Hydropower generator and power system interaction

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ACTA UNIVERSITATIS

UPSALIENSIS UPPSALA

Digital Comprehensive Summaries of Uppsala Dissertations

from the Faculty of Science and Technology 978

Hydropower generator and

power system interaction

JOHAN BLADH

ISSN 1651-6214 ISBN 978-91-554-8486-6

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Dissertation presented at Uppsala University to be publicly examined in Häggsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, November 16, 2012 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English.

Abstract

Bladh, J. 2012. Hydropower generator and power system interaction. Acta Universitatis Upsaliensis. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 978. 119 pp. Uppsala. ISBN 978-91-554-8486-6.

After decades of routine operation, the hydropower industry faces new challenges. Large-scale integration of other renewable sources of generation in the power system accentuates the role of hydropower as a regulating resource. At the same time, an extensive reinvestment programme has commenced where many old components and apparatus are being refurbished or replaced. Introduction of new technical solutions in existing power plants requires good systems knowledge and careful consideration. Important tools for research, development and analysis are suitable mathematical models, numerical simulation methods and laboratory equipment. This doctoral thesis is devoted to studies of the electromechanical interaction between hydropower units and the power system. The work encompasses development of mathematical models, empirical methods for system identification, as well as numerical and experimental studies of hydropower generator and power system interaction. Two generator modelling approaches are explored: one based on electromagnetic field theory and the finite element method, and one based on equivalent electric circuits. The finite element model is adapted for single-machine infinite-bus simulations by the addition of a network equivalent, a mechanical equation and a voltage regulator. Transient simulations using both finite element and equivalent circuit models indicate that the finite element model typically overestimates the synchronising and damping properties of the machine. Identification of model parameters is performed both numerically and experimentally. A complete set of equivalent circuit parameters is identified through finite element simulation of standard empirical test methods. Another machine model is identified experimentally through frequency response analysis. An extension to the well-known standstill frequency response (SSFR) test is explored, which involves measurement and analysis of damper winding quantities. The test is found to produce models that are suitable for transient power system analysis. Both experimental and numerical studies show that low resistance of the damper winding interpole connections are vital to achieve high attenuation of rotor angle oscillations. Hydropower generator and power system interaction is also studied experimentally during a full-scale startup test of the Nordic power system, where multiple synchronised data acquisition devices are used for measurement of both electrical and mechanical quantities. Observation of a subsynchronous power oscillation leads to an investigation of the torsional stability of hydropower units. In accordance with previous studies, hydropower units are found to be mechanically resilient to subsynchronous power oscillations. However, like any other generating unit, they are dependent on sufficient electrical and mechanical damping. Two experimentally obtained hydraulic damping coefficients for a large Francis turbine runner are presented in the thesis.

Keywords: Amortisseur windings, applied voltage test, automatic voltage regulators, damper

windings, damping torque, empirical modelling, equivalent circuits, excitation control, finite element method, hydropower generators, power system restoration, power system stability, synchronous machines, self excitation, shaft torque amplification, short circuit test, single machine infinite bus, slip test, standstill frequency response test, subsynchronous oscillations, synchronising torque, synchronous generators, torsional interaction.

Johan Bladh, Uppsala University, Department of Engineering Sciences, Electricity, Box 534, SE-751 21 Uppsala, Sweden.

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List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

List of Papers

I Bladh, J., Wallin, M., Saarinen, L. and Lundin, U., “Standstill frequency response test on a synchronous machine extended with damper bar measurements”, submitted to IEEE Transactions on

Energy Conversion, September 2012.

II Bladh, J., Sundqvist, P. and Lundin, U., “Torsional stability of hydro-power units under inﬂuence of subsynchronous oscillations”, submit-ted to IEEE Transactions on Power Systems, July 2012.

III Bladh, J. and Lundin, U., “Synchronised measurements on hydropower units during power system startup”, submitted to Hydro

Review Worldwide (HRW), July 2012.

IV Lidenholm, J. and Lundin, U., “Estimation of hydropower generator parameters through ﬁeld simulations of standard tests”, IEEE

Transac-tions on Energy Conversion, vol. 25(4), pp. 931-939, December 2010.

V Ranlöf, M., Bladh, J. and Lundin, U., “Use of a ﬁnite element model for the determination of damping and synchronizing torques of hy-droelectric generators”, International Journal of Electrical Power and

Energy Systems, vol. 44, pp. 844-851, 2013.

VI Lidenholm, J., Ranlöf, M. and Lundin, U., “Comparison of ﬁeld and circuit generator models in single-machine inﬁnite-bus system simu-lations”, in proceedings of the XIX International Conference on

Electri-cal Machines, September 2010.

VII Lidenholm, J., Ranlöf, M. and Lundin, U., “Effects of including au-tomatic excitation control in transient ﬁeld simulations of hydrogen-erators”, in proceedings of the 44th International Universities’ Power

Engineering Conference, September 2009.

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The author has also contributed to the following paper, not included in this thesis:

A Ranlöf, M., Wolfbrandt, A., Lidenholm, J. and Lundin, U., “Core Loss Prediction in Large Hydropower Generators: Inﬂuence of Rotational Fields”. IEEE Transactions on Magnetics, vol. 45, no. 8, pp. 3200-3206, 2009.

B Wallin, M., Bladh, J. and Lundin, U., “Damper winding inﬂuence on unbalanced magnetic pull in synchronous machines with rotor eccentricity”, submitted to IEEE Transactions on Magnetics, October 2012.

_{The author changed his surname from Lidenholm to Bladh in September}

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Nomenclature

This chapter contains a selection of abbreviations and symbols that are used repeatedly in the thesis. Subscript k is used in place of natural num-bers. Bold symbols are used to denote vector ﬁelds.

Abbreviations

Abbreviation Description

AC Alternating current

BOP Bipolar operational ampliﬁer

CD Continuous damper winding

d-axis Direct (pole) axis

DAQ Data acquisition

DC Direct current

DTC Direct torque control

EC Equivalent circuit

EMF Electromotive force

FE Finite element

FEM Finite element method

GPS Global positioning system

HVDC High voltage direct current

IEEE Institute of electrical and electronics engineers

IM Induction machine

MMF Magnetomotive force

NCD Non-continuous damper winding

NTP Network time protocol

OCC Open-circuit characteristic

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Abbreviations continued

Abbreviation Description

PRBS Pseudo random binary signal

RMS Root mean square

pu Per unit

q-axis Quadrature (interpole) axis SCC Short-circuit characteristic

SMIB Single-machine inﬁnite-bus

SSFR Standstill frequency response

SSO Subsynchronous oscillation

SSR Subsynchronous resonance

SVC Swedish hydropower centre (Svenskt vattenkraftcen-trum in Swedish)

