5 0,5 x x x x x x x x x x x x 2,0:100 2,0:101 2,0:102 2,0:101 2,5:10 2,5:10 2,5:10 2,5:10 2,5:100 2,5:100 2,5:100 2,5:100 3,58E+00 3,96E+00 4,12E+00 4,44E+00 2,000E-02 2,01E-02 2,000E-02 2,00E-02 4,063E-02 2,789E-02 3,382E-02 2,856E-02 1,272E-01 1,241E-01 1,082E-01 1,296E-01 5,676E-02 7,074E-02 5,048E-02 5,067E-02 3,671E-02 3,692E-03 3,55E+00 3,355E-02 2,987E-03 3,58E-03 3,556E+00 3,818E-02 3,49E-03 3,309E ,665 0,538 0,645 0, ,2 3,64E+00 4,769E ,1 640,1 595,6 6 maximum stress [MPa] 1,578E+02 1,501E+02 1,140E+02 1,637E+02 1,863E+02 1,884E+02 Ntb-5 ariabili ott ,5 x 2 20 x x 2,0:101 2,5:10 2,5:100 6,86E+00 2,000E-02 2,512E-02 1,383E-01 4,433E-02 3,743E-02 3,60E-03 3,113E+00 0, ,4 Ntb-6 ariabili ott ,5 x 2 20 x x 2,0:101 2,5:10 2,5:100 9,59E+00 2,00E-02 2,500E-02 1,390E-01 5,948E-02 3,103E-02 3,20E-03 3,297E+00 0, ,5 Table 24: final rotor geometry optimization, low weight material and improed geometry, 15 degrees of freedom model

98 96 Flywheel energy storage Optimization code Ptb-1 Ptb-2 Ptb-3 Ptb-4 Ptb-5 Ptb-6 Pb-7 Cost function weights C F_scala C F_scala C F_scala C F_scala C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 2,19E ,19E ,19E ,19E ,19E ,19E ,19E-03 2 rotor mass 1 3,16E ,16E ,16E ,16E ,16E ,16E ,16E-01 3 rotor ibration 1 2,65E ,65E ,65E ,65E ,65E ,65E ,65E-05 4 frame ibration 1 0, , , , , , , rotor encumbrance 1 8,50E ,50E ,50E ,50E ,50E ,50E ,50E+02 1 General parameters rotor geometry ariabili ariabili ariabili ariabili ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry ott ott ott ott ott ott ott 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x x x 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x x x x x 4 11 dof model x x x x x x x 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry 1 Boundaries of the field length l2 [cm] 2,0:100 2,0:101 2,0:102 2,0:101 2,0:101 2,0:101 2,0:101 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 7,00E+00 4,12E+00 7,70E+00 4,23E+00 1,11E+01 9,95E+00 1,28E+01 2 optimal length l2 [m] 2,000E-02 2,000E-02 2,00E-02 2,03E-02 2,00E-02 2,02E-02 2,00E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,500E-02 2,530E-02 2,530E-02 4 optimal diameter D2 [m] 1,484E-01 1,039E-01 1,479E-01 9,616E-02 1,479E-01 9,221E-02 1,102E-01 5 optimal length l4 [m] 5,113E-02 1,143E-01 5,113E-02 4,403E-02 5,113E-02 5,042E-02 4,364E-02 6 optimal length s1 [m] 3,994E-02 4,072E-02 4,015E-02 6,753E-02 4,015E-02 6,569E-02 6,108E-02 Additional results 1 olume [m^3] 4,127E-03 3,925E-03 4,123E-03 4,246E-03 4,123E-03 3,98E-03 4,36E-03 2 mass [Kg] 3,96E+00 5,78E+00 3,96E+00 4,26E+00 3,96E+00 4,520E+00 4,150E+00 3 rotor number [---] total olume[m^3] (20 kwh) 0,495 0,471 0,495 0,510 0,495 0,478 0,523 5 total mass [kg] (20 kwh) 475,1 693,9 475,7 511,5 475,7 542,4 498,0 6 maximum stress [MPa] 2,147E+02 1,051E+02 2,133E+02 9,013E+01 2,133E+02 8,288E+01 1,185E+02 Table 25: final rotor geometry optimization, low weight material and improed geometry, higher energy stored 1, 15 degrees of freedom model

