 In today's lecture, we will introduce you to the concept of voltage stability, but before we go on to studying this phenomena, let me do a bit of a shift of what we are doing. We were studying the simulation of a two machine system and in the previous lecture I gave you a expose of angular stability and frequency stability of a system that is the stability of the center of inertia motion as well as the stability of the relative motion of the system. In today's lecture in the beginning, we will just briefly review what goes into making what you call an angular or frequency stability program. So today's lecture is initially focused on frequency in angular stability programs and thereafter I shall introduce you to a voltage stability example. Now remember that one of the objectives of this course was to introduce you to some simulation tools or analysis tools, power system analysis tools and one of the most important stability tools is what is known as a transient stability program. Now normally transient stability refers to large disturbance, the study of large disturbance angular relative angular stability. Now if you look at transient stability programs, they essentially focus on phenomena which are roughly between you know you can say 0.1 hertz to 1 hertz about 0.1 hertz to around 10 hertz kind of phenomena. So you are focusing on electromechanical oscillations of this kind of frequency range. So our modeling as we did in our two machine example was suitable for this kind of situation. If you recall the key modeling issues which are key issues which are to be considered in the modeling analysis of angular stability or frequency stability as well is that these are slow modes, they are slow patterns slowly changing patterns in the system. So you can represent the network by algebraic equation. So that is what I call as a static rate work. So recall that the network of course consists of transmission lines, transformers, compensating capacitors and so on. So that part of the network is represented by algebraic equations by effectively setting the d i by d t's and d v by d t's in the transform frame of reference. So what we are doing is the voltage, the transform voltage is d v are set equal to 0 in your equations of transmission lines and other components of the network. As a result we will get algebraic equations. So your network is represented by algebraic equations and one of the interesting things which we observed as far as transmission lines which is also true of transformers etcetera is that you can represent a balanced network compactly by complex equations. So for example, if you had a transmission line R and L, so we found out that in case you neglect these d i d by d t's for an inductive element you neglect 0. In that case an interesting the representation of the network is a compact complex representation which is nothing but R plus j omega L, omega is the frequency of the transformation which is used. We have talked of C k the transformation C k is a synchronously rotating transformation say of constant frequency in that case this will be omega naught and this is equal to the voltage across this. So this is the voltage across the inductive element. So you can actually represent this is a very compact notation and it is a familiar representation of the network. So whenever you are writing down your algebraic equations of the network you will find that effectively you can represent a network by an admittance matrix. So you can actually treat the network as if you apply k v l k c l to the individual elements or rather the topology of the interconnected network as well as the individual relationships of the elements you will effectively get that in case so I will call this i bar and v bar. So the network can be represented by an admittance matrix i are the injections these injections are due to elements like generators etcetera. So which are injecting current into the network. So the network itself can be represented simply by i bar is equal to y v i bar being the current injections. The current injections current injections of course are determined by the generator states or functions of the generator states. So what you can do is gather all the algebraic equations corresponding to the generator as well as the network. In fact if you look at even a generator if you look at the generator equations if you neglect d psi d by d t is equal to 0 and d psi q by d t is equal to 0 then to you get an algebraic equation then of course there is also an algebraic relationship which relates psi d and i d and psi q and i d i q. So these are all algebraic equations which are obtained from the generator itself. So you have got algebraic equations of the network then you have got algebraic equations 4 algebraic equations per generator obtained simply by the relationship between psi d and i d and psi q and i q as well as because you have said d psi d by d t and d psi q by d t equal to 0. So these are the algebraic equations which you have to combine with the network. Now loads as well so if you for example have loads they also may be represented by algebraic in some cases for example if you have got a purely resistive load then the load current is simply i q plus j i d into R L is equal to v q plus j v d this is the bus voltage this is the load current. So you can actually get various algebraic equations depending on the loads and of course the network is in the study of slow modes slow patterns is represented by simply an admittance matrix. Remember that i bar is nothing but the d q component of the current injections these current injections are due to generators and other dynamical elements voltage is the voltage at buses. So remember that each element of the network will be represented by some equation of this kind then all the currents and voltages are related to each other depending on the topology of the network by Kirchhoff voltage and current laws and therefore you are going to get what we what we use as a commonly used as the admittance matrix of the network to represent its behavior. Remember that do not use admittance matrix notation for the network do not use the admittance matrix representation of a network in case you are interested in fast transients. So anything faster than a few tens of hertz or 20 30 hertz you may have to represent the network by its differential equations and of course if you are really interested in things like switching in lightning transients have to represent it even in more detail which is the partial differential equations or the travelling wave equations which is the solution of the partial differential equations. So this is something you need to keep in mind an angular stability program will represent the network by algebraic equations. So this is something you should remember and also stator