 as well as a you know a simulation wherein we had given a three phase fault. And one of the key phenomena which we observed was that in the especially that it was observed in the rotor speed and the angle the phenomena which was observed was the loss of synchronism whenever there is a very large disturbance I mean after for a certain large disturbance also there is a center of inertia motion or you can say the common speed of the system changes in case there is a load generation imbalance which of course can be corrected by governors. In fact there are two things which affect the motion of the center of inertia the in fact the load generation imbalance is affected by the governor characteristics as well as the frequency dependence of loads of course the frequency dependence of loads is generally weak and it is not a good idea to rely completely on the load frequency dependence to get you to an equilibrium speed the center of inertia speed which is an equilibrium. But if you depend on the load frequency characteristics to get you to an equilibrium that perhaps is not a very good idea because for example if there is a large load generation imbalance which occurs then you will find that the frequency settles down to a value which is very low low or high of course it is low in case there is a load is greater than the generation and vice versa. Now governors should be you know enabled governors or what is known as the primary response has to be enabled on all generators or at least the major generators in the system so that each of them in some ways contributes to trying to maintain the load generation balance. So of course there is an issue then of how to share the you know load amongst the various generators you know so for example for a particular schedule there is a load generation balance and you are operating at a certain frequency but if the there is suddenly a load change then the frequency tends to deviate and all the governors which are enabled respond. So various generators increase their mechanical input power and therefore the electrical output power and the frequency tends to an equilibrium. The sharing is dependent in fact on the gains or you know gains are in fact the inverse of the droop of the governor characteristics. So that normally governors are proportional controllers in fact the proportional gain in fact determines the load sharing. If you look at for example suppose one of the governors it changes its mechanical power based on the difference between the reference value and omega 1 the actual speed measured at that generator this is the gain. So the change in the output of mechanical output of a generator will be basically given by this. Now this is of course true of other general governors as well. So in another generator the turbine changes is mechanical power as per this rule. Now if omega f 1 and omega f 2 are equal it is obvious that this delta p 1 by delta p 2 is equal to k 1 by k 2. Why? I have first assumed here that omega f 1 and omega f 2 are equal and another thing is that if your machines are still in synchronism after load disturbance normally to create a loss of synchronism scenario you require large disturbance. In principle even a load disturbance can cause relative angle motion and possibly in certain situations a loss of synchronism situation. But we will assume that the system does the relative motion is stable and the system will retain synchronism in that case omega 1 and omega 2 in steady state are going to be equal. Therefore in steady state your delta p 1 and delta p 2 is equal to k 1 by k 2. So the gains of your governor in fact determine how much each generator shares this extra load. Remember that in an integrated system what matters as far as the center of inertia frequency or the common frequency of the system is the total load generation imbalance. So if once I have integrated all the synchronous generators that is connected them by AC lines you cannot say that a load belongs to a certain generator. So all the generators put together are pooling in their powers and meeting that load. So any load change affects all generators which have a governor and you will find that this mechanical power change is proportional to k 1 by k 2. Of course this is as I mentioned last time a nice way to implement the load change. I mean you can for example keep k 1 and k 2 so that generator share the load according to their ratings that is a reasonable thing to do. So the proportional kind of scheme of maintaining you know kind of changing the mechanical output power of the turbines is quite a good idea. In fact one consequence of having a proportional controller is of course that in case there is a load generation imbalance if I want to change the mechanical power output of a machine then there has to be some steady state error between omega ref and omega. So whenever you have these kind of governors your frequency does not come back to the original value before that is what existed before the disturbance load disturbance. It will have some frequency error but this in spite of this I mean you can choose a gains for example large enough. So if your gains are large enough in fact this error is smaller but this proportional scheme is a nice way of sharing the loads amongst various generators and one of the important thing is a completely local control all you require is the local speed of the machine. So in this way although generators may be interconnected and spread hundreds of kilometers apart so thousands perhaps so still you can have a nice scheme to control the center of inertia motion. The other phenomena which we discussed was of course the loss of synchronism. So in today's lecture let us just recap this aspect also. So today's lecture in fact is we will just think of what happens when you have got even larger systems. We of course consider two machine system in the previous class we will go for even larger systems. When you talk of relative motion there are two issues whenever you look at any