eindhoven university of technology master conceptual
TRANSCRIPT
Eindhoven University of Technology
MASTER
Conceptual design of energy exchange systems for hybrid vehicles
Hoekstra, D.
Award date:2005
Link to publication
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Conceptual design of energyexchange systems for hybrid
vehiclesDouwe Hoekstra
DCT 2005.100
Master’s thesis
Coach: Theo Hofman
Supervisor: Maarten Steinbuch
Technische Universiteit EindhovenDepartment Mechanical EngineeringDynamics and Control Group
Eindhoven, August, 2005
i
Summary
To design and analyse hybrid vehicle drivetrains, numerous software packages are available suchas ADVISOR [1]. A disadvantage of most of these packages is that they are based on a limitedset of topologies combined with a limited set of components, limiting the design space of thedrivetrain designer.
A generic design and analysis model of a hybrid vehicle drivetrain has been presented as partof the project “Impulse Drive". Based on this model, a generic design procedure is under in-vestigation. For this procedure, scalable models of hybrid components were required, meaningscalable in power performance and energy storage capacity. Further, there was a need to comparethe different conversion and storage components that are available nowadays for hybridization ofvehicle drivetrains.
In this report, six different hybrid systems that have similar behavior are defined (S1−S6). The hy-brid systems are all able to be attached directly to a conventional drivetrain to hybridize a vehicle.The components of these systems are described and modeled such that they can be implementedinto the design process, as well as to be able to compare the performance of these hybrid systems.
The S systems are scaled by design parameter scaling to deliver a target power profile. Thespecific properties of the hybrid systems could then be compared, being mass, volume, price andmean efficiency.
As a result, the Nickel-Metal Hydrid battery with Permanent Magnet motor is a device that isthe best performing in most situations. For a hybrid system that should be able to deliver highswitching powers and that has a low energy storage capacity requirement, flywheel storage de-vices are also possible solutions.
ii
Samenvatting
Hybride aandrijflijn simulatiesoftware is in ruime mate voorhanden om een hybride aandrijflijnte ontwerpen en analyseren, bijvoorbeeld het software pakket ADVISOR [1]. Een nadeel voor deontwerper van een hybride aandrijflijn is dat deze software uitgaat van een vaststaande topolo-gie en een beperkte set componenten waaruit gekozen kan worden, hiermee de ontwerpvrijheidbeperkende.
Als onderdeel van het project “Impulse Drive" is er een generiek model gepresenteerd waarmeeeen hybride aandrijflijn ontworpen kan worden. Er wordt gewerkt aan een zo generiek mogelijkontwerpproces gebaseerd op dit model. Binnen dit ontwerpproces was er behoefte aan schaalbaremodellen voor de hybride componenten, dat wil zeggen schaalbaar in vermogen dat de compo-nenten kunnen leveren en in energie die de componenten kunnen opslaan. Verder bestond er debehoefte om de verschillende beschikbare hybride conversie- en opslagtechnologieën te vergelij-ken in hun voor- en nadelen en eigenschappen.
In dit verslag worden zes verschillende hybride systemen gedefinieerd (S1 − S6) die één-op-éénvergelijkbaar zijn in hun functionaliteit. De componenten van deze systemen worden in dit ver-slag beschreven en gemodelleerd om deze later in een ontwerpproces mee te kunnen nemen,alsmede om deze systemen te kunnen vergelijken op basis van prestatie. De S systemen zijnallemaal direct aan een conventionele aandrijflijn toe te voegen zodat het voertuig ge-hybridizeerdwordt.
Om de S systemen te kunnen vergelijken, worden ze allereerst geoptimaliseerd door middelvan ontwerpparameter schaling, naar een vermogensprofiel dat ze moeten kunnen leveren. Ver-volgens zijn de gemodelleerde eigenschappen massa, volume, prijs en efficiency van de systemente vergelijken en kunnen conclusies getrokken worden.
Hieruit volgt dat met name de Nickel-Metaalhydride accu in combinatie met een PermanentMagneet motor voor vrijwel alle situaties de meest effectieve oplossing biedt. Voor het geval ereen signaal met grote vermogenswisselingen wordt gevraagd en een lage energie opslag capaciteitbenodigd is, zijn vliegwiel oplossingen ook in overweging te nemen.
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Acknowledgements
This work wouldn’t have been possible without the continuous support of my close friends andfamily. I would like to thank the master students at W-Hoog -1.113 that have made the numerouscomputer hours more lively, I grew quite close to Janneke van Baalen, Paul van Dongen, ArdBommer, Michiel Wondergem and Ilona Soons.
I further thank Theo Hofman for the support and critical perspectives on this work, challeng-ing the decisions that were taken and questioning every move I made, and to Ulises Diego-Ayalafor providing me with the validation data for the analytical models. Many thanks to my examboard, consisting of Theo Hofman, Maarten Steinbuch, Roëll van Druten and Erjen Lefeber.
Further many special thanks go to my ex-girlfriend Kristie Spijker, she has supported me in-tensively for the last couple of years of my study, it is unfortunate that we couldn’t finish thisthing together as planned.
This Master’s Thesis is the final work after a long period of studying Mechanical Engineeringat Eindhoven, University of Technology. It has been a lively period, with many good and bad mo-ments but they all shaped me to the person I am today and I am grateful for that. With finishingthis thesis, a challenging period now lies ahead of me.
Douwe Hoekstra
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Contents
1 Introduction 11.1 Hybrid vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Contribution and outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Hybrid vehicle design process 52.1 Problem structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 S design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 S candidate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Efficiency and State-of-Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Optimization methods 123.1 Sequential Quadratic Programming . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Weighted Multi-Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3.1 Penalty and Barrier method . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Optimization Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Optimization models 184.1 Modeling limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Validation of analytically modeled components . . . . . . . . . . . . . . . . . . . 23
4.2.1 Subcritical Flywheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.2.2 Supercritical Flywheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.3 Compressed Air Energy Storage . . . . . . . . . . . . . . . . . . . . . . . 24
5 Results 265.1 20kW Power profile optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.2 Power and Energy sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2.1 Storage devices only power and energy sensitivity . . . . . . . . . . . . . . 335.3 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5.3.1 Sensitivity to the initial design . . . . . . . . . . . . . . . . . . . . . . . . 345.3.2 Sensitivity to the initial State Of Charge . . . . . . . . . . . . . . . . . . . 345.3.3 Sensitivity to α and Mmax, Vmax and €max . . . . . . . . . . . . . . . . . 35
6 Conclusion 376.1 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Bibliography 41
v
Acronyms and symbols 42
A Component modeling 44A.1 Flywheel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
A.1.1 Super-critical flywheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.1.2 Sub-critical flywheel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
A.2 Battery Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.3 Ultracapacitor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58A.4 Pressurized gas storage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A.5 Continuously Variable Transmission . . . . . . . . . . . . . . . . . . . . . . . . . 66A.6 Electrical Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
A.6.1 Induction Motor/ Generator . . . . . . . . . . . . . . . . . . . . . . . . . 68A.6.2 Permanent Magnet Motor/ Generator . . . . . . . . . . . . . . . . . . . . 72A.6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A.7 Hydraulic Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74A.7.1 Gear Pump . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A.8 Power Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
B Model Sensitivity 78B.1 Sensitivity to x0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.2 Sensitivity to alpha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B.3 Sensitivity to Mmax, Vmax, €max . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
C Validation References 99
D Scripts 100
E Paper submitted to VPP conference 130
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Chapter 1
Introduction
The polutional effects of cars and solutions to reduce the environmental load have been the focusof research for an extensive time. A zero emission vehicle is the design goal to many vehicle de-velopers, and research of the last thirty years has focussed much on completely electric vehicles.However, with current energy storage solutions, the driving range is still too limited and the costsof these vehicles are too high.
A novel power source solution currently being investigated is the fuel cell with hydrogen tankattached (combined in short usually mentioned to as the fuel cell). This is a promising technology,that is expected to be commercially available in ten to fifteen years time. The fuel cell is a devicethat stores and converts hydrogen gas into water and electricity, that can be used, e.g., to propelthe vehicle. However, the technology has not matured yet.
As an intermediate solution, while fuel cell technology is developed further, fuel use and emissionreductions can be accomplished using hybrid vehicles.
1.1 Hybrid vehiclesHybrid technology is the key solution to improve the driving functions of a vehicle. The hybridtechnology developed today will proliferate into future drivetrains, including fuel cell technology.Nowadays, hybrid vehicles have besides a conventional Internal Combustion Engine (ICE) as firstpropulsion device, a second propulsion device, usually an Electric Motor (EM). By coupling thesetwo power sources, the size of the ICE can be adjusted to the "mean" power needs for the vehicle.The secondary propulsion source can assist the ICE when the drivetrain power need is higherthan the mean need and store energy when the power need is below. A secondary propulsionsource allows regenerative braking, storing energy in an accumulator for later use. Additionally,the power flows between the different thermal, mechanical and electric paths can be optimizedthrough optimal control design.
Overall efficiency of the hybrid vehicle can be increased substantially compared with a conven-tional drivetrain vehicle, in the order of 40% fuel use reduction. No infrastructural changes areneeded as petrol is usually still the only fuel added to the hybrid vehicle, extra (electrical) chargingis also possible but not a necessity.Disadvantages of hybrid configurations are an increased complexity of the vehicle drivetrain and
1
higher maintenance and purchase cost, the latter is often compensated by government subsidies.Several hybrid vehicles are currently commercially available, e.g., the Toyota Prius and the HondaCivic IMA (Integrated Motor Assist).
Hybridization of a vehicle powertrain implies adding a Secondary power source (S, e.g., batteryand electric motor) to a Primary power source (P , e.g., internal combustion engine and petroltank). Usually, the storage component S is rechargeable to be able to recuperate braking energy,for example. Rechargeability is however not strictly required from a definition point of view ofhybrids. It is possible to combine different storage solutions in S, e.g. a battery for energy needsand an ultra capacitor for peak power demands. Technologies considered as storage solutions forS are batteries, fuel cells, ultra-capacitors, flywheels and pressure accumulators, with fuel cellscurrently the most promising, yet immature technology (and as such not considered further).
For analysis and comparison purposes, a distinction is conventionally made between a series hy-brid, a parallel hybrid and a series-parallel hybrid topology as indicated in Figure 1.1. Basically, a
ICE GE EM TR Wheels
BATT
ICE
EM
TR Wheels
BATT
ICE
GE EM
TR Wheels
BATT
Figure 1.1: From top to bottom: Series hybrid, Parallel hybrid and parallel-series drivetraintopologies. ICE: Internal Combustion Engine, GE: Generator, BATT: Battery, EM: ElectricMachine, TR: Transmission. [2]
series hybrid allows no direct power flow from ICE to the vehicle wheels through a mechanicalpath, but through an electro-mechanical path. A parallel hybrid allows powerflow from ICE di-rectly to the vehicle wheels, but contrary to the series-parallel hybrid configuration, only throughone mechanical path. Each topology has its own advantages and disadvantages.
In most cases, hybrid drivetrains that need analysis can be represented by one of the three topolo-
2
gies. However, for a more generic view at hybrid vehicle drivetrains, another analysis modelfor a hybrid vehicle drive train has been proposed (Figure 1.2) in [3]. In this model, the circles
P
S V
T
C
Figure 1.2: Generic hybrid vehicle model. P = Primary power source (e.g. ICE+ petrol tank),S = Secondary power source (e.g. EM + battery), T = Transmission technology (e.g. CVT),C = Control of operating points of P ,S and T . V represents the vehicle wheels.
represent drivetrain subsystems, the lines indicate interaction between the different subsystems.P represents the Primary power source of the vehicle, usually an Internal Combustion Engine(ICE) including fuel tank and exhaust gasses after treatment, V represents the Vehicle wheels,S represents the Secondary power source in the vehicle (e.g. electric motor and battery), T theTransmission technology and C the systems Control strategy. By combining these subsystems,a complete hybrid drivetrain design can be acquired. The hybrid drivetrain design challenge isto choose and design technologies for each subsystem, such that performance demands are metand the drivetrain as a whole is optimal with respect to design goals.The use of the model can be adapted to modeling needs, e.g. a series or parallel hybrid topologycan be constructed and analyzed in this manner. For examples of the analyzing powers of thismodel, the reader is referred to [3].
1.2 Contribution and outline of this thesisHybrid drivetrain design optimization is a complex design process. Many researchers have fo-cussed on developing software packages that can be used as a design tool (e.g., ADVISOR [1],SIMPLEV [4], CarSim [5], HVEC [6], CSM HEV [7], V-Elph [8], QSS toolbox [9]) [10]. An optimalhybrid powertrain design is obtained by optimizing a selected hybrid powertrain configuration(e.g. series, parallel, series-parallel), to components and control by minimizing fuel consumptionalong a predefined trajectory (drive cycle) and other performance constraints (e.g., accelerationperformance, gradability). These software packages are based on a discrete set of (existing) pow-ertrain components, fixed in size (power, energy capacity), and predefined topologies. This way,optimization of a hybrid drivetrain design, incorporates the optimization of the use of the compo-nents (system control) and not of the components and topologies themselves. An integral controloptimization process is typically characterized by large computation times, complex design for-mulations, multiple subsystem simulations and non-smooth and non-continuous modeling. [11]To reduce the computation time, the QSS-toolbox uses scalable models for the EM and the ICE
3
based on an affine relationship between torque losses and torque output for a certain componentsize. Furthermore, the QSS-toolbox has the advantage of building easily different powertrain ar-chitectures using scalable blocks, that represent the subsystems such as the EM and the ICE. Inthis Master’s thesis, in addition to scalable models for the EM, scalable models will be constructedand evaluated for ultracapacitor, battery, gas-pressurized tank, sub- and supercritically operatedflywheel storage systems.
The goal of this research is to analyse, compare and find a sizing strategy for current techno-logical solutions to the hybridization of conventional vehicles. This will provide insights into:
1. The solutions of the possible set of components for S fulfilling certain power and energystorage capacity requirements.
2. The quantification of the design trade-offs (between volume, mass, cost price, efficiency) inachieving the design objectives (power and energy storage capacity).
3. The use of scalable component models. The size of the components can be added to ahybrid vehicle design process as an optimization parameter.
4. The optimization techniques that give solutions to a non-linear constrained multi-objectiveoptimization problem.
5. The practical operation limits of the S components that are under consideration
A design framework in which this thesis fits is presented in Chapter 2 and the chosen strategy isexplained. Optimization techniques for the problem at hand are discussed in Chapter 3 and theoptimal design problem is translated into minimizing a weighted sum objective function value.In order to be able to compare and evaluate performance of the hybrid powertrain componentdesigns, models will be generated of the efficiency, mass, volume and cost of candidate hybridassistance systems. This is discussed in Chapter 4. The models that are analytical in nature, willbe validated in this chapter for their power losses. Finally, the optimization algorithms are put touse in Chapter 5 and the results are evaluated. General validation of the applied modeling is partof this evaluation process.
The work presented in this thesis fits within the framework of the NWO1 research programme“Impulse Drive", that currently focuses on determining the required design specifications of thesystems components for a hybrid vehicle, aiming at significant reduction in fuel usage and emis-sions (50-75%).
1The Netherlands Organization for Scienti�c Research
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Chapter 2
Hybrid vehicle design process
In this chapter, the design framework of the project “Impulse Drive" is explained and the solutionstrategy for the design of the S systems is explained, this strategy is the start for the conceptualdesign of S in the next chapters.
The design of a hybrid vehicle drivetrain is a complex task. Many subsystems (P ,S,V ,T ,C, Fig-ure 1.2) are present in the drivetrain that all have an influence on the final performance. Theobjectives of a design process are summarized [3]:
1. Determine the steps of design order
2. Determine sensitivity of the design specifications in order to select and design new tech-nologies for S and T
3. Optimization of models for S, T and vehicle control C to targets
4. Find an optimal hybrid vehicle drivetrain.
A single, integral design effort to determine the subsystem designs is conventionally adopted fora hybrid drivetrain design. This is done by selecting the topology of the drivetrain and sizing thedrivetrain subsystem all at the same time. However, several reasons exist to choose for a so-calledcascaded design approach, thus scaling the subsystems sequentially [12]:
1. Insights into the design problem at hand are lost when a single final design proposal ispresented as a result of the integral complex design process. Interactions between thedifferent drivetrain subsystems are difficult to investigate.
2. Computational times are generally large of an integral design process, because of the manyinteraction possibilities between the subsystems and the large number of design variables.In a cascaded design approach, the optimization strategy of the subsystems can be stream-lined and thus speeded up.
3. Since significant fuel use reduction is one of the main hybrid vehicle design objectives,system component efficiency plays an important role in the design process. Efficiency ofthe P subsystem is in the order of 10-30%, of the S subsystem is expected to be around 60-95%, of the T to be around 70-98%. It seems plausible to design and optimize the worstperforming (in terms of efficiency and emissions) subsystem first (P ), then S, followed byT (the cascaded design approach), to achieve a "best" drivetrain design.
5
2.1 Problem structureThe design of the drivetrain is translated into a cascaded design process; the problem can be par-titioned into a hierarchical set of subproblems, each with systems and components (Figure 2.1).Each subsystem design has its own set of specifications, starting with the Vehicle system level
Hybrid Drivetrain
Primary Power P Transmission T
Vehicle level
Technology choice level
(system level)
Technology sizing level
(subsystem level)
- Petrol / Diesel choice - Combustion cycle
- Vehicle Class - Front/ rear wheel drive - P-S-T configuration (topology) - Forward/ Backward model
Power
- Engine size - Number of cilinders
Launch elements
Ratio change
Secondary Power S
Storage Conversion
- Electrochemical - Mechanical - Hydropneumatic - Electrostatic
THIS THESIS
- Storage type: Electrochemical/ Mechanical/ Hydropneumatic/ Electrostatic (Conversion type follows)
- Electric motor - Pump - Power electronics - CVT
Control C
Figure 2.1: Coordination scheme of the design problem. Indicated with the thick arrows isthe main design path to acquire design speci�cations for S. Each block represents a designmodel. P , S, T are explained in Figure 1.2.
where vehicle performance goals are set, cascading down to the Technology level where principaldesign decisions are taken to select technologies, down to the Sizing level where the optimal di-mensions of the technologies are determined. The result of each subsystem design is the inputto a lower level subproblem. By using this cascading design process, each subsystem design canbe handled sequentially, as is the case with designing and sizing subsystem S, which is the mainscope of this report as indicated in Figure 2.1. First step of the design cascade, beyond the scopeof this thesis, is the design of P [3]. In this design step, S is taken constant at a set efficiency of80% and as a result, a preliminary Control C can be determined such that the power distributionbetween P , S and V is optimal. In this first design step, P , is already sized for the problem athand, as a result the C and consequently the power flows from and to P , S and V can be calcu-lated using Dynamic Programming. Design constraints are then obtained for the S design (Table2.1), after which the two sublevel design steps can be taken (technology choice and sizing), theseconstraints are input to the S design problem.
Concluding, the design constraints coming from the control C and the primary power sourceP consist of the parameters in Table 2.1. After both P and S have been designed and sized, thetransmission T can be designed based on the Optimal Operating Line (OOL) properties of P and
6
Source Input to S SymbolHybrid Drivetrain Mass limit Mmax
Volume limit Vmax
Price limit €max
E�ciency limit ηmin
Control C and Primary power source P Power �ow PS(t)
Table 2.1: S design constraints
S. It is expected that OOL operation will not be possible and DP optimization techniques couldbe necessary to optimize the use of P , S and T .
2.2 S design
The design of the S system consists of two design levels, a Technology level design and a Sizinglevel design. One advantage of this two-level design approach is elimination of unfeasible S de-signs quickly before proceeding to the sizing of the components, thus reducing calculation time.As an example, when a high storage capacity is required from the Topology design level, a designbased on an Ultra Capacitor (UC) is not feasible from a mass point of view since an UC has lowspecific energy. Another advantage is that feedback towards the Topology choice level about thedesign requirements can be given quickly, without the need to calculate and size the componentsof S.
I. Technology Level design choice is based on the applicable type of buffer and the topology ofthe components. Input to this design level are generic power rating specifications based on thedrivetrain power demand PS(t) and a choice of design constraints, e.g. volume V , mass M , ef-ficiency η, price €, etc. The mass constraint Mmax is determined by the vehicle mass associatedwith the vehicle class and the mass of the engine, further drivetrain components and parts of thecar used. Mmax, the maximum design mass of S, is then simply the mass remaining (e.g. citycar 900 kg, 800 kg for engine, further drivetrain components and parts of the car, leaving 100 kgfor the mass of S). Other constraints can be obtained analogously. The choice of the designconstraints is solely based on the design problem at hand and depends on the vehicle class underconsideration.In case no technologies are capable of delivering required performance, the design space {V, M, η, €}must be relaxed or the power demand PS must be altered from the Topology level to be able tofind a feasible technologie.
II. Sizing Level design is the actual sizing of the topology components towards maximizing thedesign goals of minimum volume, mass, price and maximum efficiency. In this way, technolo-gies can be compared on a specific problem basis and with comparable performance, based onthe mass, volume, efficiency and price of the S systems.
For the remainder of this report, all candidate hybrid systems will be sized and the first designlevel has been omitted, however in practice this design step is always present.
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2.3 S candidate systemsTo identify candidate systems that will be analyzed further, the limiting properties of these sys-tems for this particular thesis are summarized:
• sufficiently compact and light for hybrid vehicle application,
• capable of storing and releasing energy from and to the rest of the drivetrain,
• one mechanical in-/ output shaft,
• control input has to be able to control in-/ output power.
The set of S system thus consists of systems that can be added to an existing drivetrain to obtaina hybrid drivetrain design. Several candidate S systems have been identified, the set is limited tosystems in Table 2.2 for the following reasons:
• models with sufficient detail could be derived and, if necessary, measurement data waspresent of the systems,
• the systems are representative for current technical developments in storage and conver-sion technologies.
Design choice is based on the choice of buffer(s) and consequently the related choice of energyconvertor(s). The choice for the buffer technologie has been limited to the following:
• Supercritical electrically coupled Flywheel (fw_suprcr or HS FW)
• Subcritical mechanically coupled Flywheel (fw_subcr or LS FW)
• Nickel-Metal Hydride battery (batt_nimh or NiMH)
• Valve-Regulated Lead-Acid Battery (batt_vrla or VRLA)
• Ultra Capacitors (uc or UC)
• Compressed Air Energy Storage (caes_oil or CAES).
The technologies are explained in more detail in Chapter 4. When considering two or more buffertechnologies in one topology, a control strategy for the power distribution is also required. For themoment, only one buffer will be considered, for multiple buffer solutions the reader is referredto [13].
For all these technologies, it is possible to generate models that describe the typical propertiesfor the technologies, like typical volume V , mass M , efficiency η, price €, energy contents E,etc. Technology specific properties such as specific volume and specific mass can be generated fromthese properties to compare different technologies. Based on the required performance PS(t),several technologies will prove to be not feasible, these technologies can then be eliminated fromthe sizing procedure in the Sizing design level.
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Table 2.2: S system design concepts
Si S system Components and Modeling Layout
S1
Flywheel: analytical model, FW
Electric C
VT
��EM
EM
INV
Super-critical �ywheel scalable to power pro�lewith electrical CVT Electrical CVT: scalable to
power, empirical model
S2
Sub-critical Flywheel: analytical model, FW
reduction +CVT
���ywheel scalable to power pro�le.with mechanical CVT: discrete, empiricalV-belt CVT model
S3
Nickel Metal Battery: scalable to power NiMH batt
EM
INV
Hydrid (Ni-MH) pro�le, empirical modelbattery with Motor: scalable to power,PM motor and Inv empirical model
S4
Valve Regulated Battery: scalable to power VRLA batt
EM
INV
Lead Acid (VRLA) pro�le, empirical modelbattery with Motor: scalable to power,PM motor and Inv empirical model
S5
Super capacitor: scalable to Super Capa citor
EM
INV
Super capacitor with power pro�le, empirical modelPM motor and Inv Motor: scalable to power,
empirical model
S6
Compressed air Pressure tank: analytical
pump
volum. valve
GPT
energy storage (CAES) model, scalable to powerwith hydraulic control pro�levalve and gear pump Pump: scalable to power
PM = Permanent Magnet, EM=Electric Machine, FW=Flywheel, CVT=Continuously Vari-able Transmission, GPT=Gas-Pressurized Tank, INV=Inverter
2.4 E�ciency and State-of-ChargeIn this section the definitions of efficiency and State Of Charge (SOC) will be given that will beused further throughout this report.Efficiency of a component K is defined as
η =Pout
Pin. (2.1)
For the general model depicted in Figure 2.2a this can be written as the mean efficiency over apowercycle
η =
∫ t0 Psdt∫ t0 Pdt
. (2.2)
9
With PL the losses power, PS the power from the buffer and P the power coming from thecomponent K. For a discretized signal and loosing time index t:
η =∑
Ps∑P
. (2.3)
Now, one charge state and two discharge states can be defined based on the value and sign of thesignals (Figure 2.2b).
P K S
P s
P L
in out
(a) E�ciency model
d1 d2 c
(II) (III) (I)
(b) States
Figure 2.2: Power �ow model and states
(I) charging state c (P < 0 and − Pª ≥ PL)For this situation the mean efficiency can be written as:
ηc =∑
Ps∑P
=∑
(Pª + PL)∑Pª =
∑ |P | −∑PL∑ |P | = 1−
∑PL∑ |P | , (2.4)
with ª indicating an all-negative sign signal, P (t) < 0 ∀t. Note that the efficiency ηc is 0 for∑Pª =
∑PL and 1 for
∑PL = 0.
(II) discharging state d1 (P > 0 and P⊕ ≥ PL)For this situation the mean efficiency can be written as:
ηd1 =∑
Ps∑P
=∑
(P⊕ + PL)∑P⊕ =
∑ |P |+ ∑PL∑ |P | = 1 +
∑PL∑ |P | . (2.5)
Now, ηd1 > 1, this would cancel out efficiencies< 1 for a power signal that has all three charge/dischargestates. The correct way to handle this is by inverting the in- and outputs, e.g.
1ηd1
=∑
P∑Ps
=1
1 +P
PLP |P |. (2.6)
However, compared with the efficiency ηc, ηd1 will never reach 0. To obtain comparable efficiencyresults, another definition for this state is proposed that overcomes this problem and makesefficiencies of all states comparable:
η∗d1 = 1−∑
PL∑ |P | . (2.7)
10
(III) discharging state d2 (P < 0 and − Pª < PL)This is a special case, since energy is flowing from the drivetrain ("in" side) to component K andstill the S system is being discharged. This is for example the case when the losses PL are greaterthan the output power, e.g., if a flywheel has constant losses of 1 kW and is being charged at0.5 kW , the flywheel has to spin down while being charged. The mean efficiency can then bewritten as:
ηd2 =∑
Ps∑P
=∑
(Pª + PL)∑Pª =
∑ |P | −∑PL∑ |P | = 1−
∑PL∑ |P | . (2.8)
However, since PL > |P |, the situation occurs that the mean efficiency will be smaller than zero,this is logical since the device is being charged from the drivetrain side and discharged from thestorage side. For the optimization routines used later on, that minimize 1− η, this is not a prob-lem.
Now, for all three situations the mean efficiency formula is the same and this will be used further.Characteristic property of using this definition of the mean efficiency, is that mean efficiencyη ≤ 1:
η = 1−∑
PL∑ |P | ≤ 1. (2.9)
This equation is valid for∑
PL ∈ R and∑
P ∈ R/{0}.
State Of Charge (SOC) of a buffer can be defined as:
SOC =Current energy contents [J]
Maximum energy contents [J](2.10)
and SOC ∈ [0 1].
Now, the different candidate S systems have been defined and definitions have been set. Nextstep is to optimize the S systems to a predefined output power profile and finally compare thesystems properties.
11
Chapter 3
Optimization methods
In the last chapter, six S system concepts have been presented that will be modeled and analyzedfurther. These S systems have a general layout:
• inputs: SOC0 the state of charge at time = t0, x the set of design parameters and PS(t) thepower profile.
• outputs: M(x) the mass, V(x) the volume, €(x) the price and η(x,P(t),SOC0) the mean effi-ciency.
Next steps to compare the different S systems is to make them comparable in performance (opti-mize to a given power output profile) and compare the properties of the resulting systems. In thischapter, optimization techniques will be presented for the first task.
The optimization problem can be formulated as being:
"Find the design parameter set x∗ such that M,V and € are minimal as well as η is maximal for agiven power profile PS(t) and a begin State of Charge SOC0".
Generally, all S systems have constraints with respect to x and since multiple objectives (min(M),min(V), min(€), max(η)) are under investigation. the optimization problem can be described asa constrained, multi-objective nonlinear optimization problem and can be written as:
minx
(M), minx
(V ), minx
(€),maxx
(η), ... (3.1)
subject to g ≤ 0
with g the design constraints, the number and types of constraints depends on the S system.Several optimization techniques exist that can be used to optimize a constrained problem (e.g.0th order methods such as Complex method or approximation methods such as Sequential Lin-ear/ Quadratic Programming) or translate constrained problems into unconstrained problems(Penalty and Barrier Method, Augmented Lagrangian Multiplier Method). Since comparison ofthese techniques is not an issue in this research, merely a choice for the most used techniqueshas been made.Some of the models that are introduced further in the next chapter are based on discrete valuessuch as number of battery modules. A Branch-and-Bound optimization method could be used tofind the discrete minimizer for the optimization problems at hand. However, to not complex the
12
design process further, this technique has not been used and non-discrete values for the designparameters can be found.The methods used for the optimization of the S systems are explained further in this chapter.
3.1 Sequential Quadratic ProgrammingOne of the most efficient minimization methods of recent years for nonlinear constrained opti-mization problems is the Sequential Quadratic Programming (SQP) method. In this method, anapproximation is made of the Hessian of the Lagrangian function using a quasi-Newton updatingmethod [14]. This is then used to generate a Quadratic Programming (QP) subproblem whosesolution is used to form a search direction for a line search procedure. At the new found iterationpoint, this procedure is repeated. Based on a standard problem formulation,
minx
f(x) (3.2)subject to gpx = 0, p = 1, ..., me
gpx ≤ 0, p = me, ..., m,
the first order Taylor expansion of the Lagrangian can be written as
L(x, λ) = f(x) +m∑
p=1
λp · gp(x). (3.3)
This can be regarded as being the Karush-Kuhn-Tucker (KKT) conditions for optimality of thesolution of the quadratic optimization problem
mind
q(d) =12dT Hd + cT d (3.4)
subject to Apd = bp, p = 1, ..., me
Apd ≤ bp, p = me, ...,m.
When solving this problem a search direction and stepsize to obtain the next iteration can beobtained using line search optimization techniques.The SQP method will find a global optimum if the objective function is convex, however thiscan not be guaranteed in this case since convexity tests need a differentiable objective functionas well as differentiable boundary conditions that have to be expressed in the design variables x.Since, for several models, look-up tables are used in the objective function, it is not differentiableat the iteration brake points. A ‘feeling’ for the locality of the solution can be found by choosingdifferent starting points x0.The SQP method is considered to be an efficient and robust general purpose algorithm that canstart from an unfeasible design. Through the algorithm fmincon.m with option "Largescale=off"the SQP optimization method is available in the optimization toolbox of MATLAB.
3.2 Weighted Multi-Objective FunctionGenerally, optimization algorithms minimize a function value that can be considered to be acost function. In this particular case, multiple cost functions are present in the cost functiondescription (3.1). Minimizing each cost function separately and comparing the results is one
13
way of solving this multi-objective problem, convergence problems can then be expected (such asminimal mass of the flywheel system will converge to an infinitely large inner and outer radiusof the flywheel). For this reason, a single objective function including all objectives in (3.1) isproposed:
T = minx
n∑
q=1
αq · fq(x)
subject to g ≤ 0. (3.5)
In which the weighted objective design function T is the proposed cost function. In this equation,α represents the weight factor of the design criterion q under consideration, f the cost functionfor this particular criterion and x the design variables for the topology components.Possible choices for q are visible in Table 3.1, this is merely a possible set of criteria that can beconsidered. Several of these criteria can be determined by modeling the drive train and optimiz-ing the technologies for this drive train (e.g. η, M , V ), other criteria are more economical innature (e.g. €, E.F., lt).
q Criterion scaling factorη E�ciency 1− ηM Mass M/Mmax
V Volume V/Vmax
€ Price €/€max
lt Lifetime not consideredmi Maintenance intervals not considered
E.F. Environmental Friendliness not considered
Table 3.1: Possible choice of criteria to optimize
The factor α is the weight factor that determines what the importance of the aspect to the de-signer is. It might be necessary that for a budget, city hybrid vehicle the price of the solution ismore interesting than the efficiency itself from a marketing point of view. For a passenger bussolution, volume of the design may be of less importance than the other criteria. To investigatee.g. efficiency and compare efficiencies of different topologies it is possible to set all αq to zeroexcept for αη.The sum of αq is set to be:
n∑
q=1
αq = 1. (3.6)
The design targets M ,V and € must be scaled so that they have comparable order (∼ 1) in theobjective function. This scaling is done using Mmax,Vmax and €max. In this way, these valuesalso contribute to the importance of the criteria in the objective function. Elimination of infeasibledesigns based on the maximum values for mass, volume, price and the minimum value forefficiency, is thus done after the optimization step.
