 Today, I will show you a simulation example to illustrate slow voltage collapse which can occur if you have got a load which is controlled in the sense that it is the amount of power drawn is kind of made kind of maintained constant by the load irrespective of the voltage which appears at the terminals. How is that done? In the previous class we saw that you know if you use for example, tap changing transformers one can kind of overcome the effect of falling voltages and still draw the same amount of power by increasing the tap ratios. So, what in effect we have is a kind of a selfish load which tries to keep its power constant even if the voltage drops. So, the load effect for example, if you have got a resistive kind of load you connected wire tap changing transformer and the tap ratio is changed. So, that irrespective of what happens in the primary of the transformer is connected to the distribution supply. The secondary voltage that appears across a resistive load remains constant. So, how do you do that is by changing the tap ratios of the transformer. So, this is an example of a selfish load. Now, we know that selfish load may cause problems in the sense that in case the system impedance is large it becomes quite large. Then you may have a situation where in attempting to maintain the power output constant or the power drawn constant by the load by the mechanism say of tap changers you may end up doing exactly the opposite thing that is the voltage may decline to such an extent that decrease in R effective decrease in R causes the voltage to decline further to such an extent that the power actually reduces with decreasing R. So, this kind of you know phenomena is something I qualitatively discussed in the previous lecture. Let us do a simulation example in which what I am trying to say becomes more clear. So, today's lecture we will divide between having a simulation example of voltage stability and also begin a discussion on some of the tools and phenomena associated with fast transients in process in something which we have kind of not considered much in this course so far. Now, if you recall quickly going back to what we did in the previous class if you have a system of this kind of voltage source feeding say a resistive load in that case there is a maximum power point when R becomes equal to x we see that a decrease in resistance in fact causes a decrease in the power dissipated at the load. So, this is called a maximum power point and if you see the load power as a function of the resistance R you will find that it actually reaches a maximum when x is equal to R. So, this is something we saw last time and the same thing the reason why that happens is that the load voltage dramatically starts falling with decrease in R and eventually v square by R actually drops reduces with decrease in R at after R is equal to after R is decrease beyond point 2 which is equal to x. So, this is something we saw in the previous class now normally we do not have a you know a variable R kind of load is not is quite rare, but effectively we can have a load which is a variable R in this fashion the R is fed via a tap changing transformer the load is fed via a tap changing transformer and the tap is changed. So, that the voltage at the resistance is practically constant. So, this is a selfish load which draws the same amount of power irrespective of the voltage which is there on the primary of the transformer. So, at the secondary you try to keep the voltage constant. So, effectively we have a you know a resistance which is a function of a the resistance presents itself to the system as a function of a and this a is changed. So, that the power dissipated by the load is constant. So, this is a selfish load which irrespective of the voltage at the terminals tries to dissipate the same amount of power. So, the a is changed to keep the voltage at this secondary almost constant. So, in this fashion you have got a variable resistance. So, what we are going to do today is a kind of an example of a voltage collapse scenario in which I will have a synchronous generator. A synchronous generator is in steady state and if saliency is neglected can be represented as a voltage source behind the synchronous reactance. So, we assume here that x d is equal to x q then in steady state the generator can be represented in this fashion. Of course, this voltage behind this synchronous reactance can be controlled by the field voltage applied to the field winding. The field voltage is continuously controlled. So, as to keep the terminal voltage in this example at 1.025 per unit. So, the terminal voltage is maintained constant. So, for all practical purposes this generated terminal voltage is actually constant. So, e is changed. So, as to keep this a constant remember that this can be done up to a point if the field voltage hits its limit. In fact, the field current hits its limit field current in field voltage are directly proportional to each other because of heating we may not be able to increase this e beyond a certain point. Of course, one thing which I missed telling you last time was this field heating limits may actually be time dependent. So, when I say that the field hits its limit what I mean is that you cannot continually operate the machine with the field exceeding its limit. So, actually there are short term and long term limits of the field current. Remember that the field winding may