small disturbance given to a multi machine system you will find that one of the prominent patterns there are of course other patterns as I mentioned in the previous class but one of the prominent patterns for small disturbance is a low frequency oscillation which occurs around 1 hertz. So that is essentially you know associated with the motion of the you know or rather the states delta and omega of the machines and that is of course a relative motion between the machines. So what we call as power swings or low frequency oscillations etc are in fact electromechanical oscillations which involve relative motion between the machines. Now this manifests as an oscillation. So you will find that you know you give a disturbance and you will find that machines tend to oscillate amongst themselves like a spring mass system and if your small disturbance if the machine is machines are small disturbance stable the system is small disturbance stable these oscillations die out and the machines kind of remain in synchronism. So that is an important point about the relative motion. However for large disturbances last time we saw a large fault large duration fault. So for large disturbance there is a possibility of loss of synchronism of course depends on the disturbance itself and what operating point you are. So loss of synchronism also is a phenomena associated with the relative motion of machines. Suppose you have got the machines not coming back into synchronism but accelerating or decelerating with respect to each other then you would say that the relative motion is unstable and loss of synchronism has occurred. Now loss of synchronism is not acceptable it occurs for certain large disturbance does not occur for most large disturbance and it is a endeavor of an operator to ensure that the system has enough margin it is operated a bit conservatively. So that you do not lose synchronism one of the things which we perhaps did not discuss much in detail but something which you can do is try to increase the power flow bit on transmission line. For example in the two machine system if you increase the initial power flow through the transmission lines and then subjected to a particular disturbance it becomes more susceptible to loss of synchronism. So one of the endeavors of a system operator is to operate the system a bit conservatively so that the system can withstand certain credible disturbance. In fact even while planning the system planner will try out few credible contingencies and ensure that at least for loss of a credible kind of disturbance that is a fault with the loss of a transmission line the system remains in synchronism. What we discussed sometime back of course was this loading generation imbalance which is associated with the center of inertia motion that is the overall movement of the system. So that may or may not involve this center of inertia motion can be thought somewhat as something distinct from the relative motion. So of course as I concluded in the previous lecture it does not mean that there are no other phenomena which can occur dynamical phenomena which can occur. We are focused on two particular phenomena this is relative motion stability as well as the center of inertia motion stability which is associated mainly with the electromechanical states but you can have other modes of instability involving for example poorly designed controls etcetera. What happens when you go from two machines to three machines? So to just kind of give you a crude analogy again of a system which is kind of will give us at least some idea of how the electromechanical modes behave of a process. Remember whatever analogy I am showing you here of a spring mass system is actually not completely does not completely reflect our system. Remember our system is many other modes as well the order of our system is quite high it involves not only delta and omega states but several others too. So whenever I am talking of this analogy it is only highlighting or kind of it is an analog of only the electromechanical pattern which you see in a power system behavior. So if you look at this multi mass multi spring system what really holds true here is that the center of inertia motion that is the motion of this that is you can talk of M 1 V 1 plus M 2 V 2 plus M 3 V 3 divided by M 1 plus M 2 plus M 3 this is the center of inertia speed it is proportional to the sum of the external forces on the system that is F g 1 plus F g 2 minus plus F g 3 minus F l 1 minus F l 2 minus F l 3. So this aspect of the motion in fact remains the same we just add up all the torque equations or the what you call the swing equations of all the synchronous machines just add them up you will get this equation of the motion of center of inertia that is of course something which what we saw in the two machine system will still be valid for a three machine system and so on. So even a 500 machine system the center of inertia motion is going to be equal to the sum of the mechanical power input minus the electrical power output that is the sum of loads and the losses electrical losses. But a much more interesting point which you may consider at this point is what about the relative motion in case of a single machine connected to infinite bus we considered of course the relative motion consisted of the motion of the machine with respect to the voltage source. When I got a two machine system what we saw is the relative motion between the two machines delta 1 minus delta 2 what happens when you have got 3 or 4 or 5 machines connected to each other. Now looking at this analogy of course gives you a picture of what is going to happen. Now what you have now is 3 machines so 1 2 and 3 3 machines are there and suppose I give a disturbance to this system that the kinds of patterns you will see are going to be again quite interesting one of course is the motion of the center of inertia. But if you look at a 3 machine system just to highlight a simple kind of situation suppose you have got one large