14
3.3 ConstraintsEquality constraints are denoted as h and can be used to couple or fixate design constraints:
h = 0, (3.7)
inequality constraints are denoted as g:
g ≤ 0. (3.8)
For the modeling used for the components of S, that are explained further in the next chapter,two distinct sets of inequality constraints can be identified. One of the two is the functionalconstraints to the dimensions of the components, e.g. real and positive lengths, or the outerdiameter of the flywheel should be larger than the inner diameter. These constraints can bewritten in standard notation:
g1(x) ≤ 0. (3.9)
Besides these constraints another set of constraints can be determined, e.g. SOC ∈ [0,1] orP<Pmax:
g2(x, SOC0, P (t)) ≤ 0 (3.10)
and
g = {g1, g2} ≤ 0. (3.11)
The main difference between the two constrained sets is that the first set can explicitly be writtenin the design variables x whereas the second set can not. This has consequences for the SQPalgorithm since it can only cope with constraints, explicitly written in the design variables x. Tocope with this problem, the constraints can be integrated into the objective function using Penaltyand Barrier methods. Another method to deal with this problem is to increase the number ofdesign variables.
3.3.1 Penalty and Barrier methodPenalty and barrier methods are used to translate a constrained optimization problem into anunconstrained optimization problem. In this case, the technique is used to eliminate constraintsthat can not explicitly be written in the design variables, but that can be calculated. An example ofthis is State of Charge, it is dependent on the design variables x but on the initial state of chargeSOC0 and the power profile P as well. In other words:
minx
f(x) s.t. g1(x) ≤ 0, g2(x, SOC0, P (t)) ≤ 0 ⇒ minx
T (x, g2, r) s.t. g1(x) ≤ 0. (3.12)
In the penalty method the function T is constructed as:
Tp(x, r) = f(x) +1r
∑(max[0, g2(x)])2, (3.13)
which implies that the inequality constraints g will only be taken into account when the boundshave been crossed. The optimal solution (minimizer x∗) can be found through iteration:
r ↓ 0 ⇒ x∗(r) → x∗. (3.14)
15
The barrier method is constructed as:
Tb(x, r) = f(x)− 1r
∑log[−g2(x)], (3.15)
which also shows iteration by minimizing r.In the barrier method, the design remains feasible throughout since the objective function valueincreases just before reaching the bound, opposed to the Penalty method.Generally, the penalty method is more robust and does not need a feasible design as a startingpoint, the disadvantage of the penalty method is that the resulting design might be unfeasible[14].
3.4 Optimization OverviewNow, based on the problem description in Chapter 2 and the techniques introduced in this chap-ter, the general overview scheme for the optimization can be presented (Figure 3.1). The finalminimization objective function based on the penalty function method is given as:
T = minx
{αη(1− η) + αM
M
Mmax+ αV
V
Vmax+ α€
€
€max+
1r
∑(max[0, g2(x, SOC0, P (t))])2
},
s.t. h(x) = 0, g1(x) ≤ 0 (3.16)
and is the objective function used for all S systems.
The next step in the procedure is the determination of the model functions for the mass, volume,price and efficiency of the S systems such that the objective function value can be calculated andthe systems can be optimized. This is presented in the next chapter.
16
Is resulting S i system feasible?
START i:=1 P(t)
choose S i system from
library
Optimize S i component sizes
for P(t)
Yes
No
Store performance
All S i systems in library
processed?
i<6
i=6
Compare Performance of
S i systems
i:=i+1
M max V max EUR max
(a) Overview
Desired accuracy reached?
P(t)
Yes
No
Output x opt = x *
Set (next) design x (through SQP)
Calculate ETA(P, x )
Calculate M( x )
Calculate V( x )
Calculate EUR( x )
Calculate weighted
objective function
OPTIMIZATION BLOCK
(b) Optimization
Figure 3.1: Design procedure
17
Chapter 4
Optimization models
The next step in the procedure is to describe the S systems in the objective function form (3.16).For this, it is necessary to describe for each component of all S systems the mass, volume, priceand efficiency (Table 4.1).
Model Input OutputE�ciency x, P (t) ηMass x MVolume x VPrice x €
Table 4.1: Submodels necessary for each S component
The combination of these building blocks renders an S system, the submodels are described indetail in the appendices, see Table 4.2
Appendix Submodel MATLAB �le nameA.1 Flywheel fw_mech.mA.2 Battery batt_vrla.m, batt_nimh.mA.3 Ultra Capacitor uc.mA.4 Compressed air energy storage caes_oil.mA.5 Continuously Variable Transmission cvt.mA.6 Electric Motor motor_ac.m, motor_pm.mA.7 Pump pump.mA.8 Power electronics -
Table 4.2: Paragraphs describing the modeling of the S storage and conversion subsystems
All models have been modeled using a single track strategy, meaning that all dynamics can re-spond instantanuously to speed and torque changes within the simulation time resolution of 1second. No drivetrain elasticity terms are thus considered either.
Each building block is constructed of a MATLAB function file with the syntax:
[SOCnew, P loss,M, V, price, poss] = component(x0, x, SOCold, P, t), (4.1)with
18
SOCnew new State of Charge at time T+t,Ploss Power loss at time T,M Mass [kg] of component comp,V Volume [m3] of component comp,price Price [€] of component comp,poss Feasibility of component comp, 0 for not feasible, 1 for feasible,x0 vector of variable design parameters,x vector of fixed design variables,SOCold old State of Charge at time T,P Power demand at time T (=PS(T )),t step time duration (1 [sec] throughout this report).
Scripts of the components can be found in appendix D as well as an example optimization script.
4.1 Modeling limitationsUsing all introduced component models, the S systems can be constructed. For all Storage mod-els, the power loss can be difficult to determine, e.g. the power entering the storage is constructedof (in charging state):
Pstorage = PS − PLoss. (4.2)
In general the power Ploss is a function of:
Ploss = Ploss(SOCold, Pstorage), (4.3)
so a feedback loop is introduced in this way. It is possible to implement the feedback loop in thescripts, however for speed of calculations the power losses at moment t = ta are penalized fromthe storage device at time t = ta +1, a time shift of 1 s. As a result, it is possible that major powerlosses at the last time step are not taken into account at the efficiency calculation, a solution is toadd a power demand of zero after the last time step to do take these into account. Static losses of1 s duration are then introduced, however these are normally significantly lower than operatinglosses. Further, the longer the duration of the time simulation, the smaller the error is.
Several properties and limitations of the specific models and procedures in use are discussedfurther.
S1- Supercritical flywheel with electrical CVT.
FW
Electric C
VT
��EM
EM
INV
The CVT model consists of two electric machines electrically coupled, withthe axis of the second electric machine coupled to the supercritical flywheel.This is necessary because —in general— the vibrations of the flywheel atthe first natural frequency are so intense that a mechanical connection is notpossible [15]. By using an Optimal Operating Line (OOL) efficiency modelof the electric machine, it is not possible for the model to describe motor speed efficiency forarbitrary speeds of the flywheel as is needed in this case. By using a Willans line model technique,the efficiency map based on torque and speed can be be generated relative to a base electricmachine. This has however not been done in this case, because of time limitations. The efficiencyof the electric CVT is thus higher then the model should be. However two reasons exist such thatthe OOL operation assumption is still made:
19
1. One electric motor is operated in near vacuum at 5 mbar at very high operating speeds.The generic design of the electric machine is not meant for this type of operating envi-ronment, however it is assumed that some kind of electric machine with similar operatingbehavior can be constructed, with specific design and higher performance for this kind ofenvironment.
2. Specifically designing of two coupled electric machines into a CVT construction is expectedto have better overall peformance than the coupling of two electric machines, meant as asingle device. In other words, in CVT construction the motors are expected to be optimizedfor this purpose and have higher efficiency.
The resulting efficiency figures are further assumed correct under the given assumptions, how-ever a generic high-speed electric machine model, operated in a vacuum environment, should bedeveloped for more accurate results.A further limitation of the design is the lack of a model that describes the stresses on the fly-wheel axis, resulting from spinning up and down the flywheel. A minimum axis diameter isset to 0.03 m as a lower bound, this bound is always found in the optimization routine and thediameter of the axis is set to this size. The axis diameter is an important factor in the design ofthe flywheel in general, since it partly determines the maximum power the flywheel can generate.For hybrid vehicle purposes however, with maximum powers of roughly 50 kW , the diameter isassumed less important and no stress model for the axis (nor for the hub) has been included.
S2- Subcritical flywheel with mechanical V-belt CVT.
FW
reduction +CVT
��The mechanical CVT has fixed size and is not scalable to optimal efficiency,mass, volume and price. The CVT is limited in input speed, because at agiven power demand and maximum torque, the input speed has a mini-mum (e.g. power demand of 10 kW , max 100 Nm torque, speed withoutlosses ω = P/T = 10e3/100 = 100 rad/s). Since flywheel losses are mini-mal at lowest turning speeds, these two properties counteract each other and a larger CVT mightbe preferred because of lower flywheel speed possibilities and thus lower losses. Also, reductiongears can be used for this purpose and clutches could be introduced to uncouple the flywheelfrom the CVT. This is not considered further in this thesis.
S3/S4- NiMH/ VRLA Battery with Electric Motor and inverter.
NiMH batt
EM
INV
VRLA batt
EM
INV
Motors are operated at OOL, this implies that the drivetrain transmissioncomponent T can cope with an arbitrary output speed of the S systems.This is a constraint on the design of the T component in a later stage of theproject.
20
S5- UltraCap Battery with Electric Motor and inverter.
Super Capa citor
EM
INV
The price of UltraCaps is very high at the moment and is expected to dropconsiderabely the following years [16]. The resulting price properties of theUCs are to be interpretated carefully if longterm developments are consid-ered.The modeling of the UltraCap pack has foreseen the use of UltraCaps inparallel operation, as well as in series operation (Appendix A.3). However, resulting from theoptimization runs, the ultracaps are practically always placed all in a series configuration. Thismakes sence, since power losses are minimal with lowest current (since Ploss = I2R), so a par-allel configuration modeling was not strictly necessary. However, in practice, high voltages willresult in thick insulated cables which is also not practical, the model can then still be used topredict the efficiency behavior for parallel configurations.
S6- Compressed Air Energy Storage with hydraulic gear pump and lossless volumetric valve.
pump
volum. valve
GPT The storage tank is modeled such that losses are zero, because it is assumedthat the process is adiabatic. This means there is no heating of the oil in thetank and no heat losses from the tank to the direct environment. The pumphowever is modeled with power losses, the volumetric valve is assumed loss-less. The volumetric valve is a device that can control the flow to and fromthe pump, thus the power. It is constructed such that the pump rotates in one direction only, bothfor the charge- and discharge situation.The pump can only operate between certain pressure limits from the tank and between a mim-imum and maximum operating speed. As a result, the design of the S6 system has many con-straints from the pump.
21
Mod
elLa
yout
Designspace
xBo
unds
forx
Constraintsg
Remarks
S1_pm
/S1
_ac
FW
Electric CVT ��EM E
M
INV
x 1=fw
_suprcr_d
x 1∈
[0.031]
g 1=
0.5x
1−
x3≤
0OOLop
erationof
electric
machine
sis
assumed
.
x 2=fw
_suprcr_Ro
x 2∈
[0.001
1]g 2
=−S
OC≤
0x 3=fw
_suprcr_Ri
x 3∈
[0.001
1]g 3
=S
OC−
1≤
01/
r=1e8(
1)
x 4=fw
_suprcr_h
x 4∈
[0.001
1]Max
Iter=20
0(1)
x 5=motor_pm
/ac_kW
x 5∈
[max|P
s|2
00]
S2F
W
redu
ctio
n +
CV
T ��x 1
=fw
_subc
r_d
x 1∈
[0.031]
g 1=
0.5x
1−
x3≤
0FixedV-belt
CVT
size,
nots
calable.
x 2=fw
_subc
r_Ro
x 2∈
[0.001
1]g 2
=−S
OC≤
01/
r=1e8(
1)
x 3=fw
_subc
r_Ri
x 3∈
[0.001
1]g 3
=S
OC−
1≤
0Max
Iter=30
0(1)
x 4=fw
_subc
r_h
x 4∈
[0.001
1]S3
_pm
/S3
_ac
NiM
H
batt
EM
INV
x 1=ba
tt_nimh_
nx 1∈
[11e3]
g 1=
Pnim
h,in−
Pnim
h,m
ax≤
0-
1/r=
1e8(
1)
x 2=ba
tt_nimh_
Qmax
x 2∈
[1093
]g 2
=−S
OC≤
0Max
Iter=10
0(1)
x 3=motor_pm
/ac_kW
x 3∈
[max|P
s|2
00]
g 3=
SO
C−
1≤
0S4
_pm
/S4
_ac
VR
LA
batt
EM
INV
x 1=ba
tt_vrla_n
x 1∈
[11e3]
g 1=
Pvrla
,in−
Pvrla
,max≤
0-
1/r=
1e8(
1)
x 2=ba
tt_vrla_Qmax
x 2∈
[1210
4]g 2
=−S
OC≤
0Max
Iter=10
0(1)
x 3=motor_pm
/ac_kW
x 3∈
[max|P
s|2
00]
g 3=
SO
C−
1≤
0S5
_pm
/S5
_ac
Sup
er
Cap
a ci
tor
EM
INV
x 1=uc
_Ns
x 1∈
[11e4]
g 1=
Puc,
in−
Puc,
max≤
0Pr
icemod
elno
tvalid
forlon
gert
erm.
x 2=uc
_Np
x 2∈
[110
]g 2
=−S
OC≤
01/
r=1e8(
1)
x 3=uc
_cell_
Cx 3∈
[500
1e4]
g 3=
SO
C−
1≤
0Max
Iter=20
0(1)
x 4=motor_pm
/ac_kW
x 4∈
[max|P
s|2
00]
S6_p1
/S6
_p2
pum
p
volu
m.
valv
e
GP
T
x 1=pu
mp_
vgx 1
,pum
p1∈
[20.9e-6
45.2e-6]
g 1=−S
OC≤
0Lo
ssles
smod
elforG
PTan
dforv
olum
etric
valve.
x 2=pu
mp_
desig
n(∈
{1,2})
x 1,p
um
p2∈
[40e-6
100e-6]
g 2=
SO
C−
1≤
0x 3=caes_oil_
Rx 3∈
[0.0510
]g 3
=m
ax(p
caes
)−
ppum
p,m
ax≤
0x 4=caes_oil_
tx 4∈
[0.001
0.5]
g 4=
ppum
p,m
in−
min
(pca
es)≤
01/
r=1e3(
1)
x 5=caes_oil_
hx 4∈
[0.0510
]g 5
=ωpum
p−
ωpum
p,m
ax≤
0Max
Iter=15
0(1)
g 6=
max(|P
s|−
Pcv
t)≤
0
Table4.3:
Sde
signmod
eling
overview
.(1) :
thevalueMax
Iter
determ
ines
themax
imum
numbe
rofo
ptim
izatio
nite
ratio
nsbe
fore
the
optim
izatio
nisterm
inated
.Itisa
nem
peric
alvalue,
determ
ined
bythegene
ralspe
edof
theop
timiza
tionfort
hespeci�cS
considered
andisaba
lanc
ebe
tweenspeedan
daccu
ratesse.
rha
salso
been
determ
ined
empe
rically
fore
achsystem
,too
low
avaluewi
llresultin
very
high
pena
ltyfunc
tionvalues
( 3.3.1)a
ndassocia
tedconv
ergenc
edi�cu
lties.H
owever,too
high
avaluean
dinfeasible
desig
nswi
llresult.
EM=ElectricMachine
,FW
=Flyw
heel,
CVT=
Continuo
usly
Varia
bleT
ransmiss
ion,
GPT
=Gas-P
ressurize
dTa
nk,INV=Inverter
22
4.2 Validation of analytically modeled componentsValidation of the empirical models is most of the time not necessary, since they are based onactual designs of the components (ac+pm motor, batteries, pumps, CVT). It is however possiblethat programming errors have occured in the model scripting, therefore the resulting storagesystems properties will be compared with literature data in Chapter 5.2. Analytical models havebeen used for the flywheel and the CAES storage models, a validation effort to test these analyticalmodels is necessary and presented in this section.
4.2.1 Subcritical FlywheelFor the flywheel, experimental data is available for a small flywheel from Imperial College, Me-chanical Engineering Department, London. The dimensions of the flywheel and housing areknown, it is a supercritical flywheel that is operated in athmospheric pressure and the experi-ments have been done subcritically. The subcritical flywheel storage model has been put to thetest, and a figure can be created of the power losses relative to the flywheel rotating speed (Figure4.1). The simulation model of the flywheel predicts power losses that are about half the measured
0 100 200 300 400 500 600 7000
20
40
60
80
100
120
140
speed [rad/s]
Pow
er lo
ss [W
]
experiment
simulation
Figure 4.1: Flywheel simulation and experimental data. Flywheel model rotor: inertia0.10 kg/m2, experimentally determined: 0.11 kg/m2, the di�erence is due to a di�erent hubdesign. Maximum subcritical model storage capacity: 95 Wh
power losses, the shape of the power loss curve does match.
Another reference for validation of the subcritical flywheel is [15]. In this thesis, the power lossesfor a 20 kg rotor are evaluated, resulting in power losses of a little over 0.5 kW for a flywheel withspecific energy of 2 Wh/kg. When checking the subcritical flywheel model with this data, thepower loss at for this design are 1 kW , thus approximately double the expected.These two validation data sets contradict each other, and since the model behaves in betweenthese two data sets, it is assumed the model is representative to the losses present in a subcritical
23
flywheel.
4.2.2 Supercritical FlywheelFor the supercritically operated flywheel, the losses for a 20 kg rotor mass flywheel at 50 Wh/kg(3.6 MJ ), are calculated to be around 1.9 kW [15]. When simulating this flywheel with the modelin this thesis, the losses correspond exactly with this.
4.2.3 Compressed Air Energy StorageMeasurement data was obtained from VEUNAM, a hybrid passenger bus built by the Universi-dad Nacional Autonoma de Mexico (UNAM). The VEUNAM comprises a medium size passengerbus chassis, 52 lead-acid batteries, a 22 kW DC motor and the hydro-pneumatic system. In thispneumatic system, a 68.5 liter nitrogen tank is pressurized using a fixed-displacement oil pump.The system is highly comparable with the concept storage system analyzed in this thesis.For the experiment, the passenger bus was braked to a full stop from a certain speed. Brakingwas performed partly by the CAES storage system and partly by the brakes on the wheels, the DCmotor was not used for braking. The pressure in the gas tank over time was recorded, as wellas the vehicle speed, thus the power profile Ps(t) into the gas tank can be determined (assum-ing Ps ∝ dv
dt and a fraction of kinetic energy is transferred into the gas tank). The dimensionsof the simulation tank were adapted to the experimental tank sizes, and the power profile wastested (Figure 4.2). The experimental tank does give more or less the correct response, takinginto account that the experimental peak pressure reading is caused by the high flow in the pipesleading to the tank, that decrease slightly after the flow in the piping has stopped at the end of theexperiment (the pressure sensor is located at the end of the entrance tube to the tank, so thesetransient effects occur). The slow pressure decrease in the tank is further caused by a balancingflow into a second tank, not modeled, that is used to increase the storage capacity, thus this effectcan be neglected too.
0 10 20 30 40 50 6012.8
13
13.2
13.4
13.6
13.8
time [s]
Pre
ssur
e [M
Pa]
0 10 20 30 40 50 60−8000
−6000
−4000
−2000
0
2000
Pow
er P
s [W]
power profile Ps
experiment pressure
model pressure predicted
Figure 4.2: Validation of CAES storage tank.
24
Now that all models have been introduced in this chapter and the techniques to optimize thesize of the S systems in Chapter 3, the S systems can be compared with each other which is thesubject of the next chapter.
25
Chapter 5
Results
Using the optimization techniques in Chapter 3 the S systems can be sized to have minimal mass(M ), volume (V ), price (€) and maximum efficiency (η) for a given power output profile Ps. Howimportant each property is to the designer can be set using the αn weight factors, with
∑αn = 1,
and by using the limits Mmax, Vmax, €max and ηmin. In this chapter, the S systems will be sizedand the properties of these S systems are compared based on several test cases:
1. Actual S power profile of 20 kWResulting from a Dynamic Programming optimization, a power profile for the S systemhas been determined, representative for the use of a S system. The signal is based on:
• A Mid-sized passenger car
• Mass 1360 kg, Frontal area 2.0 m2, Wheel friction coefficient 0.013 −, Air resistancecoefficient Cw 0.33 −, Wheel radius 0.31 m.
• New European Drive Cycle (NEDC, 1180 s)
• Efficiency for S and T for initial Dynamic programming cycle: 100 %
All S systems will be optimized to this profile to see if the optimization solver can cope withthis problem as this is the intended use of the simulation models.
2. Power and Energy sensitivityThe S systems are evaluated for their sensitivity of the optimized designs to varying powerand energy requirements to these systems.Then, the energy conversion elements are omitted and the energy storage componentsare evaluated for their sensitivity of the optimized designs to varying power and energy.Resulting from these last optimizations, storage properties such as specific power W/kgand specific energy Wh/kg can be deducted for the storage systems.
3. Sensitivity AnalysisSensitivity analysis of the optimized candidate S systems is performed with a 50 s testprofile. In this profile, a continuous discharge and charge power of 10 and 5 kW is presentplus a rapid switching component with a peak at 12 kW . The following parameters areevaluated for each S system:
• Influence of the start design x0 on the optimized systems. The most extreme valuesof the design space will be used as a starting point for the optimization procedure toinvestigate de global validity of the minimum x∗.
26
• Influence of SOC0 on optimized designs, this gives more or less an optimal startingpoint for each S system that can be used as a starting point for each optimization,found optimization solutions are then better comparable because they are used attheir most optimal operating point.
• Influence of α on optimized designs.
• Influence of Mmax, Vmax, €max on optimized designs.
Principally eleven candidate S systems are under investigation (Table 4.3),these will all be takeninto account at the optimization sequences.
5.1 20kW Power pro�le optimizationOptimal power distribution between P , S and V (Figure 1.2) has been obtained by minimizingfuel consumption, maintaining SOC within a certain range and accomplish any drive powerdemand using a Dynamic Programming optimization technique. The vehicle is assumed to bea mid-sized passenger car (1360 kg) with a 1.6 l engine operated at the OOL, with NEDC drivecycle. For the DP optimization, the efficiencies of S and T have been assumed 100 %. Theresulting power profile Ps is pictured in Figure 5.1. As a result, the ICE is operated solely on itsmost efficient operating points, thus a highly dynamic Ps power profile results. The candidate
0 200 400 600 800 1000 1200−20
0
20
Ps (
kW)
0 200 400 600 800 1000 1200−50
0
50
Pv (
kW)
0 200 400 600 800 1000 12000
10
20
30
Pp (
kW)
Time [s]
Figure 5.1: Power signals for P , S and V , resulting from an optimal power �ow DynamicProgramming computation. Topology T and transmission T are assumed lossless and optimal(for an explanation of S, P , T , V and C see Figure 1.2).
S systems have been optimized for this power profile with an amplitude of 20 kW charge- anddischarge power. The properties of the resulting systems can be represented in a spider diagramfor comparison, Figure 5.2. For all eleven systems, only the best performing for each storage type
27
0.53
2553
1.1
149
1−η [−]
M [kg]
V [m3]
Cost [kEUR]
S1pm
(HS FW) T=2S
2 (LS FW) T=8
S3pm
(NiMH) T=0.57S
4pm (VRLA) T=0.44
S5ac
(UC) T=18S
6p1 (CAES) T=4.1
sub−critical flywheel
Ultra−Capacitor
Compressed air energy storage tank
super−critical flywheel
Ni−MH battery
VRLA battery
Figure 5.2: Optimization of S systems to 20kW NEDC optimized power cycle (SOC0=0.9,Mmax = 150, Vmax = 0.5, €max = 2000, αM = 0.2, αV = 0.2, α€ = 0.2, αη = 0.4. Thesmaller the square, the better the system performs in terms of e�ciency, mass, volume andprice.
is shown, thus the PM motor systems and the CAES system with pump design 1. Immediately, itcan be seen that the sub-critical flywheel system (S2), the Compressed Air Energy Storage Tank(S6p1) and the Ultra-Capacitor (S5ac) are not feasible solutions, caused by too high mass, volumeor cost respectively. The VLRA battery (S4pm) and the super-critical flywheel (S1pm) system arerejected caused by too low average efficiencies of 50.1% and 52.5% respectively. Therefore, thecombination of a NiMH battery with an energy capacity of approximately 9 kWh and a 20 kWPM motor (S3pm) is the most suitable solution fulfilling the design targets.The SOC evolution and the power losses of this system are shown in Figure 5.3. The discrepancybetween the SOC levels at the end of cycle between the imaginary S simulated without losses(blue line) from DP and the S consisting of the Ni-MH and PM motor (magenta line) is caused bythe power losses. The mass, volume, cost price and average efficiency are optimized to be 141 kg,31 dm3, 2.9 k€ and 81.8% respectively for the complete system. The battery solely has a mass of100 kg and an output power of 22.4 kW corresponding to a 0.22 kW/kg and 90 Wh/kg for thepower and energy density. It should be noted that the maximum allowable battery voltage andthe battery configuration (series or parallel connected) is of influence on the amount of requiredbattery energy capacity and nominal power rating. In this report, the batteries are assumed to beconnected in series reducing the internal battery losses due to lower current losses. Furthermore,no constraints are set to the maximum battery voltage.
28
0 200 400 600 800 1000 1200−20
0
20
Ps [k
W]
0 200 400 600 800 1000 12000.88
0.9
0.92
0.94
SO
C [−
]
0 200 400 600 800 1000 12000
2000
4000
6000
Time [s]
Plo
ss [W
]
efficiency is 81.8%
Optimal EMS from DP (blue)
Resulting SoC evolution (magenta)
Figure 5.3: Optimized NiMH battery with a PM motor (S3), Ps, SOC and Ploss are shown asfunction of time.
29
5.2 Power and Energy sensitivityAll S systems are evaluated on a varying peak power demand and on varying storage capacity.The power is varied from 5 to 25 kW in five steps and the storage capacity from 0.1 to 2 MJ infive steps, resulting in a grid of 25 power profiles. These power profiles consist of a continuousdischarge profile with power and capacity, followed by a same continuous charge profile, as shownin Figure 5.4. By varying the duration time Td, the storage capacity of the storage device canbe varied. The optimization start parameters are a SOC0 = 1, Mmax = 150, Vmax = 0.1,
P s [kW]
0
nominal power rating
t d
0.5*t d time [s]
Figure 5.4: Ps signal for power and energy variation experiment.
€max = 2000, αM = 0.2, αV = 0.2, α€ = 0.2, αη = 0.4.1 The resulting optimized systemobjective function values are shown in Figure 5.5. The objective function values of the batterysystems S3 and S4 are relatively insensitive to variations of the energy capacity opposed to theother systems. They are however highly sensitive to power variations, indicating that the powerrequirements of the batteries determine the optimal design of the batteries. Storage capacity is thedesign factor for the rest of the S systems. The battery systems are the best solution to most of thecases, however for high power, low storage capacity, the flywheel hybrid systems are competitivecompared with the battery systems. The design of the S5 system shows infeasible designs forenergy capacities of over 0.5 MJ , this is due to limits in the design space, that result in too largea solution. The objective function value is a weighed sum of efficiency, mass, volume and price.Because of the very low production cost of the VRLA battery system (S6), this is reflected in alower objective function value. If one is interested in the efficiency variations over the varyingpower and energy domain, this is pictured in Figure 5.6 The efficiency of the S4 system (VRLA)is very poor and is below 60 % for power over 6 Kw, deteriorating fast with increasing power. Inthis respect, S3 is a far better performer with an average efficiency over the entire working rangeof 81 % and practically no variation. The flywheel systems grow more efficient with increasingpower, the other systems decrease in efficiency. The contour plots of the systems for mass andvolume are pictured in Figures 5.7 and 5.8
1Noteworthy is the time it took to �nish these calculations, 20 hrs on a 3 Ghz Athlon XP.
30
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S1 pm
0.5
0.7
1
1.5
2
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S2
0.50.7
1
2
48
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S3 pm
0.25
0.3
0.35
0.4
0.45 0.
50.
55 0.6
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S4 pm
0.3
0.35
0.4
0.45
0.5
0.55
0.6
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S5 pm
0.81
2
INFEASIBLE DESIGN
5 10 15 20 25
0.5
1
1.5
2
Power [kW]E
nerg
y [M
J]
S6 pump 2
0.81
1.2
1.5
2
2.5
Figure 5.5: Objective function values T for the optimized S systems with varying power andenergy capacity.
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S1 pm
0.6
0.65
0.7
0.72 0.73
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S2
0.1
0.75
0.75
0.8
0.8
0.820.84
0.86
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S3 pm
0.790.8
0.81
0.82
0.82
3
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S4 pm
0.25
0.3
0.35
0.4
0.5
0.6
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S5 pm
0.70.74
0.76
INFEASIBLE DESIGN
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S6 pump 2
0.86
0.88
0.88 0.88
0.9
0.9
0.91
0.920.92
Figure 5.6: E�ciency for the optimized S systems with varying power and energy capacity.
31
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S1 pm
0.6
0.65
0.7
0.72 0.73
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S2
0.1
0.75
0.75
0.8
0.8
0.820.84
0.86
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S3 pm
0.790.8
0.81
0.82
0.82
3
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S4 pm
0.25
0.3
0.35
0.4
0.5
0.6
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S5 pm
0.70.74
0.76
INFEASIBLE DESIGN
5 10 15 20 25
0.5
1
1.5
2
Power [kW]E
nerg
y [M
J]
S6 pump 2
0.86
0.88
0.88 0.88
0.9
0.9
0.91
0.920.92
Figure 5.7: Mass (kg) of the optimized S systems with varying power and energy capacity.
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S1 pm
0.6
0.65
0.7
0.72 0.73
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S2
0.1
0.75
0.75
0.8
0.8
0.820.84
0.86
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S3 pm
0.790.8
0.81
0.82
0.82
3
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S4 pm
0.25
0.3
0.35
0.4
0.5
0.6
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S5 pm
0.70.74
0.76
INFEASIBLE DESIGN
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S6 pump 2
0.86
0.88
0.88 0.88
0.9
0.9
0.91
0.920.92
Figure 5.8: Volume (m3) of the optimized S systems with varying power and energy capacity.
32
5.2.1 Storage devices only power and energy sensitivityWhen omitting the conversion devices for the S systems, such as the motor and CVT, the sys-tems can be optimized to the same varying power and energy. From these systems, data for thestorage devices alone can be obtained such as specific power and energy (Figure 5.9). In coloredrectangles, data is presented that was found in literature for these storage devices as listed inappendix C. These data is not comparable directly, since specific power and specific energy areconventionally defined as the maximum power and energy a device can deliver/store. In this case,all optimized storage devices are shown for the varying power and energy profile in Figure 5.4 andspecific power and energy numbers are generated based on this signal. The devices are thus notoptimized to maximum specific energy and maximum specific power, as is the case for mostdata found in literature. The data can be used however to put the systems in a perspective, andused as validation of the systems. The specific power data for the battery systems correspondsto literature, but the specific energy is much lower. This is due to the sizing of the batteries topower needs, the entire capacity of the batteries is not used completely in any case. The powercapacity for the NiMH battery is a fixed number, due to the modeling applied to it. The UltraCapstorage also scores lower for the capacity (about half the values found in literature) just as thebattery systems, however the storage capacity of this system is used completely in the optimizeddesigns. This difference is probably due to optimistic numbers found in literature, the storagecapacity of UltraCap devices is expected to grow the following years and this is already reflected inthe literature data. For the flywheel systems, the data corresponds nicely with literature data. Noliterature data was available for the Compressed Air Energy Storage, however the specific powerand energy resulting are very low.
10−1
100
101
102
100
101
102
103
104
Specific energy [Wh/kg]
Spe
cific
pow
er [W
/kg]
HS FWLS FWNiMHVRLAUCCAES
Figure 5.9: Speci�c power and speci�c energy for the storage devices resulting from literature(squares) and the systems resulting from the Storage devices only optimization.
33
5.3 Sensitivity analysisThe sensitivity to variation of x, x0, Mmax, Vmax, €max and α will be investigated in this section.
5.3.1 Sensitivity to the initial designThe exact optimization procedure is explained in detail in appendix B. The lower bounds (xmin)for the S system optimization parameters have been chosen such that negative values are notpossible, the upper bounds are chosen with an engineering feeling for the expected size of theparameters and to prevent infinite values. Major conclusions from the optimization runs are thatthe initial start design x0 is preferably a feasible design. Further, multiple minima are found forall S systems indicating that comparing the different solutions must be done very carefully andwith an engineering feeling of optimality of the solutions. An optimal design is expected to notbe on the upper bounds for all problems presented, caution is needed for interpretation of theresulting designs because of local minima.The start design, based on an engineering feeling and that lies roughly in the middle between theupper and lower bound, was shown to have generally the best results and will be used further.
5.3.2 Sensitivity to the initial State Of ChargeThe begin State-of Charge SOC0 is varied and optimal designs to the short powerful charge/dischargeprofile are determined. The resulting design properties, being the minimizer value T , volume,price, efficiency and mass, are displayed in Figure 5.10. As can be seen, the ultracapacitor ishighly dependent on the begin SOC, this is due to the dependency of the efficiency on the SOC.The best performing for this problem set is the VRLA battery combined with the PM motor, asthis system has the lowest value for T for most values of SOC0. For a SOC0 value of 0.9, the bestperfoming is the NiMH battery system with PM motor. In general, the PM motor systems allperform better because of higher efficiency of the motor system.