exceed its continuous limit continuous time limit. It can for short while have a transient over heating because the field winding takes a little bit of time to heat up. So, this is one aspect we will not consider in this particular example. We will assume that we do not make a distinction between continuous or continuous limits and short term limits. So, short term limit of a synchronous generator field winding can be quite high. You can in fact increase the field current to more than 1.5 times its rated value for a short while, but in the long term you will have to get back the field winding to its rated value. You cannot exceed the rating otherwise the field winding will get hot. So, we will not in this example let me reiterate consider the distinction between short term and long term limits. We will assume that there is one limit and the field winding cannot exceed that limit. So, as I was saying this synchronous generator for all practical purposes as long as you can vary the field voltage and the field current you can maintain the terminal voltage a constant, but if in case this hits its limit then we should not treat this as if it is a constant voltage bus, but treat take the synchronous machine as an equivalent E behind the synchronous reactants. In fact it is equivalent E behind the synchronous reactants, but E is variable. So, when it hits the limit E becomes a constant. So, as long as you know your field winding is within the field winding current is within the limit this is practically a stiff voltage source otherwise this becomes a constant voltage source. Once it hits this limit this becomes a constant voltage source, but it is behind the large reactants. So, remember we have for all practical purposes this is a voltage constant magnitude voltage source provided E is within limits. You have got a generator transformer whose reactants is quite small compared to the synchronous reactants and the transmission line reactants is also quite small they are not very large. This is the equivalent kind of system we assume we have got two transmission lines. Then you have got a load transformer and then you have got this load. This load of course is the voltage V is the voltage at the secondary of a tap changing transformer. When I say this load is this is V and in fact load admittance the load admittance that is g plus j B is 0.75 minus j 0.375 this is a 1 upon this is suppose this is R and this is x. So, B is equal to minus 1 by x. So, basically this is a inductive load. So, that is why you get 0.75 minus j 0.375. So, this is the resistance of the load. Remember that the tap is changed so, basically this is a inductive load. So, that is why you get 0.75 minus j 0.375. So, this is the resistance of the load remember that the tap is changed. So, as to keep this V constant. So, if the voltage here is V the voltage here becomes V by A is it ok. So, we will try to keep this voltage V by changing the tap at 0.875 per unit irrespective of what voltage appears here. So, we will you know if the voltage here goes below 0.875 we will try to increase the tap. There is a minor correction here instead of 0.875 what I meant was 0.985. If you look at the system you will find that the equivalent R dash and x dash are such that you will get the admittance that is g dash plus j B dash is nothing but 0.75 minus j 0.375 into A square. So, this is the reflected load admittance as seen by the system. So, when I am doing the analysis remember that as far as the system is concerned the voltage here is V by A V is the load side voltage. Remember that the load admittance 0.75 is actually much smaller than the system impedance under these conditions. The system impedance is 0.145 plus 0.j08 plus the equivalent impedance of this line. Now, if you look at the admittance rather the equivalent admittance it is much greater than 1 upon the system impedance. So, we are nowhere near the condition of voltage you know you can just check this out you can do a power versus admittance study and you can really see that with this much source relatively low source admittance impedance you are not near the maximum power points. So, if you add up the load admittance from this point onwards why this point onwards this voltage is practically maintained at 1.025. So, you can at this point assume it is a voltage source and this is the system admittance the system impedance effective impedance which is not very large it is nowhere near the maximum power point of this load. So, this is the initial operating condition of course. Now, what I will do is I will simulate this system using R K method with the time step of 1 second. Now, one of the important things regarding the modeling what we are going to study is relatively a slow phenomena we are not going to study fast transients here. In fact, the tap changing action is over several seconds if not minutes. So, what we really are going to do here is we can actually assume that the system is in sinusoidal the network especially we do not have to model by you know we do not have to model the network transients a network d i by d t etcetera d i d by d t d i q by d t etcetera is not modeled you can treat the network as if it is in sinusoidal steady state the synchronous machine also we represent by its steady state model that is a voltage source behind the reactance why we why do we model it that way because these transients are very slow. So, the synchronous machine