TSO Transmission system operator

UMP Unbalanced magnetic pull

UPS Uninterrupted power supply

WD Without damper winding

Latin symbols

Symbol Unit Description

A Wb/m Magnetic vector potential

Az Wb/m z-component of the magnetic vector

po-tential

B T Magnetic ﬂux density

D Nm/(rad/s) Damping constant

Den Nm/(rad/s) Modal damping constant (electrical)

Dn Nm/(rad/s) Modal damping constant (mechanical)

Dt Nm/(rad/s) Turbine damping constant

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Latin symbols continued

ea, eb, ec V Terminal phase voltages

ed, eq pu EC model terminal voltages

ef d V (pu) Field voltage

eDk V EMF induced in damper bar k

ef L(R) V EMF induced on the left (right) coil side of the ﬁeld winding

ew V EMF induced in an arbitrary stator

wind-ing

EB d, EB q pu Inﬁnite-bus voltages

f0 Hz Nominal electrical system frequency

fen± Hz Sub- and supersynchronous frequency

on the stator side

fer Hz Subsynchronous electrical natural

fre-quency

fn Hz Mechanical oscillation (or modal)

fre-quency

fosc Hz Oscillation frequency

fr± Hz Sub- and supersynchronous frequency

on the rotor side

H A/m Magnetic ﬁeld strength

Hr A/m Radial component of the magnetic ﬁeld

strength

H_ϕ A/m Tangential component of the magnetic

ﬁeld strength

H s Inertia constant

Hg s Generator rotor inertia constant

Hn s Modal inertia constant

Ht s Turbine runner inertia constant

ia, ib, ic A Stator phase currents

id, iq pu EC model stator currents

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Latin symbols continued

if L(R) A Field winding current source on the left

(right) coil side

iDk A Current in damper bar k

IF G A Field current corresponding to base

ar-mature voltage on the airgap line

IF N L A Field current corresponding to base

ar-mature voltage on the no-load curve

IF SI A Field current corresponding to base ar-mature current on the short-circuit sat-uration curve

J A/m2 Current density

J kg m2 Moment of inertia

kf w Winding factor of the ﬁeld coil

Kd pu/(rad/s) Damping torque coefﬁcient

Ks pu/rad Synchronising torque coefﬁcient

le m Effective axial length of a machine

ln H Nominal load inductance

ls H End-winding inductance (FE)

Lad, Laq pu Saturated magnetising inductances

Lad u, Laqu pu Unsaturated magnetising inductances

Ld, Lq pu Synchronous inductances

Ld(s), Lq(s) pu Operational inductances

LE pu Inductance of the external network

Lf f d, Lkkd, Lkkq pu Rotor winding self inductances

Lf d, Lkd, Lkq pu Rotor winding leakage inductances

Lf 12d pu Peripheral leakage inductance

Ll pu Stator leakage inductance

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Latin symbols continued

n Inertia ratio

Na f o Stator-to-ﬁeld turns ratio

NaDd (q)o Stator-to-damper turns ratio

Np Number of poles

Npp Number of pole pairs

Pe W (pu) Electrical power

Pm W (pu) Mechanical power

rDa Ω Damper winding intrapole resistance

(FE)

rDb Ω Damper winding interpole resistance

(FE)

rf Ω Field winding resistance (FE)

rn Ω Nominal load resistance (FE)

rs Ω Stator resistance (FE)

Ra Ω (pu) Armature resistance

RDd, RDq Ω (pu) Equivalent damper circuit resistances

Rf d, Rkd, Rkq Ω (pu) EC model rotor resistances

RE pu Resistance of the external network

Re f f pu Induction machine equivalent effective

resistance

Rr pu Induction machine equivalent rotor

re-sistance

s Slip (or a complex argument in

frequency-domain functions)

Sbase VA Three-phase apparent power base

sH (s) pu Stator-to-damper current transfer

func-tion

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Latin symbols continued

T_d, T_d s Transient and subtransient d-axis time constants

T_{d o} , T_{d o} s Open-circuit transient and subtransient d-axis time constant

Te Nm (pu) Electrical torque

Tm Nm (pu) Mechanical torque

ua, ub, uc V Phase voltages

uab, ucb V Line-to-line voltages

Ubase kV Line-to-line RMS voltage base

Un V Nominal line-to-line RMS voltage

Ut V Terminal line-to-line RMS voltage

V V Electric potential function (FE)

Xd, Xq Ω (pu) Saturated synchronous reactances

Xd u, Xqu Ω (pu) Unsaturated synchronous reactances

X_d, Xq Ω (pu) Transient reactances

X_d, X_q Ω (pu) Subtransient reactances

zD Ω (pu) Damper bar impedance

Za f o(s) pu Stator current to ﬁeld voltage transfer function

ZaDo(s) pu Stator current to damper voltage transfer function

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Greek symbols

δ el.rad Rotor (load) angle

δ0 el.rad Initial rotor angle

δn Logarithmic decrement

μ H/m Total permeability

μ0 H/m Permeability of free space

μr H/m Relative permeability

ν m/H Magnetic reluctivity

Φ Wb Magnetic ﬂux

ϕ el.rad Power-factor angle

Ψ Wb-t Magnetic ﬂux linkage (FE)

ψad,ψaq pu Mutual ﬂux linkages (EC)

ψd,ψq pu Stator ﬂux linkages (EC)

ψf d,ψkd,ψkq pu Rotor ﬂux linkages (EC)

σ S/m Electric conductivity

σn 1/s Decrement factor

τ s Delay time constant

τD s Damping time constant

ωm mech.rad/s Mechanical angular velocity of the rotor

ωm0 mech.rad/s Synchronous mechanical angular veloc-ity

Δωm mech.rad/s Deviation from synchronous mechani-cal rotor velocity

ω el.rad/s Electrical angular velocity of the rotor

ω0 el.rad/s Electrical synchronous angular velocity

Δω el.rad/s Deviation from synchronous electrical velocity

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1. Introduction

1.1 Development of hydropower in Sweden

The early exploitation of hydropower was vital to the foundation of the Swedish welfare state. The large-scale expansion commenced in 1909 by the establishment of the Swedish state power board (Kungliga vattenfallstyrelsen in Swedish). The ﬁrst large power plant Olidan, located in the south-west of Sweden, was built between 1910 and 1921. The ﬁrst northern power station Porjus was taken into operation in 1915. Its primary purpose was to electrify the railway between the iron mines and the harbour in Narvik, Norway. By the end of the 1930s, the hydropower potential in southern Sweden was almost fully utilised, but the large hydropower resource in the north was still largely unexploited. The reason for this was the lack of transmission capacity.

The topology of the Swedish power system with the large hydropower re-source in the northern rivers and the centre of consumption approximately 1000 km further south has always been a challenge. The ﬁrst transmission line from the north to the south, rated 220 kV, was taken into operation in 1936. However, in just a few years, the transmission capacity again pre-vented further hydropower expansion. In 1952, the ﬁrst 400 kV transmission line (in the world) was built between Harsprånget and Hallsberg [2]. This, in turn, allowed the most extensive development of hydropower in the history of Sweden during the 1950s and 1960s [3].

