fulltext - diva

F11028

Examensarbete 30 hpJuli 2011

Evaluation of Finite Element Method Based Software for Simulation of Hydropower Generator - Power Grid Interaction

Gustav Persarvet

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Evaluation of Finite Element Method Based Softwarefor Simulation of Hydropower Generator - Power GridInteractionGustav Persarvet

The accuracy, ease of use, and execution time of the finite element method basedsoftware Maxwell coupled to the system simulation software Simplorer was evaluatedfor simulation of hydropower generator - power grid interaction. A generator test rigwere modelled in Maxwell and coupled to Simplorer with a strong circuit coupling asa single machine infinite bus system. The accuracy of the model was measured bycomparing the simulated output power oscillation frequency and dampingcharacteristics to the measured ones after a torque step. The result shows that thedifference in output power oscillation frequency between the model and thegenerator test rig was small, and that the difference in damping characteristics wassignificant. The usability of the software package was found to be fair, as were theexecution times.

Sponsor: Vattenfall Research & DevelopmentISSN: 1401-5757, UPTEC F11 028Examinator: Tomas NybergÄmnesgranskare: Urban LundinHandledare: Johan Bladh

Sammanfattning

Sverige har en lång tradition av att använda sig av vattenkraft och år 2009 stodden för 49% av den producerade elkraften. Vattenkraftverken har i stort settbyggts och använts på samma sätt under hela 1900-talet.

Nu står vi inför en stor omsällning av energisystemet där vattenkraften i alltstörre utsträckning ska fungera som balanskraft för annan förnybar generering.Samtidigt står vi inför omfattande renoveringar av befintliga kraftverk som falitför åldersstrecket. Det finns därför anledning att i detlj studera den komplexainteraktionen mellan generator och kraftnät för att se om etablerade sanningarfortfarande gäller. För att kunna göra detta är det intressant att undersöka ommodern simuleringsprogramva kan bidra med ny kunskap.

Vattenkraftgeneratorers elektromagnetiska egenskaper beskrivs vanligen medenkla kretsekvivalenter. Dessa har fördelen av att vara numeriskt effektiva ochfungerar i många fall alldeles utmärkt, speciellt vid simuleringar av hela kraft-system där varje generator ses som en komponent bland många, och fokus liggerpå hur systemet beter sig som helhet. Om intresset istället rör generatorn självoch kraftsystemet ses som en komponent ansluten till dess terminaler räckerinte alltid noggrannheten i kretsmodellen till.

Den här rapporten fokuserar på att ta reda på huruvida kommersiell finita-element-programvara kan användas för simulering av generator - kraftnätsinter-aktion då kraftnätet ses som en infinite bus. Det finns ett antal olika program-varupaket för att genomföra finita element-beräkningar av roterande elektriskamaskiner. Tre av dessa undersöktes för att avgöra vilken programvara som skullegås vidare med. Dessa var Ansoft Maxwell tillsammans med Simplorer, CedratFlux2D tillsammans med Portunus, och Comsol Multiphysics. Ansoft Maxwelltillsammans med Simplorer valdes ut, och en experiment-rigg vid institutionenför ellära vid Uppsala Universitet modellerades i programvarupaketet.

Simuleringar genomfördes som jämfördes med mätningar gjorda på tidigarenämnda generator. Utöver detta undersöktes även användarvänlighet och exe-kveringstid. Resultatet visar att frekvensen hos effektpendlingarna hos systemetöverensstämmer väl mellan modell och experiment-rigg, men att dämpningska-rakteristiken inte överensstämmer väl mellan modell och experiment-rigg. Bådeexekveringstid och användarvänlighet bedömdes vara tillräckligt bra.

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Innehåll

1 Introduction 6

1.1 Project Background . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Aim of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Simulating Generators using FEM 8

2.1 2D Field Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 The Finite Element Method . . . . . . . . . . . . . . . . . 8

2.1.2 Calculating the Magnetic Field Vector Potential . . . . . 8

2.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . 9

2.1.4 Materials and Sources . . . . . . . . . . . . . . . . . . . . 9

2.1.5 The Sliding Mesh Technique . . . . . . . . . . . . . . . . 10

2.2 Rotational Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Modeling the Power Grid as an Equivalent Circuit . . . . . . . . 11

2.3.1 Infinite Bus . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.2 Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Strong Circuit Coupling . . . . . . . . . . . . . . . . . . . 12

2.4 Calculating the Initial Condition Angles . . . . . . . . . . . . . . 12

3 Initial Study 14

3.1 Criteria for the Software Candidate . . . . . . . . . . . . . . . . . 14

3.2 Results of the Initial Study . . . . . . . . . . . . . . . . . . . . . 14

4 Modeling and Simulation of the SMIB System in the ChosenSoftware 16

4.1 Criteria for the Software Candidate . . . . . . . . . . . . . . . . . 16

4.2 Modeling the Synchronous Generator - Infinite Bus System . . . 16

4.2.1 Software Package Components . . . . . . . . . . . . . . . 16

4.2.2 Generator Data and Measurements . . . . . . . . . . . . . 16

4.2.3 Geometry of the Generator . . . . . . . . . . . . . . . . . 17

4.2.4 Stator and Rotor Windings . . . . . . . . . . . . . . . . . 17

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4.2.5 Damper Winding . . . . . . . . . . . . . . . . . . . . . . . 18

4.2.6 Mechanical Transients . . . . . . . . . . . . . . . . . . . . 18

4.2.7 The Infinite Bus and Transformer . . . . . . . . . . . . . 20

4.2.8 Boundary Conditions and Initial Values . . . . . . . . . . 21

4.2.9 Load Angle . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Transient Simulations and Validation . . . . . . . . . . . . . . . . 22

4.3.1 Damping of Initial Transients . . . . . . . . . . . . . . . . 22

4.3.2 Finding the Torque From the Experiment Data . . . . . . 23

4.3.3 Torque Step Simulation Setup . . . . . . . . . . . . . . . . 24

4.3.4 Analysis of Data . . . . . . . . . . . . . . . . . . . . . . . 25

5 Results and Discussion 27

5.1 Execution Times . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Ease of Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 Validation Simulations . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3.2 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . 31

6 Conclusion 33

7 Suggested Future Work 34

A Generator Measurement and Data 37

B MATLAB Initial Angle Script 43

C MATLAB Experiment Torque Script 45

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List of Symbols and Abbreviations

Symbol Unit DescriptionA Wb/m Magnetic vector potentialAz Wb/m z-component of magnetic vector potentialf Hz Curve fit parameter, frequency of oscillationD C/m² Electric displacement fieldEq V Generator internal electromotive forceETh V Thévenin equivalent circuit electromotive force

Fx1, Fx2

, Fy1 , Fy2 N Force componentsi, I A Current in stator

ia, ib, ic A Current in stator phase a, b and cJ A/m² Current densityJ kg m² Moment of inertiaKD Nm s rad−1 Damping torque coefficientLs H Stator winding inductance

np, ns Number of turns in primary and secondary sidesP W Electric powerP0 kW Curve fit parameter, steady-state output powerp Number of pole pairspa kW Curve fit parameter, amplitude of power oscillationReq Ω Combined stator and Thévenin equivalent circuit resistanceRs Ω Stator resistanceRTh Ω Thévenin equivalent circuit resistancer m Stator radiusT Nm Total torqueTe Nm Electrical torqueTm Nm Mechanical torquet s TimeUt V Terminal voltageV V Electric potential

va, vb, vc V Voltage over stator phase a, b and cXTh Ω Thévenin equivalent circuit reactanceXq Ω Quadrature axis reactanceZe Ω Transformer equivalent impedance

Zp, Zs Ω Transformer primary and secondary side impedance

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Greek Symbol Unit Descriptionβ el. rad Angle between Thévenin equivalent

electromotive force and terminal voltageδ el. rad Rotor (load) angleδ0 el. rad Initial rotor angleθ rad Curve fit parameter, phase of oscillationν m/H Magnetic reluctivityξ s−1 Curve fit parameter, damping factor of

the oscillationσ S/m Electric conductivityϕ rad Power factor angleωms mec. rad/s Synchronous mechanical angular velocity

of the rotorωm mec. rad/s Mechanical angular velocity of the rotor

∆ωm mec. rad/s Deviation from synchronous mechanicalrotor velocity

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1 Introduction

1.1 Project Background

Sweden has a long tradition of using hydropower to generate electricity, and itis one of our most important sources of electrical energy. In 2009, hydropoweraccounted for 49% of the produced electric energy[1]. It is also the primaryproducer of balancing power due to the relative short time it takes to changethe amount of power produced by the plants[2]. During the 1900’s, hydropowerplants have been built in the same way[4].