need not be modeled by any you know dynamical equation except the dynamical equation corresponding to the variation of E. So, that is something I will discuss shortly in this particular example we do not have one synchronous machine connected to another voltage source or another synchronous machine it is simply a synchronous machine connected by a transmission system or a distribution system to an impedance. So, relative angles or relative angle stability is not an issue here. So, all the loss of synchronism swings etcetera is not at all an issue here. So, we are looking at a distinct phenomena in which these things are not coming into the picture because it is simply have one synchronous machine connected to a load. So, the disturbances we will consider are the loss of one transmission line we will assume of course, the system is initially at steady state then we will consider the loss of one transmission line at 1000 seconds and the step change in the load admittance a small step in the load admittance at 3000 seconds. So, there are two disturbances. So, I will first show you what happens when you have the first disturbance. Now, remember what we are going to do here is trip one of these lines tripping one of the lines will reduce the source impedance or rather I am sorry you will increase the source impedance. So, because of that the voltage here of course, will change in fact, it will drop down a bit because the impedance is increased the tap will be changed the tap or the of the tap changing transformer being changed will change the reflected load. Now, so the amount of obviously, when you trip a transmission line eventually the current flowing through this system will change once the system current changes the stator current changes you will find that the terminal voltage tends to vary, but luckily since I have got a field voltage which can vary I adjust it. So, that this terminal voltage is back to its original value. So, this is what we will try to simulate and see remember that we need to simulate a tap changing transformer action will assume that it is a dynamical system this is one possible logic for tap changing this is of course, not a standard logic or anything of that kind it just reasonable logic which you know person in the first cut could think of. So, what you we want to regulate the load voltage to 0.985 per unit. So, what I do is give the set point compare it with the actual prevailing voltage at the load then I filter it. So, I do not respond to fast changes and noise. So, I filter it then I pass it through a dead band. So, that if they are very small changes in voltage I mean for example, if the voltage V is 0.986 I may not think of actually adjusting the tap it is very small. So, I this error is very small. So, for very small errors nothing is done, but if the error is reasonable beyond this dead band what I take it as an indication that I should either increase the tap or reduce the tap, but I do not do it right away what I do is I pass this signal say 1 says that you increase the tap. Now, what is done is that 1 is pass through an integrator or a counter. So, why once this thing is integrated say if this is 1 then you can notice that it will take about 1000 seconds for this to increment to the output of this to become 1 because this gain here is 0.01. Similarly, if this is 1 for 100 seconds if this is 1 for 100 seconds this will increment eventually by this will eventually increase increment by 0.1 after 1000 seconds. So, this is kind of I did I say the right thing if this is 1 for 1000 seconds the output here will become increment by 1 since this is a plane integrator. So, of course, the output of the integrator is a continuous value a kind of vernier value it is a continuous value we the tap settings are discrete. So, we need a quantizer quantizer or a discretizer to actually tell you what the tap is. So, the tap of course, normally is 1. So, you start off with the tap of 1 1 is to a let us assume a is initially 1. So, anytime this exceeds the limit exceeds we rather there is a error between this which is greater than the dead band this will start incrementing or decrementing depending on whether the error is positive or negative and reduce or increase the tap, but the tap is a discrete value. So, this is one possible way you can have a tap controller of course, this is not the way which is you know implemented may be in a real setup, but this seems to be a reasonable starting point of how a tap changing controller should look like. We now look at the field, field controller of a synchronous machine. So, synchronous machine is trying to keep its terminal voltages 1.05 per unit you measure the terminal voltage the error is passed through a integral controller. You can have a PI controller a P controller with a large gain any of these possibilities exist and of course, the important point is that the field voltage is not allowed to increase beyond 2.75 and minus 2.75. So, this is the absolute limit of the field voltage actually in steady state 1 does not expect the voltage to be negative anyway. So, actually this could have been need not have been given minus it could have been just a 0 or slightly more than 0 positive direction. So, the thing which I was trying to tell you again let us come back to that this terminal voltage is being regulated by an AVR in this case the AVR is assumed to be an integral controller and the field voltage limits