Control of Hydrostatic Transmission Wind Turbine

Transcription

1 Sa Jose State Uiversity SJSU ScholarWorks Master's Theses Master's Theses ad Graduate Research all 14 Cotrol of Hydrostatic Trasissio Wid Turbie Daop Rajabhadharaks Sa Jose State Uiversity ollow this ad additioal works at: Recoeded Citatio Rajabhadharaks, Daop, "Cotrol of Hydrostatic Trasissio Wid Turbie" (14). Master's Theses This Thesis is brought to you for free ad ope access by the Master's Theses ad Graduate Research at SJSU ScholarWorks. It has bee accepted for iclusio i Master's Theses by a authorized adiistrator of SJSU ScholarWorks. or ore iforatio, please cotact scholarworks@sjsu.edu.

2 CONTROL O HYDROSTATIC TRANSMISSION WIND TURBINE A Thesis Preseted to The aculty of the Departet of Electrical Egieerig Sa José State Uiversity I Partial ulfillet of the Requireet for the Degree Master of Sciece by Daop Rajabhadharaks Deceber 14

3 14 Daop Rajabhadharaks ALL RIGHT RESERVED

4 The Desigated Thesis Coittee Approves the Thesis Titled CONTROL O HYDROSTATIC TRANSMISSION WIND TURBINE by Daop Rajabhadharaks APPROVED OR THE DEPARTMENT O ELECTRICAL ENGINEERING SAN JOSÉ STATE UNIVERSITY Deceber 14 Dr. Pig Hsu Dr. Peter Reischl Dr. Burford J. ura Departet of Electrical Egieerig Departet of Electrical Egieerig Departet of Mechaical Egieerig

5 ABSTRACT CONTROL O HYDROSTATIC TRANSMISSION WIND TURBINE by Daop Rajabhadharaks I this study, we proposed a cotrol strategy for a wid turbie that eployed a hydrostatic trasissio syste for trasittig power fro the wid turbie rotor via a hydraulic trasissio lie to a groud level geerator. Wid turbie power curve trackig was achieved by cotrollig the hydraulic pup displaceet ad, at the other ed of the hydraulic lie, the hydraulic otor displaceet was cotrolled so that the overall trasissio loss was iiized. Steady state respose, dyaic respose, ad syste stability were assessed. The axiu trasissio efficiecy obtaied raged fro 79% to 84% at steady state whe the proposed cotrol strategy was ipleeted. The leakage ad frictio losses of the hydraulic copoets were the ai factors that coproised the efficiecy. The siulatio results showed that the syste was stable ad had fast ad well-daped trasiet respose. Double wid turbie syste sharig hydraulic pipes, a hydraulic otor, ad a geerator were also studied. The hydraulic pipe diaeter used i the double-turbie syste icreased by 7% copared to the sigleturbie syste i order to ake the trasissio coefficiet coparable betwee both systes. The siulatio results suggested that the leakage losses were so sigificat that the efficiecy of the syste was worseed copared with the sigle-turbie syste. uture studies of other behavioral aspects ad practical issues such as fluid dyaics, structure stregth, aterials, ad costs are eeded.

6 ACKNOWLEDGEMENTS irst ad foreost, I would like to express y very great appreciatio to Dr. Pig Hsu, y thesis advisor, for his guidace, kidhearted ispiratio, ad valuable techical support of this work. His carig of y work ad his willigess to spedig tie so geerously has bee very uch appreciated. I feel extreely fortuate to have hi as y thesis advisor. I would like to thak Dr. Peter Reischl ad Dr. Burford J. ura for their willigess to serve o y thesis coittee. Their isightful coets ad supports for this auscript are very uch appreciated. I would like to gratefully ackowledge Sa Jose State Uiversity for fiacial support of y trip to preset this work at the IEEE Coferece o Techologies for Sustaiability (Sustech 14) i Portlad, Orego. ially, I wish to thak y faily ad frieds for their love ad costat ecourageet throughout y pursuit of the Master of Electrical Egieerig. Special thaks go to y o for those delicious eals that keep e stuffed durig y study tie. v

7 TABLE O CONTENTS I. INTRODUCTION... 1 A. Wid Eergy... 1 B. Covetioal Wid Turbie Basics... C. Proble Stateet... 3 D. Literature Review... 4 E. Study Objective ad Report Structure... 8 II. HYDROSTATIC TRANSMISSION WIND TURBINE MODEL... 9 A. Itroductio... 9 B. Aerodyaic Power ad Torque Model... 1 C. Variable Displaceet Hydraulic Pup... 1 D. Hydraulic Trasissio Lie E. Variable Displaceet Hydraulic Motor Sychroous Geerator III. CONTROL STRATEGY A. Itroductio B. Regio Cotrol Strategy ) Maxiizatio of Wid Turbie Power Coefficiet Strategy ) Maxiizatio of Trasissio Coefficiet Strategy... IV. SIMULATION RESULTS IN MATLAB/SIMULINK... 5 A. Steady State Respose... 5 vi

8 B. Syste Respose... 3 C. Syste Stability Aalysis V. DOUBLE TURBINE CONIGURATION A. Itroductio B. Hydrostatic Trasissio Modelig for Double Turbies ) Aerodyaic Power ad Torque Model ) Variable Displaceet Hydraulic Pup ) Hydraulic Trasissio Lie ) Variable Displaceet Hydraulic Motor ad Sychroous Geerator C. Regio Cotrol Strategy ) Maxiizatio of Wid Turbie Power Coefficiet Strategy... 4 ) Maxiizatio of Trasissio Coefficiet Strategy... 4 D. Siulatio Results i MATLAB/Siulik ) Steady State Respose ) Syste Respose ) Syste Stability Aalysis... 6 VI. CONCLUSION BIBLIOGRAPHY APPENDIX A: MATLAB/SIMULINK OR SINGLE TURBINE A. Siulik Base Model B. MATLAB Code for Assigig Syste Paraeters C. MATLAB Code ad Siulik for Steady State Respose vii

9 D. MATLAB Code ad Siulik for Syste Respose E. MATLAB Code ad Siulik for Liearizatio... 8 APPENDIX B: MATLAB/SIMULINK OR DOUBLE TURBINE A. Siulik Base Model B. MATLAB Code for Assigig Syste Paraeters C. MATLAB Code ad Siulik for Steady State Respose D. MATLAB Code ad Siulik for Syste Respose E. MATLAB Code ad Siulik for Liearizatio viii

10 TABLE O IGURES igure 1: Net electricity geerated by wid eergy i the U.S. fro 4 to igure : Structure of a typical wid turbie... igure 3: Hydrostatic trasissio wid turbie... 6 igure 4: Hydrostatic trasissio wid turbie odel... 1 igure 5: Siplified sychroous geerator odelig igure 6: Wid turbie syste operatioal regios igure 7: Trasissio coefficiet (C T ) as a fuctio of otor displaceet (V )... igure 8: Optial otor displaceet (V,opt ) as a fuctio of wid speed (U) ad its liear approxiatio equatio... 4 igure 9: Overall syste level block diagra icludig cotrol strategy to axiize C p ad C T i roud-dotted boxes... 4 igure 1: Rotor speed ( r ) ad otor speed ( ) as fuctios of wid speed (U)... 7 igure 11: Pup differetial pressure (p p ) ad otor differetial pressure (p ) as fuctios of wid speed (U)... 7 igure 1: Pup displaceet (V p ) ad otor displaceet (V ) as fuctios of wid speed (U)... 8 igure 13: Geerator torque load ( load ) as a fuctio of wid speed (U)... 8 igure 14: Wid power (P wid ), rotor rotatioal power (P rotor ), ad output power (P out ) as fuctios of wid speed (U)... 9 ix

11 igure 15: Trasissio coefficiet (C T ), power coefficiet (C p ), ad overall syste efficiecy (C T C p ) as fuctios of wid speed (U)... 9 igure 16: Wid speed (U) as a fuctio of tie igure 17: Rotor speed ( r ) ad otor speed ( ) as fuctios of tie igure 18: Differetial pup pressure (p p ) ad differetial otor pressure (p ) as fuctios of tie... 3 igure 19: Pup displaceet (V p ) ad otor displaceet (V ) as fuctios of tie.. 3 igure : Geerator torque load ( load ) as a fuctio of tie igure 1: Wid power (P wid ), rotor rotatioal power (P rotor ), ad output power (P out ) as fuctios of tie igure : Trasissio coefficiet (C T ), power coefficiet (C p ), ad overall syste efficiecy (C T C p ) as fuctios of tie igure 3: Hydrostatic trasissio wid turbie with shared hydraulic pipes, hydraulic otor, ad geerator igure 4: Motor displaceet (V ) as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 5: Overall syste level block diagra icludig cotrol strategy to axiize C p1, C p, ad C T i roud-dotted boxes igure 6: Rotor speed 1 ( r1 ) i rad/s as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 7: Rotor speed ( r ) i rad/s as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) x

12 igure 8: Motor speed ( ) i rad/s as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 9: Differetial pup pressure (p p ) i MPa as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 3: Differetial otor pressure (p ) i MPa as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )... 5 igure 31: Pup displaceet 1 (V p1 ) i 3 /rad as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )... 5 igure 3: Pup displaceet (V p ) i 3 /rad as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 33: Motor displaceet (V ) i 3 /rad as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 34: Geerator torque load ( load ) i kn- as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )... 5 igure 35: Wid power 1 (P wid1 ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )... 5 igure 36: Wid power (P wid ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 37: Rotor rotatioal power 1 (P rotor1 ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 38: Rotor rotatioal power (P rotor ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) xi

