power system dynamics prof. m l kothari department...

Power System Dynamics

Prof. M L Kothari

Department of Electrical Engineering

Indian Institute of Technology, Delhi

Lecture - 16

Modelling of Excitation Systems

Friends, in this lecture we shall discuss about modeling of excitation systems.

(Refer Slide Time: 01:07)

While talking about the modeling we will study non reciprocal per unit system then we will talk

about the modeling of excitation system components because any excitation system has several

components therefore first we will discuss about modeling of the components of the excitation

system one by one. The components which we will be model, we will start with the separately

excited dc exciter, self-exciter, dc exciter, ac exciters and rectifiers as we know that in the ac

excitation system the output of the ac exciter is rectified and supplied to the field winding of

synchronous generator and therefore there is necessary to model the three phase rectifiers which

are commonly used.

The other components which require to be model are the amplifiers, excitation system stabilizing

circuit, wind up and non-wind up limits. As we will see that there exist certain limits that the

output of say AVR cannot actually certain limit like this therefore, we will have certain limits

which are imposed and they are two different categories of limits wind up and non-wind up

limits. Then to implement the under excitation limit and the over excitation limit you will have

gaiting functions and terminal voltage traducer. Over the years a variety of excitation system

models have been developed. We will confine our discussion to a few typical excitation system

models which have been documented by IEEE in the standards published by IEEE.



We will discuss the excitation models type DC1A exciter model, type AC1A exciter model, type

ST1A exciter model, type ST A exciter model. These are the 4 models which we will discuss

however, number of models are available in the IEEE standards. The over the years the IEEE has

developed the excitation system models for power system stability studies. The 4 important

references references which cover the excitation system models are the first report that is first

IEEE committee report.



This was published in the year 1968 and the IEEE transaction on power apparatus systems then

subsequently subsequently the the models were upgraded taking care of the new developments

which have taken place then the next IEEE committee report was published in the year 1981 and

when you look into the this IEEE committee report, you will see that they have 3 different

models for DC excitation systems, 4 different models for ac excitation systems and 3 different

models for static excitation systems. Therefore, in all they have discussed 10 different models

excitation systems. Now these different models were needed to cope up with the variety of

excitation systems manufactured by different manufacturers and existing in the power system

today.

Recently in 1996 another, you know a committee report came in where they have discussed

specifically the digital based excitation systems because today the excitation systems have micro

process based controls and with this digital controls right there was a necessity to model the

excitation systems and give standard excitation system models. Then one very comprehensive

standard which is IEEE standard 425.5 published in 1992, it discusses in detail the different

models which have been developed the material which we will be presenting here is drawn from

these 4 references.

Now here when we talk about the excitation system models, we have to develop the per unit

system for excitation systems. We have developed earlier the per unit system for synchronous

generator models and while developing the synchronous generator model in order to simplify the

synchronous generator equations, we had made we had made certain assumptions and the model

developed is normally known as the reciprocal model right.

Now if we use this reciprocal model there are some problems particularly if we use the reciprocal

model then then the per unit values of the exciter output voltage becomes very low. The typical

value is which is mentioned is that it may be as low as .001 per unit, if we use the same same per

unit system and therefore the another per unit system is proposed for the modeling the excitation

systems and that is now called as non-reciprocal per unit system.



In this non-reciprocal per unit system the the definition goes like this 1 per unit exciter output

voltage is required to produce rated synchronous machine armature terminal voltage on air gap

line as the definition goes that one per unit exciter output voltage is required to produce produce

the rated terminal voltages of the synchronous machine on the air gap line not on that saturation

curve but on the air gap line. Now here we define the one per unit exciter output current is the

corresponding current corresponding current which flows in the field winding of the synchronous

generator that is when you apply certain voltage at the terminal of the synchronous generator

which produces the the rated voltage at the terminal of the synchronous generator right.

Then at that time whatsoever is the current which flows in the field winding, we call that current

as 1 per unit I will just take the example suppose you have a synchronous generator whose

terminal voltage is say 11KV okay now to produce the 11 KV voltage on the air gap line if you

require say 600 volts to be applied to the synchronous generator field winding right then 600

volts will be considered as 1 per unit volts per unit voltage of the excitation system.