transients are neglected. So you have got 4 sets of rather 4 algebraic equations as a result of a generator. Now of course I because of time limitations of course I cannot cover all dynamically elements which are there in a power system you have got things like HPDC links and you could have fax devices like static wire compensators, thyristor control series compensators and so on. But remember the guiding principle in all of these is that represent the system by a model which is compatible with what transients are there of your interest. So for example if you are interested in very slow transients you may even representing devices like HPDC links etcetera you can use a simplified notation as a controlled power flow you know between 2 nodes. So you have to really take that call depending on what you want to really see. So this is often a matter of engineering judgment and of course one way of verifying that what assumptions we make or what modeling simplifications we make are correct or not is to actually compare at least for a few benchmark examples. The response we get by considering all details and the response which we get by modeling simplification this kind of thing we did in fact in this course. Now getting back to our original problem that is talking of how you would make a transience stability program or large scale or transience stability program what you would do first of course is start off with some base case or what case or what you want to study some base case or base example. Remember that the whole idea of doing this kind of stability analysis is to answer questions like what if what if this happens does the system lose stability in case there is a large disturbance. So this kind of what if questions if you have to answer what you need to do first is define first what is your initial operating condition. So you usually whenever you do a transience stability program you do not start from you know you do not really start from the scratch in the sense that you do not do the simulation of synchronizing of each individual generator to the grade and then you know lighting it up what we usually assume is that the system is operating at equilibrium at some 50 hertz kind of condition at you know everything is normal and we are operating the system at equilibrium this is what is the usual assumption which is made it does not mean that you cannot model things like synchronization etcetera you can, but remember one thing is that if you are modeling you know a phenomena or rather you know the behavior of the system during synchronization of generators remember that your frequency will be quite off nominal. So some of the assumptions we have made at omega almost equal to omega mean may not hold at that for that kind of study. But typically if you are doing a transience stability study what we would like to do is assume that you are operating at a system a certain equilibrium system condition where certain load is being made by certain generation and then you subject the system to some credible contingency and see how the system behaves. So that is what typically a stability program would would like you to do. So first thing is think of a base case scenario that is do a load flow load flow of course is a steady state kind of study which tells you the flows and the the voltage is the voltage phases at various points in the network corresponding to certain specifications. So if there is certain load generation configuration or a scenario you can do a load flow and obtain all the voltage phases at all the buses from the voltage phases of all buses the first step is compute the equilibrium values of all the states. We did this in a single machine infinite bus example when we studied the AVR. So you can look back few few lectures to really see how you can calculate initial conditions. So equilibrium condition of the states is computed from the load flow. The next step is of course define your contingency define what are the disturbances you would like to study. After that remember that your system is represented in general by equations of this kind x dot is equal to f x f x y and u and algebraic equations because we have simplified and neglected several transients we get algebraic relationships x y and u usually you will not appear in this usually but it can and there is no reason you can always have a situation where inputs you could appear in this algebraic equations. What are these x y x x other states? So they are all the generator fluxes deltas omegas of all the generators then all the states corresponding to the excitation system governors and turbines so excitation. So all these states are there depends you can also have other dynamical elements in your system you know you can have other controllers and so on. So all that has to be substituted in what are known as the states they are basically differential equations. Why are the algebraic variables? Algebraic variables in the sense these are related to x through algebraic equations. So examples are the voltage at all buses if you neglected the stator and network transients the voltage at all buses the currents through all branches you can also say power flows through all the branches and q's they all related. So if you know v and i at all buses you also know p and q flows in the network. Then so why is usually consisting of voltages currents or p's and q's then the dq fluxes in case you have neglected stator transients what else? Yeah these are the these are the basic algebraic variables. So this is basically how your equations look like if you are at equilibrium of course you will find that x dot if you have computed the equilibrium conditions correctly you will find that x dot will be equal to 0 and everything stays where it is. So this is what really are the equations is the form of the equations. Now what you need to do of course is to integrate these equations and numerically integrate them using some integration method. Now one of the things which if you have formulated your equations and simplified the model by neglecting the fast transients in some sense you are not faced with the problem of stiffness. We have discussed what is a stiff system in the first the fifth to tenth lecture of this course you would have seen that a stiff system is a system in which they are fast and slow transients. Now the point is that if I neglect or I make modeling simplifications and ensure that fast transients have been removed the faster patterns in the are not visible in that case fast dynamic patterns are not visible in that case this system will have will not have fast patterns and as a result you can use explicit methods to integrate this system. If you feel you can integrate these set of equations using implicit integration methods as well but explicit methods are easier easier