Influence of damping winding, controller settings and exciter on …398778/... · 2011. 2. 18. ·...

UPTEC E 11 002

Examensarbete 30 hpFebruari 2011

Influence of damping winding, controller settings and exciter on the damping of rotor

angle oscillations in a hydroelectric generator

The testing of a mathematical model

Jonathan Hanning

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

Influence of damping winding, controller settings andelectrical feeders on the damping of rotor angleoscillations in a hydroelectric generatorJonathan Hanning

This thesis has been performed in the university context for Master thesis 30 credits,which is a compulsory exercise in order to gain a degree in electrical engineering.

The thesis main objectives were to investigate how the damping and the stiffness of ahydroelectric generator changed depending on different parameter values, and to testa new mathematical model to calculate the damping and stiffness constants Kd and Ks.The work has been performed at the request of VG Power, but has been performedat the division for electricity at Uppsala University. The reason for undertaking thisthesis was to ensure that generators are robust. But also when building future modelsfor generators, to have a system that can be used to compute robustness.

During this thesis a power cabinet has also been constructed to be able to test thesimulated model on a real generator. Under the first five weeks a power cabinet wasconstructed in the laboratory at the division for electricity. The tests were thenperformed at a generator with a rated power of 75 kVA.

ISSN: 1654-7616, UPTEC E 11 002Examinator: Nora MassziÄmnesgranskare: Urban LundinHandledare: Martin Ranlöf

Sammanfattning

Detta examensarbete har utförts vid avdelningen för elektricitetslära vid Uppsala universitet.

Uppgiften var att undersöka hur dämpning och styvhet påverkas av olika faktorer i en

generator. En del av arbetet bestod i att jämföra skillnaden mellan kontrollerad dämpning med

hjälp av en automatisk spänningsregulator tillsammans med en PSS, mot ett system som

använder kopparskenor för dämpning.

Den viktigaste slutsatsen som kan dras av detta examensarbete är att om man vill ange

riktlinjer för tillverkare av generatorer, när systemet endast består av en generator och en

regulator, bör riktlinjerna bestämmas utifrån massan. Eftersom denna faktor är den viktigaste

för robustheten i systemet. Tanken med systemet skulle vara att för varje viktig variabel, så

skulle ett värde erhållas och skulle sedan kunna kontrolleras mot en tabell för att säkerställa

att inga farliga värden erhålls. Att konstruera denna tabell är ett annat examensarbete, som

skulle kräva fler simuleringar på många fler maskiner, och därför bör utföras av någon med en

bakgrund inom beräkningsvetenskap.

Den matematiska modellen som testats i detta arbeta behöver lite mer justering på grund av att

den inte verkade matcha helt den nuvarande accepterade modellen. Det måste dock sägas, till

den nyares försvar, att med vissa inställningar, så korrelerade den mycket bra med den äldre

modellen. Men det kommer att behöva ändras och anpassas lite mer, särskilt i beaktande vid

beräkningen av den synkrona vridmomentskoefficienten, som nästan alltid verkade vara 10

till 30 procent för låg.

Abbreviations AVR Automatic Voltage Regulator

D-Q-axis Direct and Quadrature axis

DAE Differential-Algebraic Equation

Et Terminal Voltage

H An inertia constant

Ka/Kp Gain constant in the feedback system

Kd (1) Damping constant in electric torque equation

Kd (2) Derivative constant in the feedback system

Ki Integrating constant in the feedback system

Ks Synchronous constant in electric torque equation

ODE Ordinary Differential Equation

Pf Power Factor

PhD “Philosophiæ Doctor” or Doctor of Philosophy

PSS Power System Stabilizer

P.U. Per Unit

Re Resistance in the tie-line

SMIB Single Machine, infinite bus

St Power output

Td Foresight of the time step

Te Electrical torque

Xe Reactance in the tie-line

UU Uppsala University

Conclusion This master thesis has been conducted at the division of electricity at Uppsala University. The

task was to conduct research about the damping and stiffness of a generator. One part was to

compare the difference with controlled damping with the help of an automatic voltage

regulator, together with a power system stabilizer. And also a system which used copper bars

for damping.

The main conclusion that can be drawn from this thesis is that if you want to provide

guidelines for manufacturers of generators, when the system contains only of the generator

and a regulator, the guidelines should be determined by the mass. Since this factor is the most

important one for the robustness of the system. The idea of the system would be that for each

important variable, a number is acquired and could then be checked against a table to ensure

that no dangerous values are obtained. To construct a table like this is another thesis, which

would need a lot more machines to be simulated on, and therefore should be performed by

someone with a background in scientific computing.

The mathematical model tested in this thesis need some more adjusting, due to the fact that it

did not seem to match entirely to the current accepted model. It must be said, though the

latter’s defense, that with some settings, the altered mathematical model matched very well.

But it will need to be modified and tuned some more, especially in regard to the calculation of

the synchronous torque coefficient, which almost always seemed to be 10 to 30 percent to

low.

2010-12-31

Influence of damping winding, controller settings and exciter on the damping of

rotor angle oscillations in a hydroelectric generator

2

Foreword This thesis has been performed in the university context for Master thesis 30 credits, which is

a compulsory exercise in order to gain a degree in electrical engineering.

The thesis was to investigate how the damping and the stiffness of a hydroelectric generator

changed depending on different parameter values. And also to test an altered mathematical

model to calculate the damping and stiffness constants Kd and Ks. The work has been

performed at the division for electricity at Uppsala University, as a joint operation together

with VG Power. The reason for undertaking this thesis was to ensure that generators are

robust. But also when building future models for generators, to have a system that can be used

to compute robustness.

I would like to especially thank my supervisor Martin Ranlöf for the time he has spent helping

me over the threshold incurred during my work. My thanks are also directed to the people in

the same working group, Johan Lidenholm, whose thesis has been very helpful, but who has

also helped with understanding some problems in Matlab. Thanks also to Mattias Wallin for

much practical instruction during the construction of the power cabinet. Also big thanks to

Urban Lundin for the hydropower course and for making this thesis possible and I would also

like to thank Kjartan Halvorsen for the help with the automatic control. Also thanks to Stefan

Pålsson for this knowledge with Matlab. Last but absolutely not least, all the teachers who has

put in much effort in my education so that the courses I have read has become much more

interesting, thanks also to my examiner, Nora Masszi.

Jonathan Hanning

January 2011

Uppsala

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1 Introduction ............................................................................................................................. 4

1.1 Background .................................................................................................................. 4 1.2 Method ............................................................................................................................. 5 1.3 Demarcation ..................................................................................................................... 5

1.4 Objectives ......................................................................................................................... 5 2 Theory ..................................................................................................................................... 7

2.1 Synchronous generator ..................................................................................................... 7 2.2 The damping and synchronous coefficient ....................................................................... 7 2.3 System analysis ................................................................................................................ 8

2.4 The mathematical model .................................................................................................. 9 2.4.1 Direct and Quadrature axis ........................................................................................ 9

2.4.2 Per unit representation ............................................................................................. 10 2.4.3 State-space representation ....................................................................................... 10 2.4.4 Automatic control .................................................................................................... 10 2.4.5 The nine basic equations ......................................................................................... 11

2.4.6 Ordinary differential equations solver ..................................................................... 12 2.4.7 Standard parameters ................................................................................................ 12

2.5 Rotor angle oscillation ................................................................................................... 12

3 Method and construction ....................................................................................................... 14 3.1 Mathematical model in matlab for simulation ............................................................... 14

3.1.1 The introducing of state space representation ......................................................... 14

3.2 The construction of the power cabinet ........................................................................... 16

3.2.1 Modifying the generator with damper bars ............................................................. 16 3.3 Operating the generator ...................................................................................................... 17

4 Results ................................................................................................................................... 18 4.1 Simulation results ........................................................................................................... 18

4.1.1 Single machine without regulators .......................................................................... 18

4.1.2 Single machine with an automatic voltage P-regulator ........................................... 19 4.1.3 Single machine with an automatic voltage PD-regulator ........................................ 20

4.1.4 Single machine with an automatic voltage PID-regulator ...................................... 21 4.1.5 Single machine with both PID-regulator and PSS .................................................. 22 4.1.6 The mathematical model ......................................................................................... 22

4.1.7 Unstable systems ..................................................................................................... 23 4.2 Laboratory tests .............................................................................................................. 23

4.2.1 Connecting the generator to the grid ....................................................................... 23 5 Discussion ............................................................................................................................. 25

5.1 The simulation model ..................................................................................................... 25 5.2 The constructed power cabinet ....................................................................................... 25 5.3 The results ...................................................................................................................... 25 5.4 Future work .................................................................................................................... 26 5.5 Confounding ................................................................................................................... 26

6 References ............................................................................................................................. 27 6.1 Literature ........................................................................................................................ 27

7 Appendix ............................................................................................................................... 28

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1 Introduction

A different model, [1] compared with the accepted model, has been expanded and tested to

calculate the synchronizing and damping components of electrical torque developed in a

synchronous machine. The method is based on the numerical analysis of system response

time, using least squares adjustment.

1.1 Background

Since the introduction of synchronous generators in the late 19th

century, the way of operating

a system with several generators has significantly improved. In the early years, it was not

unusual to have power black-outs over a huge area of the grid. But when the regulation was

modernized, it has become more and more unusual with power failure. Nowadays it is almost

required a storm which destroys a cable to receive a power failure.

The stability of a power grid is depending both on the total grid, but also on its individual

components. Usually in a grid there are power consumers, power producers, power

transmission and power control. And since the producers are depending on the consumers,

there has always been an interest of how the producing unit reacts to changes in consuming.

For example how the electrical torque changes when a huge load is connected to the grid. The

electrical torque is built up by the synchronous and damping constants of the generator.

Therefore these constants have been of interest for some time.

The ability to calculate the damping and synchronizing constants has been an important

problem since the expansion of power system interconnections. And since the improvement

of digital computers and modern control theory, a better control of power systems has been

gained. However, the method how to calculate these torque components has not improved at

the same rate. This new approach is thus based on the time-domain analysis of system

response. Precision depends on how good the accuracy was of the time response.

Due to imperfection in the system, a couple of oscillations will occur. The most interesting

and important one is the rotor angle oscillation. This oscillation occurs when the power is

raised or lowered, and the generator is trying to find its new equilibrium, the equilibrium

between the torque from the turbine and the electrical torque. This oscillation gives a few

other oscillations, which will be studied in this thesis. For example the oscillation in the

power produced. The power produced is connected to the swing equation, equation 1b, which

is connecting the rotor angle acceleration, the mechanical torque, and the electrical torque.

This is further described in chapter 2.2.

When measuring is performed on a generator, different variables are calculated in order to be

able to compare the results. It is usual to use either damping time constant Td, which is the

time required for the amplitude to decrease to a new value from its original value. Another

value that is often use is the damping torque coefficient, Kd, which is used to identify the rotor

angle stability of the system. Yet another constant that is interesting is the damping capacity

b, which is the ability to absorb vibration by internal friction. In the current situation, there is

some confusion about what requirements that should be set on the generator supplier,

concerning the damping qualities.

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The purpose of this thesis is to show how the damping and synchronous constant is affected

by the electrical design, selection of exciters, adding damping parameters and inertia. The

equation relationship between attenuation and these factors are certainly well known, but are

found in various places in the literature. The study of damping has been performed on a two-

axle model of a generator.

1.2 Method

This thesis will analyze this problem in two different ways; first the expanded mathematic

model which will be tested in Matlab. In this thesis are also included some testing on a real

generator. To make this possible, a synchronization unit will be built to link up one of the

division of electricity´s generator to the electrical grid. Since Uppsala does not have any great

waterfall, the generator is driven by a motor which is connected to the rotor, instead of a

turbine. Therefore there will be the possibility to perform quick torque changes. The idea is

then that the natural damping of the generator, which occurs due to the copper in the rotor

windings, together with the quick torque change, will give rise to oscillating revolutions per

minute. This oscillation should continue until the generator has found its new steady-state

level of operation. This should also provide an oscillating power curve. Another test will be to

connect a network of copper bars to get a stronger damping, due to the current that will be

induced in these which will counteract the change in torque.