99 Adanced single-rotor optimization 97 Optimization code Rtb-1 Rtb-2 Rb-3 Rtb-4 Rtb-5 Cost function weights C F_scala C F_scala C F_scala C F_scala C F_scala 1 kinetic energy decade rate 1 2,19E ,19E ,19E ,19E ,19E-03 2 rotor mass 1 3,16E ,16E ,16E ,16E ,16E-01 3 rotor ibration 1 2,65E ,65E ,65E ,65E ,65E-05 4 frame ibration 1 0, , , , , rotor encumbrance 1 8,50E ,50E ,50E ,50E ,50E+02 1 General parameters rotor geometry ariabili ariabili ariabili ariabili ariabili 2 initial rotor elocity [rpm] stored energy [J] initial rotor geometry ott ott ott ott ott 5 rotor density [kg/m^3] perceptual eccentricity 0,5 0,5 0,5 0,5 0,5 7 rotation axis x x x x x 8 number of bearings (per side) ehicle speed [m/s] Forcing and soler forcing for rotor dynamics 2 forcing for ground roughness 3 forcing for ehicle acceleration x x x x x 4 11 dof model x x x x x 5 15 dof model 6 imposed rotor symmetry 7 simplified rotor geometry 1 Boundaries of the field length l2 [cm] 2,0:100 2,0:101 2,0:102 2,0:101 2,0:101 2 diameter D1 [cm] 2,5:10 2,5:10 2,5:10 2,5:10 2,5:10 3 diameter D2 [cm] 2,5:100 2,5:100 2,5:100 2,5:100 2,5:100 Results 1 total cost function 8,27E+00 9,66E+00 9,16E+00 6,23E+00 1,23E+01 2 optimal length l2 [m] 2,002E-02 2,00E-02 2,000E-02 2,02E-02 2,00E-02 3 optimal diameter D1 [m] 2,500E-02 2,500E-02 2,500E-02 2,551E-02 2,501E-02 4 optimal diameter D2 [m] 1,912E-01 1,912E-01 1,799E-01 1,474E-01 1,733E-01 5 optimal length l4 [m] 5,060E-02 5,060E-02 9,074E-02 7,302E-02 2,992E-02 6 optimal length s1 [m] 3,071E-02 3,071E-02 2,333E-02 3,924E-02 5,143E-02 Additional results 1 olume [m^3] 5,051E-03 5,051E-03 4,66E-03 4,072E-03 6,04E-03 2 mass [Kg] 4,217E+00 4,22E+00 4,724E+00 5,14E+00 3,968E+00 3 rotor number [---] total olume[m^3] (20 kwh) 0,455 0,455 0,419 0,366 0,543 5 total mass [kg] (20 kwh) 379,5 379,5 425,2 462,5 357,1 6 maximum stress [MPa] 3,561E+02 3,561E+02 3,156E+02 2,117E+02 2,928E+02 Table 26: final rotor geometry optimization, low weight material and improed geometry, higher energy stored 2, 15 degrees of freedom model 4.3 Ealuation of the results In this section an in-depth analysis of some of the most significant cases between those introduced is proided. The analysis of at least one rotor resulting from eery optimization setting is reported, and this is selected as the one producing the better results. The configurations at issue are the one highlighted in blue in the results tables.

100 98 Flywheel energy storage Two kinds of parameters are inestigated: firstly, some quantities related to the dynamics of the single rotor system are analyze and plot with respect to time. These quantities are the kinetical energy stored in the system, the rotor ibration, and the frame ibration. Secondly, parameters related to the mass end encumbrance of the single rotor and multirotor solution are illustrated together with the number of rotors necessary to store 20 KWh. It is to be noted that in this section a linear kinetical energy decay model is used. This model tents to oerestimate the oerall kinetical energy decay rate, and a more proper model to depict this trend would be an exponential one. Howeer, with the introduction of the kinetical energy decay rate, it is not intended to forecast how rapidly the system dissipates energy, but it is used only for comparing the different results. Additional data on these optimal configurations are presented in appendix F.

101 Adanced single-rotor optimization Rotor Jtb-2 General information is here listed for the optimal rotor configuration Jtb-2. Figure 59: graphical representation of optimal rotor Jtb-2 Quantity Unit Value Energy stored per rotor J Number of rotors (multirotor) Kinetical energy decay rate J/s -270,68 Kinetical energy decay rate %/s -0,0677% Single rotor mass Kg 4,5825 System mass (multirotor) Kg 824,8412 Single rotor olume m 3 0,0034 System olume (multirotor) m 3 0,6072 Table 27: optimal rotor Jtb-2 characteristics Figure 60: time eolution of the kinetical energy stored in the rotor, configuration Jtb-2

102 100 Flywheel energy storage Rotor Ltb-1 General information is here listed for the optimal rotor configuration Ltb-1. Figure 61: graphical representation of optimal rotor Ltb-1 Quantity Unit Value Energy stored per rotor J Number of rotors (multirotor) Kinetical energy decay rate J/s -585,74 Kinetical energy decay rate %/s -0,1464% Single rotor mass Kg 3,0767 System mass (multirotor) Kg 553,8126 Single rotor olume m 3 0,0009 System olume (multirotor) m 3 0,1708 Table 28: optimal rotor Ltb-1 characteristics Figure 62: time eolution of the kinetical energy stored in the rotor, configuration Ltb-1