Presently, we face big changes in the electricity system, where hydropower plantsmore and more need to act as balancing power to enable a large expansionof other sustainable power generation sources. At the same time, we are alsofacing large refurbishment activities for hydropower plants that are becomingold.[4] Therefore there exists a need to, in detail, study the complex interactionsbetween hydropower generators and the power grid in order to verify whetheror not established truths still hold. In order to accomplish this, it is interestingto see if modern simulation software can be used to produce new knowledge.

The electromagnetic properties of hydropower generators are commonly descri-bed mathematically as a set of ordinary differential equations extracted fromsimple equivalent circuits. This approach has the advantage of being numeri-cally efficient and is in many cases able to produce good results when studyingthe dynamics of generators with a disturbed steady-state operating conditioncoupled to the power grid. This is especially true for simulations of whole powersystems where the generator is seen as a component among many others, andwhere focus is on how the system behaves as a whole. If the interest instead liesin the generator itself and the power grid is seen as a component connected toits terminals, this approach is often not accurate enough[4].

Another method is to calculate the electromagnetic field in a domain represen-ting the geometry of the system, and from the field solution calculate otherquantities, such as current, etc. This is the finite element method (FEM). Thisapproach is useful when the component studied is the generator, and the powergrid is seen as a component coupled to its terminals. FEM is very resource in-tensive, and its usefulness thus diminishes fast with increasing number of FEMcomponents.

Several mature FEM based software packages exist. The hydropower group atthe Division of Electricity at Uppsala University are currently using a FEMbased software called Ace to simulate hydropower generators for their research.This software has for example been used to predict core loss[5], and for tran-sient electromechanical analysis[4]. However, the group is interested in similarcommercial software as a complement to Ace. Modern FEM based packages arebelieved to be a feasible substitute to Ace in many cases. Advantages of anin-house developed software includes the ability to alter the code to support

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non-standard features for research projects. Advantages for a commercially de-veloped software includes easy setup of simulations and commercial support anddocumentation.

1.2 Aim of the Thesis

The aim of this thesis is to evaluate the capabilities of commercial FEM ba-sed software packages to see where they stand both in terms of what can andwhat cannot be modeled, and also ease of use of the program and accuracy ofthe solution. Three different commercial software packages - Comsol Multiphy-sics, Cedrat Flux2D together with Portunus and Ansoft Maxwell together withSimplorer - will be tested briefly. One of them will be chosen for additional tes-ting, which involves modeling of an experimental generator test rig available tothe author[6], and comparison of simulation results to measurements.

1.3 Outline of Thesis

Chapter 2 contains theory on how to model the system. In chapter 3 the initialstudy to choose among the programs is presented. In chapter 4, further criteriaon how to evaluate the software candidate is presented together with a descrip-tion of how the system was modeled and how the validation simulations wereset up. In chapter 5, results are presented and discussed. In chapter 6 the con-clusions drawn from the results are presented, and finally, in chapter 7, futureresearch topics are discussed.

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2 Simulating Generators using FEM

2.1 2D Field Problem

2.1.1 The Finite Element Method

FEM is a numerical tool for obtaining field quantities in a computational do-main. The method was first proposed in the 1940’s, and was at first used inaeronautical and structural analysis. Over the years, it has been adopted in al-most all physical and mathematical problems, and it has been used in electricalmachine analysis since the 1970’s.[7][8]

The solution of the finite element problem is obtained by finding an approximatesolution to the studied field quantity that lives in the same space as the basisfunctions. The basis functions are functions that have the value one in theirnode, and zero in all other nodes. Often, they are piecewise linear functions, butother types of basis functions also exists. The nodes are obtained from a mesh,often with triangular elements. An example mesh is shown in Figure 1. The finiteelement formulation is transformed into a system of equations, which togetherwith boundary and initial conditions can be solved using linear algebra.[9]

Figur 1: The analysis domain taken from Ansoft Maxwell (one generator pole)with an example mesh.

2.1.2 Calculating the Magnetic Field Vector Potential

In 2D simulation of generators the field problem is often simplified by assumingthat J = [0, 0, Jz]. This is a reasonable assumption if eddy-currents are neg-

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lected, i.e., the current density is only nonzero in the copper windings. Anotherassumption that is made is that there is no displacement current ∂D

∂t in Ampè-re’s law. This assumption can be made since the displacement current is smallat frequencies common to electric power systems (50-60 Hz). The 2D problemcan then be stated as

∂

∂x

(ν∂Az(x, y, t)

∂x

)+

∂

∂y

(ν∂Az(x, y, t)

∂y

)= σ

∂Az(x, y, t)

∂t+ σ

∂V

∂z(1)

where ν is the magnetic reluctivity, σ is the electric conductivity, V is the electricpotential and Az is the magnetic vector potential. Since the problem involvesthe magnetic vector potential, a gauge must be chosen. Coulomb’s position isgenerally adopted, which is ∇ •A = 0.[7][4]

2.1.3 Boundary Conditions

Two types of boundary values are generally used when simulating rotatingelectric machines, the homogeneous Dirichlet boundary condition Az = 0 andthe periodic boundary condition Az(Γ1) = ±Az(Γ2), where Γ1 and Γ2 are twodifferent parts of the boundary. The Dirichlet condition is commonly used at thestator back and at the rotor hub. This is equivalent to the condition that theoutside material is a magnetic insulator, i.e. no flux lines crosses the bounda-ry between the materials. A periodic boundary condition is used to shrink thecomputational domain by using symmetries in the generator model. It is appli-ed along the radii from the origin located in the rotor center to the ends of thestator back circle segment. Periodic conditions are used for an even number ofpoles in the computational domain, and antiperiodic conditions are used for anodd number of poles. The boundaries can be seen in Figure 6, where they aremarked with a C.[7]

2.1.4 Materials and Sources

The FEM analysis domain can be divided into three different regions with dif-ferent characteristics due to their material properties. The rotor ring, pole andstator are assigned iron as material, and are thus modeled with nonlinear mag-netic properties. It is assumed that there are no currents in these regions. Con-ducting regions, i.e., field winding and stator winding, are given the materialcopper and modeled with non-zero currents and linear magnetic properties. Thelast type of region is assigned the material air and is modeled with linear mag-netic properties and zero current.[4]

In equation 1, the right hand side terms are source terms, contributing to themagnetic vector potential. The first term, σ ∂Az(x,y,t)

∂t , represents the inducedcurrent density, and the second term, σ ∂V∂z , represents the applied current den-sity from external sources.[4]

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2.1.5 The Sliding Mesh Technique

Remeshing the entire domain during transient simulations for each time step,due to the rotor rotating, might take considerable time. One way of avoiding toremesh the entire domain is to have two static meshes that interacts with eachother in a small boundary layer, so that only the boundary layer needs to beremeshed in each time step. This boundary layer is placed in the air gap of thegenerator. This is the sliding mesh technique. An example of this can be seenin Figure 2.[10]

Figur 2: Two static meshes that are connected by a mesh that is remeshed eachtime step to handle the rotation of the rotor.

