 One of the interesting things which we were discussing at the end of the lecture the previous lecture was that the rate of you know magnification or increase of the unstable oscillation predicted by Eigen analysis is somewhat different from what is obtained by our simulation. Now, simulation we see a larger rate of change. So, that of course, was not very surprising. In fact, the reason why that probably what was true was because we used Euler method for numerically integrating the equations which tends to show an unstable system to be even more unstable than it actually is. So, that is one of the reasons why the rate of growth in the simulation seem to be much much higher than what we saw in the from the Eigen analysis. Just to you know show you the results again I will do that in a short while from now. So, today's lecture of course, we will continue with our discussion of what this behavior the simulation of the AVR automatic voltage regulator regulated synchronous generator which is connected to an infinite bus, but we will probably have more time and we can now shift in this particular lecture to another topic which is the issue of load models. So, there is another component or you know a class of components which need to be considered which are the loads themselves. So, what we will do now in today's lecture was the new topic of course, would be a load modeling which we should begin with in this particular lecture. We will of course, discuss some of the remnant issues in our discussion of the automatic voltage regulated synchronous generator. Now, to see the results as I mentioned sometime back again I will run a silo program which really displays the behavior which I was talking of. So, what I was telling you in the previous class was the operating point corresponding to the torque being equal to 1 per unit appears to be unstable because the system when given step change in mechanical torque does not seem to be going and settling to an operating point is a growing oscillation. The rate of growth of the oscillation is much higher than that was predicted by Eigen analysis. Remember here we are giving step changes in mechanical torque and in some previous simulations we also saw step changes in the voltage reference of the automatic voltage regulator. Remember that the final operating point to which it has to settle to must be small signal stable in order that the system go and settle to that operating point. This is not what is happening in the case of the operating point corresponding to mechanical torque being 1 per unit. In fact, the operating point corresponding to T m is equal to 0.5 is in fact stable. So you will find that this first disturbance which in which we gave a step change of mechanical torque from 0 to 0.5 was indeed stable the system was kind of settling down to an operating point. But the second disturbance which takes the system to a new operating point is does not settle down because the new operating point is not small signal stable. So, this is what really we were discussing in the previous lecture. So, just to make things even more clear we just look at the same disturbance with change in. So, our first disturbance of mechanical power resulted in the electrical power settling to 0.5 that was the first step change. The second step change taking it to 1 does not seem to settle because the new operating point of electrical torque equal to the mechanical torque equal to 1 is not a stable 1. So, that is what we saw in the previous class. Now, when we did the Eigen analysis we will do the Eigen analysis around the operating point corresponding to this. What we saw then was indeed the system is not small signal stable because the real Eigen value has got a positive real part. The issue we were talking of was of course, that the rate of growth of oscillation of the oscillation as observed in the state variables etcetera was not as was much higher in the simulation than was predicted by Eigen analysis. So, that is one point which we discussed and we attributed that to the numerical method. I encourage you to actually try out other methods like trapezoidal rule etcetera to see how much what is the rate of you know the magnification of the oscillation whether it correlates better with the Eigen values. So, this is something I leave to you as an exercise. There are a few more points which we need to examine. If you look at the plot of power we will just do this again we will have to simulate it because the variables have got cleared. In fact, it is taking a very long time because I have used Euler method with a very small time step. So, this is not the thing to do in real programs. Now, one more thing you notice is that is oscillation not just growing with time. If it was a exponential growing oscillation as was predicted by Eigen analysis we would eventually find that this just blows up in this fashion and goes off to infinity practically. So, this is what our you know your Eigen analysis predicts, but of course, Eigen value analysis remember is of a linearized model it is not valid for large disturbances. So, after a point you know our linearized analysis is not valid in the sense that as the oscillation grows the deviations from the equilibrium become large and it is no longer correct to assume that our oscillation should behave the way it is predicted by Eigen analysis. In fact, what we find actually is it is kind of the oscillation is kind of saturating to a steady oscillation. Now, this is an interesting behavior which cannot be explained by the linearized analysis it is purely a non-linear phenomena and what you see is that this is something which is obtained only by simulation it is something we cannot predict by