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Friends, today we will study the dynamics modeling of hydro turbines and hydro governors.
We have studied the dynamics models for steam turbines and a turbo governors, the characteristic
of the hydraulic turbine are different from that of a steam turbine and therefore different
dynamic models are required for hydro turbines the the physical arrangement for a hydro power
plant is shown in this diagram.
This is the hydro turbine, this is the water storage tank, we will call it fore bay and
this is the penstock through which the water flows from storage tank to the hydro turbine.
The transient characteristic of the hydro turbine is determined by the dynamics of water
flow in the penstock and we shall determine the transfer function of the hydro turbine.
Now when we consider small perturbations about a steady state operating condition then we
can relate the various parameters by linear equations.
Now this equation that is q equal to a11 h plus a12 n plus a13 g and another equation
m equal to a21 h plus a22 n plus a23 g. Now this these two are the algebraic equations
in this equation q stands for per unit deviation in flow, h stands for per unit deviation in
head, n is per unit deviation in speed, g is the per unit deviation in gate position,
m is the per unit deviation in torque.
There we can say here therefore small perturbations we can relate the per unit deviation in flow
to per unit deviation in head, per unit deviation in the speed, per unit deviation in gate position.
Similarly the per unit deviation in torque is related to per unit deviation in head,
per unit deviation in n that is the speed and per unit deviation in gate position. The
now these are the two algebraic equations which relate the small change in the water
flow water flow following following change in head speed and gate position.
Similarly this is the change in torque produced following the small changes in head speed
and gate. Now using these equations and using the dynamics of the water flow in the penstock
a transfer function model for the hydro turbine is obtained. The transfer function relating
the mechanical power output of the hydro turbine to the gate position or position of the gates
that is PGV is the position of the gates is related by these constants which were shown
in those algebraic equations and an important term Tw.
Now this Tw is the water time constant or water starting time constant of the hydro
turbine. We will discuss about this time constant Tw in detail, further these coefficients a
one one a 11, a 13, a21, a22, a23. These can be interpreted as the partial derivatives
of q with respective to head partial derivative of q with respect to speed deviation and so
on now for ideal turbine which is normally considered to be a loss less turbine, the
n for rated speed ideal turbine at rated speed these coefficients are having their values
equal to a11 is .5, a12 is 0, a13 is 1, a21 is 1.5, a22 is minus 1 and a23 is 1. Therefore,
if we make use of these values and obtain a transfer function relating the mechanical
power developed to the valve position.
Now this transfer function comes out to be a very simple transfer function of the form
one minus STw divided 1 plus 0.5 STw. Now let us understand what are the factors on
which this water starting time or time constant Tw depends upon.
Now this time constant Tw is associated with in write Tw is associated with the acceleration
time for water in the penstock between turbine inlet and forebay but basically represent
the acceleration time. Now the equation for this time constant Tw is L into V divided
by HT into g, now here here L is the length of the penstock in feet, V is the velocity
of the water in feet per second, HT is the head in feet and g, g here is the acceleration
due to gravity, g here is the acceleration due to gravity its value is feet per second
square its units are like that that is if you put actually value of g it comes out to
be 32.2 feet per second square 32.2.
Okay now what we do is that there exist a relationship between the power developed and
the various parameters of the system therefore in this equation the velocity will be replaced
by power and other terms that is the power generated is given by this equation V into
HT into A into e divided by 11.8, here P is the power in kilowatts, V is the velocity
in feet per second velocity of the water in the penstock, HT is the head, A is the cross
sectional area of the penstock in feet square or square feet and e is the combined efficiency
of the generator and turbine is the total efficiency divided by one therefore what we
do the from this equation we express the value of V in terms of power, head, A and e and
then substitute this value of the expression of V in this equation 24.2.
When we make this substitution we get the expression for water starting time constant
Tw equal to 0.366 PL divided by HT square A into e. Here the value of g has been substituted
as 32.2. Now this time constant Tw is a very important parameter. So for the model of the
hydro turbine is concerned the hydro turbine the transfer function has some special characteristic
and we will discuss the special characteristic of this hydro turbine transfer function.
The transfer function of the hydro turbine which we have just now derived is 1 plus TwS
1 plus 0.5 TwS this is minus here, correct thank you, this is minus 1 minus TwS 1plus
0.5 Tw okay is this, okay is minus. Now we can write down here mechanical power output
Pm and the gate position PGV.
Now this transfer function has one special characteristic that is the if I look at the
poles in 0s of this transfer function then it has 10 in the right half of the S plane
and a transfer functions or the systems the systems which have at least 10 or one pole
in the right half of the S plane are known as the non minimum phase transfer functions
or non-minimum phase systems.