13 igure 39: Output power (P out ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 4: Power coefficiet 1 (C p1 ) as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 41: Power coefficiet (C p ) as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 4: Trasissio coefficiet (C T ) as a fuctio of wid wpeed 1 (U 1 ) ad wid speed (U ) igure 43: Wid speeds (U 1 ad U ) as fuctios of tie igure 44: Rotor speeds ( r1 ad r ) ad otor speed ( ) as fuctios of tie igure 45: Differetial pup pressure (p p1 =p p p p ) ad differetial otor pressure (p ) as fuctios of tie igure 46: Pup displaceets (V p1 ad V p ) ad otor displaceet (V ) as fuctios of tie igure 47: Geerator torque load ( load ) as a fuctio of tie igure 48: Wid power 1 (P wid1 ) ad rotor rotatioal power 1 (P rotor1 ) as fuctios of tie igure 49: Wid power (P wid ) ad rotor rotatioal power (P rotor ) as fuctios of tie... 6 igure 5: Total wid power (P wid1 + P wid ), total rotor rotatioal power (P rotor1 + P rotor ), ad output power (P out ) as fuctios of tie... 6 xii

14 igure 51: Trasissio coefficiet (C T ) ad power coefficiets (C p1 ad C p ) as fuctios of tie igure 5 (Appedix A): Wid Turbie sybol ad detailed odel igure 53 (Appedix A): Pup Syste sybol ad detailed odel... 7 igure 54 (Appedix A): Motor Syste sybol ad detailed odel... 7 igure 55 (Appedix A): Pressure Syste sybol ad detailed odel igure 56 (Appedix A): Sychroous Geerator sybol ad detailed odel igure 57 (Appedix A): MATLAB code for assigig syste paraeters igure 58 (Appedix A): MATLAB code for steady state respose igure 59 (Appedix A): Workspace (part 1) for OeTurbie_SteadyState.dl igure 6 (Appedix A): Workspace (part ) for OeTurbie_SteadyState.dl igure 61 (Appedix A): MATLAB code for syste respose igure 6 (Appedix A): Workspace (part 1) for OeTurbie_Dyaics.dl... 8 igure 63 (Appedix A): Workspace (part ) for OeTurbie_Dyaics.dl igure 64 (Appedix A): MATLAB code for liearizatio... 8 igure 65 (Appedix A): Workspace (part 1) for OeTurbie_Liearizatio.dl igure 66 (Appedix A): Workspace (part ) for OeTurbie_Liearizatio.dl igure 67 (Appedix B): Pressure Syste sybol ad detailed odel (part 1) igure 68 (Appedix B): Pressure Syste sybol ad detailed odel (part ) igure 69 (Appedix B): MATLAB code for assigig syste paraeters igure 7 (Appedix B): MATLAB code for steady state respose (part 1) igure 71 (Appedix B): MATLAB code for steady state respose (part ) xiii

15 igure 7 (Appedix B): Workspace for TwoTurbie_SteadyState.dl... 9 igure 73 (Appedix B): MATLAB code for syste respose igure 74 (Appedix B): Workspace for TwoTurbie_Dyaics.dl... 9 igure 75 (Appedix B): MATLAB code for liearizatio igure 76 (Appedix B): Workspace for TwoTurbie_Liearizatio.dl xiv

16 LIST O TABLES Table 1: Wid turbie paraeter values used i this study... 6 Table. Loss aalysis coputed fro Equatio (4) to (31). The values are calculated i percetage copared with rotor rotatioal power Table 3. Jacobia atrices ad eigevalues of the syste at specified operatig poit.. 37 Table 4. Jacobia atrices ad eigevalues of the syste at specified operatig poit.. 65 xv

17 NOMENCLATURE Sybol Defiitio Uit air Air Desity kg/ 3 R Wid Blade Radius A Wid Blade Swept Area U Wid Speed /s P wid Aerodyaic Power W C p Power Coefficiet diesioless Tip-Speed Ratio diesioless Pitch Agle rad P rotor Rotor Rotatioal Power W r Rotor/Pup Rotatioal Speed rad/s rotor Rotor Torque N- C q Torque Coefficiet diesioless pup Hydraulic Pup Torque N- J r Pup/Rotor Iertia kg- V p Pup Displaceet 3 /rad p p Differetial Pressure at Pup Side MPa ech,p Pup Mechaical Efficiecy diesioless Q p Pup low Rate 3 /s k leak,p Pup Leakage Coefficiet 3 /(s Pa) k HP,p Pup Hage-Poiseuille Coefficiet (/s) kg/pa V p,ax Maxiu Pup Voluetric Displaceet 3 /rad p,o Noial Pup Speed rad/s v,pup Pup Voluetric Efficiecy diesioless p,o Noial luid Kieatic Viscosity at Pup Side /s p,o Noial Pup luid Desity kg/ 3 p p,o Noial Differetial Pressure at Pup Side Pa p x Differetial Pressure at Ceter of Straight Pipe Pa fluid luid Bulk Modulus Pa V fluid Pipe Volue 3 Q Motor low Rate 3 /s p Pressure Loss i Pipe Pa f rictio actor diesioless L pipe Pipe Legth fluid luid Desity kg/ 3 D pipe Pipe Diaeter A pipe Pipe Cross-sectioal Area Re Reyolds Nuber diesioless r pipe Roughess of the Pipe xvi

18 Sybol Defiitio Uit fluid luid Kieatic Viscosity /s p Differetial Pressure at Motor Side MPa ech, Motor Mechaical Efficiecy diesioless V Motor Displaceet 3 /rad J Motor Iertia kg- Motor Rotatioal Speed rad/s load Geerator Load Torque N- k leak, Motor Leakage Coefficiet 3 /(s Pa) k HP, Motor Hage-Poiseuille Coefficiet (/s) kg/pa V,ax Maxiu Motor Voluetric Displaceet 3 /rad,o Noial Motor Speed rad/s v,otor Motor Voluetric Efficiecy diesioless,o Noial luid Kieatic Viscosity at Motor Side /s,o Noial Motor luid Desity kg/ 3 p,o Noial Differetial Pressure at Motor Side Pa agle Geerator Phase Agle Differece rad grid Phase of Grid Voltage rad Sychroous Geerator s Electrical Agle rad syc Geerator Sychroous Speed rad/s K s Sychroizig Torque Coefficiet N-/rad K d Dapig Torque Coefficiet N-/(rad/s) P out Output Power Geerated by Sychroous Geerator W C p,ax Maxiu Power Coefficiet diesioless opt Optial Tip-Speed Ratio diesioless C T Trasissio Coefficiet diesioless Loss ech,pup Pup Mechaical Loss diesioless Loss leak,pup Pup Leakage Loss diesioless Loss ech,otor Motor Mechaical Loss diesioless Loss leak,otor Motor Leakage Loss diesioless Loss fric rictio Loss diesioless Loss total Total Hydrostatic Trasissio Loss diesioless xvii

19 1 I. INTRODUCTION A. Wid Eergy The eergy fro wid has bee haressed for servig huas for a log tie. I early history, it was widely used for sailig ships, gridig grai, ad pupig water i differet locatios i the world [1]. The use of wid eergy for geeratig electricity becae apparet i 194s whe the largest wid turbie (1.5 MW) of that tie was developed ad operated to geerate electricity [1]. The icrease i the price of oil ad the cocer over liited fossil fuel resources triggered a uber of goveret fuded wid eergy projects worldwide [1]. By far, wid eergy is the fastest growig eergy source [] copared with the other reewable eergy sources. I the Uited States, the et wid eergy geeratio icreases at a rate approxiately, thousad MWh per year as show i igure 1 fro 7 to 13. igure 1: Net electricity geerated by wid eergy i the U.S. fro 4 to 13 [3]

20 B. Covetioal Wid Turbie Basics A wid turbie is a device that trasfors the kietic eergy of ovig air (i.e., wid) ito electrical eergy. igure shows the structure of a typical wid turbie. It is coposed of turbie blades, a tower, ad a acelle that houses a gearbox, a geerator, a power coverter, ad a cotroller. Air flow over the blades causes the turbie shaft to rotate. The gearbox is eployed to step up the rotatioal speed to a higher speed at which the geerator ca produce electrical power ore efficietly. The power coverter ad trasforer covert the frequecy ad the voltage level so that the power ca be fed ito the power grid [4]. The cotroller esures that the wid turbie is operated safely ad effectively. igure : Structure of a typical wid turbie

21 3 C. Proble Stateet The weight of a wid turbie sigificatly affects the costs of productio, trasportatio, istallatio, ad aiteace [5]. The heavier the weight, the higher the cost. or istace, a asychroous geerator with a squirrel cage rotor (ABB, ilad; rated power=.3mw) is reported by Jeppsso et al to weigh 658 kg [6]. The gearbox i a Siees.3 MW Mk II wid turbie (Wiergy; ratio=1:91), reported by Jeppsso et al, weighs 3 tos [6]. A power coverter (ABB, ilad; up to 9 MVA) is reported by ABB to weigh betwee 5 to 6 kg [7]. A.5 MVA vacuu cast coil trasforer (ABB, ilad) weighs 6 kg [8]. The acelle of Siees.3 MW Mk II wid turbie weighs 8 tos i total [6]. The cost of costructio ad structure aterials for supportig such a assive acelle at about 1 eter height o top of the tower is substatial. If the weight of the acelle could be lowered, these costs could be reduced. Reliability is oe of the ost iportat qualities for a wid turbie. Poor reliability ca lead to expesive aiteace ad uecessary operatioal loss due to dowtie, which ca sigificatly icrease the cost of electrical eergy productio. I a covetioal egawatt wid turbie, the gearbox by far is oe of the ost expesive copoets [9] ad the weakest lik of the etire structure [1]. Misaliget of the gearbox due to wid gusts ad turbulece is a coo cause for failure [1]. It is ot ucoo for a gearbox to fail o average every 5 years while the desiged lifetie of a wid turbie is typically about years [1]. The gearbox replaceet is usually a part of prevetive aiteace every 5 years; therefore, it cotributes sigificatly to the operatig cost [1].