2.2 Rotational Dynamics

When modeling the rotational dynamics of the rotor, it is convenient to look atdeviations from synchronous operation, i.e.,

∆ωm = ωm − ωms (2)

where ∆ωm is the difference between the angular velocity of the rotor ωm andthe synchronous angular velocity ωms. Additionally, the rotor angle, δ, can bemodeled as

δ = δ0 + p

ˆ∆ωmdt (3)

where p is the number of pole pairs, and δ0 is the initial offset in electrical degreesfrom the rotating reference axis, which rotates at synchronous angular velocitypω. δ is also known as the load angle and is, if losses in the iron regions areneglected, the angle between the internal electromotive force in the generatorand the terminal voltage[11]. Taking the derivative of (3) yields

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dδ

dt= p

d

dt

ˆ∆ωmdt = p∆ωm (4)

Combining (4) with Newton’s second law of motion in rotational form yields

Jdωmdt

= Tm − Te, (5)

where J is the moment of inertia, Tm is the mechanical torque on the turbineand Te is the electromagnetic torque on the rotor. Eq. (4) and (5) are calledthe swing equations. Sometimes, a damping term proportional to the deviationof the synchronous angular velocity, −KD∆ωm, is added to (5) to representdamping torque attributable to frictional losses.[12]

2.3 Modeling the Power Grid as an Equivalent Circuit

2.3.1 Infinite Bus

When modeling a single generator coupled to a comparativly strong power grid,it is convenient to consider the grid as an Thévenin equivalent circuit. It isthen assumed that the dynamics of the single generator will cause virtually nochange in the system to which it is supplying its power. It is therefore modeledas an ideal voltage source, i.e., a voltage source of constant amplitude, frequencyand phase. This is called an infinite bus. The whole system, which consists ofa generator interacting with the whole power grid, is called a single machineinfinite bus (SMIB) system.[12]

2.3.2 Transformer

An ideal transformer can be modeled as two windings, each with an impedance,that are coupled together by a mutual inductance. A varying current in theprimary side induces an electromotive force in the secondary side, which isproportional to ns

npwhere np is the number of turns in the primary side of the

transformer, and ns is the number of turns in the secondary side. The reciprocal,np

ns, is usually called the turns ratio. An example can be seen in Figure 3a. The

impedance on the primary side can be transferred to the secondary side after

scaling it with the impedance scaling factor(ns

np

)2

which yields the equivalentcircuit in Figure 3b. The equivalent impedance is thus

Ze = Zs +

(nsnp

)2

Zp (6)

where Zs and Zp are the secondary and primary side impedances.[12]

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(a) Equivalent circuit for ideal transformer. (b) Equivalent circuit after transferringthe primary side impedance to the secon-dary side.

Figur 3: Equivalent circuits for modeling ideal transformers.

2.3.3 Strong Circuit Coupling

When the field problem and circuit problems are coupled together and solvedsimultaneously the connection between them is called strong, as opposed to aweak coupling where the different solvers for field and circuit problems exchangesolutions with each other. This means that the field and circuit equations arecombined in a system of equations for each time step and solved.[10][13]

2.4 Calculating the Initial Condition Angles

The steady-state angle difference between the infinite bus and the generatorinternal electromotive force can be calculated from the phasor diagram of thecircuit. The generator equivalent circuit can be seen in Figure 4a, where Rs isthe stator resistance, Xq is the quadrature axis reactance, Eq is the generatorinternal electromotive force, Ut is the terminal voltage, and I is the currentin the stator winding. Combining this equivalent circuit with the transformerand the infinite bus gives the equivalent circuit seen in Figure 4b, which can berepresented by the phasor diagram seen in Figure 4c. From the phasor diagramit can be deduced that the load angle δ can be calculated as

δ = arctan

(XqI cos (ϕ)−RsI sin (ϕ)

Ut +XqI sin (ϕ) +RsI cos (ϕ)

)(7)

and that the angle between the terminal voltage and the infinite bus electromo-tive force, β, can be calculated as

β = arctan

(RThI sin (ϕ)−XThI cos (ϕ)

Ut −RThI cos (ϕ)−XThI sin (ϕ)

)(8)

where cos (ϕ) is the power factor, RTh is the grid and transformer resistance andXTh is the grid and transformer reactance. At steady state the generator will berunning at rated power, rated terminal voltage and rated power factor. Theseparameters together with known stator resistance, quadrature axis reactance

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and grid plus transformer impedance, allows the calculation of the total anglebetween the infinite bus and the generator internal voltage, β + δ.[12]

(a) Generator as an equivalent circuit.

(b) An equivalent circuit of the generator together with the grid Thévenin equiva-lent.

(c) A phasor diagram of the generator and grid Théveninequivalent.

Figur 4: The generator represented by equivalent circuits.

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3 Initial Study

In this chapter, the initial study used to choose among the software packages ispresented. The first section describes the criteria used for choosing among theprograms, and the second section presents the results of the study.

3.1 Criteria for the Software Candidate

When starting the project, it was decided that the three softwares Comsol Mul-tiphysics, Cedrat Flux2D together with Portunus and Ansoft Maxwell togetherwith Simplorer should be evaluated. Each one would be tested briefly duringthe first two weeks of the project to assess their ease of use, because a full studyof each of the programs would take too long. The following criteria were set up(in no particular order) to choose among them. The program should:

1. be able to couple the field and circuit problems in a strong sense, i.e.the program should solve both the field problem and the circuit problemsimultaneously.

2. be able to use the sliding mesh technique so that the mesh can be reusedfor each time step, unless it has to be refined.

3. be able to couple the swing equation to the rest of the problem to simulatemechanical transients.

4. be able to make parameter optimization runs.

5. be able to import CAD geometries.

6. be able to model block programmed control functions.

7. be able to generate the geometry of a salient pole three-phase synchronousgenerator from a template.

8. have support in a language spoken by the potential user group.

3.2 Results of the Initial Study

The result of the initial study can be seen in Table 1. It can be seen that oneof the programs fulfills all of the criteria, which is Ansoft Maxwell. Flux2D andComsol Multiphysics fulfills 7 out of 8 criteria. Therefore it was decided thatMaxwell would be studied more thoroughly. This does not imply that the othertwo programs are either fit or unfit to model hydropower generators, but ratherthat these programs were not tested further in this project.

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Tabell 1: The results from the initial studyCriterianum-ber

Comsol Multiphysics Flux2D Maxwell

1 Yes[14] Yes[17] Yes[13]2 Yes[15] Yes[18] Yes[13]3 Yes Yes[18] Yes[13]4 Yes[16] Yes[18] Yes[13]5 Yes[16] Yes[18] Yes[20]6 With Simulink With

Portunus/Simulink[19]With Simplorer[20]

7 No No* Yes8 Swedish/English English/French Swedish/English

* Templates are available for other types of rotating electrical machines.

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4 Modeling and Simulation of the SMIB Systemin the Chosen Software

This chapter describes the additional testing done on the software candidate.The first section discusses additional test criteria for the software candidate, thesecond section describes the modeling procedure, and the third section describesthe setup of the validation simulations.