Eigen analysis. In fact, the origin of this itself is an interesting enough dynamical phenomena. In fact, if you look at the field voltage what we see that the field voltage seems to also oscillate and it is clipped to a certain value why is it clipped because the static excitation system has got limits. So, that is the reason why it is getting clipped. Now, in case I do not have these this clipping action. So, if I remove the limiters in the model in that case what happens now if I do that. So, I will just do that for you what I will do is I will make the limits very very large which in fact practically disables the limits I may instead of 7 times the terminal voltage E f d is limited to 70 times which is of course, a very large value. So, for all practical purposes this excitation system is no longer limited. So, I rerun the simulation without with the excitation system unlimited having unlimited voltage field voltage capability and in that case this is what I get. We will plot the phase angle the angle delta the rotor angle and now you are getting an altogether different behavior. So, if I remove the field voltage limits you are getting altogether different non-linear behavior delta is going on increasing. So, instead of having. So, if you look at what is predicted by Eigen analysis around this equilibrium point is for any disturbance the deviation grow with time and go to infinity this is what small signal analysis predicts. What you got is why the non-linear simulation was with Euler method was a higher rate of growth initially with saturates this is in case exciter limits are model exciter limits are model and in case you assume that excitation system does not have any limits. We find that this movement of delta eventually it becomes a monotonic movement as seen in this figure. In fact, if you expand this this will become very clear. Do you see this? So, what you are seeing is the classical loss of synchronism phenomena. So, what happens is that you get different non-linear behavior depending on the kind of excitation system model you have used. In fact, excitation system model with limit seems to be giving us a sustained oscillation and not just a growth towards the loss of synchronism. This is a classical case of loss of synchronism. Remember loss of synchronism is also a non-linear phenomena it is not something which cannot be predicted by Eigen analysis and it because it is a kind of monotonic increase in the rotor angle after a point it just goes and loses synchronism. So, this is these are interesting non-linear behaviors observed in the synchronous machine. In fact, if a machine is small signal unstable there are two things which will happen you will have a sustained kind of oscillation because and it does not settle down to the equilibrium point or you have got this loss of synchronism phenomena. So, both these things are in fact possible and I will now show you a real life example or real life measurement of small disturbance instability. This was observed on 28th August 2010. What you see in this figure is the frequency of two places two locations which are quite far apart. In fact, one is in Mumbai and the other is Ahmedabad both in the western region of our country. What we see is that the frequency which is a near about 49.9 hertz is in fact not constant you know the out here there seems to be an operating point change and thereafter you see that the frequency at Ahmedabad seems to grow with time and then settles to a sustained oscillation which in fact lasts for more than a few minutes. So, you just see how it is just going on and on. So, this is an example of small disturbance stability which has been observed and if you look at the frequency of swings of these swings it is roughly you know corresponding to one hertz. So, this is in fact a electro mechanical swing which is observable in the frequency measured at Ahmedabad and you see that the frequency does not damp out. In fact, it does damp out, but only after the operating point appears to change. So, it just goes on and somewhere here the operating point changes which is seen by a slight change in the frequency and then this oscillation peters out. So, this is an example of small disturbance oscillations you know of poorly damped small low frequency oscillations of swings and this was actually observed in our Indian grade. So, we will just have a look at it again we will just run through it again. So, these are the frequencies measurement and there is a frequency and finally, the oscillation dies down after a change in operating point. So, you have a sustained oscillation for almost 5 minutes. So, shows that that particular operating point was in fact small disturbance unstable. Of course, if the system tends to lose synchronism or in some cases when you have got sustained oscillation there is a good chance that some relay or predictive relays which are present in the system may pick up. They will see that something abnormal is happening and if it fits in the logical conditions which are given for relay tripping operation they may actually trip out the machine. So, we should of course, take our results is not the full story. In fact, once a system is unstable it is possible that there may be relay operation at some point of time with the responses playing out the unstable responses playing out. So, this is something you should keep in mind. So, the things which you now should remember about our AVR modeling and simulation what are the issues? What is the special issue? We have of course, done the modeling and behavior of the AVR and the special behavior when it becomes unstable under certain circumstances. So, these are the special issues these are not the issues