22 4 D. Literature Review May solutios have bee proposed to solve the gearbox reliability issue. Oe of the ost direct solutios is to fid a way to etirely reove the gearbox without usig ay other eas for replacig its fuctio. This idea leads to a direct-drive or gearless wid turbie, pioeered i 1993 by the Eerco copay of Geray ad the ScaWid copay of Norway [1]. To reove the gearbox, the geerator iside the acelle eeds odificatios. Equatio (1) [1] shows the relatioship betwee the geerator rotatioal speed, the uber of agetic pole pairs, ad the frequecy f i Hz. rp f (1) 1 or a 4 pole-pair achie, i order to geerate 6 Hz electrical power, rp rp 18 the geerator has to operate at 18 rp. Assuig the rotor of the wid turbie is rotated at 18 rp, the gear ratio required to step up the rotatioal speed would be 1. The cocept of the direct-drive wid turbie is to icrease the uber of agetic pole pairs. Usig Equatio (1) agai to obtai the sae 6 Hz operatig frequecy with the geerator rotatig at 18 rp, the uber of agetic pole pairs has to be icreased to 4 pairs. This idea sees reasoable ad straightforward to solve the gearbox issue. However, the direct-drive geerator requires a large aout of a expesive rare earth eleet aget, which leads to high cost, large size, ad heavy weight acelle [11]. Aother proble with the direct-drive geerator is that, i order to obtai high flux

23 5 desity, the sall air gap betwee the rotor ad the stator ust be kept withi a few illieters. Achievig such a high precisio ca be a aufacturig challege [11]. Aother proposed solutio is to eploy hydrostatic trasissio to step up rotatioal speed ad trasit power i place of the gearbox. A hydrostatic trasissio syste cotais a hydraulic pup, a hydraulic otor, ad a hydraulic pipe/hose syste. Hydrostatic trasissio has bee utilized i autoobile applicatio [1, 13] ad recetly itroduced to wid turbie applicatio. Hydrostatic trasissio ca offer ay positive beefits. or istace, it allows the use of cotiuously variable trasissio (CVT) that is capable of varyig the iput-output speed ratio cotiuously ad soothly [11, 14, 15], ulike the covetioal gearboxes that offer a liited uber of gear ratios. Because the low-speed shaft ad high-speed shaft are decoupled, sychroous geerators ca potetially be used istead of iductio geerators because the fluctuatios are ore tolerable with hydrostatic trasissio ad the eed of the power electroic coverters ca be eliiated [11, 13]. I additio to the aforeetioed beefits, the weight of the acelle ca be sigificatly lowered with the hydrostatic trasissio cofiguratio. A echaical gearbox is about 3 to 4 ties heavier tha the hydrostatic trasissio syste (a gearbox reported by Jeppsso et al weighs 3 tos [6] ad that reported by NewScietist weighs 15 tos [16] versus a hydrostatic trasissio reported by NewScietist weighs 6 tos [16]). The geerator ca be situated o the groud [13] (a asychroous geerator reported by Jeppsso et al weighs 658 kg [6]). Thus, the weight of the geerator does ot cotribute to the total weight of the acelle. Moreover, the eed for the power

24 6 electroic coverter (oe reported by ABB weighs 5 to 6 kg [7]) ca be eliiated because a sychroous geerator ca be used. The trasforer (oe reported by ABB weighs 6 kg [8]) ca also be oved to the groud (a trasforer is usually istalled iside the acelle for offshore wid fars to avoid I R losses [5]). I this exaple, the total acelle weight reductio ca be reduced by reovig the gearbox (15 tos [16]), addig hydrostatic trasissio (6 tos [16]), reovig the geerator (6.58 tos [6]), reovig the power electroic coverter (5 tos [7]), ad reovig the trasforer (6. tos [8]). The total acelle weight reductio is 6.78 tos. Copared with the total acelle weight (reported by ABB [8]) of 8 tos for the covetioal wid turbie, the reductio ca be 6.78/8=.33 or 33%. igure 3 illustrates hydrostatic trasissio wid turbie ad its copoets. igure 3: Hydrostatic trasissio wid turbie

25 7 SeaAgel [17] is a livig exaple of a offshore hydrostatic trasissio wid turbie, developed sice 11 by Mitsubishi Power Systes Europe (MPSE). It cotais o gearbox. It uses Digital Displaceet Trasissio (DDT) [18], developed by Arteis Itelliget Power (Ediburgh, Scotlad), for its drive syste. DDT cosists of a variable Digital Displaceet Pup (DDP) ad a variable Digital Displaceet Motor (DDM) i place of a covetioal gearbox. A brushless sychroous geerator is also eployed istead of a popular asychroous geerator for a covetioal wid turbie ad o power electroic coverter is used. A advatage of DDT copared with other variable displaceet achies, such as Bet Axis Pup ad Axial Pisto Pup is that its efficiecy is fairly uifor at differet operatig poit as show i igure 1.1 by Yige [19] ad, thus, this techology allows the SeaAgel to be efficiet ad copetitive copared with the covetioal gearbox [16]. Curretly, the SeaAgel is still uder testig, accordig the ilestoes reported by Mitsubishi Power Syste [17]. Modelig ad cotrol aspects of the hydrostatic trasissio wid turbie have bee reported i literature. or istace, Pusha et al [, 1], Hazehlouia et al [-8], Deldar et al [9], Izadia et al [3] developed a hydraulic trasissio odel ad cotrol strategy techique for wid turbie usig a fixed displaceet pup ad a fixed displaceet otor. Varpe [31] ad Dola et al [13] developed cotrollers to cotrol variable displaceet otor for axiu eergy capture objective. Both of the used fixed displaceet pup ad a variable displaceet otor. A differece betwee Varpe [31] ad Dola et al [13] was that Dola et al eployed sychroous geerator but Varpe eployed asychroous geerator. Dutta et al [3] preseted a ethod to use hydraulic

26 8 accuulators for eergy storage to iprove eergy capture efficiecy. Wag et al [33] proposed the odel predictive cotrol (MPC) to icrease the respose tie of the rotor speed to reach the optial tip-speed ratio. Lagua et al [34] studied overall efficiecy ad dyaics of the odel havig the fixed displaceet hydraulic pup i the acelle ad the variable displaceet hydraulic otor ad the sychroous geerator o the groud. Despite the aboveetioed potetial advatages of the hydrostatic trasissio wid turbie, thorough studies o cotrol aspects usig a variable displaceet hydraulic pup ad a variable displaceet hydraulic otor i a odel ad havig the otor ad a sychroous geerator o the groud are still lackig. E. Study Objective ad Report Structure The objective of this study is to develop a cotrol strategy for a wid turbie with hydrostatic trasissio that trasits the wid power to a groud level geerator usig a variable displaceet hydraulic pup ad a variable displaceet hydraulic otor. The cotrol objective is to axiize aerodyaic eergy capture ad overall efficiecy. We study two syste cofiguratios: a sigle turbie with a dedicated hydrostatic trasissio syste ad a double turbie cofiguratio sharig a hydrostatic trasissio syste ad a geerator. Modelig ad siulatio are coducted i MATLAB/Siulik. The atheatical odels of the wid turbie with hydraulic copoets are defied i Chapter. The strategies to axiize aerodyaic eergy capture for the wid turbie are proposed i Chapter 3. MATLAB/Siulik siulatio odels ad results are discussed i Chapter 4. Two Turbie cofiguratio aalysis is discussed i Chapter 5. Coclusio is preseted i Chapter 6.

27 9 II. HYDROSTATIC TRANSMISSION WIND TURBINE MODEL A. Itroductio igure 4 shows the overall hydrostatic trasissio wid turbie odel used i this study. The syste is coposed of turbie blades, a variable displaceet hydraulic pup, hydraulic trasissio pipes, a variable displaceet hydraulic otor, ad a sychroous geerator o the groud level. I a actual syste, there are other auxiliary copoets for hydrostatic trasissio. or istace, a auxiliary hydraulic pup ad two check valves are required to esure that the pressure value at ay locatio of the hydraulic trasissio pipe is ever lower tha, typically, MPa [35] to prevet cavitatio i the syste. There are also pressure relief valves i the syste to prevet the pressure value exceedig the syste liit. or siplicity, those auxiliary copoets are excluded i this study. Oly the copoets i igure 4 are cosidered.