are 2.75 per unit. So, you cannot increase E beyond 2.75 per unit. Now, of course, if you recall in our AVR discussion of AVR I had once told you with the field voltage ceiling limits are sometimes even as high as 6 or 7 per unit. But remember that that is the transient variation which you can have normally you cannot increase the field winding beyond roughly these kind of values this is practically the limit of the field current. So, continuous rating cannot be more than 2.5 or 2.6 or in this case 2.75 per unit. So, this is one important thing we should keep in mind let me repeat it that the transient field voltage limits can be quite high you can have transient field voltage to be quite high and for a short while. But in steady state your field voltage limits are likely to be around this range. In fact, in steady state there is no almost no chance that you will apply a negative field voltage. So, in fact this is a kind of an error I should keep the in steady state if these are the steady state limits the lower limit should be actually a positive value not a negative small positive value not a negative value. So, just remember these important points we can actually kind of make a distinction between we need not make a distinction between fast and slow limits because the phenomena we are going to study here is very slow. So, we can talk in terms of just the continuous time limit of the area not worry about transient limit of the area which can be actually quite high. For a short while you can have a large field voltage and in fact exceed even the current rating of the field winding for a short while. Now, of course important thing is that this is the field voltage is applied to a synchronous machine. It will manifest as the E F D after the field voltage the voltage at the stator will kind of manifest after some time. So, how it manifests in transient is dependent on the dynamical equations of a synchronous machine. What we will assume here is that you know if I give a change in the field voltage it takes about a second before it manifests as the you know voltage or rather the effect is seen only after some time. So, this is some kind of very rough modeling of a synchronous machine. So, this is something which is if you want to study slow transients, but this is certainly not if you want to study transient stability or angular stability and so on. So, this is just a very rough model of a synchronous machine. So, this delay of one second this time constant of one second in this transfer function is kind of representing all the cumulative effects faster effects which are there which were neglected. So, this is a very rough kind of model, but it is if you are going to study slow transients like the one we are doing here. So, what happens if you lose a transmission line? So, if you look at what is seen here in this response you see that the voltage of course, initially is 0.985 1000 seconds one of the transmission line trips. So, one of the transmission line trips and the voltage dips suddenly at the load because the load directly you know there is a larger source impedance and the voltage drops. Now, the voltage drops to a value of 0.965, but what you are seeing is practically like steps which are occurring are because of the fact that the tap is incrementing. So, the voltage is dropped beyond 965 below 965. So, a tap changer will try to change the tap remember the tap changing controller we discussed sometime ago it will try to increment the tap. So, of course, the tap is incremented you know there is a definite time period which is required before the taps increment and another thing is of course, there is also a discrete value of the tap. So, there is a fixed time delay before any tap is increased and also the tap values are discrete in nature it is not a vernier smooth control. So, you find that the tap increases, but shifting the tap by 0.01 does not really bring the voltage back to 0.985. So, after sometime again another tap is changed the rather the tap is incremented again then again, but beyond this point you see tap is not incrementing though we have not reached the value 0.985 the reason is of course, we have got a dead band which allows you to settle down at a voltage which is slightly less than the reference value. So, beyond this point in fact you see that there is no increment in the tap and the voltage settles to a value which is almost the same this 0.98 may be 0.982 or so which is almost the same as the reference value. So, this is a acceptable behavior there is nothing wrong in this. If you look at the tap ratio of course, as I mentioned sometime from the initial tap I think the initial tap was 1.1 which I mistakenly said was 1 initially it is 1.1 initially it goes on increasing in steps rather it increases gradually in steps to this value in a discrete fashion. So, the word gradual may not be appropriate here I would say in a discrete fashion it increases to a new value this basically action of the load is ensuring that the voltage which appears at the load itself is almost 0.985 985. So, this voltage which you see here on this sheet of paper is actually 0.985. So, if you look at this here it is 0.98 it is brought almost back to 0.985 by adjusting this tap A. So, this is what is done if you look at the generator terminal voltage. In fact, the voltage is maintained at 0.1.025 almost strictly there is a terminal voltage of the generator it is not E