Similarly, suppose at that 600 volts applied the field current comes out to be say 200 amperes

right then this 200 ampere will be considered as 1 per unit field current or 1 per unit excitation

current excitation system current. Okay now with this definition, now there is a necessity to

establish the relationship between the reciprocal per unit system with the non-reciprocal per unit

system. We know that the excitation system is is interfaced with the synchronous generator both

at the field winding as well as at the synchronous machine terminals that is for controlling the

output of the excitation system, the input control signals are derived from the terminals of the

synchronous generator that is you change the terminal voltage you use actually the load

compensator and obtain a voltage VC, okay this is obtained from the terminally synchronous

generator.

Similarly you may sense the speed of the synchronous generator, okay or you sense the power

output of the synchronous generator all these things right real power, reactive power all these

quantities are fed to the AVR of the excitation system right. Therefore, so far the input to the

excitation system is concerned is obtained from terminal of synchronous generator while the

output of the excitation system is fed to the field winding of the synchronous generator therefore,

excitation system is interfaced interfaced at if with the field winding of the synchronous

generator and the terminal of the synchronous generator right and therefore we have to have the

the arrangements. So that these 2 different forms of per unit systems are properly interfaced.


Now in order to achieve this interfacing we first look at the basic open circuit equations that is

even the synchronous generator is open circuited right. We have the equations that ed, ed is equal

to 0 and eq equal to psi d which is equal to Lad into ifd. Now this equation these two equations are

the basic stator winding equations in terms of dq axis components okay.

Now we we know also that under open circuit condition this voltage or this quadrature axis

component is also equal to the terminal voltage of the synchronous generator. This was already

established, okay therefore now if I plot a characteristic relating eq or et as a function of ifd that is

the field current right then the characteristic will look like this. This is open circuit characteristic

of the synchronous generator. On this axis we have marked the field current in per unit ifd. Now

this ifd when I use this symbol this is actually the symbol used in the reciprocal system of per

units or per unit reciprocal system and we represent the per unit terminal voltage on the y axis.

Now this is the open circuit characteristic if you draw a transient to the characteristic passing

through origin, okay that is the initial portion then this is your air gap line right.

The slope of this air gap line slope of this air gap line is the mutual inductance unsaturated Ladu

that is the mutual inductance unsaturated that is the slope of this characteristic right. Therefore

now to produce on this axis I have put this ah current in per unit of ifd that is reciprocal system

and for producing 1 per unit voltage here, the current required will be current required will be

that is ifd required will be equal to that is to produce et equal to 1 that is seem same as eq right

therefore, ifd required will be 1 upon Lad and since we are considering here the air gap line we are

considering the unsaturated value of mutual inductance and therefore the the current required to

produce 1 per unit terminal voltage in reciprocal system of units right is one upon Ladu.



Now this current will be denoted as 1 per unit in the non-reciprocal system of units that is same

quantity. Okay will be said as 1 per unit okay therefore, using this information we can establish

relationship between the between the ah non-reciprocal per unit system and reciprocal per unit

system because we have seen actually that the ifd required to produce one per unit terminal

voltage is one upon Ladu right and since this should be equal to 1 per unit in the non-reciprocal

system and therefore the relationship which is established between established between the non-

reciprocal system of unit.



We call the current in non-reciprocal we will denote by ifd capital. Okay in the reciprocal system

we have been using this symbol ifd therefore, the ifd will be equal to Ladu ifd this you can check

yourself that the relationship between between the reciprocal per unit system of field of the field

current and non- reciprocal per unit system of the field current they are related by this quantity

Ladu.


Okay, that is similarly when we talk about the field voltage okay the efd the field voltage efd is

equal to Rafd ifd okay and I replace this ifd by one upon Ladu. So that the efd efd if applied it

becomes Rfd upon Ladu in per unit right and this is denoted as 1 per unit, so far actually the non-

reciprocal per unit system is concerned and hence the relationships which we establish are like

this that a non-reciprocal per unit system, the field voltage efd is equal to Ladu divided by Rfd efd in

the reciprocal system of unit therefore, these two equations that is equation 16.5 and 16.6 relate

relate the field current and field voltage in reciprocal system of per unit reciprocal system of

units to non-reciprocal per unit systems.