Today there are 1800 hydropower plants in Sweden out of which 200 are large (>10 MW). Their total capacity is 16.2 GW and the average annual pro-duction is 65 TWh, 45 percent of the total Swedish electric energy demand [4]. Four large rivers are unexploited and protected on account of politi-cal and environmental considerations. Vattenfall AB (former Swedish state power board) is one of the largest energy companies in Europe employing 34700 people (2011) [5].

1.2 New challenges

Clearly, hydroelectric energy has been a cornerstone in the Swedish elec-tricity system for more than a hundred years and it is not an understate-ment to say that the technology is mature. However, the hydropower in-dustry is facing new challenges. With the majority of the plants built in

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Figure 1.1: Horizontal-axis synchronous generators at the ﬁrst large Swedish

hyd-ropower plant Olidan [1]. Four units were commissioned in 1910, followed by an-other four in 1914 and anan-other ﬁve in 1921. Ten out of the thirteen units are still operational.

the 1950s and 1960s, extensive refurbishment and upgrading work has just commenced. Dams and spillways are being upgraded to meet new safety requirements and obsolete plant components are being rebuilt or replaced. At the same time, the role of hydropower as a regulating resource is accen-tuated with the large-scale installation of other renewable power sources [6].

Power system operators are interested in short term and long term bility of the electrical system. Distinction is made between rotor angle sta-bility, frequency stability and voltage stability. The latter two are related to the balance between generation and load on both short and long term ba-sis. Rotor angle stability is the ability of synchronous machines of an in-terconnected power system to remain in synchronism after being subject to a disturbance [7]. High rotor inertia, fast-acting excitation systems and short water ways are generally considered to be good characteristics from a power system stability perspective.

Owners of the generators have a slightly different focus. They are inter-ested in reliable operation, low cost in maintenance and the integrity of

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their machines [8]. In addition, there is always a strive to reduce the costs associated with the initial investment.

However, at the doorstep of a new era, there is a need to take a compre-hensive view of the system. Coupled systems must be modelled, analysed and understood together as a whole.

1.3 Project background and related work

The work presented in this doctoral thesis is part of a research programme devoted to hydropower generator technology at Uppsala University, initiated by the Swedish Hydropower Centre (Svenskt Vattenkraftcentrum, SVC). SVC is a national competence centre for education and research in the hydropower field. The vision of SVC is to secure knowledge and competence for efficient and reliable hydropower production and for maintaining safety in dam operation [9]. This will be accomplished by creating high-quality long-term competence at five selected technical universities and by establishing close co-operation between power suppliers and distributors, manufacturers of hydropower equipment, consultants, the Swedish Energy Agency and the universities.

To meet some of the challenges described above, an initiative was taken by SVC to study more closely the mutual electromechanical interaction between hydropower generators and the power system. Two doctoral projects were launched, one at Uppsala University and one at Vattenfall Research & Development AB. The ﬁrst, intended to study electromechanical interaction from a generator perspective, commenced in February 2007 and was ﬁnalised in May 2011 when Martin Ranlöf defended his thesis Electromagnetic Analysis of Hydroelectric

Generators [10]. The second, intended to study electromechanical

interaction from a grid perspective, commenced in November 2007 and ends with the defence of this thesis. A large part of both these projects has been the development of mathematical models for numerical studies as well as development of experimental equipment and methods.

In 2008, another doctoral project was started with the aim to study the unbalanced magnetic pull (UMP) resulting from rotor eccentricity. The PhD candidate, Mattias Wallin, has had the primary responsibility for the me-chanical design and construction of the hydrogenerator experiment setup described in Section 6.1.

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Figure 1.2: Overview of a hydropower plant [1]. Large dams trap the water in

reser-voirs to create the necessary potential energy. The water is led in conduits through a hydraulic turbine and further to a downstream reservoir. A hydraulic pressure and speed reduction over the turbine creates a mechanical torque on the turbine runner, which in turn drives the generator rotor. Electricity is produced in the gen-erator stator and transmitted to the power grid via a transformer.

1.4 Hydropower

generator

and

power

system

interaction

This section introduces the subjects studied in the thesis and reviews pre-vious work. Section 1.4.1 is an overview of the studied system intended for non-expert readers. Section 1.4.2 introduces electromechanical modelling which is central in the work presented. Section 1.4.3 is an introduction to empirical identiﬁcation of generator model parameters which is another central part of the thesis. Section 1.4.4 introduces the subject torsional sta-bility.

1.4.1 Overview

A hydroelectric generator is a machine that converts the mechanical power supplied by a hydraulic turbine into electrical power on a transmission net-work. Figure 1.2 is an overview of a hydroelectric power plant. Large hyd-ropower generators (>10 MW) are usually vertical-axis synchronous

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ma-chines. They are typically characterised by many poles (12-72), a short axial length (1-2 m) and a large diameter (4-16 m).

The mechanical and electrical subsystems of an electrical machine exchange energy via a magnetic field in the airgap between the stationary and rotating parts (the stator and the rotor). In a synchronous machine, the magnetic field is generated by supplying direct current to the rotor field winding. As this magnetic field rotates, voltage is induced in the stator windings, which drives a current if the machine is connected to a load. If the power supplied to the turbine exceeds the electrical power (the load), the machine will speed up. Conversely, it will slow down if the inverse power relation prevails.

A large transmission system is operated so that the supplied and consumed electric energy is balanced. Consequently, all synchronous machines rotate on average at a ﬁxed angular frequency proportional to the electric frequency (nominally 50 Hz in the Nordic power system). Each machine has only a small1 inﬂuence on the network frequency. Hence, although hydropower generators can be very large machines, their kinetic energy at rated speed is small in relation to the aggregated kinetic energy of all the machines in a synchronous transmission system.

It may be realised from the above that the coupled electromechanical system of study can be viewed in different ways. In power system analysis, each generator is seen as a small component in a large dynamic system and the focus of study is the interplay between them. In machine analysis, the generator is in focus and the whole power system is lumped together into one large component with very simple dynamics. This work is intended to bridge the gap between these two worlds.

1.4.2 Electromechanical modelling

Stability studies of large power systems require simple mathematical models of individual components to remain computationally effective. Very useful simple mathematical models of synchronous machines have been developed on the basis of the two-axis theory [11, 12], where all the electric and magnetic properties of the machine are lumped together in a set of equivalent electric circuits [13]. Such equivalent circuit (EC) models are also popularly employed in studies of the electromechanical interaction between a single generator and the power system [14].

Numerical models based on electromagnetic field theory and the finite element method (FEM) are more suited for analysis of the individual machine. However, electromagnetic field models (henceforth referred to

1_{The largest hydropower generator in the Nordic power system, rated 500 MVA, can achieve}

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as field models or finite element (FE) models2) have traditionally not been used in dynamic studies due to the relatively large computational effort associated with solution of the field equations.