4.1 Criteria for the Software Candidate

Once the software candidate was chosen, more criteria were added to evaluate itsperformance. First and foremost, it must give a correct answer, i.e., it must giveapproximately the same answer as measurements on the system it models, giventhat the model approximates the system. To test this, a model of a researchgenerator available at the Division of Electricity at Uppsala University wasbuilt in Maxwell. This was then used to simulate a torque step to see howthe generator responded. The same test was done on the research generator, inorder to compare the results. In addition to this it should complete simulationsin an reasonable time. In this case this means that a simulation time of 20hours instead of 18 hours does not matter, since these simulations are executedduring non-office hours. An execution time of 48 hours though would impactproductivity.

4.2 Modeling the Synchronous Generator - Infinite BusSystem

4.2.1 Software Package Components

The software package tested contained two programs, Ansoft Maxwell and An-soft Simplorer. Maxwell is the FEM based software in which the generator wasmodeled. There are three different model type choices, 2D, 3D, and RMxprt.The RMxprt model is an equivalent circuit based model, which can be used as atemplate for a Maxwell 2D geometry. The power grid equivalent circuit and themechanical system was modeled in Ansoft Simplorer, which is a multi-domainsystem simulation software.

4.2.2 Generator Data and Measurements

When setting up the model of the generator, it was necessary to find both phy-sical and electrical parameters of the generator in order to model it correctly.The physical information came mainly from two sources. The first one is me-asurements of the generator itself, which was done with a caliper and a tape

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measure. The second one was a 3D CAD construction model created at the timewhen the generator was acquired to the Division of Electricity. The electricalparameters were mainly obtained by researchers at the hydropower group atUppsala University. In Appendix A there is a full list of parameters and howthey were measured.

4.2.3 Geometry of the Generator

When setting up the geometry of the research generator, two methods were used.At first a 2D CAD model of the generator was setup. This model was not usedfor simulations though, since a faster method became available with the choiceof Maxwell, which was converting an RMxprt model available in Maxwell into a2D geometry. A RMxprt model is an equivalent circuit model of the generatorand is built up using input parameters, such as geometrical properties (statorouter radius, stator inner radius, slot dimensions, etc.), winding information(winding type, wire sizes, etc.) and electrical properties (rated power, ratedline-line voltage, load type, etc.). This model could then be transformed intoa Maxwell geometry without human intervention (except that required to tellthe program to do so). When setting up the model in this way, Maxwell looksfor symmetries in the model and reduces the size of the model automatically tosave computational time. The resulting geometry is thus a circle sector of the 2Dslice of the generator as seen in Figure 1. Depending on what type of symmetrythat exists in the model, the cut is either made in between poles (quadratureaxis) or in the middle of a pole (direct axis). The number of poles included inthe sector is also dependent on symmetries in the generator model. The wholegenerator is simulated by using periodic or antiperiodic boundary conditions asdescribed in section 2.1.3.

4.2.4 Stator and Rotor Windings

The winding information was first given in the RMxprt model, and then expor-ted into the Maxwell model. Then the field winding were changed into external.This gives access to the winding in Simplorer as two pins.

The stator windings were also changed into external, i.e., coupled to the externalcircuit model, so that they could be coupled to the transformer and infinite busin Simplorer. The winding type was changed to solid. That meant that instead oftwo parallel strands in each conductor, each conductor was modeled as a singlesolid conducting path. The reason for this decision was that it made it possibleto change the resistance of the stator winding by changing the conductance ofthe material, which was not possible with the stranded winding. This resistancecould be modeled by adding extra resistance to the transformer resistance inthe grid, but since the load angle is in part dependent on the stator resistance,it was decided that it would be modeled by changing the conductance.

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The right conductance was found by virtual experiment on the generator model.A DC voltage of 1 mV was sent through a 1 mΩ resistance in the grid into thestator winding. The rotor was held still by assigning it a moment of inertia of750000 kgm², and assigning it an initial speed of 0 rad/s. The current througha RL circuit with an applied DC voltage is

i (t) =EThReq

(1− e−RLst)

where i is the current, ETh is the applied electromotive force, Req is the combi-ned resistance of the grid and the stator winding, and Ls is the inductance ofthe stator winding. When time increases, the initial transient in the current diesout and approaches that of the resistive only circuit. Since the voltage is known,the total resistance (for grid and stator) can then be obtained as Req = ETh

i(t→∞) .The conductance was then set so that the stator resistance agreed with themeasurements done on the research generator[21].

Worth noticing is that positive direction for the stator current is out of thegenerator, while positive direction of the stator voltage is into the machine, asseen from the infinite bus. For the field winding, positive current direction is fromthe in-pin into the machine. Positive voltage direction is reversed compared tothe current, and is positive from the out-pin into the machine. For a descriptionof the pins, see section 4.2.7.

4.2.5 Damper Winding

The damper bars were modeled in the RMxprt model, and were thus setupautomatically when the model was transformed into a Maxwell 2D model. Theresearch generator have poles with three damper bars that are connected by acopper bar on top and bottom of the pole (incomplete damper winding). Thesecopper bars can also be connected by copper wires in between adjacent polesto make a complete damper winding. The damper winding can also be removedaltogether. A non-standard feature of the generator is that one of the damperbars is located 1.2 times the distance to the other damper bar as seen fromthe middle damper bar. This could not be modeled with the RMxprt model,which instead used 1.1 times the distance for both. This can be modeled directlyon the Maxwell 2D model though. The RMxprt settings used can be found inAppendix A.[6]

4.2.6 Mechanical Transients

There are two different ways of setting the mechanical load, depending on ifthe model is coupled to Simplorer or not. If only Maxwell is used, the mecha-nical torque is set within Maxwell. The options are either constant speed ormechanical transient simulation. If, on the other hand, the model is coupled

18

to Simplorer, the mechanical part is handled by Simplorer via two pins thatconnects to the FEM component. These are labeled T4 and T8 in Figure 5, andcan be seen in the area marked D.

In this project, Simplorer was used to model the mechanical system. This was do-ne by using an ideal torque source for mechanical load and another for couplingto the control loop (K ) together with control functions (H ), a rotating mass (F ),and a mechanical ground component (G). The ideal torque source was coupledto the rotating mass component, which in turn was connected to the in-pin ofthe mechanical pins. The mechanical out-pin of the FEM simulation componentwas connected to the mechanical ground. The rotating mass components initialangular velocity was set to 500 rpm, and the moment of inertia to 59 kgm². Itis modeled with the equation

ωm =1

J

ˆT dt (9)

where T is the accelerating torque. Taking the time derivative and rearrangingthe expression yields equation (5).[22]

In some simulations a mechanical damping term was added that was proportio-nal to the deviation from the synchronous angular velocity. It was implementedusing a proportional control loop (K ) that used the synchronous angular velocityas setpoint.

19

Figur 5: The components as seen in Simplorer. The three main parts are theFEM component (D,E ), the power grid Thévenin equivalent circuit (A,B,C ),and the mechanical system (F,G,H,K ).

4.2.7 The Infinite Bus and Transformer

The infinite bus was modeled in Simplorer using three ideal voltage sourceswith constant amplitude and phase, which can be seen in the area marked A inFigure 5. The phase difference between them was +0 degrees for phase A, +240degrees for phase B, and +120 degrees for phase C. The order of the phasesare dependent on the order of the windings in the FEM component (E ). Thesevoltage sources were connected by the positive side (through the transformer) tocorresponding the out-pins (D, labeled T5, T6 and T7) of the phases in the FEMcomponent. The negative sides were coupled together into a return conductor

20

that were connected to the Wye-connection of the phases. This were in turnconnected to the in-pins (D, labeled T1, T2 and T3) of the FEM component. Aresistor (C ) was connected on the return path.