of an automatic voltage regulation system. In fact, if the system is stable and designed well you may find that you will not come across instability that often. In fact, it may happen once in a while for situations which are not envisaged when you are doing your design. But more often your voltage regulator will behave normally and you will have a stable response and you will have good voltage regulation and so on. So, we should not take instability as the thing which always occurs. It occurs under special circumstances. Now, the thing of course is that it does occur sometimes that is something which you should keep in mind. Now, the issues which we have seen in this first part of our lecture today was that there are two kinds of non-linear behavior which may manifest itself themselves and those are will not be predicted by small signal analysis. They are sustained oscillations and the loss of synchronism. Loss of synchronism incidentally can take place under other circumstances as well. If there is a large enough disturbance you may also lose synchronism. We saw the situation in the latter simulation which we did today in which the system lost synchronism because the rotor angle went on increasing with time and after a point it just split from the infinite bus to which it is connected to. Split I mean the system may still be connected, but your rotor angle has become very large and keeps on becoming large and the machine kind of slips against the infinite bus. There is a pole slipping taking place and as I mentioned some time back as you have pole slipping and other unstable responses it is very likely some relays also would trip. So, in that sense our simulation is not really complete. So, we saw loss of synchronism because of small signal instability this also can occur or you can have sustained oscillations this also can occur. Now, one of the other issues which are special to our discussion of automatic voltage regulation system was our definition of q v and angle during transient nothing to do with automatic voltage regulation systems per se, but as we analyze it we did come across definition of reactive power voltage magnitude and angle during transient. So, just to these are important things which you should remember these are definitions it is difficult to assign a physical meaning to voltage magnitude reactive power and phase angle during transient that is difficult to do. So, these are definitions. So, once the definitions which you should remember as voltage magnitude is root of v d square plus v q square this is a definition of voltage magnitude during transient conditions as well reactive power can be defined as this is a reactive power fed into the terminal generator terminals from the generator this is q this something is consistent with our definition of reactive power in steady state. So, it is up to you to prove that our definition of reactive power in steady state confirms to this definition in transient. So, this is an interesting thing which you should try to prove similarly phase angle theta during transient can be defined as v d by v q these are these are all interesting things which are which do not have the difficult to assign physical meaning to them during transient, but they confirm they can be used and confirm to the steady state definitions in steady state. So, in steady state you will indeed find that our intuitive idea of what voltage magnitude of a sinusoid is what is being given by v d square plus v q square the square root of that. Similarly, the this definition confirms to the definition of reactive power in steady state. So, this is something you prove. So, this is also something you should prove. The other special issue which I did not I have not tackled here is that you have other kinds I remember when we discussed our automatic voltage regulation system or the excitation system in general you had basically the excitation system power apparatus then you have got the control system which is essentially the regulator, but the automatic voltage regulation function is not sacrosite you know in case in case any limit of the synchronous generator is hit. Suppose the field current limit of the synchronous machine has been exceeded in the sense that the field current has become too large and it causes excessive heating of the field winding what we would like to do is once this limit is hit we can sacrifice what we need to do of course, is reduce the field voltage. So, what one can do is suppose this is your a v r suppose and this is your static excitation system a simple model which is just a limit. This is the voltage limits at the output of the static excitation system, but on the other hand if a field current limit is exceeded what you need to do is reduce reduce the field voltage to reduce the field voltage we will have to reduce the field the signal given to the AC to DC converter the thyristor converter by the a v r. So, one thing you can do is have a summing block here plus and minus in case so I will just write the logic here in case field current is exceeded override override or modify the order given by the automatic voltage regulator and reduce the field voltage. So, that is the thing you ought to do so I will not spend much time on what logic you can use in fact, you can instead of reducing the field the control order given to the static excitation system you can also instead modify the field voltage reference itself you can reduce the field voltage reference in case the field current limit is exceeded. Remember of course, this is a point of practical concern in a designing a logic is that when the field current is exceeded it exceeds its steady state limit it is not necessarily to necessarily