It is now interesting to see the variation of the design variables x for each optimized S sys-tem. If the values vary uniformly, meaning without erratic variation along the SOC range, it canbe concluded that the minimum x0 shifts for varying SOC and, as a consequence, the minimumhas a global character over the SOC variation range. It is however still not guaranteed that thefound minimum is the global minimum, only that the location of the minimum shifts with vary-ing SOC0 and with this particular problem.The variation of the design parameters is visualized in Figure 5.11, the design variables vary uni-formly indicating a global character of the minima.
34
0 0.2 0.4 0.6 0.8 1
0.5
1
1.5
2
2.5T
[−]
SOC [−]
S1 acS1 pmS2S3 acS3 pmS4 acS4 pmS5 acS5 pmS6 p1S6 p2
0 0.5 10.4
0.5
0.6
0.7
0.8
0.9
1
Effi
cien
cy [−
]
0 0.5 10
50
100
150
200
Mas
s [k
g]0 0.5 1
0
0.05
0.1V
olum
e [m
3 ]
0 0.5 10
500
1000
1500
2000
2500
3000
Pric
e [E
UR
]SOC [−]SOC [−]
SOC [−] SOC [−]
Figure 5.10: Objectove function values (T ) and system properties of optimized S systemswith varying SOC0. For battery systems, SOC0 of 0.4�0.6 is preferred, for the other systemsa SOC0 of 0.9. (Mmax = 150, Vmax = 0.1, €max = 2000, αM = 0.2, αV = 0.2, α€ = 0.2,αη = 0.4)
5.3.3 Sensitivity to α and Mmax, Vmax and €max
Some of the systems show sensitivity to these factors, see Table 5.1 that summarizes the sensi-tivities of the designs to all the input parameters in this section. Sensitivity of the optimized Ssystems to α and Mmax, Vmax and €max is again explained in detail in appendix B.
35
0 0.5 110
−4
10−2
100
102
S1 ac
SOC [−]
X [v
ario
us]
0 0.5 110
−3
10−2
10−1
100
S1 pm
SOC [−]0 0.5 1
10−2
10−1
100
S2
SOC [−]0 0.5 1
101
102
S3 ac
SOC [−]
0 0.5 110
0
101
102
S3 pm
SOC [−]0 0.5 1
101.1
101.2
S4 ac
SOC [−]
X [v
ario
us]
0 0.5 110
1.02
101.08
101.14
S4 pm
SOC [−]0 0.5 1
100
101
102
103
S5 ac
SOC [−]
0 0.5 110
0
102
104
S5 pm
SOC [−]0 0.5 1
10−5
100
105
1010
S6 p1
SOC [−]0 0.5 1
10−5
100
105
1010
S6 p2
SOC [−]
X [v
ario
us]
Figure 5.11: Optimal design parameters x∗ for each S system with varying SOC0. (Mmax =150, Vmax = 0.1, €max = 2000, αM = 0.2, αV = 0.2, α€ = 0.2, αη = 0.4)
System x0 SOC0 α Mmax, Vmax and €max
S1ac ++ + 0 0S1pm + + 0 0S2 ++ + + +S3ac ++ 0 0 0S3pm + 0 0 0S4ac ++ 0 + +S4pm + 0 + +S5ac ++ + 0 0S5pm + + 0 0S6p1 + + + ++S6p2 ++ + 0 +
Table 5.1: Sensitivity of the objective function values to input parameters. 0: relativelyinsensitive, +: sensitive, ++: highly sensitive.
36
Chapter 6
Conclusion
In this report, a design tool for choosing and sizing of S, the hybrid system in a hybrid vehi-cle drivetrain, has been presented. It can be used to conceptually design and evaluate differentconcept designs to hybridize a vehicle. The components used for S have been described and mod-eled. The S systems have been optimized given a target power output, using SQP optimizationalgorithms and with the aid of a weighted sum objective function and Penalty and Barrier tech-niques. Several optimization cases have been evaluated, and it has been shown that the resultingstorage components show properties very similar to literature values.
The sizing of the battery systems depends solely on the power requirements from the storagedevice, since storage capacity of batteries is generally sufficient for single-storage hybrid vehiclepurposes. Batteries are favorably used in their most efficient SOC range (SOC ∈ [0.4, 0.6]). Theother storage components (flywheels, ultracapacitors, compressed air energy storage) are sizedbased on storage capacity and not on power ratings, and generally use the full storage capacityavailable to deliver the required power profile most efficiently (SOC ∈ [0, 1]).
The NiMH battery pack has high efficiency over its complete working range and low sensitiv-ity to the efficiency with varying power- and energy capacity needs. The VRLA battery systemis often presented as the most optimal S system for the optimization runs, however efficiencyis generally very low compared to all other S systems (about 60 % vs. 80 %). The flywheel sys-tems might be favorable for situations of high peak power demands and low storage capacityneed since performance is on par with the battery systems. The CAES system is generally toolarge and heavy to be used as a single-storage S system, combined with a secondary system (e.g.NiMH battery) the performance can be expected to improve drastically.
For the electric motors, the PM motor always outperforms the AC motor due to higher efficiencyand lower mass and volume. The AC motor is cheaper, however. Scaling the PM motor can bedone based on the peak power requirements since OOL efficiency is uniform over the power out-put range. The AC motor is most efficiently operated (in terms of efficiency, mass, volume, price)if it is sized slightly larger then the peak power (1.2 to 1.8 times larger).
Some aspects are not taken into account in this research, such as lifetime expectancy and en-vironmental life-time-cycle loads. The models can however be expanded easily with these aspectsif desired.
37
Next step in the design of hybrid vehicles, is the transmission component T . Input to this com-ponent are the speed/ torque from P , S and V , an ideal transmission would be able to convertany arbitrary speed from P and S to the desired rotation speed of V , the vehicle wheels.
6.1 RecommendationsThe optimization time of the analytically modeled systems (FW, CAES) is considerably lowercompared to the empirical systems. This is due to look-up tables that were used to find the sys-tem properties (in the order of 10 times faster). If speed of optimization is an issue, an effort tofind fit functions and replace the look-up tables is advisable.
Using the proposed optimization techniques, Pareto optimality can not be guaranteed, so multi-ple minima can exist in the solution space. To find a globally valid minimum, other optimizationtechniques could be employed such as Dynamic Programming.
The S systems that were investigated, are either suited for a high-power output or a high stor-age capacity. Combination of two different systems might render a dual-hybrid storage device(e.g. UC combined with NiMH battery and PM motor) that has far better properties than thesingle-storage solutions. A decision algorithm that determines the use of these devices is thenneeded, very much similar to the algorithms used for the power split between S and P . DynamicProgramming or heuristic optimization techniques can be applied for optimal power distribu-tion. The power split can be translated into two power profiles, one for the UC and the otherfor the NiMH battery, and each storage device can subsequently be optimized using the designtechniques presented in this paper or using Dynamic Programming.
The modeling of the motors in this paper has been done on OOL operation, whereas for thedetermination of the transmission T it can not be guaranteed that the motor is operated at theOOL. Therefore, if it proves necessary, another scalable motor model that can describe the effi-ciency for an arbitrary output speed and torque, is required. This is possible using the Willansscaling technique that can be applied to the ICE and motors.
38
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41
Acronyms and symbols
Acronyms
AC motor Alternating Current (Induction) motor, an electric motor with induction coils togenerate the magnetic field.
accumulator Energy storage component, capable of being charged and discharged.C Control of power flows between P , S, T and V in a vehicle drivetrain.CAES Compressed Air Energy Storage.CVT Continuously Variable Transmission.DP Dynamic Programming, a discrete optimization technique.drive cycle Predefined speed profile, representative to normal use of a vehicle.EM Electric Motor, converts electric energy to kinetic and vice versa (generator func-
tion).FW FlyWheel energy storage.ICE Internal Combustion Engine.Inv Inverter.NEDC New European Drive Cycle.NiMH Nickel Metal Hydrid battery.OOL Optimal Operating Line, component operating points resulting in the lowest
power losses.P Primary power source, usually an ICE, in a vehicle drivetrain.PM motor Permanent Magnet motor, an electric motor design with magnets to generate the
magnetic field.S Hybrid system in a vehicle drivetrain (storage + conversion components).SOC State of Charge.T Transmission system in a vehicle drivetrain.topology Combination of drivetrain components or subsystems.UC, UltraCap Ultra Capacitor.V Vehicle wheels in a vehicle drivetrain.VRLA Valve Regulated Lead Acid battery.
42
Symbols
α [-] Objective function weights, with∑
αi = 1.€max [€] Maximum price of S.€ [€] Price of S.ηmin [-] Minimum efficiency of S.η [-] Mean efficiency of S over a power cycle (e.g. PS(t)).f [-] Objective function.g [-] Constraints to the objective function f .h [-] Equality constraints.Mmax [kg] Maximum mass of S.M [kg] Mass of S.ω [rad/s] Rotational speed.p [Pa] Pressure.PL [W] Power losses.PS(t), P (t) [W] Power profile, the S system must be able to deliver.SOC0 [-] SOC at time t0.r [-] Penalty and Barrier function penalty factor.t [s] Time.T [-]/[Nm] Objective function of S/ Torque.Vmax [m3] Maximum volume of S.V [m3] Volume of S.x [div] Vector of design variables of S.x0 [div] Initial guess for design variables of S.x∗ [div] Values of the variable design variables of S at the minimum.
43
Appendix A
Component modeling
A.1 Flywheel modelFor flywheels, principally two designs are available, a sub-critical and a super-critical flywheel [15].The two types differ in operating speed and construction:
• sub-critical: operated between zero and safely below first critical speed ωcr, no vacuumoperation, mechanical output. (low energy storage per kg)
• super-critical: operated above first critical speed and below maximum allowable rotationspeed due to rotor stresses, vacuum operation needed for minimal power loss, electricaloutput. (high energy storage per kg)
The first critical speed (for flywheels with inertia around the z-axis Jf much greater than theinertia around x-or y axis Jt) ωcr is
ωcr =√
Crad
mrotor, (A.1)
with Crad the total stiffness felt in radial direction in N/m.Since —generally speaking— power is not a limiting factor in flywheel design, a maximum al-lowable power model is not constructed in the modeling of the flywheel function (this wouldincorporate analysis of maximum spin-down time from maximum to minimum speed and theresulting stresses on each flywheel component). However, the drive train components attachedto the outgoing axis of the flywheel, will limit the power capacity of the flywheel storage system(e.g. CVT).
A.1.1 Super-critical �ywheelA reference super-critical flywheel model is schematically depicted in Figure A.1. In this refer-ence design, the flywheel is a cilinder with inner and outer radius, connected to the drive shaftby means of some light-weight, very stiff construction (the hub). It is assumed that only theflywheel cilinder has mass, inertia terms from the shaft-flywheel construction and the shaft areassumed negligible compared with the flywheel inertia. The housing is a cilinder, meant as a
44
safety measure as well as a vacuum chamber. The varying design variables x̃ are:
d Diameter of flywheel shaft and bearings [m],Ro Outer radius of flywheel [m],Ri Inner radius of flywheel [m],h Rotor thickness [m].
(A.2)
Other design variables x̃0 are fixed:
σmax Maximum stress of rotor material [N/m2] (Table A.1),ρFWrotor Rotor density [kg/m3] (Table A.1),ρFWhousing Housing density [kg/m3] (8000),ρair,5mbar Air density [kg/m3] (6.32·10−3),ηair Air dynamic viscosity [Pa · s] (17.1·10−6),ν Poisson’s Ratio [−] (fixed at 0.3),S Stress safety factor [−] (fixed at 1.2),t Circumference air gap [m],s Side planes air gap [m],Crad Bearing radial stiffness [N/m] (106 for supercritical flywheel),n Dynamic load to static load of the bearings ratio [−].
(A.3)
Power and energy model
FLYWHEEL
NEAR VACUUM
VACUUM SEAL R o
R i
h ���������������������d
ROTOR
BEARINGS ���������������sigma r
FLYWHEEL SIDE PROJECTION
omega
t
s
g
sigma theta ���������� omega
R o
R i
r
HUB
Figure A.1: Reference design for a super-critical �ywheel.
The super-critical flywheel is operated between two operating turning speeds:
ωcr < ω ≤ ωmax. (A.4)The energy contents of a flywheel is described by the equation:
E =12Jω2, (A.5)
45
with J = 12MFWrotor(R2
o + R2i ) and MFWrotor the mass of the flywheel rotor. Maximum and
minimum energy contents Emax are described by
Emax =12Jω2
max, (A.6)
Emin =12Jω2
cr (A.7)
and maximum speed ωmax of the flywheel is determined by the stresses occurring in the flywheelmaterial. From [17]:
σr(r) =3 + ν
8
(R2
i + R2o − r2 − R2
i R2o
r2
)ρFWrotorω
2, (A.8)
σθ(r) =3 + ν
8
(R2
i + R2o −
1 + 3ν
3 + νr2 +
R2i R
2o
r2
)ρFWrotorω
2, (A.9)
in which σθ has the highest value ([17] fig. 8.9). To determine the maximum stress on thematerial the Von Mises stress criterium is applied, specifically for this situation:
σmax = S ·max
(√σ2
θ + σ2r − σθσr
), (A.10)
for which σmax/S may not exceed the maximum stress of the flywheel material. Maximum stressoccur at r = Ri and the maximum flywheel turning speed is then:
ωmax =
√√√√ 8σmax
ρFWrotor (3 + ν)(R2
i + 2R2o − 1+3ν
3+ν R2i
) . (A.11)
The flywheel State of Charge can then be described by:
SOCFWsuprcr =E −Emin
Emax −Emin=
12J(ω2 − ω2
cr)12J(ω2
max − ω2cr)
=ω2 − ω2
cr
ω2max − ω2
cr
. (A.12)
Power losses modelPower losses for a super-critical flywheel system are composed from different subsystems (seeappendix):
Ploss,FW = Pvacuumpump + Psealfriction + Pbearings +Psideplanefriction + Pcircumferenceairdrag. (A.13)
The vacuum system consists of a pump, to lower the pressure in the flywheel housing gaining inless friction losses but with a penalty towards power consumption of the pump. There exists anoptimum in lower friction losses due to a lower pressure versus a higher power consumptionof the pump itself, for further reading on this subject see [1]. For modeling purposes, a powerconsumption of 100 W and a housing pressure of 5 mbar will be considered as was achieved forthe design of a flywheel system in [18]:
Pvacuumpump = 100. (A.14)
46
Material σa ρf Emax,stress/mf Emax,stress/V[106N/m2] [kg/m3] [Wh/kg] [Wh/dm3]
Ferro 360 120 7800 1.3 10.1Aluminium 173 2800 5.2 14.6Wrought steel 550 7800 5.9 46.3Maraging steel 765 8000 8 64.4E-glass 220 2000 9.3 18.5Titanium 662 4500 12.4 55.7High modulus aramid 700 1400 42 58.9High strength carbon 1600 1550 86.9 134.7
Table A.1: Maximum speci�c energy and energy density for di�erent materials according tomaximum allowable stress of the rotor (107 cycles for disc �ywheel).
The sealings used on the outgoing axle also contribute to the friction losses of the flywheel system.A linear relationship to the axle speed and friction losses can be assumed according to: [18]
Tseal = 1.71d,
Pseal = Tsealω. (A.15)
Bearing friction can be modeled as [19]:
Tbearings = 0.5µFdynd (A.16)
with
Fdyn = Fradial + Faxial = 0 + MFWrotorg = gρFWrotorhπ(R2o −R2
i ) (A.17)
and µ the coefficient for the bearing type, for a simple groove bearing this is 0.0015[−]. Assuminga linear relationship with the flywheel speed ω:
Pbearings = Tbearingsω = 0.5ωµFdynd. (A.18)
Side plane and circumference friction torques can be defined as:
Tcirc = Ccirc12πρhR4
oω2, (A.19)
Tside = Cside12ρR5
o.ω2 (A.20)
The torque coefficients Ccirc and Cside depend on the Reynolds number for the type of vortexdevelopment:
Rec =vθt
ν=
ωR0tρ
η, (A.21)
Res =ωR2
o
ν, (A.22)
Rec,crit =[ωRot
ν
]
crit
≈ 41.2
√√√√1 + t2Ro
tRo
. (A.23)
47
Regime k Circumference torque coe�cient Application range 1
I Ccirc = 3.6 ·(
tRo
)−0.25· (1+ t
R)2
2+ tRo
·Re−1c Rec < Rec,crit
II Ccirc = 10const1·log(Rec)+const2 2 Rec,crit < Rec < 400
III Ccirc = 0.920 ·(
tRo
(1 + t
Ro
))0.25·Re−0.5
c max(Rec,crit, 400) < Rec < 104
IV Ccirc = 0.146 ·(
tRo
(1 + t
Ro
))0.25·Re−0.3
c Rec > 104
Table A.2: Circumference torque coe�cients. 1: as the friction models are based on theassumption of incompressible �uid �ow, the application range is valid for subsonic �ows only.2: const1 and const2 are empirical values, for details see [18] appendix A.3.
m Side plane torque coe�cient Application range Res,m→m+1
I Cside = 2π ·(
sRo
)−1·Re−1
s 0 < Res < Res,I→II Res,I→II =
2.9 · ( sR
)− 115
II Cside = 3.7 ·(
sRo
) 110 ·Re
− 12
s Res,I→II < Res < Res,II→III Res,II→III =
4.6 ·106 · ( sR
) 1615
III Cside = 8.0 · 10−2 ·(
sRo
)− 16 ·Re
− 14
s Res,II→III < Res < Res,III→IV Res,III→IV =7.8 · 10−3 ·(
sR
)− 163
IV Cside = 1.0 · 10−2 ·(
sRo
) 110 ·Re
− 15
s Res > Res,III→IV
Table A.3: Side planes torque coe�cients.
The circumference torque coefficients can be described as in Table A.2, the side plane frictioncoefficients as in Table A.3.
In practice it suffices to take the maximum friction coefficient instead of determining the flowregime first, e.g.
Cmax = max {CI , CII , CIII , CIV } .
Again, having a linear relationship between torque and power:
Pcirc =12CcircπρhR4
oω3, (A.24)
Pside =12CsideρR5
oω3. (A.25)
Volume modelThe volume of the rotor can be determined by:
VFWrotor = πh(R2o −R2
i ). (A.26)
The volume of the flywheel is determined by the volume of the housing and of the pump:
VFW = VFWhousing + VFWpump. (A.27)
48
The volume of the housing is set to be twice the rotor size and the pump volume is set to be equalto half the housing volume so:
VFW = 3πR20h. (A.28)
Mass modelThe mass of the flywheel system is composed of several components:
MFW = MFWrotor + MFWhousing + MFWpump + MFWperipherals. (A.29)
The mass of the rotor can be determined by the volume of the rotor times its specific massρFWrotor:
MFWrotor = VFWrotor · ρFWrotor = hπ(R2
o −R2i
) · ρFWrotor. (A.30)
The mass of the housing is expressed as:
MFWhousing = VFWhousing · ρFWhousing = VFWrotor · ρFWhousing. (A.31)
The mass of the pump (vacuum pump+ driving motor) is set fixed to 25 kg, which seems a rea-sonable estimate based on existing vacuum pump technologies, MFWpump = 25. The mass ofthe flywheel peripherals (hoses, mountings, etc.) is set to 0 kg, as this is expected to be negligiblecompared with the other components.
Price modelThe supercritical flywheel can be set to 25000 €/kWh, from [20].
Optimization constraintsConstraints to the super-critical flywheel system are summarized:
−ω − ωcr ≤ 0 Equivalent to SOC ≥ 0,ω − ωmax ≤ 0 Equivalent to SOC ≤ 1.
(A.32)
A.1.2 Sub-critical �ywheel
A reference design for the sub-critial flywheel is depicted in Figure A.2, compared with the super-critical flywheel the vacuum seals are not present and the vacuum pump is not needed. Thevarying design variables x̃ are:
d Diameter of flywheel shaft and bearings [m],Ro Outer radius of flywheel [m],Ri Inner radius of flywheel [m],h Rotor thickness [m].
(A.33)
49
Other design variables x̃0 are fixed:
σmax Maximum stress of rotor material [N/m2] (Table A.1),ρFWrotor Rotor density [kg/m3] (Table A.1),ρFWhousing Housing density [kg/m3] (8000),ρair Air density [kg/m3] (1.29)),ηair Air dynamic viscosity [Pa · s] (17.1·10−6)),ν Poisson’s Ratio [−] (fixed at 0.3),S Stress safety factor [−] (fixed at 1.2),t Circumference air gap [m],s Side planes air gap [m],Crad Bearing radial stiffness [N/m] (108 for subcritical flywheel),n Dynamic load to static load of the bearings ratio [−].
(A.34)
R o
R i
h ����������������������������d
ROTOR
BEARINGS ���������������sigma r
FLYWHEEL SIDE PROJECTION
omega
g
sigma theta ���������� omega
R o
R i
r
SAFETY HOUSING
Figure A.2: Reference design for a sub-critical �ywheel.
Power and energy modelThe maximum speed of the sub-critical flywheel can be limited by either the maximum allow-able rotor material stress ( (A.1.1) and following) or by the first maximum allowable rotor speedaccording to
ωmax = ωcr
√n− 1
n, (A.35)
with n the maximum dynamic load to static load of the bearings ratio. The State of Chargeequation then reduces to:
SOCFWsubcr =E
Emax=
12Jω2
12Jω2
max
=nω2
(n− 1)ω2cr
. (A.36)
50
In [15] it is concluded that the rotor material maximum stress is not the limiting factor, for a sub-critically designed flywheel (nonetheless, the stress equations will be embedded in the sub-criticalflywheel model). Maraging steel is therefore chosen to be the material the optimized flywheel isconstructed of, because of relatively high energy density.
Power losses modelThe power losses of the sub-critical flywheel are:
Ploss,FW = Pbearings + Psideplanefriction + Pcircumferenceairdrag, (A.37)
so vacuum related power losses are emitted, compared with the super-critical flywheel.
Volume modelThe volume of the housing is set to be twice the rotor size so:
VFW = 2 · VFWrotor. (A.38)
Mass modelThe mass of the flywheel system is composed of several components:
MFW = MFWrotor + MFWhousing, (A.39)
resulting in the total mass:
MFW = VFWhousing(ρFWhousing + ρFWrotor). (A.40)
The mass of the flywheel peripherals (hoses, mountings, etc.) is set to 0 kg, as this is expected tobe negligible compared with the other components.
Price modelThe subcritical flywheel price can be set to 300 €/kWh, from [20].
Optimization constraintsConstraints to the sub-critical flywheel system are summarized:
−ω − ωcr ≤ 0 Equivalent to SOC ≥ 0,ω − ωmax ≤ 0 Equivalent to SOC ≤ 1.
(A.41)
51
A.2 Battery ModelDifferent types of batteries are currently under investigation for application in EV and HEV, TableA.4 lists the types that are viable to be used or are being used for (H)EVs. A brief description of
Speci�c Energy Energy Density Speci�c Power Cycle Life Projected cost(Wh/kg)a (Wh/L)a (W/kg)b (Cycles) (US$/kWh)
VRLA 30-50 60-100 200-400 400-600 120-150Ni-Fe 30-55 60-110 25-110 1200-4000 NANi-Zn 60-65 120-130 150-300 100-300 NANi-Cd 40-50 80-100 150-350 800-2000 300-350Ni-MH 50-70 100-140 150-300 800-2000 150-200Zn/Cl2 65 90 60 200 NAZn/Br2 65-75 60-70 90-110 300 150Fe/Air 75 100 60 300-600 NAAl/Air 190 190 16 NAc NAZn/Air 230 269 105 NAc 100Na/S 100 150 120 800 250-500
Na/NiCl2 86 149 150 1000 160-300Li-Al/FeS 130 220 240 1000 NALi-Al/FeS2 180 350 400 1000 NA
Li-Po 155 220 315 600 125Li-Ion 120-140 240-280 200-300 1200 150-180
USA BC 200 300 400 1000 100
Table A.4: Battery types and their characteristics [21].a At 80% depth-of-dischargeb At 3-h discharge ratec Mechanical recharge
these battery types now follows.
Lead-Acid battery (VRLA). The lead acid battery is composed of lead plates of grids suspendedin an electrolyte solution of sulphuric acid and water. These batteries can be ruined by com-pletely discharging them, thus it is necessary to set a lower charge limit. The advanced Lead-Acidbattery has a longer cycle life, an example is the Valve-Regulated Lead Acid (VRLA) battery.
Nickel-Cadmium. The nickel-cadmium battery is composed of a nickel hydroxide cathode anda cadmium anode in an alkaline electrolyte solution. Nickel-Cadmium batteries suffer from theso-called memory-effect, meaning that the batteries will not charge to full capacity when they arenot discharged completely first. Batteries suffering from this effect can be repaired by a few dis-charge/ charge cycles. Ni-Cd batteries are typically used to power small appliances such as mobiletelephones. Batteries made from Ni-Cd cells offer high currents at relatively constant voltage andthey are tolerant to physical abuse.
Nickel-Metal Hydride. Environmentally friendly, the Ni-MH battery is composed of a hydro-gen storage metal alloy, a nickel oxide cathode and a potassium hydroxide electrolyte. Quicklyrechargeable, used up till recently in mobile phones, laptops and power flashlights. This type of
52
battery has been used in the Honda Civic IMA and the Toyota Prius.
Lithium-Ion. Lithium is the lightest metal having the highest electric potential of all metals, mak-ing it ideal for HEV applications. Lithium is however also an unstable metal so batteries basedon lithium must be made using lithium ions (e.g. lithium-thionyl chloride). Many of the compo-nents of the battery and its casing are destroyed by lithium ions and in contact with water, lithiumwill react to create the explosive hydrogen. If the lithium melts (around 180 ◦C) it may come indirect contact with the cathode causing violent chemical reactions. Lithium batteries are appliedmostly in small devices such as mobile phones and laptops. Li-ion batteries have no memoryeffect and are environmentally friendly. Recent research shows that a lithium pack based on 36VL12P Saft cells can achieve, under laboratory conditions, over 200.000 charging cycles (at 25Wh cycling profile and intelligent charging system), over 15 years of expected calendar life and aspecific power of around 1500 W/kg [22].
Generally, modeling of a battery is a non-trivial task since the behavior of batteries depends ona large quantity of variables in a nonlinear fashion. For this reason, a lot of battery models havebeen developed in the past, ranging from very basic and simple models, neural network basedmodels to empirical models to highly complex models based on fundamental electrochemicalprocesses. For this research, a fairly simple battery model will be chosen based on varying inter-nal resistances, this type of model is also used in the ADVISOR software package. Advantages ofusing this type of model is simplicity in the model description and availability of measurementdata for different battery types. Disadvantages are no aging modeling, no heat influence on per-formance, no change of capacity with different discharge profiles (e.g. Peukert relation).
Only the Nickel-Metal Hydride (NiMH) and Valve-Regulated Lead-Acid (VRLA) batteries are mod-eled and analyzed further.
Power and energy modelFor simulation purposes, a battery could be modeled as in Figure A.3. A battery is composedof modules that are themselves composed of cells. Battery modules are approximately 12V for allbattery types.Design variables are:
n Number of battery modules [−],Qmax Module capacity [Ah], the discharge current depleting the battery in 1 hrs.
(A.42)
The battery module has two internal resistances, being the charging resistance (Rc) representingthe energy losses while charging and the discharging resistance (Rd) representing the energylosses while discharging, I represents the current and Vcell the module voltage. Battery voltage Vand battery capacity Q are, assuming equal battery modules:
Vbatt = n · Vmodule, (A.43)Qbatt = n ·Qmodule, (A.44)
Rc and Rd can depend on the battery SOC, temperature (T ) and capacity (Ah), for example.Energy losses because of the internal resistances account for a battery power loss
Ploss = nI2R. (A.45)
53
R d
R c
+ -
+
-
V module
...
V batt Module n
Module 4
Module 3
Module 2
Module 1
I
V int
Figure A.3: Reference design for a battery storage.
This power loss always has negative results on the overall efficiency of the battery. Also, to min-imize power losses generally, the current I should be as low as possible, the voltage Vbatt ispreferably high (thus the need for a high voltage electric circuit between the battery and motor).The net power output of the battery:
Pbatt = nI(Vint − IR). (A.46)
To determine the State of Charge of a battery, consider a completely discharged battery. WithI+(t) a charging current, the charge delivered to the battery is
∫ tt0
I(τ)dτ . With Qmax =∫∞t0
I+(τ)dτthe total charge the battery can hold, the SOC of the battery is:
SOC(t) =
∫ tt0
I(τ)dτ
Qmax× 100. (A.47)
Typically, the SOC of a battery is held strictly between certain operation limits (SOC≈ 0.4 − 0.6)to optimize the performance, efficiency and lifespan of the battery. The Voc can be approximatedlinearly based on the SOC:
Voc(t) = a1SOC(t) + a0, (A.48)
SOC(t) =Voc(t)− a0
a1. (A.49)
Estimating the SOC of a battery under working conditions is not easy, applying the pure integra-tion is subject to biases. Another approach is to measure the voltage of the battery under opencircuit conditions, however this condition never occurs during driving conditions. In practice,the SOC can be estimated by algorithms, that take into account the current power demands onthe battery and the measured voltage. Empirically determined look-up tables can also be used.For the four different battery types, empirical parameter fits have been made to describe the elec-tric dynamic behavior, e.g. R(SOC, Ah, T, charge/discharge) and Voc(SOC)
The empirical battery models are based on data from the software package ADVISOR, on theinternal resistances models. For the VRLA, the NiCad and the NiMH battery types, the ADVISOR
54
data does not depend on temperature, for the LiIon battery it does. Since the temperature ofthe battery modules heavily depends on the actual design of the modules (e.g. forced cooling,location of the battery) it is assumed the temperature is fixed at 25◦C. Further, ADVISOR datais incomplete such that it lacks measurement data for Rc. In Table A.5 the ADVISOR batterymodels are depicted, unfortunately for the LiIon and the NiCd case only one data set is availablethus a reliable empirical model with a varying battery capacity could not be generated from thesedata sets. To be able to include the LiIon and NiCd data, it is assumed that the internal resistancesvary linearly with Capacity, in the same way as in the VRLA case.
Capacity [Ah] Temperature Rc Rd ADVISOR nameVRLA 12 No Yes Yes ESS_PB12
16 No Yes Yes ESS_PB1618 No Yes Yes ESS_PB1865 No Yes Yes ESS_PB65_FocusEV85 No No Yes ESS_PB8591 No No Yes ESS_PB91104 No No Yes ESS_PB104
NiMH 6 No Yes Yes ESS_NIMH628 No No Yes ESS_NIMH28_OVONIC45 No No Yes ESS_NIMH45_OVONIC60 No No Yes ESS_NIMH60_OVONIC80 No No Yes ESS_NIMH80_EV1_draft90 No No Yes ESS_NIMH90_OVONIC93 No Yes Yes ESS_NIMH93
NiCd 102 No Yes Yes ESS_NICAD102LiIon 7 Yes Yes Yes ESS_LI7_temp
Table A.5: ADVISOR internal resistance battery models used
55
The modeling procedure for the batteries can be best illustrated for the VRLA battery. In Fig-ure A.4 the internal resistances for varying capacity are plotted, both for the charging and thedischarging case. When fitting a plane through these data points, the resulting fit function isan approximation for the internal resistance based on Capacity and State of Charge for VRLAbatteries. In fact, this plane is assumed to be an approximation to a large set of VRLA batterieswith different design geometries, thus the predictive accuracy for a specific design of battery isexpected to be very low. However, the fit function is assumed to be indicative for the performanceof VRLA batteries in general and can be used to predict the behavior roughly.
For the open-circuit voltage Voc, Mass M and Volume V the same trick is applied, based on
0
0.5
10
50
100
0
0.05
0.1
Cap [Ah]
Rdfit resistance
SOC
R
0
0.5
10
50
100
0.05
0.1
0.15
0.2
0.25
Cap [Ah]
Rcfit resistance
SOC
R
Figure A.4: Resistances �t for VRLA battery. Error least squares �t, third order polynomialin the SOC direction and �rst order polynomial in the Capacity direction. Solid: �tted plane,white: data from ADVISOR
ADVISOR data (Figure A.5). The same can be done for the NiMH battery, Figure A.6 and A.7.The resulting function fits are thus approximations for the battery characteristics, with varying
capacity. For the NiCd and Li-Ion batteries, too little data sets (1 each) are available to be able tobuild an empirical battery model with varying capacity.
Power capabilities of the batteries is set according to Table A.4, meaning 225 W/kg for the NiMHbattery and 300 W/kg for the VRLA battery.
Volume modelThe Volume of the battery depends also on the number of modules used according to:
Vbatt = n · Vmodule. (A.50)
Since Vmodule depends on the module capacity, the battery is also dependant on the capacity ofthe modules used.
56
0 50 100 1500
5
10
15
20
25
30Module Mass
Module capacity [Ah]
Mod
ule
Mas
s [k
g]
0 50 100 1500
1
2
3
4
5x 10
−3 Module Volume
Module capacity [Ah]
Mod
ule
Vol
ume
[m3 ]
0 0.2 0.4 0.6 0.8 111
11.5
12
12.5
13
13.5Voc
SOC
Voc
[V]
Figure A.5: Mass, Volume and Voc (Open Circuit Voltage) �t for VRLA battery, all linear.Dots: data points from ADVISOR, line: linear least squares �t used as empirical approxima-tion
Mass modelThe Mass of the battery depends on the number of modules used according to:
Mbatt = n ·Mmodule. (A.51)Price modelModeling of the price of different technologies can be related to power sizing of the drivetraincomponents, according to [23] and Table A.4. The following battery prices can be generated TableA.6.