In the 1990s, the coupling of finite element models and external elec-tric circuits where explored [15–18] which made simulations of generators in network operation possible. A model of a turbine-generator infinite-bus system comprising both an excitation control system and angular dynamics was developed by Sturgess, Zhu and Macdonald [19]. The simulations pre-sented are compared with test data and show remarkable agreement. The same machine is then modelled recurrently in the work of Escarela-Pérez and his colleagues, both for calculation of machine characteristics [20–22], and for simulation of machine [23] and controller dynamics [24]. Compar-isons with EC models of the same system are presented in [20,21,25]. Darabi and Tindall focused on development of an automatic voltage regulator [26] and a brushless exciter [27] model for use in finite element simulations of a small synchronous generator on load [28, 29]. Previous work where finite element software have been used for simulation of hydrogenerator infinite-bus systems is scarce. The work of Schlemmer [30] is an exception.

A central part of this doctoral work is devoted to development and analy-sis of models and methods for transient simulation of grid-connected hyd-ropower generators. Both FE and EC generator models are used and com-pared to assess their performance and applicability.

1.4.3 Identification of equivalent circuit parameters

The validity of EC models is largely dependent on a representative set of parameters. These parameters can be calculated analytically from geomet-rical data [31, 32], or obtained by empigeomet-rical test methods [33, 34].

Common methods for determining the transient and subtransient parameters are the three-phase short-circuit test and the field decrement test. The former exposes the unit to severe mechanical stress and is regularly performed on commissioning of hydropower units, not only to obtain model parameters, but also to test its mechanical integrity. The risk of damaging a unit for pure model identification purposes however, motivated the development of simulation-based testing. The simple field decrement test was simulated using FEM by Hannalla and Macdonald in 1977 [35]. Simulation of symmetrical and unsymmetrical short-circuit tests [36–40] and other FEM based methods to compute the stationary and transient reactances [41, 42] have been explored recurrently since the mid 1980s.

Standstill frequency response (SSFR) testing emerged as an attractive al-ternative to short-circuit testing since it can be performed at a low cost 2_{Although finite element refers to the solution technique rather than the model, this is an}

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and without the risk of damaging the machine [33]. The method was devel-oped for round-rotor machines, but with a few additional practical consid-erations, it can equally be applied to salient-pole machines [43–46]. SSFR data can also be obtained from finite element simulations [21,47–49]. Much of the previous work focused on the model identification process, e.g., on the choice of model structure [50, 51] or the numerical model fitting tech-nique [52, 53].

Recognised disadvantages of the SSFR test are that the low ﬂux levels and the lack of centrifugal forces on wedges and joints produce a model which does not correctly represent the machine at rated load operation [33,53]. Experience shows that short-circuit tests as a rule yields more accu-rate models. A combination of test methods is therefore suggested to obtain the best possible result [33].

In this thesis, both simulation-based and experimental testing is used to identify generator models. Paper IV explores finite element simulation of the classical standard tests described above. In addition to the common short-circuit and field-decrement tests, also the slip and applied-voltage tests are simulated, which has not been done in the past as far as the author is aware. Paper I deals with experimental SSFR testing with special attention given to the damper winding configuration. This work encompasses two unique features: SSFR measurements for three physically different damper winding configurations on the same machine and inclusion of measured damper bar quantities in the SSFR analysis scheme.

1.4.4 Torsional stability of hydropower units

Paper II and Paper III discuss torsional stability of hydropower units, where torsion refers to the angular displacement between the generator rotor and the turbine runner. Due to their design, thermal power production units risk torsional instability as a result of electromechanical interaction with se-ries capacitors or power system controllers. This interaction occurs at sub-synchronous frequencies (<50 Hz) and can be of both electrical and me-chanical nature. These phenomena are grouped under the term subsyn-chronous oscillations (SSO). Numerous papers have been published on var-ious aspects of this topic, see for instance [54–58]. To facilitate a common understanding of SSO, the IEEE has proposed a terminology [59], developed benchmark cases for numerical simulations [60, 61] and summarised pre-vious work [62].

Hydropower units are generally considered to be at low risk of torsional instability due to their comparatively high generator-to-turbine inertia ra-tio and the supposedly large viscous damping torque acting on the turbine runner [14, 63]. Data recorded during a full-scale startup test of the Nordic power system, presented in Paper III, shows a subsynchronous power os-cillation in close range of normal hydropower unit torsional mode

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frequen-cies. This observation and the fact that it was recorded during very special operating conditions is the background to Paper II which assesses the tor-sional stability margins of hydropower units.

A literature survey revealed that despite the conviction of high viscous damping of torsional oscillations in hydropower units, many authors [54, 63–66] had observed that numerical data is scarce. Further motivation for the work reported in Paper II is the fact that the inertia ratio of modern (or modernised) hydropower units in general is lower than that of old ones.

1.5 Structure of the thesis

The thesis encompasses three theory chapters (2-4) and two method chap-ters (5-6). Chapter 2 outlines the fundamental theory of electrical and me-chanical modelling of hydropower units. Chapter 3 summarises the empiri-cal test methods used for identiﬁcation of equivalent circuit model parame-ters. Chapter 4 outlines the topic subsynchronous oscillations which is rel-evant to the work on torsional stability. Chapter 5 and Chapter 6 describe the computational and experimental methods used in the various works.

Chapter 7 and 8 summarises the results and conclusions grouped in four sections. Results related to model identiﬁcation are outlined ﬁrst since they constitute input data to the dynamic studies reviewed in the second sec-tion. The third section discusses some results from the Nordic startup test measurement, which constitute input data to the fourth section on tor-sional stability. Chapter 9 reviews on-going activities and suggests future work.

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2. Electromechanical modelling of

hydropower generators

This chapter outlines the fundamental theory of synchronous generator modelling. Section 2.1 reviews electromagnetic modelling and the ﬁnite element technique. Section 2.2 goes through modelling by equivalent circuits. Section 2.3 discusses mechanical modelling. Section 2.4 outlines the concept of synchronising and damping torques.

2.1 Electromagnetic ﬁeld modelling

All macroscopic electromagnetic phenomena can be modelled using Maxwell’s equations together with the Lorentz’s force equation and the principle of conservation of charge [67]. The electromagnetic field theory can be used to derive a model of an electrical machine for use in computer aided machine analysis based on its geometry and material properties. One suitable numerical technique for solving the electromagnetic field problem in the relatively intricate geometries associated with rotating machinery is the finite element method [68].

This section treats the fundamentals of electromagnetic field modelling and analysis using the finite element method. The terms field model and FE model will be used interchangeably as abridgements for electromagnetic field model. The section is based on material from [31, 67, 68].

2.1.1 Formulation of the 2D ﬁeld problem

The electromagnetic field model used in this work is two-dimensional, meaning that the magnetic field distribution is calculated for an axial cross section of the machine and then extrapolated to its axial length. Non-negligible effects that are disregarded in such a 2D approach must be accounted for by other means. The computational effort needed to solve the same problem in three dimensions is significantly higher and can not be motivated for the purpose of this work.