103 Adanced single-rotor optimization Rotor Ntb-5 General information is here listed for the optimal rotor configuration Ntb-5. Figure 63: graphical representation of optimal rotor Ntb-5 Quantity Unit Value Energy stored per rotor J Number of rotors (multirotor) Kinetical energy decay rate J/s -0,2375 Kinetical energy decay rate %/s -0, % Single rotor mass Kg 3,1132 System mass (multirotor) Kg 560,3780 Single rotor olume m 3 0,0036 System olume (multirotor) m 3 0,6480 Table 29: optimal rotor Ntb-5 characteristics Figure 64: time eolution of the kinetical energy stored in the rotor, configuration Ntb-5

104 102 Flywheel energy storage Rotor Ptb-3 General information is here listed for the optimal rotor configuration Ptb-3. Figure 65: graphical representation of optimal rotor Ptb-3 Quantity Unit Value Energy stored per rotor J Number of rotors (multirotor) Kinetical energy decay rate J/s -0,3003 Kinetical energy decay rate %/s -0, % Single rotor mass Kg 3,9286 System mass (multirotor) Kg 471,4317 Single rotor olume m 3 0,0011 System olume (multirotor) m 3 0,1302 Table 30: optimal rotor Ptb-3 characteristics Figure 66: time eolution of the kinetical energy stored in the rotor, configuration Ptb-3

105 Adanced single-rotor optimization Rotor Rtb-2 General information is here listed for the optimal rotor configuration Rtb-2. Figure 67: graphical representation of optimal rotor Rtb-2 Quantity Unit Value Energy stored per rotor J Number of rotors (multirotor) Kinetical energy decay rate J/s -1,1265 Kinetical energy decay rate %/s -0, % Single rotor mass Kg 4,1808 System mass (multirotor) Kg 376,2698 Single rotor olume m 3 0,0018 System olume (multirotor) m 3 0,1631 Table 31: optimal rotor Rtb-2 characteristics Figure 68: time eolution of the kinetical energy stored in the rotor, configuration Rtb-2

106 104 Flywheel energy storage 4.4 Final rotor selection From the data related to the rotor characteristics and dynamics reported in the preceding section, and from the data related to other rotors that for breity are not here reported, the following conclusion can be drawn. 1. The selected material for the rotors in the flywheel energy storage system is a low-density composite material. Reason of this is mainly related to the high stresses that the system must bare as a consequence of the centrifugal load. A high-density material flywheel like an iron flywheel for instance not only generates higher stresses on the rotor axis, but also is capable of a lower ultimate tensile stress. Finally, a low-density flywheel is the best alternatie for safety reasons. 2. For what concerns the rotor mass and encumbrance, the best solutions are those obtained with the improed rotor geometry. At equal mass, or equal olume, and considering all other parameters unchanged, these are the solutions that ensure the higher energy stored. It is also stated that the adantages related to considering such a geometry oercame those of considering a higher rotational speed of rpm. 3. The ibrations of both the frame and the rotor are considerably reduced when the improed rotor geometry is implemented. Consequently, less energy is dissipated on the rotor and the frame suspension systems, leading to a drastic reduction of the rotor kinetic energy decay rate. The decay rate is reduced of two orders of magnitude when considering similar optimal configurations for the simple disc and the improed geometry rotors. 4. Additional adantages, both in the reduction for the oerall mass and encumbrance, and in the reduction of the kinetic energy decay rate, can be found when considering a higher energy stored in a single rotor. These adantages, howeer, are not as critical as those related to the improed geometry, moreoer, they come at the cost of making a bigger rotor, and so more difficult to manufacture and to control. For these reasons the rotors selected for the multi-rotor optimization in the following chapter are all characterized by the improed geometry. Moreoer, the three alternaties selected differ on the base of the energy stored. Finally, to choose one option oer the others in the same energy stored category, the best solution in terms of oerall mass for the 20 KWh system is considered. Therefore, the selected rotors for the multi-rotor optimization are Ntb-5, Ptb-3 and Rtb-2.

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109 5 Multi-rotor optimization In this chapter a multi-rotor model representing a ehicle equipped with a flywheel-based energy storage system is defined. For this system the three different rotors that were selected in chapter 4 are considered. A set of different configurations for eery rotor choice is simulated and studied to obtain the best multi-rotor configuration. The possibility of performing the rotor positioning task with an optimization process is analyzed. The dynamic behaior of the selected configurations is simulated, and the results are presented. Finally, the data collected in this chapter, together with those from the preceding ones, are used in chapter 6 to draw final conclusions on the feasibility of such a solution.

