 we will embark on the last part of this course that is the various methods for stability improvement. Before we do that, let us just have a quick look at where we ended last time, we had basically discussed the phenomena of sub synchronous resonance. There is one case study which I ran. So, I will just quickly relook at that and then we will move on to the next part of this course that is on stability improvement. So, today's lecture will be primarily on stability improvement. In the previous example where we studied sub synchronous resonance, we considered the adverse interaction of a series compensated transmission network on the shaft of a generator turbine system. So, we had done a case study of a steam turbine generator and where the rotor was represented by a six mass system connected by shafts. So, we had seen that for a certain level of series compensation, there was adverse interaction and you had growing oscillations for one particular torsional mode. So, if you look at in fact the Eigen values which we got for this particular system, we see that torsional mode 3 is very badly affected. In fact, that is what you see in the simulation study as well. We see that particular mode growing with time. Now, in this particular study of course, I should mention that we did no damping was no mechanical viscous damping was considered. So, the only damping which was possible was due to the interaction with the electrical network and you see that in fact, all the torsional mode seem to have positive real parts. In actual practice, there will be some viscous damping and some of these modes actually torsional mode 1, 2, 4 and 5 would be stable if one considered that viscous damping typically, but torsional mode 3 has got a very large positive real part and you see that it could cause a real problem in the system. So, if you have got this kind of situation, it under that is you have got a sub synchronous network mode due to series compensation and typically if you have a situation where the network mode frequency coincides with the torsional mode frequency, this problem could occur. Now, of course, there are a few more qualifications to this statement. It is assumed that the electrical torque can affect the torsional mode, all torsional mode this need not be true. In fact, the electrical torque is actually applied at the generator mass that may not actually you know kind of control or affect all torsional modes. In fact, in this system torsional mode 5 it turns out is hardly affected by the presence of the network mode, sub synchronous network mode. So, in fact, although this is not really we have not proved this or I am not really proved this to you, but it actually turns out that in this particular case this is true. So, it is not certainly true that if you have got a series compensated network, it should affect all torsional modes equally and you know it depends on how much and the electrical torque affects a particular mode. So, that actually kind of analysis can be done using our Eigen vector Eigen vector and Eigen value analysis. Now, there are some other situations which we need to the last word has not been said on this. For example, in a practical situation you can have several series compensated lines this is one possibility you know you got 1, 2, 3, 4, 5, 6 series compensated lines in this network and several generators and each generator what I have shown here is generators these round things are generators themselves may be actually several units of a power plant need not be one generator here. So, in fact, you can have many sub synchronous network modes and you can have many, many torsional modes and you actually have to do a kind of a linear a linearized analysis to check out the actual effect of all these network modes on individual turbine generator shafts. Normally, as I mentioned sometime back hydro turbines have a relatively large generator a very large generator mass compared to the turbine mass this is not true in fact, for steam turbine generators. So, in such a situation one one can the what really happens is that the electrical torque does not have that much controllability over the torsional modes these are all statements which of course, I am making without much proof in this course it is out of the scope of this course to discuss these things in too much detail will take a very long time. So, I request you to go through the literature on this subject especially the kind of discussions which are given in Padyaar and Padyaar's book and Kundur's book. In fact, for this particular system where I have shown 6 series compensated lines just to give you an idea you have got 6 series sub synchronous network modes of course, there are other network modes there are other modes as well, but I have just written down the sub synchronous network modes. Now, this kind of analysis can be done with the tools which I have already you know discussed in this course. So, in principle you should be able to formulate the equations of this multi machine system each generator having you know several rotor masses and it should be possible to do a kind of linearized analysis remember it is a non-linear system. So, you can simulate the non-linear system, but if you want to do a linear analysis you have to linearize the system around an equilibrium point. So, this is one of the things which I would like I wanted to discuss before we go on before we take on the new topic. So, with this I kind of end my discussion of sub synchronous resonance