The analysis domain

The geometry of the model can be reduced to two dimensions by using the planar symmetry of the machine, i.e., by considering the magnetic

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phe-C o n d u c t o r

Air

Iron

Figure 2.1: Example analysis domain of a synchronous machine where the three

main subdomains: iron, conductor and air have been marked.

nomena to be the same on each plane (x,y) perpendicular to the rotor axis. By this assumption, effects around the ending edges are omitted from the ﬁeld part of the model1.

Further, by utilising symmetries around the axis, it is normally sufﬁcient to solve the ﬁeld equations for a tangential fraction of the total cross section [68]. The analysis domain, shown in Figure 2.1, can roughly be divided into three types of subdomains characterised by their electric conductivity,σ, and magnetic permeability,μ, as follows:

Iron Currents are assumed zero (σ = 0) and the magnetic properties are non-linear.

Conductor Currents are non-zero and the magnetic properties are linear (μ=constant).

Air Currents are zero and the magnetic properties are lin-ear.

Currents may alternatively be allowed to ﬂow in the iron region, but the above assumptions are made for all the work presented here.

Derivation of the 2D transient magnetic ﬁeld equation The differential form of Faraday’s law of induction,

∇ × E = −∂B

∂t , (2.1)

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together with the two magnetostatic equations,

∇ × H = J (2.2)

∇ ·B = 0, (2.3)

govern the magnetic phenomena in the analysis domain. H is the magnetic field strength, J is the current density, B is the magnetic flux density and E is the electric field. Maxwell’s extension to (2.2), the displacement current density,−∂D/∂t, is neglected because it is very small at fundamental AC frequencies (50-60 Hz).

A generalised form of Ohm’s law relates the current density to the electric ﬁeld as

J= σE. (2.4)

Their magnetic counterparts, B and H , are related through the expression

B= μ0μrH , (2.5)

whereμ0 is the permeability of free space andμr is the material-speciﬁc relative permeability. It follows from Helmholtz’s theorem that since B is divergence free, it can be expressed as

B= ∇ × A, (2.6)

where A is a magnetic vector potential ﬁeld measured in Weber per meter (Wb/m). Hence, if A of a known current distribution can be found, B can be calculated [67]. Substituting (2.6) for B in (2.5) and introducing the mag-netic reluctivity,ν = 1/μ0μr, yields

H= ν(∇ × A). (2.7)

Further, substituting (2.7) for H in (2.2) yields

∇ × (ν∇ × A) = J. (2.8)

An expression for J can also be related to the magnetic vector potential though the laws of Ohm and Faraday. A combination of (2.1) and (2.6) re-sults in

E= −∇V −∂A

∂t , (2.9)

where V is a scalar electric potential function added for the expression to be completely general. Further, by using (2.4), it is found that

J= −σ∇V − σ∂A

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Before substituting this expression for J in (2.8), it is useful to note that if the cross section of the machine is in the x y-plane, then J and A only have

z-components, Az and Jz. Thus, combining (2.10) and (2.8) ﬁnally yields the expression ∂ ∂x ν∂Az(x, y, t ) ∂x + ∂ ∂y ν∂Az(x, y, t ) ∂y = σ∂Az(x, y, t ) ∂t + σ ∂V ∂z. (2.11)

This is the transient 2D magnetic ﬁeld equation that describes the spatial and temporal evolution of the magnetic vector potential, Az, in the cross section of the machine. The termσ∂A_∂t represents the induced current den-sity and σ∂V_∂z represents the applied current density in conductors con-nected to external sources. The right-hand side of (2.11) constitutes the in-terface to the external electric network.

Magnetic saturation 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Magnetic ﬁeld strength (kA/m)

M a g ne ti c ﬂux de ns it y (T)

Figure 2.2: Virgin magnetisation curve for iron of the magnetic steel grade 250-50.

Air and conductor materials like copper and aluminium exhibit linear magnetic properties with constant relative permeabilities close to one. This means that the ﬂux density in such materials is proportional to the applied ﬁeld strength. Hence, in both the air and the conductor subdomains, the re-luctivity,ν, in (2.11) can be assumed constant (ν = 1/μ0), which was utilised

in Section 2.1.1.

For iron on the other hand, the relationship between the applied mag-netic ﬁeld, H , and the density of the resulting ﬂux, B is non-linear; the iron is said to saturate for high values of H . In effect, the reluctivity in (2.11) is a function of Azin the iron subdomain. Figure 2.2 shows the virgin

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magneti-Figure 2.3: Mesh for a partition of the example geometry.

sation curve for iron typically used in the stator and rotor core of electrical machines. It can be seen that when the field strength is increased over a cer-tain value, the flux density levels out fast. In the unsaturated region (for flux densities below 1 T in the figure), the characteristic is approximately linear.

2.1.2 The ﬁnite element method

The finite element method is a numerical technique based on partition of the analysis domain into a fixed number of small non-intersecting subdo-mains called finite elements. The method can be applied not only to elec-tromagnetic problems; it is in fact the most diffused method for numerical analysis of physical and mathematical vector field problems today [68].

For two-dimensional problems, the elements are usually triangular and the vertices of the triangles are called nodes. The complete body of ele-ments is called mesh. A ﬁner mesh resolution generally yields a more accu-rate solution at the expense of increased computational effort and memory space required. A reasonable compromise is to increase the resolution of the mesh in regions of special interest, where the ﬁeld is time varying, or near elaborate parts of the geometry. The enlarged part of Figure 2.3 shows the mesh generated for a partition of the example geometry.

With the mesh in place, the next step is to chose interpolating functions (or base functions) for approximation of the unknown function,Az(x, y, t ), within each element. If the elements are small enough, a first or second order polynomial is usually sufficient. The field problem can then be re-solved using either the Residual method (also known as Galerkin’s method) or the Variational method (also known as the Reyleigh-Ritz’s method) and the solution of the field problem is found when the coefficients of the base polynomials have been found for each element.

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2.1.3 Boundary conditions

In order for the field problem to be completely specified, it is necessary to assign appropriate constraints on the boundaries of the analysis domain. A homogenous Dirichlet condition, Az= 0, is assigned along the inner and outer circumference of the machine, which is equivalent to considering an external material with null magnetic permeability (a perfect magnetic in-sulator). The magnetic flux is thereby not allowed to cross these boundary lines.

The size of the analysis domain is reduced by using the repetitive nature of slots, poles and windings. Numerically, this is done by assigning even or odd periodic boundary conditions on the radially oriented boundary lines. Rotor motion is accounted for by means of a sliding mesh technique where separate meshes are generated for the stator and rotor geometries with equidistant nodes along a line in the middle of the airgap. Instead of remeshing at each time step, which would be time consuming, rotor mo-tion is mimicked by cyclic permutamo-tion of Az at the connection points.