110 108 Flywheel energy storage 5.1 Rotor positioning The rotor positioning task is the last problem to be addressed in order to define a flywheel based energy storage system. Before deeloping a procedure to define the rotor positioning, the main characteristics that are desired for the solution are now listed. 1. The solution must proide good results in terms of the elementary cost functions preiously introduced, in that the kinetical energy decay rate must be minimized together with the frame and rotor ibrations. Moreoer, the total olume occupied by the multirotor energy storage system must be as low as possible. 2. During the rotor acceleration and deceleration phases and so during the loading and unloading of the energy storage system the pitch, roll and yaw torques must be minimized. This characteristic is of crucial importance to guarantee good dynamics for the moing ehicle. 3. The oerall system center of mass must be as low as possible and, moreoer, the solution must hae encumbrance characteristics that are compatible with the internal olume of a priate road car. In general, the solution of this problem is not an easy task because of the many alternaties aailable, howeer, by analyzing the characteristics of the rotors selected in chapter 4, some simplifications can be introduced. Firstly, a ertical rotation axis characterizes these rotors, and so for eery rotor only four coordinates must be selected. These are the three displacements for the rotor global reference origin, and one coordinate that represents if the rotation axis is facing upwards or downwards, and so if the rotor is rotating clockwise or counterclockwise. Secondly, these solutions are already designed to minimize the elementary cost functions, and so this task should not be the main priority in this phase. In accordance to these obserations, the multi-rotor ealuation procedure illustrated in the scheme of Figure 69 is introduced. Figure 69: multi-rotor ealuation procedure With the first step the rotors global reference origins are positioned, this is done according with the third point of the desired solution characteristics. This means that a small fraction of the frame olume is assigned to the flywheel energy storage system. This region is a thin parallelepiped-shaped layer that lies at the bottom of the frame and stretches until the ery end of it. The layer thickness is enough to accommodate the rotors with ertical rotation axis. In this olume the largest possible number of rotors are placed symmetrically with respect to the ehicle symmetry planes. A minimum distance between two rotors and between rotor and frame boundaries is imposed. No constraints on the total energy this system must store and so on the

111 Multi-rotor optimization 109 number of rotors is set. On the contrary, the total energy such a system, composed of a single layer of rotor, can store is one of the parameters that are to be considered once the positioning phase is completed. The result of this phase is the number of rotors that can fit in the assigned olume and a table containing the coordinates of the origin of each rotor global reference, and so the position of eery rotor in the frame. As an example, it is reported in Figure 70Figure 70: rotor positioning example the plot of y rot and z rot coordinates for the case of rotor Ntb-5. Figure 70: rotor positioning example In the second step the direction of the rotors rotation axis must be selected, and so it must be decided if the single rotor is rotating clockwise or counterclockwise. The rotation direction of eery single rotor is determined in order to minimize the pitch, roll and yaw torques that arise during the flywheel acceleration and deceleration phases. Since all three of the selected rotors are characterized by haing a ertical rotation axis, the pitch and roll torques are expected to be null, while the yaw torque must be minimized. In other words, with this step the desired characteristic of the solutions listed at point 2 are meet. The details on how the procedure is performed are proide in the following subsection. Finally, as a third step, the model with the just obtained characteristics is created and a simulation is run. The elementary cost functions are ealuated and the accordance between these results and the solution desired characteristics listed at point 1 is ealuated. This phase is performed in the next section together with all the details regarding the multirotor system Selection of the rotation direction As preiously stated, the rotation direction of the rotors is set to minimize the yaw torque that arises from the flywheel acceleration and deceleration phases. For this reason, a model capable of measuring the yow torque during the flywheel acceleration phase is deeloped. All the different combinations of rotation directions are possible solution for this problem. Despite the huge number of alternaties, and before deeloping an optimization process to assess this problem, some peculiar configurations are ealuated. These are the ones where the rotation direction is the same for all the rotors on a line perpendicular to the ehicle progress direction, and where the rotation direction is the same for all

112 110 Flywheel energy storage rotors on transerse lines. These configurations are represented in the following plots of Figure 71 for rotor Ptb-3, and are referred to as the horizontal and the transersal configurations. Figure 71: the two alternaties initially considered for the rotors rotation direction. In the first one, the setup is composed of alternated rows of rotors rotating in the opposite direction, arranged diagonally. In the second one the rows are arranged horizontally. The flywheel acceleration for this test is set to 2000 rpm/s. The yaw torque that arises from the acceleration of the energy storage systems obtained with rotor Ptb-3, in the two preiously represented configurations, is represented in the two following graphs of Figure 72 and Figure 73. It can be noted that in both cases the yaw torque is ery little and of not particular importance for the system dynamics. In particular the configuration where the rotation direction is the same for all rotors on transerse lines the transersal configuration proides the best results. Gien the high quality of the results, an optimization process to obtain better ones is considered not necessary, and the transersal configuration is selected as the solution of the rotor rotation problem. It is to be noted that the results here reported are for the Ptb-3 rotor, because for this an uneen number of flywheel can fit in the bottom layer of the frame. In fact, for cases where the number of rotors is een, the same results are achieed but with an een lower torque magnitude. Figure 72: yow torque during rotors acceleration for the transersal configuration

113 Multi-rotor optimization 111 Figure 73: yow torque during rotors acceleration for the horizontal configuration 5.2 Final solutions In this section the results obtained by applying the preiously defined procedure are listed. Two kinds of parameters are inestigated: firstly, some quantities related to the dynamics of the multi rotor system are analyze and plot with respect to time. These quantities are the kinetical energy stored in the system and the frame ibration. Secondly, parameters related to the mass end encumbrance of the multirotor solution are presented together with the number of rotors that can fit in a single layer. Moreoer, also the system representations are presented. With these, different rotor colour represents the different rotation directions. As it was stated for the result presented in the preious chapter, it is to be noted that in this section a linear kinetical energy decay model is used. This model tents to oerestimate the oerall kinetical energy decay rate, and a more proper model to depict this trend would be an exponential one. Howeer, with the introduction of the kinetical energy decay rate, it is not intended to forecast how rapidly the system dissipates energy, but it is used only for comparing the different results. A further analysis of these data is proided in the next chapter together with a comparison between this technology and other kind of energy storage systems.