to do anything right away. The thing is that the since the time constants for heating up of the rotor are much higher or larger you can wait for a few seconds it is not going to the machine is not going to get immediately it will not exceed its temperature limits. So, for a short while you can you can in fact, sustain a slightly larger value of field current than the continuous limit. So, in fact for a short while say for few seconds you can in fact, exceed the field current by say 1.2 or 1.3 times the steady state limit there after you can actually start dropping down the current. So, what basically is important is that the temperature of the rotor should not rise the rotor winding should not rise beyond the rate. So, for that you can sustain for a very short while a larger field current than the continuous maximum limit a maximum limit under continuous conditions. So, it we can call it some kind of transient rating of a synchronous machine may be higher than the continuous rating of the synchronous generator. So, your whatever logic which you use in your designing of the excitation control system should use the fact that the system has got higher transient ratings than the steady state rating. So, for short while you can in fact, use the field forcing capability of a synchronous generator to improve the response of a synchronous generator. The other thing which is used in automatic voltage regulation system or rather I should say the excitation control system which also includes the regulation system as one the major component is the stabilizing function. The stabilizing function or the stabilizer is something which does not act in steady state at all. So, those in steady state you should do voltage regulation if some limit is being hit modify the regulation function override it slightly. But both these limiting functions and the regulation functions are to a large extent steady state functions, but if you want to use improve a dynamic response. For example, we saw that for a particular operating point your oscillations were unstable there was small signal unstable predicted by linearized analysis as well to be small signal unstable. In such a case you can modify your basic control system. So, for example, you could this is the automatic voltage regulator is giving EFT. You could for example, put in a modulating signal or an additional signal to modulate the aviar reference in a certain way. So, as to improve your stability. So, what you are doing is take a signal in which this oscillation is observable take some signal for example, the speed power etcetera design a control system design a transfer function appropriate transfer function and modulate the voltage this modulate V ref at this summing point. So, that the oscillation which would possibly grow with time for certain operating points is in fact stable. So, this is in some sense a controller design problem you are designing an additional loop additional stabilizing loop in your system which already exists. So, as to make your system stable. So, this is something you could do of course, the question which may be asked which is very natural is that instead of having this particular loop this particular loop here why cannot we just modify the transfer function here. So, instead of just having k upon 1 plus s t a the thing is that the parameters of the aviar in fact, do affect the you know the response you know the stability of the swing mode. If I change this k this is something I did not actually simulate and show you but this is something you can try to do it try to do it yourself you change the aviar gains or instead of a simple proportional gain you have a proportional gain in addition to a lead lag block. So, if you have got all these things it is possible in fact, for you to modify the stability of various equilibrium points, but normally this is not what is done in fact, you will have this extra stabilizing loop which is provided here. So, you have got something which modulates the v ref and by doing that you try to improve the response. So, let me just put it this way. So, what you are trying to do here just take a minute yeah. This is your system this is your v ref this is your synchronous machine excitation system etcetera. You have got a variable like speed or delta or power in which this oscillation is visible delta is a bit difficult to measure using just local measurements, but speed and power you will certainly see these oscillations along with delta. So, you can take speed as a feedback signal design a controller and modulate this voltage reference itself this is what I really mean this controller is also called a stabilizer because it has no steady state function. So, it in fact should not you know override or interfere with the voltage regulation function too much. So, you will have to limit this another thing is that during steady state it should not affect your voltage regulation function. So, a controller of this kind will always have some high pass filter component which will prevent any output from coming out in steady state condition. So, this is the generic way how people try to stabilize oscillations the other option as I mentioned is do not have this loop, but you play around with the gains of the AVR also the structure of the AVR itself. This is also conceivable, and this is found to be a better way of doing things because you can actually choose variables in which the oscillatory mode is more observable. And therefore, I have you can have a much more effective stabilizer this way. So, this is something of course stabilizers is something we will just do briefly when we understand methods of enhancing system instability. Now, let us just go on to another