Figure 2.4 shows appropriate boundary conditions for the example do-main introduced in Section 2.1.1.

Homogenous Dirichlet

condition (A

z

=0

)

Even periodic

boundary condition

Sliding mesh

Figure 2.4: Boundary conditions for the example geometry.

2.1.4 Calculation of the induced EMF

From Faraday’s law of induction it follows that the electromotive force (EMF) induced in an arbitrary winding of N turns can be calculated as

ew= − dΨ

dt = −N dΦ

dt , (2.12)

whereΨ is the magnetic flux linkage and Φ is the magnetic flux. Calculating the EMF is hence a matter of calculating the net flux crossing the surface effectively covered by the winding.

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The ﬂux through a surface S is given by Φ =

S

B·dS, (2.13)

which by (2.6) and Stoke’s theorem can be rewritten as Φ = S B·dS = S (∇ × A) ·dS = l A·dl , (2.14)

where l is the contour that encloses S. Further, it can be shown that in-tegration of a potential ﬁeld along a non-closed curve equals the potential difference between the end points of the curve. Hence, in the cylindrical ge-ometry of the electrical machine, the ﬂux through a surface of length lethat is parallel to an arbitrary line on the x y-plane between the points (x1, y1)

and (x2, y2) is given by

Φ = le·(Az(x1, y1)− Az(x2, y2)). (2.15)

The magnetic vector potential can thus be seen as the magnetic ﬂux per unit length that crosses a given area [31]. If the end points are replaced by surfaces on the x y-plane, such as the cross sections of a winding conductor, the magnetic vector potential must be integrated over the surface area. It follows that for a winding of N turns and with the total cross section area, Γ, the ﬂux linkage can be calculated as

Ψ =le Γ N+ Γ+AzdΓ − N− Γ−AzdΓ , (2.16)

where N+and N−are the total number of positively and negatively oriented winding conductors respectively, andΓ+andΓ−are the corresponding con-ductor areas. It has been assumed thatΓ+=Γ−=Γ.

2.1.5 No-load simulation of a synchronous generator

The induced phase EMFs, ea, eb, and ec, constitute the link between the electromagnetic field equations and the circuit equations. In a finite ele-ment simulation of the no-load operation of a synchronous generator, the field current if d is introduced as an additional unknown. By adding a

re-quirement

e2a+ e2_b+ e2c

_b

e

c

Figure 2.5: Circuit diagram of the generator connected to an impedance load.

potentials. In no-load operation, the internal voltages are seen directly at the terminals.

2.1.6 On-load simulation of a synchronous generator

When the generator is delivering power to an external load, the three stator currents, ia, iband ic, are introduced as additional unknowns. Hence, three supplementary circuit equations are needed. Figure 2.5 shows the equiva-lent circuit of a machine feeding power to a resistive and inductive load.

One equation is obtained from Kirchhoff’s current law

ia+ ib+ ic= 0. (2.18)

With current ﬂowing in the stator windings, the voltage drop over the sta-tor resistance rsand the phase shift caused by the end-winding inductance

ls must be subtracted from the induced EMFs to obtain the line-to-line ter-minal voltages uab = ea− eb+ rs(ib− ia)+ ls d ib d t − d ia d t (2.19) ucb = ec− eb+ rs(ib− ic)+ ls d ib d t − d ic d t . (2.20)

The terminal voltages must also satisfy the external part of the circuit which is written uab = rn(ia− ib)+ ln d ia d t − d ib d t (2.21) ucb = rn(ic− ib)+ ln d ic d t − d ib d t , (2.22)

where rnand ln is the load resistance and inductance respectively. Substi-tuting the right-hand side of (2.21) and (2.22) for uaband ucbin (2.19) and (2.20) yields the additional two circuit equations.

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The ﬁeld equations and the circuit equations are then solved simultane-ously. This is referred to as strong (or direct) ﬁeld-circuit coupling [15, 16]. With the generator being the only frequency-keeping device in the system thus far, only island conditions can be studied. It will be described in Sec-tion 5.1 how a power system load can be implemented.

2.1.7 Calculation of airgap torque

The electromagnetic torque exerted on the rotor of an electrical machine can be calculated using either Maxwell’s stress tensor method or the Virtual-work method [68]. The latter is computationally less effective and has therefore not been used in the work covered by this thesis.

The tangential force caused by a magnetic ﬁeld H on an inﬁnitesimal part

d S of a surface is given by

d Fϕ= μ0HrHϕd S, (2.23) where Hr and Hϕare the radial and tangential components of the ﬁeld re-spectively [68]. The surface is assumed to be placed in air whereμr= 1. For the cylindrical geometry of an electrical machine, the torque on the rotor can then be obtained by integration of the tangential force around the air-gap and then multiplied by the effective length leof the machine, i.e.,

Te= leμ0

2π 0

HrHϕr dϕ, (2.24)

2.1.8 Representation of the damper winding

The damper winding is a short-circuited cage of thick solid copper or brass conductors mounted in the rotor pole faces. Its purpose is to counteract all2 changes of the magnetic airgap flux to obtain better stability properties of the machine and to protect the field winding from excessive currents in case of severe electrical disturbances on the stator side. The damper winding is built up of three fundamental parts: damper bars located in axial slots in the pole faces, upper and lower short-circuiting conductors (plates) which connect the ends of the bars within the pole, and interpole conductors (also called end-ring segments) which connect the bars of adjacent poles. Some-times, no physical connection is in place to conduct current between the poles. Instead, the interpole current finds its way along ill-defined paths in the rotor rim. Such an arrangement is difficult to model correctly.

In the FE model, the damper winding is represented by a resistance cir-cuit as shown schematically in Figure 2.6, where eDi−Dl denotes the EMF induced in the damper bars and zDsymbolises the bar impedance which is implicitly represented in the ﬁeld equations. rDa is the resistance of the in-2_{Except for the slot ripple.}

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2.2 Equivalent circuit model

The electromagnetic ﬁeld model presented in the previous section is not suitable for power system studies where each generator is seen as one com-ponent out of many. The computational cost of solving the ﬁeld equations would simply be too high. On the other hand, an accepted opinion is that a high-order model is not needed in power system stability analysis, where only the characteristics of the machine as seen from its terminals is of inter-est [14]. Much effort has therefore been put into developing reduced-order equivalent circuit models where the magnetic properties of the generator is represented by constant inductances. In the strive for simplicity, a number of approximations are made.

The equivalent circuit models used in this thesis are based on standard synchronous machine model formulations from [13, 69]. This section de-scribes the fundamentals of these model formulations and outlines some of the underlying assumptions. It is not the ambition to give complete de-scriptions of the models and hence the reader is referred to the references for details.