the phenomena of sub synchronous resonance remember one thing of course, how can you avoid sub synchronous resonance? One of the ways it appears you can do that is to prevent a coincidence of network mode with a torsional mode remember that if there is no coincidence the effect on the other torsional modes is quite low. See torsional mode 1, 2 and 4 the adverse interaction is quite low and one would expect that if mechanical viscous damping is present this these modes would actually be would be stable. Problem comes in torsional mode 3 where there is a good coincidence. So, it would make sense to limit the series compensation in a transmission line such that it avoids coincidence of you know coincidence with the torsional mode and this usually can be done if you try to design your series compensation. So, that the network mode frequency is higher significantly higher than the highest torsional mode frequency which is controllable by the electrical torque. So, in this case you should you should limit fixed series compensation in the form of fixed capacitors such that the network mode frequency is higher than this 202 radian per second. So, you need to you cannot have a very large xc it you can easily prove that this in sub synchronous network mode frequency reduces this is of course, the equations of formulated in the dq reference frame remember the sub synchronous frequency reduces if the equations of formulated in the dq frame. If xc that is the series compensation the reactance corresponding to the series compensation increases. So, one may actually have to limit the amount of series compensation so that there is no torsional mode and network mode coincidence. In fact, this actually tells you an important thing that stability issues in certain conditions can restrict what you can do with the system. In fact, if you recall I had mentioned that series compensation is something you do to reduce the reactance the series reactance of the transmission network. This in turn you know improves the power transmission capability of a transmission line because it reduces x of the transmission line. If it reduces x you can improve you know you can have better large disturbance stability characteristics. So, by because of this SSR problems under certain situations you may have to limit the amount of series compensation you can have. So, stability problems sometimes can be a limiting factor in the design of your system. So, now let us end the discussion on this topic and move on to what our next and last topic is that is improvement of the stability of a system. So, the whole idea of doing this analysis was to see how it affects a real power system. So, just to keep things in perspective the first part of a course dealt with analysis a generalized analysis of dynamical systems. Then we spent quite a bit of time on modeling of synchronous machines, excitation systems, loads, prime movers you would like probably to go back and have a look at these some of the simulations and all which I carried out when I was discussing these topics. Then we also discussed how you can model a transmission line. The part B of this course we discussed several stability issues one was the motion of common frequency that is the center of inertia frequency of the system which is in fact affected by the load generation balance in the system. Then we also discussed the most probably the most important problem in a synchronously connected systems say that is when you have got synchronous machines connected by AC transmission lines the most important problem stability problem is in fact the relative angle or the relative frequency motion and we discussed things like loss of synchronism etcetera under the topic of relative angle movement and frequency movement. Then we also discussed voltage stability we took an example of voltage stability we in fact studied slow voltage instability caused by the action of tap changing transformers and reactive power limits in a synchronous generator. The last topic which we just kind of ended the discussion sometime back was on shaft torsion the stability of shaft torsional oscillations. We also actually merged we did not consider this as actually although I had listed it as a separate topic we have in fact discussed or actually done a few simulations and there are two kinds of simulations we did one in which we considered network transients and stator flux transients and the other in which we did not consider it. The studies in which we consider do not consider network transients the stator flux transients are called transient stability programs because these are really low frequency phenomena the rather suited for the study of low frequency phenomena and EMTP kind of programs in fact have to model fast transients in detail. We also did side by side and Eigen value analysis in many cases but Eigen value analysis remember requires linearization of the system in case your system is non-linear. We now move on to enhancing system stability and in this context will be I think appropriate to consider this from T s written by a kind of code written by T s Eliot. We shall not cease from exploration and the end of all are exploring will be to arrive where we started and to know the place for the first time. So, in fact when I am talking about the topic of stability improvement when I will be discussing this topic we have done all these phenomena before the first phenomena in fact I will be consider considering is under frequency relaying stability