is the voltage which appears at the terminal of the generator. The generator terminal voltage is regulated quite well the reason of course, is that none of the limits are hit. So, that is one of the reasons why this happens. In fact, if you look at the internal voltage E of the generator what I call as E F D you see that with every tap change see what happens initially is that when the transmission line is tripped the voltage at the load decreases and eventually it turns out that the generator the voltage which is required the generator internal voltage which is required to maintain the voltage at 1.025 per unit actually becomes smaller that is because the load probably drops because of the trip of the transmission line. But as the load recovers it tries to draw the same amount of power by increasing the tap you find that the reactive power loading or the current loading of the synchronous machine increases and probably the voltage at the terminal is tending to drop. So, what happens is that as every tap is increased the requirement of the synchronous machine increases the E requirement in order to maintain the terminal voltage of the machine constant increases. So, you see with every tap change the internal voltage increases, but of course it is much it is still lower than the limiting value 2.75. Now what we will do is very interesting example is where we increase the load voltage increase the load admittance or rather you change the load admittance you increase the load if you look at this load here this r is suddenly decreased or g is increased from 0.75 to 0.85. So, what you have done is increase the load on this system by changing r if I decrease r suddenly your voltage here will change the load voltage will dip because you have increased the load resistance. Now if you increase your sorry decrease the load resistance if you decrease your load resistance the voltage here tends to drip and what happens is your tap changing action starts increasing a in order to keep this voltage constant. Now this would be a normal and natural thing to happen and well there is no problem in a way this is what one would expect a tap changer to do and probably we think it would be doing the thing quite right, but actually the opposite thing happens when you look at this step change in the admittance the surprising thing is that the voltage dips and the tap tends to as the tap is increased you look at this blue blue graph as the tap actually increases the tap ratio is actually increasing as I will show you in the next slide I will show you that first perhaps the tap ratio is increased to try to maintain the voltage via constant. But, the surprising thing is I am sorry yeah the surprising thing is as the tap a is increased the voltage actually starts dipping every time I increase the tap instead of expecting instead of having the voltage increase as I would have normally expected the voltage actually dips increasing a voltage dips that is a very surprising thing the reason why that happens is that will be apparent in some in the next slide. So, if you just right now we will concentrate on the blue graphs we will come back to the green graphs a bit later. So, blue graphs show that a tap ratio increases in an attempt to keep the load voltage constant. However, an increase in tap further brings down the voltage. So, the taps go on increasing. So, it is a kind of a unstable situation till the tap limits are reached. So, you go on increasing the tap still tap ratio becomes 1.2 which is the limit now why is that happening now what you notice here is the generated terminal voltage when you increase this load admittances concentrate on the blue graph right now we will discuss the green graph a bit later. If you look at this blue plot as because of the load increase the load admittance has been increased or the load resistance has been decreased the load on the synchronous machine has actually increased. And because of that the voltage drops and every time the tap increases the generator terminal goes on voltage goes on decreasing now why is it actually increase decreasing the generator terminal voltage after all is regulated right. The reason why it is going on decreasing the terminal voltage is going on decreasing is because the generator the internal voltage of the generator or the field voltage which is proportional to the field voltage e has gone and hit 2.75 the upper limit the continuous time upper limit has got has been hit. So, because of this reason the generator is not able to regulate the terminals terminal voltage magnitude a constant because the internal voltage is hit its limit. So, the correct equivalent circuit now is the e rather the correct equivalent of a synchronous machine which is unable to maintain its terminal voltage constant would be a large reactance such a synchronous reactance almost 2 per unit on its own base behind e which is around 2.75 although e is 2.75 remember now it is a constant e. So, when you are analyzing the system you will have to take the system as e which is constant and the system impedance now has to include that 2 per unit reactance of the synchronous generator. In some sense when the field voltage had not actually hit its limit and you had this closed loop voltage regulation the synchronous machine was effectively a stiff voltage source because we could adjust the field in order to keep the terminal voltage constant, but this cannot be done