We shall analyze the special characteristic of this turbine hydro turbine by considering
unit step input to this turbine that is we change the valve position suddenly and once
we give a any step input to this hydro turbine transfer function right then we can write
down the expression for mechanical power developed.
Now here instead of considering a PGV, we will prefer to consider this deviations deviations
in the gate position okay and deviation in the mechanical power developed
therefore when I consider the deviations I can write down the model in this form that is delta PGV is the
input and output is delta P, okay.
Now when we consider the unit step change in the position of valve or gate of this hydro
turbine then delta PmS can be written as 1 by S 1minus TwS over 1 plus 0.5 TwS. Okay
because we have considered the unit step change in gate position. So that the transfer function
ah not the transfer function where the Laplace transform of unit step input is 1 by S.
We shall write down this function in this form 1 by S take this Tw out, so that we can write here
1by Tw minus S divided by take the 0.5 Tw. So it write down 2 by Tw, okay plus S that
is we are writing in the in the form S plus A or here actually is the S minus A this is
the numerator directly it with not going to come actually in our expression for the response.
This can be written as 2 by S 1 upon Tw minus S divided by 2 by Tw plus S.
Okay, we obtain the time response by taking the inverse transform of this transfer function
that is we can obtain delta Pm(t) by taking the inverse transform that is Laplace inverse
of this whole function which I can write down here as 2 by 2 by S1 by Tw minus S divided
by 2 by Tw plus S, okay. Now we obtain the partial fraction of this and the partial fractions
will come out to be like this and this will come out to be equal to Laplace transform
of Laplace inverse of 1 by S minus 3over 2 by Tw plus S that is if you simplify this
expression you will get this expression you can just check it.
Okay therefore now we can say that delta Pm(t) is equal to 1 minus 3 to the power minus 2
by Tw into T this is the this is the response of the hydro turbine to step unit step change
in valve position, okay. Now if I put t equal to 0 that is at time equal to 0 that is delta
Pm(o) is how much is equal to minus 2 and if I put now t equal to infinity in this equation
I will get delta Pm infinity equal to 1. Now we can you can appreciate actually or the
difference in the normal response which we get for the transfer function for a system
suppose there is a system and we give a some input we except that the output should start
following the input.
Now here you find that when at time t equal to 0 when I give a step change in valve position
then immediately the change in mechanical power is minus 2. We have given a unit step
input but the output is negative and this is as high as minus 2 and under steady state
condition the output is same as input right. Now this is the main difference or many special
characteristic of the hydro turbine.
Now if I plot the response that is time t delta Pm, we start at minus 2 and t equal
to 0 its value is minus 2 and it settles to value equal to 1and it is very exponentially,
so that response is of this valve the time constant is time constant is Tw by 2 right
that is if you if you draw a transient to this response curve at t equal to 0 then the
curve will be the this transient will be like this and this time can be identified as Tw
by 2, the meaning is that if the if the change in mechanical power changes at the initial
rate then it will reach its steady state value in time equal to Tw by 2 right, otherwise
it is very exponentially.
Now to understand what is the implication of this special characteristic of the hydraulic
turbine on the type of governor which will be required. Now let us first consider that
we use a simple governor no special features, a simple governor and let us see the what
will be the requirement for stable operation for this system.
To understand this the a special requirement let us start with the simple governor we can
represent the governor by a simple transfer function 1by R the transfer function of the
hydro turbine or hydraulic turbine is 1minus TwS, 1plus 0.5, TwS, output of this hydro
turbine will act on the inertia of the turbine generator system is 1 upon 2 HS, I am neglecting
the damping term this becomes my speed deviation delta omega R and here is a negative feedback.
Okay this is negative this is change in reference this this term is delta omega reference I
give as and is a plus.
Now to analyze the stability of this system what we will do is we will consider, we will
consider the H equal to so let us say H equal to 5, okay. Let us take Tw equal to what was
the taking 22. Okay now with this parameters we shall find out that what are the requirements
on speed regulation parameter? Our basic requirement is it is a closed loop system and this system
should have high degree of stability it should have a required stability margin.
To analyze this requirement we write down the characteristic equation of this system,
the characteristic equation can be written as 1 plus 1by R, 1 minus TwS over 1 plus 0.5TwS
into 1 upon 2 HS. Now here here this this is the forward loop transfer function 1 by
R, 1 by R into this transfer function into this this is the forward loop transfer function
and the the feedback is unity feedback and the of the characteristic equation is 1 plus
gH which is the standard formula and it comes out to be equal to that is 1 plus 1 by R.