improvement measures. We have actually discussed this issue of under frequency or over frequency which results because of load generation imbalance. So, we have done all these things before now we shall focus on trying to see how we can actually improve stability of the system. So, let us take the simplest phenomena that is of frequency stability. If frequency stability is easy to understand we saw that the moment of the center of inertia of a multi machine system is dependent on the load generation balance and the controls which affect this are effectively the speed governors which may be present on the turbine of a synchronous generator. So, these the main control by which we can actually affect the frequency of a system or ensure that the frequency remains within limits is essentially a governing action of steam or hydro turbines or gas turbines. Now, in addition to governors remember that the loads themselves are frequency dependent but that frequency dependence is not very strong. So, if you do not have governors and as a load generation imbalance the frequency changes and it settles down because the loads themselves are a function of frequency. But say in our system which is around 150 gigawatt it is a 150 gigawatt system of installed capacity a 100 megawatt change may cause say 0.1 hertz change in the frequency. In case there is there is no governors if there are no governors just because of load general load frequency dependence you can come to an equilibrium which is reasonably which is acceptable if the load generation imbalance is low. But if there is a say a sudden generation trip of 1000 megawatts then just depending on load frequency dependence to get you to an equilibrium is not appropriate because the frequency deviation can be very large. And remember we had discussed this before that if frequency deviation is large many turbines will have to be tripped out because typically under off of nominal frequency there is a danger of exciting shaft torsional vibration not shaft torsional vibration sorry blade vibrations in the turbine. So, if there is a frequency deviation of more than 1 and a half hertz or so you may have to quickly trip out your turbine. So, actually a blackout can ensue in case of frequency of a grid falls below say 47.5 in a 50 hertz system. So, this is in fact an important consideration. So, in fact along with governors and effect of load you know load frequency you know flow frequency dependence another way of ensuring that frequency does not you know go to you know perilous kind of levels is having some kind of under frequency or you know these are emergency controls if the primary control in the form of governors etcetera is not able to arrest frequency drop. In that case you may consider using under frequency relays or in fact rate of change of frequency relays in trying to arrest this decline. So, let me just show you an actual event which occurred in our Indian grid several years say about a year ago this 8th August 2009 there was a generation trip at generator at Anpara that is in Uttar Pradesh and the effects of it in surprisingly were immediately failed not surprisingly we know that frequency everywhere in the grid is practically the same it is not really surprising. But something to wonder about it it really works quite a bit of wonder that immediately you see a frequency decline in Mumbai, Sungly and so on because these parts of the grid are synchronously connected to the grid in Uttar Pradesh which is a more than 1000 kilo kilometers away. So, there was a generation trip in the synchronous grid and the frequency drops everywhere in the grid including the western region and you see this quite a precipitous drop in frequency very high rate of change of drop frequency. And if you just allow this to go on then the frequency can go down right down and cause a tripping of all turbines all the steam turbines in this in the system and it could be a blackout. So, what you have is some kind of control action here seems to have you know reduced the rate of drop and at around 48.8 there seems to have been some load relief in the form of under frequency relaying. So, at this point the frequency is drop down and in the system you have got certain loads which are designed to trip or rather there are you know protect relays which are designed to trip out these loads in case of frequency falls and this is exactly what is occurred at this point here. And of course, after the frequency the load is stripped the frequency rises again. So, in fact if you look at this a bit carefully there is a generation trip occurring here there appears to be another generator trip out here this is an event which occurred about a year back. And it seems that there has been if you look at this there is a sudden slope change at this point. So, there was some load relief even at this point. So, there are actually two you have got a load relief somewhere here and you have got under frequency load relief even at this point. So, this is what is probably been designed in the grid to it is a kind of emergency control in the grid. Now, you can instead of having under plain under frequency relaying in case there is a drop trip in generation you will find that the frequency drops you can kind of you know measure the rate of change of frequency at various buses. In fact, this is the rate of change of frequency at various buses after a generated trip this is a simulated response of a multi machine system. And you see that there is a this the