2.2.1 Fundamental assumptions

The stator windings are distributed in slots around the stator periphery. However, by assuming that the stator windings are sinusoidally distributed along the airgap and that the stator slots cause no variation of the rotor inductances with rotor position, the distributed windings can be concen-trated into three electric coils positioned 120 electrical degrees apart. For generators, a positive sign is conventionally used for phase currents ﬂow-ing out of the machine.

Also the rotor (ﬁeld and damper) windings are treated as concentrated windings3in a two-axis reference frame. One reference axis, called the di-rect axis (or d-axis), marks the centre line of the poles and the other, called the quadrature axis (or q-axis) marks the centre line between the poles. The two axes are hence displaced by 90 electrical degrees and rotates with the angular speedω = Nppωmel.rad/s, where Nppis the number of pole pairs andωmis the angular rotor speed. By convention, the q-axis leads the d-axis.

By assuming linear magnetic material properties, the inductances of the coils described above can be represented by constant parameters. As a result, hysteresis and saturation effects are omitted. Hysteresis effects are truly negligible; however, saturation is important and must be accounted for. Saturation is treated separately in Section 2.2.4.

3_{The physical ﬁeld winding is concentrated around the pole axis, whereas a continuous}

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Model structure

Hydropower generators are usually salient-pole synchronous machines with laminated rotors and damper windings. Such machines are normally represented by a model of order 2.1 (notation according to [13, 70]), meaning two rotor circuits (ﬁeld and damper) on the d-axis and one damper circuit on the q-axis. For salient-pole machines with solid iron poles it is motivated to include one additional d-axis circuit to represent eddy currents in the pole faces.

The model developed here is of order 3.2 which is relevant for the work presented in Paper I. A graphical representation of this model connected to an external network equivalent is provided in Figure 2.7. The rationale for the choice of this unusually high order is that the effective model structure can equally be determined by the parameters. Redundant rotor circuits are effectively disabled if their resistance is large.

d e fd e fd R fd L d L1 d R1 ad L l L d p\ a R d f L12 Z\q d R2 d L2 Bd E E R E L q e q L1 q R1 aq L l L q p\ a R d Z\ q R2 Bq E E R E L q L2 Machine Grid

Figure 2.7: Equivalent circuit representation of a synchronous generator connected

to an inﬁnite network bus. p denotes the operator d /d t

The series inductance Lf 12dis included to model the so-called peripheral leakage ﬂux that links all of the d-axis rotor windings, but not the armature [14]. This parameter is often found positive for round-rotor machines and negative for salient-pole machines [13, 44, 45].

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dq0-transformation of stator quantities

It can be realised that with the stator coils ﬁxed in space and the rotor coils expressed in a rotating reference frame, the mutual inductances between them depend on their relative positions. To avoid this angular (time) depen-dence, the well-known Park transformation [11] is used to express the stator circuits in the rotor (dq0) coordinates. Details on this transformation can be found in any textbook on equivalent circuit modelling of synchronous ma-chines (see for instance [14, 71]).

2.2.2 Machine equations

If balanced conditions4 are assumed, the electrical equations of a synchronous machine can be written:

Stator voltage equations ed= 1 ωbase dψd d t − ψqω − Raid (2.25) eq = 1 ωbase dψq d t + ψdω − Raiq. (2.26)

Rotor voltage equations ef d= 1 ωbase dψf d d t + Rf dif d (2.27) 0= 1 ωbase dψ1d d t + R1di1d (2.28) 0= 1 ωbase dψ2d d t + R2di2d (2.29) 0= 1 ωbase dψ1q d t + R1qi1q (2.30) 0= 1 ωbase dψ2q d t + R2qi2q. (2.31) Stator ﬂux linkage equations

ψd= −(Lad+ Ll)id+ Ladif d+ Ladi1d+ Ladi2d (2.32)

ψq= −(Laq+ Ll)iq+ Laqi1q+ Laqi2q. (2.33)

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Rotor ﬂux linkage equations ψf d= Lf f dif d+ Lm f 12di1d+ Lm f 12di2d− Ladid (2.34) ψ1d= Lm f 12dif d+ L11di1d+ Lm f 12di2d− Ladid (2.35) ψ2d= Lm f 12dif d+ Lm f 12di1d+ L22di2d− Ladid (2.36) ψ1q= L11qi1q+ Laqi2q− Laqiq (2.37) ψ2q= Laqi1q+ L22qi2q− Laqiq, (2.38) where the following mutual and self inductances have been introduced

Lm f 12d= Lf 12d+ Lad (2.39) Lf f d= Lf d+ Lm f 12d (2.40) L11d= L1d+ Lm f 12d (2.41) L22d= L2d+ Lm f 12d (2.42) L11q= L1q+ Laq (2.43) L22q= L2q+ Laq. (2.44)

More detailed speciﬁcations of the parameters can be found in for instance [13, 14, 69]. In the above equations, all quantities are given in per unit5(pu) except for time which is given in seconds. The airgap torque is calculated as

Te= ψdiq− ψqid. (2.45)

2.2.3 External network

A single-machine inﬁnite-bus (SMIB) system is accomplished by adding two equations to represent the external network,

ed= EB d+ REid+ LE d id d t (2.46) eq= EB q+ REiq+ LE d iq d t , (2.47)

where EB d and EB q are the d- and q-axis components of the inﬁnite-bus voltage. A resistance RE and an inductance LE are added to represent a transmission line or a transformer.

5_{Details on per unit calculations can be found in any textbook on electrical engineering. The}

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2.2.4 Magnetic saturation

Saturation of the magnetic ﬂux paths have a non-negligible effect on the transient response of the machine. Several more or less advanced meth-ods to account for the saturation characteristics in the machine model are available. In this thesis a very simple approach based on constant satura-tion factors has been adopted from [14]. The method rely on the following assumptions:

1. The leakage inductances are independent of saturation since their asso-ciated ﬂuxes are largely in air.

2. Leakage ﬂuxes do not contribute to iron saturation because they are rel-atively small and because they have little effect on the main ﬂux due to their short common path. This assumption implies that onlyψad and

ψaqare subject to saturation.

3. The saturation relationship is the same under loaded conditions as un-der no-load conditions. This assumption allows the saturation factor to be calculated from the open-circuit saturation curve, which is usually the only one available.

4. No magnetic coupling between the d- and q-axes is introduced as a re-sult of saturation.

In view of these assumptions, the saturated mutual d- and q-axes induc-tances can be estimated from their respective unsaturated values Lad uand

Laquas follows:

Lad= KsdLad u (2.48)

Laq= KsqLaqu. (2.49)

Ksd for a given operating point can be calculated from the open-circuit characteristic as

Ksd=

IF G

IF N L

, (2.50)

where IF G denotes the field current on the airgap line and IF N Lis the field current on the no-load curve needed to produce rated terminal voltage re-spectively. For a salient-pole machine, it may be assumed that the quadra-ture axis does not saturate, i.e., Ksq = 1, since the q-axis flux is largely in air and the iron portion of the path does not vary considerably with satura-tion [14]. This view has been adopted here. Methods to calculate the q-axis saturation factor more accurately can be found in [72].

