 hinted to you in the previous class that the oscillations associated with the torsion of the shaft of a turbine generator system can be adversely affected by the behavior of a electrical network. That is the electrical network under certain circumstances can destabilize the torsional oscillations. From a modeling perspective, shaft torsional transients are faster than the transients associated with say relative angle swings, low frequency swings and frequency stability phenomena which we discussed sometime back or the voltage stability phenomena which we discussed sometime back. Therefore, the most important modeling difference is that for study of torsional transients and the interaction with the electrical network, one has to consider network transients and stator transients of a synchronous machine. Today's lecture, we will continue our modeling, our effort in the previous class was directed towards trying to understand how electrical network with series compensation in the form of a fixed series capacitor can what kind of characteristics it has. So, today's lecture, we will try to complete that analysis and the icing on the cake as far as the analysis is concerned is a startling phenomena that is when we have a series compensated network with a series capacitor, fixed series capacitor, you can actually cause shaft torsional oscillations to grow, we can cause shaft torsional oscillations to grow. This phenomena in fact is called sub synchronous resonance, it will become clear in this lecture why it is called why the word resonance is used. After all, we are talking of a transient phenomena not kind of a force response phenomena, but still you will understand why we use the word resonance under these situations. So, today's lecture, we will complete our discussion on sub synchronous resonance and we will also I will just give you a flavor of what we shall do in the next 3 of our lecture 2 or 3 lectures that is the last part of the course on stability improvement. So, let us continue where we left off in the previous class, just to recap torsional transients require the modeling of multi mass modeling. Remember that whenever we are studying torsional transients the speed of all the turbines and the generator is not the same, though in steady state they are the same, the transient differences in speed are usually caused by these oscillations torsional oscillations. We consider network transients because as we shall see the interaction with the electrical network under certain circumstances is very important, but remember that we have to consider network and stator transients, but the frequencies involved are between say 10 hertz to around 50 hertz. So, often we can understand this phenomena using lumped models. So, this is why we left off last time I just gave you a kind of a motivational example to show you that you can this particular system in fact displays the phenomena of shaft torsional oscillations. Then the frequency of course depends on the shaft stiffness and the mass of the generator and the turbine. Now, we will try to understand the interaction with the electrical network. I hinted to you last time that this problem arises often due to series compensation that is whenever you got a capacitor in the network, a fixed capacitor in the network this electrical interaction becomes very prominent in the sense that there could be instability of these torsional oscillations. So, let us just study this phenomena why it occurs and you know take a case study. Now, remember of course torsional natural oscillations the natural oscillations of the shaft and mass and the shaft and turbine and generator system is effectively like a spring mass oscillation, but importantly the friction windage is very low it is not much the bearing friction etcetera is very low. So, the damping which we associate with these torsional oscillations is actually quite small. Now, electrical torque also affects the torsional oscillations one can look at it as an input though it is not strictly speaking an input as we shall see. The damping due to electrical torque can be even negative. So, that is very important negative damping means that the oscillations can grow. So, this how this can occur is something we will try to understand in this lecture. So, before we try to understand you know the properties of a electrical network in the previous class we have in fact modeled a network we shall just go through that again. Let us just take an example of a 6 mass system 6 mass there is a high pressure turbine there is a intermediate pressure turbine. So, low pressure there are two stages LPA and LPB there is a generator there is a rotating excitation system say a brushless excitation system and the shaft sections which interconnect the HP and IP windings and various other turbines. So, sorry not windings the turbines the HP and IP turbine the IP and LPA turbine the LP and LPB and so on. So, they are connected rather I should say they are coupled via the shaft in this fashion the HP is connected coupled to the IP to the LPA and so on. So, these this is a typical data given in the IEEE first benchmark model which is available in the literature. Now, how do you for example take out the frequencies just let us look at this turbine you know generator system in isolation we say it is not connected to the electrical network. Let us assume that damping is 0 it is not 0, but it is a bit difficult to estimate it is quite small. So, damping from purely mechanical sources assume to be 0. So, let us see what is the what are the model properties of this particular system. So, if I am going to try to analyze this system remember what are the states the states are the angular position of all the masses and the speeds of the masses. Remember that if