1.3 Demarcation

The idea of this thesis is to develop a functional program for the mathematic model in Matlab,

which can simulate different machines, with different parameter values. This program will

then be modified so that you can connect an automatic voltage regulator in front of the

generator, and the final version should also include a power system stabilizer. This is done to

be able to compare the stability and robustness of a system, due to variation in settings on the

control systems and various types of generators. A proposal will also be included of how the

value of the included parameters in the regulator should be, to ensure stable systems. If the

results of the simulations in Matlab give a distinct and unambiguous picture, a proposal for

recommendation of generator design in terms of robustness and torque stability will be made.

Primary focus will be to investigate the influence of the automatic voltage regulator’s

parameters on stability of the system, while the power system stabilizer will more or less, if

not time permits, just be implemented. The simulations will only be on a single machine

infinite bus. No simulation will be tested on island grids (weak bus). Primary focus will be on

the synchronous coefficient and the damping coefficient, described more closely in chapter

2.2. Secondary focus will be on the changes in rotor angle velocity, and torque change and

also how the electric angle between rotor and stator magnetic axes difference. Most units in

this program will be measured in per unit, and the reason is that it gives more comparable

data.

1.4 Objectives

The primary objective is to construct a program for analysis of the damping and synchronous

coefficient. Secondary is to build a functional synchronizing unit to be able to connect a

generator to the grid. This to make it possible to try the theory in reality, but also for further

experiments being performed by other students and PhD under the division of electricity.

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After completing the main objective, a series of add-ons is desirable. For example, the

possibility to use an automatic voltage regulator and a power system stabilizer to increase the

robustness and stability of the system. Another sub objectives there would be to investigate

how the different parameters change the system. Another sub objective is to analyze how the

new mathematical model works, compared to the old accepted model, with other words, if

they correlate. Another objective is to try to make up a system so companies who design

generators can have some kind of model for robustness and stability when designing the

generators. It would also be desirable to look into the stability from an automatic control

perspective, due to the fact that both the automatic voltage regulators as well as the power

system stabilizers are feedback systems. A desirable and maybe final objective would be if

the simulated results would correlate with those which can be measured in the laboratory,

primary the change in rotor speed when a disturbance in torque is being done. This thesis is

being done because there is a gap in the literature concerning how the stability is inflicted by

an automatic voltage regulator and its parameters.

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2 Theory

In this chapter we will familiarize ourselves with the theory underlying this thesis, if one

wants to look further into this theory, reference [2] is recommended.

2.1 Synchronous generator

A large part of the power production in the world originates from power stations using

generators directly connected to the grid, synchronous generators. A synchronous generator

rotates with a speed that is proportional to the frequency of the current in the armature. The

magnetic field that is created by the armature currents, rotates with the same speed as that

created by the current on the rotor, the field current. If it is a strong grid, special preparations

have to take place to meet the demands. In general there are five conditions that are required

before synchronizing a generator to the grid, but phase sequence and waveform should be

fixed by the construction of the generator and its connection to the system. But voltage,

frequency and phase angle must be controlled each time a generator is to be connected to a

grid.

1. The generator frequency is equal to the system frequency.

ω1 = ω2.

2. The generator voltage is equal to the system voltage.

E = V (Generator E = Grid V)

3. The generator voltage is in phase with the system voltage.

α = 0 (phase difference)

A voltage difference will result in a steady flow of reactive power and when this coincident

with a frequency difference a substantial reactive and active power will flow back and forth to

the grid under a short interval of time that could damage the generator.

2.2 The damping and synchronous coefficient

The equation that is the focus of this thesis is described below, equation 1a, chapter 2.2 in

reference [2]. This describes the changes in the electrical torque ▲Te, depending on the

synchronous coefficient, Ks, which is multiplied with the change in the electrical rotor angle

▲δ. Then it is the damping coefficient, Kd, which is multiplied with the change in rotor angle

velocity ▲ω. The change in electrical torque is taken from its context where it usual belongs,

the swing equation, equation 1b. Where J is the total moment of inertia of the rotor mass, Tm

is the mechanical torque supplied by the prime mover. Te is the electrical torque output of the

alternator, and θ is the angular position of the rotor in radians.

[Equation 1a]

[Equation 1b]

It is around this equation the thesis is built. But also about how to calculate Ks and Kd. The

procedure to calculate the Ks and Kd works by using the time response of the torque, speed

and angle. And then by applying the least squares adjustment to obtain the electric torque to

the two signals [3], equation 1. Then the error can be determined by equation 2:

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[Equation 2]

And to be able to see the summation of error over time, we are forced to integrate over the

interval of oscillations, equation 3.

[Equation 3]

Damping and synchronizing torque coefficients Kd and Ks are calculated to minimize the

integral of the least squares adjustment. They must fulfill these two equations, equation 4 and

equation 5.

[Equation 4 (top)]

[Equation 5 (below)]

Kd and Ks are time-independent, hence also the differential equations and integration

parameters are time-independent. This means that it is possible to change the differential

equation and integrating the system. This makes it possible to rewrite equation 4 and equation

5, to equation 6 and equation 7:

[Equation 6]

[Equation 7]

By using the above equations, Kd and Ks could be estimated, the other values in the equations

are calculated numerically in the simulation, which can be further studied in chapter 2.4.

2.3 System analysis

This thesis has been tested on several known systems. Which was given an electric or a torque

disturbance, and then with help from equations 6 and 7, with the stated time integrations

performed numerically, on the supplied data from a simulation and the resulting algebraic

equations solved for Ks and Kd. The Equations can then be used with the three time responses

to calculate the damping and synchronizing constants. The different settings on the machines

can be found in appendix 1. The system was tested on a single machine, infinite bus.

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2.4 The mathematical model

One of the main objectives of this thesis was to try an altered mathematical model, with the

help of Matlab. The basic mathematical model was constructed in such a way that nine

equations, see chapter 2.4.5, were needed to be met, in order to calculate Ks and Kd, both

with the analytic and square alignment. Much more information can be found in appendix 2,

which is the code for the Matlab-model. Here the mathematic equations are put into its

context. Which may simplify the understanding on how they are used, therefore, they will be

less described here in the text. These nine basic equations are divided into two different kinds

of equations. The first five equations will return an actual value, with help from Matlab. This

value will be obtained with help of numerical analysis. The other four equations are

calculated to become zero. The reason why nine equations are needed is simply due to the fact

of the numbers of unknown variables, later on in this thesis, more equations will be added to

satisfy the need when new variables arose as a result of more complicated and automation

systems.

2.4.1 Direct and Quadrature axis

Direct and quadrature axis is more known as d-q-axis. They are often used when calculating

on a generator, instead of traditional x-y-axis. This due to the simplified equations received,

when looking at an electrical point of view. They are simplified because the magnetic circuits

and all rotor windings are symmetrical with respect to both polar axis and the inter-polar axis.

The direct (d) axis is defined by that it is centered magnetically in the centre of the north pole,

and the quadrature (q) axis is defined by that it is 90 (electrical) degrees ahead of the d-axis.

Further on, the position of the rotor, relative to the stator, is measured by the angle theta,

which is the angle between the d-axis and the magnetic axis of phase a winding, seen from

above. The choice of the q-axis leading the d-axis is based on the IEEE standard definition

[4]. Throughout this thesis the d-q-axis is used if nothing else is specified, for a graphical

representation, see figure 1.

[Figure 1: shows the relationship between the rotor and stator-current together with the d-q-

axis]

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2.4.2 Per unit representation

It is very common in power systems calculations to use per unit system, due to the fact that it

becomes much easier to compare results when you do not need to remember real values, the

per unit system is functioning according to equation 8a. In this thesis, per unit system is

standard for all units, such as voltage, current, resistance, and so on. One example is that since

the simulations are performed on different machines, it would be hard to compare the results

if you constantly would need to check-up the basic value of each machine, it is easier now to

see the percentage difference instead and compare that way. So for each quantity that is used

in the simulation, a base value is chosen, then use your actual quantity value and divide with

your base value, to receive your quantity in the per unit system. An example is given in

equation 8b, which will represent Machine I in appendix 1. This example will show the rated

active power.

[Equation 8a]

[Equation 8b]

2.4.3 State-space representation

As mentioned earlier, in chapter 2.3, the need for more equations due to more variables will

be discussed here. The way of solving the problem with more complex regulation system,

machine 5 in Appendix 1, is to add more equations to the system. First, the state-space

representation should be introduced, chapter 12.2.6 in reference [2], equation 9, which shows

a state space representation for a time-domain solution. A, B, C and D are just constants, but

with higher order systems, they will become rows and columns with constants. These

constants can represent different matrices. In this thesis, A will be the state matrix. B is the

input matrix. C is the output matrix, and D is the feedthrough, or feedforward, matrix. If the

system only contains an automatic voltage regulator, then x is the automatic voltage regulator

state-vector, and u is the difference between the wanted signal and the feedback connected

signal.

[Equation 9]

This system is a time-domain representation of the transfer function, described further in

chapter 3.1.1, obtained through inversed Laplace-transformation. The derivative dx/dt will be

a column by changing constants when the system becomes more complex, these columns will

provide the need for new equations as new variables appear.

2.4.4 Automatic control

In order not to totally rely on the strong grid, in maintaining the frequency, phase and voltage

within permissible limits, various devices may be connected in front of the generator to help

the grid. In this thesis, the AVR will be closely studied, but also a PSS will be investigated.

Both of these devices use feedback connection for control. The automatic voltage regulator

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can be of different complexity, from as simple as just a gain, that means that the error is

amplified by a factor k, to quickly eliminate the difference between desired and actual value.

But the regulator might be of the degree to have gain, integrating step, and derivation step,

together with limits on the signals, to ensure no transients disturbs the system. More

information about the different regulators and power stabilizers can be found in appendix 3.

2.4.5 The nine basic equations

The number of equations that were needed was determined by the number of unknown

variables, and without a regulator, there were nine unknown. That led to the need for nine

equations that could determine nine unknowns. The equations represent state variables, which

will receive a new value for every discrete time step that the simulation takes. Since many of

the variables in equations 10-18 contains other variables which also need new values for

every discrete time step, there will be more lines of code executed than these nine, see

appendix 2 for more information. All equations are represented in the per unit scale, and also

in d-q-axis. The equations were derived, chapter 13.3 in reference [2], and here they are

presented in their final versions, equation 10-18, as they were used in the code. Equation 10

and 11 is part of the swing equation, which describes how the rotor speed is inflicted by the

unbalance of the mechanical torque and the braking torque. Equation 10 shows how the rotor

speed changes, and to get the right unit, it is multiplied with the base for angular velocity.

Equation 11 shows what will happen to the mass equation when the torque is changed, H is an

inertia constant (described in equation 20), and Pm0 is the mechanical torque, and Pe is the

electrical torque. For equation 12 to 18, mentioned should be that Psi stands for flux linkage,

d and q is the axis, L is the inductance, R stands for resistance. Efd is the measured voltage

going out from the regulator, and i stand for current. Ed and Eq is the terminal voltage of the

d and q axis, while Ebd and Ebq is the d and the q components for the field voltage Eb. X

represents the reactance in the tie line.

[Equation 10]

[Equation 11]

[Equation 12]

[Equation 13]

[Equation 14]

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[Equation 15]

[Equation 16]

[Equation 17]

[Equation 18]

2.4.6 Ordinary differential equations solver

In order to function with sufficient precision in the calculations an ode-solver is used, which

produces new values for each discrete time step the simulation takes. It is for the nine

equations described in chapter 2.3.5, that the ODE-solver obtains new values in every time

step. But since the variables in the nine equations difference as well, there will be more than

nine lines of code which will be executed on every discrete time step. The ODE-solver used

in these simulations were Matlabs ODE23t, since it is quite fast but still with high accuracy. It

is also good to use when you got differential algebraic equations, which the last four

equations of the nine basic ones are.