28 1 igure 4: Hydrostatic trasissio wid turbie odel B. Aerodyaic Power ad Torque Model Power of the wid blowig over the circular area swept by the blades at a speed of U, P wid, ca be characterized by Equatio () [1]. P wid 1 3 air AU () air is the desity of air ad A is the blade swept area. Sice wid power coversio is ot 1% efficiet, the available rotatioal power at the rotor (P rotor ) ca be characterized usig Equatio (3) [1].

29 11 Protor Pwid C p (, ) (3) P rotor is the power available as echaical rotatioal power at the low-speed shaft. C p (, ) is the characteristic power coefficiet of the wid turbie, whose value is depedet o the tip-speed ratio ad blade pitch agle. The tip-speed ratio is defied i Equatio (4) [1]. R r (4) U r is the rotor rotatioal speed ad R is the blade legth. The aerodyaic torque at the low-speed shaft ( rotor ) ca be expressed as [1] rotor P 1 rotor ARC (, ) U air q. (5) C q is the characteristic torque coefficiet ad is defied as [1] r C p (, ) Cq (, ). (6) or this study, C p odel is obtaied fro SiPowerSystes (The MathWorks, Natrick, MA) ad is odeled as c C (, ) 1 p c c3 c i i 4 e c5 i c 6 (7) where c 1 =.5176, c =116, c 3 =.4, c 4 =5, c 5 =1, ad c 6 =.68.

30 1 C. Variable Displaceet Hydraulic Pup The variable displaceet hydraulic pup coverts the turbie rotor rotatioal otio to fluid flow. Usig Newto s d law, the first-order dyaic equatio ca be expressed as Equatio (8) [13]. rotor pup J rr (8) J r is the oet of iertia of the rotor ad pup is the resistat torque iposed by the hydraulic pup o the turbie rotor ad ca be expressed as [13] pup V pp p ech, p. p p is the differetial pressure betwee high ad low side of the pup. ech,p is the pup echaical efficiecy. V p is the pup displaceet which ca be cotrolled directly accordig to a per-defied cotrol strategy. Equatio (8) ca be expressed as [13] r 1 J r ( rotor V pp p ech, p ). luid flow rate at the pup side ca be characterized as [8] Q p V. p r kleak, pp p (9) Q p is the fluid flow rate at the pup side. k leak,p is the leakage coefficiet at the pup side, which ca be expressed as the followig equatios [8].

31 13 k k leak, p HP, p k HP, p V fluid p,ax fluid p, o (1 p v, pup p, o ) p, o p, o (1) fluid is the fluid kieatic viscosity. fluid is the fluid desity. k HP,p is the Hage-Poiseuille coefficiet for the pup, which depeds o axiu pup voluetric displaceet V p,ax, oial pup agular speed p,o, pup voluetric efficiecy v,pup, oial pup fluid kieatic viscosity p,o, oial pup fluid desity p,o, ad oial pup differetial pressure p p,o. D. Hydraulic Trasissio Lie I this study a odel of a straight, cylidrical pipe syste is used. Differetial pressure ad flow rate dyaics ca be expressed as [8] fluid p x Q p Q. (11) V fluid p x is the pressure differece betwee the high ad the low pressure side at the ceter of the straight pipe syste. fluid is the hydraulic fluid bulk odulus. V fluid is the volue of the fluid i the pipe or hose. Q p ad Q are the flow rate at the pup ad otor side, respectively. Pressure loss i the pipe syste due to frictio ca also be sigificat ad should be cosidered. The Darcy equatio ad Haalad approxiatio suggest that [36] Lpipe fluid p f Q. D A (1) pipe pipe

32 14 p is the pressure loss alog the pipe due to frictio. L pipe is the legth of the pipe. D pipe is the cross-sectioal diaeter of the pipe. A pipe is the cross-sectioal area of the pipe. Q is the flow rate i the pipe. f is the frictio factor, which is odeled as i Equatio (13) [36]. f 64 / Re f f 4 f Re rpipe / D 1.8 log 1 Re f L Re pipe 1.11 for Re for Re 4 for Re 4 (13) r pipe is the height roughess of the pipe iteral surface ad Re is the Reyolds uber which ca be expressed as [36] QDpipe Re. (14) A Pup differetial pressure (p p ) ad otor differetial pressure (p ) ca ow be expressed as the followig equatios. pipe fluid p p p x p (15) p p x p (16)

33 15 E. Variable Displaceet Hydraulic Motor A variable displaceet hydraulic otor, at the receivig ed of the lie, coverts fluid flow back to rotatioal otio. The otor s dyaics ca be expressed by a firstorder dyaic equatio as show i Equatio (17) [8]. ech, V p 1 ( J J ech, V p load load ) (17) ech, is the otor echaical efficiecy. V is the otor voluetric displaceet. p is the differece i pressure betwee the high ad low side at the otor side. J is the oet of iertia of the high-speed shaft ad that of the geerator rotor. is the otor rotatioal speed. load is the torque produced by the load (i.e., sychroous geerator). luid flow rate at the otor side ca be odeled as Equatio (18) [8]. Q V kleak, p (18) Q is the fluid flow rate at the otor side. k leak, is the leakage coefficiet at the otor side, which ca be expressed as the followig equatios [8]. k k leak, HP, k V fluid,ax HP, fluid, o (1 p v, otor, o ), o, o (19) k HP, is Hage-Poiseuille coefficiet for the otor, which ca be calculated fro axiu otor displaceet V,ax, oial otor agular speed,o, otor

34 16 voluetric efficiecy v,otor, oial otor fluid kieatic viscosity,o, oial otor fluid desity,o, ad oial otor differetial pressure p,o.. Sychroous Geerator A sychroous geerator is used i this study. Coected to the hydraulic otor, the sychroous geerator geerates electricity at the costat sychroous speed syc. or siplicity, the sychroous geerator is odeled by a secod-order syste as show i igure 5 ad Equatio (). agle is the phase agle differece betwee the phase of the grid voltage ad the sychroous geerator s electrical agle. Sychroizig torque coefficiet (K s ) ad dapig torque coefficiet (K d ) are chose so that the geerator odel has a fast ad stable respose. agle load grid K s agle K d agle () igure 5: Siplified sychroous geerator odelig [39]

35 17 Output power (P out ) geerated by the sychroous geerator is give by Equatio (1) eglectig geerator losses. Pout load (1)

36 18 III. CONTROL STRATEGY A. Itroductio Typically, there are three ai operatioal regios for variable-speed wid turbies [37]. igure 6 shows the output power vs. wid speed curve (i.e., the power curve) ad the operatioal regios of a variable-speed wid turbie. Regio 1 refers to a coditio whe the turbie is ot producig output power due to low wid speed uder its threshold value or durig start-up. The rotor rotates freely util the rotor speed reaches the threshold value, ad the turbie oves ito the regio operatig regio. The ai cotrol objective whe operatig at regio is to axiize wid eergy capture. To achieve this, the rotor speed of the wid turbie is cotrolled such that the characteristic power coefficiet (C p ) is axiized. Detailed techiques for axiizig the wid eergy capture for covetioal variable-speed wid turbies are discussed by Johso et al [37]. Regio 3 refers to a coditio whe the turbie operates at high wid speed ad the output power is costat at the axiu regardless of wid speed. A typical cotrol objective i this regio is to prevet the output power exceedig the rated value i order to protect the syste fro power overloadig, which ca cause daage to the wid turbie. This cotrol objective ca be realized by adjustig the blade pitch agle to lower the power coefficiet. If the wid turbie cotiues to experiece high wid speed beyod the cutout speed, the wid turbie will be shutdow. I this study, regio 1 operatio is ot discussed.

37 19 igure 6: Wid turbie syste operatioal regios B. Regio Cotrol Strategy I this sectio the variable-speed hydrostatic trasissio wid turbie cotrol strategies are developed with the goal of axiizig wid power capture. All variables ad their uits are defied i Noeclature at the begiig of this thesis. 1) Maxiizatio of Wid Turbie Power Coefficiet Strategy Power coefficiet (C p ) is the efficiecy of a wid turbie covertig wid power (P wid ) to rotor rotatioal power (P rotor ), i.e., C p =P rotor /P wid. The cotrol law [37] for regio ca be applied so that C p is optiized at steady state by settig the last ter i Equatio (8) to V p p p ech, p 1 K K air AR r 3 C p,ax 3 opt ()

38 where C p,ax is the axiu power coefficiet ad opt is the optial tip-speed ratio that reders the axiu power coefficiet. A detailed aalysis of this regio cotrol law is discussed by Johso et al [37]. Rearragig Equatio () so that V p K r p ech, p p. (3) It ca be see that this cotrol strategy ca be realized by cotrollig V p accordig to Equatio (3). As a result, at steady state, the rotor speed coverges to the speed that gives optial tip-speed ratio ad axiu eergy coversio is achieved (i.e., C p =C p,ax ). ) Maxiizatio of Trasissio Coefficiet Strategy The Trasissio Coefficiet (C T ) is defied as the efficiecy of power trasfer i hydrostatic trasissio syste (i.e., C T =P out /P rotor ). By idetifyig all losses associated with the hydrostatic trasissio ad iiizig the cobied loss, C T ca be axiized. Whe both C p ad C T are axiized, the overall syste efficiecy (C p C T ) is axiized. There are five ai losses i a hydrostatic trasissio syste as discussed below. Pup Mechaical Loss (Loss ech,pup ): ro Equatio (8), at steady-state,. ech, p rotor Vpp p (4) ro Equatio (4), it ca be show that the pup echaical loss (Loss ech,pup ) at steady state ca be characterized as