Technology Price [23] Price1 [21] Price UnitLead Acid 100 90-120 100 €/kWhNiMH 350 115-150 240 €/kWh
Table A.6: Battery types and their prices.1at a rate 1€=1.3US$
Optimization constraintsConstraints to the battery system:
−SOC ≤ 0 Equivalent to SOC ≥ 0,SOC − 1 ≤ 0 Equivalent to SOC ≤ 1,max|PS | − Pmax,batt ≤ 0 Maximum power throughput limited.
(A.52)
57
0
0.5
1 2040
6080
100
0
0.02
0.04
0.06
Cap [Ah]
Rdfit resistance
SOC
R
0
0.5
120
4060
80100
0.015
0.02
0.025
0.03
0.035
0.04
Cap [Ah]
Rcfit resistance
SOC
R
Figure A.6: Resistances �t for NiMH battery. Error least squares �t, third order polynomialin the SOC direction and �rst order polynomial in the Capacity direction. Solid: �tted plane,white: data from ADVISOR
A.3 Ultracapacitor ModelA super- or ultracapacitor behaves like a high-power, low- capacity battery. Where batteries storeenergy chemically, capacitors store energy by accumulating and separating opposite charges phys-ically. Ultra capacitors (Ultracaps) have a very long lifecycle, compared with batteries, and full ca-pacity can be utilized without negative impact on ultracap performance. Specific energy howeveris significantly lower as opposed to batteries, in the order of 5 Wh/kg. Further, output voltage isproportional with the SOC thus, for a high voltage power line as preferred in hybrid vehicles, theultracap can only be used in the higher SOC range or voltage control is needed to compensate thewidely varying voltage. Power density of ultracaps is in the order of 2 kW/kg.
Power and energy modelModeling an Ultracap is very similar to modeling a battery. Ultracap packs can be built from cells,connected in parallel and/or in series, each cell generally is built with 2.5 V output at maximumcharge. A first-order approximation model (RC circuit) is depicted in Figure A.8.From [24] it is clear that a higher-order capacitor model is unwanted for vehicle simulation whenusing a one second time step, because of possible higher-order oscillatory responses to a stepwisechange of the input. E.g. when using a time step of 0.1 seconds, a second order model is pre-ferred for dynamic accuracy and prevention of oscillations in the response.Design variables are:
NS Number of series connected cells [−],NP Number of parallel connected cells [−],C Cell capacitance [F ],Vmax Cell maximum Voltage (generally 2.5V) [V ].
(A.53)
58
0 20 40 60 80 1000
5
10
15
20Module Mass
Module capacity [Ah]
Mod
ule
Mas
s [k
g]
0 20 40 60 80 1000
0.5
1
1.5
2x 10
−3 Module Volume
Module capacity [Ah]
Mod
ule
Vol
ume
[m3 ]
0 0.2 0.4 0.6 0.8 112
12.5
13
13.5
14
14.5
15
15.5Voc
SOC
Voc
[V]
Figure A.7: Mass, Volume and Voc (Open Circuit Voltage) �t for NiMH battery, all linear.Dots: data points from ADVISOR, line: linear least squares �t used as empirical approxima-tion
R cell
+ -
+
-
V cell V UC
I
V int
Cell
N p
N S
Figure A.8: Reference design for an Ultracap pack
The internal resistance of the Ultracap package can be calculated from the number of parallel(NP ) and series (NS) connected cells:
RUC =NS
NP·Rcell, (A.54)
59
and for the capacitance the same can be done:
CUC =NP
NS·Rcell. (A.55)
The pack voltage is:
VUC = NS · Vcell. (A.56)
Current can be calculated from:
I = C
(dV (t)
dt−RUC
dI(t)dt
). (A.57)
Cell voltage is linear with the charge q:
Vcell,OC = VR + VC = IRcell +q
C, (A.58)
with q = q0 +∫
Idt. And since SOC is linear with cell voltage:
SOC(t) =Vcell,OC
Vmax, (A.59)
for which Vmax is usually 2.5 V .Energy contents of the pack:
W =12CUCV 2
UC =12qVUC . (A.60)
Efficiency of the Ultracap pack is not solely dependent on the internal Resistance, it also dependson how well its time constant matches the expected pulse duration [16]. With a constant currentI the efficiency ηc for charging and ηd for discharging:
ηc =1
1 + 2 IRVmax
, (A.61)
ηd = 1− 2IR
Vmax. (A.62)
The Ultracap model is, similar to the battery models, an empirical model based on manufac-turer’s data from Maxwell Technologies, available from their website.For the Ultracapacitor empirical model, use of manufacturers data of Maxwell Technologies hasbeen used, see Table A.7. The resulting linear function fits that describe the empirical model areplotted in Figure A.9. From Figure A.10 the dependency of the efficiency, resulting from (A.61)and (A.62), is illustrated.
Volume modelVolume is the number of Ultracap cells times the number of cells, based on manufacturer’s data:
V ol = NS ·NP · V olcell. (A.63)
Mass modelMass is determined analogous to the volume:
m = NS ·NP ·mcell. (A.64)
60
Name Capacitance [F] Max. capacity [kJ] VoltageBCAP0350 350 1.0 2.5BCAP0013 450 1.4 2.5BCAP0008 1800 5.6 2.5
BPAK0350-15E 58 6.5 15BCAP0010 2600 8.1 2.5PC2500 2700 8.4 2.5
Table A.7: Used data for Ultracapitor devices for empirical model
0 2 4 6 8 100.5
1
1.5
2
2.5
3
3.5x 10
−3
Ultracap size [kJ]
Res
ista
nce
[ohm
]
Cell internal resistance fit
R dataR fit
0 2 4 6 8 100
0.2
0.4
0.6
0.8
Ultracap size [kJ]
Mas
s [k
g]
Cell mass fit
M dataM fit
0 2 4 6 8 100
2
4
6x 10
−4
Ultracap size [kJ]
Mas
s [k
g]
Cell volume fit
Vol dataVol fit
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
SOC [−]
OC
Vol
tage
(fo
r i=
0) [V
]
Open circuit cell voltage for i=0
Figure A.9: Mass, Volume, Voc (Open Circuit Voltage) and internal resistance �t for UC. Voc
is linear with SOC. Dots: data points from ADVISOR, line: linear least squares �t used asempirical approximation
Price modelPrice for the ultracapacitor is set to €30 /kJ , the price for the Maxwell 1 kJ BCAP0350. The8.4 kJ BCAP2500 costs €270, this is the same price level (€32). Price is assumed linear withstorage capacity.
Optimization constraintsConstraints to the ultracapacitor system:
−SOC ≤ 0 Equivalent to SOC ≥ 0,SOC − 1 ≤ 0 Equivalent to SOC ≤ 1.
(A.65)
61
0 10 20 30 40 50 60 70 80 90 1000.75
0.8
0.85
0.9
0.95
1
Current [A]
disc
harg
e ef
ficie
ncy
[−]
1 kJ device 3 kJ device
9 kJ device
Figure A.10: Ultracap e�ciencies as a function of cell capacity
A.4 Pressurized gas storage modelThe pressurized gas storage energy storage takes place by pressurizing gas in a closed gas-tank,several variants to this idea can be generated. For example, filtered air can be compressed usinga gas compressor (Compressed Air Energy Storage, [25]) or N2 (nitrogen gas) can be compressedusing an oil pump and oil (Compressed Air Energy Storage (CAES), [26]). The latter will beinvestigated here, the storage device is modeled as a cylindrical shaped tank as depicted in FigureA.11. The volume of the gas changes and when the gas is pressurized, energy is stored. Powerto and from the drivetrain can be controlled using a valve, capable of controlling oil flow. Designvariables are:
R Radius of gas tank [m],t Thickness of gas tank [m],h Height of gas tank [m],p0 Initial gas pressure [Pa].
(A.66)
And for the non-varying parameters:
cp Specific heat at constant pressure [J/(kg ·K)] (29.1006 for N2),cv Specific heat at constant volume [J/(kg ·K)] (20.7865 for N2),σmax Maximum stress of tank material [N/m2] (340·106 for steel),ρtank Tank material density [kg/m3] (8000 for steel),ρoil Oil density [kg/m3] (950 for oil),S Stress safety factor [−] (1.3),ppump,max Maximum pressure in tank because of pump capacity [Pa] (pump specific),n Gas behavior constant (Adiabatic/ Isothermal) [−] (1.4 for N2).
(A.67)
The pump is described in more detail in Section A.7, it is assumed the pump can generate a flowQ from the input ω, T , so Q = Q(ω, T ). Inertia effects of the oil entering and leaving the tank are
62
������������������OIL RESERVOIR
GAS TANK
VOLUME V PRESSURE P
OIL
PUMP
Q
omega, T
R
h
t
r
VOLUMETRIC VALVE
charge discharge
Figure A.11: Reference design for compressed gas energy storage.
neglected. The gas tank without pump is assumed to be 100 % efficient, meaning no pressurelosses over time and a quasi static process.
Power and energy modelFor a quasi static process the work can be formulated as [27]:
W =∫
pdV . (A.68)
For a polytropic process:
pV n = Constant, (A.69)
for which
n = cp/cv = γ For adiabatic processes,n = 1 For isothermal processes.
(A.70)
In practice, gas compression or expansion is never either adiabatic nor isothermal, processes canbe fairly well approximated by a polytropic process with 1 < n < γ. Based on the design of thegas tank (a well isolated system design results in nearly adiabatic behavior), n can be chosen. Nowfor the work of a polytropic process:
W1→2 =∫ 2
1pdV =
1n− 1
(p1V1 − p2V2). (A.71)
The heat interaction with the surroundings can be calculated from the first law of Thermodynam-ics:
Q = ∆U + W = mcv
[n− γ
n− 1
](T2 − T1). (A.72)
63
For a purely Adiabatic process, Q = 0 (tank has thermal isolation) and a theoretical maximumenergy contents of the gas tank storage can be determined:
Emax = Umax = −Wmax = − 1γ − 1
(p0V0 − pmaxVmin). (A.73)
For a purely isothermal process, work would be zero and energy storage is not possible. Maxi-mum allowable tank pressure can be calculated from the stresses in the tank material. The Stateof Charge for the Adiabatic case of the gas storage system may be expressed as:
SOC =E
Emax=
pV − pmaxVmin
p0V0 − pmaxVmin. (A.74)
From the last equation and from (A.69) the pressure and volume for a given SOC can be calcu-lated:
V =(
p0V0(1− SOC) + pmaxVmin · SOC
p0V n0
) 11−n
, (A.75)
p =p0V
n0
V n, (A.76)
with p0 and V0 the pressure and volume at no pressure conditions in the tank (so p0 = patm).Maximum power Pmax of the tank principally depends on the size of the tubings used, howeverit is assumed that the pump attached to these tubings is the limiting factor in the determinationof Pmax. For the tank material stresses the following applies [17]:
tt+R > 0.1 Thick-walled stress theorie,
tt+R < 0.1 Thin-walled stress theorie.
(A.77)
For a thin-walled cilinder subject to internal pressure the tangential stress may be regarded aconstant with thickness and reads:
σθ =pR
t, (A.78)
σr = −p. (A.79)
For thick-walled cilinders the stress equations are:
σθ =a2p
b2 − a2
(1 +
b2
r2
), (A.80)
σr =a2p
b2 − a2
(1− b2
r2
), (A.81)
with a = R and b = R + t. Maximum stresses occur at r = a [17], the Von Mises stress criterium:
σmax = S ·√
σ2θ,r=a + σ2
r,r=a − σθ,r=aσr,r=a, (A.82)
σθ,r=a = pb2 + a2
b2 − a2, (A.83)
σr,r=a = −p. (A.84)
σmax/S may not exceed the maximum stress of the tank material (for steel ≈ 340MPa), nor thepressure pmax in the tank may exceed ppump,max, whichever comes first.
64
Volume modelThe volume of the gas storage system, excluding the pump, is expressed as:
VCAES = Vgastank + Voiltank. (A.85)
The volume of the gas tank:
Vgastank = π · (R + t)2 · (h + 2t). (A.86)
The volume of the oil tank depends on the amount of oil needed to achieve maximum pressurepmax and is thus a function of the smallest volume Vmin. The tank volume is set equal to 1.1 timesthe oil volume to compose the (non-pressurized) oil tank volume:
Voiltank = 1.1 · Voil = 1.1 · (π ·R2 · h− Vmin). (A.87)
Mass modelMass of the CAES system without pump:
MCAES = Mgastank + Moil,
= πρtank((R + t)2h−R2(h− 2t)) + ρoil(πR2h− Vmin), (A.88)
where it is assumed that the mass of the gas and oil tank material are negligible.
Price modelHydraulics and pneumatics are very mature technologies with non-exotic materials and produc-tion methods. It is assumed that the price of the technology is on par with the rule-of-thumb steelprice of €10/kg.
Optimization constraintsConstraints to the pressurized gas storage system:
−SOC ≤ 0 Equivalent to SOC ≥ 0,SOC − 1 ≤ 0 Equivalent to SOC ≤ 1.
(A.89)
65
A.5 Continuously Variable TransmissionA Continuously Variable Transmission (CVT) is a gear box with a continuously variable transmis-sion ratio:
r =ωin
ωout(A.90)
opposed to a manual gearbox or gear with one or more fixed transmission ratios (r1, r2, ..). Asa result, the shaft speed from the flywheel for example can be transformed into any given shaftspeed connected to the rest of the drivetrain. The CVT can further be used as a device that cancontrol the spinning up or down of the flywheel and thus as a device that can control a chargingor discharging state of the S flywheel system.The CVT model is a fixed model, not scalable in power capabilities and based on the ADVISORCVT TX_CVT50_SUBARU, this is done since the power capabilities of the CVT are sufficient forthe required function of the S system with a maximum power throughput capability of 50 kWand more static data is currently not available.
T fw J fw w fw
T cvt,fw
T cvt,ds J ds w ds
T ds
r=w fw /w ds
Figure A.12: CVT model
Equations describing the dynamic behavior of the CVT are:
ω̇fw =Tfw − Tcvt,fw
Jfw, (A.91)
ω̇ds =Tcvt,ds − Tds
Jds. (A.92)
One problem now arises when the CVT is to be used as a charge/discharge controller, that isthe dynamics of the CVT and thus the operating point of the CVT and flywheel system dependon the inertia terms Jfw (flywheel) and Jds (driveshaft and the rest of the drivetrain coupled toit, e.g. wheels and vehicle). For Jfw the inertia is simply the inertia of the flywheel plus inertiaof the primary cvt pulley, however the inertia terms of Jds consist of the cvt secondary pulleyinertia plus the rest of the drivetrain inertia terms. Since the drivetrain inertia is unknown, theoperating points of the CVT can not be determined through dynamic analysis. Instead, staticoperating points of the CVT are assumed for calculation of the efficiency of the CVT. For a givenωfw, the most efficient operating point (Optimal Operation Line, OOL) of the CVT is calculatedfrom a static efficiency map and the output speed can then be calculated, see Figure A.13. Outputpower of the CVT is known (Ps) and taking into account the loss terms of the CVT the outputpower of the flywheel can be determined:
Ps = Pfw − Pcvt,loss (A.93)
66
0 10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
OO
L E
ffici
ency
[−]
0 10 20 30 40 50 60 70 80 90 1000.5
1
1.5
2
2.5
Torque [Nm]
OO
L G
ear
ratio
r [−
]
0200400500520
Figure A.13: CVT OOL operation points for di�erent input speeds (ωfw))
and
Pfw = ωfwTfw. (A.94)
The efficiency maps are used from ADVISOR. From efficiency map analysis, the maximum inputspeed of the CVT is set to 471 rad/s since efficiency drops dramatically beyond this speed.Another problem is the throughput of power at lower speed ranges, see Figure A.14. This prop-erty puts extra constraints on the storage system attached to it.
Volume modelVolume of the CVT is not known, volume is assumed to be 0.036 m3 (40x30x30 cm guess).
Mass modelMass of the CVT is 57.6 kg.
Price modelPrice is assumed €576 based on a machine price of €10/kg.
Optimization constraintsConstraint to the CVT system:
max|PS | − Pmax,cvt ≤ 0 Maximum power throughput limited. (A.95)
67
0 50 100 150 200 250 300 350 400 450 5000
10
20
30
40
50
60
70
80
90
ωcvt
Tcv
t
5
5
5
5
10
10
10
10
15
15
15
20
20
20
25
25
30
30
35
35
40
45
Figure A.14: CVT isopower curves [kW ]. For the CVT to operate within its power range,the speed and/or torque range is limited, e.g. for a throughput of 10 kW , the speed must behigher then 110 rad/s (with a maximum of 471 rad/s) and the torque be higher then 19 Nm.
A.6 Electrical MotorsAn Electrical Motor (Motor) converts electricity into a rotating drive shaft and can also converta rotating drive shaft into electricity. Of electric motors, a lot of different types exist and all canbe more or less subdivided into two different types: AC and DC motors. The application of DCmotors is very common in vehicles, e.g. for driving fans and the starter motor, this is mainly dueto the 12V DC power supply present.Developments in cheaper high power semiconductors have led to the application of AC motorsin DC situations, essentially converting an AC motor into a DC one, e.g. a "brushless DC" isactually a synchronous machine coupled with a set of semiconductor switches controlled by rotorposition. The advantage is that a brushless motor operates without brushes, decreasing servicelevels drastically. AC motors can be broadly categorized into two classes; induction and syn-chronous. The Permanent Magnet (PM) motor is a special type of synchronous motor, where thefield excitation is provided by permanent magnets opposed to electromagnets in the conventionalsynchronous motor. The induction motor works by magnetic fields imposed (induced) by themoving rotor through the stator magnetic field. For variation of the turning speed of these ACmotors, the frequency has to be varied. This involves electronics to convert the DC power supply(battery) or AC power supply (flywheel) to a desired frequency.
A.6.1 Induction Motor/ GeneratorAn effort has been made to model the AC motor analytically. Main result is that analytical mod-eling all loss terms in the motor, can be done by using a finite element analysis of all power lossterms in the motor stator and rotor. However, for the purpose of sizing and scaling the motor for
68
optimal component design, this is beyond the scope of this particular research (not mentioningthe computational difficulties of iterating the sizing parameters and calculating performance ofthe motor for each optimization step). Instead, an empirical approximation of the motor proper-ties will be generated. This has two major advantages;
• The motor models are based on actual measurement data, validation is thus not necessary
• Short calculation times as the functions approximating the motor performance are rela-tively simple
In ADVISOR several AC motors are present, ranging in size from 25 kW power to 187 kW(continuous power). The motors will be operated in their OOL (optimal operating line) pointas is visualized in Figure A.15. Now, when plotting the OOL efficiency and OOL speed/torque
0 200 400 600 800−200
−150
−100
−50
0
50
100
150
200
ω [rad/s]
Tor
que
[Nm
]
0.7
0.70.7
0.7
0.7
0.7 0.70.7
0.8
0.8 0.8
0.80.8
0.8
0.85
0.85
0.85
0.85
0.850.85
0.9
0.9
0.90.9
−100 −50 0 50 1000
50
100
150
200
250
300
350
400
450
Power [kW]
ω [r
ad/s
]
−100 −50 0 50 100−200
−150
−100
−50
0
50
100
150
200
Tor
que
[Nm
]
OOL
Torque
speed
Figure A.15: ADVISOR AC59 motor/generator e�ciency map and OOL properties. Left:e�ciency map and OOL, right: speed and torque operating points corresponding to OOLoperation.
characteristics in one figure (A.16), one can find consistencies among the different AC motors:
• The motors have different speed ranges, but the usable range does not depend on the sizeof the motor.
• Efficiency, as well as speed and torque over the OOL have the same general shape for allsizes
Now, to develop an OOL AC motor model, the ADVISOR OOL models are all scaled to a 10 kWmotor through power/torque scaling:
Poriginal = α · P10kW ,
Toriginal = α · T10kW , (A.96)ωoriginal = ω10kW ,
69
−200 −150 −100 −50 0 50 100 150 2000.7
0.75
0.8
0.85
0.9
OOL Efficiency η
power [kW]
η [−
]
AC187AC150 Focus draftAC124 EV1 draftAC83AC75AC62AC59AC30AC25
−200 0 2000
500
1000
1500OOL ω
power [kW]
ω [r
ad/s
]
−200 0 200−300
−200
−100
0
100
200
300OOL T
power [kW]
T [N
m]
Figure A.16: OOL e�ciency and speed/torque characteristics for ADVISOR AC motors
η1 1.0208 ω1 1314.7η2 3.2742 ω2 0.00047938η3 0.77719 ω3 1.5248η4 0.013493 ω4 65.57
Table A.8: Values for (A.97) for the 10 kW scaled AC motor
with α the scaling factor. The speed operating range is unaltered.To describe the resulting efficiency and speed characteristic, a completely arbitrary efficiency fitfunction is proposed, composed of four parameters, as well as a speed fit function:
η10kW (P ) = η1 +1
(−η2 · (|P |+ η3))− η4 · |P |,
ω10kW (P ) = ω1 +1
(−ω2 · (|P |+ ω3))− ω4 · |P |. (A.97)
Through least square error minimization and use of the function fminunc, values for η1−4 andω1−4 can be found that describe the mean, approximated behavior of the scaled ADVISOR motormodels (Figure A.17). The function approximation values for the 10 kW motor are listed in TableA.8.Thus a description for a "mean", scalable AC motor (∼ 5−180 kW ), for OOL operation purposeshas been obtained. Efficiency does not differ more than 5 % for all original models with respectto the efficiency description of this scalable model.
Volume and Mass modelVolume and mass are interpolated between different sized motors: Unfortunately, no volume in-formation was available for the motors used to determine efficiency and mass, data from motors
70
−10 −5 0 5 100.7
0.75
0.8
0.85
0.9
scaled OOL Efficiency η
power [kW]
η [−
]
−10 0 100
500
1000
1500scaled OOL ω
power [kW]
ω [r
ad/s
]
−10 0 10
−20
−10
0
10
20
scaled OOL T
power [kW]
T [N
m]
Figure A.17: In thick black, approximations of the AC motors, scaled to 10 kW . In thick red,the mean behavior, of the scaled AC motors
0 50 100 150 20050
60
70
80
90
100
110Mass
Continuous power [kW]
Mas
s [k
g]
0 50 100 1500
0.005
0.01
0.015
0.02
0.025
0.03
0.035Volume
Continuous power [kW]
Vol
ume
[m3 ]
Figure A.18: Volume and Mass �ts for AC motor model, dots: discrete designs from ADVI-SOR, line: linear approximation of mass and volume with respect to size.
from Siemens, BRUSA an MES were used for this purpose [28].
Price modelPrice was set to 15 €/kW [23]. This is lower then proposed in [23] (18 €/kW is proposed),it is assumed that 18 €/kW has been assumed in the mentioned report for the more costlierPermanent Magnet motor devices.
71
η1 0.97494 ω1 1046.7η2 4.3141 ω2 0.00015293η3 0.51984 ω3 7.1738η4 0.004349 ω4 20.466
Table A.9: Values for (A.97) for the 10 kW scaled PM motor.
A.6.2 Permanent Magnet Motor/ GeneratorAnalogue to the AC motor, the PM motor will be modeled based on available ADVISOR models.The ADVISOR models vary in power range from 8 kW power to 100 kW (continuous power).Again, the motor model derived will be operated only in the OOL. The same conclusions can bedrawn from the OOL operation characteristics for the different PM motors (Figure A.19). The
−60 −40 −20 0 20 40 600.7
0.75
0.8
0.85
0.9
0.95
1OOL Efficiency η
power [kW]
η [−
]
PM100PM58PM49PM33PM32PM25PM16PM15PM8
−50 0 500
200
400
600
800OOL ω
power [kW]
ω [r
ad/s
]
−50 0 50−400
−200
0
200
400OOL T
power [kW]
T [N
m]
Figure A.19: OOL e�ciency and speed/torque characteristics for ADVISOR PM motors
ADVISOR OOL models are all scaled to a 10 kW motor through power/torque scaling accordingto (A.6.1). The fit function of (A.97) is again applied resulting in Figure A.20). The function ap-proximation values for the 10 kW motor are listed in Table A.9. Thus a description for a "mean",scalable AC motor (∼ 5 − 100 kW ), for OOL operation purposes has been obtained. Efficiencydoes not differ more than 5% for all original models with respect to the efficiency description ofthis scalable model.
Volume and Mass modelVolume and mass are interpolated between different sized motors (Figure A.21). Unfortunately,no volume information was available for the motors used to determine efficiency and mass, datafrom motors from UQM technologies and Lynx motion technologie were used for this purpose.
Price model
72
−10 −5 0 5 100.7
0.75
0.8
0.85
0.9
0.95
1OOL Efficiency η
power [kW]
η [−
]
−10 0 100
200
400
600
800OOL ω
power [kW]
ω [r
ad/s
]
−10 0 10−60
−40
−20
0
20
40
60OOL T
power [kW]
T [N
m]
Figure A.20: In thick black, approximations of the PM motors, scaled to 10 kW . In thick red,the mean behavior, of the scaled PM motors.
10 20 30 40 50 6010
20
30
40
50
60
70
80
90
100
110Mass
Continuous power [kW]
Mas
s [k
g]
0 20 40 60 800
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Volume
Continuous power [kW]
Vol
ume
[m3 ]
Figure A.21: Volume and Mass �ts for PM motor model, dots: discrete designs from ADVI-SOR, line: linear approximation of mass and volume w.r.t. size
Price was set to 18 €/kW [23].
Optimization constraintsNo constraints are set for the EM.
A.6.3 ConclusionFor PM and AC motors, operation models for the OOL operation have been derived, as expectedthe efficiency of the PM motors is slightly better than the efficiency of the AC motors and hasalso a wider high-efficiency speed region.
73
A.7 Hydraulic PumpThe hydraulic pump converts a rotating drive shaft to output oil flow. The according powers:
Pin = T · ω, (A.98)Pout = Q · p, (A.99)
with P the power, T input torque, ω input speed, Q output flow and p the pump pressure differ-ence. The pump can also act as a generator when the flow is reversed:
Pin = Q · p, (A.100)Pout = T · ω. (A.101)
Different pump types exist, for hydraulic purposes the axial piston pump, the radial piston pumpand the gear pump can be considered. Modeling of a pump consists of a mechanical model ofthe pump system and a dynamic flow model of the oil that is pumped, efficiency modeling thendescribes the power losses in these submodels.Instead of constructing these models which is considered too much detail for this thesis, empiri-cal models were derived based on manufacturer’s data.
A.7.1 Gear PumpA gear pump is constructed of two gears, several different designs exist (Figure A.22). Characteristic
Figure A.22: Di�erent gear pumps, left: internal gear pump, right: external gear pump (fromhttp://www.pumpschool.com).
properties of the gear pump construction are:
• simple and robust construction,
• small,
• self-suctioning,
• up till 200 bar pressure.
Two internal gear pump models have been derived from manufacturers data, the MannesmannRexroth G3 gear pump line with a maximum flow of 115 L/min (40 kW top power at 200 bar) andthe Mannesmann Rexroth G4 gear pump line with a maximum flow of 145 L/min (55 kW top
74
power at 200 bar). A one-pump solution is preferred in this particular design problem, opposedto pumps connected in series or parallel. Design variables for the internal gear pump design:
Vg Theoretical pump volume per turn [m3],G3/G4 Pump principal design series [−].
(A.102)
The pump volume equation can be defined as:
Q = ηV Vgω
2π= Q(p, Vg, ω), (A.103)
with ηV the volumetric efficiency of the pump (compensates for the internal volume leakage,depends on pressure difference over pump), Vg the theoretical (ideal) pumped volume per turn,ω the axis speed. Input torque is a function of the pressure difference:
T = T (Vg, p). (A.104)
When calculating the efficiencies of the pumps relative to the internal pump volume Vg accordingto (A.98) – (A.101), Figure A.23 can be generated.
As can be seen, the efficiency of these pumps is poor under low pressures, thus the need to
0 50 100 150 200 250 3000.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Pressure [bar]
Effi
cien
cy [−
]
Efficiency of MRG3
Vg=20 232629323845
0 50 100 150 200 2500.4
0.5
0.6
0.7
0.8
0.9
1
Pressure [bar]
Effi
cien
cy [−
]
Efficiency of MRG4
Vg=4050637080100
Figure A.23: E�ciency of the used internal gear pumps, left: G3 series (pump1), right: G4series (pump2).
operate within a defined State of Charge region of the pressurized gas tank is certainly needed(e.g. operate > 50 bar). For intermediate sizes between the discrete Vg values, interpolation willbe applied to predict the performance.
Volume modelVolume is interpolated between different Vg designs, from manufacturers data.
Mass modelMass is interpolated between different Vg designs, from manufacturers data.
75
Price modelPrice is assumed linear with mass based on a machine price of €10 /kg.
Optimization constraintsConstraints to the pump system:
max|p| − pmax,pump ≤ 0 Maximum pressure limited,pmin,pump −min|p| ≤ 0 Minimum pressure limited,ω − ωmax ≤ 0 Rotating speed of pump limited.
(A.105)
76
A.8 Power ElectronicsPower electronics can be used to convert one type of electrical energy into another, e.g. low-voltageDC to high-voltage AC current. Specific requirement to this element is high efficiency, this canbe achieved by specifically designing the electronics for a given problem, taking into account allloss terms e.g. Hysteresis and Eddy current losses. For the efficiency modeling of the powerelectronics, a fixed efficiency of 98 % is assumed.
Volume modelVolume of the Power electronics is not known, volume is assumed to be 0.001 m3 (10x10x10 cmguess).
Mass modelMass is assumed fixed at 10 kg.
Price modelPrice is assumed €100 based on a machine price of €10 /kg.
Optimization constraintsNo constraints.
77
Appendix B
Model Sensitivity
Sensitivity of the initial design of the models will be investigated in this chapter. For this purpose,the models will be evaluated based on the 50 sec power signal in Figure B.1
0 5 10 15 20 25 30 35 40 45 50−15
−10
−5
0
5
10
15
P [k
W]
time [s]
Figure B.1: Power signal Ps for sensitivity analysis analysis.
B.1 Sensitivity to x0
With fixed αn and Mmax, Vmax, €max. All models will be dealt with separately. The initial param-eters are stated:
αM 0.2αV 0.2α€ 0.2αν 0.4Mmax 150Vmax 1€max 2000
78
And r is set to 1/1e8, unless stated otherwise. Each optimization model is evaluated at threeinitial design points, being
1. the initial design used further in this report as a starting point. This starting point is theresult of numerous trial runs and the consequent results, aiming at the lowest value for Tfor the problem set as stated above.
2. the upper bound values for x, to test convergence and the existence of local minima.
3. the lower bound values for x, to test convergence and the existence of local minima.
It is noted that most optimizations will fail for the latter test, as the starting design is unfeasibleand the Penalty and Barrier function values are very high, thus iteration to a minimum is not tobe expected. It is also noted that the maximum number of iterations possible is usually set tobe 100, further iteration steps normally produce only marginally better systems but at the costof great calculation times, this value has also been determined experimentally. The number ofstarting points for this sensitivity experiment is low, and for a complete picture of the sensitivityof the optimized solutions it should be expanded. However, the experiment is merely intendedto investigate the globality of the found minima and have for these starting points already beenshown to be very poor, extra starting points would not contribute further to this insight.
79
S1_ac_opt.m Supercritical Flywheel with ac motor CVTSOC0 = 0.9;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
fw_suprcr_d_0 [m] 0.03 1 0.05 0.0333 1 0.0591 0.03 0.934fw_suprcr_Ro_0 [m] 0.001 1 0.3 0.172 1 0.0873 0.001 0.944fw_suprcr_Ri_0 [m] 0.001 1 0.1 0.131 1 0.0592 0.001 0.934fw_suprcr_h_0 [m] 0.001 1 0.05 0.00912 1 0.350 0.001 0.035motor_ac_kW_0 [kW] max|Ps| 200 max|Ps| 12.0 200 200 max|Ps| 15.1
T [-] 0.696 1.61 3.11Mass [kg] 181 235 574Volume [dm3] 34.5 81 181Price [€] 2165 8131 11981Efficiency [%] 57.7 19.9 -95.8
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500
0.5
1
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
2000
4000
6000
time [s]
Plo
ss [W
]
efficiency is 57.7%
The optimal designs all differ, indicating a solution space with multiple local minima. It seemscritical to choose a representative start design, that is close to the expected optimal design forthe problem at hand. Caution is required when interpreting results, the optimal design found isexpected to be not the globally optimal design.For the parameter fw_suprcr_d_0, the lower bound value is (almost) reached, this is due to thefact that the diameter of the axis is not restricted by stress equations on the axis. It is thus ex-pected to always reach the value of 0.03 m.The resulting SOC and power loss profile show that the flywheel has been dimensioned such that
80
SOC= 0 will be reached. The flywheel is thus dimensioned such that the energy capacity is justenough, to reduce size. Further, efficiency is highest at the lower SOC values, the lower the powerdemands the lower the power losses due to losses in the ac motors.