2.4.7 Standard parameters

Since this thesis has been engaged by a company, standard parameters may differ a bit from

the manufactory to the university. The parameters you insert in this program are the standard

parameters for companies, which are based on inductances. In the chosen Per Unit system, the

inductances are equal to the corresponding reactances. To be able to present numbers which

both can be satisfied with, a transfer-script was needed, to see these scripts, appendix 2 should

be studied, StandardParam.m and StandardParamTieLine.m. An example can be found below,

in example 1. Describing the saturated synchronous q-axis reactance, this is the standard

parameter, while Laq and L1 are parameters taken from the generator manufacturer. These are

the inductance in the phase a, on the q-axis, Laq and the leakage inductance, Ll.s

Xq = Laq + Ll;

[Example 1]

2.5 Rotor angle oscillation

In electric power engineering, there is a concept of rotor angle oscillation of a generating unit,

consisting of generator, shaft and turbine, which slowly oscillate around the synchronous

speed. Usually these oscillations occur when the driving torque is different from the braking

torque. Fluctuation is not desirable and therefore there are several different measures to be

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taken in order to remove the fluctuations. Among the measures that are most common there is

damper windings which provides natural attenuation, which add damping contribution from

the current that is induced. The damping of rotor angle oscillations depends on the generator’s

electrical design, with the reactance’s and time constants, inertias, the operating point,

controller settings and more.

Due to that synchronous generators are directly connected to the grid, change of load on the

grid directly influence the frequency. For example, if a huge load connects to the grid, the

frequency tends to lower a little bit from its stable operation point, until more power is

produced. Either by letting more water through or connecting another hydropower station

onto the grid. In this thesis the thought is to give the turbine, which actually is a motor, order

to raise its revolutions per minute. And then hopefully get a rotor angle oscillation, which can

be detected with the help of measuring equipment. The change in rotor angle should give rise

to a few other oscillations, for example in power production, which is based on voltage and

current.

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3 Method and construction

In this chapter we will familiarize ourselves with the method and construction that has been

performed during this thesis, this chapter is divided into two parts, the first part will address

the theoretical part. While the second part will explain how the practical testing and

construction of the power cabinet were built up.

3.1 Mathematical model in matlab for simulation

This part of the thesis is a continued study of Johan Lidenholm’s thesis [5], which included a

Matlab-program, which several parts of this program has been aided by. The program sets up

a generator with different parameters, which is connected to an infinite bus. To be able to

compare results more easily, all parameters are converted to the per unit system. The

generator is then running at nominal speed and nominal voltage level for a time t1. After that,

either an electrical error occur, or a torque disturbance, this for a short period t2. At last, the

voltage or torque disturbance is restored, and the generator is trying to get back to normal

state of operation, during the time t3. And it is the behavior during the time t3 that is studied,

and the behavior can be altered by changing parameters, or connect different kind of

automatic control units in front of the generator. The interesting data is then saved from the

simulation, and then treated in order to compare between different simulations to obtain a

behavior that correlates with the changes in settings or parameters. Even without any

automatic controller unit, the generator is supposed to have a moment of inertia in itself

because of the mass, and the damping also gets a contribution from the resistance is the

windings. Also the strong grid is supposed to try to get the generator back to normal steady

state. Therefore a number of simulations were made on just plain generators connected to an

infinite bus, to see how much the generators parameters would change the stability of the

system. And then especially of interest would be the change in the electrical torque.

3.1.1 The introducing of state space representation

In order to use automatic voltage regulator, the feedback system needs to be reversed laplace-

transformed back to the time-domain. Since its representation is in the frequency-domain. The

complete systems can be found in appendix 3, but below in figure 2, one system is

represented.

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[Figure 2: shows the second automatic voltage used in this thesis, Kp is the gain-factor, Ki/s

is the integrating part, and Kd*s/(1+sTd) is the derivative part]

As can be seen in equation 9, y is what comes out of the system, efd in figure 2. The equation

9 is a classical way of representing a state space equation. Therefore there is a function in

Matlab which could be used. It works in the manner of that you insert the transfer function,

from figure 2, which can be found in equation 19, and the function in matlab gives in return

values for A, B, C and D. The system whose transfer function can be seen in equation 19 is a

regulator with, gain, derivative and integrating part.

[Equation 19]

After retrieving these constants, they are used in the differential-algebraic equations. Used for

calculating a new value on dx/dt, appendix 2, in simulation3.m. Since a new value must be

obtained in every discrete time step, dx/dt is one of the time-dependent variables that the

ordinary differential equations solver produces for each time step. But as for all automatic

control units, the system is always using the last value, to control its next behavior. So the

need for short discrete time step is inevitable. The change in e, is actually Vref (reference)

minus the actual voltage value et. This is the furthest left summation in figure 2.

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3.2 The construction of the power cabinet

In order to be able to try out the theory of this thesis, as well as trying to see if the results

from the simulation could be seen in reality, a power cabinet had to be constructed in order to

be able to connect one of the division of electricity’s generators onto the Swedish grid. This

then could be seen as a single machine infinite bus. The construction work took an estimated

three weeks of this thesis. All parts were ordered and the work of assembling them was done

in the laboratory hall. The final result of the construction can be seen in appendix 4. The

power cabinet contained a synchronization unit, and the interior can be studied below in

figure 3a (to the left) and the exterior is shown in figure 3b (to the right).

[Figure 3a (to the left) shows the inside of the power cabinet, which housed the

synchronization unit, at the top one can see the cables going out to the connection to the grid,

and at the bottom, is the cables that is connecting with the generator]

[Figure 3b (to the right) shows the outside of the power cabinet, with its voltmeter, ampere

meter and inductive meter. The buttons is for connecting and disconnecting the unit]

3.2.1 Modifying the generator with damper bars

As one test out of many in this thesis, a series of damper bars were connected to the generator,

these damper bars were made out of copper, and were connected to each others, by bridges

between the bars. The reason for this modification was the idea of that the copper bars would

increase the generators damping, due to the fact that the copper would induce a current which

would result in a magnetic field that would oppose changes. The copper bars were low into

cavities in the rotor steel, and then connected with bridges, see appendix 5.

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3.3 Operating the generator Once the generator was connected to the grid, a series of testes with different characters were

performed. These tests were to try to show the oscillation in rotor angle, by changing the

amount of power inserted to the motor. In normal case, if it would not have been connected to

the grid, it would have raised its revolutions per minute. But now instead it would raise its

torque. But before it reached its new steady state, an oscillation should be possible to

measure. The measuring of revolutions per minute was done by using laser equipment and

different color stripes on the rotor.

First of all, the generator had to be connected to the grid, this was done by using the earlier

described synchronization unit. The speed of the motor, which acted as a turbine, was set a

little bit higher then which were required to gain the 50 hertz. After that, the speed were set to

a value, which would represent lower than 50 hertz, and hopefully in the transition in-between

these values, a connection to the grid could be made. Otherwise one had to do the procedure

again, the other way around, first to low speed, and then raise the power to the motor, and

hope that the three requirements, described in chapter 2.1, would be met.

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4 Results

This part of the thesis will not be presented in its entirety, since all the data that was obtained

during the work is impossible to present. Therefore, just the part that was considered

important to present is represented here, the rest of the data will be available in the appendix

or in some sort of link attachment.

4.1 Simulation results

In this chapter the results of the simulation will be presented, due to the amount of data

received during these simulations, most of the figures will be in appendix 6-12. The values

chosen for these simulations were based upon recommendation, chapter 12.4 in reference [2]

and from manufacturers [6], but sometimes interesting results were followed up by

simulations with values outside those boundaries.

4.1.1 Single machine without regulators

The first part of the thesis was to investigate different type of machines which should be

considered to be equal real generators. The simulations were performed by first running with

the standard settings, and then the parameters were changed. Just to try to get a grip of how

much influence the different parameters had for the overall performance in concern of

stability. The original settings for each machine can be found in appendix 1. Some parameters

were changed to see the impact of stability, the parameters that were changed were:

Et: This is the terminal voltage

H: This is an inertia constant, which is based on equation 20. And it shows how

H, which is used in equation 11, is depending on J, which is mass moment of inertia, wm ,

which is the rated mechanical angular frequency and S which is the apparent power in VA

Pf: This is the power factor

Re: This is the resistance in the tie-line

Xe: This is the reactance in the tie-line

St: This is the apparent Power output

[Equation 20]

The result of this simulation can be found in appendix 6-9, together with the results for the

other machines. Conclusions that can be drawn from these simulations are that the most

important factor for stability and robustness, which is shown in figure 4, of the system was

when the parameter H was changed. This was done in such a manner that it was set to a value,

and hence the equation 20 was overwritten, the values was in the per unit system. Important

things to note from Figure 4 is that at lower H, the system becomes sensitive to disturbance,

but also more rapid to return to stable operation after the disturbance. One can generally say

that when the value of H is higher, the system is simply slower. And since H is dependent on

the mass of the rotor, the mass should be the factor that sets the guidelines of other

parameters.

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[Figure 4: shows how the torque reacts when a disturbance is made, at time t1 = 1 sec, the

red color is H=2, green 3, blue 4, magenta 5, yellow 6]

If mass is a parameter that could be changed during construction, it should be adapted in a

way that if one want a robust system, the mass should be maximized, and if one want a

system that can response to quick oscillations, the mass should be reduced.

4.1.2 Single machine with an automatic voltage P-regulator

This simulation was made with a P-regulator, which is a simple regulator with only a gain

step which increases the error between the wanted signal and the actual signal. It should be

said that this simulation were made without regards for stability criteria for the feedback

system. A stable system is a system which has only negative poles, the poles are obtained by

the solutions to the denominator roots. That means that some of the results might be on

unstable systems, which will give some strange results, as can be seen in figure 5, Kd (the

damping constant) is negative for higher value of Ka (gain constant in the feedback system).

This unstable feedback system may be the reason why suddenly the value 1000 on Ka seems

to increase the damping constant, all data is consolidated in appendix 10. The machine used

with the P-regulator is machine V, which can be closely studied in appendix 1, the main

feature of machine V is that R1d and R1q is set to 1 p.u. which should represent a machine

without damping.

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[Figure 5: shows how the damping constant change with increasing Ka (gain value in the

feedback system), negative value of Kd might represent an unstable feedback system]

4.1.3 Single machine with an automatic voltage PD-regulator

This simulation was made with a PD-regulator, which both has a gain and a derivative step.

The machine used with the PD-regulator is machine V, which can be closely studied in

appendix 1, the main feature of machine V is that R1d and R1q is set to 1, which should

represent a machine without damping. Except changing the gain and derivative constants, also

the time constant Td was altered to see the impact on the synchronizing and damping

constant. Td is the foresight of the time step. One thing that should be kept in mind when

reading through the results, which can be found in appendix 11, is that some of the

combinations, of the values of gain, the derivative and time step constants, might create

unstable feedback systems. This might be the reason for why in figure 6, it is not entirely

conclusive, but one should still be able to see the trends. One interesting and maybe alarming

trend is that the result seems to vary very little with a low value on Ka (gain constant), blue

line below. Every value which calculate Ks has been analyzed, and they all seems to be in the

per unit system. But even so, a larger impact could be expected when you multiply the error

between the wanted signal and the actual signal. But further analyzes are needed to

distinguish if any errors has been done, and to investigate if the feedback systems are stable.

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[Figure 6: shows how Ks change with Ka (gain) and Kd (derivative, not damping constant)]

4.1.4 Single machine with an automatic voltage PID-regulator

This simulation was made with a PID-regulator, which has a gain, a integrating and a

derivative step. The machine used with the PID-regulator is machine V, which can be closely

studied in appendix 1, the main feature of machine V is that R1d and R1q is set to 1, which

should represent a machine without damping. Except changing the gain, integrating and

derivative constants, also the time constant Td was altered. The possibilities to alter different

settings made the amount of data enormous, for intense studies of specific cases, see appendix

12, below in figure 7, the synchronous constant can be studied, with different Kd and Ki.

[Figure 7: shows how Ks change with Kd (derivative part of regulator, not damping constant)

and Ki (integrating part of regulator)]

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The trend seems to say that with increasing integrating constant, the synchronous constant is

lowered. The strange thing is that it does not seem to be linear, due to the fact of the order of

the lines, which can be seen in figure 7. One could think that they should be in rising or in

dropping order, but it seems more randomly then that, maybe due to unstable feedback

systems.