39 1 Loss 3 (1 ech, p ) K. (5) ech, pup r Pup Leakage Loss (Loss leak,pup ): ro Equatio (9), it ca be show that the pup leakage loss (Loss leak,pup ) at steady state ca be odeled as Loss k p leak, pup leak, p p. (6) Motor Mechaical Loss (Loss ech,otor ): Recall Equatio (17). At steady state,. ech, V p load (7) ro Equatio (7), it ca be show that the otor echaical loss (Loss ech,otor ) at steady state ca be characterized as Loss ech, otor ( ech, 1 ) V p. (8) Motor Leakage Loss (Loss leak,otor ): Recall Equatio (18). It ca be show that the otor leakage loss (Loss leak,otor ) at steady state ca be characterized as Loss k p. (9) leak, otor leak, rictio Loss (Loss fric ): ro Equatio (1) ad (13), it ca be show that the frictio loss (Loss fric ) at steady state ca be odeled as Loss fric L pipe fluid 3 f Q. (3) D A pipe pipe Total Hydrostatic Trasissio Loss (Loss total ): Total hydrostatic trasissio loss (Loss total ) ca be suarized as show i Equatio (31).

40 Loss total... Loss Loss leak, otor ech, pup Loss fric Loss leak, pup Loss ech, otor... (31) Trasissio coefficiet (C T ) ca be characterized as C T Pout Losstotal 1. (3) P P rotor rotor igure 7 shows C T as a fuctio of V at three differet wid speeds (U=4, 8, ad 1/s), assuig rotor speed is related to the wid speed by the optial tip-speed ratio (Paraeters fro Table 1 are used i geeratig these three curves). It ca be see that C T peaks at differet V ad therefore C T ca be axiized by a properly chose V. igure 7: Trasissio coefficiet (C T ) as a fuctio of otor displaceet (V ). Note, U is the wid speed.

41 3 C T Optiizatio Process: The axiu C T ca be foud by takig the partial derivative of Equatio (3) with respect to V ad settig it to zero as show i Equatio (33). V C T V Loss P rotor total (33) Sice P rotor is idepedet of V, Equatio (33) ca be siplified as V Loss total. (34) By solvig Equatio (34) for V, optial V (V,opt ) ca be obtaied (i.e., V =V,opt ). igure 8 shows V,opt for each wid speed U. A liear approxiatio equatio ca be obtaied fro the graph as r R V, opt..3. opt (35) C T ca ow be axiized by cotrollig V =V,opt accordig to Equatio (35). igure 9 is the overall syste-level block diagra for the proposed syste.

42 4 igure 8: Optial otor displaceet (V,opt ) as a fuctio of wid speed (U) ad its liear approxiatio equatio. Note that U = r R/ opt. igure 9: Overall syste level block diagra icludig cotrol strategy to axiize C p ad C T i roud-dotted boxes

43 5 IV. SIMULATION RESULTS IN MATLAB/SIMULINK I this sectio, steady state ad trasiet resposes of the hydrostatic trasissio wid turbie are studied by coputer siulatio accordig to the odel ad cotrol strategies proposed i Chapter ad Chapter 3. Table 1 shows all variables ad their values used i all siulatios i this study. A. Steady State Respose Steady state resposes as a fuctio of U are show i igure 1 to igure 15. is costat at rad/s at steady state regardless of U because the 6-pole sychroous geerator is used i this study. r icreases liearly as a fuctio of U because the cotrol law guaratees that r is proportioal to U at steady state. p p ad p icrease as a fuctio of rotor speed ad, therefore, as a fuctio of U. The differece betwee p p ad p is due to loss fro frictio p. V p ad V also icrease as a fuctio of U accordig to cotrol strategy give by Equatio (3) ad (35). load fro the sychroous geerator icreases as a fuctio of U. P wid, P rotor, ad P out icrease as a fuctio of U. Note that P rotor /P wid =C p, P out /P rotor =C T, ad P out /P wid =C p C T. As show i igure 15, C p is costat regardless of U because of the cotrol strategy proposed i Equatio (3). C T, optiized by Equatio (35), decreases as a fuctio of U (5% decrease fro ed to ed). Table shows all losses associated with C T, suggestig that Loss leak,pup ad Loss fric are higher as U is higher ad, thus, worse C T. The overall syste efficiecy (C p C T ) decreases as fuctio of U, followig C T.

44 6 Table 1: Wid turbie paraeter values used i this study Sybol Defiitio Uit Value air Air Desity kg/ R Wid Blade Radius 63 A Wid Blade Swept Area R Pitch agle Deg C p,opt Optial Power Coefficiet diesioless.48 opt Optial Tip-Speed Ratio diesioless 8.1 fluid luid Bulk Modulus Pa 1.443e9 D pipe Pipe Diaeter.15 L pipe Pipe Legth 1 r pipe Roughess of the Pipe 1.5e-5 fluid luid Desity kg/ fluid luid Kieatic Viscosity /s e-6 J r Pup/Rotor Iertia kg ech,p Pup Mechaical Efficiecy diesioless.95 v,pup Pup Voluetric Efficiecy diesioless.95 V p,ax Maxiu Pup Voluetric Displaceet 3 /rad.3 p,o Noial Pup Speed rad/s 1 p,o Noial luid Kieatic Viscosity at Pup Side /s e-6 p p,o Noial Pressure Differet at Pup Side Pa e7 J g Motor Iertia kg ech, Motor Mechaical Efficiecy diesioless.95 v,otor Motor Voluetric Efficiecy diesioless.95 V,ax Maxiu Motor Voluetric Displaceet 3 /rad 8e-4,o Noial Motor Speed rad/s ,o Noial luid Kieatic Viscosity at Motor Side /s e-6 p,o Noial Pressure Differet at Motor Side Pa e7 syc Sychroous Speed rad/s K s Sychroizig Torque Coefficiet N-/rad.67e5 K d Dapig Torque Coefficiet N-/(rad/s) 1.7e4

45 7 igure 1: Rotor speed ( r ) ad otor speed ( ) as fuctios of wid speed (U) igure 11: Pup differetial pressure (p p ) ad otor differetial pressure (p ) as fuctios of wid speed (U)

46 8 igure 1: Pup displaceet (V p ) ad otor displaceet (V ) as fuctios of wid speed (U) igure 13: Geerator torque load ( load ) as a fuctio of wid speed (U)

47 9 igure 14: Wid power (P wid ), rotor rotatioal power (P rotor ), ad output power (P out ) as fuctios of wid speed (U) igure 15: Trasissio coefficiet (C T ), power coefficiet (C p ), ad overall syste efficiecy (C T C p ) as fuctios of wid speed (U)

48 3 Table. Loss aalysis coputed fro Equatio (4) to (31). The values are calculated i percetage copared with rotor rotatioal power. Loss Type Wid Speed (U) 4 /s 8 /s 1 /s Loss ech,pup 5.% 5.% 5.% Loss leak,pup 3.6% 4.% 5.3% Loss ech,otor 4.4% 4.3% 4.% Loss leak,otor 1.% 1.3% 1.6% Loss fric.4% 4.3% 4.9% Loss total 16.5% 19.%.9% B. Syste Respose The syste resposes are show i igure 16 to igure. The iput U is a uit step fuctio with a iitial coditio of 4/s at t<5s ad the agitude of 6, 1, ad 1 at tie=5, 1, ad 15s, respectively. As U chages, reais costat at rad/s ad r tries to keep up with U so that it operates at r opt U/Rwhere opt =8.1. p p, p, ad load also chage accordig to U. V p ad V are cotrolled accordig to the cotrol law so that the syste reaches the optial efficiecy at steady state. igure shows that C T, C p, ad C T C p track their optial poits as U varies.

49 31 igure 16: Wid speed (U) as a fuctio of tie igure 17: Rotor speed ( r ) ad otor speed ( ) as fuctios of tie

50 3 igure 18: Differetial pup pressure (p p ) ad differetial otor pressure (p ) as fuctios of tie igure 19: Pup displaceet (V p ) ad otor displaceet (V ) as fuctios of tie

51 33 igure : Geerator torque load ( load ) as a fuctio of tie igure 1: Wid power (P wid ), rotor rotatioal power (P rotor ), ad output power (P out ) as fuctios of tie

52 34 igure : Trasissio coefficiet (C T ), power coefficiet (C p ), ad overall syste efficiecy (C T C p ) as fuctios of tie C. Syste Stability Aalysis I this sectio the dyaic equatio of the turbie syste is liearized ad its stability is assessed [38]. Let X (t) be systestatespace variable defied as ( t) ( ) ( ) r t X t. p ( t) x ( t) (36) X(t) ca be writte i ters of a operatigsystetrajectory X ΔX(t) as ad a sall chage

53 35. ) ( ) ( ) ( ) ( ) ( ) (,,,, t t p t t p t X X t X x r x r (37) the followig equatio. ad defied as be systeiput let Siilarly, (t) I ) ( ) ( ) ( ) ( t t t U t I grid syc (38) as ad the variatio value the u - perturbed thesu of as ca be expressed ΔI(t) I (t) I. ) ( ) ( ) ( ) ( ) (,, t t t U U t I I t I grid syc grid syc (39) The oliear syste state space represetatio of the syste ca be writte as ) ( ), ( ) ( t I t X t X dt d (4) where is the 4x4 diesioal vector fuctio. If sall chages are cosidered, the oliear syste ca be liearized at a operatig poit as ) ( ) ( ) ( ) ( ) (,, t I B t X A t I I t X X t X dt d I X I X (41) where A ad B represet the Jacobia atrices ad are give by the followig equatios.