point we will not really going to stabilizer design at this stage that is the topic of doing other kinds of simulation. Now, we have really done in this particular past couple of lectures two simulations we have done simulations as well as Eigen value analysis of disturbances like step change in V ref. In fact step change in V ref is very much doable you go to a power plant and you go to an excitation system there will be a provision for you to which will allow you to give step changes to an automatic voltage regulator. So, doing this step change is very much possible it is a realistic disturbance which you could do give. In fact, a system operator may wish to change the V ref why is this facility given not just to test, but V ref may be changed by a system operator or plant operator based on the reactive power output which he wants out of a synchronous generator. So, V ref in fact is decided by us if we are operating the power system or a particular generator, but the other disturbance which we consider that step change in mechanical torque is not really very realistic we cannot really give step change in mechanical torque it is not an easy thing to do the prime more systems are much less amenable to this kind of step changes. In fact, to give a step change in torque what conceivable you would have to do is suddenly increase the you know in some sense the steam force on the turbine which is not really feasible, but we nonetheless use that as a disturbance just to show a certain phenomena. So, just we will keep the realistic things in mind a more important disturbance which does occur which we could have tried to simulate using this simulator you know this simulation which we did was simulation of a fault simulation of a disturbance in the form of a fault. So, fault is a short circuit due to loss of insulation the system. So, you could have a transmission line in which the insulation is broken down on some insulator as a result is a flash over between a conductor and a tower and that is a short circuit effectively. So, that is a fault and a fault is usually cleared by protective action by tripping of the faulted line. So, there is a system which actually senses this and trips it out. So, you can in fact so what we did was we studied very benign disturbances in fact some of them unrealistic like a step change in torque. We can in fact give other disturbances for example, we could have model instead of one line two lines and assume that at some point of time there would be a fault here that is v d v q at the terminal of the generator would become 0. And as a result of which some protective relay would act and this line would be out. So, you would need to change your equations. So, whenever there is a sudden disturbance like a fault your equations of the system change. So, your simulation in fact would require you to not change the inputs by like in the case of v ref and t m, but in in case of a fault you would need to change your mathematical equations which describe the system itself. So, if you got a three phase fault for example, you would need to put v d v q equal to 0 for the faulted duration. Once the fault was cleared you would have to write your equations describing the interconnection only one line would be present after clearing of this line. So, you need to modify the equations. So, this something is not very difficult to do you you can try to do it yourself. We try to give a small disturbance rather a rather a large disturbance in the form of a fault and then you would trip the one of the transmission lines on which you have a fault. So, this something you can try to attempt to do of course, one point you know it is important to raise it at this juncture is that in case you have got an unbalance fault you have got an unbalance fault. There is one problem which you will come across is that your model of the network or interconnection using d q variables may be of less value because when you do the d q transformation of an unbalance system unfortunately you will not get time way invariant differential equations you will have differential equations with vary with time. So, please chew on this you just think over this and contemplate what would happen in case you did have an unbalance system. If you apply it for example, d q transformation on an unbalance star connected R load a resistive load just a star connected resistive network. You will find that the equations we describe them would suddenly become time variant in the d q frame when you in the new variables. So, this is an important point which you should remember. So, you need to handle this in when you try to simulate practical disturbances because faults most of the times are in fact single phase faults. Single phase faults which are the most common type of faults single line to ground fault and so on. So, when you use d q frame first thing you will of course, have to consider the 0 sequence equations. The second thing is that the d q equations the equations in the d q variables will become time variant. So, when you simulate you will have time variant components in your system. So, let us go now just we have kind of concluded our discussion of automatic voltage regulation systems and excitation systems. Let us just look at where we stand a kind of bird's eye view we have completed the modeling of excitation systems right now. We have also done modeling of synchronous machines and the general analysis general principles for the analysis of dynamical systems. What we need to do in the next couple of lectures or maybe three lectures is the modeling of loads prime viewers and transmission line. These are important components which we should consider in our discussion. Before