4.1.5 Single machine with both PID-regulator and PSS

The results from this study should not be presented, since either the mathematical model or

the computer power was not enough to calculate this model with accurate accuracy. Due to

the many T-values (foresight in time step) in the PSS, the model in Matlab could never finish,

even though it was given a 24 hour time span. Even so, the model is probably right, but need

more computer power or other limits in the mathematical model, the PSS-code can be found

in appendix 2 (DAE).

4.1.6 The mathematical model

The new mathematical model [1] was proven to work quite well, even thou it did not entirely

match, it seems that it still need some folding to be a perfect match, as can be seen in figure 8,

it is close to a perfect match. The black colored lines are the ones that is the new mathematical

model, and the different colors are the old model. For a closer look how well they matched,

appendix 13 is recommended which is a table of data from study I, machine I.

[Figure 8: shows how Ks and Kd changes when a disturbance is made, at time t1 = 0 sec, the

red color is Xe 0.0 green 0.15 blue 0.30, magenta 0.45, yellow 0.6, the black is the new

mathematic model]

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4.1.7 Unstable systems

During some simulations, unexpected results occurred. And many of those might be due to

unstable feedback systems. It is not always easy to see from the response on Ks and Kd, but it

can be seen from the other graphs, see figure 9 for an example of an unstable feedback

system. And as can be easily spotted, when the integrating part of the feedback system is

given a very high value, the whole system gets unstable.

[Figure 9: shows how the angle reacts when a disturbance is made, at time t1 = 1 sec, Ka is

100 and the red color is Ki 0.005, green 0.1 blue 0.4, magenta 1.0, yellow 2.0]

4.2 Laboratory tests

Due to the resources at the division of electricity at Uppsala University, this thesis could be

tested with a generator. The test equipment is described in chapter 3.2. And pictures can be

found in appendix 4 and 5. A set-up with different tests was performed with this equipment.

To try and establish if the simulation result could be transposed into the laboratory generator.

The generator used in these tests was a synchronous generator with rated power 75kVA.

4.2.1 Connecting the generator to the grid

The first test that were performed and recorded were the connection of the generator to the

grid, to be able to perform this, the right voltage, the right frequency and the right phase was

needed to be obtained. Instead of a river through the laboratory, a motor acted as the turbine,

this motor could simulate different kind of disturbance. Mostly used was to try to change the

revolutions per minute, but due to the connection to the strong grid, it was not possible to

raise the RPM. Instead a power increase occurred, sadly due to the strong net, it was hard to

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see the mechanical oscillations. They were so small that they were lost is the accuracy of the

measurement. But the oscillations in power could be perfectly seen, see figure 10.

[Figure 10: Shows when a torque change is performed on the motor what happened to the

total power output. The oscillations can be perfectly seen directly after the change, before

stabilizing on a level, two changes are made in this figure]

Time (second)

Power

(MVA)

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5 Discussion

This thesis has been testing a new mathematical model for calculating the electric torque in a

generator, with the help from the synchronous constant Ks and the damping constant Kd.

5.1 The simulation model

This thesis has provided a method in a program for using advanced regulators and PSS

together with a generator. The PSS could not be implemented in a satisfying way, due to lack

of computer power or more likely, the input to the ordinary differential equation solver in

matlab is in such a way that the condition never could be meet. It was tested to let the

program run for 24 hours, and still it had not produced a single value. That is if you think of

the ten seconds it was supposed to run as a long chain of discreet data points. So the model

probably needs new conditions that are put into the solver. Therefore it is recommended that

someone with a background in scientific computing continue forward and investigates if the

feedback system is stable, and also continue with the mathematical model and investigate if it

needs folding, since it does not entirely match the common used model.

Also the results should perhaps be more consolidated with someone with scientific

computing, since my own knowledge in the subject is limited. The program which was

programmed to solve this task should be streamlined, since there is a need for many changes

and clicks to produce one result. Perhaps even the layout of the program should be changed in

order to make it more easily used by a third part person.

5.2 The constructed power cabinet

Except for some minor misses and flaws during construction, most of the work went without

problems. The one thing I want to recommend to someone making their first power cabinet is

to drag cable corridors. Even though it seems to work out in the beginning, there will be a lot

of cables in the end. As do not be stingy when it comes to the use of cable, or you will regret

it later. Now afterwards, I know how I should have constructed my first power cabinet.

As far as to this date, its functionality has worked flawlessly, the process of connecting it to

the grid I would recommend is a two person job, since it is hard to both keep the control

voltage level and the right revolutions per minute at the same time as trying to connect the

generator to the grid.

5.3 The results

The results show clearly some trends which could be expected when tuning on the regulator.

For example, you get a faster system with a higher gain value on the regulator, but at the same

time higher transients. But one should be alarmed when the tuning constants on the regulator

are getting high values, since the probability of an unstable system is imminent. None the less

this thesis could work as guidelines, both for manufacturers and for further research into this

unexplored territory. Most of the results is quite expected, for example when the gain is

raised, you get a faster system, but with higher transients and oscillations. A trivial but

confusing matter is that both the damping constant and the derivative part of the feedback

system has the abbreviation Kd, and maybe one should be altered, but they are both very

standard to use. When using derivation and integration steps, one could think that the

transients should get to a lower altitude. And sometimes that was the case, but far from

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always, which might again be due to an unstable system. The reason for having integrating

and derivation step is to try to optimize the return to steady state after a disturbance. And

sometimes it helped with these steps, but not always, and for high values on these constants,

Kd and Ki, it even had the reverse effect. The results concerning the automatic voltage

regulator with a derivative step came up with some problems which also were encountered in

reference [5]. Which was that sometimes it seemed like the system swung twice around a

center point, see Appendix 11, full version for the figures.

5.4 Future work

I would recommend letting someone with a scientific computing background, maybe as a

master thesis, look over the model and run some simulations with confirmed stable systems.

In order to make the results interesting for manufacturers of generators, the compared system

all should be stable and tested on real models of generators. Also the model should be altered

so that the PSS could be implemented, since it is an important part of the total system. Also it

would be interesting to compare simulation on the same machine with and without different

kind of regulators. To investigate how important they are, and how the generator changes its

response due to different disturbances.

5.5 Sources of errors

Mainly I would suspect that many of the simulation had parameter values which gave an

unstable system, which would produce results which is not trustworthy. Another thing that

could be worth investigated is what limits the time step should be in, because with a value to

big, the regulator had no influence. Probably because the changes are faster than the regulator

could handle. The question is then what value is small enough to get a reliable result, because

when you lower the time step, the amount of data for every simulation grows. And due to that,

the simulation with a PSS could not be performed. For those tests performed with a real

generator, the errors should strictly be bound to the inaccuracy of the measurement

equipment.

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6 References Below are the references which have been used in this thesis.

6.1 Literature

[1] R.T.H. Alden and A.A. Shaltout, ”Analysis of Damping and Synchronizing Torques Part I

– A General Calculation Method”, IEEE Transactions on Power Apparatus and Systems, vol.

PAS-98, Sept./Oct. 1979.

[2] P. Kundur “Power System Stability and Control”, McGraw-Hill inc. 1994.

[3] B.P. Lathi, “An introduction to Random Signals and Communication Theory”,

International Textbook Company, Chapter 1, 1968.

[4] ANSI/IEEE Standard 100-1977, “IEEE Standard Dictionary of Electrical and Electronic

Terms”.

[5] J Lidenholm “Power System Stabilizer Performance” ISSN 1401-5757, UPTEC F07 109

[6] M. Wahlén, “Transfer function for excitation system and automatic voltage regulator”,

VG Power AB, 2004.

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7 Appendix

These versions of the appendix concerning the results are a limited edition, the complete

result appendixes can be given at a request.

Appendix 1 Settings on the the different machines

Appendix 2 the matlab-code.

Appendix 3 example of AVR and PSS

Appendix 4 Pictures of the power cabinet that was built

Appendix 5 Pictures of the generator and damping bars

Appendix 6 Results from the simulations, study 1, machine 1




Appendix 10 Results from the simulations, study 2, AVR 1 (simple gain)

Appendix 11 Results from the simulations, study 2, AVR 2 (gain and derivative)

Appendix 12 Results from the simulations, study 2, AVR 3 PID-regulator (gain

integrating and derivative)

Appendix 13 Results from study I, machine I. A table of Ks and Kd with the

new mathematical model compared to the old model.

Appendix 1

Settings on the the different machines, Appendix 1

% FUNDAMENTEL PARAMETRES IN [P.U] (weak damper) MACHINE I Lad = 0.65; Ra = 0.003; Laq = 0.35; Lfd = 0.1; Rfd = 0.0003; L1d = 0.05; R1d = 0.005; L1q = 0.2; R1q = 0.01; Ll = 0.15; Lad_u = 0.75;

% FUNDAMENTEL PARAMETRES IN [P.U] (strong damper) MACHINE II Lad = 0.65; Ra = 0.003; Laq = 0.35; Lfd = 0.1; Rfd = 0.0003; L1d = 0.05; R1d = 0.005; L1q = 0.05; R1q = 0.005; Ll = 0.15; Lad_u = 0.75;

% FUNDAMENTEL PARAMETRES IN [P.U] (weak damper, large synchronous

reactance) MACHINE III Lad = 0.95; Ra = 0.003; Laq = 0.55; Lfd = 0.1; Rfd = 0.0003; L1d = 0.05; R1d = 0.005; L1q = 0.2; R1q = 0.01; Ll = 0.15; Lad_u = 1.05;

% FUNDAMENTEL PARAMETRES IN [P.U] (strong damper, large synchronous

reactance) MACHINE IV Lad = 0.95; Ra = 0.003; Laq = 0.55; Lfd = 0.1; Rfd = 0.0003; L1d = 0.05; R1d = 0.005; L1q = 0.05; R1q = 0.005; Ll = 0.15; Lad_u = 1.05;

% FUNDAMENTEL PARAMETRES IN [P.U] (machine without damping) MACHINE V Lad = 0.65; Ra = 0.003; Laq = 0.35; Lfd = 0.1; Rfd = 0.0003; L1d = 0.05; R1d = 1; L1q = 0.2; R1q = 1; Ll = 0.15; Lad_u = 0.75;

DAE.m

Appendix 2

The Matlab-code

% File: DAE_3.m

% Formulation of the system of differential-algebraic equations % of the SMIB-system before and after the fault.

function [dae] = DAE_3(T,X_Y,PARAM);

H = PARAM(1); Pm0 = PARAM(2); w_base = PARAM(3); Rfd = PARAM(4); Lfd = PARAM(5); R1d = PARAM(6); L1d = PARAM(7); R1q = PARAM(8); L1q = PARAM(9); Xdb = PARAM(10); Xqb = PARAM(11); Ll = PARAM(12); Ra = PARAM(13); efd0 = PARAM(14); EB = PARAM(15); RE = PARAM(16); XE = PARAM(17); noStAE = PARAM(18); Et = PARAM(19); Lad = PARAM(20); noStpss = PARAM(21); AE = PARAM(22:30,1); %cheating code, find length of AE pss = PARAM(23+8:end,1); %cheating code, find length of pss

A_ae = zeros(noStAE,noStAE); B_ae = zeros(noStAE,1); C_ae = zeros(1,noStAE); D_ae = 0;

for i=1:noStAE A_ae(1,i) = AE(i); A_ae(2,i) = AE(noStAE + i); end

for i=1:noStAE B_ae(i,1) = AE(2*noStAE + i); end

for i=1:noStAE C_ae(1,i) = AE(3*noStAE + i);

DAE.m

end

D_ae = AE(4*noStAE + 1);

A_pss = zeros(noStpss,noStpss); B_pss = zeros(noStpss,1); C_pss = zeros(1,noStpss); D_pss = 0;

for i=1:noStpss A_pss(i,1) = pss(noStpss*(i-1)+1); A_pss(i,2) = pss(noStpss*(i-1)+2); A_pss(i,3) = pss(noStpss*(i-1)+3); A_pss(i,4) = pss(noStpss*(i-1)+4); A_pss(i,5) = pss(noStpss*(i-1)+5); end for i=1:noStpss B_pss(i,1) = pss(5*noStpss + i); end

for i=1:noStpss C_pss(1,i) = pss(6*noStpss + i); end

D_pss = pss(7*noStpss + 1);