114 112 Flywheel energy storage Solution 1: from rotor Ntb-5 Figure 74: graphical representation of the multirotor system based on the Ntb-5 rotor Quantity Unit Value Stored energy per rotor J Total stored energy J Number of rotors Decay rate J/s -24,5741 Decay rate %/s -0, % Rotor mass Kg 3,1132 System mass Kg 305,0947 System encumbrance m 3 0,4043 Energy density KJ/Kg 128,5 Energy density MJ/m 3 97,0 Table 32: characteristics of the multi-rotor configuration base on rotor Ntb-5 Figure 75: frame ibration FFT, x T direction

115 Multi-rotor optimization Solution 2: from rotor Ptb-3 Figure 76: graphical representation of the multirotor system based on the Ptb-3 rotor Quantity Unit Value Stored energy per rotor J Total stored energy J Number of rotors Decay rate J/s -32,0551 Decay rate %/s -0, % Rotor mass Kg 3,9645 System mass Kg 360,7722 System encumbrance m 3 0,4037 Energy density KJ/Kg 151,3 Energy density MJ/m 3 135,2 Table 33: characteristics of the multi-rotor configuration base on rotor Ptb-3 Figure 77: frame ibration FFT, x T direction

116 114 Flywheel energy storage Solution 3: from rotor Rtb-5 Figure 78: graphical representation of the multirotor system based on the Rtb-5 rotor Quantity Unit Value Stored energy per rotor J Total stored energy J Number of rotors Decay rate J/s -65,6273 Decay rate %/s -0, % Rotor mass Kg 4,2167 System mass Kg 278,3018 System encumbrance m 3 0,4038 Energy density KJ/Kg 189,7 Energy density MJ/m 3 130,8 Table 34: characteristics of the multi-rotor configuration base on rotor Rtb-5 Figure 79: frame ibration FFT, x T direction

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119 6 Conclusions In this last chapter, a detailed study of the final configurations defined in chapter 5 is performed. Therefore, the solutions here analyzed are results of the two optimization processes. To gie an oerall ealuation of these results, a comparison between the characteristics of the obtained ehicle and the ones related to an electrical ehicle selected between those currently aailable on the market is presented. In this ealuation, fie main characteristics are analyzed. These are: the oerall storage system energy density both in terms of energy oer mass, and energy oer olume the encumbrance and the weight of the storage system and, finally, the energy decay rate. Moreoer, additional thoughts regarding the effects that such a flywheel-based energy storage system cause on the ehicle during motion are presented. General conclusions are drawn on the possibility of realizing a road automobile, for priate or public use, that relies on a flywheel based energy storage system with mechanical support for the rotors as primary energy source.

120 118 Flywheel energy storage 6.1 Analysis of the results Mass and mass-based energy density The oerall mass of the constructie solutions for the flywheel-based energy storage systems identified in chapter 5 is analyzed in this section. A comparatie representation of the weight of these systems is presented in Figure 80. Howeer, it must be considered that the releant mass related parameter in the construction of an energy storage system for ehicle use is not the oerall mass, but its mass energy density. A representation of this quantity for the three solutions is proided in Figure 81. In this same picture, the energy density of the identified solutions is compared with the energy density of some of the most significant lithium-ion batteries of comparable sizes. These are the Tesla Powerwall 2 a domestic energy storage deice and the battery deeloped for Tesla Model S 6, and so an energy storage for on ehicle use. As it can be noted from the graph, increasing the energy stored in each indiidual rotor the oerall energy density of the system is improed. In any case, some additional data must be collected to fully assess whether such a trend could be identified. Howeer, when it comes to energy density, much better results are obtained with the two lithium-ion batteries with respect to those obtained with a flywheel based system. Rtb-2 based system Ptb-3 based system Ntb-5 based system mass [Kg] Figure 80: oerall mass of the energy storage systems Tesla Powerwall Tesla Model S Ntb-5 based system Rtb-2 based system Ptb-3 based system Energy density [KJ/kg] Figure 81: mass energy density of the energy storage systems 6 The data presented are related to Tesla Model S 75D.