you have got two masses connected by a shaft this is something we did last time you will have suppose this is the HP turbine this is the HP turbine and this is the intermediate pressure turbine of a steam turbine generator system. So, you will have dHP by dt is equal to this is the change of the angular position and if you look at in per unit if you write this in per unit it will look like this is equal to dHP. Remember now we have got several turbine stages and the mechanical torque in fact is created in all these. So, for the HP turbine the mechanical torque may be dHP minus the you do not have any electrical torque right at this mass right because this is a turbine what instead you will write here is k times a k is the shaft stiffness delta HP minus delta IP. So, similarly for the IP turbine we will have 2HIP by omega B d omega naught by dt this is in per unit. So, TIP minus k times. So, this is k HP IP. So, this is k HP IP. So, it will be plus delta HP minus delta IP minus k IP LP A delta IP minus delta LPA. So, if the LPA turbine is connected here then this is LPA and so on. So, the turbine equation the equations of the masses the motion of the turbine masses in fact d delta IP by dt is equal to omega IP minus omega naught. So, this k of course is the shaft stiffness. So, the forces are there is a driving force due to the mechanical torque created in the turbine and if the shaft is slightly twisted you also have a torque due to that twisted I mean it is there is a torsion torsional force. So, you will have equations for all the turbine masses in this fashion and for if you look at the generator itself suppose this is the generator and this is the exciter the equations are d delta by dt I will not use any subscript for generator you will call it delta itself and d I am sorry here in the previous thing it should be just omega naught omega naught. So, this should be omega HP and this should be omega IP. So, d omega by dt omega and delta the speed of and angular position of the generator in the previous class I had written a subscript gg, but you know for simplicity we will just not put any subscript. So, this will be 2 h generator by omega b. So, in this for a generator of course there is no there is no mechanism there is the driving force actually comes from the shaft in fact. So, you will have minus of k delta minus delta exciter plus k times delta LPB which is the shaft connect which is the mass connected here the turbine minus delta minus the electrical torque for the time being we will assume it to be 0 actually because the electrical torque is there is a coupling with the electrical system. So, this is the expression for the generator similarly you can have an expression for the excitation system. So, our states eventually will be for a you know you will have delta HP right up to delta X and omega HP right up to omega X. So, you will have actually for 6 mass system you will have 12 states. So, d by d t of this will be a matrix A into the states again delta HP to omega X. So, we can actually in isolation if you take the turbine generator system in fact we can obtain the Eigen values of A. So, find the Eigen values of A. So, what we will do is first try to see that we do actually get an oscillatory kind of you do get these torsional oscillations we will just have a quick we will run a psi lab program to really get this. So, what we will do is I have already run it before. So, I will just show it to you again now what you have is you look at this program Eigen value makes this C i. So, open. So, if you look at this program what I have done it may not be absolutely visible on your screen depend. So, what I will just quickly also read through it this is the data from the I triple E first benchmark model. We will assume omega is 2.2 60 it is a 60 hertz system and the data corresponding to the turbine is given here the H the inertia constant value as well as the shaft constants. Now, I write this A matrix is very simple. So, I write this the states are arranged of course, here as delta HP omega HP delta I P omega I P and so on. So, of course, this is not required here this part of the program this is basically T m is equal to 1 T is equal to 1 and T m actually is made out of components you know in a steam turbine with many turbine stages the mechanical power actually is obtained as a sum of all the mechanical powers generated at the individual turbine. So, for example, the HP turbine could be 30 percent of the total mechanical power may be being may be generated at the high pressure turbine 30 percent at the I P turbine and 40 percent in the 2 LP turbine stages. So, anyway this is not really important as far as our analysis is concerned this particular analysis is concerned you have got the A matrix let us just take out its Eigen value. So, I will do a limited analysis here. So, we will run this program execute and form the A matrix. So, this is just forming the A matrix mechanical system this is the mechanical system in isolation assuming that electrical torque is not is 0 or it is a constant. So, what you will find is if you take out the Eigen values the Eigen values are as follows. So, this is in radian per second of course. So, if you look at this there are basically 6 pairs of Eigen values 5 of them are torsional modes oscillatory modes you see this. So, if I in fact do spec A divided by 2 divided by 3.14 this is going to give you the frequency of these oscillations as well. So, this is the frequency frequencies of the oscillations which will be seen you see 2 0 Eigen values that 2 0 Eigen values because you have got an isolated system there is a absolutely isolated system and there is no friction. So, in fact if you have got a completely isolated system that is a spring mass system of this or