mass connected to two other small masses and they are connected by relatively a stiff spring. So, you have got a small mass these are m 1 and m 2 are small masses and this is the mass 3 which is a large mass and this is a weak spring. Now if you look at the relative motion in this case forget the center of inertia motion we will talk only of the relative motion. It you can guess of course this is something I do not prove here now I do the mathematical analysis of it. But you can just guess that in case this spring is very stiff that is difficult to stretch it. Whereas this spring is a bit loose I mean the sense you can stretch it quite easily this mass is large these two masses are small. Then you will find if you do a Eigen value analysis you can actually write down the equations of the states you know the differential equation corresponding to x 1, x 2, x 3 and v 1, v 2, v 3 which are the velocities. In that case if you look at the Eigen values in the modes of the system this something you can do with typically you can choose some simple small values of m 1, m 2 a large value of k small value of k large value of m 3. In this particular system in this particular system what you can expect are two actually two oscillatory modes corresponding to relative motion of the machines. So, if you take a three machine three mass spring system you have got you will have two oscillatory modes which actually if you look at the Eigen vectors corresponding to these oscillatory modes that is the complex pair of Eigen values which will come about when you do the Eigen value analysis of this system. Two oscillatory modes one would be of high frequency mode one would be a low frequency mode. So, I encourage you to just do this Eigen value analysis for this system suppose this is 1 kilogram, this is 1 kilogram, this is 100 kilogram or 10 kilogram you can take something extreme. So, that you know what I am trying to tell you will come out very clearly. Let us say this is k is you know 10 Newton per meter and this is k is equal to 1 Newton these are just fictitious values which I have asked you to just consider to do the Eigen value analysis. Now, if you do the Eigen values of the system as I told you will find if there is no friction the Eigen values will come out to be this. In fact, plus or minus is relevant here there will be two Eigen values zero Eigen values these two zero Eigen values of course, are associated with the motion of the center of inertia of the system the motion and actual x 1 x 2 x 3 values corresponding to the center of inertia motion or the common pattern of the system. If you have got friction then you will have a Eigen values of this kind and you will have one zero Eigen value and one sigma 1 sigma 2 and sigma 3 are less than 0. So, this is in case you have got friction, friction is provided that is if you make the external forces a function of speed. So, this of course, in a in a multi machine system or in a actual power system is provided by governors and load frequency characteristics of the system. Now, the two frequencies are in fact these are corresponding to the relative motion and if you look at the relative motion it is easy to guess this is only a guess I am not really proving anything to you can actually do an Eigen value analysis and show it is that in this particular system not saying this first for all systems is that the low frequency oscillation you will find that these masses look at Eigen vectors the information that will give you is that the masses m 1 and m 2 whenever only the low frequency mode is excited the masses m 1 and m 2 move together against the big mass m 3. So, these are the two masses m 1 and m 2 and this is m 3. So, you find that there is a low frequency oscillation associated with these two masses moving against the large mass there is hardly any relative motion between the smaller masses whenever the another pattern of motion is when this m 3 mass does not move much, but there is some relative motion between m 1 m 2 of relatively high frequency. So, you have got these three patterns which you will rather three patterns one with the center which corresponds to center of inertia and the relative motion there are two patterns two oscillatory patterns one is a high frequency one involving masses m 1 and m 2 and low frequency one involving masses m 3 moving against masses m 1 and m 2 which move together. So, this is the pattern which you can infer by doing an Eigen value analysis. Now, the important point which of course I wish to make here is that whenever you have got a three machine three mass spring system you have got two relative modes of oscillation. Now, in general whenever you give a disturbance if it is a general disturbance you will find that all the modes are excited. So, a general disturbance no special initial conditions you will find that the masses move together there is a this high frequency mode and a low frequency mode. So, you can have all this. So, this kind of behavior you can expect. So, if you take a 500 machine system you will find that there are you know 499 relative modes of oscillation yes that is true not all of them are excited for all disturbances. If you give a disturbance or a initial condition which is not the equilibrium condition you will find that certain modes are excited others are not. So, remember that in the n machine system the n minus 1 relative modes of electromechanical oscillations. Now, one more issue which you should remember is that whenever you go from us you know the relative motion although in the spring mass system where it is a linear system a power system is a non-linear system. So, for large disturbances again you may have relative motion going you know is you can have a loss of synchronism phenomena in which machines separate out from each other that is you have got machines one of the machines accelerating in one machine decelerating this is a non-linear phenomena which cannot be predicted by Eigen value analysis. So, you can have a