Now will let us substitute the value of Tw equal to 2 and H equal to 10 and let us see
1 plus 1 by R,1 minus 2S, 1 plus S, 1by 10s okay. We can write this as 10 SR plus is it
okay or we can write this is in the form now 10 R S square plus 10 R minus 2 into S plus
1 equal to 0. Therefore this is a second order characteristic equation and for this second
order characteristic equation to have its roots to lie in the left half of the S plane.
Our requirement is that these coefficients should be positive therefore from this consideration
that these coefficients to be positive one requirement is now the 10 R should be greater
than 0 or we can say R should be greater than 0 the second requirement here is the 10 R
minus 2 should be greater than 0 this puts the requirement that R should be greater than
.2, it means from the consideration of stability the minimum value of R which we can choose
for this governor is 20 percent.
Generally the the speed drop which we choose out or which is the governor parameter or
2 parameter of the governor is taken around 5 percent. Now if I take the R equal to 5
percent then this system is going become unstable, okay. Now suppose I want actually the response
to be critically damped then for critically damped response our requirement is 10 R minus
2 whole square minus 10, 4, 40 R equal to 0 okay that is b square minus 4 ac should
be equal to 0 and if you solve this equation the value of R which is required comes out
to be .736 and another value of R is 0.05, what is the value 536 okay.
It means we get 2 roots of this equation one root says that the value of R or if I call
it R1 we call this as R2 one root says that it should be R should be .736, the meaning
of .736 means it is a 73.6 percent right and when I say R2 equal to .0536 it is 5.36 percent
but if you assume R2 equal to .0536 then it does not made this requirement let R is greater
than 0.2 and if you take this value of R equal to .0536 this will come out or this will result
into a damping is so equal to minus 1 that is negative damping and with R1 equal to .736
the it will have a critically damped response where will be equal to 1 right but this is
too high and therefore in order to overcome this problem the the hydro governors which
are provided have to have different characteristics. Now we will discuss the characteristic of
the hydro governor which is required for hydro turbines. Now this is the block diagram of
a hydro governor that is the governor for hydro hydraulic turbine. The the building
blocks of this governor or a speed control mechanism or the hydro governor is similar
to that we use in a steam turbine, the only difference which we will see here is that
this block dashpot okay.
Now let us just discuss what are the main blocks, the first block we can see here is
the speed governor, the speed governor sensor the speed and it gives you a position which
is which is proportional to the speed deviation or speed right there is the speed governor
position. This first block pilot valve and servomotor, the input signal to this pilot
valve and servomotor are the speed governor, speed changer position which is the signal
which comes from AGC of the system and another is the position of the governor speed governor
position. These two are the input signal to the pilot valve and servomotor, the output
of this pilot valve and servomotor goes to another servomotor hydraulic servomotor which
is named as distributor valves and gate servo motor.
Now these 2 servo motors are required to amplify the power which is required to move the gates
and the output of this hydraulic servomotor control the governor control valves or gates
of the hydro hydraulic turbine and therefore output is the gate position, okay. Now this
dashpot is to provide feedback signal and this feedback is a derivative feedback this
dashpot is to realize if derivative feedback and you can see actually that there are 2
input signals coming one is directly coming from the position of the servomotor another
is coming through dashpot right. Now these 2 these two feedback signals are required
to obtain the required characteristic for the hydro governor.
Now this shows or this block diagram shows the transfer function of various building
blocks of the hydro turbine governor or hydraulic turbine governor we start like this, this
is the this is the transfer function of the pilot valve and servomotor this is the time
constant of the distributor valve and servomotor TG stand for time constant of the distributor
valve and servomotor. Here we show the rate limits then this transfer function one by
S to represent the integration function and the another limits which we have is the position
limit right, therefore these two limits are shown in this transfer function diagram and
whenever we are studying the small perturbation dynamics or whenever the system is subjected
to small perturbation then these these limits particularly may not be touched particularly
the the position limit may not touch but the rate limits may have to incorporated. Again,
you can see here there exist a non-linear function to relate the gate position to the
hydraulic servomotor position that is when the hydraulic servomotor piston moves right
it moves the gates and the movement of gates and the movement of the servomotor they are
again related by a non-linear function. Therefore this show the nonlinear function the transfer
function of the dashpot is delta STR over 1plus STR.
Now this transfer function you can see here that it is the derivative feedback, this is
a derivative feedback because we have a a term S in the numerator that is STR or 1 plus
STR this becomes the derivative feedback then this term delta delta is called transient
drop delta is called transient drop then the another feedback which was shown in the block
diagram is through this sigma.