81
S1_pm_opt.m Supercritical Flywheel with PM motor CVTSOC0 = 0.9;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
fw_suprcr_d_0 [m] 0.03 1 0.05 0.0332 1 0.03 0.03 0.030fw_suprcr_Ro_0 [m] 0.001 1 0.3 0.173 1 0.04 0.001 0.532fw_suprcr_Ri_0 [m] 0.001 1 0.1 0.133 1 0.03 0.001 0.030fw_suprcr_h_0 [m] 0.001 1 0.05 0.00873 1 0.846 0.001 0.001
T [-] 0.493 1.41 0.778Mass [kg] 79.1 95.3 80.3Volume [dm3] 19.8 25.0 19.9Price [€] 2127 9275 3520Efficiency [%] 66.3 24.1 30.2
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500
0.5
1
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 501000
2000
3000
4000
time [s]
Plo
ss [W
]
efficiency is 66.3%
Again, all optima differ, further all observations made for S1_ac_opt.m are valid, however theefficiency is substantially higher due to the use of the PM motors, which have higher efficiencyover their working range.
82
S2_opt.m Subcritical Flywheel with mechanical CVTSOC0 = 0.9;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
fw_subcr_d_0 [m] 0.03 1 0.05 0.0329 1 0.981 0.03 infeasiblefw_subcr_Ro_0 [m] 0.001 1 0.3 0.0930 1 0.991 0.00101 infeasiblefw_subcr_Ri_0 [m] 0.001 1 0.1 0.0403 0.9 0.981 0.001 infeasiblefw_subcr_h_0 [m] 0.001 1 0.05 0.140 1 1 0.001 infeasible
T [-] 0.501 66.72 infeasibleMass [kg] 113 25254 infeasibleVolume [dm3] 43.6 6211 infeasiblePrice [€] 2020 164121 infeasibleEfficiency [%] 84.4 -954 infeasible
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500
0.5
1
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
1000
2000
3000
time [s]
Plo
ss [W
]
efficiency is 84.4%
Because of the mechanical CVT that can not pass through power at zero speed, a zero turningspeed is not reached. The optimizations starting at the outer limits of the design space, do notproduce a useful result.
83
S3_ac_opt.m NiMH battery with ac motorSOC0 = 0.5;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
batt_nimh_n [-] 1 1000 10 10.8 1000 57.9 1 infeasiblebatt_nimh_Qmax [Ah] 10 93 80 35.1 93 10.0 10 infeasiblemotor_ac_kW_0 [kW] max|Ps| 200 max|Ps| 12.7 200 200 max|Ps| infeasible
T [-] 0.522 0.999 infeasibleMass [kg] 144 177 infeasibleVolume [dm3] 25.1 56.7 infeasiblePrice [€] 1522 4968 infeasibleEfficiency [%] 68.0 61.6 infeasible
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 50
0.49
0.5
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
2000
4000
6000
time [s]
Plo
ss [W
]
efficiency is 68%
x is again the better starting point. The capacity of the battery is more then enough, the batteryhas been sized such that the power demand of 10 kW has been met. It seems that the NiMHbattery has to be sized according to power capacity.
84
S3_pm_opt.m NiMH battery with PM motorSOC0 = 0.5;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
batt_nimh_n [-] 1 1000 60 50.4 1000 50.4 1 1000batt_nimh_Qmax [Ah] 10 93 20 10 93 10 10 93
T [-] 0.428 0.428 55.5Mass [kg] 90.0 90.1 16439Volume [dm3] 24.2 24.2 1747Price [€] 1940 1942 300195Efficiency [%] 83.7 83.7 88.1
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500.49
0.495
0.5
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
1000
2000
3000
time [s]
Plo
ss [W
]
efficiency is 83.7%
Both the x0 and x0,max starting points result in the same system, the x0,min start value will resultin the upper bound values for the design parameters, which appears to be a local minimum.The efficiency of the battery is maximum in the latter, and since efficiency generally increases forhigher battery systems, this is an upper bound for the maximum efficiency that can theoreticallybe achieved for this power profile Ps. Power losses depend highly on the power output of thesystem, opposed to the flywheel systems where the losses depend on the SOC.
85
S4_ac_opt.m VRLA battery with ac motorSOC0 = 0.5;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
batt_vrla_n [-] 1 1000 60 10.5 1000 14 1 1000batt_vrla_Qmax [Ah] 12 104 15 12.0 104 12 12 104motor_ac_kW_0 [kW] max|Ps| 200 max|Ps| 18.4 200 200 max|Ps| 200
T [-] 0.478 0.947 62.31Mass [kg] 135 170 29636Volume [dm3] 34.8 59.7 4754Price [€] 531 3280 131590Efficiency [%] 56.2 31.6 66.7
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500.46
0.47
0.48
0.49
0.5
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
2000
4000
6000
8000
time [s]
Plo
ss [W
]
efficiency is 56.2%
Only x0 will result in an optimal design, local minima are found for the other cases. The VRLAbattery is sized for the power requirements, since the SOC variation is minimal.
86
S4_pm_opt.m VRLA battery with PM motorSOC0 = 0.5;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
batt_vrla_n [-] 1 1000 25 10.3 1000 10.3 1 1000batt_vrla_Qmax [Ah] 12 104 15 12.0 104 12.0 12 104
T [-] 0.376 0.376 61.8Mass [kg] 82.2 82.2 29558Volume [dm3] 26.3 26.3 4724Price [€] 468 468 128806Efficiency [%] 58.2 58.2 87.9
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500.46
0.47
0.48
0.49
0.5
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
2000
4000
6000
8000
time [s]
Plo
ss [W
] efficiency is 58.2%
Maximum theoretical efficiency is now 87.9 −, the PM motor is a significantly more efficient.The differences in efficiency between the S4_pm and S4_ac systems is however very low, but thesize and mass of the motor contribute greatly to a lower objective function value T .
87
S5_ac_opt.m UltraCap with ac motorSOC0 = 0.9;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
uc_Ns [-] 1 10000 1000 472 10000 499 1 10000uc_Np [-] 1 10 1 1.00 10 1.10 1 10uc_cell_C [J] 500 10000 1000 501 10000 501 500 9990motor_ac_kW_0 max|Ps| 200 max|Ps| 18.7 200 200 max|Ps| 200
T [-] 1.06 1.63 3216Mass [kg] 112 142 74233Volume [dm3] 39.3 64.9 59903Price [€] 7475 11330 29973100Efficiency [%] 77.6 56.7 69.6
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500
0.5
1
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
5000
10000
time [s]
Plo
ss [W
]
efficiency is 77.6%
Power losses depend on the SOC heavily, local minima are again found. The optimum found forthe start value of x0,min is the upper bound of the design space, which in this case is certainly notan upper bound of the efficiency of S5_ac. The system is sized according to the energy capacity,opposed to the battery systems.
88
S5_pm_opt.m UltraCap with PM motorSOC0 = 0.9;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
uc_Ns [-] 1 10000 1000 463 10000 448 1 infeasibleuc_Np [-] 1 10 1 1.00 10 1.04 1 infeasibleuc_cell_C [J] 500 10000 1e3 501 10000 501 500 infeasible
T [-] 0.962 0.963 infeasibleMass [kg] 59.3 59.3 infeasibleVolume [dm3] 30.9 30.9 infeasiblePrice [€] 7305 7304 infeasibleEfficiency [%] 77.2 77.2 infeasible
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500
0.5
1
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
5000
10000
time [s]
Plo
ss [W
]
efficiency is 77.2%
The PM solution is again more efficient, and the same minima are found for x0 and x0,max.
89
S6_opt_pump1.m CAES with pump design 1SOC0 = 0.9; r = 1e− 3;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
pump_vg [m3] 20.9e-6 45.2e-6 40e-6 36.7e-6 45.2e-6 39.1e-6 20.9e-6 infeasiblecaes_oil_R [m] 0.05 10 5 0.420 10 0.44 0.05 infeasiblecaes_oil_t [m] 0.001 0.5 0.3 0.0352 0.5 0.0190 0.001 infeasiblecaes_oil_h [m] 0.05 10 5 0.815 10 1.28 0.05 infeasible
T [-] 5.39 7.07 infeasibleMass [kg] 1365 1455 infeasibleVolume [dm3] 1065 1705 infeasiblePrice [€] 13702 14608 infeasibleEfficiency [%] 84.3 81.6 infeasible
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500.4
0.6
0.8
1
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
1000
2000
3000
4000
time [s]
Plo
ss [W
]
efficiency is 84.3%
The objective function values indicate that the minima found are not very well suited for thispower profile, opposed to the S1 to S5 systems. Two very different systems are found for x0 andx0,max, in terms of dimensions. The first is a thick-walled gas tank with small hight, the last isa high thin-walled gas tank. The mass, price and efficiency do not differ much, only the volumeis higher of the latter, resulting in a higher objective function value. Masses can hardly be calledfeasible for building the system into a hybrid vehicle.
90
S6_opt_pump2.m CAES with pump design 2SOC0 = 0.9; r = 1e− 3;
x xmin xmax x0 x∗ x0,max x∗ x0,min x∗
pump_vg [m3] 40e-6 100e-6 40e-6 45.5e-6 100e-6 80.0e-6 40e-6 infeasiblecaes_oil_R [m] 0.05 10 5 0.429 10 0.950 0.05 infeasiblecaes_oil_t [m] 0.001 0.5 0.3 0.0360 0.5 0.0830 0.001 infeasiblecaes_oil_h [m] 0.05 10 5 0.827 10 0.127 0.05 infeasible
T [-] 5.82 17.38 infeasibleMass [kg] 1461 4643 infeasibleVolume [dm3] 1130 1373 infeasiblePrice [€] 14662 46461 infeasibleEfficiency [%] 90.7 86.8 infeasible
0 5 10 15 20 25 30 35 40 45 50−20
−10
0
10
20
P [k
W]
0 5 10 15 20 25 30 35 40 45 500.4
0.6
0.8
1
SO
C [−
]
0 5 10 15 20 25 30 35 40 45 500
500
1000
1500
time [s]
Plo
ss [W
]
efficiency is 90.7%
The two feasible solutions are again very different, one with a very thick tank (8.3 cm) and highmass, the other with a thickness of 3.6 cm. The efficiency with the pump design 2 is higher thanwith pump design 1. Masses are again infeasible for a 1300 kg vehicle.
91
B.2 Sensitivity to alphaTest cases are:
αi Range Test descriptionα1 [0.2;0.2;0.2;0.4] Medium values, slightly high efficiencyα2 [0.4;0.4;0.1;0.1] Low mass and volumeα3 [0.1;0.1;0.1;0.7] High efficiencyα4 [0.1;0.1;0.7;0.1] Low price
With αi = [αM ; αV ; α€;αη].
S1 Supercritical Flywheel with electric motor CVTFrom the table it is clear that the designs hardly change with changing αi.
αi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]α1 S1_ac [0.0333 0.172 0.131 0.00912 12.0] 0.696 181 34.5 2166 57.7α2 S1_ac [0.0333 0.172 0.131 0.00913 12.0] 0.771 181 34.5 2165 57.7α3 S1_ac [0.0335 0.164 0.121 0.00945 12.0] 0.557 181 34.4 2154 57.9α4 S1_ac [0.0333 0.176 0.135 0.00899 12.0] 0.957 181 34.5 2170 57.5α1 S1_pm [0.0332 0.173 0.133 0.00873] 0.492 79.2 19.8 2127 66.3α2 S1_pm [0.0332 0.172 0.132 0.00874] 0.430 79.2 19.8 2126 66.3α3 S1_pm [0.0300 0.165 0.124 0.00903] 0.412 79.0 19.8 2129 66.8α4 S1_pm [0.0331 0.177 0.138 0.00859] 0.853 79.3 19.9 2132 66.2
Table B.1: Sensitivity to α, α1 = [0.2; 0.2; 0.2; 0.4], α2 = [0.4; 0.4; 0.1; 0.1], α3 =[0.1; 0.1; 0.1; 0.7], α4 = [0.1; 0.1; 0.7; 0.1] (SOC0=0.9, Mmax = 150, Vmax = 0.1, €max = 2000)
S2 Subcritical Flywheel with mechanical V-Belt CVTThe α2 design, low mass and volume, does show a significant other design. Also, the α4 design,
αi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]α1 S2 [0.0329 0.0928 0.0404 0.142] 0.501 113 43.7 2015 84.8α2 S2 [0.0330 0.0932 0.0404 0.0919] 0.532 93.9 41.0 2025 83.9α3 S2 [0.0329 0.0928 0.0403 0.147] 0.328 115 44.0 2015 84.9α4 S2 [0.0329 0.0924 0.0402 0.406] 0.917 215 57.8 2005 85.8
Table B.2: Sensitivity to α, α1 = [0.2; 0.2; 0.2; 0.4], α2 = [0.4; 0.4; 0.1; 0.1], α3 =[0.1; 0.1; 0.1; 0.7], α4 = [0.1; 0.1; 0.7; 0.1] (SOC0=0.9, Mmax = 150, Vmax = 0.1, €max = 2000)
low price, does show an increased mass however not a substantially lower price.
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S3 NiMH battery with electric motorFrom the table it is clear that the designs hardly change with changing αi.
αi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]α1 S3_ac [10.9 35.1 12.7] 0.522 144 25.1 1522 68.0α2 S3_ac [10.9 35.2 12.0] 0.594 144 25.0 1519 67.4α3 S3_ac [10.8 34.6 17.7] 0.408 143 25.5 1563 70.2α4 S3_ac [10.9 35.4 12.0] 0.686 145 25.0 1519 67.4α1 S3_pm [50.4 10.0] 0.428 90.0 24.2 1940 83.7α2 S3_pm [50.4 10.0] 0.450 90.0 24.2 1940 83.7α3 S3_pm [50.3 10.0] 0.295 90.0 24.2 1939 83.7α4 S3_pm [50.4 10.0] 0.780 90.0 24.2 1940 83.7
Table B.3: Sensitivity to α, α1 = [0.2; 0.2; 0.2; 0.4], α2 = [0.4; 0.4; 0.1; 0.1], α3 =[0.1; 0.1; 0.1; 0.7], α4 = [0.1; 0.1; 0.7; 0.1] (SOC0=0.5, Mmax = 150, Vmax = 0.1, €max = 2000)
S4 VRLA battery with electric motorFor the high efficiency case (α3) the optimization algorithms find a substantially better performer
αi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]α1 S4_ac [10.5 12.0 18.4] 0.478 135 34.8 531 56.2α2 S4_ac [9.42 12.0 12.0] 0.545 129 32.2 420 48.6α3 S4_ac [19.1 12.0 21.1] 0.413 179 48.9 700 70.0α4 S4_ac [9.45 12.0 12.0] 0.317 129 32.3 420 48.7α1 S4_pm [10.3 12.0] 0.376 82.2 26.3 468 58.2α2 S4_pm [8.87 12.0] 0.365 75.0 24.0 447 53.5α3 S4_pm [19.4 12.0] 0.352 128 41.0 604 72.2α4 S4_pm [8.87 12.0] 0.277 75.0 24.0 447 53.5
Table B.4: Sensitivity to α, α1 = [0.2; 0.2; 0.2; 0.4], α2 = [0.4; 0.4; 0.1; 0.1], α3 =[0.1; 0.1; 0.1; 0.7], α4 = [0.1; 0.1; 0.7; 0.1] (SOC0=0.5, Mmax = 150, Vmax = 0.1, €max = 2000)
compared with the other cases. This seems due to an increased size of the AC motor for the S4_acsystem to increase motoring performance and more battery modules for S4_ac and S4_pm.
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S5 UltraCap with electric motorThe high efficiency case (α3) does find higher efficiency solutions due to a larger AC motor
αi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]α1 S5_ac [472 1.00 501 18.7] 1.06 112 39.3 7476 77.5α2 S5_ac [467 1.00 502 16.8] 0.848 111 38.9 7397 73.6α3 S5_ac [492 1.00 501 20.3] 0.632 113 40.4 7798 82.0α4 S5_ac [465 1.00 501 17.8] 2.72 111 38.9 7349 67.4α1 S5_pm [463 1.01 501] 0.963 59.3 30.9 7305 77.2α2 S5_pm [466 1.00 501] 0.670 59.4 30.9 7318 77.5α3 S5_pm [492 1.00 501] 0.585 61.0 32.1 7710 81.9α4 S5_pm [455 1.00 501] 2.616 58.8 30.4 7382 67.4
Table B.5: Sensitivity to α, α1 = [0.2; 0.2; 0.2; 0.4], α2 = [0.4; 0.4; 0.1; 0.1], α3 =[0.1; 0.1; 0.1; 0.7], α4 = [0.1; 0.1; 0.7; 0.1] (SOC0=0.9, Mmax = 150, Vmax = 0.1, €max = 2000)
(S5_ac) and more cells (S5_ac and S5_pm). Low price case (α4) has high impact on the efficiencywhile the price of S5_pm for α4 is not even the lowest of all cases, this could be a local minimum.
S6 CAES storage with pumpThe overall performance of all systems do not differ much from each other, the optimizations
αi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]α1 S6_p1 [45.2e-6 0.366 0.0115 0.246] 0.779 162 158 1680 86.6α2 S6_p1 [29.1e-6 0.416 0.0123 0.228] 1.45 202 186 2074 80.4α3 S6_p1 [45.2e-6 0.452 0.0131 0.195] 0.556 228 191 2338 86.2α4 S6_p1 [37.8e-6 0.459 0.0106 0.438] 1.52 259 362 2651 83.2α1 S6_p2 [55.0e-6 0.208 0.0175 0.213] 0.443 112 64.8 1165 88.8α2 S6_p2 [55.0e-6 0.210 0.0177 0.209] 0.635 113 65.0 1177 88.8α3 S6_p2 [55.0e-6 0.207 0.0174 0.215] 0.278 112 64.7 1161 88.8α4 S6_p2 [55.0e-5 0.206 0.0173 0.220] 0.556 112 65.1 1158 88.8
Table B.6: Sensitivity to α, α1 = [0.2; 0.2; 0.2; 0.4], α2 = [0.4; 0.4; 0.1; 0.1], α3 =[0.1; 0.1; 0.1; 0.7], α4 = [0.1; 0.1; 0.7; 0.1] (SOC0=0.9, Mmax = 150, Vmax = 0.1, €max = 2000)
are relatively insensitive to variation of αi.To obtain a higher efficiency, the size of the AC motor is increased to about twice the peak powerneeded from the power profile. Further, the battery and ultracap systems are increased in sizeby adding extra modules and cells respectively. Other test cases did sometimes produce differentdesigns but with the same average performance (Mass, Volume, price, efficency). The test set forthe rest of the optimizations will be α = [0.2; 0.2; 0.4; 0.2]
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B.3 Sensitivity to Mmax, Vmax, €max
Test cases are:
Maxi Range Test descriptionMax1 [50;0.1;2000] Maximum mass lowMax2 [400;0.1;2000] Maximum mass highMax3 [150;0.05;2000] Maximum volume lowMax4 [150;0.5;2000] Maximum volume highMax5 [150;0.1;200] Maximum price level lowMax6 [150;0.1;5000] Maximum price level high
With Maxi = [Mmax;Vmax; €max].
S1 Supercritical Flywheel with electric motor CVTThe found minima are all insensitive to the variations, the only varying parameter is the objective
Maxi System x∗ T [-] Mass [kg] Volume[dm3]
price [€] η [-]
Max1 S1_ac [0.0333 0.171 0.129 0.00919 12.0] 1.18 181 34.5 2163 57.7Max2 S1_ac [0.0333 0.172 0.131 0.00912 12.0] 0.545 181 34.5 2166 57.7Max3 S1_ac [0.0333 0.172 0.131 0.00912 12.0] 0.765 181 34.5 2165 57.7Max4 S1_ac [0.0333 0.172 0.131 0.00912 12.0] 0.641 181 34.5 2166 57.7Max5 S1_ac [0.0333 0.176 0.135 0.00899 12.0] 2.65 181 34.5 2170 57.5Max6 S1_ac [0.0334 0.167 0.125 0.00934 12.0] 0.565 181 34.5 2158 57.8Max1 S1_pm [0.0332 0.172 0.132 0.00877] 0.703 79.1 19.8 2125 66.3Max2 S1_pm [0.0332 0.173 0.133 0.00873] 0.427 79.2 19.8 2127 66.3Max3 S1_pm [0.0332 0.173 0.133 0.00874] 0.532 79.2 19.8 2127 66.3Max4 S1_pm [0.0332 0.173 0.133 0.00873] 0.461 79.2 19.8 2127 66.3Max5 S1_pm [0.0331 0.177 0.138 0.00858] 2.41 79.3 19.9 2133 66.2Max6 S1_pm [0.0333 0.167 0.126 0.00895] 0.364 79.0 19.8 2119 66.5
Table B.7: Sensitivity to Maxi (SOC0=0.9, α = [0.2; 0.2; 0.2; 0.4])
value T .
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S2 Subcritical Flywheel with mechanical V-Belt CVT
Maxi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]Max1 S2 [0.0314 0.0872 0.0549 0.0520] 0.657 73.5 38.5 2069 80.3Max2 S2 [0.0329 0.0926 0.0402 0.211] 0.425 140 47.4 2009 85.3Max3 S2 [0.0329 0.0928 0.0403 0.130] 0.580 109 43.0 2017 84.7Max4 S2 [0.0329 0.0928 0.0403 0.156] 0.438 119 44.4 2014 84.9Max5 S2 [0.0329 0.0924 0.0402 0.602] 2.58 292 68.3 2003 85.9Max6 S2 [0.0314 0.0877 0.0540 0.0514] 0.336 73.7 38.5 2068 80.5
Table B.8: Sensitivity to Maxi (SOC0=0.9, α = [0.2; 0.2; 0.2; 0.4])
S3 NiMH battery with electric motor
Maxi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]Max1 S3_ac [10.9 35.1 12.3] 0.909 144 25.1 1520 67.7Max2 S3_ac [10.8 35.2 13.1] 0.402 144 25.1 1523 68.2Max3 S3_ac [10.9 35.2 12.6] 0.573 144 25.1 1521 67.9Max4 S3_ac [10.9 35.1 12.9] 0.482 144 25.1 1522 68.1Max5 S3_ac [10.9 35.4 12.0] 1.89 144 25.0 1518 67.4Max6 S3_ac [10.8 35.0 13.7] 0.428 144 25.2 1527 68.7Max1 S3_pm [50.4 10.0] 0.668 90.0 24.2 1941 83.7Max2 S3_pm [50.4 10.0] 0.353 90.0 24.2 1940 83.7Max3 S3_pm [50.4 10.0] 0.476 90.0 24.2 1940 83.7Max4 S3_pm [50.4 10.0] 0.389 90.0 24.2 1940 83.7Max5 S3_pm [50.4 10.0] 2.17 90.0 24.2 1940 83.7Max6 S3_pm [50.4 10.0] 0.311 90.0 24.2 1940 83.7
Table B.9: Sensitivity to Maxi (SOC0=0.5, α = [0.2; 0.2; 0.2; 0.4])
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S4 VRLA battery with electric motor
Maxi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]Max1 S4_ac [9.23 12.0 14.3] 0.821 128 32.2 451 50.2Max2 S4_ac [13.1 12.0 15.8] 0.359 148 38.6 531 61.2Max3 S4_ac [9.26 12.0 16.3] 0.543 128 32.5 482 51.4Max4 S4_ac [12.0 12.0 21.0] 0.424 143 37.5 593 60.4Max5 S4_ac [9.41 12.0 12.0] 0.861 129 32.2 419 48.5Max6 S4_ac [11.0 12.0 22.7] 0.448 138 36.0 603 58.0Max1 S4_pm [8.87 12.0] 0.579 75.0 24.0 447 53.5Max2 S4_pm [13.0 12.0] 0.303 95.9 30.7 508 64.3Max3 S4_pm [9.07 12.0] 0.427 76.0 24.4 450 54.3Max4 S4_pm [11.7 12.0] 0.332 89.4 28.6 489 61.8Max5 S4_pm [8.87 12.0] 0.781 75.0 24.0 447 53.5Max6 S4_pm [10.8 12.0] 0.348 84.7 27.1 475 59.6
Table B.10: Sensitivity to Maxi (SOC0=0.5, α = [0.2; 0.2; 0.2; 0.4])
S5 UltraCap with electric motor
Maxi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]Max1 S5_ac [504 1.00 524 133] 1.78 131 55.4 10010 63.3Max2 S5_ac [472 1.00 501 18.6] 0.972 112 39.3 7471 77.4Max3 S5_ac [472 1.00 501 18.4] 1.14 112 39.3 7469 77.3Max4 S5_ac [472 1.00 501 18.9] 1.00 112 39.4 7482 77.7Max5 S5_ac [465 1.00 501 17.9] 7.70 111 38.9 7354 68.9Max6 S5_ac [475 1.01 501 19.3] 0.613 112 39.8 7613 80.1Max1 S5_pm [462 1.01 501] 1.12 131 55.4 10010 63.3Max2 S5_pm [465 1.00 501] 0.913 112 39.3 7471 77.4Max3 S5_pm [464 1.00 501] 1.02 112 39.3 7469 77.3Max4 S5_pm [465 1.00 501] 0.913 112 39.4 7482 77.7Max5 S5_pm [455 1.00 501] 7.45 111 38.9 7354 68.9Max6 S5_pm [478 1.00 501] 0.522 112 39.8 7613 80.1
Table B.11: Sensitivity to Maxi (SOC0=0.9, α = [0.2; 0.2; 0.2; 0.4])
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S6 CAES storage with pump
Maxi System x∗ T [-] Mass [kg] Volume [dm3] price [€] η [-]Max1 S6_p1 [45.2e-6 0.390 0.0120 0.224] 1.39 178 165 1839 86.5Max2 S6_p1 [44.2e-6 0.496 0.0142 0.167] 0.957 269 200 2746 85.8Max3 S6_p1 [38.9e-6 0.352 0.0140 0.175] 0.922 161 120 1667 85.4Max4 S6_p1 [37.4e-6 0.442 0.0108 0.371] 0.781 233 296 2387 83.2Max5 S6_p1 [45.2e-6 0.101 0.00617 1.21] 1.00 67.0 73.3 726 90.4Max6 S6_p1 [45.2e-6 0.320 0.0112 0.265] 0.565 138 137 1438 87.4Max1 S6_p2 [55.0e-6 0.207 0.0173 0.216] 0.745 111 64.5 1154 88.8Max2 S6_p2 [55.0e-6 0.206 0.0173 0.218] 0.347 111 64.6 1154 88.8Max3 S6_p2 [55.0e-6 0.207 0.0173 0.217] 0.571 111 64.6 1155 88.8Max4 S6_p2 [55.0e-6 0.194 0.0162 0.246] 0.320 105 63.9 1093 88.8Max5 S6_p2 [55.0e-6 0.189 0.0158 0.259] 1.39 102 63.2 1066 88.8Max6 S6_p2 [54.7e-6 0.199 0.0167 0.234] 0.360 107 64.1 1116 88.9
Table B.12: Sensitivity to Maxi (SOC0=0.9, α = [0.2; 0.2; 0.2; 0.4])
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Appendix C
Validation References
Component Speci�c Power[W/kg]
Speci�c Energy∗[Wh/kg]
Price+ [€/kWh] E�ciency[-]
Source
VRLA battery 150-400 50 [29]350 30 [30]200-400 30-50 90-120 [21]75-100 161 [31]
27 290 0.75 [32]150-400 20-60 [2]400 25 120-200 [33]
100 [23]NiMH battery 250-1000 45-70 [30]
150-300 50-70 115-150 [21]300 140-200 [31]170-200 80 [2]250-1000 46-66 300-550 [33]
350 [23]Ultra Capacitor 350-2400 2.5-4 [30]
1000 5 [21]1000 6 [2]
100 [€/Wh] Manuf.data
Subcritical �ywheel 300 [20]1.54 [15]
Supercritical �ywheel 400-4504 7-154 [21]25000 [20]
7.54 [15]Flywheel 500-1000 20 [2]AC motor - - 30-60 [€/kW] [33]
- - 15♦ [€/kW] [23]PM motor - - 35-70 [€/kW] [33]
- - 18♦ [€/kW] [23]motor - - 4-5 [€/kW] [34]inverter - - 550-1000 [€] [33]
Table C.1: Properties of components from di�erent literature sources. ∗: at three-hour dis-charge for batteries. +: at $1.30=1€. ♦: at high production volumes >100.000/year. 4:�ywheel properties are usually given for the rotor only, not for the accompanying housing andpump. Therefore, the values in literature are halved to have a housing mass equal to the rotormass.
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Appendix D
Scripts
Several scripts will be presented with which the simulations in this thesis can be done. Anoverview of the presented scripts is given:
MATLAB name Descriptioneline.m E-line calculation (OOL) based on an efficiency mapfw_subcr.m Subcritically operated flywheel modelfw_suprcr.m Supercritically operated flywheel modelbatt_nimh.m Nickel Metal Hydride battery modelbatt_vrla.m Valve Regulated Lead Acid battery modeluc.m Ultra Capacitor pack modelcaes_oil.m Compressed Air Energy Storage modelcvt.m Continuous Variable Transmission modelmotor_ac.m Induction motor modelmotor_pm.m Permanent Magnet motor modelpump.m Gear pump modelS1_ac_opt.m Example of one of the optimization routines, fw_subcr.m with motor_ac.mS1_ac.m Example of the accompanying objective function calculation
S1_ac_opt.m and S1_ac.m are representative for the S2-S6 systems optimization routines.
————————————
eline.m
function [eline] = eline(mc_map_spd,mc_map_trq,mc_eff_map,PWR,PWRpts)
% E-Line calculation for efficiency maps.%% This function calculates the optimum efficiencies for a given range of Power demands% based on an efficiency map. It operates by first calculating the efficiencies for an% iso-power curve and finding the maximum efficiency value on this curve.% Input can be e.g. an efficiency map for an electromotor, where the X-axis is speed (rad/s)% versus Y-axis is Torque (Nm). Plotting of the E-line can be done by using the function% contour and plotting the E-line on top of it.%% Syntax: [EL] = eline(X,Y,eff_map,PWR,PWRpts)%% X X-axis vector corresponding to Efficiency Map
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% Y Y-axis vector corresponding to Efficiency Map% eff_map Matrix with Efficiency points% PWR (optional, default=(0.05:0.05:1)*max(X)*max(Y)),% Vector with Power points (X*Y) for which efficiencies will be evaluated% Entries should be nonzero, otherwise a power curve can not be calculated% PWRpts (optional, default=20) Number of efficiency calculations% per power point defined in PWR.%% Result is matrix [EL]%% 1st column Power% 2nd column X-value corresponding to maximum efficiency% 3rd column Y-value corresponding to maximum efficiency% 4th column Maximum efficiency value%% See also CONTOUR%% Written by D.Hoekstra, version 2004.10.14a (yyyy.mm.dd)% Technical University of Eindhoven
if nargin==3, PWR=(0.05:0.05:1)*max(mc_map_spd)*max(mc_map_trq); PWRpts=20; endif nargin==4, PWRpts=20; end
Wtmp=[mc_map_spd(1):max(mc_map_spd)/(PWRpts):max(mc_map_spd)];
r=1;
for p=1:length(PWR)if PWR(p)==0
warning(['WARNING: one or more elements in PWR is zero, termination of function eline']);return
endfor q=1:PWRpts-1
if Wtmp(q)==0;PWRmem(q,:)=[PWR(p) NaN NaN NaN];
elseTmem=PWR(p)/Wtmp(q);PWRmem(q,:)=[PWR(p) Wtmp(q) Tmem interp2(mc_map_spd,mc_map_trq,mc_eff_map,Wtmp(q),Tmem)];
endend
[a,b]=max(PWRmem(:,4));eline(r,:)=PWRmem(b,:);r=r+1;clear PWRmem;
end
————————————
fw_subcr.m
function [fw_subcr_SOC_new,fw_subcr_w_new,fw_subcr_wmax,fw_subcr_Ploss,fw_subcr_M,fw_subcr_V,fw_subcr_price,fw_subcr_poss]=fw_subcr(fw_subcr_x0,fw_subcr_x,fw_subcr_SOC_old,fw_subcr_P,t)
% Mechanical subcritical flywheel operation calculation%% NOTE: FW losses at time T are substracted from flywheel energy at time T+t to% prevent algebraic loops
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%% Syntax: [fw_subcr_SOC_new,fw_subcr_w_new,fw_subcr_wmax,fw_subcr_Ploss,fw_subcr_M,fw_subcr_V,% fw_subcr_price,fw_subcr_poss]=fw_subcr(fw_subcr_x0,fw_subcr_x,fw_subcr_SOC_old,fw_subcr_P,t)%% fw_subcr_x0 vector of variable design parameters of flywheel:% [fw_subcr_d_0; fw_subcr_Ro_0; fw_subcr_Ri_0; fw_subcr_h_0]% fw_subcr_x vector of fixed design parameters of flywheel:% [fw_subcr_sigmax; fw_subcr_rhofwrotor; fw_subcr_rhohousing; fw_subcr_rhoair;% fw_subcr_etaair; fw_subcr_eta; fw_subcr_S; fw_subcr_t; fw_subcr_s; fw_subcr_Crad];% fw_subcr_SOC_old State of Charge of previous time step% fw_subcr_P Power demanded from flywheel system at t=tnew% t Simulation time step size%% Result is [fw_subcr_SOC_new,fw_subcr_w_new,fw_subcr_wmax,fw_subcr_Ploss,fw_subcr_M,fw_subcr_V,% fw_subcr_price,fw_subcr_poss]%% fw_subcr_SOC_new State of Charge of flywheel (0-1) [-]% fw_subcr_w_new Flywheel rotating speed [rad/s]% fw_subcr_wmax Maximum flywheel speed [rad/s]% fw_subcr_Ploss Power losses at fw_subcr_SOC_old [W]% fw_subcr_M Flywheel mass [kg]% fw_subcr_V Flywheel volume [m^3]% fw_subcr_price Flywheel price [�]% fw_subcr_poss Flywheel is capable of delivering or storing power fw_subcr_P (1=feasible% design, 0=unfeasible design)%% Use fw_subcr_ini for fast generation of fw_subcr_x0 and fw_subcr_x%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)% Technical University of Eindhoven
% calculate volumefw_subcr_V_rotor=pi*(fw_subcr_x0(2)^2-fw_subcr_x0(3)^2)*fw_subcr_x0(4);fw_subcr_V_housing=pi*fw_subcr_x0(2)^2*fw_subcr_x0(4);fw_subcr_V = 2*fw_subcr_V_housing;
% calculate massesfw_subcr_M_rotor = fw_subcr_V_rotor*fw_subcr_x(2);fw_subcr_M_housing = fw_subcr_V_housing*fw_subcr_x(3);fw_subcr_M = fw_subcr_M_rotor + fw_subcr_M_housing;
% calculate maximum speed fw_subcr_wmaxfw_subcr_Js = 0.5*fw_subcr_M_rotor*(fw_subcr_x0(2)^2+fw_subcr_x0(3)^2);fw_subcr_wmax_stress = sqrt((fw_subcr_x(1)*8)/(fw_subcr_x(2)*(3+fw_subcr_x(6))*(fw_subcr_x0(3).^2+
2*fw_subcr_x0(2).^2.-fw_subcr_x0(3).^2.*(1+3*fw_subcr_x(6))/(3+fw_subcr_x(6)))));fw_subcr_wmax_crit = 0.9*sqrt(fw_subcr_x(10)/fw_subcr_M_rotor);fw_subcr_wmax = min([fw_subcr_wmax_crit fw_subcr_wmax_stress]);fw_subcr_Emax = 0.5*(fw_subcr_Js)*fw_subcr_wmax^2;fw_subcr_Pmax = fw_subcr_Emax/fw_subcr_x(6);fw_subcr_wmin = 0;
% calculate old spinning speedif fw_subcr_SOC_old<0 %to prevent a bug that speed would turn out imaginary number.