situation where the machines you know m 1 m 2 for example move together and they go out of synchronism with m 3 that is the springs break you can look at it that way the springs are non-linear than the springs can break. So, you can have loss of synchronism phenomena even in multi machine systems larger systems also have this phenomena of loss of synchronism. So, what we have seen in the single machine infinite bus system is also seen in a two machine system it is also seen in a multi machine system when you go from graduate from a single machine connected to a voltage source to a two machine system the additional phenomena which you have to consider is the motion of the center of inertia when you go from a two machine system to a n machine system the center of inertia motion of course remains there, but you have got n minus 1 swing modes or patterns, but again the loss of synchronism corresponding to relative motion is still there you can have loss of synchronism for large disturbance only of course is an important point which you should remember is that in multi machine systems groups of machines may lose synchronism with respect to other groups of machines. So, that is the major difference which you should note and as I mentioned some time in case you lose synchronism that is one set of machines has a frequency which is which tends to be different from other center of machines then you will have oscillations which is something we saw in the previous class you have got oscillations in the electrical power as well as several at several places for example, in this example the midpoint voltage tends to oscillate this is a 50 hertz waveform, but you see the magnitude goes on going up and down. So, in case two machines which are interconnected lose synchronism in that case you have got wild oscillations in voltage and other quantities and usually distance relays will think this is the fault whenever you got a voltage 0 any distance relays which observes this kind of voltage and current will tend to trip and then you have got a separation of the two systems and as I mentioned some time back whenever you have got a loss of synchronism phenomena you have to separate or you inadvertently or in an uncontrolled fashion you separate and once you separate you are left with the other problem that is maintaining load generation balance in the separated systems. If you remain connected you will have see a very large these kind of oscillations and variations which cannot be tolerated because they can lead to equipment damage. So, loss of synchronism is never tolerated for more than you do not allow the machines to slip poles as they say. Several years ago the western part of our country was a synchronous grid. In fact now it has got integrated to several other grids in the country, but this is the scenario which was prevalent about 4 or 5 about 5 to 6 years back when western region was kind of an independent system it was consisting of the states which are shown here. It was seen that for large disturbances groups of machines on the eastern part of the western grid there are large machines here you know at Vindhya Chalkorba and so on. The large number of coal base generating plants here they separated they normally used to separate out if the system was subjected to large disturbances. So, there are several instances where there are fairly large disturbances and the system used to separate out in the sense that these machines used to accelerate with respect to these machines while still remaining connected. The center of inertia motion would have been not really prominent, but the machines would be separating out with respect to each other and because of which the distance relays on these lines used to trip and cause islanding of the western part in the eastern part of the western region. And once this used to happen this used to be a load deficient or other generation deficient area. So, the frequency in this islanded system would suddenly drop and often the system used to collapse. Interestingly in Mumbai we had a controlled islanding we have in fact a controlled islanding system which kind of isolates itself from the rest of the system in case such a situation. In several they have certain checks if this happens they separate out Mumbai from the rest of the western grid. So, this is an example of a controlled islanding, but you can have uncontrolled system separation as well. So, if you look at relative angular stability I should use the word relative angular stability because there is also corresponding phenomena of the center of inertia motion is also an electromechanical phenomena. So, we will talk here in these two points which are here are relating to angular stability. Remember as I mentioned sometime back small disturbances you generally see an electromechanical behavior which is oscillatory which is a mixture of n minus 1 modes. Remember that you can analyze small disturbance stability by Eigen value analysis of the linearized system around an equilibrium point. Remember the power system is a non-linear system. So, if you want to do small signal analysis you have to linearize around an equilibrium we have done this for a study of the AVR some lectures back. You can show of course that for certain operating points the linearized system has got Eigen values which are positive real parts. So, sometimes you can have in fact small disturbance behavior unstable too. So, this is something you should remember large disturbances on the other hand are disturbance dependent they are also operating point dependent and quite inappropriately power system engineers refer to the phenomena of large disturbances as transient stability. Transient instability or transient stability refers to large disturbance. So, if some power system you know confronts you with the word transient stability what he really means is large disturbance instability which results in loss of synchronism. So, transient instability means that it is disturbance dependence and remember