anymore and now the machine is like a constant voltage source yes, but behind a large reactance. So, it is a constant voltage source of 2.75 per unit behind a large impedance and because now the impedance of the synchronous of the generator is in series with a large impedance that is in series with the transmission system and distribution system impedances you have actually gone beyond the maximum power point of this system. So, if in fact the R value or the load value has gone beyond the maximum power point. So, any now attempt of the load to effectively to effectively draw the same amount of power by effectively presenting to the system as a decreased reactance because the action of the tap is bound to fail because increase in the tap increase in the tap decreases the effective R seen by the system which in fact results in lower power output or in effect the load voltage is not able to we cannot maintain the load voltage a constant. So, that is a very interesting point that under certain circumstances the load tap changer in fact does some kind of harm because every time the load tap changes the voltage actually decreases and this happens because the system impedance has become really very large I am sorry. Now, let us focus on the green plot the green plot is a plot of if there was no tap changing action that is the tap changer has been disabled and kept at whatever value it was before this disturbance the interesting thing is that if tap had not operated one would have settled down to a value of 0.965 or so per unit at the load. So, interestingly if we had just disabled the tap things would not be so bad, but because the tap changing action is there we are ending up causing a voltage drop or a voltage collapse scenario because the tap changing action after the maximum power point is exactly the opposite of what is actually should be done. So, decreasing the tap or rather increasing the tap actually results in lower load voltage the voltage V reduces. So, the same thing you can see with the terminal voltage of a synchronism if the tap was not operated at all then we would actually have been we would still be in an acceptable kind of scenario the voltage would have been lower, but you would not have allowed voltage to collapse to an unacceptable value. So, this is what really this example tells you that sometimes the effect of load controllers like load online tap changers can sometime cause some detriment to the system because under stress condition they may effectively present the load to the system as a very selfish load is drawing or trying to maintain its power no matter what kind of voltage is there at the terminal and if you try to do that you end up achieving exactly the opposite of what you intend to do. So, voltage actually decreases instead of increasing as you change the tap of course, this happens only when the effective system impedance becomes large and in this case it becomes large because the field winding of a synchronous machine has set its limit and you can no longer regulate the terminal voltage. Therefore, thereby the effect of synchronous reaction or the armature reaction of a synchronous machine really comes into full force. So, this is a example of slow voltage collapse caused by tap changing action resulting in a selfish load. Now, I told you sometime back I told I discussed it in the previous class that you can have under such other situation where this faster voltage collapse as well especially when you have got induction machines you know kind of stalling in case the voltage goes down. The voltage for example, if you have got a bus in which there are several induction machines and the voltage drops the slips of the machine change they draw more current cause a further drop in voltage and eventually you can land up in a fast voltage collapse even when you have got these kind of situations. Of course, if the system is strong that is the system impedance is large or rather system impedance is small the source impedance is small less likely you are going to have voltage collapse. Now, so remember now in your mind try to this phenomena actually looks quite different from the kind of phenomena we have seen you know when we are studying angular instability and so on. Angular instability was different it pertain to the kind of phenomena which occur when you connected two synchronous machine or many synchronous machines to each other and they do not stay in synchronism. Of course, one interesting point which you can just experiment taking the two machine example which you know we had discussed before coming to voltage stability is that in case a two machine or a multi machine system loses synchronism the voltage does very very wildly when you have got a loss of synchronism scenario. We have discussed this sometime back when you have two voltage sources of two different frequencies connected together by an AC transmission system somewhere the other midpoint of this transmission system the electrical center you can say the voltage undergoes very very large variations in case you lose synchronism. So, voltage variation or voltage dip after loss of synchronism can occur, but it is a distinct phenomena than this kind of system this kind of phenomena. This voltage stability phenomena as you see is not really associated with relative angles of synchronous machines it is completely associated with the dynamics of the load operating in a low or larger