center of inertia frequency goes on dipping. Now, you can use this rate of change information also to do some quick control action. Now, if you even look at plain under frequency relaying what is plain under frequency relaying if the frequency falls below a certain value you trip out certain loads that is basically what happens here frequency falls you trip out load. Now, if you look at the issues behind which have to be considered when designing an under frequency scheme they are as follows first thing how much load do you trip at what frequency do you trip remember load comes in chunks. So, after trip out a certain amount of load if the frequency falls below a certain value which value should you choose another point is how much you trip if you trip too much you know you trip too many loads say you trip out the loads at several small towns or cities small cities in the country say you to get a kind of load relief of say 600 or 700 megawatt then the point is how much you know how much load do you actually trip if you trip too much you will find that they may you may have an over frequency situation. So, the point really is that this kind of design of an emergency control system is often carried out based on offline studies. So, you kind of do an offline study based on the parameters of the system and see you know if I trip this much load how much you know frequency rise will be there, but this is obviously since it is an offline study it is based on some you know our knowledge of the parameters of the system, but if you do not have adequate knowledge of the parameters of the system design of this may be quite tricky. In fact, under normal operation you will find that the operating condition which you have under a normal say today right now may be difficult to anticipate during planning, but roughly you have got you can get a operating point close to what the situation when I say what is the operating point today basically the the generation schedule the load at various points actually changes from day to day from hour to hour and depends on the season as well. So, during planning you have to kind of anticipate the kind of load pattern and the generation pattern in the grid and you know design how much load would need to be tripped in case the frequency fell below a certain value. So, this is how you design your under frequency relinks you know obviously it is not a perfectly a perfect science. So, as I was saying instead of plane under frequency relaying you can even think of d f by d t relaying d f by d t relaying is basically checking out the derivative or the rate of frequency drop at a bus and then take a decision of whether to trip or not to trip. So, these schemes also have to be designed at a planning stage, but remember d f by d t is a kind of a anticipatory scheme in the sense that if the rate of change of frequency exceeds a certain value then you trip out certain predetermined loads. So, it is a kind of a anticipatory, but it is again designed offline and the things you need to really consider in your design is what exactly affects the rate of change of frequency. In fact, if you know if you remember the center of inertia frequency of the system the rate of change of the center of inertia frequency of the system is inversely proportional to the sum of the machine inertia. The other thing which determines d f by d t is the amount of load generation imbalance. So, if the load generation imbalance is large you will have a larger rate of change of course. So, by looking at d f by d t at a particular bus the bus frequency as it falls you can roughly make out whether the disturbance is large or small. Now, there are of course, again design issues of how much load to trip where do you trip and so design of this aspect is usually done offline, but if you really look at some import let us just look at some important aspects which you need to consider when you design a d f by d t relaying scheme. Remember that although I have said that the center of inertia frequency depends on the load generation imbalance. Remember that local frequency of frequency of a individual generator is often contaminated or it contains swings as well. So, what you see for example, is the frequency of all generators or the frequency at various buses in the system you will see that not only is there a kind of a common motion there is also these swings. And interestingly there are many many swings you see that there is one relatively low frequency swing here then there is a higher frequency swing here. So, there seems to be a mixture of swings along with superimposed with the center of inertia or the common motion of the system. So, these of course, these swings are something we have discussed before they arise due to the power oscillation or the motion of the relative motion between generators. So, in fact, we had done simulation of a single machine connected to an infinite bus then we had also done simulation of a two machine system. And what you notice there was in addition to the common motion which depends on the load generation balance there is also the motion a relative motion of speeds and angles rotor angles of individual machines. If you have got a large multi machine system this is not a large multi machine system, but just to illustrate in a large multi machine system you can have oscillations between the machines within a plant or the machines the oscillation between a machine in the same area or a machine nearby against the machines in a plant