54 36 I X x r x r x r x r I X p p p p A X, , (4) I X grid syc grid syc grid syc grid syc I X U U U U B I, , (43) MATLAB coad liod is eployed to evaluate these. Jacobia atrices ad the eigevalues of the syste are calculated at wid speed U=4, 8, ad 1 /s as show i Table 3. As show, all the eigevalues are o the left-half plae. Therefore, the syste is stable at these operatig poits.

55 37 Table 3. Jacobia atrices ad eigevalues of the syste at specified operatig poit Operatig Poit Jacobia Matrices ad Eigevalues e e5,.513e , / 4 T X I s U i i.687 -, 5.88, e e e Eigevalue B A e5,.513e , / 8 e X I s U T i i.1373, 5.177, e e e Eigevalue B A e5,.513e , / 1 e X I s U T i i.6 -, 5.65, e e e8.6-1 Eigevalue B A

56 38 V. DOUBLE TURBINE CONIGURATION A. Itroductio The cost of a ulti-turbie syste could be reduced if soe copoets ca be cobied. I this sectio, a double-turbie syste with cobied hydraulic pipe, hydraulic otor, ad geerator, as show i igure 3, is aalyzed. igure 3: Hydrostatic trasissio wid turbie with shared hydraulic pipes, hydraulic otor, ad geerator

57 39 B. Hydrostatic Trasissio Modelig for Double Turbies I this sectio, the hydrostatic trasissio wid turbie odel discussed i Chapter, Equatio () to (1), is reviewed ad applied for a double-turbie cofiguratio. I the followig aalysis, the subscript (1 or ) of a variable deotes the turbie (#1 or #) that the variable is associated with. 1) Aerodyaic Power ad Torque Model ro Equatio (), aerodyaic wid powers (P wid1 ad P wid ) for a doubleturbie cofiguratio ca be characterized by the followig equatios. P P wid1 wid 1 3 AU (44) air air AU (45) ro Equatio (3), rotor rotatioal powers (P rotor1 ad P rotor ) ca be characterized by the followig equatios. P P C, ) (46) rotor1 wid1 p1( 1 1 P P C, ) (47) rotor wid p ( ro Equatio (5), aerodyaic torques ( rotor1 ad rotor ) ca be expressed as the followig equatios. P 1 U (48) rotor1 rotor1 air A1 R1C q1( 1, 1) r1 1

58 4 P 1 U (49) rotor rotor air A RCq (, ) r ) Variable Displaceet Hydraulic Pup ro Equatio (8), the first-order dyaic equatio for double turbie cofiguratio ca be expressed as the followig equatios. 1 V p1p p1 r1 ( rotor1 ) (5) J r1 ech, p1 1 Vpp p r ( rotor ) (51) J r ech, p ro Equatio (9), fluid flow rates at the pup side (Q p1 ad Q p ) ca be characterized as the followig equatios. Q V k p (5) p1 p1 r1 leak, p1 p1 Q V k p (53) p p r leak, p p ro Equatio (1), leakage coefficiets at the pup side (k leak,p1 ad k leak,p ) ca be expressed as the followig equatios. k k leak, p1 HP, p1 k V HP, p1 fluid p,ax 1 fluid p, o1 (1 p v, pup1 p, o1 ) p, o1 p, o1 (54)

59 41 k k leak, p HP, p k V HP, p fluid p,ax fluid p, o (1 p v, pup p, o ) p, o p, o (55) 3) Hydraulic Trasissio Lie Sice the fluid flow fro the hydraulic pup of wid turbie #1 ad # are cobied to a sigle pipe syste, the odel of this part is differet fro that i the sigleturbie cofiguratio. Differetial pressure ad flow rate dyaics ca be expressed as Q (56) fluid p x Q p1 Q p V fluid where Q p1 ad Q p are the flow rate at the pup of turbie #1 ad # ad Q is the flow rate at the otor side. All other sybols have the sae defiitio as those defied i Chapter. Pressure loss equatio ad frictio factor for the double-turbie cofiguratio ca be characterized usig Equatio (1) ad (13), respectively, fro the sigle-turbie cofiguratio. 4) Variable Displaceet Hydraulic Motor ad Sychroous Geerator cofiguratio. Equatio (17) to (1) fro Chapter ca be applied to the double-turbie

60 4 C. Regio Cotrol Strategy 1) Maxiizatio of Wid Turbie Power Coefficiet Strategy Siilar to Equatio (3), this cotrol strategy ca be realized by idepedetly cotrollig V p1 ad V p for the double-turbie cofiguratio as show below. V p1 K 1 r1 p ech, p1 p1 (57) V p K r p ech, p p (58) At steady state, the rotor speeds ( r1 ad r ) ca idepedetly coverge to the speed that gives optial tip-speed ratio for axiu eergy coversio efficiecy (i.e., C p1 =C p,ax1 ad C p =C p,ax ). ) Maxiizatio of Trasissio Coefficiet Strategy Siilar to the defiitio give i Chapter 3, the trasissio coefficiet (C T ) is defied as the efficiecy of power trasfer i hydrostatic trasissio syste (i.e., C T =P out /(P rotor1 +P rotor )). To optiize C T, five ai losses are idetified for the doubleturbie cofiguratio as show below. Pup Mechaical Loss (Loss ech,pup1 ad Loss ech,pup ) Loss 3 ech, pup1 ( 1ech, p1) K1r1 (59) Loss 3 ech, pup ( 1ech, p) Kr (6) Pup Leakage Loss (Loss leak,pup1 ad Loss leak,pup )

61 43 Loss Loss k p leak, pup1 leak, p1 p1 k p leak, pup leak, p p (61) (6) Motor Mechaical Loss (Loss ech,otor ) Loss ech, otor ( 1ech, ) Vp (63) Motor Leakage Loss (Loss leak,otor ) Loss k p (64) leak, otor leak, rictio Loss (Loss fric ) Loss fric L pipe fluid 3 f Q (65) Dpipe Apipe Total Hydrostatic Trasissio Loss (Loss total ) Loss total... Loss Loss leak, pup ech, pup1 Loss Loss ech, otor ech, pup Loss Loss leak, otor leak, pup1 Loss fric... (66) Siilar to the sigle-turbie cofiguratio, the trasissio coefficiet (C T ) ca be characterized as C T Pout Losstotal 1. (67) P P P P rotor1 rotor rotor1 rotor C T Optiizatio Process: The axiu C T ca be foud by takig the partial derivative of Equatio (67) with respect to V ad settig it to zero as show i Equatio (68).

62 44 V C T V Loss Protor 1 P total rotor (68) Sice P rotor1 ad P rotor are idepedet of V, Equatio (68) ca be siplified as V Loss total. (69) By solvig Equatio (69) for V, optial V (V,opt ) ca be obtaied (i.e., V =V,opt ). igure 4 shows V,opt as a fuctio of wid speed 1 ad (U 1 ad U ) i a cotour plot at steady state. It ca be show that V,opt depeds o the rotor speed of both turbies ( r1 ad r ), as show i Equatio (7). V f ( r1, ), opt r V,opt for the double-turbie cofiguratio ca be ipleeted usig a lookup table. igure 5 is the overall syste-level block diagra for the proposed double-turbie syste. (7)

63 45 igure 4: Motor displaceet (V ) as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ). Note, whe copared with the sigle-turbie syste, igure 8, V is o loger liear.

64 46 igure 5: Overall syste level block diagra icludig cotrol strategy for axiizig C p1, C p, ad C T (i roud-dotted boxes) D. Siulatio Results i MATLAB/Siulik I this sectio, steady state ad trasiet resposes of the hydrostatic trasissio wid turbie i the double-turbie cofiguratio are siulated accordig to the odel ad cotrol strategies proposed above. Paraeters i Table 1 are used for both turbies except D pipe =.19 (7% icrease fro.15 used i the sigle turbie), which is selected so that C T for the sigle- ad double-turbie are coparable whe U 1 =U.

65 47 1) Steady State Respose Steady state of the syste as a fuctio of wid speed U 1 ad U are show i igure 6 to igure 4. The otor speed is costat at rad/s at steady state regardless of U 1 ad U, siilar to the case of sigle turbie cofiguratio. r1 is a fuctio of U 1 ad ot depedet o U. r is a fuctio of U ad ot depedet o U 1. p p (p p1 =p p =p p ) ad p icrease as a fuctio of U 1 ad U. V p1, V p, ad V icrease as a fuctio of U 1 ad U accordig to cotrol strategy give by Equatio (57), (58), ad (7). load fro the sychroous geerator icreases as a fuctio of U 1 ad U. P wid1 ad P rotor1 are fuctios of U 1 ad idepedet of U. P wid ad P rotor are fuctios of U ad idepedet of U 1. P out icrease as a fuctio of U 1 ad U. Note that P rotor1 /P wid1 =C p1, P rotor /P wid =C p, ad P out /(P rotor1 +P rotor )=C T. As show i igure 4 ad igure 41, C p1 ad C p are costat (C p1 =C p =.48) regardless of U 1 ad U because of the cotrol strategy proposed i Equatio (57) ad (58). C T, optiized by Equatio (7), decreases as a fuctio of U 1 ad U.