121 Conclusions Encumbrance and olume-based energy density In this section the olume occupied by the energy storage systems defined in chapter 5 is ealuated. A comparatie representation of the olume of these systems is proided in Figure 82. As it was done for the mass, in Figure 83 the olume energy density of the obtained solutions is presented, together with the same data for Tesla Powerwall 2 and Tesla Model S. The three flywheel-based solutions do not present any particular trend for what concerns the olumetric energy density at the change of the amount of energy stored in each indiidual rotor. In any case, some additional data must be collected to fully assess whether such a trend could be identified. Finally, from the same graph, it must also be noted that the alues for the olume energy density of the flywheel energy storage systems are much smaller than the those obtained with the two lithium ion batteries. Een if this difference is more significant than the one obsered for the mass energy density, this latter one represents a more crucial problem in the deelopment of a flywheel-based energy storage system. Rtb-2 based system Ptb-3 based system Ntb-5 based system Volume [m 3 ] Figure 82: oerall olume of the energy storage systems Tesla Powerwall Tesla Model S Rtb-2 based system Ntb-5 based system Ptb-3 based system Energy density [MJ/m 3 ] Figure 83: olume energy density of the energy storage systems Self-discharge In this section the storing efficiency of the flywheel-based systems is analyzed and compared with the results for the already presented lithium ion batteries solutions. Two different decay rates are considered: the first

122 Energy stored [J] 120 Flywheel energy storage one is the kinetical energy decay rate obsered during the ehicle motion at the constant speed of 20 m/s, and so subject to the forcing introduced by the ground roughness, the second one is the kinetical energy decay rate that can be ealuated for the ehicle at rest. Both of these quantities are presented in Figure 84. In Figure 85 and Figure 86 the decay of the energy stored in the system is forecasted on the base of the data in Figure 84. These trends are related to a hypothetical case where the initial energy stored for each system is equal to 50 MJ. In the first figure, the data related to the ehicle during motion are presented while, in the second one, the once for the ehicle at rest are considered. In the same figures, the time eolution of the energy stored for the flywheel based solutions is compared with the one from a Tesla Model S. In this latter case, the self-discharge effect is considered for both the stationary and the dynamic ealuations. It can be noted that, for the ehicle at rest, a much better storing efficiency is obtained with a flywheel-based system. Ptb-3 based system Ptb-3 based system Ntb-5 based system -1,E-06-1,E-06-1,E-06-8,E-07-6,E-07-4,E-07-2,E-07 0,E+00 Kinetical energy daced rate [%/s] Figure 84: kinetical energy decay rate for the ehicle during motion (in orange) and at rest (in blue) 6,E+07 5,E+07 4,E+07 3,E+07 2,E+07 1,E+07 Ntb-5 Ptb-3 Rtb-2 Tesla Model S 0,E Time [h] Figure 85: forecasted eolution of the energy stored, ehicle in motion

123 Energy stored [J] Conclusions 121 6,E+07 5,E+07 5,E+07 4,E+07 4,E+07 Ntb-5 Ptb-3 Rtb-2 Tesla Model S 3,E Time [h] Figure 86: forecasted eolution of the energy stored, ehicle at rest Effects on the ehicle dynamics From the ealuations performed in section 5.1.1, it can be noted that, for the rotor configuration proposed in the same section, the effects of the flywheel energy storage system charging and discharging phases on the ehicle dynamics are negligible. Moreoer, a further reduction of these effects can be achieed selecting an een number of rotors, and imposing for them the rotation directions detailed in the same section. A second improement could also be obtained by stacking two layers of rotors one oer the other. In this case the two rotors sharing the same rotations axis must be designed to rotate in opposite directions. 6.2 Final conclusions and future deelopments As it is detailed in the preceding sections, the mass and the olume energy density for the deeloped flywheel-based energy storage systems are considerably lower than what is obtained for some commercially aailable lithium ion battery solutions. Moreoer, it has also to be considered that the mass and the olume of the complete kinetical energy storage systems are negatiely affected by all necessary secondary systems. These include a continuously ariable transmission, a control unit for the energy storing and many others. To improe the mass energy density, a higher rotational speed for the flywheels can be inestigated. This opportunity must be carefully ealuated because of the problems arising from imposing an een higher rotational speed for the rotors. In this case, in fact, it might be hard to achiee a good dynamic for the flywheel without reducing at really low alues the percental eccentricity, and without introducing some additional stiffening element in parallel to the bearings supporting the rotor. If these precautions are not taken, high ibrations might arise, leading to a high energy density decay rate and to a short life for the bearings. Finally, additional problems connected to a higher rotational speed of the rotor regard the structural stress of the rotor itself. In these case, a in depth structural analysis of the spinning rotor must be performed. A second alternatie for improing the mass energy density of the flywheel based energy storage is to inestigate the trend that can be noted in Figure 81. In this graph, in fact, an increase in the mass energy density is obsered when a solution characterized by higher energy per single rotor is considered. Anyway,