all moving together. So, you have got in an inter plant inter plant mode only the machines within a plant participate and they swing against each other they swing against each other then you can have this local mode in which these two machines swing together with respect to this machine in the same area or you can have all these machines in an area swinging together against the machines in another area. So, in fact if you look at this simulated response you will find in fact if you look in the closely enough you have got some machines which are swinging together against machines in another area. If you look at this real real life picture of how the frequency changes you see that the frequency is measured at Mumbai Sangli Surat which are all in one area seem to swing together here you see this small oscillation here the minor oscillations here swings which you see. These are actually for these this particular mode which is visible oscillatory mode which is visible all these machines swing together this is not actually a surprise this disturbance occurred very far away from these locations which are in the western system and therefore, all the machines in this area of swing together. So, this intuitively makes sense. So, whenever you are going for d f by d t relaying you have to face one problem that is in addition to the common motion they also these swings. So, that may actually give a completely complicated picture of. So, if I take the derivative of the frequency this is what I see. So, if I see this I may actually have spurious tripping because you know because the slope at some points may be large because of the swings and not because of the you know common motion. So, this is one problem which you may face if you are going to have local frequency measurement. So, if I take a frequency at one particular point and try to take a local decision whether to trip a load or not I may actually have a problem because the local frequency actually has got these swings and because of that I may get a wrong information or spurious information about load generation imbalance in the system. So, this is one of the problematic issues when you try to apply a stability improvement method like under frequency or d f by d t relaying. But one may suggest at this point that why do not we actually compute the center of inertia speed remember we had got this formula. So, we had actually derived this formula sometime back in our course that the center of inertia motion depends on is proportional to the sum of the mechanical total mechanical input to all generators minus the total load minus the losses that is because sigma p e is equal to sigma p l plus losses this of course, with network transients neglected this is true if network transients are neglected electrical power output of a generator is equal to under sinusoidal steady state conditions sigma p l plus losses we are considering a slow phenomena. So, we do not have to consider network transients in our analysis. So, this is certainly true. So, the point is that suppose there is some disturbance and now you want to trip load to control the drop in the center of inertia frequency or the common motion you want to prevent a precipitous drop. So, you want to do some load shedding now what one can do is try to compute omega c o i and then because you know omega c o i you can compute this rate of this particular term and because of this you can estimate what this imbalances. So, I will call this imbalances. So, depending on this imbalances you can then decide to trip certain loads to bring back balance in case there is a drop in frequency but what is the catch in all this the good thing of course, in doing this in doing this particular this kind of computation is unlike frequency local frequency measured at every bus or at every generator which is contaminated by these oscillations which may kind of blur the common motion a minor error here the y axis is the rate of change of the center of inertia frequency it is wrongly indicated in this figure as the center of inertia frequency itself. So, remember the y axis label should read rate of change of center of inertia frequency. So, what you are seeing in this graph is in fact the rate of change of center of inertia frequency. The center of inertia speed of you know is somewhat incomponent you see there is relatively lesser than what you will see here. So, you can use the rate of change of the center of inertia speed, but what is the catch the catch is that this omega c o i is not a local signal first of all you will have to take out the speeds of all the generators then use the speeds of all the generators to compute this. Remember that speed of generators is time varying. So, you have to get the speed of the generators at you know kind of to a common point and then compute this particular thing. So, this requires some communication of signals similarly, if there is imbalance. So, if you estimate the imbalance you will have to send a trip signal to a remote location and trip out the thing. So, this seems nice I mean as engineers it sounds like a nice challenge to really make a wide area measurement scheme of a frequency or speeds of all generators compute this and then do this kind of thing. So, this is just of course, this is not implemented anywhere in the world, but this is an interesting thing you can do and as students also you can easily conceptualize this kind of thing. But remember things are not as easy as they look remember that the power system