The transformer (B) was modeled as an ideal transformer with a resistance andan inductance for each of the phases, and it was put in between the electricalout-pins of the FEM component and the infinite bus. The electromotive force onthe infinite bus was divided by the turns ratio to take into account the voltagetransformation by the transformer.

4.2.8 Boundary Conditions and Initial Values

The boundary conditions for the FEM simulation were set by Maxwell duringthe conversion from the RMxprt model to the Maxwell FEM model. There werethree conditions, one for the outer radius of the stator, where the magneticvector potential A were set to zero. The other two were along the radii fromthe origin to the ends of the stator outer radius circle segment. These wereset to antiperiodic for the magnetic flux density B. The edges with boundaryconditions can be seen in Figure 6, where they are marked C.

The initial current in the field winding (A) were either set from Simplorer orwithin Maxwell depending on if a regulator was coupled to the voltage sourceof the field winding or not. If the regulator was not used, the field windingwas fed by constant voltage. An extra 3.025 Ω were added to the coil end ofthe field winding to get the resistance in agreement with measurements. Initialcurrent was generally set around 50% to 80% of steady-state field current, sincesetting it to either zero or 100% would give large transients. These transientsgenerally took long time to dampen out, and with a feasible simulation timeof 10 simulated seconds on the computer used, these transients needed to beavoided.

If, on the other hand, the regulator was used, the winding was set to “external”.The extra resistance (3.025 Ω) was then added as a resistance component inbetween the voltage source and the generator in-pin. The initial current couldnot be set when this configuration was used, but since there was a regulator,transients were dampened out quickly enough.

The initial voltage and current of the stator windings (B) were set to zero.This was done in part because Maxwell would not allow the user to set thesewhen co-simulating with Simplorer, and in the case of current, because an initialcurrent (when set in Simplorer) did not give smaller initial transients.

21

Figur 6: The FEM model geometry in Maxwell.

4.2.9 Load Angle

The infinite bus electrical angle was adjusted so that it would give the approx-imately correct steady-state load angle in the generator at the beginning of thesimulation. Three things had to be taken into account when this was done. The-se were the initial mechanical position of the rotor, the impedance in the grid(transformer), and the actual steady-state load angle. To compute this, a MAT-LAB script was written that computed the total angle from the phasor diagramshown in Figure 4c combined with the initial offset from the mechanical initialposition. The procedure is described in section 2.4. The script can be found inAppendix B.

The mechanical initial position was found by putting open ideal switches inbetween the generator phases and the infinite bus (no-load operation), and re-moving the inductances in the grid. The difference in electrical degrees betweenthe generator and the infinite bus could then be obtained by plotting the phasevoltage in the same plot as the infinite bus voltage, measuring the time dif-ference between the signals, and finally multiplying it with the rotor angularvelocity times the number of pole pairs.

4.3 Transient Simulations and Validation

4.3.1 Damping of Initial Transients

Initial transients were damped out using two different methods. The first were tosimply choose the initial field current in such a way that the machine dampenedout the transients by itself. The procedure is described in section 4.2.8. The othermethod was to use a regulator to add a damping term in the mechanical equation

22

proportional to the difference between the angular velocity and the synchronousangular velocity. This was the same damping term as the one described at theend of section 2.2. It was turned off before the torque steps were made.

4.3.2 Finding the Torque From the Experiment Data

The research generator is equipped with strain gauges attached to the statoras shown in Figure 7 to measure the total force on the stator. These were usedto find the torque acting on the generator before and after the torque step.The data collected could not be used directly though, since the strain gaugeswere not calibrated before the experiment. Instead, it was calculated from therelations

P = ωmTe (10)

Te = r ((Fx2− Fx1

) + (Fy2 − Fy1)) (11)

where P is the power, ωm is the mechanical angular velocity, Te is the electro-magnetic torque, and r is the stator outer radius. The power produced by thegenerator could be calculated as

P = vaia + vbib + vcic (12)

where vx and ix refers to the voltage and current in phase x. In practice, this wascalculated by taking the average over the measurements done during 20 seconds,and only for the torque after the step. The torque before the step was calculatedby using the data obtained from the strain gauges. It was assumed that thecalibration error was constant. Thus, by subtracting these DC signals from thestrain gagues’ data, so that the force acting on each strain gauge before the stepaveraged to zero, and inserting them into equation (11), a difference betweenthe torque before and after the step could be obtained, and the torque beforethe step could be calculated. This method can be used under the assumptionthat the electromagnetic and mechanical torques are equal, which they are if therotor angular velocity is constant. A MATLAB script was created to computethe torque from the experiments. It can be found in Appendix C.[6]

23

Figur 7: Orientation of strain gauges relative to each other. The gray circlerepresents the stator.

4.3.3 Torque Step Simulation Setup

Four different torque step simulations were performed and compared to mea-surements on the research generator. The first two were done with completedamper winding, and the other two had an incomplete damper winding. Foreach of the damper winding configurations, two different voltages were fed tothe field winding. The different quantities used in each simulation can be seenin Table 2. Quantities that were not changed between simulations can be seenin Table 3. The components mentioned in the first column can be seen in Figure5.

The mesh generation was handled by Maxwell. During setup of the 2D Maxwellgeometry, six mesh refinement operations were added to the model. One for thestator winding, one for the field winding, one for the steel parts in the generatorand one for the damper bars, each restricting triangle edge length. The fifthand sixth were placed on the steel parts and damper bars to restrict the surfacedeviation of the triangle edges, i.e., the maximum length from a curved surfaceto the triangle edge that is supposed to coincide with it. During the analysis,Maxwell also refines the mesh if the a posteriori error in the computation is toolarge, and repeats this step until the error is small enough.

24

Tabell 2: Simulation Quantities

# DamperWinding

FieldVoltage

InitialTorque

FinalTorque

InfiniteBus Angle

1 Complete 45.4 V 29 Nm 371 Nm -33° electric2 Complete 36.3 V -21 Nm 364 Nm -31° electric3 Incomplete 45.4 V 39 Nm 373 Nm -33° electric4 Incomplete 36.3 V 67 Nm 368 Nm -34° electric

Tabell 3: Common Quantities

Component(see Fig. 5) Quantity Value Notes

R1,R2,R3Grid/Transformer

Resistance 8.5 mΩ

L1,L2,L3Grid/Transformer

Inductance 18 μH

R4Return

ConductorResistance

105 Ω

E1,E2,E3Infinite Bus

EMF 89.84 VRMSE

ns/np

- Field WindingResistance 3.025 Ω

MchRMas1Rotating MassMoment ofInertia

59 kgm2

K,H Damping torquecoefficient KD

36

Used during thefirst threeseconds todampen out

transients beforethe torque step

4.3.4 Analysis of Data

To compare the simulated results to the experimental results, an exponentiallydamped sine function were fitted to the power output by the generator just afterthe torque step, i.e

pae−ξt sin (2πft+ θ) + P0 (13)

where pa, ξ, f , θ and P0 are constants to be found. pa will determine theamplitude at the initial time. ξ determines how fast the system returns to steadystate, which means that it is the damping factor of the oscillation. f is the

25

frequency of the oscillation, and θ models the phase of the damped sine curve.The latter is not very interesting, since it depends on which part of the data isused for the curve fit. It is necessary though for a good curve fit. P0 determinesthe value that the oscillation centers around, i.e., the power generated at thenew steady state. The power was first averaged over a whole electrical period,to smooth out the curve and reduce outliers.