or a weakened transmission or generation system. So, this is what you should keep in mind as far as this phenomena is concerned. Now, if you are studying slow voltage collapse like this one on load tap changing you can consider very simplified models remember we have considered synchronous machine model which is very very simplified. It is more of a reasonable model rather than a derived model in the sense that the field the synchronous machine is represented by steady state E F D behind a synchronous reactance. So, it is a kind of a steady state model E is practically proportional to the field voltage, but all the dynamics kind of has been gathered together and represented as a simple transfer function 1 upon 1 plus s. So, this is a very very kind of rough model, but using this you can actually show that this kind of phenomena can occur. If you are going to if you are studying for example, fast voltage collapse for example, especially associated with induction machines in that case of course, you may have to model you know things in much more detail. For example, if you have got one synchronous machine feeding several induction machines and under that circumstances circumstance you are studying actually fast voltage collapse in that case you may have to model the synchronous machine and its excitation system in more detail. So, this is something you should keep in mind, but when you are studying OLTC kind of dynamics it is not necessary. Now, we move on to another aspect which some in some sense in our last part of our course we have been kind of neglecting this aspect because some of the very exciting and common phenomena and commonly seen phenomena in a synchronous grid relate to angular stability and you know stability of the frequency. Our focus has been to a large extent in this course on slower transients you know this what we saw as in this OLTC kind of transient tap changer transients are ultra slow loss of synchronism and angular stability relative angular motion relatively faster, but we have of course, faster transients compared to that we have other transients which are faster than that which we are not really analyzed or I am not actually told you of any phenomena which involves say transients which are faster rates of change which are faster than say 10 hertz or so. We are talking of swings power swings etcetera we have talked of that so far relative angular swings etcetera they are much slower they are on 1 hertz and so on. Now, if you are going to study faster transients let us just have a look at what are the transients fast transient phenomena which we have not really studied very much in detail in this course, but of course they are important is a dynamics course. So, any transient is important for example, the phenomena associated with lightning and switching these are fast very fast transients in which you may have to model a transmission line by distributed parameter model traveling wave phenomena really important here. You may have you may want to study the dynamics of power electronic controllers for example, an HVDC system you want to actually see how the firing angular controllers firing angle controllers the phase lock loops synchronization system that is the synchronization system current regulators etcetera work and how they interact with the electrical AC network these kind of transients also are fast and require you to model relatively faster phenomena you cannot use the approximation of you know for example, neglecting stator and network transients. So, network transients become important another reason why you would be interested in studying fast phenomena was for example, how do equipment protection systems behave how do differential protection schemes or you know distance protection schemes behave for example, if there is a sudden fault in that case you will have to model the network in relatively more detail. In fact, many of these disturbances will be unbalanced disturbances something we have not really dealt too deeply into in this course. So, you can have unbalanced disturbances nothing in a course of course, prevents you from applying the modeling principles to fast phenomena as well, but remember that this is one aspect we have not spent. So, too much time in this course probably it will require another course to study the fast transients as well specific transients relating to fast phenomena. Now, equipment protection schemes need to be checked out. So, you may actually need to see whether your protection logic or relay logic protective relay logic is working fine or not. So, that kind of studies those kind of studies may require you to model fast transients you also have another kind of transients torsional transients which may be a relatively higher frequency phenomena more than 10 hertz kind of phenomena. Now, if you look at the basic tools which are required for studying fast phenomena they are kind of classified the generic name for a simulation tool which studies faster transients like the ones I have mentioned right now is called electromagnetic transients program this is generic name E M T P these are the class of programs which study faster transients they model the networking in detail in the sense that the d by d t's you know which are not you know in when we are studying relative angular stability or slower phenomena we often neglect the in the d q reference frame the d by d t's associated with the voltages and