124 Mass energy density [KJ/Kg] 122 Flywheel energy storage further data must be collected in order to check this trend, since the three alues presented in this figure are not sufficient to draw final conclusions. In case the opportunity of realising such a solution is exploited, a compromise between improing the mass energy density and not reducing excessiely the kinetical energy decay rate must be done. In fact, from Figure 84 it is noted that with a solution characterized by higher energy stored in the single rotor, a higher kinetical energy decay rate is obtained. To improe the olume energy density, the opportunity of increasing the flywheel speed can be once more inestigated. The obserations introduced when dealing with the mass energy density must be considered. Anyway, it is here reminded that the problem of reducing the mass energy density is more critical for ehicle application of energy storage systems of any kind, with respect to the reduction in the olume energy density. When it comes to the storing efficiency, ery good results can be obsered from Figure 86, especially when these are compared with those for a lithium ion battery. Howeer, it has to be reminded that the model used for deeloping these results do not take into account of all the effects that might influence the rotor dynamics. A more complete model could be deeloped for these purpose, een if ery good results are expected. To further improed these, low damping bearings can be introduced. Additional deelopment must be performed before drawing final conclusions on the feasibility of a road ehicle equipped whit a kinetical primary energy storage. These deelopments must mainly deal with the problematics related to the design of a continuously ariable transmission and with an oerall cost analysis. Een if the results obtained in terms of mass energy density might seem disappointing, it is considered possible to perfect these with the techniques yet introduced. Moreoer, these results were compared with some of the best lithium ion battery solutions aailable on the market in This technology had some major deelopments in the past 10 years, and if the results for the flywheel based solutions were compared with the best batteries aailable on the market only 10 years ago, these would look much more encouraging. [16] Figure 87: history of deelopment of secondary batteries in iew of energy density [16] Furthermore, for the sake of completeness, one additional study that must be performed to assess the feasibility of such a solution is the one related to the gyroscopic effects arising from the rotors during turning. It must be ealuated weather the gyroscopic torques arising in this phase are significant enough to affect the ehicle dynamics.

125 Conclusions 123 Finally, gien the excellent loading and unloading properties of a flywheel energy storage, it is belieed that such a solution could proide good functionality when coupled with other main energy sources in hybrid ehicles. These energy sources could be both a chemical battery or an internal combustion engine. The kinetical energy of a ehicle could efficiently be recoered during breaking and, as shown in these work, solutions that minimize the effects of the rotor acceleration and deceleration on the ehicle dynamics can be obtained.

126

127 Bibliography Bibliography [1] G. Genta, Politecnico di Torino, Kinetic energy storage, Butterworth & Co. (Publishers) Ltd., [2] E. J. H. a. A. R. Hall, History of Technology, C. Singer, [3] G. N.V., Flywheel Engines, Mashinostroyeniye Press, [4] C. Posthumus, Fathers of Inention, Hamlyn / Phoebus, [5] D. Castelecchi, Spinning into control: High-tech reincarnations of an ancient way of storing energy, Science News, [6] E. Beedham, Flywheel Generators for the JET Experiment, GEC Journ. of Sc. and Tech., [7] K. I. e. a. T. Hattori, Rotating Strength of Glass-Carbon Fiber Reinforced Hybrid Composite Discs, Bull, of Japan Soc. of Mech. Eng., [8] R. Miller, Flywheels Gain as Alternatie to Batteries, DataCenter Knowledge, [Online]. Aailable: [9] Beacon Power Corporatio, Flywheel-based Solutions for Grid Reliability, Wayback Machine, [10] FIA, Federation Internationale de l'automobile, [Online]. [11] Formula One race technology to power buses in Oxford, BBC News, Oxford, [Online]. Aailable: [12] Oerlikon, The Oerlikon Electrogyro, Automobile Engineer, [13] M. Trend, The GYROBUS: Something New Under the Sun?, [14] M. Mboka, Leopoldille 1954 Transports en Commun de Leopoldille hits the streets, [Online]. Aailable: [15] D. M. Henderson, Euler Angles, Quaternions and Transformation Matrices, McDonnel Douglas Technical Serices Co. - NASA, [16] H. Kawamoto, Trends of R&D on Materials for High-power and Large-capacity Lithium-ionBatteries for Vehicles Applications, Quarterly Reiew, no. 36, 2010.

128 126 Flywheel energy storage 8 Appendix 8.1 Appendix A The results from the simulations of the system dynamics performed imposing the initial conditions reported in table 6 are here presented. The units used in these plots are those of the international system. The simulations are numbered from one to six according to the numbering introduced in the same table. The accordance with the expected results is discussed. Simulation 1

129 Appendix 127 The results correspond to the expected one of a second order system in the x G direction, characterized by a stiffness equal to two times the bearing stiffness and a damping equal to two times the bearing damping. All the other degrees of freedom were expected not to be excited. Simulation 2 The results correspond to the expected one of a second order system in the y G direction, characterized by a stiffness equal to two times the bearing stiffness and a damping equal to two times the bearing damping. All the other degrees of freedom were expected not to be excited.

130 128 Flywheel energy storage Simulation 3 The results correspond to the expected one of a second order system in the z G direction, characterized by a stiffness equal to the bearing stiffness and a damping equal to the bearing damping. All the other degrees of freedom were expected not to be excited.

131 Appendix 129 Simulation 4 The results correspond to the expected one. No excitation was expected on the three displacement degrees of freedom, while a transitory ibration was expected on the two angles representing the rotor orientation.