66 48 igure 6: Rotor speed 1 ( r1 ) i rad/s as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 7: Rotor speed ( r ) i rad/s as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )

67 49 igure 8: Motor speed ( ) i rad/s as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 9: Differetial pup pressure (p p ) i MPa as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )

68 5 igure 3: Differetial otor pressure (p ) i MPa as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 31: Pup displaceet 1 (V p1 ) i 3 /rad as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )

69 51 igure 3: Pup displaceet (V p ) i 3 /rad as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 33: Motor displaceet (V ) i 3 /rad as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )

70 5 igure 34: Geerator torque load ( load ) i kn- as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 35: Wid power 1 (P wid1 ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )

71 53 igure 36: Wid power (P wid ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 37: Rotor rotatioal power 1 (P rotor1 ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )

72 54 igure 38: Rotor rotatioal power (P rotor ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 39: Output power (P out ) i MW as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )

73 55 igure 4: Power coefficiet 1 (C p1 ) as a fuctio of wid speed 1 (U 1 ) ad wid speed (U ) igure 41: Power coefficiet (C p ) as a fuctio of wid speed 1 (U 1 ) ad wid speed (U )

74 56 igure 4: Trasissio coefficiet (C T ) as a fuctio of wid peed 1 (U 1 ) ad wid speed (U ) ) Syste Respose The syste resposes of the double turbie cofiguratio are show i igure 43 to igure 51. Two iputs, wid speed U 1 ad U, are uit step fuctios as show i igure 43. As U 1 ad U chage, reais at rad/s ad r1 ad r try to keep up with U 1 ad U so that both wid turbies operate at optial tip-speed ratio, siilar to the sigle turbie cofiguratio. p p, p, ad load also chage accordig to U 1 ad U. V p1, V p, ad V are cotrolled accordig to the cotrol law so that the syste reaches the optial efficiecy at steady state. igure 51 shows that C T, C p1, ad C p track their optial poits as U 1 ad U vary.

75 57 igure 43: Wid speeds (U 1 ad U ) as fuctios of tie igure 44: Rotor speeds ( r1 ad r ) ad otor speed ( ) as fuctios of tie

76 58 igure 45: Differetial pup pressure (p p1 =p p p p ) ad differetial otor pressure (p ) as fuctios of tie igure 46: Pup displaceets (V p1 ad V p ) ad otor displaceet (V ) as fuctios of tie

77 59 igure 47: Geerator torque load ( load ) as a fuctio of tie igure 48: Wid power 1 (P wid1 ) ad rotor rotatioal power 1 (P rotor1 ) as fuctios of tie

78 6 igure 49: Wid power (P wid ) ad rotor rotatioal power (P rotor ) as fuctios of tie igure 5: Total wid power (P wid1 + P wid ), total rotor rotatioal power (P rotor1 + P rotor ), ad output power (P out ) as fuctios of tie

79 61 igure 51: Trasissio coefficiet (C T ) ad power coefficiets (C p1 ad C p ) as fuctios of tie It is worth otig that whe oe turbie is operatig at a uch higher speed tha the other oe, the syste pressure will stay at a high value (followig the higher wid speed turbie) because V is adjusted accordig to Equatio (7) to axiize C T. As a result, the leakage flow rate of the hydraulic pup at the lower speed wid turbie (k leak,p p p ) will be sigificat coparig to the puped flow rate due to the rotor speed (V p r ). I fact, if V p r caot overcoe k leak,p p p, the aerodyaic power fro the low speed turbie does ot cotribute to the power geeratio. I this case, two idividual wid turbies operatig at U 1 =1/s ad U =4/s will produce ore power tha what the double-turbie cofiguratio wid turbie ca produce.

80 6 3) Syste Stability Aalysis I this sectio, the dyaic equatio of the double-turbie cofiguratio is liearized ad its stability is assessed [38]. Let (t) be systestatespace variable defied as X. ) ( ) ( ) ( ) ( ) ( ) ( 1 t t p t t t t X x r r (71) as chage ad a sall a operatigsystetrajectory i ters of ca be writte ΔX(t) X (t) X. ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1,,, 1,, t t p t t t p t X X t X x r r x r r (7) the followig equatio. ad defied as be systeiput let Siilarly, (t) I ) ( ) ( ) ( ) ( ) ( 1 t t t U t U t I grid syc (73) as ad the variatio value the u - perturbed thesu of as ca be expressed ΔI(t) I (t) I

81 63 ) ( ) ( ) ( ) ( ) ( ) ( 1,,,, 1 t t t U t U U U t I I t I grid syc grid syc (74) The oliear syste state space represetatio of the syste ca be writte as ) ( ), ( ) ( t I t X t X dt d (75) where is the 5x5 diesioal vector fuctio. If sall chages are cosidered, the oliear syste ca be liearized at a operatig poit as ) ( ) ( ) ( ) ( ) (,, t I B t X A t I I t X X t X dt d I X I X (76) where A ad B represet the Jacobia atrices ad are give by the followig equatios. I X x r r x r r x r r x r r x r r I X p p p p p A X, , (77)

82 64 I X grid syc grid syc grid syc grid syc grid syc I X U U U U U U U U U U B I, , (78) MATLAB coad liod is eployed to evaluate these. Jacobia atrices ad the eigevalues of the syste are calculated at wid speed (U 1 =4, U =4); (U 1 =4, U =8); (U 1 =4, U =1); (U 1 =8, U =8); (U 1 =8, U =1); ad (U 1 =1, U =1) as show i Table 4. As show, all the eigevalues are o the left-half plae. Therefore, the syste is stable at these operatig poits.

83 65 Table 4. Jacobia atrices ad eigevalues of the syste at specified operatig poit Operatig Poit Jacobia Matrices ad Eigevalues , / 4 / 4 1 e e X e I s U s U T i i , , e e e7 8.86e Eigevalue B A , / 8 / 4 1 e e X e I s U s U T i i , , e e e8.975e Eigevalue B A , / 1 / 4 1 e e X e I s U s U T i i , , e e e e Eigevalue B A

84 66 Operatig Poit Jacobia Matrices ad Eigevalues , / 8 / 8 1 e e X e I s U s U T i i , , e e e e Eigevalue B A , / 1 / 8 1 e e X e I s U s U T i i , , e e e e Eigevalue B A , / 1 / 1 1 e e X e I s U s U T i i , , e e e e Eigevalue B A

85 67 VI. CONCLUSION The dyaics ad cotrol of a wid turbie syste with a hydrostatic trasissio were studied i this work. I this syste, the acelle cotaied a variable displaceet hydraulic pup. A hydraulic trasissio pipe coected the pup to a variable displaceet otor o the groud level. The hydraulic otor was echaically coected to a sychroous geerator that coverted the rotatioal power to electrical power. A double-turbie syste was also icluded i the study. The objective of this research was to develop a cotrol strategy that axiized the power coversio efficiecy fro captured aerodyaic power to the productio of the electrical power. The pup displaceet was cotrolled to achieve axiu wid power capture ad the otor displaceet was cotrolled to iiize the losses i the hydraulic trasissio syste. The stability of the proposed cotrol strategy was established by liearizatio aalysis ad was validated by coputer siulatio usig MATLAB/Siulik. The results showed that the syste was stable ad had fast ad well-daped respose. This thesis provides a theoretical basis for the cotrol of a wid turbie syste usig hydraulic trasissio. While the study icludes all the basic eleets i the syste, there are ay aspects of the syste behavior (e.g., fluid dyaics) ad practical issues such as structure stregth, aterials, ad costs that eed further study. Cosiderig the beefits of such a wid turbie syste, the realizatio ad ipleetatio of such a wid turbie syste will be a fertile research area for years to coe.