is spread over tens of thousands of square kilometers and you need to communicate quickly you know you do not want too many delays in all this. But to quickly communicate this signal to a common place where this computation is done and this computation has to be done and then you have to estimate the load generation imbalance and then you have to choose a suitable chunk of load remember load comes in chunks. So, you have to choose a suitable chunk of load which you can trip to get the frequency back to normal. Now of course, there are several important points which have to be considered at this point is that when you have got a synchronous grid typically this very precipitous kind of drop in frequency occurs when you have got an interconnected grid suppose you have got an interconnected grid which has got connections with AC line there are two areas suppose one area has got say 10 gigawatt generation and 11 gigawatt load and the other generation region has got 11 gigawatt generation and 10 gigawatt load will assume losses are negligible. In that case if because of say some disturbance there is a loss of synchronism and you separate out these two systems then this system will be an under generated island. There is a load generation imbalance and you will have to trip out this load some load to get this load generation balance. So, one of the ways you can really do it is compute the center of inertia of the now once you separate out you just have you have to compute the center of inertia speed of this particular system. So, something which your central computer which is computing the center of inertia needs to know is which are the generators in this island. So, you need to know that. So, this is one issue the other thing is the speed of all the generators in this island need to be computed in a synchronous way in the sense that they should be time tagged and sent to a common center at which they are all you know after all omega c o i t is sigma h i omega i t by sigma h i you need to know the generator speeds of all generators in that island at the same instant of time in order to get the center of inertia at that point of time. So, you need to time tag all the frequency measurements send them to a common location and compute this for every time instant. And then take out the derivative and then do d f by d t relaying. So, this is one particular scheme you can do this which you can use you know proper technology and actually compute this and had derive a good scheme for under frequency relaying. So, this particular scheme is not implemented, but it turns out that today it is feasible to do this although there issues as I mentioned sometime back you should know whether you are there is an island which generators are part of the synchronous gradient which are not and so on. So, there are still some issues which need to be you know considered before such a scheme can be actually put into action, but today it turns out that it is feasible to have time synchronized very well time synchronized measurements using GPS satellites. So, you can actually measure the frequency or generator speeds at various locations time tag them all measurement units here these are called phasor measurement units are all synchronized having the same you know kind of synchronized clocks by using GPS you know GPS signals you make all these have synchronized clocks. So, that the time stamp the frequency or generator speeds are time stamp and send to a common load dispatch center where in the center of inertia speed in principle can be calculated and then you can use these signals to trip out certain loads. So, in principle this kind of scheme is feasible the reason why I have discussed this kind of a scheme is not because I mean I have not done a kind of a rigorous analysis of the feasibility of such a scheme or anything of that kind just to spur you to think that you can actually use technology and actually implement these kind of things in principle it is possible, but of course as I have cautioned and I will caution you again there are many many engineering issues which you need to be dealt with before you kind of make this kind of scheme it is called an emergency control scheme. So, I have kind of introduce you to a way by which you can have a method for improving stability in this particular case we are talking of improving the frequency stability of a system. So, we have talked of under frequency relaying to prevent sudden drops you know which may occur when there is sudden load generation imbalance because of generator trip or sudden islanding because of loss of synchronism which is another stability phenomena we will just discuss a bit later. Now, if you have got two systems which are not you know connected synchronously with each other that is they are connected via HVDC links. So, the frequency and phase angles of system A and B are need not be the same the frequency here F A and here F B need not be the same the power through this link is regulated strictly. In this situation in case there is a precipitous fall of frequency here or in fact there is a even a rise of frequency sudden rise of frequency because of a major load trip in that case you could ramp up the power suppose this for example, a sudden very large load trip in this system because of which the frequency tends to rise and the governors here are unable to prevent a sudden you know the rate of change of frequency or the frequency rises very fast and the governors are not able to control it