%Valid assumption for SOC~0 and SOC<0fw_subcr_SOC_old=0.001; warning('error: fw_subcr_SOC_new is <0 so lower value of SOC is
set to 0.001, otherwise erronous imaginary results')end
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fw_subcr_w_old = sqrt(fw_subcr_SOC_old*(fw_subcr_wmax^2-fw_subcr_wmin^2)+fw_subcr_wmin^2);
% Efficiency calculations (based on fw_subcr_w_old)fw_subcr_Re_crit = 41.2*sqrt((1+(fw_subcr_x(8))/(2*fw_subcr_x0(2)))/(fw_subcr_x(8)/fw_subcr_x0(2)));fw_subcr_Rec = fw_subcr_w_old*fw_subcr_x0(2)*fw_subcr_x(8)*fw_subcr_x(4)/fw_subcr_x(5);fw_subcr_Res = fw_subcr_w_old*fw_subcr_x0(2)*fw_subcr_x0(2)*fw_subcr_x(4)/fw_subcr_x(5);
const1 = (log(0.92*((fw_subcr_x(8)/fw_subcr_x0(2))*(1+fw_subcr_x(8)/fw_subcr_x0(2)))^(0.25)*400^(-0.5))-log(0.92*((fw_subcr_x(8)/fw_subcr_x0(2))*(1+fw_subcr_x(8)/fw_subcr_x0(2)))^(0.25)*fw_subcr_Re_crit^(-0.5)))/(log(400)-log(fw_subcr_Re_crit));
const2 = log(0.92*((fw_subcr_x(8)/fw_subcr_x0(2))*(1+fw_subcr_x(8)/fw_subcr_x0(2)))^(0.25)*400^(-0.5))-const1*log(400);
fw_subcr_Ccirc = max([3.6*(fw_subcr_x(8)/fw_subcr_x0(2))^(-0.25)*(1+fw_subcr_x(8)/fw_subcr_x0(2))^2/
(2+fw_subcr_x(8)/fw_subcr_x0(2))*fw_subcr_Rec^(-1);10^(const1*log(fw_subcr_Rec)+const2);0.92*((fw_subcr_x(8)/fw_subcr_x0(2))*(1+fw_subcr_x(8)/fw_subcr_x0(2)))^(0.25)*
fw_subcr_Rec^(-0.5);0.146*((fw_subcr_x(8)/fw_subcr_x0(2))*(1+fw_subcr_x(8)/fw_subcr_x0(2)))^(0.25)*
fw_subcr_Rec^(-0.3)]);
fw_subcr_Cside = max([2*pi*(fw_subcr_x(9)/fw_subcr_x0(2))^(-1)*fw_subcr_Res^(-1);3.7*(fw_subcr_x(9)/fw_subcr_x0(2))^(1/10)*fw_subcr_Res^(-1/2);8e-2*(fw_subcr_x(9)/fw_subcr_x0(2))^(-1/6)*fw_subcr_Res^(-1/4);1e-2*(fw_subcr_x(9)/fw_subcr_x0(2))^(1/10)*fw_subcr_Res^(-1/5)
]);
fw_subcr_Pcirc = fw_subcr_w_old^3*fw_subcr_Ccirc*0.5*pi*fw_subcr_x(4)*fw_subcr_x0(4)*fw_subcr_x0(2)^4;fw_subcr_Pside = 2*fw_subcr_w_old^3*fw_subcr_Cside*0.5*fw_subcr_x(4)*fw_subcr_x0(2)^5;
fw_subcr_Pbearings = 2*0.5*fw_subcr_w_old*0.0015*9.81*fw_subcr_x(2)*fw_subcr_x0(4)*pi*(fw_subcr_x0(2)^2-fw_subcr_x0(3)^2)*fw_subcr_x0(1);
fw_subcr_Ploss = fw_subcr_Pcirc+fw_subcr_Pside+fw_subcr_Pbearings;
% Price modelfw_subcr_price = 25000*fw_subcr_Emax/(1000*3600);
% Calculate energy change of fw systemif fw_subcr_P < 0 %add power to fw system
if fw_subcr_Ploss > -fw_subcr_P %Ploss larger than charging power, so still spin-downfw_subcr_E = (-fw_subcr_P-fw_subcr_Ploss)*t; % <0: discharge fw with this energy
elseif fw_subcr_Ploss < -fw_subcr_P %Charging state of fwfw_subcr_E = (-fw_subcr_P-fw_subcr_Ploss)*t; % >0: charge fw with this energy
else %fw_subcr_P == fw_subcrPloss, so no change in speedfw_subcr_E = 0;
endelseif fw_subcr_P > 0 %discharge
fw_subcr_E = -(fw_subcr_P+fw_subcr_Ploss)*t; % <0: discharge fw with this energyelse
fw_subcr_E = -fw_subcr_Ploss*t; % fw_subcr_P == 0 so only Plossend
% New statefw_subcr_w_new = sqrt((2*fw_subcr_E/fw_subcr_Js)+fw_subcr_w_old^2);
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fw_subcr_SOC_new = (fw_subcr_w_new^2-fw_subcr_wmin^2)/(fw_subcr_wmax^2-fw_subcr_wmin^2);fw_subcr_torque = (fw_subcr_P+fw_subcr_Ploss)/fw_subcr_w_new;
% Limitsif isreal(fw_subcr_SOC_new)==0
fw_subcr_poss = 0; disp('error: fw_subcr_SOC_new is imaginary')elseif fw_subcr_SOC_new > 1
fw_subcr_poss = 0; warning('error: fw_subcr_SOC_new > 1')elseif fw_subcr_SOC_new < 0
fw_subcr_poss = 0; warning('error: fw_subcr_SOC_new < 0')else
fw_subcr_poss = 1;end
————————————
fw_suprcr.m
function [fw_suprcr_SOC_new,fw_suprcr_w_new,fw_suprcr_wmin,fw_suprcr_wmax,fw_suprcr_Ploss,fw_suprcr_M,fw_suprcr_V,fw_suprcr_price,fw_suprcr_poss]=fw_suprcr(fw_suprcr_x0,fw_suprcr_x,fw_suprcr_SOC_old,fw_suprcr_P,t)
% Mechanical supercritical flywheel operation calculation%% NOTE: FW losses at time T are substracted from flywheel energy at time T+t to% prevent algebraic loops%% Syntax: [fw_suprcr_SOC_new,fw_suprcr_w_new,fw_suprcr_wmin,fw_suprcr_wmax,fw_suprcr_Ploss,% fw_suprcr_M,fw_suprcr_V,fw_suprcr_price,fw_suprcr_poss]=% fw_suprcr(fw_suprcr_x0,fw_suprcr_x,fw_suprcr_SOC_old,fw_suprcr_P,t)%% fw_suprcr_x0 vector of variable design parameters of flywheel:% [fw_suprcr_d_0; fw_suprcr_Ro_0; fw_suprcr_Ri_0; fw_suprcr_h_0]% fw_suprcr_x vector of fixed design parameters of flywheel:% [fw_suprcr_sigmax; fw_suprcr_rhofwrotor; fw_suprcr_rhohousing;% fw_suprcr_rhoair; fw_suprcr_etaair; fw_suprcr_eta; fw_suprcr_S;% fw_suprcr_t; fw_suprcr_s; fw_suprcr_Crad]% fw_suprcr_SOC_old State of Charge of previous time step% fw_suprcr_P Power demanded from flywheel system at time tnew% t simulation time step size%% Result is [fw_suprcr_SOC_new,fw_suprcr_w_new,fw_suprcr_wmin,fw_suprcr_wmax,% fw_suprcr_Ploss,fw_suprcr_M,fw_suprcr_V,fw_suprcr_price,fw_suprcr_poss]%% fw_suprcr_SOC_new State of Charge of flywheel (0-1)% fw_suprcr_w_new Flywheel rotating speed [rad/s]% fw_suprcr_wmin Minimum flywheel speed [rad/s]% fw_suprcr_wmax Maximum flywheel speed [rad/s]% fw_suprcr_Ploss Power losses at fw_suprcr_SOC_old [W]% fw_suprcr_M Flywheel mass [kg]% fw_suprcr_V Flywheel volume [m^3]% fw_suprcr_price Flywheel price [�]% fw_suprcr_poss Flywheel is capable of delivering or storing power fw_suprcr_P (1=feasible% design, 0=unfeasible design)%% Use fw_suprcr_ini for fast generation of fw_suprcr_x0 and fw_suprcr_x%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)
104
% Technical University of Eindhoven
% calculate volumefw_suprcr_V_rotor=pi*(fw_suprcr_x0(2)^2-fw_suprcr_x0(3)^2)*fw_suprcr_x0(4);fw_suprcr_V_housing=pi*fw_suprcr_x0(2)^2*fw_suprcr_x0(4);fw_suprcr_V = 1.5*fw_suprcr_V_housing; %includes pump
% calculate massesfw_suprcr_M_rotor = fw_suprcr_V_rotor*fw_suprcr_x(2);fw_suprcr_M_housing = 0.5*fw_suprcr_V_housing*fw_suprcr_x(3);fw_suprcr_M = fw_suprcr_M_rotor + fw_suprcr_M_housing+25; %25 for pump
% calculate maximum speed fw_suprcr_wmaxfw_suprcr_Js = 0.5*fw_suprcr_M_rotor*(fw_suprcr_x0(2)^2+fw_suprcr_x0(3)^2);fw_suprcr_wmax = sqrt((fw_suprcr_x(1)*8)/(fw_suprcr_x(2)*(3+fw_suprcr_x(6))*(fw_suprcr_x0(3).^2+2*fw_suprcr_x0(2).^2.-fw_suprcr_x0(3).^2.*(1+3*fw_suprcr_x(6))/(3+fw_suprcr_x(6)))));fw_suprcr_Emax = 0.5*(fw_suprcr_Js)*fw_suprcr_wmax^2;
% calculate minimum speed fw_suprcr_wmin = w_crfw_suprcr_wmin = sqrt(fw_suprcr_x(10)/fw_suprcr_M_rotor);
% calculate old spinning speedif fw_suprcr_SOC_old<0 %to prevent a bug that speed would turn out imaginary number.
%Valid assumption for SOC~0 and SOC<0fw_suprcr_SOC_old=0; warning('error: fw_suprcr_SOC_new is <0 so lower value of speed is
set to zero, otherwise erronous imaginary results')endfw_suprcr_w_old = sqrt(fw_suprcr_SOC_old*(fw_suprcr_wmax^2-fw_suprcr_wmin^2)+fw_suprcr_wmin^2);
% Efficiency calculations (based on fw_suprcr_w_old)fw_suprcr_Re_crit = 41.2*sqrt((1+(fw_suprcr_x(8))/(2*fw_suprcr_x0(2)))/(fw_suprcr_x(8)/fw_suprcr_x0(2)));fw_suprcr_Rec = fw_suprcr_w_old*fw_suprcr_x0(2)*fw_suprcr_x(8)*fw_suprcr_x(4)/fw_suprcr_x(5);fw_suprcr_Res = fw_suprcr_w_old*fw_suprcr_x0(2)*fw_suprcr_x0(2)*fw_suprcr_x(4)/fw_suprcr_x(5);
const1 = (log(0.92*((fw_suprcr_x(8)/fw_suprcr_x0(2))*(1+fw_suprcr_x(8)/fw_suprcr_x0(2)))^(0.25)*400^(-0.5))-log(0.92*((fw_suprcr_x(8)/fw_suprcr_x0(2))*(1+fw_suprcr_x(8)/fw_suprcr_x0(2)))^(0.25)*fw_suprcr_Re_crit^(-0.5)))/(log(400)-log(fw_suprcr_Re_crit));const2 = log(0.92*((fw_suprcr_x(8)/fw_suprcr_x0(2))*(1+fw_suprcr_x(8)/fw_suprcr_x0(2)))^(0.25)*400^(-0.5))-const1*log(400);
fw_suprcr_Ccirc = max([3.6*(fw_suprcr_x(8)/fw_suprcr_x0(2))^(-0.25)*(1+fw_suprcr_x(8)/fw_suprcr_x0(2))^2/(2+fw_suprcr_x(8)/fw_suprcr_x0(2))*fw_suprcr_Rec^(-1);10^(const1*log(fw_suprcr_Rec)+const2);0.92*((fw_suprcr_x(8)/fw_suprcr_x0(2))*(1+fw_suprcr_x(8)/fw_suprcr_x0(2)))^(0.25)*fw_suprcr_Rec^(-0.5);0.146*((fw_suprcr_x(8)/fw_suprcr_x0(2))*(1+fw_suprcr_x(8)/fw_suprcr_x0(2)))^(0.25)*fw_suprcr_Rec^(-0.3)]);
fw_suprcr_Cside = max([2*pi*(fw_suprcr_x(9)/fw_suprcr_x0(2))^(-1)*fw_suprcr_Res^(-1);3.7*(fw_suprcr_x(9)/fw_suprcr_x0(2))^(1/10)*fw_suprcr_Res^(-1/2);8e-2*(fw_suprcr_x(9)/fw_suprcr_x0(2))^(-1/6)*fw_suprcr_Res^(-1/4);1e-2*(fw_suprcr_x(9)/fw_suprcr_x0(2))^(1/10)*fw_suprcr_Res^(-1/5)
]);
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fw_suprcr_Pcirc = fw_suprcr_w_old^3*fw_suprcr_Ccirc*0.5*pi*fw_suprcr_x(4)*fw_suprcr_x0(4)*fw_suprcr_x0(2)^4;
fw_suprcr_Pside = fw_suprcr_w_old^3*fw_suprcr_Cside*0.5*fw_suprcr_x(4)*fw_suprcr_x0(2)^5;
fw_suprcr_Seals = 1.71*fw_suprcr_x0(1)*fw_suprcr_w_old;fw_suprcr_Pbearings = 2*0.5*fw_suprcr_w_old*0.0015*9.81*fw_suprcr_x(2)*fw_suprcr_x0(4)*pi*
(fw_suprcr_x0(2)^2-fw_suprcr_x0(3)^2)*fw_suprcr_x0(1);fw_suprcr_Pvacpump = 100;fw_suprcr_Ploss = fw_suprcr_Pcirc+fw_suprcr_Pside+fw_suprcr_Seals+fw_suprcr_Pbearings+fw_suprcr_Pvacpump;
% Price modelfw_suprcr_price = 25000*fw_suprcr_Emax/(1000*3600);
% Calculate energy change of fw systemif fw_suprcr_P < 0 %add power to fw system
if fw_suprcr_Ploss > -fw_suprcr_P %Ploss larger than charging power,%so still spin-down
fw_suprcr_E = (-fw_suprcr_P-fw_suprcr_Ploss)*t; % <0: discharge fw with this energyelseif fw_suprcr_Ploss < -fw_suprcr_P %Charging state of fw
fw_suprcr_E = (-fw_suprcr_P-fw_suprcr_Ploss)*t; % >0: charge fw with this energyelse %fw_suprcr_P == fw_suprcrPloss, so no
%change in speedfw_suprcr_E = 0;
endelseif fw_suprcr_P > 0 %discharge
fw_suprcr_E = -(fw_suprcr_P+fw_suprcr_Ploss)*t; % <0: discharge fw with this energyelse
fw_suprcr_E = -fw_suprcr_Ploss*t; % fw_suprcr_P == 0 so only Plossend
% New statefw_suprcr_w_new = sqrt((2*fw_suprcr_E/fw_suprcr_Js)+fw_suprcr_w_old^2);fw_suprcr_SOC_new = (fw_suprcr_w_new^2-fw_suprcr_wmin^2)/(fw_suprcr_wmax^2-fw_suprcr_wmin^2);fw_suprcr_torque = (fw_suprcr_P+fw_suprcr_Ploss)/fw_suprcr_w_new;
% Limitsif isreal(fw_suprcr_SOC_new)==0
fw_suprcr_poss = 0; warning('error: fw_suprcr_SOC_new is imaginary')elseif fw_suprcr_SOC_new > 1
fw_suprcr_poss = 0; warning('error: fw_suprcr_SOC_new > 1')elseif fw_suprcr_SOC_new < 0
fw_suprcr_poss = 0; warning('error: fw_suprcr_SOC_new < 0')else
fw_suprcr_poss = 1;end
————————————
batt_nimh.m
function [batt_nimh_SOC_new,batt_nimh_Pmax,batt_nimh_Ploss,batt_nimh_M,batt_nimh_V,batt_nimh_price,batt_nimh_poss]=batt_nimh(batt_nimh_x0,batt_nimh_x,batt_nimh_SOC_old,batt_nimh_P,t)
% NiMH battery operation calculation%% Syntax: [batt_nimh_SOC_new,batt_nimh_Pmax,batt_nimh_Ploss,batt_nimh_M,batt_nimh_V,% batt_nimh_price,batt_nimh_poss]=batt_nimh(batt_nimh_x0,batt_nimh_x,batt_nimh_SOC_old,% batt_nimh_P,t)
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%% batt_nimh_x0 vector of design parameters of battery:% [batt_nimh_n;batt_nimh_Qmax]% batt_nimh_x vector of fixed design parameters of battery:% []% batt_nimh_SOC_old State of Charge of previous time step% batt_nimh_P Power demanded from battery system at t=tnew% t time step size%% Result is [batt_nimh_SOC_new,batt_nimh_Pmax,batt_nimh_Ploss,batt_nimh_M,batt_nimh_V,% batt_nimh_price,batt_nimh_poss]%% batt_nimh_SOC_new State of Charge of battery (0-1)% batt_nimh_Pmax Maximum power output possible [W]% batt_nimh_Ploss Power losses at batt_nimh_SOC_old [W]% batt_nimh_M battery mass [kg]% batt_nimh_V battery volume [m^3]% batt_nimh_price battery price [�]% batt_nimh_poss battery is capable of delivering or storing power% batt_nimh_P (1=feasible design, 0=unfeasible design)%% Use batt_nimh_ini for fast generation of batt_nimh_x0 and batt_nimh_x%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)% Technical University of Eindhoven
% calculate volumebatt_nimh_V = batt_nimh_x0(1)*polyval([0.0175e-3 0.1106e-3],batt_nimh_x0(2));
% calculate massbatt_nimh_M = batt_nimh_x0(1)*polyval([0.1834 -0.6464],batt_nimh_x0(2));
% energy needed time step T-t to T:batt_nimh_E = batt_nimh_P*t;
% Cell voltage based on SOC_oldbatt_nimh_Volt = batt_nimh_x0(1)*polyval([1.5603 12.6553],batt_nimh_SOC_old);
% maximum total energy contentsbatt_nimh_Emax = 3600*batt_nimh_x0(2)*batt_nimh_Volt;batt_nimh_Pmax = batt_nimh_M*225; %Comment for limited Pmax
% Efficiency calculationsSOC_R = [0:0.1:1];N_R = [10 45 56 60 80 90 93];Rdfit= [0.0435 0.0405 0.0383 0.0368 0.0360 0.0357 0.0357 0.0361 0.0367 0.0374 0.0380;
0.0287 0.0257 0.0235 0.0220 0.0212 0.0208 0.0209 0.0213 0.0219 0.0226 0.0232;0.0241 0.0210 0.0188 0.0173 0.0165 0.0162 0.0163 0.0167 0.0173 0.0179 0.0185;0.0224 0.0193 0.0171 0.0156 0.0148 0.0145 0.0146 0.0150 0.0156 0.0162 0.0168;0.0139 0.0109 0.0086 0.0072 0.0063 0.0060 0.0061 0.0065 0.0071 0.0078 0.0084;0.0097 0.0066 0.0044 0.0029 0.0021 0.0018 0.0019 0.0023 0.0029 0.0035 0.0041;0.0084 0.0054 0.0031 0.0017 0.0008 0.0005 0.0006 0.0010 0.0016 0.0023 0.0029];
Rcfit= [0.0363 0.0350 0.0340 0.0334 0.0329 0.0328 0.0330 0.0334 0.0341 0.0351 0.0364;0.0270 0.0258 0.0248 0.0241 0.0237 0.0235 0.0237 0.0241 0.0248 0.0258 0.0271;0.0241 0.0228 0.0219 0.0212 0.0208 0.0206 0.0208 0.0212 0.0219 0.0229 0.0242;0.0230 0.0218 0.0208 0.0201 0.0197 0.0196 0.0197 0.0202 0.0209 0.0219 0.0231;
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0.0177 0.0165 0.0155 0.0148 0.0144 0.0143 0.0144 0.0149 0.0156 0.0166 0.0178;0.0151 0.0138 0.0129 0.0122 0.0118 0.0116 0.0118 0.0122 0.0129 0.0139 0.0152;0.0143 0.0130 0.0121 0.0114 0.0110 0.0108 0.0110 0.0114 0.0121 0.0131 0.0144];
SOC=batt_nimh_SOC_old;
if SOC>1SOC=1;
elseif SOC<0SOC=0;
elseif isreal(SOC)==0end
if batt_nimh_P < 0 %chargebatt_nimh_I = abs(batt_nimh_P)/(batt_nimh_Volt);batt_nimh_R = interp2(SOC_R,N_R,Rcfit,SOC,batt_nimh_x0(2));batt_nimh_Ploss = batt_nimh_x0(1)*batt_nimh_I^2*batt_nimh_R;batt_nimh_SOC_new = SOC+(t*(abs(batt_nimh_P)-batt_nimh_Ploss))/(batt_nimh_Emax);
elseif batt_nimh_P > 0 %dischargebatt_nimh_I = abs(batt_nimh_P)/(batt_nimh_Volt);batt_nimh_R = interp2(SOC_R,N_R,Rdfit,SOC,batt_nimh_x0(2));batt_nimh_Ploss = batt_nimh_x0(1)*batt_nimh_I^2*batt_nimh_R;batt_nimh_SOC_new = SOC-(t*(abs(batt_nimh_P)+batt_nimh_Ploss))/(batt_nimh_Emax);
elsebatt_nimh_Ploss = 0;batt_nimh_SOC_new = SOC;
end
% Price modelbatt_nimh_price = batt_nimh_Volt*batt_nimh_x0(2)*240/1000;
% Limitsif max(abs(batt_nimh_P)) > batt_nimh_Pmax
batt_nimh_poss = 0; warning('error: max(abs(batt_nimh_P)) > batt_nimh_Pmax')elseif batt_nimh_SOC_new > 1
batt_nimh_poss = 0; warning('error: batt_nimh_SOC_new > 1')elseif batt_nimh_SOC_new < 0
batt_nimh_poss = 0; warning('error: batt_nimh_SOC_new < 0')else
batt_nimh_poss = 1;end
————————————
batt_vrla.m
function [batt_vrla_SOC_new,batt_vrla_Pmax,batt_vrla_Ploss,batt_vrla_M,batt_vrla_V,batt_vrla_price,batt_vrla_poss]=batt_vrla(batt_vrla_x0,batt_vrla_x,batt_vrla_SOC_old,batt_vrla_P,t)
% VRLA battery operation calculation%% Syntax: [batt_vrla_SOC_new,batt_vrla_Pmax,batt_vrla_Ploss,batt_vrla_M,batt_vrla_V,batt_vrla_price,% batt_vrla_poss]=batt_vrla(batt_vrla_x0,batt_vrla_x,batt_vrla_SOC_old,batt_vrla_P,t)%% batt_vrla_x0 vector of design parameters of battery:% [batt_vrla_n;batt_vrla_Qmax]% batt_vrla_x vector of fixed design parameters of battery:% []
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% batt_vrla_SOC_old State of Charge of previous time step% batt_vrla_P Power demanded from battery system at t=tnew% t time step size%% Result is [batt_vrla_SOC_new,batt_vrla_Pmax,batt_vrla_Ploss,batt_vrla_M,batt_vrla_V,% batt_vrla_price,batt_vrla_poss]%% batt_vrla_SOC_new State of Charge of battery (0-1)% batt_vrla_Pmax Maximum power output possible [W]% batt_vrla_Ploss Power losses at batt_vrla_SOC_old [W]% batt_vrla_M battery mass [kg]% batt_vrla_V battery volume [m^3]% batt_vrla_price battery price [�]% batt_vrla_poss battery is capable of delivering or storing power% batt_vrla_P (1=feasible design, 0=unfeasible design)%% Use batt_vrla_ini for fast generation of batt_vrla_x0 and batt_vrla_x%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)% Technical University of Eindhoven
% calculate volumebatt_vrla_V = batt_vrla_x0(1)*polyval([3.3794e-005 0.0012],batt_vrla_x0(2));
% calculate massbatt_vrla_M = batt_vrla_x0(1)*polyval([0.2660 1.8647],batt_vrla_x0(2));
% energy needed time step T-t to T:batt_vrla_E = batt_vrla_P*t;
% Cell voltage based on SOC_oldbatt_vrla_Volt = batt_vrla_x0(1)*polyval([1.4796 11.6151],batt_vrla_SOC_old);
% maximum total energy contentsbatt_vrla_Emax = 3600*batt_vrla_x0(2)*batt_vrla_Volt;batt_vrla_Pmax = batt_vrla_M*300;
% Efficiency calculationsSOC_R = [0:0.1:1];N_R = [12 16 18 65 85 91 104];Rdfit= [0.0688 0.0532 0.0423 0.0352 0.0313 0.0297 0.0298 0.0307 0.0318 0.0322 0.0312;
0.0672 0.0516 0.0406 0.0336 0.0297 0.0281 0.0282 0.0291 0.0302 0.0306 0.0296;0.0663 0.0508 0.0398 0.0328 0.0289 0.0273 0.0274 0.0283 0.0294 0.0298 0.0288;0.0475 0.0319 0.0210 0.0139 0.0100 0.0084 0.0085 0.0094 0.0105 0.0109 0.0099;0.0394 0.0239 0.0129 0.0059 0.0019 0.0004 0.0004 0.0014 0.0024 0.0029 0.0019;0.0370 0.0215 0.0105 0.0035 0 0 0 0 0 0.0004 0;0.0318 0.0162 0.0053 0 0 0 0 0 0 0 0];
Rcfit= [0.0416 0.0434 0.0436 0.0438 0.0458 0.0515 0.0624 0.0805 0.1073 0.1447 0.1944;0.0415 0.0433 0.0434 0.0436 0.0457 0.0513 0.0623 0.0803 0.1071 0.1445 0.1942;0.0414 0.0432 0.0433 0.0435 0.0456 0.0512 0.0622 0.0802 0.1070 0.1444 0.1941;0.0394 0.0412 0.0413 0.0416 0.0436 0.0492 0.0602 0.0782 0.1051 0.1424 0.1921;0.0386 0.0404 0.0405 0.0407 0.0428 0.0484 0.0594 0.0774 0.1042 0.1416 0.1913;0.0383 0.0401 0.0402 0.0405 0.0425 0.0482 0.0591 0.0771 0.1040 0.1414 0.1910;0.0378 0.0395 0.0397 0.0399 0.0420 0.0476 0.0586 0.0766 0.1034 0.1408 0.1905];
SOC=batt_vrla_SOC_old;
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if SOC>1SOC=1;
elseif SOC<0SOC=0;
elseif isreal(SOC)==0disp('SOC is NaN')
end
if batt_vrla_P < 0 %chargebatt_vrla_I = abs(batt_vrla_P)/(batt_vrla_Volt);batt_vrla_R = interp2(SOC_R,N_R,Rcfit,SOC,batt_vrla_x0(2));batt_vrla_Ploss = batt_vrla_x0(1)*batt_vrla_I^2*batt_vrla_R;batt_vrla_SOC_new = SOC+(t*(abs(batt_vrla_P)-batt_vrla_Ploss))/(batt_vrla_Emax);
elseif batt_vrla_P > 0 %dischargebatt_vrla_I = abs(batt_vrla_P)/(batt_vrla_Volt);batt_vrla_R = interp2(SOC_R,N_R,Rdfit,SOC,batt_vrla_x0(2));batt_vrla_Ploss = batt_vrla_x0(1)*batt_vrla_I^2*batt_vrla_R;batt_vrla_SOC_new = SOC-(t*(abs(batt_vrla_P)+batt_vrla_Ploss))/(batt_vrla_Emax);
elsebatt_vrla_Ploss = 0;batt_vrla_SOC_new = SOC;
end
% Price modelbatt_vrla_price = batt_vrla_Volt*batt_vrla_x0(2)*100/1000;
% Limitsif max(abs(batt_vrla_P)) > batt_vrla_Pmax
batt_vrla_poss = 0; warning('error: max(abs(batt_vrla_P)) > batt_vrla_Pmax')elseif batt_vrla_SOC_new > 1
batt_vrla_poss = 0; warning('error: batt_vrla_SOC_new > 1')elseif batt_vrla_SOC_new < 0
batt_vrla_poss = 0; warning('error: batt_vrla_SOC_new < 0')else
batt_vrla_poss = 1;end
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uc.m
function [uc_SOC_new,uc_Ploss,uc_Pmax,uc_M,uc_V,uc_price,uc_poss]=uc(uc_x0,uc_x,uc_SOC_old,uc_P,t)
% UltraCap operation calculation%% Syntax: [uc_SOC_new,uc_Ploss,uc_Pmax,uc_M,uc_V,uc_price,uc_poss]=% uc(uc_x0,uc_x,uc_SOC_old,uc_P,t)%% uc_x0 vector of design parameters of ultracap:% [uc_Ns;uc_Np;uc_cell_C]% uc_x vector of fixed design parameters of ultracap:% [uc_V_max]% uc_SOC_old State of Charge of previous time step [-]% uc_P Power demanded from ultracap system at time tnew [W]% t Time step size [s]%% Result is [uc_SOC_new,uc_Ploss,uc_Pmax,uc_M,uc_V,uc_price,uc_poss]%
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% uc_SOC_new State of Charge of ultracap (0-1)% uc_Ploss Power losses at uc_SOC_old [W]% uc_Pmax Maximum power output possible [W]% uc_M ultracap mass [kg]% uc_V ultracap volume [m^3]% uc_price ultracap price [�]% uc_poss ultracap is capable of delivering or storing power uc_P (1=feasible% design, 0=unfeasible design)%% Use uc_ini for fast generation of uc_x0 and uc_x%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)% Technical University of Eindhoven
% data validity checkuc_poss = 1;if uc_x0(3) < 0.5e3
disp('Design not feasible, 0.5e3 < UC capacity < 10e3')uc_poss = 0;return
elseif uc_x0(3) > 10e3disp('Design not feasible, 0.5e3 < UC capacity < 10e3')uc_poss = 0;return
end
% calculate volumeuc_V = uc_x0(2)*uc_x0(1)*polyval([5.8314e-008 1.6072e-005],uc_x0(3));
% calculate massuc_M = uc_x0(2)*uc_x0(1)*polyval([7.1506e-005 0.0269],uc_x0(3));
% energy needed time step T-t to T:uc_E = uc_P*t;
% pack voltage based on SOC_olduc_Volt = uc_x0(1)*uc_x(1)*uc_SOC_old;
% maximum total energy contentsuc_Emax = uc_x0(1)*uc_x0(2)*uc_x0(3);uc_Pmax = uc_M*3000; %Comment for unlimited Pmax, 3 kW per kg
% Efficiency calculations