26

5 Results and Discussion

5.1 Execution Times

It took about 18 to 19 hours to complete a simulation of 10 simulated seconds onthe machine used, which had an Athlon 64 X2 4400+ processor, 2 GB of RAM,and a magnetic hard drive. The simulations were run single threaded due to thefact that the author needed to work on the machine during simulations. Maxwellcan however divide the work to several threads placed on several cores. Bettersimulation times can be expected on a dedicated machine. This system is gettingold, and better simulation times can also be expected on newer hardware.

5.2 Ease of Use

When it comes to ease of use, no truly objective answer can be given. Therefore,the authors subjective view is presented here.

Maxwell is generally easy to use, and it does not take all too long to learn thebasics of the program using tutorials. The RMxprt to Maxwell 2D geometryfeature is intuitive, except for the fact that the design have to be solved beforeit is converted. The Maxwell/Simplorer simulation link is easy to setup, and it iswell documented in the Simplorer documentation. The support have been helpfuland easy to contact. The biggest problem with Maxwell is the documentation.Many pages describe settings and options by listing them without describinghow they influence the simulations or what the option physically describes, i.e.,many pages lacks pictures or a more descriptive explanation than option name- option name written in a slightly different way. This makes it hard to makesure that the problem is modeled correctly, and it might take much time to findout and/or verify what the parameters actually do.

Simplorer, on the other hand, has better documentation. Most components havetheir mathematical description along with an example on their documentationpage, which makes it easier to make sure that the system modeled is the oneintended. There is also a getting started guide in the documentation to helplearn the basics of the program. It is relatively easy to learn the basics of theprogram and to setup a simple system.

In both the programs the graphical user interfaces is functional.

5.3 Validation Simulations

5.3.1 Results

In Figures 8 to 11 the results from the validation simulations and their corre-sponding experiments can be seen. Figures 8 and 9 depicts the two simulations

27

with a complete damper winding, while Figures 10 and 11 depicts the two si-mulations with an incomplete damper winding. The x-axis in the figures havethe unit seconds, and the y-axis of the figures have the unit watts. The firstsecond of the figures shows the power output by the generator before the torquestep, which is described in section 4.3.3, and after that the torque step and thesubsequent attenuation of the oscillation caused by the torque step. To the datashown in Figures 8 to 11 an exponentially damped sine function, described insection 4.3.4, were fitted to extract numerical measures. The results from thecurve fitting can be seen in Table 4.

The parameter pa represents the amplitude of the oscillation in kW , and isrelated to the overshoot as compared with the steady-state output power. Itis generally larger for the simulated data than for the experimental data. Thismight be due to the fact that in the simulation, the torque is stepped up instan-taneously, something that the motor that drives the research generator cannotdo.

The parameter ξ represents the damping factor of the sine curve in s−1, andis thus interesting to look at when comparing data with a complete damperwinding versus data from an incomplete damper winding. It can be seen thatthe experimental data with a complete damper winding have a better dampingfactor by three to five times the data with the incomplete damper winding.

(a) Measured (b) Simulated

Figur 8: Power oscillation following a torque step from 29 Nm to 371 Nm. Thedamper winding is complete, and a field voltage of 45.4V is applied.

28


Figur 9: Power oscillation following a torque step from -21 Nm to 364 Nm. Thedamper winding is complete, and a field voltage of 36.3V is applied.


Figur 10: Power oscillation following a torque step from 39 Nm to 373 Nm. Thedamper winding is incomplete, and a field voltage of 45.4V is applied.

29


Figur 11: Power oscillation following a torque step from 67 Nm to 368 Nm. Thedamper winding is incomplete, and a field voltage of 36.3V is applied.

Tabell 4: The results from the simulations and experiments presented as curvefitting parameters.

# Type pa (kW ) ξ (s−1) f (Hz) P0 (kW ) %Deviation,

ξ

%Deviation,

f

1 Experiment 10.740 0.35 2.39 19.370 - -1 Simulation 18.550 1.4 2.39 19.860 400 0.002 Experiment 10.310 0.41 2.23 19.020 - -2 Simulation 21.080 1.4 2.20 18.880 341 1.363 Experiment 17.820 0.077 2.36 19.550 - -3 Simulation 18.080 1.3 2.38 19.930 1688 0.844 Experiment 16.360 0.13 2.21 19.190 - -4 Simulation 16.420 1.4 2.20 19.090 1077 0.45

The simulations overestimates the damping of the machine in all cases by quitea large amount, as seen in the column “% Deviation, ξ” in Table 4. It is wellknown that the damper winding impedance has a large impact on the transientcharacteristics of the machine[12], so this is probably due to the fact that thedamper winding impedance was not altered from the standard value, whichMaxwell calculates from the geometry of the damper winding. This is also hard

30

to measure, in the case of resistance because the resistance is lower than 1 mΩ,which is the smallest resistance that can be measured with the instrumentsavailable to the author. A too low resistance would give rise to larger currentsin the damper windings, which would dampen out the transients quicker than ina damper winding with higher resistance, unless the inductance in the damperwinding is such that resonance phenomena with the oscillating frequency exists.

The damping parameter does not increase at the same rate for simulations asfor experiments when switching from incomplete to complete damper winding.This gives an indication that the damper winding is not modeled well enoughin the FEM model.

The frequency of the damped sine curve, measured in Hz, is modeled by theparameter f . As can be seen in column “% Deviation, f ”, the deviation is ge-nerally under two percent from the the curve fit of the experimental data. Thisindicates that the magnetic circuit and the mechanical system of the machineis modeled well in Maxwell.

The parameter P0 determines the output power at steady state in kW , andis thus a measure of how good the torque estimate is. It is within a couple ofhundred watts, which means that the deviation in mean torque after the torquestep is less than three percent.

5.3.2 Sources of Error

There are a number of factors that may decrease the accuracy of the model.The generator FEM model is in 2D, which means that the generator is seenas infinity long, and end effects are ignored, unless they are explicitly modeledas with the field winding coil end resistance. Maxwell also sets up a simplifiedgeometry, where some curved surfaces, such as the stator back and pole face,are approximated with a number of straight lines. The fact that the stiffness ofthe model is good indicates that the errors in the modeling of the mechanicalsubsystem and the magnetic circuit are small.

The damper winding was modeled with standard values for inductance andresistance, which Maxwell calculated from the geometry of the damper winding.Since these values have not been measured because a lack of accurate enoughinstruments, it is hard to estimate the error. Just calculating the resistancefrom the dimensions of the winding might not give an accurate result, since thecontact resistance would not be included. There is also a significant error in thedamping characteristics of the model. This would need to be looked into more.

The method used to obtain the mechanical torque for the simulations seems togive about a three percent error compared to the measurement. The measure-ment itself also contains an error, but the author believes that it is small.

There are also inevitable errors in the numerical simulation. Maxwell makes surethat the solution converges by refining the mesh if the computational error is

31

too large, as described in section 4.3.3. The author has made a small qualita-tive study of the effects of shorter time step lengths. The conclusion was thatchoosing a shorter time step than the one chosen automatically by Maxwell didnot give better results. This indicates that the solution has converged in time.The time step is 50 times shorter than the Nyquist criterion requires for a 50 Hzsignal, so it might be possible to make the time step longer. This was howevernot studied.

32

6 Conclusion

From the curve fit parameters comparison in Table 4, it can be concluded thatMaxwell models the magnetic circuit and the mechanical system of the machi-ne well, since the difference in output power oscillation frequency is small. Itcan also be concluded that the damping characteristics of the model signifi-cantly differs from that of the experimental rig. This is probably due to thedamper windings not being modeled sufficiently accurate, due to the fact thatthe damper windings have a major impact on the damping characteristics ofthe machine. This can probably be solved by doing a more through study ofthe damper winding characteristics. Overall, Maxwell coupled with Simplorerseems to be able to model the experimental generator test setup coupled to aninfinite bus.