even if you take a isolated spring mass system you will find 2 0 Eigen values if we do not have friction if you consider friction then 1 of these Eigen values 0 Eigen values will disappear. So, remember that the 0 Eigen values are generally related to the motion common motion where all the masses move together this is something we also did when we considered the analysis of a 2 machine system. So, things kind of keep popping out in different situations the similar things keep popping out. So, if since you have no friction you have 2 0 Eigen values and both these Eigen values are associated with the common motion. Now, in fact this can be even inferred from the Eigen vectors. So, if for example, if you look at if you take a run this command. So, if you look at the Eigen vector corresponding to the first Eigen value I am sorry. So, if you look at v colon 1. So, what you will notice is remember that the states are arranged as delta H p omega H p delta I p omega I p and so on. What you see is delta and omega the terms which you get are of course, you will find in j this is complex number then this is a real number. This is if this is a complex this is a purely imaginary number then this is a real number that kind of behavior you see this is not very surprising whenever you have got an oscillation and delta is the derivative of the derivative of delta is omega you know it is proportional to omega. In that case it is obvious that an oscillation if you have then delta and omega will be 90 degrees phase shifted and that is the reason why this delta Eigen vector component corresponding to delta is 90 degrees to the Eigen vector component corresponding to this corresponding to the speed. So, this is why we have it also you notice that different masses have different observability as far as this mode is concerned. For example, this mode is will be hardly visible or rather is not very well visible in the last 4 states that is delta generator delta omega generator delta exciter and omega exciter. So, this is the property of this mode also you notice that the delta component here is negative while for the I p component it is positive. So, it means that you know when one of the masses or rather the displacer if one of the masses is going ahead the other is going behind. So, there is a kind of a this is very common in any the mode shape in any kind of oscillatory mode. So, this is basically giving the nature or the relative movement of all the states given that the first mode alone is excited. So, if you look at the second mode will have some other property and so on. Now, one thing of our interest of course is the zeroth mode. So, if you look at the zeroth mode that is 1 2 3 4 for fifth mode an interesting you know thing you notice is that this particular mode appears to be equally observable in all the states all the deltas. So, this is as far as our analysis of the mechanical system taken alone you know we without worrying about the electrical system if you just take the mechanical system in isolation these are the Eigen values. What about the electrical system if you recall in the previous lecture we had gone through the modeling of the electrical system which consists of the synchronous machine connected to a transmission line connected to which is compensated by a fixed series capacitor. So, we will just quickly go through this. So, this is the synchronous generator model, but we did make some assumptions to make our analysis of it simplified. So, the assumptions were of course, that the line is a lumped is represented by a lumped inductor. The second assumption we made was we of course, we took e the infinite bus voltage is 1 and omega naught the frequency of the infinite bus as a rated speed it was just a simplification it need not always be so that the infinite bus to which this synchronous machine is connected to its speed should be equal to the rated speed or its frequency is equal to the rated frequency. This is just a simplification we are doing for this analysis it is likely to be very close to the rated frequency. If resistances are small we can combine the flux and the current equations remember that there was a redundancy of states because flux and currents are related by an algebraic equation. So, we do not have to write separately write you know differential equations for the stator flux and the current. So, we combine those equations and made one equation and if resistances are small this can be easily done. Thereafter we made another assumption that the rotor fluxes are assumed to be constant. So, we can define e 1 and e 2 in this fashion and it is possible to substitute for psi in terms of i psi d and psi q in terms of i d and i q and get the first two equations this is something we did in the previous class. So, I will not spend too much time on revising that again we then converted these differential equations into the uppercase d and q variables the advantage of doing that of course, was the infinite bus voltages will become constant if I use this particular transformation of variables. Now, the capacitor equations were these in the a b c frame of reference then we converted them to the uppercase d and q variables remember the uppercase d and q variables are obtained by using a transformation like parks transformation, but instead of theta we take a fixed omega naught t theta being the position actual angular position of a synchronous machine. Now, if you combine all these equations you get a state space equations of these this form. So, you have got a linear state space equation in fact, this a matrix is constant, but remember that these are the two sources even in e 1 dash and e 2 dash or remember dependent on delta as well as the fluxes the rotor fluxes which for the simplified analysis we