% State variables DELTA = X_Y(1); OMEGA = X_Y(2); PSI_fd = X_Y(3); PSI_1d = X_Y(4); PSI_1q = X_Y(5); xAE = X_Y(6:6+noStAE-1); xpss=X_Y(7:7+noStpss-1); PSI_ad = X_Y(8+noStAE+noStpss-2); PSI_aq = X_Y(9+noStAE+noStpss-2); ed = X_Y(10+noStAE+noStpss-2); eq = X_Y(11+noStAE+noStpss-2);

Ladsb = Xdb - Ll; Laqsb = Xqb - Ll;

% Infinite bus voltage in machine reference frame EBd = EB*sin(DELTA); EBq = EB*cos(DELTA);

% Subtransient voltage sources Edb = Laqsb*(PSI_1q/L1q); Eqb = Ladsb*(PSI_fd/Lfd + PSI_1d/L1d);

% CALCULATION OF STATOR CURRENTS id and iq (Kundur sid. 783) RT = Ra + RE; XTd = XE + Xdb; XTq = XE + Xqb; D = RT^2 + XTd*XTq;

DAE.m

EdN = Edb + EBd; EqN = Eqb - EBq; id = (XTq*EqN - RT*EdN)/D; iq = (RT*EqN + XTd*EdN)/D;

% Air-gap (breaking) power Pe = PSI_ad*iq - PSI_aq*id;

Vref = 1;

% Differential equations: xdot = f(x,y)

f_xy = [w_base*OMEGA; (1/(2*H))*(Pm0 - Pe); w_base*(efd0 + (Rfd/Lad)*(C_ae*xAE + ... D_ae*(Et-sqrt(ed*ed+eq*eq))) + (PSI_ad - PSI_fd)*Rfd/Lfd);

%PSI_fd/dt (general solution) w_base*(PSI_ad - PSI_1d)*(R1d/L1d);

%PSI_1d/dt w_base*(PSI_aq - PSI_1q)*(R1q/L1q);

%PSI_1q/dt A_ae*xAE + B_ae*(Et-sqrt(ed*ed+eq*eq)+C_pss*xpss+D_pss*OMEGA);

%dxAE/dt A_pss*xpss+B_pss*OMEGA];

%dxpss/dt

g_xy = [PSI_ad + Ladsb*(id - PSI_fd/Lfd - PSI_1d/L1d); % PSI_ad PSI_aq + Laqsb*(iq - PSI_1q/L1q); % PSI_aq ed - EBd - RE*id + XE*iq; % ed eq - EBq - XE*id - RE*iq]; % eq

dae=[f_xy; g_xy;];

ExtraktionsKsKd.m

% FILE: ExtraktionKsKd.m

% Created by Martin Ranlöf 2010-08-27

% Extract Ks and Kd from the simulated responses of deltaTE, deltaOMEGA and % deltaDELTA following a small disturbance.

% A least-square fitting approach is used.

% The script is designed to operate on data output from the MATLAB script % "Simulering.m".

% Subscript P = "Post"

start = 200;

time = T_P(start:end,1) - T_P(start,1); % [s] DELTA_res = X_Y_P(start:end,1); % [rad] dOMEGA = w_base*X_Y_P(start:end,2); % [rad/s] PSI_fd_resP = X_Y_P(start:end,3); PSI_1d_resP = X_Y_P(start:end,4); PSI_1q_resP = X_Y_P(start:end,5); PSI_ad_resP = X_Y_P(start:end,13); PSI_aq_resP = X_Y_P(start:end,14);

ifd_resP = (PSI_fd_resP-PSI_ad_resP)/Lfd; % [p.u] field

current i1d_resP = (PSI_1d_resP-PSI_ad_resP)/L1d; % [p.u] D-damper

current i1q_resP = (PSI_1q_resP-PSI_aq_resP)/L1q; % [p.u] D-damper

current

id_resP = -(PSI_ad_resP - Lad*(ifd_resP + i1d_resP))/Lad; % [p.u] d-axis

current iq_resP = -(PSI_aq_resP - Laq*i1q_resP)/Laq; % [p.u] q-axis

current

Te_anp = PSI_ad_resP.*iq_resP - PSI_aq_resP.*id_resP; % [p.u] moment som

skall anpassas

% Calculate dTe and dDELTA Te_mean = Te0; dTe = Te_anp - Te_mean; DELTA_mean = delta0; dDELTA = DELTA_res - DELTA_mean;

% Create time-interval vector time2 = time(2:end); dT = time2 - time(1:end-1);

% Calculate least square integrals (see paper by Alden and Shaltout) B1 = sum(dTe(1:end-1).*dDELTA(1:end-1).*dT); B2 = sum(dTe(1:end-1).*dOMEGA(1:end-1).*dT); A1 = sum(dDELTA(1:end-1).*dDELTA(1:end-1).*dT);

ExtraktionsKsKd.m

A2 = sum(dDELTA(1:end-1).*dOMEGA(1:end-1).*dT); A4 = sum(dOMEGA(1:end-1).*dOMEGA(1:end-1).*dT);

% Solve system of equations to find Ks and Kd Ks_anp = (A4*B1 - A2*B2)/(A1*A4 - A2^2); % [p.u./rad] Kd_anp = (B1 - A1*Ks_anp)/A2; % [p.u./rad/sec]

dTe_cntrl = Ks_anp*dDELTA + Kd_anp*dOMEGA;

figure(8) plot(time,dTe,'y') hold on plot(time,dTe_cntrl,'k') xlabel('time [s]'); ylabel('Ks+Kd') hold on

InitiateSimulation.m

% FILE: InitiateSimulation.m

% INITIATE DYNAMIC SMIB SIMULATION WITH STEADY-STATE VALUES

Pt = St*PF; % [p.u] Rated active power Qt = St*sqrt(1-PF^2); % [p.u] Rated reactive power It = sqrt(Pt^2 + Qt^2)/Et; % [p.u] Rated terminal current phi = acos(PF); % [rad el.] power factor angle

% Internal rotor angle delta_i0 = atan((Xq*It*cos(phi) - Ra*It*sin(phi))/... (Et + Ra*It*cos(phi) + Xq*It*sin(phi)));

% Terminal voltage d-axis component ed0 = Et*sin(delta_i0); % Terminal voltage q-axis component eq0 = Et*cos(delta_i0); % Line current d-axis component id0 = It*sin(delta_i0 + phi); % Line voltage q-axis component iq0 = It*cos(delta_i0 + phi);

% Initial values stator flux linkages (wr = 1 [p.u] at steady state) Psi_d0 = eq0 + Ra*iq0; Psi_q0 = -ed0 - Ra*id0;

% Field current ifd0 = (Psi_d0 + (Xad + Ll)*id0)/Xad; % Field voltage efd0 = Rfd*ifd0;

% ROTOR CIRCUIT FLUX LINKAGES Psi_fd0 = (Lad + Lfd)*ifd0 - Lad*id0; Psi_Dd0 = Lad*(ifd0-id0); Psi_Dq0 = -Laq*iq0;

% MUTUAL FLUX LINKAGES Psi_ad0 = -Lad*id0 + Lad*ifd0; Psi_aq0 = -Laq*iq0;

% DAMPER CIRCUIT CURRENTS iD0 = 0; iQ0 = 0;

% ELECTRIC TORQUE Te0 = Pt + Ra*It^2;

% MECHANICAL POWER INPUT Pm0 = Te0;

% BUS VOLTAGE EBd0 = ed0 - RE*id0 + XE*iq0; % [p.u] EBq0 = eq0 - RE*iq0 - XE*id0; % [p.u] EB = sqrt(EBd0^2 + EBq0^2); % [p.u]

InitiateSimulation.m

% INITIAL ROTOR ANGLE IN NETWORK REFERENCE FRAME delta0 = atan(EBd0/EBq0); % [p.u]

% EXCITER OUTPUT VOLTAGE Efd0 = Lad_u*ifd0; % [p.u]

% STEADY-STATE INTERNAL EMF Eq0 = Xad*ifd0 - (Xd-Xq)*id0; % [p.u]

KdKsAnalytiskt.m

% KdKsAnalytiskt.m

SS_analys

for i=1:length(Eig) aa = Eig(i,i); if imag(aa) ~= 0 break end end

w_eig = imag(aa)/(2*pi*FREQ); % [p.u] Frequency of oscillation f_eig = w_eig/(2*pi);

% CALCULATE STANDARD PARAMETERS WITH TIE-LINE IMPEDANCE INCLUDED StandardParamTieLine

% Initial values stator flux linkages (wr = 1 [p.u] at steady state) Psi_d0 = -Xd*id0 + Lad*ifd0; Psi_q0 = -Xq*iq0;

% TIME CONSTANT CONVERSION (from seconds to p.u) ss = 2*pi*FREQ;

Td0p_pu = ss*Td0p; % [p.u] Tdp_pu = ss*Tdp; % [p.u] Td0b_pu = ss*Td0b; % [p.u] Tdb_pu = ss*Tdb; % [p.u] Tq0b_pu = ss*Tq0b; % [p.u] Tqb_pu = ss*Tq0b*(Xqb/Xq); % [p.u]

% CALCULATION OF OPERATIONAL PARAMETERS AT OSCILLATING FREQUENCY Xd_op = Xd*(1+j*w_eig*Tdp_pu)*(1+j*w_eig*Tdb_pu)/... ((1+j*w_eig*Td0p_pu)*(1+j*w_eig*Td0b_pu)); Xq_op = Xq*(1+j*w_eig*Tqb_pu)/(1+j*w_eig*Tq0b_pu); Zd_op = Ra + RE + j*w_eig*Xd_op; Zq_op = Ra + RE + j*w_eig*Xq_op; D_op = Zd_op*Zq_op + Xd_op*Xq_op;

% PARK'S ELECTRICAL TORQUE-ANGLE RELATIONSHIP Te_op = ((Psi_d0 + id0*Xq_op)*((Et*sin(delta0) + Psi_d0*j*w_eig)*Zd_op + ... (Et*cos(delta0) + Psi_q0*j*w_eig)*Xd_op) + ... (Psi_q0 + iq0*Xd_op)*((Et*cos(delta0) + Psi_q0*j*w_eig)*Zq_op - ... (Et*sin(delta0) + Psi_d0*j*w_eig)*Xq_op))/D_op;

Ks_Park = real(Te_op); % [p.u./rad] Kd_Park = imag(Te_op)/(w_eig*w_base); % [p.u./(rad/s)]

% RE-SET STANDARD PARAMETERS E.T.C TO VALUES WITHOUT INCLUSION OF THE TIE-LINE

IMPEDANCE StandardParam Psi_d0 = eq0 + Ra*iq0; Psi_q0 = -ed0 - Ra*id0;

Simulering.m

% FILE: Simulering.m

% Dynamisk simulering och/eller småsignalanalys av vattenkraftgenerator.

% Synkronmaskinmodellen förutsätter en dämplindning såväl d-som q-axeln. % Statortransienter försummas. Förändringarna i rotorhastigheten antas % vara små under det transienta förloppet.