that in a synchronous grid that is synchronous machines connected by AC lines it is always found to be an issue. There will always be some large disturbance for which a system loses synchronous. So, you take any multi machine system connected by AC lines synchronous machines connected by AC lines there will always be a large enough disturbance for which the machines lose synchronism. So, just to make this now absolutely clear I hope we will not have to recap this again. Suppose you have got this is a plot of generator speeds under various situations in a multi machine system of course, it is a multi machine system, but I have plotted actually only three machines three speeds here. This is a typical situation if there is a sudden load throw off in the system the generator speeds tend to accelerate. So, if there is a general load throw off you will find that swings you will see these power swings corresponding to the oscillatory motion, but more prominently you will see the center of inertia speed is increasing and of course, if you have got governors or load frequency dependence loads of frequency dependence eventually the system settles down to an equilibrium somewhere at a higher speed. Of course, if you have got one machine with an integral type of governing system or also called an isochronous governor you may find even that the frequency goes down back to the original speed, but most governing systems in a multi machine setup rather all in a multi machine setup you cannot have you know many machines having a proportional controller an integral controller at most one can have an integral controller why that something you try to think over, but typically all machines have proportional type of droop characteristics that is you will have some steady state error in frequency the governor will not get the frequency back to 0 back to the previous speed you may have to do some extra work in order to get the frequency back to the original frequency that is called secondary control. So, that something will not do right now in this course, but you can look at any undergraduate text book they will talk of governor action coupled with slower secondary control. If there is a generator trip on the other hand you will find that the frequency drops and although relative motion is also excited the most important thing here you see of course, is that the frequency tends to the center of inertia frequency will tend to drop and settle down. So, this is a situation where the common that is the center of inertia motion as well as the relative motion is stable if this occurs and this frequency deviation is not too much nobody will be worried too much everybody people will finally say fine we have settle down to a frequency which is slightly different from the original frequency usually power system engineers cannot tolerate anything more than half a hertz or so. So, that is the absolute maximum of frequency deviation of course, in our you normally this the amount you can deviation you can allow depends on what the turbine blades in a steam turbine can you know tolerate looking at the figures at the bottom what I see here what the plots which I have shown here is two machines the speed of two machines is kind of tending to deviate from the machine which is shown in red generator speed which is shown in red. This is an example of a large disturbance angular instability or what I termed as transient instability this is a loss of synchronism phenomena and you will find that it is relating to the relative motion between machines. You also can have we have seen this in the single machine infinite bus with an AVR you can also under certain circumstances have the machine the synchronous machines having small disturbance instability this can happen if for example your control systems like an AVR have very high gains and they adversely affect the oscillatory modes this electromechanical modes that can happen we have seen that happen in the single machine infinite bus system. You find that you have got this motion which does not die down usually it is manifest is growing oscillation and is triggered by any disturbance big and small. So, what you find is that we do have situations in which a system operator is operating the system and he finds that after sometime he notices that there are oscillations which simply do not seem to be dying down. This is occurred in the past and what he is effectively seeing here is a situation of small disturbance instability where there is no large disturbance which is occurred in the system, but you have entered an operating point or a new equilibrium point of your system is such that the system is small disturbance unstable. So, you may actually have these electromechanical modes going unstable small disturbance unstable under certain situations you cannot operate this way you will have to do some corrective measure. So, that would involve at least at the planning and non real time operating stage tuning of controllers. So, that this does not occur of course remember that something which should always keep in mind is what I have told you right now is a specific aspect the electromechanical oscillations are a specific and an important phenomena which occur in synchronous grids, but there are other you know kind of phenomena and we should have some idea of the time scales which are involved when studying them. Luckily there are there is a clear separation of you know the time scales which are associated various transient phenomena. For example, if you look at the scale at the bottom the fastest transients are those associated with lightning and switching. Then you have got the slower network transients associated with the electrical network as well as what you call as torsional transients associated with the shaft of the rather the shaft and mass system corresponding to turbines and generators. These are also phenomena which are manifest, but these are faster phenomena. Relative rotor angle dynamics are in fact what we were