if uc_P < 0 %chargeuc_I = abs(uc_P)/(uc_Volt);uc_R = uc_x0(1)*polyval([-2.8777e-007 0.0031],uc_x0(3))/uc_x0(2); %R=Ns*Rcell/Npuc_Ploss = uc_I^2*uc_R;uc_SOC_new = uc_SOC_old+(t*(abs(uc_P)-uc_Ploss))/(uc_Emax);
elseif uc_P > 0 %dischargeuc_I = abs(uc_P)/(uc_Volt);uc_R = uc_x0(1)*polyval([-2.8777e-007 0.0031],uc_x0(3))/uc_x0(2); %R=Ns*Rcell/Npuc_Ploss = uc_I^2*uc_R;uc_SOC_new = uc_SOC_old-(t*(abs(uc_P)+uc_Ploss))/(uc_Emax);
else
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uc_Ploss = 0;uc_SOC_new = uc_SOC_old;
end
% Price modeluc_price = 30*uc_Emax/1000;
% Limitsif max(abs(uc_P)) > uc_Pmax
uc_poss = 0; warning('error: max(abs(uc_P)) > uc_Pmax')elseif uc_SOC_new > 1
uc_poss = 0; warning('error: uc_SOC_new > 1')elseif uc_SOC_new < 0
uc_poss = 0; warning('error: uc_SOC_new < 0')else
uc_poss = 1;end
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caes_oil.m
function [caes_oil_SOC_new,caes_oil_pnew,caes_oil_Ploss,caes_oil_M,caes_oil_V,caes_oil_price,caes_oil_poss]=caes_oil(caes_oil_x0,caes_oil_x,caes_oil_SOC_old,caes_oil_P,t)
% Compressed Air Energy Storage operation calculation%% Syntax: [caes_oil_SOC_new,caes_oil_pnew,caes_oil_Ploss,caes_oil_M,caes_oil_V,caes_oil_price,% caes_oil_poss]=caes_oil(caes_oil_x0,caes_oil_x,caes_oil_SOC_old,caes_oil_P,t)%% caes_oil_x0 vector of design parameters of battery:% [caes_oil_R;caes_oil_t;caes_oil_h]% caes_oil_x vector of fixed design parameters of battery:% [caes_oil_cp;caes_oil_cv;caes_oil_sigmax;caes_oil_rho_tank;% caes_oil_rho_oil;caes_oil_S;caes_oil_p_pumpmax;caes_oil_n]% caes_oil_SOC_old State of Charge of previous time step [-]% caes_oil_P Power demanded from battery system at ime tnew [W]% t Time step size [s]%% Result is [caes_oil_SOC_new,caes_oil_pnew,caes_oil_Ploss,caes_oil_M,caes_oil_V,% caes_oil_price,caes_oil_poss]%% caes_oil_SOC_new State of Charge of battery (0-1)% caes_oil_pnew Pressure in tank [Pa]% caes_oil_Ploss Power losses at caes_oil_SOC_old [W]% caes_oil_M battery mass [kg]% caes_oil_V battery volume [m^3]% caes_oil_price battery price [�]% caes_oil_poss battery is capable of delivering or storing power caes_oil_P (1=feasible% design, 0=unfeasible design)%% Use caes_oil_ini for fast generation of caes_oil_x0 and caes_oil_x%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)% Technical University of Eindhoven
% calculate maximum pressure in tank caes_oil_pmaxif caes_oil_x0(2)/(caes_oil_x0(2)+caes_oil_x0(1)) > 0.1 %thick-walled stress theory
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alpha = ((caes_oil_x0(1)+caes_oil_x0(2))^2+caes_oil_x0(1))/((caes_oil_x0(1)+caes_oil_x0(2))^2-caes_oil_x0(1));
else %thin-walled stress theoryalpha = caes_oil_x0(1)/caes_oil_x0(2);
endcaes_oil_pmax = min([sqrt(((caes_oil_x(3))/(caes_oil_x(6)))^2/(alpha^2+alpha+1)) caes_oil_x(7)]);caes_oil_V0 = pi*caes_oil_x0(1)^2*caes_oil_x0(3);caes_oil_Vmin = ((1e5/caes_oil_pmax)*(caes_oil_V0^caes_oil_x(8)))^(1/caes_oil_x(8));caes_oil_Emax = -1/(caes_oil_x(8)-1)*(1e5*caes_oil_V0-caes_oil_pmax*caes_oil_Vmin);caes_oil_Eold = caes_oil_SOC_old*caes_oil_Emax;caes_oil_E = -caes_oil_P*t;
%Calculate old stategamma = caes_oil_SOC_old*1e5*caes_oil_V0+caes_oil_pmax*caes_oil_Vmin*(1-caes_oil_SOC_old);caes_oil_p1 = (1e5*(caes_oil_V0^caes_oil_x(8))/(gamma^(caes_oil_x(8))))^(1/(1-caes_oil_x(8)));caes_oil_V1 = gamma/caes_oil_p1;
%Calculate new statecaes_oil_SOC_new = caes_oil_SOC_old + caes_oil_E/caes_oil_Emax;gamma = (1-caes_oil_SOC_new)*1e5*caes_oil_V0+caes_oil_pmax*caes_oil_Vmin*caes_oil_SOC_new;% caes_oil_V0% caes_oil_x(8)% gamma% (1/(1-caes_oil_x(8)))caes_oil_pnew = (1e5*(caes_oil_V0^caes_oil_x(8))/(gamma^(caes_oil_x(8))))^(1/(1-caes_oil_x(8)));if isreal(caes_oil_pnew)==0
caes_oil_pnew=caes_oil_p1; %bugfixendcaes_oil_Vnew = gamma/caes_oil_pnew;
%Losses of 2% of powercaes_oil_Ploss = 0.00*abs(caes_oil_P);
% calculate volumecaes_oil_V = pi*(caes_oil_x0(1)+caes_oil_x0(2))^2*(caes_oil_x0(3)+2*caes_oil_x0(2)) +
1.1*(pi*caes_oil_x0(1)^2*caes_oil_x0(3)-caes_oil_Vmin);
% calculate masscaes_oil_M = pi*caes_oil_x(4)*((caes_oil_x0(1)+caes_oil_x0(2))^2*caes_oil_x0(3)-
caes_oil_x0(1)^2*(caes_oil_x0(3)-2*caes_oil_x0(2))) +caes_oil_x(5)*(pi*caes_oil_x0(1)^2*caes_oil_x0(3)-caes_oil_Vmin);
% Price modelcaes_oil_price = 10*caes_oil_M;
% Limitscaes_oil_Pmax = 250e5; %determined by pumpcaes_oil_SOC_new;if max(abs(caes_oil_P)) > caes_oil_Pmax
caes_oil_poss = 0; warning('error: max(abs(caes_oil_P)) > caes_oil_Pmax')elseif caes_oil_SOC_new > 1
caes_oil_poss = 0; warning('error: caes_oil_SOC_new > 1')elseif caes_oil_SOC_new < 0
caes_oil_poss = 0; warning('error: caes_oil_SOC_new < 0')else
caes_oil_poss = 1;end
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cvt.m
function [cvt_Ploss,cvt_Pfw,cvt_Pmax,cvt_M,cvt_V,cvt_price,cvt_poss]=cvt(w_fw,cvt_P,t)
% CVT operation calculation%% Syntax: [cvt_Ploss,cvt_Pfw,cvt_Pmax,cvt_M,cvt_V,cvt_price,cvt_poss]=% cvt(w_fw,cvt_P,t)%% w_fw Turning speed on the IN side [rad/s]% cvt_P Power demanded on OUT side [W]% t Time step size [s]%% Result is [cvt_Ploss,cvt_Pfw,cvt_Pmax,cvt_M,cvt_V,cvt_price,cvt_poss]%% cvt_Ploss Power losses at t=t_new% cvt_Pfw Power demanded from IN side of cvt [W]% cvt_Pmax Maximum power throughput corresponding to IN speed [W]% cvt_M cvt mass [kg]% cvt_V cvt volume [m^3]% cvt_price cvt price [�]% cvt_poss cvt is capable of delivering or storing power cvt_P (1=feasible% design, 0=unfeasible design)%% Use cvt_ini for fast generation of cvt_x0 and cvt_x%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)% Technical University of Eindhoven
% INPUT torque vector corresponding to columns of efficiency & loss mapsgb_map_trq=[0 10 20 30 40 50 60 70]*4.448/3.281; % (N*m)
% INPUT speed vector corresponding to rows of efficiency & loss mapsgb_map_spd=[0 1500 2000 2500 3000 3500 4000 4500 5000]*2*pi/60; % (rad/s)
% Error checkingif w_fw>max(gb_map_spd)
warning(['w_fw>max(gb_map_spd), w_fw=',num2str(w_fw),' rad/s,max(gb_map_spd)=',num2str(max(gb_map_spd)),' rad/s'])cvt_poss=0;return
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LOSSES AND EFFICIENCIES%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% (--), efficiency map that goes with the 1st largest pulley ratiogb_map1_eff_raw=flipud([0.8228 0.8228 0.8228 0.8228 0.8228 0.8243 0.8243 0.8243 00.8228 0.8228 0.8228 0.8228 0.8228 0.8243 0.8243 0.8243 00.8228 0.8228 0.8228 0.8228 0.8228 0.8243 0.8243 0.8243 00.8458 0.8458 0.8458 0.8507 0.8079 0.8325 0.8325 0.8325 00.8440 0.8509 0.8114 0.8465 0.8244 0.7807 0.7807 0.7807 00.8085 0.8177 0.7823 0.8248 0.7747 0.7747 0.7747 0.7747 00.7525 0.7525 0.7525 0.7525 0.7525 0.7525 0.7525 0.7525 0
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0 0 0 0 0 0 0 0 0])';
% (--), efficiency map that goes with the 2nd largest pulley ratiogb_map2_eff_raw=flipud([0.8781 0.8781 0.8781 0.8844 0.8300 0.8300 0.8300 0.8300 00.8781 0.8781 0.8781 0.8844 0.8300 0.8300 0.8300 0.8300 00.8889 0.9078 0.8563 0.9003 0.8329 0.8717 0.8167 0.8167 00.8692 0.8606 0.8668 0.8654 0.8286 0.8351 0.8211 0.8211 00.8655 0.8694 0.8448 0.8655 0.8337 0.8002 0.8002 0.8002 00.8248 0.8451 0.8066 0.8425 0.8051 0.8051 0.8051 0.8051 00.7464 0.7464 0.7464 0.7464 0.7464 0.7464 0.7464 0.7464 00 0 0 0 0 0 0 0 0])';
% (--), efficiency map that goes with the 3rd largest pulley ratiogb_map3_eff_raw=flipud([0.8991 0.8991 0.8991 0.8917 0.8842 0.8768 0.8768 0.8768 00.9026 0.9026 0.9026 0.8943 0.8533 0.8747 0.8635 0.8635 00.9208 0.9208 0.8786 0.8886 0.8464 0.8415 0.8151 0.8151 00.8894 0.8894 0.8797 0.8752 0.8491 0.8438 0.8208 0.8208 00.8906 0.8906 0.8720 0.8789 0.8586 0.8200 0.8200 0.8200 00.8774 0.8774 0.8326 0.8558 0.8236 0.8236 0.8236 0.8236 00.8774 0.8774 0.8326 0.8558 0.8236 0.8236 0.8236 0.8236 00 0 0 0 0 0 0 0 0])';
% (--), efficiency map that goes with the 4th largest pulley ratiogb_map4_eff_raw=flipud([0.9227 0.9227 0.9227 0.9059 0.8891 0.8724 0.8768 0.8768 00.9241 0.9241 0.9241 0.9033 0.8840 0.8840 0.8882 0.8882 00.9177 0.9177 0.9002 0.9050 0.8699 0.8677 0.8344 0.8589 00.9194 0.9194 0.9043 0.8948 0.8872 0.8546 0.8315 0.8315 00.9227 0.9227 0.9103 0.8896 0.8702 0.8539 0.8539 0.8539 00.9169 0.9169 0.8907 0.8786 0.8619 0.8619 0.8619 0.8619 00.9169 0.9169 0.8907 0.8786 0.8619 0.8619 0.8619 0.8619 00 0 0 0 0 0 0 0 0])';
% (--), efficiency map that goes with the 5th largest pulley ratiogb_map5_eff_raw=flipud([0.9230 0.9230 0.9230 0.8907 0.8907 0.8907 0.8907 0.8907 00.9230 0.9230 0.9230 0.8907 0.8907 0.8907 0.8907 0.8907 00.9156 0.9156 0.8912 0.8757 0.8757 0.8757 0.8757 0.8757 00.9379 0.9379 0.9099 0.8573 0.8573 0.8573 0.8573 0.8573 00.9471 0.9471 0.9099 0.8529 0.8529 0.8529 0.8529 0.8529 00.9097 0.9097 0.8908 0.7952 0.7952 0.7952 0.7952 0.7952 00.9097 0.9097 0.8908 0.7952 0.7952 0.7952 0.7952 0.7952 00 0 0 0 0 0 0 0 0])';
% vector of gear ratios corresponding to the above efficiency mapsgb_ratio=[2.5 2 1.5 1 0.5];
%eff_map_w_fw rows:r columns:torqueeff_map_w_fw=[interp2(gb_map_trq,gb_map_spd,gb_map1_eff_raw,gb_map_trq,w_fw);
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interp2(gb_map_trq,gb_map_spd,gb_map2_eff_raw,gb_map_trq,w_fw);interp2(gb_map_trq,gb_map_spd,gb_map3_eff_raw,gb_map_trq,w_fw);interp2(gb_map_trq,gb_map_spd,gb_map4_eff_raw,gb_map_trq,w_fw);interp2(gb_map_trq,gb_map_spd,gb_map5_eff_raw,gb_map_trq,w_fw)];
%Find OOLfor m=1:length(gb_map_trq)
%eff_map_w_fw(:,m),m[nu(m),x]=max(eff_map_w_fw(:,m));%nu,xr(m)=gb_ratio(x);
end
cvt_P_w_fw=nu.*w_fw.*gb_map_trq;
P_fw=w_fw*gb_map_trq;
%charge stateif cvt_P>0
r_opt=interp1(cvt_P_w_fw,r,cvt_P,'linear','extrap');nu_opt=interp1(cvt_P_w_fw,nu,cvt_P,'linear','extrap');w_out=w_fw/r_opt;T_out=cvt_P/w_out;P_fw_w_fw=cvt_P/nu_opt;T_fw=P_fw_w_fw/w_fw;Ploss=(1-nu_opt)*cvt_P;
%discharge stateelseif cvt_P<0
r_opt=interp1(cvt_P_w_fw,r,-cvt_P,'linear','extrap');nu_opt=interp1(cvt_P_w_fw,nu,-cvt_P,'linear','extrap');w_out=w_fw/r_opt;T_out=cvt_P/w_out;P_fw_w_fw=cvt_P*nu_opt;T_fw=P_fw_w_fw/w_fw;Ploss=(1-nu_opt)*-cvt_P;
%zero stateelse
r_opt=1;nu_opt=0;w_out=w_fw/r_opt;T_out=cvt_P/w_out;P_fw_w_fw=cvt_P*nu_opt;T_fw=P_fw_w_fw/w_fw;Ploss=P_fw_w_fw-cvt_P;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% OTHER DATA%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%cvt_M=127/2.205;% = 57.6 ~= 58 kg, mass of CVT and control boxes
cvt_V=0.036; %40x30x30 cm guess
cvt_Ploss=Ploss;
cvt_Pfw=P_fw_w_fw;
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cvt_price=cvt_M*10; %�10/kg guess
cvt_Pmax=max(cvt_P_w_fw);
cvt_T_max=max(gb_map_trq);
cvt_T=T_fw;
% Error checkcvt_poss=1;if w_fw>max(gb_map_spd)
warning('CVT maximum speed exceeded, use larger reduction gear')cvt_poss=0;
elseif abs(T_fw)>max(gb_map_trq)warning(['CVT maximum torque exceeded, T_fw= ',num2str(T_fw),',
max(gb_map_trq)= ',num2str(max(gb_map_trq))])cvt_poss=0;
elseif isequal(num2str(Ploss),'NaN')==1warning('warning: CVT Ploss==NaN, something is wrong... try modifying reduction gear
between FW and CVT')cvt_poss=0;
elseif abs(cvt_P)>max(cvt_P_w_fw)warning(['abs(cvt_P)>max(cvt_P_w_fw), cvt_P= ',num2str(cvt_P),',
max(cvt_P_w_fw)= ',num2str(max(cvt_P_w_fw))])cvt_poss=0;
end
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motor_ac.m
function [motor_ac_Ploss,motor_ac_M,motor_ac_V,motor_ac_price,motor_ac_poss]=motor_ac(motor_ac_x0,motor_ac_x,motor_ac_P,t)
% AC (induction) motor operation calculation%% Syntax: [motor_ac_Ploss,motor_ac_M,motor_ac_V,motor_ac_price,motor_ac_poss]=% motor_ac(motor_ac_x0,motor_ac_x,motor_ac_P,t)%% motor_ac_x0 vector of variable design parameters of ac motor:% [motor_ac_Pmax]% motor_ac_x vector of fixed design parameters of ac motor:% []% motor_ac_P Power demanded from ac motor system at time tnew [W]% t Time step size [s]%% Result is [motor_ac_Ploss,motor_ac_M,motor_ac_V,motor_ac_price,motor_ac_poss]%% motor_ac_Ploss Power losses [W]% motor_ac_M AC motor mass [kg]% motor_ac_V AC motor volume [m^3]% motor_ac_price AC motor price [�]% motor_ac_poss AC motor is capable of delivering or storing power% motor_ac_P (1=feasible design, 0=unfeasible design)%% Use motor_ac_ini for fast generation of motor_ac_x0 and motor_ac_x%
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% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)% Technical University of Eindhoven
% calculate volumemotor_ac_V = polyval([0.00012248 0.014645],motor_ac_x0(1));
% calculate massmotor_ac_M = polyval([0.1437 69.314],motor_ac_x0(1));
% energy needed time step T-t to T:motor_ac_E = motor_ac_P*t;
% Efficiency calculationsmotor_ac_eta = -0.013493*abs(10*motor_ac_P/motor_ac_x0(1))+1.0208+1/(-3.2742*(abs(10*
motor_ac_P/motor_ac_x0(1))+0.77719));motor_ac_Ploss = ((1/motor_ac_eta)-1)*abs(motor_ac_P);
% Price modelmotor_ac_price = 15*motor_ac_x0(1);
% Limitsif max(abs(motor_ac_P)) > motor_ac_x0(1)
motor_ac_poss = 0; warning('error: max(abs(motor_ac_P)) > motor_ac_Pmax')else
motor_ac_poss = 1;end
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motor_pm.m
function [motor_pm_Ploss,motor_pm_M,motor_pm_V,motor_pm_price,motor_pm_poss]=motor_pm(motor_pm_x0,motor_pm_x,motor_pm_P,t)
% PM (Permanent Magnet) motor operation calculation%% Syntax: [motor_pm_Ploss,motor_pm_M,motor_pm_V,motor_pm_price,motor_pm_poss]=% motor_pm(motor_pm_x0,motor_pm_x,motor_pm_P,t)%% motor_pm_x0 vector of variable design parameters of pm motor:% [motor_pm_Pmax]% motor_pm_x vector of fixed design parameters of pm motor:% []% motor_pm_P Power demanded from pm motor system at time tnew [W]% t Time step size [s]%% Result is [motor_pm_Ploss,motor_pm_M,motor_pm_V,motor_pm_price,motor_pm_poss]%% motor_pm_Ploss Power losses [W]% motor_pm_M PM motor mass [kg]% motor_pm_V PM motor volume [m^3]% motor_pm_price PM motor price [�]% motor_pm_poss PM motor is capable of delivering or storing power% motor_pm_P (1=feasible design, 0=unfeasible design)%% Use motor_pm_ini for fast generation of motor_pm_x0 and motor_pm_x%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)
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% Technical University of Eindhoven
% calculate volumemotor_pm_V = polyval([0.00016842 0.0067788],motor_pm_x0(1)); %size in kW
% calculate massmotor_pm_M = polyval([1.3789 3.6362],motor_pm_x0(1)); %size in kW
% energy needed time step T-t to T:motor_pm_E = motor_pm_P*t; %P in W
% Efficiency calculationsmotor_pm_eta = -0.004349*abs(10*motor_pm_P/motor_pm_x0(1))+ 0.97494+1/
(-4.3141*(abs(10*motor_pm_P/motor_pm_x0(1))+0.51984));motor_pm_Ploss = ((1/motor_pm_eta)-1)*abs(motor_pm_P); %Ploss in kW
% Price modelmotor_pm_price = 18*motor_pm_x0(1); %size in kW
% Limitsif max(abs(motor_pm_P)) > motor_pm_x0(1)
motor_pm_poss = 0; warning('error: max(abs(motor_pm_P)) > motor_pm_Pmax')else
motor_pm_poss = 1;end
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pump.m
function [pump_Ploss,pump_pmax,pump_pmin,pump_M,pump_V,pump_price,pump_poss]=pump(pump_x0,pump_x,pump_P,pump_p,t)
% Gear pump operation calculation%% Syntax: [pump_Ploss,pump_pmax,pump_M,pump_V,pump_price,pump_poss]=% pump(pump_x0,pump_x,pump_P,pump_p,t)%% pump_x0 vector of design parameters of pump:% [pump_vg;pump_design]% pump_x vector of fixed design parameters of pump:% []% pump_P Power demanded from pump system at time t_new [W]% pump_p Pressure in the tank at time t_old [Pa]% t Time step size [s]%% Result is [pump_Ploss,pump_pmax,pump_pmin,pump_M,pump_V,pump_price,pump_poss]%% pump_Ploss Power losses at time t_new [W]% pump_pmax Maximum allowable pump pressure (tank cannot go higher) [Pa]% pump_pmin Minimum allowable pump pressure [Pa]% pump_M pump mass [kg]% pump_V pump volume [m^3]% pump_price pump price [�]% pump_poss pump is capable of delivering or storing power pump_P (1=feasible% design, 0=unfeasible design)%% Use pump_ini for fast generation of pump_x0 and pump_x
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%% Written by D.Hoekstra, version 2005.06.23 (yyyy.mm.dd)% Technical University of Eindhoven
pump_poss = 1;
% Pump dataif pump_x0(2)==1 %MRG3 design maps are defined as pump function, e.g. input w,T output Q,p.
pump_press=1e5*[0 30 180 210 220 240 260]; % (Pa)
% Volume displacement range over which data is definedpump_speed=[0 700 2600 2900 3100 3200 3600 3900]*(2*pi/60); % (rad/s)
% Displacement Vg (theoretical volume per turn) range over which data is definedpump_vg=[20.9 23.4 25.9 30.1 32.6 37.6 45.2]*1e-6; % [m^3]
% Input torque map indexed by pump_press (columns) and pump_speed (rows) [Nm]pump_in_torque(:,:,1)=[
[0 13.4200 66.8300 77.5200 81.0800 88.2000 95.3200];[0 13.4200 66.8300 77.5200 81.0800 88.2000 95.3200];[0 13.4200 66.8300 77.5200 81.0800 88.2000 95.3200];[0 13.4200 66.8300 77.5200 81.0800 88.2000 95.3200];[0 13.4200 66.8300 77.5200 81.0800 88.2000 95.3200];[0 13.4200 66.8300 77.5200 81.0800 88.2000 95.3200];[0 13.4200 66.8300 77.5200 81.0800 88.2000 95.3200];[0 13.4200 66.8300 77.5200 81.0800 88.2000 95.3200]];
pump_in_torque(:,:,2)=[[0 16.1000 74.0300 85.6200 89.4800 97.2000 104.9000];[0 16.1000 74.0300 85.6200 89.4800 97.2000 104.9000];[0 16.1000 74.0300 85.6200 89.4800 97.2000 104.9000];[0 16.1000 74.0300 85.6200 89.4800 97.2000 104.9000];[0 16.1000 74.0300 85.6200 89.4800 97.2000 104.9000];[0 16.1000 74.0300 85.6200 89.4800 97.2000 104.9000];[0 16.1000 74.0300 85.6200 89.4800 97.2000 104.9000];[0 16.1000 74.0300 85.6200 89.4800 97.2000 104.9000]];
pump_in_torque(:,:,3)=[[0 17.9500 81.8300 94.6000 98.8600 107.4000 115.9000];[0 17.9500 81.8300 94.6000 98.8600 107.4000 115.9000];[0 17.9500 81.8300 94.6000 98.8600 107.4000 115.9000];[0 17.9500 81.8300 94.6000 98.8600 107.4000 115.9000];[0 17.9500 81.8300 94.6000 98.8600 107.4000 115.9000];[0 17.9500 81.8300 94.6000 98.8600 107.4000 115.9000];[0 17.9500 81.8300 94.6000 98.8600 107.4000 115.9000];[0 17.9500 81.8300 94.6000 98.8600 107.4000 115.9000]];
pump_in_torque(:,:,4)=[[0 20.0200 95.9700 111.2000 116.2000 126.3000 NaN];[0 20.0200 95.9700 111.2000 116.2000 126.3000 NaN];[0 20.0200 95.9700 111.2000 116.2000 126.3000 NaN];[0 20.0200 95.9700 111.2000 116.2000 126.3000 NaN];[0 20.0200 95.9700 111.2000 116.2000 126.3000 NaN];[0 20.0200 95.9700 111.2000 116.2000 126.3000 NaN];[0 20.0200 95.9700 111.2000 116.2000 126.3000 NaN];[0 20.0200 95.9700 111.2000 116.2000 126.3000 NaN]];
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pump_in_torque(:,:,5)=[[0 21.7000 104.7000 121.2000 126.8000 NaN NaN];[0 21.7000 104.7000 121.2000 126.8000 NaN NaN];[0 21.7000 104.7000 121.2000 126.8000 NaN NaN];[0 21.7000 104.7000 121.2000 126.8000 NaN NaN];[0 21.7000 104.7000 121.2000 126.8000 NaN NaN];[0 21.7000 104.7000 121.2000 126.8000 NaN NaN];[0 21.7000 104.7000 121.2000 126.8000 NaN NaN];[0 21.7000 104.7000 121.2000 126.8000 NaN NaN]];
pump_in_torque(:,:,6)=[[0 23.1400 120.5000 140.0000 NaN NaN NaN];[0 23.1400 120.5000 140.0000 NaN NaN NaN];[0 23.1400 120.5000 140.0000 NaN NaN NaN];[0 23.1400 120.5000 140.0000 NaN NaN NaN];[0 23.1400 120.5000 140.0000 NaN NaN NaN];[0 23.1400 120.5000 140.0000 NaN NaN NaN];[0 23.1400 120.5000 140.0000 NaN NaN NaN];[0 23.1400 120.5000 140.0000 NaN NaN NaN]];
pump_in_torque(:,:,7)=[[0 28.7400 142.3000 NaN NaN NaN NaN];[0 28.7400 142.3000 NaN NaN NaN NaN];[0 28.7400 142.3000 NaN NaN NaN NaN];[0 28.7400 142.3000 NaN NaN NaN NaN];[0 28.7400 142.3000 NaN NaN NaN NaN];[0 28.7400 142.3000 NaN NaN NaN NaN];[0 28.7400 142.3000 NaN NaN NaN NaN];[0 28.7400 142.3000 NaN NaN NaN NaN]];
% Output flow map indexed by pump_press (columns) and pump_speed (rows) [m^3/s]pump_out_flow(:,:,1)=[
0 0;14.48 12.36;52.74 50.71;58.78 56.77;62.81 60.81;64.82 62.82;72.88 70.9;NaN NaN;]/(60e3);
pump_out_flow(:,:,2)=[[0 16.1 59.4 66.23 70.79 73.07 NaN NaN]'...
[0 14.36 57.6 64.43 68.98 71.25 NaN NaN]'...]/(60e3);
pump_out_flow(:,:,3)=[[0 18.69 65.58 72.98 NaN NaN NaN NaN]'...
[0 16.27 63.94 71.46 NaN NaN NaN NaN]'...]/(60e3);
pump_out_flow(:,:,4)=[[0 21.09 77 85.82 91.71 94.65 106.4 115.2]'...
[0 18.04 74.62 83.55 89.51 92.49 104.4 113.3]'...]/(60e3);
pump_out_flow(:,:,5)=[
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[0 22.75 82.97 92.47 98.81 102 114.7 NaN]'...[0 20.34 81.05 90.63 97.02 100.2 113 NaN]'...
]/(60e3);
pump_out_flow(:,:,6)=[[0 25.58 97.39 108.7 116.3 NaN NaN NaN]'...
[0 22.75 95.59 107.1 114.8 NaN NaN NaN]'...]/(60e3);
pump_out_flow(:,:,7)=[[0 33.13 119.1 NaN NaN NaN NaN NaN]'...