% I den dynamiska simuleringen initieras rotorpendlingen av ett steg i det % drivande momentet Pm.

clear all

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DEL A: SPECIFICERA MASKINEN, VÄLJ DRIFTPUNKT, % % BERÄKNA INITIALVÄRDESTILLSTÅND % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%% % I % %%%%% % GENERATOR DATA S_NOM = 73; % [MVA] U_NOM = 13.8; % [kV] line-to-line I_NOM = S_NOM/(sqrt(3)*U_NOM)*1000; % [A] FREQ = 50; % [Hz] P = 14; % no. of pole pairs

J = 900000; % [kg m^2] Mass moment of inertia

%%%%%% % II % %%%%%% % TIE-LINE DATA RE = 0.01; XE = 0.1;

%%%%%%% % III % %%%%%%% % DRIFTPUNKT Et = 1; % [p.u] Terminal voltage St = 1; % [p.u] Apparent power output PF = 0.9; % Power factor

%%%%%% % IV % %%%%%% % FUNDAMENTALA PARAMETRAR I [p.u] Lad = 0.65; Ra = 0.003; Laq = 0.35; Lfd = 0.1; Rfd = 0.0003; L1d = 0.05; R1d = 1; L1q = 0.2; R1q = 1; Ll = 0.15; Lad_u = 0.75;

Simulering.m

%%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % V % % EXJOBB HT 2010 % %%%%% % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% AVR Transfer function

s=tf('s');

% PID - AVR

KP = 5; KI = 0.4; KD = 1; TD = 0.01;

a2 = KP/KD*TD; a1 = (KP + KP*KI*TD)/KD; a0 = KP*KI/KD; b0 = 1/KD;

G_avr_exc = (a2*s^2 + a1*s + a0)/(s^2 + b0*s);

% CALCULATE STATE-SPACE MODEL % dx/dt = Ax + Bu % y = Cx + Du

SS_ae=ss(G_avr_exc); [A_ae, B_ae, C_ae, D_ae]=ssdata(SS_ae); dimA_ae=size(A_ae); noStAE=dimA_ae(1); xAE_0 = [zeros(noStAE, 1);]; AE = [];

for i=1:noStAE AE = [AE A_ae(i,:)]; end AE = [AE B_ae' C_ae D_ae];

% pss Transfer function

s=tf('s');

% PSS - Del2 (example-values from page 7, reference [6])

K1 = 0.1; K2 = 0.075; KS = 3.0; T1 = 0.1; T2 = 0.1; T3 = 0.1; T4 = 0.1; T5 = 0.1; T6 = 0.1; T7 = 0.1;

Simulering.m

a4 = (T1*T3*T5*T7*KS*K1); a3 = (T1*T5*T7*KS*K1+T3*T5*T7*KS*K1+T1*T3*T5*K1*KS+T1*T3*T5*K2*KS); a2 = (T5*T7*KS*K1+T1*T5*KS*K1+T1*T5*KS*K2+T3*T5*KS*K1+T3*T5*K2*KS); a1 = (T5*KS*K1+T5*KS*K2);

b5 = (T2*T4*T5*T6*T7); b4 = (T2*T4*T5*T6+T2*T4*T6*T7+T2*T4*T5*T7+T2*T5*T6*T7); b3 = (T2*T4*T6+T2*T4*T5+T2*T5*T6+T4*T5*T6+T2*T4*T7+T2*T6*T7+T4*T6*T7... +T2*T5*T7+T4*T5*T7+T5*T6*T7); b2 = (T2*T4+T2*T6+T4*T6+T2*T5+T4*T5+T5*T6+T2*T7+T4*T7+T6*T7+T5*T7); b1 = (T2+T4+T6); b0 = 1;

G_pss_exc = (a4*s^4 + a3*s^3 + a2*s^2 + a1*s)/... (b5*s^5 + b4*s^4 + b3*s^3 + b2*s^2 + b1*s + b0);

% CALCULATE STATE-SPACE MODEL % dx/dt = Ax + Bu % y = Cx + Du SS_pss=ss(G_pss_exc); [A_pss, B_pss, C_pss, D_pss]=ssdata(SS_pss); dimA_pss=size(A_pss); noStpss=dimA_pss(1); xpss_0 = [zeros(noStpss, 1);];

pss = [];

for i=1:noStpss pss = [pss A_pss(i,:)]; end pss = [pss B_pss' C_pss D_pss];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Beräkning av bas-storheter i det valda p.u-systemet VAbase = S_NOM*10^6; % [VA] e_base = sqrt(2)*U_NOM*1000/sqrt(3); % [V] i_base = sqrt(2)*I_NOM; % [A] Z_base = e_base/i_base; % [Ohm] f_base = FREQ; % [Hz elect.] %w_base = 2*pi*FREQ; % [rad/s elect.]

% Beräkning av "inertia constant", H wm0 = 2*pi*FREQ/P; % [rad/s mech.] Rated mech. angular

frequency H = (1/2)*J*wm0^2/VAbase;

% Base frequency w_base = 2*pi*FREQ; % [rad/s el.] Base angular frequency

% Calculate standard parameters from fundamental parameters StandardParam % Calculate initial values of generator state variables InitiateSimulation

Simulering.m

Ks_anp = 0; Kd_anp = 0; Ks_Park = 0; Kd_Park = 0; w_eig = 0;

Te0 = Psi_ad0*iq0 - Psi_aq0*id0;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DEL C: DYNAMISK SIMULERING AV GENERATORN %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Input parameters to the DAE-formulation function DAE_1 and DAE_1f PARAM = [H Pm0 w_base Rfd Lfd R1d L1d R1q L1q Xdb Xqb Ll Ra efd0 EB RE XE... noStAE Et Lad noStpss AE pss]';

t0 = 0; t1 = 2; t2 = 2.02; tfin = 10; deg = 180/pi; TSPAN1 = [t0 t1]; TSPAN2 = [t1 t2]; TSPAN3 = [t2 tfin]; % Tolerance tole = 1e-9; % Mass matrix for DAE M = [eye(5+noStAE+noStpss) zeros(5+noStAE+noStpss,4); zeros(4,5+noStAE+noStpss) zeros(4,4)]; % Initial conditions vector % X_Y_0 = [delta0 0 Psi_fd0 Psi_Dd0 Psi_Dq0 ... % Psi_ad0 Psi_aq0 ed0 eq0]'; X_Y_0 = [delta0 0 Psi_fd0 Psi_Dd0 Psi_Dq0 xAE_0' xpss_0' ... Psi_ad0 Psi_aq0 ed0 eq0]';

% Create Options-structure for ODE solver options =

odeset('Mass',M,'RelTol',tole,'AbsTol',ones(1,9+noStAE+noStpss)*tole);

% Pre-disturbance simulation [T_S,X_Y_S] = ode23t(@DAE_3,TSPAN1,X_Y_0,options,PARAM); DIM=size(X_Y_S); % Initial conditions for disturbed system X_Y_1 = [X_Y_S(DIM(1),:)]'; % During-disturbance simulation PARAM(15) = 0.7; [T_D,X_Y_D] = ode23t(@DAE_3,TSPAN2,X_Y_1,options,PARAM); % Initial conditions for post-disturbance system DIM=size(X_Y_D); X_Y_2 = [X_Y_D(DIM(1),:)]'; PARAM(15) = EB; % Post-disturbance simulation [T_P,X_Y_P] = ode23t(@DAE_3,TSPAN3,X_Y_2,options,PARAM);

clear TSPAN1 TSPAN2

T_S = [T_S; T_D; T_P;]; X_Y_S = [X_Y_S; X_Y_D; X_Y_P;];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DEL D: DÖP OM UTDATA FRÅN DYNAMISK SIMULERING, PLOTTA STÖRNINGEN %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Simulering.m

time = T_S(:,1); delta_res = deg*X_Y_S(:,1); Domega_res = w_base*X_Y_S(:,2); PSI_fd_res = X_Y_S(:,3); PSI_1d_res = X_Y_S(:,4); PSI_1q_res = X_Y_S(:,5); xae_res1 = X_Y_S(:,6); xae_res2 = X_Y_S(:,7); xpss_res1 = X_Y_S(:,8); xpss_res2 = X_Y_S(:,9); xpss_res3 = X_Y_S(:,10); xpss_res4 = X_Y_S(:,11); xpss_res5 = X_Y_S(:,12); PSI_ad_res = X_Y_S(:,13); PSI_aq_res = X_Y_S(:,14); ed_res = X_Y_S(:,15); eq_res = X_Y_S(:,16);

et_res = abs(ed_res + j*eq_res); Efd_res = (C_ae*[xae_res1 xae_res2]')' + D_ae*(Et-et_res+... ((C_pss*[xpss_res1 xpss_res2 xpss_res3 xpss_res4

xpss_res5]')'+D_pss*Domega_res));

ifd_res = (PSI_fd_res-PSI_ad_res)/Lfd; % [p.u] field current i1d_res = (PSI_1d_res-PSI_ad_res)/L1d; % [p.u] D-damper current i1q_res = (PSI_1q_res-PSI_aq_res)/L1q; % [p.u] D-damper current

id_res = -(PSI_ad_res - Lad*(ifd_res + i1d_res))/Lad; % [p.u] d-axis current iq_res = -(PSI_aq_res - Laq*i1q_res)/Laq; % [p.u] q-axis current

Te = PSI_ad_res.*iq_res - PSI_aq_res.*id_res; % [p.u] torque

figure(1) plot(time,Te,'y') xlabel('time [s]'); ylabel('Torque [p.u]') hold on figure(2) plot(time,delta_res,'y') xlabel('time [s]'); ylabel('Angle [degree]') hold on figure(3) plot(time,Domega_res,'y') xlabel('time [s]'); ylabel('Angle velocity [rad/s]') hold on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % DEL E: BERÄKNING DÄMP- OCH STYVHETSKONSTANTER (KD OCH KS) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%

Simulering.m

% TILLVAL 1 % %%%%%%%%%%%%% % EXTRAHERA DÄMP- OCH STYVHETSKONSTANTER FRÅN SIMULERADE SYSTEMSVARET % ExtraktionKsKd3

%%%%%%%%%%%%% % TILLVAL 2 % %%%%%%%%%%%%% % BERÄKNA DÄMP- OCH STYVHETSKONSTANTER ANALYTISKT % UTTRYCKEN GÄLLER FÖR EN GENERATOR UTAN AVR OCH PSS % KdKsAnalytiskt % Td=4*H/(w_base*Kd_Park); Omegau=sqrt(w_base*Ks_Park/(2*H)); b=Omegau*Kd_Park/Ks_Park; Utskrifter %

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % EXJOBB HT 2010 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

StandardParam.m

% FILE: StandardParam.m

% Created by Martin Ranlöf

% Calculate standard machine parameters for a conventional hydroelectric % generator.

% In the chosen p.u system, inductances are equal to the corresponding % reactances.

% REACTANCES % Saturated synchronous d-axis reactance Xd = Lad + Ll; % Unsaturated synchronous d-axis reactance Xdu = Lad_u + Ll; % Saturated synchronous q-axis reactance Xq = Laq + Ll;

% d-axis mutual reactance Xad = Lad; % q-axis mutual reactance Xaq = Laq; % Transient d-axis reactance Xdp = Ll + Lad*Lfd/(Lad + Lfd);

% Subtransient d-axis reactance Xdb = Ll + Lad*Lfd*L1d/(Lad*Lfd + Lad*L1d + Lfd*L1d);

% Subtransient q-axis reactance Xqb = Ll + Laq*L1q/(Laq + L1q);

% ARMATURE TIME CONSTANT Ta = (Xdb + Xqb)/Ra*(1/2)*(1/w_base);

% D-AXIS TRANSFER FUNCTION TIME CONSTANTS T1 = (Lad + Lfd)/Rfd*(1/w_base); T2 = (Lad + L1d)/R1d*(1/w_base); T3 = (1/R1d)*(L1d + Lad*Lfd/(Lad + Lfd))*(1/w_base); T4 = (1/Rfd)*(Lfd + Lad*Ll/(Lad + Ll))*(1/w_base); T5 = (1/R1d)*(L1d + Lad*Ll/(Lad + Ll))*(1/w_base); T6 = (1/R1d)*(L1d + Lad*Ll*Lfd/(Lad*Ll + Lad*Lfd + Ll*Lfd))*(1/w_base);

% ACCURATE EXPRESSION OF REACTANCES Xdp_acc = Xd*(T4 + T5)/(T1 + T2); Xdb_acc = Xd*(T4*T6)/(T1*T3);

% D-AXIS TRANSIENT OPEN-CIRCUIT TIME CONSTANT Td0p = T1 + T2;

% D-AXIS TRANSIENT TIME CONSTANT Tdp = T4 + T5;

% D-AXIS SUBTRANSIENT OPEN-CIRCUIT TIME CONSTANT Td0b = T3*T1/(T1+T2);

StandardParam.m

% D-AXIS SUBTRANSIENT TIME CONSTANT Tdb = T6*T4/(T4+T5);

% Q-AXIS SUBTRANSIENT OPEN-CIRCUIT TIME CONSTANT Tq0b = (Laq + L1q)/(R1q)*(1/w_base);

StandardParamTieLine.m

% FILE: StandardParamTieLine.m

% Created by Martin Ranlöf

% Calculate standard machine parameters for a conventional hydroelectric % generator.