can assume to be constant, but one thing you notice here is that the electrical system is going to be affected by omega there is an omega here which is the speed of the generator and e 1 dash and e 2 dash are also functions of delta which is the rotor position of the synchronous of the synchronous generator. So, what you notice here is some kind of coupling. So, if you look at the mechanical system you are giving the torques mechanical torques are a kind of being generated at the various turbines. So, L P A L P B and your electrical network. So, delta and omega of the generator affect the electrical variables as we are seeing in these state space equations and the electrical torque affects the mechanical equations is that. So, this is cumulatively gives you T n remember that. So, this is how really the interaction between the mechanical and the electrical system takes place. So, what we will do now in an isolated fashion just like we did for the mechanical system will assume that delta and omega are constants and just see how the what are the what is the behavior of the electrical network a simplified electrical network with all those assumptions we have made what are the assumptions we have made resistances are small of the stator as well as the transmission line delta omega will assume to be constant rotor fluxes are assumed to be constant. So, these are the assumptions we are making. So, we will just look at the properties of a electrical network which is connected to electrical network which connects the synchronous generator to an infinite bus. So, we will just run write a simple program to obtain the Eigen values of this A matrix simple. If you look at this A matrix what you will notice of course, here is there is a kind of nice symmetry well it is not a symmetry really this is a kind of skew symmetry a kind of skew symmetry. So, what you notice here is minus omega b omega b here there is a minus omega b here there is a omega b this is omega by b c here you will have minus omega by x plus x double dash this should be minus here there is a small error please note it there is a minus sign here for this element 2 4. So, there should be a negative sign here. So, if you look at this is the A matrix or the state matrix corresponding to an electrical network looked at in isolation actually there is a coupling with the mechanical system which occurs because of these terms. So, if you if you just run a simple silo program to obtain the Eigen values of this system. So, if I have already programmed it. So, I will just directly run it. So, what we need to do is of course, execute a simple program in which I have for certain values of parameters I have taken out. So, if you look at this it turn out to be if you take out the Eigen values of a compensated system what you get is of course, it is a 4 by 4 matrix. So, we will get a set of Eigen values of course, if you you ought to really see the program because I am not told you what the values are you know what the total x of the system and what the x the b c etcetera is. So, I will just open this open a file. So, the electrical system if you look at the electrical system this x l is what I will be using is x l is the total x plus x dash x double dash plus any other reactance which may come in this series path it is values 0.7 and x c or rather what I just forgot capital X c. So, it is x c is 0.3. So, if you look at the compensation level it is 0.3 divided by 0.7. So, this is the percentage of compensation which is used. Now, the point is that the Eigen values of the system are these. Now, what you notice is that they are two you know modes they are two oscillatory modes. In fact, you will find that this is 623 this is a imaginary I is a imaginary number purely imaginary number square root of minus 1. So, this is effectively a complex pair and this is a complex pair there is no resistance considered. So, you get no real component. Now, these complex conjugate Eigen values are in fact super synchronous in the sense that the frequency of these is greater than 60 hertz. So, one of the frequencies is coming out to be greater than 50 hertz and the other one is coming less than 60 hertz sorry 60 hertz is the frequency used in this example. So, you have got two frequencies one is super synchronous one is sub synchronous. Now, if you look at these modes carefully one of the modes is 60 if you look at 20 20 is nothing, but 60 minus 40 and 99 is roughly 99.27 is roughly 60 this is roughly 20.72 this is 60 plus 40. So, it appears that the Eigen values are appearing in a certain form. So, you have got 60 plus some Eigen value some frequency and 60 minus some frequency this is very typical why is it typical if you take system like this suppose this is a one phase of a three phase line if the Eigen values of this system let us say r is equal to 0. So, you have just l and you have got c if you just take this single phase in isolation then write down the equations in the a b c a variables you will find that obviously there is going to be an oscillatory Eigen value which has got a frequency 1 by root l c. Now, this is in the phase domain if I convert a three phase network of this kind a balanced three phase network of this kind with r is equal to 0 of course, this r is equal to 0 is not important in this it is true even if r is not equal to 0. The point is that if I take this write the differential equations for a linear network and get an Eigen value lambda for the a phase and similarly for the b and c phases an interesting thing is that whenever reformulate these equations in the d q variables then your Eigen values will be plus minus j omega b that is an interesting thing. So, remember that for example, in this network you have got x l is equal to 0.7 in each phase and x c is