because say the limits of the turbines etcetera they are not able to control it in that case you can use for example, in HVDC link to temporarily pump some power out of this pump some power out of this and push it into this, but the frequency here will rise remember because of that, but rather than have a very large rise in one system you can use this particular HVDC link to ramp the power and push that power here and kind of distribute this frequency change. This kind of also scheme can be kind of developed remember that in an asynchronous link each area the generation is equal to some of loads plus losses and similarly P g here will be equal to some of load plus losses plus minus P HVDC and this will be plus P HVDC. So, P HVDC is completely controlled in a local fashion by this rectifier and inverter here. So, you can actually add the link itself can regulate power flow because you have got firing angle control. So, by in some cases you can change HVDC power and control the frequency but you cannot do independent frequency control in both regions if you change the HVDC power this will get affected and this will get affected. So, this is just to kind of tell you that kind of encourage you to think of all these things. So, this is when you have got an asynchronous link asynchronous connection between two regions. So, you can use HVDC power to equalize or slightly help one region by changing the HVDC power. Please note that there is a minor error in the sign here. So, this should be positive here and negative here if the direction of HVDC flow is as shown in the figure here. We now move on to a more worrisome problem or not even frequency changes a sudden frequency changes are a worrisome problem, but a more commonly felt problem for which understanding what needs to be done is not very obvious. Now, let us to you know kind of get a very clear picture of what we are talking about. Remember that just a few moments ago I was talking of this problem if there is a sudden load throw off or a sudden generator trip and governor action load frequency dependence is inadequate. You can have may be stable common motion, but the frequency change may be too high or too low or rather the frequency change may be too large in magnitude and therefore, you may think of emergency schemes like load tripping schemes or the converse of that is generator tripping schemes. Of course, generator trip is something which you have to be careful while doing because a fully loaded generator if you suddenly trip to prevent an over frequency it is it gives a big shock to the mechanical elements of your system that is the turbine generator system sorry the turbines mainly. So, generation trip is not often talked in the same breath as load tripping load tripping is somewhat more benign compared to generate tripping of fully loaded generator. So, these are the things which you need to do in case there is a large magnitude of frequency change the common frequency change, but you may have unstable relative motion as well. For example, the figure which I am pointing out here shows the generator speeds for relative angle instability in which there is a loss of synchronism that is the machines are connected, but a couple of machines here run at a different frequency compared to the other. So, you this happens whenever you give a large disturbance like a fault this can happen for large disturbance like a fault when in synchronous machines which normally run at the same speed because of the fault suddenly lose synchronism this is a large disturbance phenomena. Another problem which you may which may arise with relative motion is that the power swings or the low frequency oscillations which we see following even a small disturbance sometimes very poorly damped or even grow with time. This is a more complicated problem in the sense that even so small disturbances sometimes you see that the oscillations which are triggered take a very long to damp time to damp out or in some situations they may even be unstable. So, this kind of situation can occur under certain situations. So, let us I will just show you the phenomena and then we will end this particular lecture. We have done this simulation sometime back we have done a Sylab simulation in which if you recall this is a Sylab simulation I have actually changed the program. So, let me save this. This is a program which simulates a synchronous machine connected to an infinite bus the field voltage of the synchronous generator is actually being controlled by the automatic voltage regulator and what I am going to simulate is the response of the system for small disturbances around an equilibrium point. So, what we have got? So, the simulation which I will just show you before we end today's lecture is you have got a synchronous machine which is connected via a transmission line to a fixed voltage source. We call this an infinite bus of frequency omega naught. The machine is assumed to be operating at a mechanical power output of 0.5 which is equal to the electrical power output in per unit. This generator is equipped with an AVR which maintains the terminal voltage of the generator a means almost at a constant value. The AVR basically has a structure V ref V a proportional controller and this is the voltage which effectively is through a static excitation system applied to the field which creates the which really creates the field flux. So, you by having this kind of control system you are controlling the field voltage and therefore, you