% In the chosen p.u system, inductances are equal to the corresponding % reactances.

% REACTANCES % Saturated synchronous d-axis reactance Xd = Lad + Ll + XE; % Unsaturated synchronous d-axis reactance Xdu = Lad_u + Ll + XE; % Saturated synchronous q-axis reactance Xq = Laq + Ll + XE;

% d-axis mutual reactance Xad = Lad; % q-axis mutual reactance Xaq = Laq; % Transient d-axis reactance Xdp = XE + Ll + Lad*Lfd/(Lad + Lfd);

% Subtransient d-axis reactance Xdb = XE + Ll + Lad*Lfd*L1d/(Lad*Lfd + Lad*L1d + Lfd*L1d);

% Subtransient q-axis reactance Xqb = XE + Ll + Laq*L1q/(Laq + L1q);

% ARMATURE TIME CONSTANT Ta = (Xdb + Xqb)/(Ra+RE)*(1/2)*(1/w_base);

% D-AXIS TRANSFER FUNCTION TIME CONSTANTS T1 = (Lad + Lfd)/Rfd*(1/w_base); T2 = (Lad + L1d)/R1d*(1/w_base); T3 = (1/R1d)*(L1d + Lad*Lfd/(Lad + Lfd))*(1/w_base); T4 = (1/Rfd)*(Lfd + Lad*(XE + Ll)/(Lad + (XE + Ll)))*(1/w_base); T5 = (1/R1d)*(L1d + Lad*(XE + Ll)/(Lad + (XE + Ll)))*(1/w_base); T6 = (1/R1d)*(L1d + Lad*(XE + Ll)*Lfd/(Lad*(XE + Ll) + Lad*Lfd + (XE +

Ll)*Lfd))*(1/w_base);

% ACCURATE EXPRESSION OF REACTANCES Xdp = Xd*(T4 + T5)/(T1 + T2); Xdb = Xd*(T4*T6)/(T1*T3);

% D-AXIS TRANSIENT OPEN-CIRCUIT TIME CONSTANT Td0p = T1 + T2;

% D-AXIS TRANSIENT TIME CONSTANT Tdp = T4 + T5;

% D-AXIS SUBTRANSIENT OPEN-CIRCUIT TIME CONSTANT

StandardParamTieLine.m

Td0b = T3*T1/(T1+T2);

% D-AXIS SUBTRANSIENT TIME CONSTANT Tdb = T6*T4/(T4+T5);

% Q-AXIS SUBTRANSIENT OPEN-CIRCUIT TIME CONSTANT Tq0b = (Laq + L1q)/(R1q)*(1/w_base);

Utskrifter.m

% Utskrifter.m

% Skriver ut parametrar för den simulerade modellen och vissa % simuleringsresultat i ett EXCEL-ark.

% FUNDAMENTAL PARAMETERS wtf = {'Machine: 5', 'Studie 2';}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Fundamental Parameters','A1'); wtf = {'Inertia (H):', H}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Fundamental Parameters','A3'); wtf = {'d-axis',' ',' '; 'Ldu', Lad_u+Ll, 'p.u';'Ld', Xd, 'p.u.';'Lfd',Lfd,

'p.u'; ... 'L1d',L1d, 'p.u';'Ll',Ll,'p.u';}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Fundamental Parameters','A5'); wtf = {'q-axis',' ',' '; 'Lq', Xq, 'p.u';'L1q', L1q, 'p.u.';}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Fundamental Parameters','E5');

% STANDARD PARAMETERS wtf = {'Machine: 5', 'Studie 2';}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Standard Parameters','A1'); wtf = {'Inertia (H):', H,' ', 'Re', RE ,' ', 'Xe', XE;}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Standard Parameters','A3'); wtf = {'d-axis',' ',' '; 'Xd', Xd, 'p.u';'Xdp', Xdp, 'p.u.';'Xdb', Xdb, 'p.u';

... 'Xl', Ll, 'p.u';}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Standard Parameters','A5'); wtf = {'Td0p', Td0p, 's';'Tdp', Tdp, 's';'Td0b', Td0b, 's'; ... 'Tdb', Tdb, 's';}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Standard Parameters','A11'); wtf = {'q-axis',' ',' '; 'Xq', Xq, 'p.u';'Xqb', Xqb, 'p.u.';'','',''; ... '', '', ''; '', '', '';'Tq0b',Tq0b,'s';'Tqb',Tq0b*Xqb/Xq,'s';}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Standard Parameters','E5');

wtf = {'w_eig','Ks_anpassat','Ks_analytiskt','Kd_anpassat','Kd_analytiskt'... ,'Td_Dämptidskonstant','b_Dämpkapabilitet';}; xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Results','A1');

Utskrifter.m

% DÄMP OCH STYVHETSKONSTANTER (NOLL OM RESPEKTIVE SCRIPT EJ KÖRTS) wtf = {w_eig,Ks_anp,Ks_Park,Kd_anp,Kd_Park,Td,b;};

xlswrite('C:\skola\universitet\jobb\exjobb\dämpning\data\s2avr3\s2avr3Td2_0.xl

s',... wtf, 'Results','A2');

Appendix 3

example of AVR and PSS

Appendix 4

Pictures of the power cabinet that was built

[Picutre 1: shows the inside of the cabinet door]

[Picture 2: shows the outide of the cabinet door, the functions of the buttons is to connect and

disconnect the generator from the grid, the three different gauges are, from left, voltage meter,

ampere meter, and cosinus γ, to the left in the middle, is the status board, and finaly the missing

piece is where the synchronometer should be]

[Picture 3: shows the external synchrony scope that were build for safety reason, so a remote

connection to the grid could be performed]

[Picture 4: shows the interior of the cabinet, complete with measuring equipment, and safety

relays]

Appendix 5

Pictures of the generator and damper bars

[Picture 1: shows the generator that has been used during the experiments]

[Picture 2: shows the motor which acted as a turbine during the experiments, the shaft going

up in the right top corner is connected to the rotor]

[Picture 3: Shows the damper bars which were inserted into cut-out tracks in the rotor plate,

in this picture one can also see the bridges which electrical connects the bars]

Appendix 6

Results from the simulations, study 1, machine 1

[Figure 1: shows how Et, terminal voltage, Pf, power factor and St, power output changes the

synchronous damping, Et, Pf and St is represented in the Per Unit system]


damping constant, Et, Pf and St is represented in the Per Unit system]

0

0,5

1

1,5

2

2,5

3

0,6 0,7 0,8 0,9 1

Syn

ch

ron

ou

s d

am

pin

g

[Per Unit representation]

Ks

Et

Pf

St

0

0,005

0,01

0,015

0,02

0,025

0,03

0,035

0,04

0,045

0,05

0,6 0,7 0,8 0,9 1

dam

pin

g c

on

sta

nt


Kd

Et

Pf

St

[Figure 3: shows how the inertia constant H changes Ks]

[Figure 4: shows how the inertia constant H changes Kd]

2,35

2,4

2,45

2,5

2,55

2,6

2,65

2,7

2,75

2,8

2 3 4 5 6

Syn

ch

ron

ou

s d

am

pin

g


Ks

H

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

2 3 4 5 6

dam

pin

g c

on

sta

nt


Kd

H

[Figure 5: shows how the resistance in the tie-line changes Ks]

[Figure 6: shows how the resistance in the tie-line changes Kd]

0

0,5

1

1,5

2

2,5

3

0 0,05 0,1 0,15 0,2

Syn

ch

ron

ou

s d

am

pin

g


Ks

Re

0

0,005

0,01

0,015

0,02

0,025

0,03

0,035

0,04

0,045

0 0,05 0,1 0,15 0,2

dam

pin

g c

on

sta

nt


Kd

Re

[Figure 7: shows how the reactance in the tie-line changes Ks]

[Figure 8: shows how the resactance in the tie-line changes Kd]

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0 0,15 0,3 0,45 0,6

Syn

ch

ron

ou

s d

am

pin

g


Ks

Xe

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0 0,15 0,3 0,45 0,6

dam

pin

g c

on

sta

nt


Kd

Xe

[Figure 9: shows how the torque reacts when a disturbance is made, at time t1 = 1 sec, the red color is

et 0.6, green 0.7 blue 0.8, magenta 0.9, yellow 1.0]

[Figure 10: shows how the angle reacts when a disturbance is made, at time t1 = 1 sec, the red color is


[Figure 11: shows how the angle velocity is changed when a disturbance is made, at time t1 = 1 sec,

the red color is et 0.6, green 0.7 blue 0.8, magenta 0.9, yellow 1.0]

[Figure 12: shows how Ks and Kd changes when a disturbance is made, at time t1 = 1 sec, the red

color is et 0.6, green 0.7 blue 0.8, magenta 0.9, yellow 1.0, the black is the new mathematic model]

Appendix 7






0

0,5

1

1,5

2

2,5

3

3,5

4

0,6 0,7 0,8 0,9 1

Syn

ch

ron

ou

s d

am

pin

g


Ks

Et

Pf

St

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,6 0,7 0,8 0,9 1

dam

pin

g c

on

sta

nt


Kd

Et

Pf

St



2,7

2,8

2,9

3

3,1

3,2

3,3

3,4

2 3 4 5 6

Syn

ch

ron

ou

s d

am

pin

g


Ks

H

0

0,02

0,04

0,06

0,08

0,1

0,12

2 3 4 5 6

dam

pin

g c

on

sta

nt


Kd

H



0

0,5

1

1,5

2

2,5

3

3,5

0 0,05 0,1 0,15 0,2

Syn

ch

ron

ou

s d

am

pin

g


Ks

Re

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0 0,05 0,1 0,15 0,2

dam

pin

g c

on

sta

nt


Kd

Re


[Figure 8: shows how the reactance in the tie-line changes Kd]

0

1

2

3

4

5

6

0 0,15 0,3 0,45 0,6

Syn

ch

ron

ou

s d

am

pin

g


Ks

Xe

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0 0,15 0,3 0,45 0,6

dam

pin

g c

on

sta

nt


Kd

Xe

Appendix 8






0

0,5

1

1,5

2

2,5

3

0,6 0,7 0,8 0,9 1

Syn

ch

ron

ou

s d

am

pin

g


Ks

Et

Pf

St

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,6 0,7 0,8 0,9 1

dam

pin

g c

on

sta

nt


Kd

Et

Pf

St



2,25

2,3

2,35

2,4

2,45

2,5

2,55

2,6

2,65

2,7

2,75

2 3 4 5 6

Syn

ch

ron

ou

s d

am

pin

g


Ks

H

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

2 3 4 5 6

dam

pin

g c

on

sta

nt


Kd

H



0

0,5

1

1,5

2

2,5

3

0 0,05 0,1 0,15 0,2

Syn

ch

ron

ou

s d

am

pin

g


Ks

Re

0

0,01

0,02

0,03

0,04

0,05

0,06

0 0,05 0,1 0,15 0,2

dam

pin

g c

on

sta

nt


Kd

Re



0

0,5

1

1,5

2

2,5

3

3,5

4

0 0,15 0,3 0,45 0,6

Syn

ch

ron

ou

s d

am

pin

g


Ks

Xe

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1

0 0,15 0,3 0,45 0,6

dam

pin

g c

on

sta

nt


Kd

Xe

Appendix 9






0

0,5

1

1,5

2

2,5

3

3,5

4

0,6 0,7 0,8 0,9 1

Syn

ch

ron

ou

s d

am

pin

g


Ks

Et

Pf

St

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0,09

0,1

0,6 0,7 0,8 0,9 1

dam

pin

g c

on

sta

nt


Kd

Et

Pf

St



2,7

2,8

2,9

3

3,1

3,2

3,3

3,4

2 3 4 5 6

Syn

ch

ron

ou

s d

am

pin

g


Ks

H

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

2 3 4 5 6

dam

pin

g c

on

sta

nt


Kd

H



0

0,5

1

1,5

2

2,5

3

3,5

0 0,05 0,1 0,15 0,2

Syn

ch

ron

ou

s d

am

pin

g


Ks

Re

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

0 0,05 0,1 0,15 0,2

dam

pin

g c

on

sta

nt


Kd

Re



0

1

2

3

4

5

6

0 0,15 0,3 0,45 0,6

Syn

ch

ron

ou

s d

am

pin

g


Ks

Xe

0

0,02

0,04

0,06

0,08

0,1

0,12

0,14

0,16

0 0,15 0,3 0,45 0,6

dam

pin

g c

on

sta

nt


Kd

Xe

Appendix 10

Results from the simulations, study 2, AVR 1 (simple gain)