[0 30.06 114.6 NaN NaN NaN NaN NaN]'...]/(60e3);
%pressure limits indexed by vg: 1e5*[0 30 180 210 220 240 260];pump_max_press=1e5*[260 260 260 240 220 210 180];pump_min_press=1e5*[30 30 30 30 30 30 30];
for a=1:length(pump_out_flow(:,1,1));for b=1:length(pump_vg);
P(a,:,b)=polyval(polyfit(1e5*[0 pump_max_press(b)], pump_out_flow(a,:,b),1),pump_press);
endendclear pump_out_flow; pump_out_flow=P; clear P;
% LIMITSpump_min_speed=0;pump_max_speed=2900*(2*pi/60);pump_max_trq=300;
% OTHER DATApump_mass=[3.9 3.9 3.9 4.3 4.3 4.3 4.3]; % (kg), indexed by pump_Vgpump_volume=1e-9*155*186*[148 154 162 166 172 182 190]; % (m^3), indexed by pump_Vgpump_price = 100;
elseif pump_x0(2)==2 %MRG4 design maps are defined as pump function, e.g. input w,T output Q,p.pump_press=1e5*[0 10 50 100 150 175 200 210]; % (Pa)pump_speed=[0 1450 2900]*(2*pi/60); % (rad/s)pump_vg=[40 50 63 70 80 100]*1e-6; % [m^3]% Input torque map indexed by pump_press (columns) and pump_speed (rows) [Nm]pump_in_torque(:,:,1)=[
1e3/(2*pi*1450/60)*[0 1.6 5 9.87 14.73 17.23 19.74 21];1e3/(2*pi*1450/60)*[0 1.6 5 9.87 14.73 17.23 19.74 21];1e3/(2*pi*1450/60)*[0 1.6 5 9.87 14.73 17.23 19.74 21]];
pump_in_torque(:,:,2)=[1e3/(2*pi*1450/60)*[0 1.9 6.68 12.7 18.75 21.86 24.9 26.12];1e3/(2*pi*1450/60)*[0 1.9 6.68 12.7 18.75 21.86 24.9 26.12];1e3/(2*pi*1450/60)*[0 1.9 6.68 12.7 18.75 21.86 24.9 26.12]];
pump_in_torque(:,:,3)=[1e3/(2*pi*1450/60)*[0 2.26 8.65 16.63 24.52 28.47 32.3 34];1e3/(2*pi*1450/60)*[0 2.26 8.65 16.63 24.52 28.47 32.3 34];1e3/(2*pi*1450/60)*[0 2.26 8.65 16.63 24.52 28.47 32.3 34]];
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pump_in_torque(:,:,4)=[1e3/(2*pi*1450/60)*[0 2.88 10 18.75 27.6 32 36.4 38];1e3/(2*pi*1450/60)*[0 2.88 10 18.75 27.6 32 36.4 38];1e3/(2*pi*1450/60)*[0 2.88 10 18.75 27.6 32 36.4 38]];
pump_in_torque(:,:,5)=[1e3/(2*pi*1450/60)*[0 3 11.2 21 31 36 NaN NaN];1e3/(2*pi*1450/60)*[0 3 11.2 21 31 36 NaN NaN];1e3/(2*pi*1450/60)*[0 3 11.2 21 31 36 NaN NaN]];
pump_in_torque(:,:,6)=[1e3/(2*pi*1450/60)*[0 3.72 13.5 25.8 37.9 NaN NaN NaN];1e3/(2*pi*1450/60)*[0 3.72 13.5 25.8 37.9 NaN NaN NaN];1e3/(2*pi*1450/60)*[0 3.72 13.5 25.8 37.9 NaN NaN NaN]];
% Output flow map indexed by pump_press (columns) and pump_speed (rows) [m^3/s]pump_out_flow(:,:,1)=[
0 0 0 0 0 0 0 0;57.5 57.5 56.5 54.6 52.7 51.2 50.3 50;2*[57.5 57.5 56.5 54.6 52.7 51.2 50.3 50]]/(60e3);
pump_out_flow(:,:,2)=[0 0 0 0 0 0 0 0;72.5 72.5 71.5 69.6 67.6 66.7 65.7 65.25;2*[72.5 72.5 71.5 69.6 67.6 66.7 65.7 65.25]]/(60e3);
pump_out_flow(:,:,3)=[0 0 0 0 0 0 0 0;91.8 91.8 90.4 88.4 86 84 82.6 82;2*[91.8 91.8 90.4 88.4 86 84 82.6 82]]/(60e3);
pump_out_flow(:,:,4)=[0 0 0 0 0 0 0 0;101.5 101.5 99.5 97.6 96.6 93.7 91.3 90.4;2*[101.5 101.5 99.5 97.6 96.6 93.7 91.3 90.4]]/(60e3);
pump_out_flow(:,:,5)=[0 0 0 0 0 0 0 0;116 116 115 112.6 110 108.5 NaN NaN;2*[116 116 115 112.6 110 108.5 NaN NaN]]/(60e3);
pump_out_flow(:,:,6)=[0 0 0 0 0 0 0 0;144.5 144.5 143 141 138 NaN NaN NaN;2*[144.5 144.5 143 141 138 NaN NaN NaN]]/(60e3);
% LIMITSpump_min_speed=0;pump_max_speed=2900*(2*pi/60);pump_max_trq=300;
%pressure limits indexed by vg:pump_max_press=1e5*[210 210 210 210 175 150];pump_min_press=1e5*[10 10 10 10 10 10];
% OTHER DATA
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pump_mass=[15 15.4 16.4 17 17.4 18.4]; % (kg), indexed by pump_Vgpump_volume=1e-9*155*186*[148 154 162 166 172 182]; % (m^3), indexed by pump_Vgpump_price = 200;
else disp(['Please choose 1 or 2 for the pump design instead of ',num2str(pump_x0(2))])pump_poss=0;
end
% eliminate unfeasible designsif pump_x0(1)>max(pump_vg)
warning('Chosen pump size is too large for pump design possibility')pump_poss=0;
elseif pump_x0(1)<min(pump_vg)warning('Chosen pump size is too small for pump design possibility')pump_poss=0;
end
%% CALCULATE INTERPOLATED PUMP DESIGN[x,y,z]=meshgrid(pump_press,pump_speed,pump_vg);
%% calculate maximum pressure:maxindex=find(pump_vg>pump_x0(1));pump_pmax=pump_max_press(maxindex(1));pump_pmin=pump_min_press(maxindex(1)-1);
%Functioning:if pump_P < 0 %charging state, so pump function
for m=1:length(pump_speed)power(m)=pump_speed(m)*interp3(x,y,z,-pump_in_torque,pump_p,pump_speed(m),pump_x0(1));
endpump_w = interp1(power,pump_speed,pump_P,'linear','extrap');if isequal(num2str(pump_w),'NaN')==1
warning('Unfeasible design, speed not achievable by pump design. Try increasing initial SOC')pump_poss=0;pump_w=max(pump_speed);
endpump_T = pump_P/pump_w;pump_Q = interp3(x,y,z,pump_out_flow,pump_p,pump_w,pump_x0(1));pump_Ploss = abs(pump_P)-abs(pump_p*pump_Q);
elseif pump_P > 0 %discharge, so generator functionfor m=1:length(pump_speed)
power(m)=pump_speed(m)*interp3(x,y,z,pump_in_torque,pump_p,pump_speed(m),pump_x0(1));endpump_w = interp1(power,pump_speed,pump_P,'linear','extrap');if isequal(num2str(pump_w),'NaN')==1
warning('Unfeasible design, speed not achievable by pump design. Try increasing initial SOC')pump_poss=0;pump_w=max(pump_speed);
endpump_T = pump_P/pump_w;
pump_Q = interp3(x,y,z,-pump_out_flow,pump_p,pump_w,pump_x0(1));pump_Ploss = abs(pump_P)-abs(pump_p*pump_Q);
else % Neither charge nor dischargepump_T = 0;pump_Q = 0;
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pump_Ploss = 0;end
%% Other propertiespump_M = interp1(pump_vg,pump_mass,pump_x0(1));pump_V = interp1(pump_vg,pump_volume,pump_x0(1));
%% Errorstatesif pump_max_press<pump_p
warning('Pressure not achievable by pump, please choose other design')pump_poss = 0;
elseif min(pump_press)>pump_pwarning('Pressure too low for pump')pump_poss = 0;
end
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S1_ac_opt.m
% Electrical flywheel optimization, AC motor%% Optimization procedure for the design of a supercritical flywheel%% Written by D.Hoekstra, version 2005.03.01 (yyyy.mm.dd)% Technical University of Eindhoven
warning off %if having problems using the procedure, put a % in front for more information on errors
% start design (must be feasible!)fw_suprcr_d_0 = 0.05;fw_suprcr_Ro_0 = 0.3;fw_suprcr_Ri_0 = 0.1;fw_suprcr_h_0 = 0.05;motor_ac_kW_0 = max([max(P_s) -min(P_s)])/1000; %first guess for motor is minimum size
%needed for power profile
S1_ac_x0=[fw_suprcr_d_0; fw_suprcr_Ro_0; fw_suprcr_Ri_0; fw_suprcr_h_0 ;motor_ac_kW_0];S1_ac_max=[M_max V_max E_max];alpha=[alpha_M alpha_V alpha_E alpha_nu];
%%%%%%%%%%%%%%%%%%%%%%%% Set boundary conditions%%%%%%%%%%%%%%%%%%%%%%%A=[0.5 0 -1 0 0;0 -1 1 0 0;1 0 -1 0 0]; B=[0 -0.01 0]';%A=[];B=[]';Aeq=[];Beq=[]';LB=[0.03 0.001 0.001 0.001 motor_ac_kW_0]; UB=[1 1 1 1 200];NONLCON=[];
%%%%%%%%%%%%%%%%%%%%%%%% Set optimisation options%%%%%%%%%%%%%%%%%%%%%%%OPTIONS=optimset('LargeScale','off','Display','iter','MaxFunEvals',200,'Diagnostics','on');%OPTIONS=optimset('Display','iter');
%%%%%%%%%%%%%%%%%%%%%%%% SQP optimization
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%%%%%%%%%%%%%%%%%%%%%%%[X,FVAL,EXITFLAG,OUTPUT,LAMBDA]=fmincon('S1_ac',S1_ac_x0,A,B,Aeq,Beq,LB,UB,NONLCON,OPTIONS,P_s,t,SOC,r,alpha,S1_ac_max,motor_ac_kW_0);
%%%%%%%%%%%%%%%%%%%%%%%% Feasibility check:%%%%%%%%%%%%%%%%%%%%%%%warning on %No warnings should occur for the optimized design X% It is even possible to achieve a feasible design by taking r larger and larger and% taking the guess X from the previous time step,% if having problem with feasibility. lim T (1/r -> 0) = f% This could also be automated by the way...[T,g,S1_ac_Ploss,S1_ac_SOC,S1_ac_M,S1_ac_V,S1_ac_price,S1_ac_eff] = S1_ac(X,P_s,t,SOC,r,alpha,S1_ac_max);Xif g(1)>0
disp(['WARNING: Infeasible design, g1>0 (g1=',num2str(g(1)),'), SOC<0 in cycle']);disp(['If the value of g is small, this might not be a problem'])disp(['If it is a problem, try increasing the value of r while taking above values as first guess'])
endif g(2)>0
disp(['WARNING: Infeasible design, g2>0 (g2=',num2str(g(2)),'), SOC>1 in cycle'])disp(['If the value of g is small, this might not be a problem'])disp(['If it is a problem, try increasing the value of r while taking above values as first guess'])
end
disp(['Mass= ',num2str(S1_ac_M),' kg'])disp(['Volume= ',num2str(1000*(S1_ac_V)),' liters'])disp(['Price= ',num2str(S1_ac_price),' euro'])disp(['Mean eff= ',num2str(100*S1_ac_eff,3),' %'])
%%%%%%%%%%%%%%%%%%%%%%%% Plotting%%%%%%%%%%%%%%%%%%%%%%%% figure;% subplot(3,1,1); plot([P_s']/1000); ylabel('P [kW]'); title('Optimized supercritical% flywheel with ac motor CVT')% subplot(3,1,2); plot(S1_ac_SOC); ylabel('SOC [-]'); V = axis; V(3)=0; V(4)=1; axis(V);% subplot(3,1,3); plot(S1_ac_Ploss'); xlabel('time [s]'); ylabel('Ploss [W]');% legend(['efficiency is ',num2str(100*S1_ac_eff,3),'%'])
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S1_ac.m
function [T,g,S1_ac_Ploss,S1_ac_SOC,S1_ac_M,S1_ac_V,S1_ac_price,S1_ac_eff] =S1_ac(S1_ac_x0,P_s,t,SOC,r,alpha,S1_ac_max,motor_ac_kW_0);
% ***************************************************% Initial SOC of flywheel system% ***************************************************fw_suprcr_SOC_old=SOC;
% ***************************************************% Define weight factors and limits% ***************************************************
alpha_M=alpha(1);alpha_V=alpha(2);alpha_E=alpha(3);
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alpha_nu=alpha(4);
M_max=S1_ac_max(1);V_max=S1_ac_max(2);E_max=S1_ac_max(3);
% ***************************************************% Fixed parameters% ***************************************************
fw_suprcr_sigmax = 1600e6; % for graphite composed rotorfw_suprcr_rhofwrotor = 1550; % for graphite composed rotorfw_suprcr_rhohousing = 8000; % density of housing (steel)fw_suprcr_rhoair = 6.32e-3; % near-vacuum air density @ 5 mbarfw_suprcr_etaair = 17.1e-6; % dynamic viscosity of airfw_suprcr_eta = 0.3; % Poissons ratio of 0.3fw_suprcr_S = 1.3; % 30% (w.r.t. fw speed) over designed construction for safetyfw_suprcr_t = 0.01; % top air gap thickness [m]fw_suprcr_s = 0.01; % side air gap thickness [m]fw_suprcr_Crad = 1e6; % critical fw speed [N/m]
fw_suprcr_x=[fw_suprcr_sigmax; fw_suprcr_rhofwrotor; fw_suprcr_rhohousing; fw_suprcr_rhoair;fw_suprcr_etaair; fw_suprcr_eta; fw_suprcr_S; fw_suprcr_t; fw_suprcr_s;fw_suprcr_Crad];
motor_ac_x=[];
% ***************************************************% Calculate fixed flywheel properties, such as mass volume price% ***************************************************
% calculate volumefw_suprcr_V_rotor=pi*(S1_ac_x0(2)^2-S1_ac_x0(3)^2)*S1_ac_x0(4);fw_suprcr_V_housing=pi*S1_ac_x0(2)^2*S1_ac_x0(4);fw_suprcr_V = 1.5*fw_suprcr_V_housing; %includes pump
% calculate massesfw_suprcr_M_rotor = fw_suprcr_V_rotor*fw_suprcr_x(2);fw_suprcr_M_housing = 0.5*fw_suprcr_V_housing*fw_suprcr_x(3);fw_suprcr_M = fw_suprcr_M_rotor + fw_suprcr_M_housing+25; %25 for pump
% calculate maximum speed fw_suprcr_wmaxfw_suprcr_Js = 0.5*fw_suprcr_M_rotor*(S1_ac_x0(2)^2+S1_ac_x0(3)^2);fw_suprcr_wmax = sqrt((fw_suprcr_x(1)*8)/(fw_suprcr_x(2)*(3+fw_suprcr_x(6))*(S1_ac_x0(3).^2+
2*S1_ac_x0(2).^2.-S1_ac_x0(3).^2.*(1+3*fw_suprcr_x(6))/(3+fw_suprcr_x(6)))));fw_suprcr_Emax = 0.5*(fw_suprcr_Js)*fw_suprcr_wmax^2;
% calculate minimum speed fw_suprcr_wmin = w_crfw_suprcr_wmin = sqrt(fw_suprcr_x(10)/fw_suprcr_M_rotor);
% calculate old spinning speedfw_suprcr_w_old = sqrt(fw_suprcr_SOC_old*(fw_suprcr_wmax^2-fw_suprcr_wmin^2)+fw_suprcr_wmin^2);
for tijd=1:length(P_s)% NOTE: motor in [kW][motor_ac_Ploss,motor_ac_M,motor_ac_V,motor_ac_price,motor_ac_poss]=motor_ac(S1_ac_x0(5),motor_ac_x,P_s(tijd)/1e3,t);motor_Ploss=2000*motor_ac_Ploss;
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motor_ac_M=2*motor_ac_M;motor_ac_V=2*motor_ac_V;motor_ac_price=2*motor_ac_price;%power conversionif P_s(tijd)>0
motor_Pinv=P_s(tijd)+motor_Ploss;elseif P_s(tijd)<0
motor_Pinv=P_s(tijd)-motor_Ploss;else
motor_Pinv=P_s(tijd);end%inverterinv_Ploss=0.02*abs(motor_Pinv); inv_M=10; inv_V=1e-3; inv_price=100;%power conversionif motor_Pinv>0
inv_Pfw=motor_Pinv+inv_Ploss;elseif motor_Pinv<0
inv_Pfw=motor_Pinv-inv_Ploss;else
inv_Pfw=motor_Pinv;end% FW SIMULATION:[fw_suprcr_SOC_new,fw_suprcr_w_new,fw_suprcr_wmin,fw_suprcr_wmax,fw_suprcr_Ploss,fw_suprcr_M,fw_suprcr_V,fw_suprcr_price,fw_suprcr_poss]=fw_suprcr(S1_ac_x0,fw_suprcr_x,fw_suprcr_SOC_old,inv_Pfw,t);S1_ac_SOC(tijd)=fw_suprcr_SOC_old;fw_suprcr_SOC_old=fw_suprcr_SOC_new;S1_ac_Ploss(tijd)=abs(fw_suprcr_Ploss)+abs(motor_Ploss)+abs(inv_Ploss);
end
S1_ac_M=fw_suprcr_M+motor_ac_M+inv_M;S1_ac_V=fw_suprcr_V+motor_ac_V+inv_V;S1_ac_price=fw_suprcr_price+motor_ac_price+inv_price;
% ***************************************************% Calculate g function values% ***************************************************g1=max([-min(S1_ac_SOC) 0]); g2=max([max(S1_ac_SOC)-1 0]); %Penalty method%g1=max([-min(S1_ac_SOC)]); g2=max([max(S1_ac_SOC)-1]); %Barrier methodg=[g1 g2]';
% ***************************************************% Calculate f function values% ***************************************************% Efficiency calculationS1_ac_eff=1-sum(S1_ac_Ploss)/sum(abs(P_s));% Calculate cost functionf=alpha_M*S1_ac_M/M_max+alpha_V*S1_ac_V/V_max+alpha_E*S1_ac_price/E_max+alpha_nu*(1-S1_ac_eff);
% ***************************************************% Calculate cost function% ***************************************************T=1/r*(g1^2+g2^2)+f; %Penalty method%T=-1/r*(log(-g1)+log(-g2))+f; % Barrier method
%eliminate and warn for infeasible designsif isequal(num2str(T),'NaN')==1
128
warning('T value is NaN'); clear T;T=9999; %to prevent NaN, high value BEWARE OF 9999 IN ANSWER AS T VALUE
elseif T<0warning('T value is <0'); clear T;T=9999; %to prevent NaN, high value BEWARE OF 9999 IN ANSWER AS T VALUE
end
if g1>0warning('g1>0, SOC<0 in cycle')
elseif g2>0warning('g2>0, SOC>1 in cycle')
end
————————————
129
Appendix E
Paper submitted to VPP conference
The following contribution has been submitted to the IEEE Vehicle Power and Propulsion (VPP)Conference, september 2005.
130
Optimal design of energy storage systems forhybrid vehicle drivetrains
T. Hofman, D. Hoekstra, R.M. van Druten, and M. SteinbuchTechnische Universiteit Eindhoven
Dept. of Mechanical Engineering, Control Systems Technology GroupNL 5600 MB Eindhoven, The Netherlands
Email: [email protected]
Abstract— Current hybrid drivetrain simulation packages arebased on discrete (existing) system components and predefinedsystem structures. Optimization of the performance of the hybriddrivetrain is then based on finding the most efficient controlstrategy of the primary and secondary power source and finallycomparing the performance of the different candidate drive-trains. In this paper, the secondary power source components,part of the energy storage system (S), are modeled continuously,i.e., scalable to power and/ or energy capacity needs. In this way,the size of the components of S can be added as an optimizationparameter to a hybrid drivetrain design procedure.
Keywords: energy storage, hybrid drivetrain, multi-objective designproblem, scalable model, SQP optimization, Penalty and Barrier
functions
I. INTRODUCTION
In recent years several drivetrain simulation softwarepackages have been developed, these packages can be used todevelop hybrid drivetrain configurations and control strategies(e.g. ADVISOR [1], SIMPLEV [2], CarSim [3], HVEC[4], CSM HEV [5], V-Elph [6]) [7]. An optimal hybriddrivetrain design is obtained by optimizing a selected hybriddrivetrain configuration (e.g. series, parallel, series-parallel),to components and control. A drawback of the mentionedsoftware packages is that they are based on a discrete setof (existing) drivetrain components, fixed in size (power,energy capacity). With the software tool QSS-Toolbox [8]it is possible to build drivetrain structures with scalablemodels for the Electric Machine (EM) and combustionengine. In this paper, in addition to scalable models for theEM, continuously scalable models will be constructed andevaluated for ultra-capacitor, battery, gas-pressurized tank,sub- and supercritical flywheel storage systems.
Hybridization of a vehicle drivetrain implies adding a Sec-ondary power source (S, mostly a battery and an electricmotor) to a Primary power source (P , usually an internalcombustion engine). The objectives of a hybrid drivetrain areto improve the driving functions of a vehicle, i.e., fuel econ-omy, emissions, driveability, comfort and safety. Hybridizationallows performing brake energy recovery, downsizing theengine and optimizing the power flows over the differentthermal, mechanical and electrical paths between the differentpower sources.
The NWO1 research programme “Impulse Drive” currentlyfocuses on determining the required design specifications ofthe systems components for a hybrid vehicle fulfilling therequired driving function improvements. The influence of thegeneric design specifications for the S on fuel economy andEnergy Management Strategy (EMS) has been investigated in[9], the required vehicle driving function improvements serveto identify the system component specifications [10] [11].
The storage and conversion components, that provide thehybrid functionality, will be modeled continuously over theirpower and/or energy range in order to find an optimal designto a given control strategy. This will provide insights into (1)the solutions of the possible set of components necessary forthe design of S, (2) quantification of the design trade-offsin achieving the objectives and into (3) the possibilities andlimitations of the technologies that are under investigation. Inorder to be able to compare and evaluate performance of thehybrid drivetrain component designs, models will be generatedof the efficiency, mass, volume and cost.
II. HYBRID S SYSTEM
The system that brings the hybrid functionality, i.e., S isdefined as a system that can store and deliver energy to thedrivetrain through one rotating mechanical drive shaft. Withinthis constraint, six different concept S system configurationshave been identified, based on current technological possibil-ities, that will be analyzed and modeled further (see Table I).To be able to compare the Si designs (i = 1, .., 6), each Si alsohas the design constraint of providing control over transmittedpower flow regardless of the rotating speed or torque. Addinga Continuously Variable Transmission (CVT) to system S2
is thus necessary to be able to change torque for a requiredoutput power at a certain angular shaft speed.
Several of these systems have been modeled based onanalytical models, a reference design was then constructedto identify the design and size scaling parameters x. Scalingof look-up table based components is done by interpolationof properties (efficiencies, mass, volume) between discreteexisting designs. A component cost model was constructedby literature investigation into manufacturing prices [12]. It isassumed that all dynamic effects of the components can be
1The Netherlands Organization for Scientific Research
TABLE IS SYSTEM DESIGN CONCEPTS
Si S system Components and Modeling Layout
S1
Flywheel: analytical model, FW
Electric C
VT
��EM
EM
INV
Super-critical flywheel scalable to power profilewith electrical CVT Electrical CVT: scalable to
power, empirical model
S2
Sub-critical Flywheel: analytical model, FW
reduction +CVT
��flywheel scalable to power profile.with mechanical CVT: discrete, empiricalV-belt CVT model
S3
Nickel Metal Battery: scalable to power NiMH batt
EM
INV
Hydrid (Ni-MH) profile, empirical modelbattery with Motor: scalable to power,PM motor and Inv empirical model
S4
Valve Regulated Battery: scalable to power VRLA batt
EM
INV
Lead Acid (VRLA) profile, empirical modelbattery with Motor: scalable to power,PM motor and Inv empirical model
S5
Super capacitor: scalable to Super Capa citor
EM
INV
Super capacitor with power profile, empirical modelPM motor and Inv Motor: scalable to power,
look-up table based
S6
Compressed air Pressure tank: analytical
pump
volum. valve
GPT
energy storage (CAES) model, scalable to powerwith hydraulic control profilevalve and gear pump Pump: scalable to power
PM = Permanent Magnet, EM=Electric Machine, FW=Flywheel,CVT=Continuously Variable Transmission, GPT=Gas-Pressurized Tank,INV=Inverter
neglected within the simulation resolution of one second, sostatic models are used.
III. DESIGN OPTIMIZATION METHOD
The design of the S systems can be regarded as a multi-objective optimization problem. S configurations will be op-timized with respect to efficiency, mass, volume and cost:
T = minx{1− η(P, x),M(x), V (x), €(x)} ,
s.t. h(x) = 0, g(x) ≤ 0 (1)
With T the objective function value,x the vector containingdesign variables of S, η the efficiency, P the output powerprofile P (t), M the total mass of S, V the total volume, € thecost, h the equality constraints and g the inequality constraints.To obtain one single objective function the weighted summethod [13] is one of the most common methods, used tosolve multi-objective design problems. The method has beenadopted in this research, the weight factors αn are thereforeintroduced together with normalization factors Mmax, Vmax
and €max:
T = minx
�αη(1− η) + αM
M
Mmax+ αV
V
Vmax+ α€
€
€max
�,
s.t. h(x) = 0, g(x) ≤ 0 (2)
The factors αn (∑
αn = 1) can be used by a designerto identify what criterium is most important, e.g., for a citybus the mass and volume are less relevant, compared with a
Is resulting S i system feasible?
START i:=1 P(t)
choose S i system from
library
Optimize S i component sizes
for P(t)
Yes
No
Store performance
All S i systems in library
processed?
i<6
i=6
Compare Performance of
S i systems
i:=i+1
M max V max EUR max
(a) Overview
Desired accuracy reached?
P(t)
Yes
No
Output x opt = x *
Set (next) design x (through SQP)
Calculate ETA(P, x )
Calculate M( x )
Calculate V( x )
Calculate EUR( x )
Calculate weighted
objective function
OPTIMIZATION BLOCK
(b) Optimization
Fig. 1. Design procedure
passenger car, so αM and αV can be set lower then α€ andαη . A Sequential Quadratic Programming (SQP) algorithmwill be used to find minima of the objective function T .Several constraints cannot be explicitly written in the designparameter set x (e.g. State-of-Charge) and a Penalty andBarrier function will be used to include these in the objectivefunction. Calculating the Hessian based on look-up table basedmodels can be problematic since the objective function is thendiscontinuous at several points. Further, introducing weightedobjectives and Penalty and Barrier techniques, Pareto optimal-ity can no longer be guaranteed regardless of a discontinuousobjective function. An optimization technique that can copewith this problem is Dynamic Programming (DP), howeverthis technique has not been explored further. Instead, the SQPsearch algorithm will be started at different starting points x0
for each Si system to obtain a robustness sensitivity of theminimized design x∗. Several minima are expected to be foundfor each Si system, depending on the choice of optimizationparameters. An overview of the design process and a flowchartfor the optimization procedure are shown in figure 1.
IV. RESULTS
Table I concepts optimizations have been performed inorder to identify the sensitivity of the solutions to the chosenweighting, i.e., αM = 0.2, αV = 0.2, α€ = 0.2 and αη = 0.4.The results will discussed in this section.
A. Design evaluation of concept designs
The influence of a continuous discharge and charge powerprofile on the average efficiency of the optimized systemsas shown in Table I has been investigated. This has beendone by sequentially changing the nominal power rating andthe duration time td (resulting in different required energy
storage capacities) (see Fig. 2). The initial SOC value is 1assumed. The results regarding the average system efficiency
P s [kW]
0
nominal power rating
t d
0.5*t d time [s]
Fig. 2. Power profile used for investigation of design responses (averageefficiency)
are shown in Fig. 3. In Fig. 3 is shown, that only S4 or theVRLA with a PM has an average efficiency lower than 60%for nominal power ratings higher than approximately 5 kW -6.5 kW . All the other systems have an average efficiencylarger than 70% for nominal powers between 5 kW -25 kWand energy storage levels of 0.1 MJ-2 MJ . It can also beseen that the efficiencies of S1, S3 and S4 are more sensitiveto nominal power rating than energy capacity compared to S2
and S5.
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S1
0.6
0.65
0.7
0.72
0.73
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S2
0.1
0.70.75
0.80.820.84
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S3
0.790.
8
0.81
0.82
0.82
3
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S4
0.25
0.3
0.35
0.4
0.5
0.6
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S5
0.70.74
0.76
INFEASIBLE DESIGN
5 10 15 20 25
0.5
1
1.5
2
Power [kW]
Ene
rgy
[MJ]
S6
0.88
0.88 0.88
0.9
0.9
0.91
0.92
0.92
Fig. 3. Contour lines of constant average efficiencies [-] as a function ofnominal power rating and energy storage capacity for the different S systems
For the rest of this section, only the storage devices fromthe S systems are considered, so conversion components areremoved (i.e., EM, CVT).
In Fig. 4, it can be seen that the efficiency of the super-criticalor High Speed (HS) and the sub-critical flywheel system orLow Speed (LS) Flywheel systems decrease with increasein duration time and decrease of nominal power rating.Additionally, the efficiencies of VRLA and the UC decreasewith increase of duration time and increase of nominalpower rating. Furthermore, the Ni-MH battery efficiency isindependent on duration time variation, because the efficiencyis dependent on SOC-level, which has a small fluctuationdue to the significant larger energy capacity compared to the
other systems. However, the battery efficiency decreases withincrease of nominal power rating.
5 10 15 20 250.4
0.5
0.6
0.7
0.8
0.9
1
Nominal power rating |Ps| [kW]
Ave
rage
effi
cien
cy η
[−]
HS FWLS FWNiMHVRLAUC
(Arrows indicate increase td)
Fig. 4. Influence of the nominal power rating and duration time td on theaverage efficiency for the different storage components. CAES storage ahsbeen omitted, the tank has 100% efficiency in the modeling.
In the Fig. 5, the specific power as a function specific energystorage capacity for the different optimized storage systems isshown. The obtained density values can be compared withaverage data available in literature (areas are indicated bythe colored rectangles). As can be seen in Fig. 5 for theLS flywheel system, the power density decreases significantlycompared to a relative small decrease of energy density withincrease of duration time. At shorter duration times and higherpower demands the LS flywheel is more efficiently (see Fig.4). The flywheel is optimized such that the bearing friction andair drag losses are minimized, which minimizes the maximumflywheel speed, which results in a larger flywheel mass.
10−1
100
101
102
100
101
102
103
104
Specific energy [Wh/kg]
Spe
cific
pow
er [W
/kg]
HS FWLS FWNiMHVRLAUCCAES
(Arrow indicates increase td)
Fig. 5. Specific power as a function specific energy storage capacity for thedifferent optimized S systems
B. Optimization of S using a 20 kW power cycle
The influence of a power demand to the S resulting fromfuel economy optimization has been used in order to se-lect a S system fulfilling the design targets as an exampleresult. The optimal power distribution between P (Primarypower source), S (Secondary power source) and V (Vehiclepropulsion power) has been obtained by minimizing the fuelconsumption, maintaining SOC within a certain range, andaccomplish any drive power demand [9]. The optimizationproblem can be described as a multi-step decision problem indiscrete-time format and therefore it can be solved using theDynamic Programming (DP) technique. The maximum powerrating of S is set to 20 kW . The vehicle is assumed to be amid-sized passenger car (1360 kg), and the engine is a 1.6 l SIoperated at the optimal operation line. The drive cycle is theNEDC. Furthermore, during these simulation the efficienciesof S and T are assumed to be 100%. The results are shownin Fig. 6. In Fig. 9, it can be seen that power flow out of theS is controlled such that P is operated as much as possibleat levels in which the engine efficiency is higher. The optimalpower demand from S has been used to optimize the differentS systems as shown in Table I using the SQP algorithm.The initial SOC level is 0.9 assumed. A spider diagram isshown in Fig. 8 for comparison of different optimized Ssystems to design criteria: efficiency, mass, volume and cost.Immediately, it can be seen that sub-critical flywheel systemor Low Speed (LS) Flywheel, Compressed Air Energy StorageTank (CAES), Ultra-Capacitor (UC) with PM EM are notfeasible solutions, caused by too high mass, volume or costrespectively. The VLRA with a PM EM and the super-criticalor High Speed (HS) flywheel system are rejected caused bytoo low average efficiencies of 50.1% and 52.5% respectively.Therefore, the combination of a Ni-MH battery with an energycapacity of approximately 9 kWh and a 20 kW PM EM is themost suitable solution fulfilling the design targets. The SOCevolution and the power losses of this system are shown inFig. 7. The discrepancy between the SOC levels at the endof cycle between the imaginary S simulated without losses(blue line) from DP and the S consisting of the Ni-MH andPM EM (magenta line) is caused by the energy losses. Themass, volume, cost price and average efficiency are optimizedto 141 kg, 31 dm3, 2.9 k€ and 81.8% respectively for thecomplete system. The battery solely has a mass of 100 kg andan output power of 22.4 kW corresponding to a 0.22 kW/kgand 90 Wh/kg for the power and energy density. The powerand energy density of the battery pack of the new ToyotaPrius is respectively increased and decreased to approximately0.52 kW/kg and 33 Wh/kg. Adapting the fitted parametersfor battery module mass and volume as a function of thecurrent capacity in the mass and volume model can easily bedone for state-of-the-art technology in mass production onceavailable. It should be noticed that the maximum allowablebattery voltage and the battery configuration (series or parallelconnected) is of influence on the amount of required batteryenergy capacity and nominal power rating. In this paper, the
batteries are assumed to be connected in series reducing the in-ternal battery losses due to lower current losses. Furthermore,no constraints are set to the maximum battery voltage.
0 200 400 600 800 1000 1200−20
0
20
Ps (
kW)
1.6 l SI ICE, Mid−sized passenger car, NEDC
0 200 400 600 800 1000 1200−50
0
50
Pv (
kW)
0 200 400 600 800 1000 12000
10
20
30
Pp (
kW)
Time [s]
Fig. 6. Optimal power distribution between P , S and V (Primary powersource, Secondary power source and Vehicle power load, respectively).Topology and transmission are optimal and are assumed lossless.
0 200 400 600 800 1000 1200−20
0
20
Ps [k
W]
Optimized Ni−MH battery system with PM EM (9 kWh)
0 200 400 600 800 1000 12000.88
0.9
0.92
0.94
SO
C [−
]
0 200 400 600 800 1000 12000
2000
4000
6000
Time [s]
Plo
ss [W
]
efficiency is 81.8%
Optimal EMS from DP (blue)
Resulting SoC evolution (magenta)
Fig. 7. Optimized Ni-MH battery with a PM EM (S3 of Table I), Ps, SOCand Ploss are shown as function of time.
V. CONCLUSION
Different S system concepts have been introduced that canprovide hybrid functionality when coupled via an transmissiontechnology to a hybrid drivetrain. Within the presented opti-mization framework, performance in terms of mass, volume,price and efficiency can be compared and evaluated betweenthe six presented S system models. In this paper, a continuouscharge and discharge power demand on the different S designhave been investigated. Most of the discussed S systems aredependent on the nominal power rating and the duration timeof the signal. Except the Ni-MH battery, which is independenton the duration time, due to its high internal capacity. The
0.53
2553
1.1
149
1−η [−]
M [kg]
V [m3]
Cost [kEUR]
S1pm
(HS FW) T=2S
2 (LS FW) T=8
S3pm
(NiMH) T=0.57S
4pm (VRLA) T=0.44
S5ac
(UC) T=18S
6p1 (CAES) T=4.1
sub−critical flywheel
Ultra−Capacitor
Compressed air energy storage tank
super−critical flywheel
Ni−MH battery
VRLA battery
Fig. 8. Spider diagram for comparison of different optimized S systems todesign criteria: Efficiency, Mass, Volume and Cost.
0 5 10 15 20 25 30 35 400
5
10
15
20
25
30
35
40
45
50
ICE power [kW]
Effi
cien
cy [%
]/Rel
ativ
e F
requ
ency
1.6 l SI ICE (74 kW), maximum power rating S = 20 kW
Fig. 9. Engine efficiency and net power frequency histogram
sizing of the battery systems can be done solely on the powerrequirements from the storage device. Since, storage capacityof batteries is sufficient for single-storage system. Batteriesare often used in their most efficient SOC range (SOC∈[30%−80%]). The other storage components (flywheels, ultracapacitors, compressed air energy storage) are sized basedon storage capacity and not on power ratings. In addition,generally the full storage capacity available to deliver therequired power profile (SOC∈ [0 − 1]) is used. From theoptimized S designs using a 20 kW power demand to the S,the battery systems are favorable, however the battery packscan be considered heavy and voluminous for hybrid vehicleapplication. Splitting up the S into two power levels (low andhigh) and designing corresponding technologies to its purposeit is expected that the overall losses, mass and volume of theS can be reduced. Nonetheless, the Ni-MH battery pack hasa high efficiency over large part of its working range and low
sensitivity to the efficiency with varying power- and energycapacity needs. Some aspects are not taken into account in thisresearch, e.g. lifetime expectancy. The models can however beexpanded easily with these aspects. Future work will focusson the applicability of the different S systems to differentvehicle classes. In addition, the optimized S system willbe used to determine the ratio specifications in combinationwith a chosen P and V following from optimal operationof these component over a defined drive cycle. Once thesespecifications are determined the transmission technology canbe selected and designed.
ACKNOWLEDGMENT
This study is part of “Impulse Drive” which is a re-search project at the Technische Universiteit Eindhoven inThe Netherlands within the section Control Systems Tech-nology of the Department of Mechanical Engineering. Theproject is financially supported by the NWO Technology Foun-dation within the Innovational Research Incentives Scheme2000/2001.
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