When it comes to program ease of use, the basics can be learned quickly throughtutorials and guides. The meshing and mesh quality control is handled automa-tically when setting up a Maxwell 2D model from a RMxprt model. It is oftennot described in the documentation how things are modeled, or how optionsinfluence the simulation result.

Simulation times are satisfactory, and if longer simulations are required, betterhardware can be bought. The hardware used for this thesis was not very new,and thus better performance should be expected if transistor density continueto follow Moore’s law.

All in all, it is the author’s view that Maxwell and Simplorer is capable ofmodeling a synchronous hydropower generator coupled to an infinte bus usingthe template for salient pole synchronous generators.

33

7 Suggested Future Work

The parameter that gave the worst result in the model was the machine dampingparameter ξ. This parameter is coupled to the damper winding model, and thusan investigation into why it was modeled so badly is appropriate.

The motor that delivers torque to research generator cannot make an instanta-neous step to another torque as the model can, and thus it would be interestingto see if the overshoot of the simulations can be reduced to the same level asfor the experiments if the torque were ramped up during several millisecondsinstead of stepped up instantaneously.

34

Referenser

[1] Energimyndigheten. El-, gas- och fjärrvärmeförsörjningen 2009. Online,http://www.scb.se/Statistik/EN/EN0105/2009A01/EN0105_2009A01_SM_EN11SM1002.pdf, February 2011.

[2] Niklas Dahlbäck. Utvecklingsbehov inom reglerkraftsområdet ur ett vat-tenkraftsperspektiv. Elforsk rapport 10:11.

[3] Energimyndigheten. Nytt planeringsmål för vindkraften år 2020. Online,http://webbshop.cm.se/System/ViewResource.aspx?p=Energimyndigheten&rl=default:/Resources/Permanent/Static/375c52d19c3f49d2b8b5821f2d937057/ER2007_45w.pdf, December 3 2010.

[4] J. Lidenholm. Transient electromechanical analysis of hydropower genera-tors using field and circuit models. Licentiate thesis, 2010.

[5] M. Ranlöf, A. Wolfbrandt, J. Lidenholm, U. Lundin. Core Loss Predic-tion in Large Hydropower Generators: Influence of Rotational Fields. IEEETransactions on Magnetics, August 2009.

[6] M. Wallin, M. Ranlöf, U. Lundin. Design and construction of a synchronousgenerator test setup. Electrical Machines (ICEM), 2010 XIX InternationalConference, 2010.

[7] N. Bianchi. Electrical Machine Analysis Using Finite Elements. Taylor &Francis Group, 2005.

[8] M. Yilmaz, P. T. Krein. Capabilities of finite element analysis and magneticequivalent circuits for electrical machine analysis and design. 2008 IEEEPower Electronics Specialists Conference, 2008.

[9] M. G. Larson, F. Bengzon. A First Course in Finite Elements, LectureNotes. Umeå University, 2007.

[10] P. Zhou, S. Stanton, Z. J. Cendes. Dynamic Modeling of Electric Machines.Online, www.ansoft.com/device/motor/Dynamic_Modeling.pdf, January11 2011.

[11] I. Boldea. Synchronous Generators. Taylor & Francis Group, 2006.

[12] P. Kundur. Power System Stability and Control. McGraw-Hill, 1994.

[13] D. Lin, P. Zhou, S. Stanton, Z. J. Cendes. A Fully Integra-ted Simulation Package for Electric Machine Design. Online,www.ansoft.com/device/motor/Electric_Machines.doc, November 162010.

[14] COMSOL. Comsol Multiphysics Help, Section Circuit Modeling. 28 Janu-ary 2011.

35

[15] COMSOL. Comsol Multiphysics User Guide. Comsol 4.0, April 2010.

[16] COMSOL. Introduction to comsol multiphysics . Online,http://www.comsol.se/shared/downloads/comsol_v4_building_model.pdf,November 20 2010.

[17] Cedrat Groupe. Flux - Portunus Co-Simulation. Online,http://www.cedrat.com/fileadmin/user_upload/cedrat_groupe/ Soft-ware_solutions/Portunus/Flux_Portunus_cosimulation.pdf, November16 2010.

[18] Cedrat Groupe. CAE Software for the analysis of electromagnetic devices.Online, http://www.cedrat.com/fileadmin/user_upload/cedrat_groupe/Software_Solutions/Flux/flux.pdf, November 30 2010.

[19] Cedrat Groupe. Flux Data Sheet. Online,http://www.cedrat.com/fileadmin/user_upload/cedrat_groupe/ Soft-ware_solutions/Flux/Flux_Data_Sheet.pdf, November 20 2010.

[20] ANSYS. Maxwell Product Datasheet. Online,http://www.ansoft.com/products/em/maxwell/datasheet.cfm?f= Max-well_Flysheet.pdf, December 9 2010.

[21] H. Young, R. Freedman. University Physics, 11th Edition. Pearson Educa-tion, 2004.

[22] Ansoft Corporation. Simplorer On-Line Help, Section Inertias. 14 January2011.

36

A Generator Measurement and Data

The parameters needed to setup the RMxprt model came mainly from threesources: direct measurement on the generator using a caliper and a tape measure,a 3D CAD construction model, and from measurements made by the researchersat the hydropower group at Uppsala University.

Tabell 5: RMxprt model stator quantities.

StatorQuantities Value Instrument Estimated

Error Notes

Stator outerdiameter 872 mm Caliper ±2 mm

Stator innerdiameter +

calipermeasurement

Stator innerdiameter 725 mm Tape measure ±1 mm

Stator length 298 mm Tape measure ±3 mmStatorstackingfactor

0.8 Caliper ±2% Stator length- air vents

Laminationsectors 1 - -

37

Tabell 6: RMxprt model slot quantities.

SlotQuantities Value Instrument Estimated

Error Notes

Number ofSlots 108 - - 3 per pole

and phaseSlot type 1 - -

Hs0 0.1 mm Eye ?

Hs2 21.75 mm Caliper ?

Total 30.5mm,

estimated byassuming

curved partsas circlesectors

Bs0 4 mm Caliper ±0.5 mmBs1 10 mm Caliper ±0.5 mmBs2 10 mm Caliper ±0.5 mm

Tabell 7: RMxprt model stator winding quantities.

StatorWindingQuantities

Value Instrument EstimatedError Notes

Windinglayers 2 - -

Winding type Whole-coiled - -Parallelbranches 1 - -

Conductorsper slot 2 - -

Coil pitch 9 - -

Number ofstrands 2 - -

Laterchanged into

solid inMaxwell 2D

design

Wire size 6.665 mm Caliper ±0.5 mm,±0.5 mm

Gauge:MIXED, W:6.5 mm, H:5.5 mm,Fillet: 1,Number: 2

38

Tabell 8: RMxprt model rotor quantities.

RotorQuantities Value Instrument Estimated

Error Notes

Number ofpoles 12 - -

Rotor outerdiameter 708.5 mm

3D CADconstruction

model±0.5 mm

Rotor innerdiameter 358 mm Caliper ?

Hard toestimate, and

of littleimportance

Rotor length 301 mm Tape measure ±3 mmRotor

stackingfactor

0.99 Estimate ?

Tabell 9: RMxprt model rotor winding quantities.

RotorWindingQuantities


Winding type Round Wire - -Parallelbranches 1 - -

Conductorsper pole 324 - - 162 turns

Number ofstrands 1 - -

Wire wrap 0.1 mm Estimate ?