[Figure 1: Shows how Ks difference with a changing Ka (gain)]

[Figure 2: Shows how Kd difference with a changing Ka (gain)]

2,04

2,06

2,08

2,1

2,12

2,14

2,16

2,18

2,2

2,22

2,24

10 50 100 500 1000

Syn

ch

ron

ou

s d

am

pin

g

Ka [Per Unit representation]

Ks

Ka

-0,004

-0,003

-0,002

-0,001

0

0,001

0,002

10 50 100 500 1000

dam

pin

g c

on

sta

nt

Ka [Per Unit representation]

Kd

Ka

Appendix 11

Results from the simulations, study 2, AVR 2 PD-regulator (gain and derivative) on machine 5


[Figure 2: shows how Kd (damping constant) change with Ka (gain) and Kd (derivative part of

regulator, not damping constant)]

1,6

1,65

1,7

1,75

1,8

1,85

1,9

1,95

2

2,05

2,1

2,15

0,4 1 5 50 200

Syn

ch

ron

ou

s d

am

pin

g

Kd

Ks with different Ka and Kd

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

-0,006

-0,004

-0,002

0

0,002

0,004

0,006

0,008

0,4 1 5 50 200

Dam

pin

g c

on

sta

nt

Kd

Kd with different Ka and Kd

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

[Figure 3: shows how Ks change with Ka (gain) and Td (foresight of the time step,)]

[Figure 4: shows how Kd (damping constant) change with Ka (gain) and Td (foresight of the time

step)]

1,7

1,75

1,8

1,85

1,9

1,95

2

2,05

2,1

2,15

0,005 0,05 0,1 0,5 1

Syn

ch

ron

ou

s d

am

pin

g

Td

Ks with different Ka and Td

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

-0,004

-0,003

-0,002

-0,001

0

0,001

0,002

0,003

0,005 0,05 0,1 0,5 1

Dam

pin

g c

on

sta

nt

Td

Kd with different Ka and Td

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

[Figure 5: shows how Ks change with Kd (derivative part of regulator, not damping constant) and Td

(foresight of the time step)]

[Figure 6: shows how Kd (damping constant) change with Kd (derivative part of regulator, not

damping constant) and Td (foresight of the time step)]

1,75

1,8

1,85

1,9

1,95

2

2,05

2,1

2,15

0,005 0,05 0,1 0,5 1

Syn

ch

ron

ou

s d

am

pin

g

Td

Ks with different Kd and Td

Kd 0.4

Kd 1

Kd 5

Kd 50

Kd 200

-0,004

-0,003

-0,002

-0,001

0

0,001

0,002

0,003

0,005 0,05 0,1 0,5 1

Dam

pin

g c

on

sta

nt

Td

Kd with different Kd and Td

Kd 0.4

Kd 1

Kd 5

Kd 50

Kd 200

[Figure 25: shows how the torque reacts when a disturbance is made , at time t1 = 1 sec, Ka is 150

and the red color is Kd 0.4, green 1.0 blue 5.0, magenta 50.0, yellow 200.0]

[Figure 26: shows how the angle reacts when a disturbance is made, at time t1 = 1 sec, Ka is 150 and

the red color is Kd 0.4, green 1.0 blue 5.0, magenta 50.0, yellow 200.0]


Ka is 150 and the red color is Kd 0.4, green 1.0 blue 5.0, magenta 50.0, yellow 200.0]

Appendix 12

Results from the simulations, study 2, AVR 3 PID-regulator (gain integrating and derivative) on

machine 5


[Figure 2: shows how Kd (damping constant) change with Ka (gain) and Kd (derivative part of

regulator, not damping constant)]

1,65

1,7

1,75

1,8

1,85

1,9

1,95

2

2,05

2,1

2,15

0,4 1 5 50 200

Syn

ch

ron

ou

s c

on

sta

nt

Kd

Ks with different Ka and Kd

Ka5

Ka 50

Ka 100

Ka 150

Ka 200

-0,0015

-0,001

-0,0005

0

0,0005

0,001

0,0015

0,002

0,0025

0,4 1 5 50 200

Dam

pin

g c

on

sta

nt

Kd

Kd with different Ka and Kd

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

[Figure 3: shows how Ks change with Ka (gain) and Td (foresight of the time step,)]

[Figure 4: shows how Kd (damping constant) change with Ka (gain) and Td (foresight of the time

step)]

1,6

1,7

1,8

1,9

2

2,1

2,2

0,005 0,05 0,1 0,5 1

Syn

ch

ron

ou

s d

am

pin

g

Td

Ks with different Ka and Td

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

-0,0025

-0,002

-0,0015

-0,001

-0,0005

0

0,0005

0,001

0,0015

0,002

0,0025

0,005 0,05 0,1 0,5 1

Dam

pin

g c

on

sta

nt

Td

Kd with different Kd and Ki

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

[Figure 5: shows how Ks change with Kd (derivative part of regulator, not damping constant) and Td

(foresight of the time step)]

[Figure 6: shows how Kd (damping constant) change with Kd (derivative part of regulator, not

damping constant) and Td (foresight of the time step)]

1,65

1,7

1,75

1,8

1,85

1,9

1,95

2

2,05

2,1

2,15

0,005 0,05 0,1 0,5 1

Syn

ch

ron

ou

s d

am

pin

g

Td

Ks with different Kd and Td

Kd 0.4

Kd 1

Kd 5

Kd 50

Kd 200

-0,0025

-0,002

-0,0015

-0,001

-0,0005

0

0,0005

0,001

0,0015

0,002

0,0025

0,005 0,05 0,1 0,5 1

Dam

pin

g c

on

sta

nt

Td

Kd with different Kd and Td

Kd 0.4

Kd 1

Kd 5

Kd 50

Kd 200

[Figure 7: shows how Ks change with Ka (gain) and Ki (integrating part of regulator)]

[Figure 8: shows how Kd (damping constant) change with Ka (gain) and Ki (integrating part of

regulator)]

0

0,5

1

1,5

2

2,5

0,005 0,1 0,4 1 2

Syn

ch

ron

ou

s c

on

sta

nt

Ki

Ks with different Ka and Ki

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

-0,005

-0,004

-0,003

-0,002

-0,001

0

0,001

0,002

0,003

0,005 0,1 0,4 1 2

Dam

pin

g c

on

sta

nt

Ki

Kd with different Ka and Ki

Ka 5

Ka 50

Ka 100

Ka 150

Ka 200

[Figure 9: shows how Ks change with Kd (derivative part of regulator, not damping constant) and Ki

(integrating part of regulator)]

[Figure 10: shows how Kd (damping constant) change with change with Kd (derivative part of

regulator, not damping constant) and Ki (integrating part of regulator)]

0

0,5

1

1,5

2

2,5

0,005 0,1 0,4 1 2

Syn

ch

ron

ou

s d

am

pin

g

Ki

Ks with different Kd and Ki

Kd 0.4

Kd 1

Kd 5

Kd 50

Kd 200

-0,016

-0,014

-0,012

-0,01

-0,008

-0,006

-0,004

-0,002

0

0,002

0,004

0,005 0,1 0,4 1 2

Dam

pin

g c

on

sta

nt

Ki

Kd with different Kd and Ki

Kd 0.4

Kd 1

Kd 5

Kd 50

Kd 200

[Figure 11: shows how Ks change with Ki (integrating part of regulator) and Td (foresight of the time

step)]

[Figure 12: shows how Kd (damping constant) change with change with Ki (integrating part of

regulator) and Td (foresight of the time step)]

0

0,5

1

1,5

2

2,5

0,005 0,05 0,1 0,5 1

Syn

ch

ron

ou

s d

am

pin

g

Td

Ks with different Ki and Td

Ki 0.005

Ki 0.1

Ki 0.4

Ki 1

Ki 2

-0,004

-0,003

-0,002

-0,001

0

0,001

0,002

0,005 0,05 0,1 0,5 1

Dam

pin

g c

on

sta

nt

Td

Kd with different Ki and Td

Ki 0.005

Ki 0.1

Ki 0.4

Ki 1

Ki 2

[Figure 28: shows how the torque reacts when a disturbance is made , at time t1 = 1 sec, Ka is 100

and the red color is Ki 0.005, green 0.1 blue 0.4, magenta 1.0, yellow 2.0]

[Figure 29: shows how the angle reacts when a disturbance is made, at time t1 = 1 sec, Ka is 100 and

the red color is Ki 0.005, green 0.1 blue 0.4, magenta 1.0, yellow 2.0]


Ka is 100 and the red color is Ki 0.005, green 0.1 blue 0.4, magenta 1.0, yellow 2.0]

Appendix 13

Results from study I, machine I. A table of Ks and Kd with the new mathematical model compared to

the old model. Where Ks_anpassat and Kd_anpassat are the new model, and Ks_analytisk and

Kd_analytiskt are representing the old model.

Ks_anpassat Ks_analytiskt Kd_anpassat Kd_analytiskt Körning

0,996268302 1,116151156 0,016978831 0,017695956 Et 0.6

1,352818503 1,476193791 0,023222525 0,022907182 Et 0.7

1,740861828 1,855373919 0,029833435 0,028226983 Et 0.8

2,168906523 2,273091847 0,036115766 0,033192919 Et 0.9

2,65082159 2,739593392 0,042191845 0,037582786 Et 1.0

2,744636288 2,832294361 0,031098819 0,027578639 H 2

2,658783888 2,747245131 0,041304361 0,036749573 H 3

2,590975196 2,68113536 0,048985668 0,043997411 H 4

2,536554142 2,628579763 0,055173757 0,049849052 H 5

2,491986348 2,586008788 0,059805368 0,054662881 H 6

2,749333151 2,954289278 0,038768315 0,036410316 Pf 0.6

2,72622353 2,904105799 0,039605372 0,036692655 Pf 0.7

2,695330015 2,83675462 0,040672288 0,037055959 Pf 0.8

2,65082159 2,739593392 0,042191845 0,037582786 Pf 0.9

2,518640745 2,478813345 0,046811172 0,039229989 Pf 1.0

2,655154846 2,716055177 0,042331676 0,037568637 Re 0.00

2,581645439 2,784831166 0,03996609 0,036280503 Re 0.05

2,394522836 2,733531706 0,034127234 0,032010315 Re 0.10

2,134158116 2,58263922 0,026402754 0,025795509 Re 0.15

1,840732227 2,365966081 0,018458258 0,018745636 Re 0.20

2,516962697 2,531226276 0,047538549 0,040677842 St 0.6

2,550084091 2,584269478 0,04618273 0,039884244 St 0.7

2,583648924 2,636959755 0,044824707 0,039098331 St 0.8

2,617160485 2,688858181 0,043492991 0,038328943 St 0.9

2,65082159 2,739593392 0,042191845 0,037582786 St 1.0

3,821085905 3,723538726 0,06915419 0,055088019 Xe 0.00

2,273634248 2,403246228 0,034260279 0,0318479 Xe 0.15

1,537054144 1,697206602 0,020421953 0,02058055 Xe 0.30

1,098620728 1,230941914 0,013190483 0,014075373 Xe 0.45

0,808892938 0,890204499 0,009688461 0,010039409 Xe 0.60

Influence of damping winding, controller settings and exciter on …398778/... · 2011. 2. 18. ·...

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