Okay now here I have shown in the form of block diagram, exciter model non-reciprocal per unit

system right. The we have a exciter model now the quantities will be Efd and Ifd. Okay

synchronous machine model reciprocal per unit system the quantities at the terminals of the field

winding are efd and ifd , okay the relationship between these 2 quantities that is efd is equal to Rfd

upon Ladu Efd and ifd is equal to Ifd upon Ladu right therefore whenever we develop the complete

model of the system right considering the synchronous generator and the excitation system right

the excitation system first will be modelled considering the non-reciprocal per unit system and

then it will be interfaced with the synchronous machine reciprocal per unit system using these

relations right.



Now now, we will devote our time for discussing the models of 3 basic component of the

excitation system that is separately excited dc exciter, self excited dc exciter, ac exciters and

rectifiers. Now as I have told you earlier that although the dc excitation systems right have have

been superseded by ac excitation systems or static excitation systems but still in the power

systems existing today there are number of dc excitation systems operating number one, second

point is that the when we develop the basic model for separately excited dc exciter the model for

ac exciters is also similar to that similar to that with some modifications. Okay therefore, we let

us start with developing the model for separately excited dc exciter.


Okay, and in order to develop the model for the separately excited dc exciter. You look at the dc

exciter this is the this is the armature of the dc exciter. Okay it is a dc exciter is a its field

winding okay, this is the field winding. We represent the field winding by a resistance Ref and

inductance Lef that is this field winding of the dc exciter right has resistance and inductance right.

Now the field winding is generally very highly inductive right and therefore, the inductance

plays very significant role in the modeling of excitation system.

Okay therefore we will start like this. Let us say that the voltage applied to the applied to the

terminal of the field winding is Eef and the current flowing is Ief okay. Now here the output

voltage of the exciter right we call this is as a Ex this is denoted by the symbol Ex okay. Now this

Ex is related to Iex okay and the relationship is a non-linear relationship. We all know actually

that I you plot the open circuit characteristic of a dc generator or separately excited dc generator

it comes out to be a non-linear characteristic saturation exist in the system. Further when this dc

exciter is loaded loaded right due to armature reaction the voltage at the excited terminals will

further drop right therefore, while modeling the dc exciter we may have to account for two

things, one is the loading effect another is the saturation effect and in fact while modeling the dc

exciter, we combine both the effects okay.


Now the basic equation for the field circuit is written as the Efd this will be slight a mistake is

there Efe. We will denote the symbol by Eef a mistake here. You make it Eef Eef is equal to Ref

into Ief plus d psi by dt where psi is the flux linkage which can be written as Lef into Ief this is the

basic circuit equation of the field winding of dc exciter.


The output voltage of the exciter is proportional to field flux linkage psi and Kx is the

proportionality constant that is output voltage at the terminal of the synchronous generator output

of the exciter dc exciter not synchronous generator is directly related to the flux linkage of the

field winding and is directly proportional okay.

Now this constant Kx depends upon the speed of the exciter it also depends upon the design of

the exciter and other parameters. Now here this is a most important point to understand about

modeling of dc exciter or separately excited dc exciter


If you plot the open circuit characteristic relating the Ex to Ief right then this open circuit

characteristic is shown here. Okay and the when we draw the transient to this we get air gap line

right and the slope of this air gap line, if you find out the slope of this air gap line right then it is

because this is a steady state characteristic, this is a steady state characteristic what will be the

slope of this? Resistance resistance of the field winding okay and since the we are considering

the air gap line we will consider this resistance as Rg shown here. Now if suppose for a certain

output Exo right, if I try to find out what will be the field current required under the open circuit

condition then the field current will be equal to the field current required corresponding to the air

gap line plus some additional current to account for the saturation.

Now in order to account for the loading effect right what we do is that we plot another saturation

characteristic and that characteristic is shown here. This saturation characteristic is the constant

resistance load saturation curve. You have to understand very carefully what we mean by the

constant resistance load saturation curve. Now this can be explained like this you have your dc

generator armature across this you put a resistance, okay and this is the field winding call this

current as Ief.