132 130 Flywheel energy storage Simulation 5 The results correspond to the expected one. No excitation was expected on the three displacement degrees of freedom, while a transitory ibration was expected on the two angles representing the rotor orientation.

133 Appendix Appendix B The dynamics of the frame, after an initial displacement off its equilibrium point of 1 cm in the x T direction, is here represented. The results correspond to the expected one of a second order system in the x T direction, characterized by a stiffness equal to four times the suspensions stiffness and a damping equal to four times the suspensions damping. All the other degrees of freedom were expected not to be excited.

134 132 Flywheel energy storage 8.3 Appendix C For completeness, some of the results of the preliminary obserations are here reported also for a new rotor configuration. The results obtained in section 3.3.1, for preliminary obseration 1, are ealuated for a different rotor configuration. In this case configuration α 2 is selected.

135 Appendix 133 This new ealuation proides similar trends to the one preiously obtained, and, for this reason, the conclusions already drawn are confirmed. The results obtained in section 3.3.1, for preliminary obseration 2, are ealuated for a different rotor configuration. In this case configuration α 2 is selected. The same results can be stated from this second configuration.

136 134 Flywheel energy storage 8.4 Appendix D General information is here listed for the optimal rotor configuration Ctb-1. configuration Ctb-1 representation Eolution of the kinetical energy stored in the rotor oer time, configuration Ctb-1 Rotor ibration oer time, configuration Ctb-1

137 Appendix 135 Frame ibration oer time, configuration Ctb-1 The rotor mass and the used olume objects of cost functions C 2 and C 5 are reported. m rot = 7,801 kg V rot = 0,00303 m 3 General information is here listed for the optimal rotor configuration Ctb-2. Eolution of the kinetical energy stored in the rotor oer time, configuration Ctb-2

138 136 Flywheel energy storage Rotor ibration oer time, configuration Ctb-2 Frame ibration oer time, configuration Ctb-2 The rotor mass and the used olume objects of cost functions C 2 and C 5 are reported. m rot = 7,801 kg V rot = 0,00337 m 3 General information is here listed for the optimal rotor configuration Ctb-3.

139 Appendix 137 Eolution of the kinetical energy stored in the rotor oer time, configuration Ctb-3 Rotor ibration oer time, configuration Ctb-3 Frame ibration oer time, configuration Ctb-3

140 138 Flywheel energy storage The rotor mass and the used olume objects of cost functions C 2 and C 5 are reported. m rot = 7,801 kg V rot = 0, m 3

141 Appendix Appendix E Additional information on optimal configuration Etb-1. Rotor ibration oer time, configuration Etb-1 Rotor ibration FFT, configuration Etb-1

142 140 Flywheel energy storage Frame ibration oer time, configuration Etb-1 Frame ibration FFT, configuration Etb-1

143 Appendix 141 Additional information on optimal configuration Etb-2. Rotor ibration oer time, configuration Etb-2 Rotor ibration FFT, configuration Etb-2

144 142 Flywheel energy storage Frame ibration oer time, configuration Etb-2 Frame ibration FFT, configuration Etb-2

145 Appendix 143 Additional information on optimal configuration Etb-3. Rotor ibration oer time, configuration Etb-3 Rotor ibration FFT, configuration Etb-3

146 144 Flywheel energy storage Frame ibration oer time, configuration Etb-3 Frame ibration FFT, configuration Etb-3

147 Appendix Appendix F Additional information on optimal configuration Jtb-2. Rotor ibration oer time, configuration Jtb-2 Rotor ibration FFT, configuration Jtb-2

148 146 Flywheel energy storage Frame ibration oer time, configuration Jtb-2 Frame ibration FFT, configuration Jtb-2

149 Appendix 147 Additional information on optimal configuration Ltb-1. Rotor ibration oer time, configuration Ltb-1 Rotor ibration FFT, configuration Ltb-1

150 148 Flywheel energy storage Frame ibration oer time, configuration Ltb-1 Frame ibration FFT, configuration Ltb-1

151 Appendix 149 Additional information on optimal configuration Ntb-5. Rotor ibration oer time, configuration Ntb-5 Rotor ibration FFT, configuration Ntb-5

152 150 Flywheel energy storage Frame ibration oer time, configuration Ntb-5 Frame ibration FFT, configuration Ntb-5

153 Appendix 151 Additional information on optimal configuration Ptb-3. Rotor ibration oer time, configuration Ptb-3 Rotor ibration FFT, configuration Ptb-3

154 152 Flywheel energy storage Frame ibration oer time, configuration Ptb-3 Frame ibration FFT, configuration Ptb-3

155 Appendix 153 Additional information on optimal configuration Rtb-2. Rotor ibration oer time, configuration Rtb-2 Rotor ibration FFT, configuration Rtb-2

156 154 Flywheel energy storage Frame ibration oer time, configuration Rtb-2 Frame ibration FFT, configuration Rtb-2