we move on to a relatively shorter discussion of stability of interconnected systems. In fact, we have studied a bit of this in the simulations which I have shown. I have shown you loss of synchronism small signal instability at certain operating points. These are phenomena which actually occur and I have tried to show them with a very simple system. In when we talk of stability of interconnected power systems later on we will move on from studying just a single machine system to multi machine systems as well. I have already kind of introduced you to how you can make stability analysis tools. You have to make basically you have to numerically integrate the differential equations or do an Eigen value analysis of the linearized systems around equilibrium points. So, these are the basic tools in our armory which we can use to attack this problem. Of course, we will I have already given you a hint of how you could improve the stability say the small signal stability of a system by making improvements or designing or constructing additional loops controller loops in your power system. So, we will discuss this aspect as well later again. Now, one of the things we will move on to today is the modeling of loads. Loads of course, are difficult to model you will appreciate that in a large power system loads. In fact, you cannot talk of an individual load I mean it if for example, if I am studying the Indian power grid the whole you know grid let us say that the southern grid of the system which is a of this country which is a synchronous grid. So, southern grid of our country is a synchronous grid. Now, if I am going to talk of load on the grid we actually have a vast or a huge diversity in the kind of loads which we have. In fact, even if you look at the loads in this studio they are in fact many and varied. In fact, you have got incandescent lamps you have got this computer load then you have got an air conditioner which is right now off because it thumbs a bit and many other loads which I have not mentioned here. So, when you are talking of modeling loads in a system do I have to model everything in anything in detail for example, do I have to model the motor in an air conditioner or do I have to consider every each and every bulk. Now, you will realize that this is will get us nowhere because when you are talking of the dynamics of the complete southern grid which has millions of loads millions of individual loads it will become impossible to do any kind of analysis of the kind I have been mentioning. You cannot really find you will not be able to come up with any useful inferences because the system will become too large to handle you will and even more worse you will not have the data of each and every load loads keep on changing depending on the day weather season and you know and so on. So, it will become an impossible situation to model loads. So, what we really need to do is aggregate loads wherever we can now aggregation by aggregation I mean I will aggregation and not only clubbing together of similar loads, but even modeling them by some gross characteristics we do not really go into very much in detail unless we feel that indeed there is a reason to really model each and every some particular component in detail. For example, if I got a few way a few large motors in the system you know of say several megawatts like our power auxiliary in some cases you may require to model depending on what study you are doing model each and every individual load in the power plant itself. So, that depends on the nature of the study you are doing, but more often than not many loads can in fact be clubbed and we will we have the flag of loads which need to be considered in much more detail. So, most of the loads will be modeled by some general and aggregate method and special loads of course, you will have to flag off and model in detail special large loads. So, that is what we need to do load at a bus the simplest way you can try to model a load is simply by describing how it is real and reactive power the amount of real and reactive power it absorbs as a function of frequency and voltage. So, you know you can just make a algebraic relationship between p and q of the load and the frequency prevalent frequency and voltage and the bus to which it is connected. Now, the best the simplest way you can do it is some kind of simple polynomial or a simple first order polynomial representation for example, p by p 0 is equal to 1 plus k p v into delta v this is a simple representation first it will be valid for practical for small changes in voltage and frequency this is a algebraic representation. So, k p v and k p f are some parameters which you have to take out a kind of for that aggregated load. So, what is p 0? So, p by p 0 is the normalized power power at the normalized. So, p is equal to p 0 when delta v and delta f which are the deviations from the normal nominal voltage and frequency take place. So, sorry p is equal to p 0 when delta v and delta f are in fact 0. So, delta v and delta f are the deviations from v naught and f naught. So, this is one representation of load and similarly q by q naught is 1 plus k q v q f delta f by f 0. So, this is a neat way of writing down the aggregate characteristics of certain loads. So, when you do not have for example, any data it may be a good or rather you do not have very detailed kind of data it may be a good idea to try to fit the characteristics roughly to these algebraic equations. So, this is the you know equations for the loads. Now of course, when you making a aggregation it would be nice to know or whenever you are deciding your these parameters k p v k p f k q f and k q v it would be good to know