studying now they are occurring in the time scale or the frequency range of 1 hertz you know half hertz 1 hertz 2 hertz half hertz you know that range you know these are relative rotor oscillations. You have got slower frequency dynamics associated with the center of inertia motion that of course depends on the overall inertia of the system the sum of inertia constants of the system. So, it is a bit slower in fact prime mover dynamics those associated with the boiler etcetera are very very slow compared to for example, rotor relative angle dynamics and of course, very at a very slow level you have got aggregate load changes load there is a kind of pattern to how load changes over the day. So, that is a very slow change. So, luckily for us all these phenomena well separated out in time. So, actually that in that sense we can analyze each phenomena separately. So, you know that is an advantage I could talk of relative angle motion without worrying too much about for example, some of the real Eigen values of the some of the patterns associated with the damper bindings of a synchronous machine. So, you could actually look at electromechanical variables in a focused way because one of the reasons that is a good time separation between various transients you need not worry about lightning switching transients whenever you are studying about you know relative motion dynamics. So, if you recall it is important to correlate what we have done before in this course and what we are doing now whenever you are studying slow transients you can assume that the network and some parts of the system are in fact, in quasi steady state. So, for example, if you are studying electromechanical transients you could afford to neglect the network transients and the stator transients because they are relatively fast. This is something we have kind of beaten to death when we were studying the in the first few first 5 to 10 lectures. On the other hand whenever you are studying lightning and switching transients you can for all practical purposes assume that the speed of the generators etcetera is constant. In fact, you can most of the times whenever you are studying such transients you can safely assume that the machines is a constant speed running at a constant speed you may even model a machine as a voltage source. So, remember that our modeling it has to be consistent with the kind of phenomena we wish to study. Now, correspondingly you can also see what are the controls you know whenever talking of transients you can also talk of in terms of controls. If you look at equipment protection and power electronic controls they occur at a very, very they are a quick acting. So, for example, the primary protection can act within a few cycles you know for example, if there is a three phase fault on a transmission line you can the state of the art is that you can switch it off in you can trip out that transmission line and isolate the fault in just two or three cycles you can have differential protection which acts even faster. So, this is the equipment protection acts at a very fast scale. So, does power electronic controls. So, if you recall for most electromechanical you know studies we could assume that a excitation system is the converter is modeled simply as a kind of a gain you give a control system and it implements the order because that is because power electronic controls also in a system are extremely fast. So, if I got HVDC systems or the static excitation systems you need not operate you know model them in excessive detail if you are studying for example, relative rotor angle dynamics or prime mover dynamics. So, just remember that modeling and analysis and in fact, the tools which you use for certain phenomena are kind of designed for that phenomena. Of course, you may take an approach that I will model all the whole system in full detail. In fact, you can model network transient state transient everything and use a variable time step in implicit method to integrate or simulate this system. This is also a viable approach. In fact, nowadays there is less of a distinction between fast and slow. You do not have different programs to study fast dynamics, slow dynamics, long term dynamics. You can in principle make a program in which you use an implicit numerical integration method and have a variable time step. So, you can capture not only the fast transient, but the slow transients as well. But at least whenever as a first step or to keep a good distinction between all the phenomena. In fact, people have written different programs with different modeling assumptions for different phenomena. One class of course, I am not you know we will be talking of rather methods for improving stability a bit later in the course. I mean we are almost at the fag end of the course, but we will still spend about a lecture discussing various ways to improve stability. You can actually think of also implementing what are known as system protection schemes. For example, let me give you an example. Suppose, there is a big load throw off. So, you can what you can do is if there is a big load throw off and the frequency falls on precipitously rather it should rise precipitously. Suppose, for example, let there is a generation trip and therefore, the frequency drops precipitously and your governors are not able to act well in time. Then you may think of deploying schemes like rate of change of frequency protection. That is if the rate of frequency drop is higher than a certain value trip out some loads. So, you can have these kind of local schemes which try to ensure that the system remains you know the frequency of the system remains within a certain value. So, you can have these are also known as emergency control schemes or rapid action schemes and so on. So, these are kind of what are known as system protection scheme which operate a slightly slower level. They are not what you call equipment protection schemes. So, they are typically act in the range of half a second, 1 second