Okay and run this machine as rated speed and now if you plot the characteristic relating the

terminal voltage Ex with respect to Ief. This characteristic will take care of two aspects saturation

as well as the ah armature reaction effect and this characteristic is called, this characteristic is

called constant resistance load saturation curve.



Okay now for a given value of Ex the field current which is required will be now written as the

field current corresponding to the air gap line plus this additional term delta Ief this delta Ief this is

additional quantity right this takes care of saturation as well as the loading effect and therefore,

now we can say that for any operating condition any operating condition the Ief will be equal to

Ex divided by Rg plus delta Ief delta Ief.


This additional current which is required to account for the saturation and the loading effect right

is function of is function of the voltage Ex which is produced and a non-linear function of Ex that

is if you, if I try to quantify what is the value of Ief required then we can easily see here actually

in this graph that this Ief is different for different values of Ex right. Therefore, we can say that

the Ief required is is equal to the voltage Ex into a non-linear function Se Ex and this Se Ex is

called the saturation function which is dependent on the voltage Ex.

In fact this has to be obtained graphically this value whenever you want you can say develop the

model for a given dc exciter this has to be obtained experimentally. Now what we do is we have

developed basic equation starting from the field circuit equation that is Eef is equal to Ref into Ief

plus plus d psi by dt, where where psi is the flux linkage.

Now what we do is that in this expression in this the first basic expression, in this equation that is

Eef equal to Ref Ief d psi by dt, you substitute the value of Ief substitute the value of psi okay and

we will get after making this substitution an equation of this form that is Eef is equal to Ref upon

Rg Ex plus Ref into Se Ex into Ex plus 1 by Kx dEx by dt that is when we have obtained this

equation, we have made use of the subsequent relations which were derived earlier.



Now in this equation, we can see is here that the field current is eliminated there is no field

current term does not appear directly. Okay and there is no flux linkages also everything is

expressed in terms of the saturation function the field resistance of field winding and this

constant Kx. Okay now this equation is converted into per unit system of equations therefore,

what we need is the base voltage for this that is Ief, we require base voltage and the base voltage

is chosen like this that is Ex base is same as the Efd base that is the field voltage which we apply

right that base value is the base for Ex and Ief base is same as Efd base divided by Rg that is the

base value of the field current as we already seen that this voltage divided by the by the air by the

air gap by the slope of the air gap line that is resistance Rg and the resistance base is equal to Rg

itself. Okay now when you when you convert this equation into per unit system of equations that

is you divide that you divide this equation by Efd base the throughout and then you can write

down the equation in the form Eef bar bar is stands for per unit and this conversion is right. Now

it is actually the non-reciprocal per unit system.



Okay now when writing this equation, okay some new terms are introduced here which are

defined as follows that is Se bar Ex bar is defined here as Se bar Ex bar is defined as delta Ief bar

divided by Ex bar is a straight forward you know definitions which you can derive and I will

suggest you to derive these things yourself okay.


Now here here again I am showing the constant resistance load saturation curve of the dc exciter

now on this axis now instead putting putting the field current in reciprocal per unit system I am

put this is field current in non-reciprocal per unit system right PU Ief and this is per unit Ex. Okay

therefore, now you can see in this equation that for any value of Ex right the field current

required will be so much right. Now I call this quantity as A and the field current required

corresponding to the air gap line that is the, if we neglect the saturation and loading effect field

current required is denoted by B okay then this saturation function which we have defined can be

written as simply A minus B divided by B.

Okay and by the basic definition of Kx the Kx can be expressed again in terms of the per unit

output voltage of the exciter and the per unit field current. Okay now here we have a term that is

Kx is equal to Rg divided by Lef Ex bar Ief bar. Okay therefore what we do is that we define

another ah term which we call as Lfu is equal to Lef Iefo Exo bar that is for a given operating

condition right you can denote this term as Lfu that is this Lef Ief divided by Ex, you can call this

term as Lfu.



Now if you denote this term by Lfu then our equation will reduce in a standard form that is Eef bar

equal to Ke Ex bar Se Ex bar Ex plus Te d Ex bar upon dt that is you can see this equation here that

you have all per unit voltages, saturation function which we as I have told you that can be

computed from the constant load resistance ah saturation curve okay.