you know for example, what are the various components of the load I mean I am not saying to the last detail you know, but for example, if I know that at IIT Bombay our loads are mainly lighting and air conditioning loads then I can get a proper roughly from you know other studies which have been carried out by others you can get this you can put in a value of k p f and k q f which fits in well with this kind of load this kind of load mixture. So, for example, it is people have carried out studies and they have found that for air conditioners for example, a window type air conditioners their power factor can be approximately 0.8 0.82 or so. So, these are obtained from certain studies obviously, they will change depending on the rating of the air conditioners and so on and this k p f k p v is for example, for air window air conditioners it has been found in certain studies that 0.47 approximately k q v is approximately 2.5 k p f is roughly 0.5 and k q f is minus 2.8. So, this is one particular load characteristic which you can which you can use you know. So, if there is a lot of air conditioning load or a certain proportion of the load is air conditioners then you can model one chunk of the load in this fashion with the appropriate values of k p v and so on. Similarly, let us just talk about for example, water heater. So, water heater is basically a voltage dependent load it has practically no you know it does not draw reactive power and it is not dependent on frequency you do not expect the heating of the element of any heating kind of load like water heaters to have any thing other than the real power dependence on voltage. So, what you will find is k p v is 2. In fact, if you just a moment so k p f and similarly k q f k q v is equal to 0 power factor is of course 1 this is the power factor. So, this is the water heater for example. So, you will find it for this particular class of loads your this is what you will get. Similarly, you have got fluorescent lighting and so on. So, you should in fact you can for example, look into the load modeling chapter of the book by Kundur which I mentioned sometime back and look at the wide variety of loads and their characteristics which are given there. One of the things which probably could be of interest is industrial motor. In an industrial motor in fact has got a certain class of industrial motors has got power factors of 0.88 k p v of 0.07 which is means it is not really voltage dependent k p f which is 2.5 which is fairly good of high frequency dependence and k q v which is 0.5 n k q f which is 1.2. So, you see this fairly large variation in these constants. So, what we need to do is once you have got a load bus it is better to first split it up into various components of loads you know like lighting loads. Lighting means whether you have to really distinguish whether it is primarily incandescent or is it fluorescent then the motors, air conditioners this could be industrial motors, air conditioners and therefore, get a kind of you know describe these individually by the appropriate k p f and k q f and so on and then get a kind of characteristic of the final load and reactive power load which appears on the system actually drawn by the system. So, this is what you need to do sometimes of course, even this when you are doing a large study even this may be very tedious to at this kind of data may be very tedious to get. So, you may find in fact, people kind of classifying loads as residential, commercial there is official office spaces etcetera and industrial usually manufacturing industries etcetera and power plant auxiliaries. So, you can have these load classes and similarly for these load classes we can actually get these k p v and k q v and so on constant. So, if you know for example, Mumbai load is this much of it is residential or domestic this much is commercial. So, you can actually get an aggregate load model for Mumbai as partly commercial partly residential partly industrial and in wherever there are power plants right at that bus you have got power plant auxiliaries remember the steam power plants require substantial amount of auxiliary power you know for their pumps etcetera. So, you will find that that itself may be 5 to 10 percent of the plant output itself the rating of all the auxiliaries. So, that itself is a quite a substantial load that also has got certain characteristics. So, I refer you to the book by Kundur which gives these characteristics in one place you will find that given in one place. Now, what I mentioned to you the load model you can say which I mentioned to you here is in fact, a static load model it is an algebraic relationship between p and b and p and f and so on. A better way of representing loads would be with a dynamic model, but as I mentioned it would be too difficult first of all to get the data for the dynamic models, but certain loads you know like very large motors you know which may be present in some places you know for them you could flag out these loads in fact, and then model them in detail for example, induction machine loads very large induction machines are there you know several megawatt rating machines are there. So, those machines you may try to flag out and model separately as dynamical equations instead of static equations. So, if you look at induction machines they also come out to become you know the modeling of induction machine may be required in another context not only have you to model large induction motor loads individually using dynamical equations very large I am talking of megawatts whenever you are studying a grid those kind of motors. The other context in which induction motors really or induction machines may manifest