equal to 0.3. So, if you look at x c is nothing but 1 by b c. So, b c is equal to 1 by 0.3. Now, if you look at just this phase in isolation if you have just written the equations of this phase the differential equations of one phase in isolation you would have said that the frequency natural frequency of this is 1 by root l c. So, this should be this is equivalent to having omega b into l into b c is you can say c is nothing but what we will do is just multiply omega b in the denominator. So, you will get this. So, this is nothing but equal to omega b divided by root of x l into b c. So, this is what we get as the natural frequency. Now, of course, if x l is given in per unit. So, x l actual is nothing but x l in per unit into z base and b c in per unit is equal b c actual value is b c per unit into y base. Now, y and z base are related like this. So, it is obvious that the natural frequency will be the same whether you express x l and b c in per unit or in actual values. So, this is the expression of the natural frequency of an electrical network considered in isolation. So, if you look at this particular equation for these values which we discussed here what we will get is omega n is equal to omega b divided by root of 0.7 into 1 upon 0.3. So, that becomes omega b into 0.3 by 0.7. So, if you look at if you just compute this omega b is nothing but around 377 for 60 hertz. So, omega b into square root of 0.7 sorry 0.3 divided by 0.7. So, the frequency comes up the radian frequency comes up to be 246. So, as I mentioned sometime back the d q in the d q reference frame if you formulate your equations and compute the Eigen values. Then your Eigen values will be suppose in the a b c or each phase taken individually you got an Eigen value of lambda in the d q you will get omega b. So, if your Eigen values in the a b c turn out to be plus or minus omega n what you will have is j omega n plus or minus j omega b. So, if for minus omega n you will have omega n plus or minus j omega b. So, you will get four Eigen values like this in the d q reference frame the 0 sequence of course, is neglected with the assumption that everything we are considering here is balanced. So, what are the. So, if you look at 246 radian per second which is appearing on the screen and actually take out its frequency. So, this is divided by 2 divided by 3.14 this is the hertz frequency it is 39.27. So, does that explain this it is not 40 exactly it is 39.27. So, the four Eigen values which we got by doing this analysis in the d q reference frame is in fact, are in fact, these Eigen values. So, remember that you get these kind of complex pairs of Eigen values. So, if you have got a series compensated network you do get these oscillatory Eigen values and interestingly some of them are sub synchronous. So, the point now is you have got an electrical network which is series compensated it has got oscillatory Eigen values or oscillatory response and the mechanical system also is an oscillatory response delta and omega. Now, you can put all we had considered these systems in isolation the electrical and the mechanical system in isolation and said that here you have an oscillatory response this in case you have got a series capacitor then you have got an oscillatory response with sub synchronous and super synchronous components. So, the thing is when you put these together what is the Eigen values of the system it becomes a coupled system remember that delta and omega affect the electrical system. If you recall if you look at our final equations here the electrical system is affected by omega and the mechanical system is affected by the electrical torque electrical torque of course, is equal to psi d i q minus psi q i d. So, we can actually compute this from the electrical states i d and i q and of course, they will also dependent on delta because here it is this i d and i q are lower case. So, you will have a dependence on delta as well. So, in fact the point is there is a of course, this equation is non-linear the product terms remember psi d is related to psi d is related to i d and psi q is related to i q. So, this is a non-linear relationship this relationship is not linear T is dependent non-linearly dependent on the terms of the electrical system the synchronous generator in the electrical network. And the electrical network also is in a non-linear way related because this comes out to be a product term you know even dash will be a function of delta and omega is also a mechanical variable. So, this kind of non-linear relationship exists what you need to do is which will not of course, pursue here is compute or rather compute the Eigen values linearized system Eigen values. So, what you need to do is the mechanical system you will have the delta and omega variable. So, I will call them x m is equal to a m into delta x m plus delta t. In fact, you will have minus delta t that is ok. Well it is not going to t is not going to affect all the states directly if you recall it will affect the generator states directly. So, you actually have to have a B matrix into delta t e and delta t e itself is equal to psi d 0 delta i q minus delta psi d into i q 0 minus psi q 0 delta i d minus delta psi q i d 0. So, this is the coupling thing. Similarly, you have got the electrical network equations electrical delta x e this you have to do a linearization this a matrix of course, this is linear you do not have to bother the non-linearity comes here because this is a non-linear this is how you have to linearize this delta t e plus plus B e into delta omega and delta delta delta delta and delta omega are in fact related to x e. So, there are some matrix C into x m. So, I will call this also as some kind of C e into delta x e and this is C into x m. So, you