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2.2.5 Simpliﬁed model for large system studies

In order to keep the computational effort on a reasonable level, some ad-ditional approximations are needed for simulations of large multi-machine systems. Although only a SMIB system is simulated in this thesis, it is in-teresting to include the simpliﬁed model for comparative purposes. This section outlines the required approximations and the structure of the sim-pliﬁed model.

Omission of network and stator transients

The derivative terms in the network voltage equations (2.46)-(2.47) and sta-tor voltage equations (2.25)-(2.26), referred to as the network and stasta-tor transients respectively, increase the model order signiﬁcantly. In addition, fast transients require small time steps in the numerical integration pro-cess. Therefore, it is considered necessary to neglect them in simulations of large power systems [14]. Note that omission of the derivative terms does not mean that the ﬂux linkages and currents are constant; on the contrary, it means that they change instantaneously.

When fundamental frequency stator quantities are viewed in the rotating reference frame, they appear to be unidirectional (DC), i.e., they have no oscillating components. Consequently, the resulting airgap torque is also unidirectional. The true airgap torque contains a fundamental frequency component due to the interaction with the rotor circuits and an additional unidirectional component, called the DC braking torque, that corresponds to the losses in these windings. Hence, omission of the network and stator transients lead to omission of some braking torque contribution from the rotor windings.

Omission of speed variations

If the stator transients are neglected as described above, it is recommended to also neglect the inﬂuence of speed variations on the stator voltages, i.e., to letω = ω0= 1 in (2.25)-(2.26). The superbar is added to emphasise that

the angular velocity is given in per unit. This does not reduce the compu-tational effort per se; the reason is that it counterbalances the effect of ne-glecting the stator transients (see [14] for a mathematical proof).

Simpliﬁed voltage equations

The simpliﬁed system voltage equations without network transients, stator transients and speed voltages can be written

EB d+ (Ra+ RE)id− XEiq+ ω0ψq= 0 (2.51)

EB q+ (Ra+ RE)iq+ XEid− ω0ψd= 0, (2.52) whereω0= 1 and XE= ω0LE= LE.

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2.3 Mechanical modelling

The shaft of a hydropower unit is rather short and the inertia of the gen-erator rotor is typically 10 to 40 times larger than that of the turbine run-ner [63]. In addition, the viscous damping torque acting on hydraulic tur-bine runners to attenuate torsional oscillations is supposedly large [14]. Therefore, it is customary in power system analysis to consider the whole rotor of a hydropower unit as one rigid rotating mass. The theory in Sec-tion 2.3.1 accounts for the dynamics of the entire rotor with respect to an external (power system) reference frame.

As can be seen in Paper III, the torsional oscillations can be of substan-tial magnitude also in a hydropower plant. To model torsional mechanics, which is necessary in Paper II, the generator rotor and the turbine runner must be regarded as individual rotating mass elements separated by a tor-sional spring. Such a spring-mass model is derived in Section 2.3.2.

2.3.1 Single-mass model

The acceleration of the rotor is determined by the net torque on the shaft according to the rotational form of Newton’s second law of motion

Jdωm

d t = Tm− Te, (2.53)

where J is the rotor mass moment of inertia,ωmis the angular velocity of the rotor, Tmis the mechanical torque on the turbine runner and Te is the electrical (airgap) torque. Multiplying (2.53) byωmyields

Jωm

dωm

d t = Pm− Pe, (2.54)

where the right-hand side now represents a power unbalance and Jωm is the angular momentum.

It can be realised from the above equations that if a generator shaft is exposed to a torque or power unbalance its speed will change and reach steady state at ﬁrst when balance is restored. The shaft speed of a grid-connected synchronous machine subject to an unbalance will be oscillat-ing around the synchronous speed (deﬁned later). A mechanical analogy can be found in the swing and therefore Equation (2.53), and variations thereof, is popularly called the swing equation.

Per unit swing equation

In power system analysis it is convenient to work with normalised (per unit) quantities. The shaft speedωmis normalised by the synchronous

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mechan-ical speed deﬁned as ωm0= ω0 Npp = 2πf0 Npp , (2.55)

where Npp is the number pole pairs,ω0 is the synchronous electrical

an-gular speed and f0 is the nominal system frequency. Introducing bars6 to

denote per unit quantities and usingω for electrical rotor speed

ωm= ωm ωm0· Npp Npp = ω ω0= ω, (2.56)

i.e., in per unit, the mechanical and electrical shaft speeds are equal. In-stead of J , it is customary to use the inertia constant

H=1

2

Jω2_m0 Sbase

, (2.57)

where Sbase is the power base, chosen as the rated apparent power of the machine. Further, introducing the following base quantities for mechanical speed and torque

ωmbase= ωm0 (2.58)

Tbase=

Sbase

ωmbase

, (2.59)

the swing equation (2.54) can be written 2Hdω

d t = Tm− Te. (2.60)

The factor 2H is the time required for the machine to accelerate from stand-still to rated speed when rated torque is applied; this time is commonly re-ferred to as the mechanical starting time.

Rotor angle

The angular speed of a grid connected synchronous machine may be ex-pressed as a deviation from the synchronous speed

Δω = ω − ω0. (2.61)

Integrating with respect to time we obtain the rotor angle

δ =

Δωdt, (2.62)

Hydropower generator and power system interaction

Digital Comprehensive Summaries of Uppsala Dissertations

from the Faculty of Science and Technology 978

Hydropower generator and

power system interaction

JOHAN BLADH

List of Papers

List of Papers

Contents

Nomenclature

Abbreviations

Abbreviations continued

Latin symbols

Latin symbols continued

Latin symbols continued

Latin symbols continued

Latin symbols continued

Greek symbols

1. Introduction

1.1 Development of hydropower in Sweden

1.2 New challenges

1.3 Project background and related work

1.4 Hydropower

generator

and

power

system

interaction

1.4.1 Overview

1.4.2 Electromechanical modelling

1.4.3 Identification of equivalent circuit parameters

1.4.4 Torsional stability of hydropower units

1.5 Structure of the thesis

2. Electromechanical modelling of

hydropower generators

2.1 Electromagnetic ﬁeld modelling

2.1.1 Formulation of the 2D ﬁeld problem

phe-C o n d u c t o r

Air

Iron

2.1.2 The ﬁnite element method

2.1.3 Boundary conditions

Homogenous Dirichlet

condition (A

=0

)

Even periodic

boundary condition

Sliding mesh

2.1.4 Calculation of the induced EMF

2.1.5 No-load simulation of a synchronous generator

r

l

u

r

l

r

l

r

l

r

l

r

l

u

i

i

i

e

e

e

2.1.6 On-load simulation of a synchronous generator

2.1.7 Calculation of airgap torque

2.1.8 Representation of the damper winding

r

r

r

r

r

r