themselves is when you are trying to model induction generators. Now, many of the for example, wind farms or wind generators are in fact induction generators and it is a good idea and they are also many very often connected to low voltage system. So, they are almost like negative loads negative induction motors. So, you may actually have to model certain large induction motors as well as large induction generators in your grid individually using differential equations. Now, how do you do that remember we will just not go very much deep into this I will just indicate how we can do it using the synchronous machine equations themselves. Now, just remember that I mentioned sometime back that you can get a simplified model of a synchronous machine using 1.1 model that is 1 winding on the d axis on the rotor and 1 winding on the q axis of the synchronous machine on the rotor. Now, so the equations which you get in fact are given on the screen we have done this in the 22nd and the 23rd lecture and the q axis equations are these. So, these are of course of a synchronous machine. Now, an interesting question to you is which I just kind of hinted in that lecture also was that using the 1.1 model in fact you can get a model of an induction machine as well. How do you get that well you need to make just remember what an induction machine looks like usually it has a round rotor configuration there is no actual distinguish between the d and q axis especially of a squirrel cage machine it is very difficult to define what is the d axis and the q axis if you got a squirrel cage it looks exactly the same in both axis. So, in such a situation what you see is that you cannot have a distinguish distinction between the parameters of the d axis and the q axis another important point which is there is that there is no field voltage applied to a particular winding on the d axis the d axis and the q axis look exactly the same even if you are modeling with two windings the parameter of the winding will look exactly the same. The E F D is 0 because you are not applying any field to the voltage of the field winding the there is no field winding. So, the field winding in fact has to be converted to a damper winding which is short circuited onto itself. So, your E F D has to be made equal to 0. So, if you want to convert the synchronous machine model to an induction machine model then what you need to do is T Q dash T D dash have to be made equal X D dash X Q dash in this 1.1 model have to be made equal and X Q and X D also have to be made equal and E F D has to be set to 0. So, what you will get is a model of an induction machine it is effectively like two damper winding one on the d axis and one on the q axis otherwise the machine looks exactly the same it is no saliency either in transient conditions or in steady state conditions. Of course, it is all very well to say now we have got the model of an induction machine, but typically if you look at induction machine theory etcetera you will get your parameters in terms of the leakage reactance is stator leakage reactance is the stator mutual reactance rather the mutual reactance mutual reactance X M you will also get the rotor leakage reactance referred to the stator X R and from that in fact, you can get the parameters X dash X and T dash which I have just mentioned using these formulae. Now, this is not difficult to prove in fact, you can take the 2.2 model of a synchronous machine set R K and H R H tending to infinity to open the two extra damper windings then you will get 1.1 model you can use the basic equations of the synchronous machine as described in the 11th 12th and the 13th lecture of this course and actually derive these X X dash and T dash. So, this of course as I do to many tedious things which are which we have to derive I leave it to you to derive this. Now, there are some other issues for which we do not have adequate time in this particular lecture we will revisit this in the next lecture. When you say I have converted the synchronous machine model to an induction machine model well what do you mean by rotor angle first thing is a synchronous machine behaves differently from a induction machine. In fact, in steady state the synchronous machine speed is equal to the voltage source to which it is connected. So, rotor the speed is in fact equal to the voltage source to which it is connected a wrote induction machine in fact, does not behave that way except when it is under no load conditions. Rotor angle is is manifest in the torque the question is it manifest in the equations of a induction machine torque the answer is no this something we will discuss in the next next lecture. In a synchronous machine yes you can show that your torque is in some way related to the angle delta, but in an induction machine it is not another thing is we of course will need to most often model induction motors. So, there is some change in convention which we may have to do because we have derived a synchronous machine equations in the generator convention currents going out of the machine the mechanical and electrical torques in certain direction with reference to the rotating direction and so on. So, we have to actually discuss this issue in the next lecture also we will also discuss the mechanical torque and speed dependence this something we will also have to discuss in the next lecture. In the next lecture we will also hint at how using the dynamical tools which we have discussed. So, far there is Eigen analysis and simulation we can actually prove the phenomena of self excitation of induction machines I will just tell you how you can actually try to prove it. So, that is of course something we will do in the next class.