are kind of almost meant if this k is large you practically are regulating this voltage V to a constant value which is almost equal to V ref. So, we have done this simulation before what I will do is assume that the system is operating at this operating point. So, when I am simulating what I need to do is specify the terminal conditions specify the power output from that you can back calculate the current if I know the reactance of the line resistance of the line you can back calculate the current output of the generator and back calculate all the initial states of the generator. Now, if the initial states of the generator are those corresponding to this kind of scenario equilibrium scenario the system will simply remain where it is. Now, thereafter what I will do is give a small disturbance here what I will do is I will increase V ref at 5 seconds for a short while and dip it back. So, I will do that and then we will see what happens now just to keep things in perspective what you need to do is of course, simulate we are what what we are going to do is simulate the system. So, you have to feed in the data we will give T m is equal to 0.5 per unit this is the gain and time constant of the AVR thereafter we specify the infinite bus voltage magnitude we assume that the infinite bus voltage angle is 0 thereafter we calculate theta which is the phase angle of the terminal voltage of the synchronous machine this is obtained by we will also specify we have already specified what T m is in equilibrium conditions P m is equal to T m is equal to P e is equal to T e there is a torque and the powers are the same with the assumption that the infinite bus voltage frequency is the same as the base frequency after that we calculate the initial conditions the steady state values and then we do a simulation the simulation of course, will remain if you have calculated the values of the states to be equal to those correspond to equilibrium values we will just stay where we are. So, we will just stay where we are and at T is equal to 5 seconds we will give a step change to v ref after 0.5 seconds we will get it back to its old value. So, if I simulate the system. So, I will just run the simulation if I run the simulation and plot time versus P e the electrical power output of the generator we will just see what it is called it is P sorry. So, this is taking some time to simulate because we are using Euler method we can afford to use Euler method because we have in this particular simulation we have kind of removed the stiffness from the equations using by neglecting the stator flux d by d t that is psi d by d psi d by d t is set equal to 0 and d psi q by d t is set equal to 0. And remember another thing is that we ought to use of course, more more accurate method like even though we are using a explicit method it would have been better if you would have used Range Kutta method which is a higher order method, but instead we have using Euler method with a smaller time step. So, do not use do this for for real life problems it is a better to use a better method like R K fourth order I have just use Euler method for illustration because easy to program is for the first time it is not a bad idea. Now, what I have done is electrical power is 0.5 and at 5 seconds I give a step change in V ref if you give a change in V ref the the actually it is a pulse change in V ref which starts at 5 and ends at 5.5, but it excites because there is a disturbance like a pulse it excites the swing mode and you see that the swing mode eventually of course, does die out with time. So, this is a this is how the swing mode would like it to behave. Now, the thing is that if I change the operating condition a very interesting thing occurs. So, I change this operating condition from 0.5 to 0.8. So, we will assume that the operating scenario is such that your output of the generator is 0.8 per unit. So, we are at a different operating point we do the simulation from that point onwards. So, I execute this program again and if you plot while this is executing let me just retrade that we are assuming a different operating point calculating the equilibrium conditions of the state corresponding to that equilibrium point then giving a pulse change in V ref. So, the system gets pushed out from its equilibrium, but it is a pulse change we get the V ref back to its old value. So, what we are doing is because of this pulse change the system gets shaken out of its equilibrium conditions the states move away from the equilibrium conditions and once a disturbance disappears the system attempts to well we have to see whether the system comes back to the equilibrium again or not. In the previous case where T m was 0.5 this actually happened, but now let us see whether it will happen when T m is 0.8. What you notice here of course is not this what you notice is that when you give a disturbance the shake the system gets shaken out of its equilibrium, but it does not come back to its original equilibrium point it just goes on oscillating and this oscillation grows with time. In fact, this occurs under certain operating conditions only and for certain AVR parameters. So, but this is a distinct possibility and this in fact has been observed in real power systems. So, in the next class we will try to see how this poorly damped oscillation or in fact growing oscillations can be damped by appropriate control systems. So, this is another aspect of improving stability which we will discuss in the next class.