have to couple these two equations and get one grand state matrix you will get it. So, I will not work this all out. So, you will get one grand state matrix of this kind we will assume delta all the inputs delta i p h p delta the mechanical inputs are constants. So, mechanical power in torque inputs to all the turbines will assume to be are constant and also the infinite bus voltage will assume to be a constant. So, this is how you will formulate your equations. What I have written here is absolutely general you can remove all the assumptions which we have you can relax all the assumptions which we have made. So, far relating to the resistance of transmission line the rotor fluxes x d double dash not be equal to x d dash x d double dash x q double sorry x t x q double dash not be equal to x d double dash. So, all these assumptions can be relaxed and this can be an absolute detailed model you need not take the simplified model which I had discussed some time back which the fourth order model that assumed that the rotor fluxes were constant resistances were small and so on. You can relax all those assumptions now one small thing which I probably a bit out of place, but I miss telling you is that there is nothing in this particular rule which I gave you relating the Eigen values when equations are formulated in a b c reference frame and the d q reference frame remember there is nothing special about this being purely complex. So, if this is sigma plus j omega n this will be sigma plus. So, this sigma will appear here also. So, there is nothing special about this being purely imaginary this kind of relationship holds to even this is a complex number with a real part it will also appear here. So, this is something which I forgot to tell you anyway. So, we can actually using more detailed models put all these things together now the point which is very important is that when I said that the electrical and mechanical system interact with each other what happens well you can actually do the you know Eigen value analysis of the mechanical and electrical system put together something what we did here put them together and just do the Eigen value analysis, but even before we do that let us have try to have some kind of insight into what will kind of happen in case the mechanical oscillatory frequency becomes close to or equal to the oscillatory frequency of the electrical network. Let us look at it in a cause and effect manner. So, this is called also the called a damping torque kind of analysis assume that the mechanical system is oscillating at a fixed frequency undamped oscillation is there suppose of the mechanical system let us say it is oscillating at a frequency omega this omega. So, because of that delta if this oscillation omega is observable in delta and omega then the electrical network this is like a input to the electrical system let us assume that the systems are kind of decoupled. So, this is a kind of input to the electrical system the electrical system will respond and you will find that in delta T e also you will find this oscillation of sigma. Now, if this delta T e the oscillation which is created is such that it actually enhances the oscillation which is already there for example, if delta T e if for example, omega of the generator is oscillating at a frequency omega this is the frequency of oscillation of the generator speed around the equilibrium value then it is so turns out that the electrical network has a torque electrical torque which is created like this. So, electrical torque is like this. So, electrical torque is in some way if you look at this these two waveforms is some way proportional to is proportional to minus of omega you know. So, if suppose this is true. So, you know what this looks like this is a kind of a situation where the mechanical system is getting an electrical torque which is proportional to negative of the speed of the generator the things are exactly in you know out of phase. In such a situation what will happen is that the electrical torque will enhance the oscillation it is almost like having negative friction viscous friction and the oscillation will increase. So, this is a kind of a cause and effect non rigorous reasoning why sometimes it is possible that the electrical network can cause electrical torque which enhance the oscillation with initially cause this electrical torque variation in the first place. So, this kind of a cause and effect analysis can reveal that we have a lot of potential problem. Now, one of the important things which you should note is that this need not be true, but it turns out that for a series compensated network that is for a network compensated by a capacitor a fixed series capacitor. If the frequency here sigma sorry this frequency of mechanical oscillation is sub synchronous then this kind of situation is likely to occur and if that occurs one can expect that there can be some problem. Now, I hope you kind of get the reason why another thing is why we can have a problem potential problem this problem will be very much enhanced in case this frequency is equal to the frequency of this in the d q frame of reference. So, if the frequency of omega delta and omega the torsional oscillation frequency is proportional is equal to a very nearly equal to the oscillatory frequency of the network whether network is kind of modeled in the d q reference frame then this t can be very large or rather very significantly large and this problem can really become very very significant. So, the electrical torque may really cause a very adverse reaction and the oscillation this oscillation may start growing with time that is why this thing is called sub synchronous resonance sub synchronous because the electrical torque is found to have this kind of or