and so on. In fact, with the availability of synchronized non-local measurements under what are known as wide area measurement systems. One can devise non-local you know when I say non-local means you are using signals which are time synchronized and obtained from various locations in the grid. You can use these signals to fine tune some of your system protection schemes. So, this some in interesting research area which some of you can think about taking up. Governor prime mover controls are somewhat slower than system protection or equipment protection schemes typical equipment protection and system protection schemes and of course, manual control is the slowest amongst them all. So, this is a good I giving you a kind of idea of various transients which you can encounter in what are their what is the bandwidth. Let us just look at some actual system. See this is what I am going to show you next is the kind of measurements we have observed in our grid. You know for example, at the present time in India you have got if you look at our country there is a approximate map. This portion of the grid is one synchronous grid that is their synchronous machines connected to each other by AC lines. So, if you have a synchronous generator here you will find AC line path right up to any generator here as well. So, this is basically a synchronized grid you have another synchronized grid this is the southern grid. These two grids are not connected to each other right now as of 2010 by AC lines they are connected by DC lines. So, you actually rectify the voltage invert it again. So, these two systems need not be of the same frequency, but generators here if they have to have a stable relative motion then all of them have to be running in synchronism in steady state. Now, what we did was we have measured frequency at fairly distance distant locations in the grid. In fact, the ones I am going to show you are approximately 400 roughly 200 300 to 400 kilometers apart. This disturbance which I am showing you is a generator in trip. So, generation trip has occurred and what you notice here of course, is that the frequency drops. So, the center of inertia motion is suddenly excited whenever there is a load generation imbalance and this is a relatively large generation trip is an expanded view actually which occurred somewhere the interesting thing is that is generation trip occurred somewhere in eastern UP. Eastern UP is a state here. So, it occurred somewhere here and what you notice is that the frequency change which is measured in the western region. Since this is a synchronous grid and relative motion was stable for this disturbance the frequency here, here, here, here would have been the same, but since there is a sudden load generation imbalance caused by a generation trip the frequency dropped here. It in fact, would drop everywhere in the grid because for this particular disturbance the system did not lose synchronism. Relative motion was stable, but the measurement was done here and interestingly for load trip here for generation trip here the effects are seen more than 1500 kilometers away in the western region and this is because they are all interconnected by AC lines synchronous machines connected by AC lines. So, you see that the frequency drops down at a certain rate then there is another frequency there is another probably generation trip here then again the slope becomes normal. So, probably there was a load tripping here. In fact, if you look at this slightly compressed graph what you notice is that the frequency drops kind of has a lower slope here and at 48.8 roughly you have under frequency relays which are system protection schemes which tend to get that the get back the center of inertia frequency back to normal. Remember that the machines or the frequency at Mumbai, Sangli and Surat which are all in the western region are practically the same forget all this noise which appears here this is measured at the distribution point. So, you have got a lot of noise, but you notice that all the frequencies are moving together you can hardly distinguish between the three locations the frequency. So, there is no loss of synchronism occurring the machines all move together, but because there is a load generation imbalance is a very prominent motion of the center of inertia of the system. This trip remember occurred 1500 kilometers roughly away from the point where these measurements were taken this is another disturbance, but this is a fault situation this is a frequency measured at four locations in the western grade for a fault which occurred in the western grade. So, what you notice of course, if you look at it carefully is that there is a very you can see relative motion oscillatory motion of one hertz. So, you will find machines in the north part of the western grade western part of the country that is at Amdabad and Surat are swinging together against machines in Mumbai and Sangli. So, this is approximately one hertz motion and they are moving out of a. So, you actually see this relative motion this relative motion is of course, stable because these oscillations eventually die out with time. So, that is an interesting really interesting phenomena which are seeing here which of course, is consistent with what we have understood. Remember that in case you have got synchronous machines say connected like this two synchronous machines in close proximity connected to each other via say a long transmission line to another machine you will find that there is a relatively high frequency local mode and in which these two machines swing against each other and there is a low frequency oscillation in which the two machines swing together against this machine. So, you can have a mixture of local and what you call inter area modes even within a power plant if there are many units you can have what are known as inter plant modes. So, I refer you to the our discussion of the two mass spring system three mass spring system where you have got two masses