Now once we have come to this level we can write this expression in the form of a transfer

function model because when I say that I want to develop the excitation system model then we

can express this model this is now the excitation system model of a dc exciter. Okay now this

can be put in a compact form in the transfer function model that is what you do is that you take

the Laplace transform of all these quantities right and when you express this in the transfer

function model form it looks like this.



We have seen you just um look at this equation if you look at this equation what is to be done is

at one summing point you have the term Ke Ex to this Ke Ex, you add this term these are the two

algebraic terms, okay you add these two terms subtract from Ief that is Eef okay and when this

whole quantity is integrated you will get Ex right. Therefore this model come something like this

in the form here is Ex therefore you this is the gain KE therefore output from this block will be KE

into Ex output from this block will be a saturation quantity that is Vx which is Ex into Se Ex okay

these two are added and subtracted from Eef.



Now whatsoever comes is multiplied by this transfer function you get the Ex therefore the

transfer function model of a separately excited dc exciter is given here these quantities all these

quantities are expressed KE of TE KE and the saturation function right they can be easily

computed from the relations which have been derived. For example Ke is equal to Ref by Rg TE is

equal to Lfu by Rg and SE Ex bar is equal to Se bar Ex bar Ref by Rg right. Therefore these

quantities are all known to us and these are obtained from the parameters of the dc exciter and

the constant resistance saturation curve.

Now this block diagram which I have just shown can be further simplified and put in the form of

a first order transfer function because here we have seen actually it is a first order transfer

function therefore, the whole thing can be simplified and you can make this model as a small

perturbation model is only small change in the field voltage applied to the exciter and small

change in the output voltage delta Ex right can be written in the form of K upon 1 plus s times T

because we are whenever we represent any system right we always try to develop a model with

input output quantities okay and the the transfer function models are always always linear

models non-linear models actually we do not have anything like transfer functions okay.

Now the model which I have developed here will be applicable for one operating condition for

once you take one particular operating condition and take small perturbations around that then a

small perturbation model is put in the form of a transfer function model right these quantities K

and ST are given by these equations.


Here, K is equal to 1 upon BEX SE EFDo plus KE, I will just explain this term Bx separately and T

is expressed as TE divided by BEX SE EFDo plus KE, where the SE Ex of the saturation curve you

know this is the non-linear saturation ah function this is modelled by this exponential function a

constant AEX e to the power BEX EFDo at any operating condition at any operating condition right

we can find out the value of SE E Exo either graphically or if you have modelled this saturation

function by this this exponential function where these terms will be known to you right then that

is why actually the the time the constant K and this T are expressed in terms of the the

parameters of saturation function and all other parameters like KE is there TE is there right. Now

next point is that we will develop the model for a self-excited dc exciter okay.





Now the self-excited dc exciter will like this you have synchronous I am sorry, the armature of

the dc exciter, this is the field winding of the dc exciter. We will have some regulating rheostat in

series with this field winding right and the output of the AVR, output of the AVR will be

connected in series with the field circuit and put across the armature that is in case suppose if it is

not a regulated one right then we will connect this field winding directly across the armature but

here what we do is that in series with the field winding we connect the output of the regulator

right. So that here here we can write the Eef equal to VR plus ExVR plus Ex that is in case it was a

separately excited exciter Eef will be as same as VR but now Eef is going to VR plus Exn. Now with

this change this is the only change with this change one can develop the complete model of the

self-excited dc exciter the that is in the equation which we had derived earlier that is 16.22 we

substituted substituting equation 16.22 in equation 16.13 right that is wherever we had this Eef

we are putting VR plus Ex other things are same because that is at the only change actually in the

circuit.



Okay when this model is simplified simplified you will find actually that the same model which

we develop for separately excited dc exciter is applicable except the definition of KE now

becomes Ref divided by Rg minus one this is only difference other things are exactly same that is

this model see this model this was the model developed for a separately separately excited dc

exciter right therefore, the KE which was defined is now modified for a self-excited dc exciter

and the value of KE comes out to be Ref by Rg minus 1 and other definitions are same.