near this kind of phase relationship or other it has got a significant component of this kind it could be like this also. So, it has got a significant component of this kind in case this frequency is sub synchronous and the network frequency comes close to this then you can show that the electrical torque has got a fairly large component which is proportional to or which is the kind of looks almost you know is out of phase with the electrical speed. So, this is a kind of cause and effect reasoning if you feel uncomfortable about this kind of reasoning no harm you can actually do the Eigen value analysis and see that this actually occurs. So, let us take a case study and in this lecture. So, if you look at this particular we will take this example of a 6 mass system. If I compensate this network by 55 percent series compensation there is one oscillatory mode which goes unstable this is what we find by simulating the system remember if you want to simulate this phenomena you will you have to use EMTP like programs in which stator transients and network transients are not neglected. So, if you look at the expanded view you can see that this oscillation has got 1, 2, 3, 4, 5 around 5 complete cycles in 0.2 seconds 5 complete cycles in 0.2 seconds would mean around 25 hertz 25 hertz would correspond to a frequency of roughly 160 radian per second. So, the thing is that if you look at this Eigen value analysis you do see this is also predicted by small signal analysis by doing a small signal analysis we can actually correlate what happens in simulation you see that the sub synchronous network mode if its frequency becomes close to one of the torsional modes we are evaluated the torsional mode sometime back remember the torsional mode frequencies have hardly changed, but the important thing is that the torsional mode damping has become negative that is the real part of this Eigen value has become positive and this happens because the coupled electrical network has got an Eigen value or has got a natural frequency in the DQ reference frame formulation which is absolutely close to this. So, whenever a network frequency comes close to this the torsional mode becomes unstable if it is sub synchronous. Now, one important thing of course, which I miss telling you is that how much effect the electrical torque has on each individual mode depends on you know the controllability of a mechanical system mode by the electrical torque. In fact, in this particular system the torsional mode 5 is hardly affected by connect by the connection or the coupling to the electrical network it shows that this mode is practically uncontrollable by anything you do in the electrical network this is something you can actually kind of analyze by looking at the Eigen vectors of the mechanical system. In fact, if you look at the Eigen vectors of this corresponding this to this particular mode the both the right and left Eigen vectors they will have very very small components corresponding to the generator mass. So, delta and omega components that is delta of the synchronous generator and omega of the synchronous generator the Eigen vector vector components corresponding to these two states are generally very small both the right Eigen vector as well as the left Eigen vector that indicates that this fifth mode is not very well controllable by anything you do on the electrical network and this is what is actually seen in this Eigen value analysis as well. So, this is what happens in case for a sub synchronous torsional mode there is a matched kind of frequency in the electrical network as well this electrical network is sub synchronous because of the fact that the natural frequency there is a oscillatory natural frequency of the electrical network because of the series capacitor. I am not shown the sub synchronous network mode or the other modes of the system I have just shown you only the torsional modes the sub synchronous network mode and very importantly the common mode. The common mode of the system is given here it is this the low frequency mode which you have already studied before. If you recall what I said last time you do not have to unlearn what we did as far as the low frequency swing oscillations of 1 hertz were concerned remember that when we had done a study of a 2 machine system we had considered the whole turbine mass system as 1 and there in we had seen that there is a low frequency oscillation of around 1 1 to 2 hertz. Now, what we have done is we have represented the turbine mechanical turbine generator system as a multi mass model. Therefore, you are getting all these torsional frequencies, but this particular mode in which all the masses of the mechanical system move together still persists and basically is manifest when you couple it to the electrical system as this common mode or swing mode. So, the 0 Eigen values which we saw in the mechanical system which is isolated from the electrical network now transforms itself into this oscillatory mode the low frequency swing mode. Now, this mode is of course, seen in this simulation you see this low frequency mode as well this simulation incidentally has been done for a step change in torque a very small step change given in the mechanical torque this is the disturbance which is given of another turbines. So, please remember just to summarize this particular phenomena you can have adverse torsional interactions at sub synchronous frequencies due to a series compensated network this is basically the crux or the important thing which you should take back with you after this lecture. I thought I would be able to start on improving stability methods in this particular lecture at least I would kind of given introduction to that we could not do that we will do it in the next class.