connected to a large mass the kind of behavior you have there. So, in fact, you can break up the motion into several modes and in some cases you can very easily distinguish between oscillations within a plant or between generators in close proximity or machines moving together against other machines which are far away these are called inter area modes. So, you can have this kind of mixture of several modes remember I told you that for a n machine system you can have n minus 1 modes. So, you have got all these mixture of modes in certain situations you can actually distinguish them you have got a high frequency inter plant modes relatively higher lower frequency local modes and much lower frequency inter area modes. So, you can have all these modes of electromechanical modes of relative motion. Interestingly for last disturbances you may lose synchronism this of course, is something which you have discussed sometime back, but it is difficult to sometimes predict instability. For example, which group of machines will lose synchronism with respect to other group of machines it is a bit difficult to predict. In fact, if you look at the movement of the machines following a large disturbance sometimes just by looking at them for a short while it may not be possible for you to tell whether the system is going to go unstable a few seconds later or not. And if it goes unstable what are the groups of machines how will the system separate out that is something which you cannot easily tell without actually doing a simulation on a system with a two machine system there is only one mode of separation the two machines fall apart. Single machine infinite bus system the single machine loses synchronism with the infinite bus in a multi machine system that may be much more tough because you can have many modes of separation just like you can have many modes of oscillation for small disturbances for large disturbances for certain large disturbances you can have many modes of separation and that may be not easy to tell which machine is going to fall within which group an accelerating group or a decelerating group. So, relative motion a large disturbance relative motion leading to loss of synchronism may be difficult to it may be difficult to predict unless you do a simulation I actually see there is no you have to do it the brute force way sometimes to really and get an answer to this question. So, just look at this graph which I have shown you here these are the relative angle plots for the motion of around 300 generators in the western region this is a kind of an academic, but realistic study not a real study, but a realistic study of 300 machine system. And this is the evolution of angles up to two seconds for a fault which occurred at 0.5 seconds which is cleared after some time cleared in the sense the line which the fault was there was stripped out. So, the rest of the system was fault free after that, but because of this disturbance the system was disturbed from the equilibria and starts evolving the transient starts evolving and you see actually several modes here you see a low frequency kind of mode and some high frequency oscillations also. So, you have got a mixture of several relative modes which are seen after this disturbance now the question is will the system lose synchronism or not is something is difficult to say just from what you have seen here. So, let us see how the system eventually evolved actually did a numerical integration of the system and what I see is here is the system breaks into three groups. So, groups of machines and our losing synchronism with another group of machines. So, this kind of thing can cause a complete you know this is a system wide kind of break up of the system and you know well the system is still interconnected, but as I mentioned some time back you may have outer step relays or distance relays actually causing a system separation. And now if each of the islands which are formed after this system separation the load generation balance is not quickly maintained using under frequency relays or by governor action or prime mover action or sometimes even generated trips you may find that each of these islands may collapse and that will cause a system wide blackout. So, these kind of things do occur though rarely it is a nightmare for a system operator if a system a disturbance leads to a system collapse of this kind. So, it starts off with a loss of synchronism then a system separation the system separation causes the islands the islands do not survive and you have got a blackout. So, you have to now start or restore the system from scratch have to re synchronize all the machines. So, that is a extremely intensive and difficult process and sometimes it may take hours if not days you know to get back the system back to normal let us re synchronize and re integrate the grid. So, this occurs rarely, but it can occur it is a nightmare really for a system operator, but for us academicians and students it is a very interesting outcome of the physical equations which govern the system. So, in that sense it is very exciting we do not have time in this course to now look at the power system tools which are used to understand these electro mechanical transients. We have actually done a two machine example and I told you how the equations are, but I really did not tell you the intricate details about how you solve these equations. Of course, you have to apply some numerical integration methods or in case of doing Eigen analysis you have to form the linearized system matrices, but we will spend a bit more time on the topics which are shown on your screen in the next lecture. This is more to do with the power system tools which are used to understand transient stability or even small signal and stability. Remember that simulation as a tool is very general it you can use it to understand fast rather small disturbance as well as large disturbance problems. So, this is what we will do in the next class. So, see you then.