Okay now with these definitions, okay we have completed the model for dc excitation system

that is self-excited as well as and what we see here is that it comes out to be a first order model

the model is a first order model. Okay now when we go for ac excitation system, ac excitation

system the in ac excitation system we have the main source of field power is ac generator, an

alternator. The output of this alternator is rectified with the help of 3 phase full wave bridge

rectifier now this 3 phase full wave bridge rectifier may be controlled or uncontrolled.

Okay now to in order to develop this model model ah we account for the demagnetizing effect or

the armature reaction effect separately in the case dc exciter. We have accounted for the armature

reaction and the saturation together while in the ac exciter the practice is to separately account

for armature reaction and and the characteristic of the ac exciter which is plotted under no load

condition there is open circuit characteristic plotted under no load condition and the open circuit

characteristic right because open circuit characteristic always under no load condition right. The

open circuit characteristic is used here and the saturation is defined making use of the open

circuit characteristic that is you plot open circuit characteristic on this axis I am putting the per

unit exciter field current and it is the per unit VE. Now this VE is is not the voltage which is

applied to the field winding of the synchronous generator but VE is the output of the exciter okay

and the saturation function which is required here in this model the saturation function is

obtained from the open circuit characteristic. Okay now to account for the armature reaction

effect we have one more block here where this the actual field current Ifd is multiplied with the

multiplying factor KD which is called demagnetizing factor and that is KD into Ifd is added at

this point therefore.



If we see the complete model the difference between the dc exciter and ac exciter is one is this

term that is we are putting additional term to account for the demagnetizing effect, second is that

this saturation function is obtained from OCC right and third thing is that the output here is the

terminal voltage of the exciter not the voltage applied to the field winding because this voltage

which is the output of the exciter is rectified and fed to the field winding therefore there is a

rectifier in between, this is all main difference between the dc exciter and ac exciter.


Now as I have told you that the this is saturation is obtained by this formula AB minus B. The

output of the rectifier is denoted by Efd that is a function Fex a multiplying factor Fex into VE, this

is very important this Fex is very complex function and as we will see actually that we have to

model the rectifier characteristic. The 3 phase rectifiers 3 phase rectifiers when they see the

system there will be the the input impedance is purely purely inductive in nature input

impedance is purely reactive in nature and this impedance has the effect of delaying the

commutation commutation and when the there is some delay in the commutation commutation

means the change of current from one valve to the another valve right and this affects the output

voltage now for modeling the 3phase rectifiers we always define 3 different modes, mode 1

mode 2, mode 3.These 3 modes are depending upon what is the load current, how much current

it is supplying right, it is very non-linear characteristic.

Therefore to obtain the characteristic of the rectifier or this is rectifier regulation model what is

done is the output of this ac exciter is multiplied with the multiplying factor which is obtained

through this loop that is you have VE multiplied some term you get Efd, this multiplying factor as

I have just now told you is Fe Fex, Fex is the multiplying factor. Now this Fex is obtained obtained

from Ief Ifd there is a field current flowing in this field winding of the synchronous generator and

the output voltage that is you obtained a non-linear obtain function IN equal to Kc Ifd by VE that is

uh using this terminal voltage which is available what is the field current which is supplying we

obtain a current IN and this Fex function is a non-linear function of IN that is Fex is equal to a

function of IN.



Now here here in mode one mode one f IN is equal to 1.0 minus .577 IN where, IN is less than

.433, this is in the mode 1 whether operating in mode 2, f IN is equal to square root of .75 minus

IN square where IN is greater than .433 and less than .75 and we are operating in mode 3 this

function f IN is equal to 1.732 into 1 minus IN where, IN is free than greater than .75 and less than

one that is this non-linear function, okay is different for different modes and they are they can be

computed using these expressions.


While IN is computed knowing knowing this Ifd and VE and this factor Kc which is the

commutation commutation reactance Kc is the, this Kc stands for a constant which depends

commutation reactance right. Now with this I conclude my presentation saying actually that we

have developed the non-reciprocal per unit system for excitation system and we have developed

the models of the dc excitations, dc exciter that is self-excited and separately excited and ac

exciter right. We will continue the discussion on modeling in our next lecture. Thank you!

power system dynamics prof. m l kothari department...

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