currents in the network remember we had in our study of the two machine system you know neglected the transients associated with the stator and network. So, what we got effectively was algebraic relationship between the voltages and currents of the network in fact we use the well known admittance matrix to describe the behavior of a network by purely algebraic equations this cannot be done if you are going to study fast transients. So, the electromagnetic transients program in fact does not model in fact does not make the kind of approximations we make often when we study slower dynamics like angular swings and so on. If you look at E M T P programs if you look at the features of E M T P programs first thing is that because if E M T P can be used to study lightning and other and switching transients which are quite fast very fast you may actually represent transmission lines by travelling wave phenomena that is the you know what I would say is the partial differential equation kind of model. You could also model it by lumped elements you can always model a transmission line by lumped L and C's if you recall when we were talking of a transmission line models we saw that for slower phenomena first on a slightly longer time scale a lumped pi equivalent of a transmission line will work just as well as distributed parameter travelling wave kind of model. But for just the few you know maybe tens of milliseconds after a disturbance you may see a significant difference between a lumped model which is an approximation of course and the more appropriate the more exact travelling wave or distributed parameter model. What is usually done whenever we have any if you look at E M T P program is you know that if you have when you are working with partial differential equations you have to solve partial differential equations. But if you look at E M T P programs they do not really they do not look actually so complicated they do not really solve partial differential equations in a classical sense the transmission we know that a lossless transmission line is a kind of has a travelling wave behavior. So, if you recall what I discussed when we were discussing transmission line the voltages at one end of a transmission line are related to the voltage and currents at the other end of the transmission line sometime ago. So the voltages at the other end of a transmission line sometime ago are you know in some way you know manifest or some way manifest on one end of the transmission line. So, there is a kind of a transmission delay associated with purely lossless elements if you have got lossy elements there is an approximate way to handle that. So, E M T P program actually take help of this lossless model and kind of make an incremental change to include losses for lumped elements trapezoidal rule is used. So, you can actually get a relationship between the value of a variable at the k th instant and the k plus 1 th instant discrete time instant. So, usually trapezoidal rule is used for the lumped transmission line network transmission line elements if then only the elements in the network use piecewise linear model you assume that the you know model is linear piecewise. So, if you are trying to model saturation for instance you do not assume it is a continuous kind of curve, but a piecewise linear kind of curve this kind of approximations are made in order to make the program more efficient. Network and stator transients cannot be neglected in any M T P kind of study. So, you have to model all the d by d t's associated with the stator flux in a synchronous machine and of course, M T P programs typically of a small duration you try to see what happens just after a fault for a short duration. So, you can in fact model things like synchronous machines and speed dynamics you can in fact freeze them you know you can assume that the speed of a synchronous machine or the rotor angle is almost a constant. If you are if a simulation is for a short duration, but you will usually use smaller time steps. The time steps in a M T P kind of program can be quite small say for if you are doing a study to of an H V D C system in which the switching is occurring every say 12th of a cycle. You may for getting adequate accuracy use time steps of a say around 50 micro second also these kind of time steps are used in M T P kind of programs. We now study one particular kind of example of fast transients it is not as fast as lightning transients, but this transient has typically a bandwidth or rates of change which are approximately greater than 10 hertz. So, it is there faster than relative angular motion, but not as fast as lightning or switching transients. So, this particular example is something we will discuss in the next lecture it is relating to torsional shaft torsional transients which do require you to model the network in detail I mean detail in the sense that you cannot assume that network and state of flux transients are neglected. So, this is something we will do in the next class it is a example of an interesting phenomena as which is relatively fast. So, we will get this balance back into the course of considering at least some one phenomena pertaining to faster transients. So, this is something we will discuss it is a very interesting thing which occurs. So, we will use the twin tools of Eigen analysis and simulation to understand this phenomena. So, that we will do of course in the next class.