Wire size 3.776 mm Manufacturerdata ?

Gauge:MIXED, W:4 mm, H: 2.8mm, Fillet: 0,Number: 1

Winding fillet 45 mm Caliper andeye ±3 mm

39

Tabell 10: RMxprt model pole quantities.

PoleQuantities

Value Instrument EstimatedError

Pole shoewidth

133.89 mm 3D CADconstruction

model

±0.5 mm

Pole shoeheight


model

±0.5 mm

Pole bodywidth


model

±0.5 mm

Pole bodyheight


model

±0.5 mm

40

Tabell 11: RMxprt model damper winding quantities.

DamperWindingQuantities


Damper slotsper pole 3 - -

Barconductor

typeCopper - -

End length 14 mm Caliper ±2 mm

End ringwidth 15 mm

3D CADconstruction

model±0.5 mm

End ringheight 3 mm

3D CADconstruction

model±0.5 mm

End ringconductor

typeCopper - -

Slot pitch 3.67° - ?

Realconfiguration:1.2τ, 1.0τ,Model

configuration:2x1.1τ

End ring type 2 - -

41

Tabell 12: RMxprt model damper winding slot quantities.

DamperWinding SlotQuantities


Slot type 1 - -

Hs0 2.12 mm3D CAD

constructionmodel

±0.5 mm

Hs01 0 mm3D CAD

constructionmodel

-Shapes theslot into a

circle

Hs2 0 mm3D CAD

constructionmodel

-Shapes theslot into a

circle

Bs0 2 mm3D CAD

constructionmodel

±0.5 mm

Bs1 8 mm3D CAD

constructionmodel

±0.5 mm

Bs2 8 mm3D CAD

constructionmodel

±0.5 mm

Tabell 13: RMxprt model setup quantities.

Setup Quantities ValueOperation type GeneratorOperation load Infinite BusRated apparent

power 20 kVA

Rated voltage 155.5 VRated speed 500 rpm

Rated power factor 0.9Winding connection WyeExciter efficiency 90%

42

B MATLAB Initial Angle Script

% generator.m%% A program that calculates phase angles for Simplorer infinte bus behind% a transformer, for a transient co-simulation with Maxwell. There are some% hardcoded valus that might not be correct for other projects than the% Svante research generator project. These values are placed in the% Input quantities and Other quantities sections.%% Written by Gustav Persarvet

clear;disp(’This program calculates angles for the infinite bus inAnsoft Maxwell/Simplorer model.’);fprintf(’\n’);

% Input quantitiesmec_torque = -57; % NewtonMetersrated_ll_voltage = 155.5; %input(’Rated line to line voltage (V) = ’);grid_inductance = 0.000018; %input(’Power grid inductance (H) = ’);grid_resistance = 0.0085; %input(’Power grid resistance (Ohm) = ’);

% Other quantitiesR_s = 0.021; % Stator resistance, phase-ground, 0.042 phase-phase, measured on SvanteX_q = 0.26; % Q-axis reactance, estimatedmec_angle = -31.87; % 148.4; for north pole % Electrical degrees,from initial mechanical displacement. Verified by no-load simulation.

% Derived quantitiespower = mec_torque*2*pi*500/60; % Power in Wattsphi = atan(100*pi*grid_inductance/grid_resistance);rated_apparent_power = power/cos(phi);rated_ll_current = rated_apparent_power/rated_ll_voltage;rated_phase_current = rated_ll_current/sqrt(3);rated_phase_voltage = rated_ll_voltage/sqrt(3);

% Phasor diagram calculationsU_t = complex(rated_ll_voltage,0); % Terminal VoltageI = complex(cos(-phi)*rated_ll_current,sin(-phi)*rated_ll_current);% Current (Reversed direction vs. voltage in Ansoft)IRth = -grid_resistance*I; % Voltage drop over grid resistanceIXth = -1j*100*pi*grid_inductance*I; % Voltage drop over grid reactanceE_b = U_t + IRth + IXth; % Grid EMFIRs = R_s*I; % Voltage drop over stator resistance

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IXq = 1j*X_q*I; % Voltage drop q-axis reactanceE_q = U_t + IRs + IXq; % Generator EMF

% Get anglesload_angle = -angle(E_q);beta = -angle(E_b);

disp([’Load angle = ’, num2str(load_angle*180/pi), ’°’]);disp([’Grid angle = ’, num2str(-beta*180/pi), ’°’]);disp([’Extra Mechanical angle = ’, num2str(mec_angle), ’°’]);disp([’Total angle = ’, num2str(load_angle*180/pi - beta*180/pi + mec_angle), ’°’]);

plot([0, real(I)],[0, imag(I)],’r’);hold on;plot([0, real(U_t)],[0, imag(U_t)],’g’);plot([0, real(E_b)],[0, imag(E_b)],’b’);plot([0, real(E_q)],[0, imag(E_q)],’m’);legend(’I (Current)’, ’U_t (Terminal Voltage)’, ’E_b (Grid EMF)’,’E_q (Generator EMF)’);title(’Phasor diagram’);xlabel(’Real axis’);ylabel(’Imaginary axis’);axis_max = 1.2*max([max(real([I,U_t,E_q,E_b])),max(imag([I,U_t,E_q,E_b]))]);axis([0,axis_max,-axis_max,axis_max]);hold off;

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C MATLAB Experiment Torque Script

% calc_torque.m%% Calculates the torque before and after the torque step, by first% calculating the difference between them, and then calculate the torque% after the step under the assumption that the rotor angular velocity is% constant.%% Written by Gustav Persarvet

steptime = 7.5; % When is the torquestepsampleRate = 1000; % samples per secondR = 0.692; % Radius of stator

figure;plot(T,TOJ1,T,TOJ2,T,TOJ3,T,TOJ4);legend(’TOJ1’,’TOJ2’,’TOJ3’,’TOJ4’);title(’Kraft, Töjningsgivare, innan DC bortdraget’);

% Remove constant error, TOJ1-4 taken from experiment dataTOJ1 = TOJ1-mean(TOJ1(1:sampleRate*(steptime-1)));TOJ2 = TOJ2-mean(TOJ2(1:sampleRate*(steptime-1)));TOJ3 = TOJ3-mean(TOJ3(1:sampleRate*(steptime-1)));TOJ4 = TOJ4-mean(TOJ4(1:sampleRate*(steptime-1)));

F = (TOJ1-TOJ2)+(TOJ3-TOJ4); % Total forceTorque = F*R;MeanTorquePre = mean(Torque(sampleRate*(steptime-5):sampleRate*(steptime-1)));MeanTorquePost = mean(Torque(sampleRate*(steptime+1):sampleRate*(steptime+21)));disp([’Torque before step = ’,num2str(MeanTorquePre)]);disp([’Torque after step = ’,num2str(MeanTorquePost)]);

figure;plot(T,TOJ1,T,TOJ2,T,TOJ3,T,TOJ4);legend(’TOJ1’,’TOJ2’,’TOJ3’,’TOJ4’);title(’Kraft, Töjningsgivare’); figure;plot(T,Torque); title(’Vridmoment, Töjningsgivare’);

t = sampleRate*(steptime+1):sampleRate*(steptime+21);length = size(t,2);P = 1/length*sum(VA(t).*IA(t)+VB(t).*IB(t)+VC(t).*IC(t)); % Active effecttau = P/(500/60*2*pi); % Torque = Active Power / Mechanical angular velocitydisp([’Framräknat vridmoment före steg = ’,num2str(tau+MeanTorquePost)]);disp([’Framräknat vridmoment efter steg = ’,num2str(tau)]);

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