Now this sigma is same as R which we use actually in the models for the hydro for the governors.
Now this sigma is known as the permanent drop, permanent drop the meaning here is that when
the system is in dynamic condition right there will be output from the dashpot and the movement
the system attain the steady state condition right there will be no output signal from
the this transfer function and therefore the the net drop will be determined by the permanent
drop sigma only, okay. The typical, the typical parameters of this hydro governor are this
TR is 5 seconds its it ranges in the range of 2.5 to 25 a wide range TG, TG is the time
constant of the distributor distribution and valve and servomotor this time constant is
small and its value is 0.2 second and its range is .2 to .4. The TP which is the time
constant of the pilot valve and servomotor this time constant is very small of the order
of .04 second and it is in the range of .03 to .05 second. The delta the temporary drop
its value is .3 and its range is .2 to 1, the sigma or R its range is its value is .05
and it varies from .03 to .06 percent. Normally it is 2 percent or one in drop is considered
and some places it may be lower than this or slightly higher than this but its the range
is very narrow or is .03 to .06.
Further further this time constant TR,TR is depends upon the water starting time constant
right, therefore the typical value is the TR should be equal to 5 times Tw this is the
thumb rule okay and suppose if Tw is 1then TR is 5, suppose Tw is 2 TR becomes 10 and
so on similarly this temporary drop is 2.5 Tw divided by 2H. Therefore you can say that
the temporary drop requirement depends upon the inertia constant of the turbine generator
system and the water starting time right because as we discussed actually that this special
feature that we have to provide a temporary drop compensation right particularly to take
care of special characteristic of the hydraulic turbine and there is a relationship which
relates a temporary drop to water starting time constant Tw and inertial constant H.
Now the question arises the how do we optimize the parameters of this hydro governor, one
way is that okay you take this thumb rules that is you set the value of delta as given
here in the range of .3 around .3, .221 or use this formula delta is 2.5 Tw divided by
2H. For example in this case when I put Tw equal to 1 and H is taken as 5 how much it
comes out to be .25, 2.3 is one which we are talking about but however in any particular
system particular system the procedure is that you you model the complete system take
the hydro turbine, take the its associated governor right and you obtain the dynamic
response of this system and we can use, we can use different techniques to set the parameters
of these governors right.
Generally the governor parameters are set under no load conditions the the general model
for a speed governing system is developed and this is shown in a compact transfer function
like this where input is delta omega the speed deviation signal in the transfer function
is in the form K into 1plus S T2 divided by 1 plus S times T1 or 1 plus S times T3. The
output of this transfer function represents the change in power delta P, now this is added
to the reference setting that is Po which is the reference setting and this output which
is the sum of Po and delta P right is the gate valve or gate valve position okay.
Now here as you know actually that this Po is the position of the speed changer, in the
AGC when we talk about this is the speed changer position. Now this time constants T1, T2,
T3 right. These time constants are can be obtained from the typical values which are
given here for this TP, TR delta sigma and TG so on that is these time constants T1,
T2, T1 and T1, T2 and T3 right are related to the parameters of this governor and this
is put in the simplified form for the purpose of simulation studies, okay.
Now another important point which we have to see here is that the there are different
manufacturers of hydraulic governors, in they are hydraulic governors have been manufactured
by the general electric company by a wasting house and so on right and therefore this manufacturers
give the parameters of the model of the governors and developed by them right therefore the
standard references are available where the parameters of different manufactures are given,
if you look into the reference material which I have given to you last time that is the
committee report right then in this report the parameters of different types of hydro
governors are are given.
Now these parameters are generally the generally the typical values however however for a any
utility for any utility has to set its own parameters by performing certain studies.
Now as we have studied actually that the excitation system and automatic voltage regulator parameters
need to be set, similarly the parameters of the governors need be set. We are so far actually
the turbine parameters are concerned they are constant only flexibility which we have
is actually in setting the parameters of the hydro, hydro governors for hydro turbine and
uh other governor for the steam turbine.
Now this constant K which I have shown here this K is the gain of the governor this gain
is basically if the reciprocal of the permanent drop that is R if R is taken as 5 percent
the gain will become 20 right and the standard techniques which are used for for setting
the parameters are by either by time domain simulation or by frequency response analysis
okay. Let me the sum up that we have studied the dynamic model for the hydro turbine.
We have also seen the special characteristic of the hydro turbine transfer function and
to meet the requirement of the hydro turbine, the hydro governor is required to have a special
feature and that feature is to have a temporary drop concept. Now when this hydro governor
with the temporary drop is provided and is the parameters are properly set it will give
the desired dynamic response for the system. Thank you!