86 68 BIBLIOGRAPHY [1] T. Burto, N. Jekis, D. Sharpe, ad E. Bossayi, Wid Eergy Hadbook, d ed. Joh Wiley & Sos Ltd, 11. [] U.S. Departet of Eergy. History of Wid Eergy. [Olie]. [3] U.S. Departet of Eergy. (14, May) Electric Power Mothly with Data for ebruary 14. [Olie]. [4] R. Garcia-Heradez ad R. Garuo-Rairez, "Modelig ad Cotrol of a Wid Turbie Sychroous Geerator," i Electroics, Robotics ad Autootive Mechaics Coferece, 11. [5] M. R. Isla, Y. Guo, ad J. Zhu, "A Trasforer-Less Copact ad Light Wid Turbie Geeratig Syste for Offshore Wid ars," i IEEE Iteratioal Coferece o Power ad Eergy, Sabah, 1, pp [6] J. Jeppsso, P. E. Larse, ad A. Larsso. (8, Sep.) Techical Descriptio Lillgrud Wid Power Plat: Lillgrud Pilot Project. [Olie]. llgrud_ pdf [7] ABB. (1) PCS 6 for Large Wid Turbies Mediu Voltage, ull Power Coverters up to 9 MVA. [Olie]. $file/pcs6wid_3bhs3517_e1_reva.pdf [8] ABB. () Distributio Trasforers. [Olie]. df [9] W. Musial, S. Butterfield, ad B. McNiff, "Iprovig Wid Turbie Gearbox Reliability," i Europea Wid Eergy Coferece, Mila, 7. [1] A. Ragheb ad M. Ragheb, "Wid Turbie Gearbox Techologies," i Iteratioal Nuclear ad Reewable Eergy Coferece, Aa, 1. [11] U.S. Departet of Eergy, "Advaced Wid Turbie Drivetrai Cocepts: Workshop Report," 1. [Olie]. [1] K. Wu, Q. Zhag, ad A. Hase, "Modellig ad Idetificatio of a Hydrostatic Trasissio Hardware-i-the-Loop Siulator," Iteratioal Joural of Vehicle Desig, vol. 34, o. 1, pp ,

87 69 4. [13] B. Dola ad H. Aschea, "Cotrol of a Wid Turbie with a Hydrostatic Trasissio - a Exteded Liearisatio Approach," i Methods ad Models i Autoatio ad Robotics (MMAR), 1 17th Iteratioal Coferece o, Miedzyzdrojie, 1, pp [14] J. Cotrell, "Assessig the Potetial of a Mechaical Cotiuously Variable Trasissio for Wid Turbies," i WidPower, Dever, 5. [Olie]. [15] C. Gorla ad P. Cesaa, "Efficiecy Models of Wid Turbies Gearboxes with Hydrostatic CVT," Balka Joural of Mechaical Trasissios (BJMT), vol. 1, pp. 17-4, 11. [16] NewScietist. Clea Tech Revolutio: Ideas for a Low-Carbo World. [Olie]. [17] Mitsubishi Power Systes Europe, Ltd. (13) The uture of Offshore Wid. [Olie]. [18] Arteis Itelliget Power LTD. Our Techology. [Olie]. [19] C. Yige, "Cotrol of a Digital Displaceet Pup," M.S. Thesis, Mechatroic Cotrol Egieerig, Aalborg Uiversity, Aalborg, Deark, May 31, 1. [Olie]. [] A. Pusha, A. Izadia, S. Hazehlouia, ad N. Girres, "Modellig of Gearless Wid Power Trasfer," i 37th Aual Coferece o IEEE Idustrial Electroics Society, 11, pp [1] A. Pusha, M. Deldar, ad A. Izadia, "Efficiecy Aalysis of Hydraulic Wid Power Trasfer Syste," i IEEE Iteratioal Coferece o Electro/Iforatio Techology, 13, pp [] S. Hazehlouia ad A. Izadia, "Adaptive Speed Regulatio of Gearless Wid Eergy Trasfer Systes," i 38th Aual Coferece o IEEE Idustrial Electroics Society, 1, pp [3] S. Hazehlouia ad A. Izadia, "Noliear State Space Model of a Hydraulic Wid Power Trasfer," i 38th Aual Coferece o IEEE Idustrial Electroics Society, 1, pp [4] S. Hazehlouia ad A. Izadia, "State-Space Represetatio of a Hydraulic Wid Power Trasfer," i IEEE Iteratioal Coferece o Electro/Iforatio Techology, 1, pp [5] S. Hazehlouia ad A. Izaia, "Modellig of Hydraulic Wid Power Trasfers," i IEEE Power ad Eergy Coferece, Illiois, 1, pp [6] S. Hazehlouia, A. Izadia, ad S. Awar, "A Eergy Storage Techique for Gearless Wid Power

88 7 Systes," i IEEE Iteratioal Coferece o Electro/Iforatio Techology, 13, pp [7] S. Hazehlouia,. A. Goodarzi, R. Hojatpaah, ad S. Awar, "Stability Aalysis of the Hydraulic Wid Eergy Trasfers Model," i IEEE Electro/Iforatio Techology, 13, pp [8] S. Hazehlouia, A. Izadia, A. Pusha, ad S. Awar, "Cotrols of Hydraulic Wid Power Trasfer," i 37th Aual Coferece o IEEE Idustrial Electroics Society, 11, pp [9] M. Deldar, A. Izadia, ad S. Awar, "Modelig of a Hydraulic Wid Power Trasfer Syste Utilizig a Proportioal Valve," i IEEE Eergy Coversio Cogress ad Expositio, 13, pp [3] A. Izadia, S. Hazehlouia, M. Deldar, ad S. Awar, "A Hydraulic Wid Power Trasfer Syste_Operatio ad Modelig," IEEE Trasactios o Sustaiable Eergy, vol. 5, o., pp , Apr. 14. [31] S. A. Varpe, "Cotrol Syste o a Wid Turbie," M.S. Thesis, Norwegia Uiversity of Sciece ad Techology, Trodhei, 8. [Olie]. [3] R. Dutta,. Wag, B.. Bohla, ad K. A. Stelso, "Aalysis of Short-Ter Eergy Storage for Midsize Hydrostatic Wid Turbie," Joural of Dyaic Systes, Measureet, ad Cotrol, vol. 136, o. 1, p. 117, Sep. 13. [33]. Wag ad K. A. Stelso, "Model Predictive Cotrol for Power Optiizatio i a Hydrostatic Wid Turbie," i The 13th Scadiavia Iteratioal Coferece o luid Power, Liköpig, 13, pp [34] A. J. Lagua, N.. B. Diepevee, ad J. W. v. Wigerde, "Aalysis of Dyaics of luid Power Drive-Trais for Variable Speed Wid Turbies: Paraeter Study," Reewable Power Geeratio, IET, vol. 8, o. 4, pp , May 14. [35] N. D. Marig, Hydraulic Cotrol Syste. Joh Wiley & So, Ic., 5. [36] MathWorks. (14) Hydraulic Resistive Tube. [Olie]. [37] K. E. Johso, L. Y. Pao, M. J. Balas, ad L. J. igersh, "Cotrol of Variable-Speed Wid Trubies: Stadard ad Adaptive Techiques for Maxiizig Eergy Capture," IEEE Cotrol Systes Magazie, vol. 6, o. 3, pp. 7-81, Ju. 6. [38] [39] Z. Gajic, Liear Dyaic Systes ad Sigals, 1st ed. USA: Pretice Hall, Ic, 3. MathWorks. (14) Siplified Sychroous Machie. [Olie].

89 71 APPENDIX A: MATLAB/SIMULINK OR SINGLE TURBINE A. Siulik Base Model igure 5 (Appedix A): Wid Turbie sybol ad detailed odel

90 7 igure 53 (Appedix A): Pup Syste sybol ad detailed odel igure 54 (Appedix A): Motor Syste sybol ad detailed odel

91 igure 55 (Appedix A): Pressure Syste sybol ad detailed odel 73

92 igure 56 (Appedix A): Sychroous Geerator sybol ad detailed odel 74

93 75 B. MATLAB Code for Assigig Syste Paraeters igure 57 (Appedix A): MATLAB code for assigig syste paraeters

94 76 C. MATLAB Code ad Siulik for Steady State Respose igure 58 (Appedix A): MATLAB code for steady state respose

95 igure 59 (Appedix A): Workspace (part 1) for OeTurbie_SteadyState.dl 77

96 igure 6 (Appedix A): Workspace (part ) for OeTurbie_SteadyState.dl 78

97 79 D. MATLAB Code ad Siulik for Syste Respose igure 61 (Appedix A): MATLAB code for syste respose

98 igure 6 (Appedix A): Workspace (part 1) for OeTurbie_Dyaics.dl 8

99 igure 63 (Appedix A): Workspace (part ) for OeTurbie_Dyaics.dl 81

100 8 E. MATLAB Code ad Siulik for Liearizatio igure 64 (Appedix A): MATLAB code for liearizatio

101 igure 65 (Appedix A): Workspace (part 1) for OeTurbie_Liearizatio.dl 83

102 igure 66 (Appedix A): Workspace (part ) for OeTurbie_Liearizatio.dl 84

103 85 APPENDIX B: MATLAB/SIMULINK OR DOUBLE TURBINE A. Siulik Base Model igure 67 (Appedix B): Pressure Syste sybol ad detailed odel (part 1)

104 86 igure 68 (Appedix B): Pressure Syste sybol ad detailed odel (part ) The Siulik base odels for two turbie cofiguratio are siilar to those of the oe turbie cofiguratio ad ca be used iterchageably, except the Pressure Syste. The Pressure Syste for the Two Turbie cofiguratio is show i igure 67 (Appedix B) ad igure 68 (Appedix B).

105 87 B. MATLAB Code for Assigig Syste Paraeters igure 69 (Appedix B): MATLAB code for assigig syste paraeters

106 88 C. MATLAB Code ad Siulik for Steady State Respose igure 7 (Appedix B): MATLAB code for steady state respose (part 1)

107 igure 71 (Appedix B): MATLAB code for steady state respose (part ) 89

108 igure 7 (Appedix B): Workspace for TwoTurbie_SteadyState.dl 9

109 91 D. MATLAB Code ad Siulik for Syste Respose igure 73 (Appedix B): MATLAB code for syste respose