to they require lesser amount of computation eventually. Now explicit methods like r k 4 5 r k fourth order are quite good. So you can use r k fourth order to integrate this kind of system for angular angular stability studies and one would expect since we have neglected fast transients and we are interested in transients which are say about 0.1 to 10 hertz you can really take time steps of the order on you know about 10 milliseconds or so 10 milliseconds to 1 millisecond. So this is the rough time step you will use if you use fixed step explicit methods if we had not neglected network transients and considered all the fast trans including the stator transients then one would have to use possibly there would not be any alternative but to use implicit methods like trapezoidal rule or backward Euler otherwise you would have a problem but since you have removed the stiffness you can actually use explicit methods. So what is commonly used and which is convenient and modular kind of way of simulating this system is to use a method like explicit method like r k fourth order and basically this involves. So you will basically discretize the equations corresponding to the differential equations as well as the algebraic equations. In fact the algebraic equations are simply something like this and suppose I had discretized our differential equation by Euler method this is what I would have had Euler method is not a very good method for integrating it is not very accurate and very poor for stiff systems but I am just showing you what the equation would look like you would basically have to solve for x k plus 1 from x k using these equations. In fact in you can in fact represent this this is also true. So if you take these two all you have to do is of course plug in the k th value of these variables and get the k plus 1th value. So this is how an explicit method like Euler would look like do not use Euler is not very accurate even though you have removed the stiffness from the system it is not very accurate. So it is good if you use an explicit method like r k fourth order. Now this is how you would do it what you can do in fact is do a kind of partition solution. So what you can do is for example if you have x k get y k from this equation. So if I have the previous value of the states I can get y k from this algebraic equation from this algebraic equation as soon as I get y k you have got both x k and y k you plug it into this in order to get x k plus 1. Similarly now get y k plus 1 and repeat this procedure. So this is called a partition solution where you are actually solving where you are solving this equation and this equation alternately. So you plug in x k get y k then use x k y k get this to get x k plus 1 and so on. So this is called a partitioned explicit method. So this is what is easy to implement very simply plug in things and get the values but there is one thing which may take some extra time is this particular equation. Now this equation if you look at it it says it is an algebraic equation relating x k and y k but in order to get y k from x k you will have to solve this algebraic equation and if this algebraic equation is non-linear you will have a problem. However if you have this so instead of having 0 is equal to g x y suppose this is your algebraic equation. If it is of this form rather I will call this g g dash of y sorry I will just rewrite this suppose this can be written as a into y plus g of g dash of x. So suppose this particular algebraic equation has this form that is the non-linear part really appears here but the equations in the y themselves are simply a matrix into y. So y appears in the equations in this form. So suppose g of x y can be written in terms of a y plus g dash x are problem to some extent is solved because if I know the k th value of x in that case all you have to do is solve a y is equal to g dash x k. So in that case you will get you can directly do a linear solution linear solution they will not require any iterations. So you can use in fact you can solve this equation a y k is equal to g dash x k minus g dash x k to directly get y k. So you can get a direct solution for y k without doing any iterative kind of solutions. Of course in a realistic situation this a can be very large. So you have to use special methods for linear system solution but the point is that the solution becomes simple you do not have to use iterative solutions in order to get y k from x k. So this happens of course only if your g x y's of this form. So this actually may be the case in some situations. For example the two machine example which we solved sometime ago if you recall what our equations were. So you just we can just quickly have a look at what the equations were. If you recall our differential equations for the generators were looking like this and algebraic equations were like this. Then we of course since we had to interface the equations of both machines we converted the interface variables i d i q v d v q and psi d psi q using a common transformation c k this is what we did and the algebraic equations of the network which are supposed to be interface with the network equations and the algebraic equations of the other machine to get the whole set of algebraic equations was looking like this. So the first second third and fourth algebraic equation of every synchronous generator look like this. In this case of course we studied two synchronous machines. So you would have similar equations here. If you look at the network equations with transients neglected they look like this. So actually when you put all the algebraic equations together in fact you would notice that A y this would actually have this form where y would have been these 14 variables which are seen on your screen can we shift to the screen. So 14 variables which are seen on the screen would be y and the differential equations would be 16. You could in fact do this in fact this form resulted because we made one important assumption here that x d double dash is equal to x q double dash. So if you look at this set of equations the algebraic equations would come out to be of this form only if x d double dash is equal to x q double dash. So it is certainly not true that your algebraic equations would always be of this form. So this is something you should keep in mind. So in a partitioned explicit solution what you would do is in case your g of x would be of this form would be to solve for this equation and the interesting thing is that if A does not change then the solution of when you are taking out the solution of this set of linear equations you need not actually you know do this solution fully every time what do I mean by that suppose I have got you have got this basically g of x and at every time step you have to get y k from x k. Now of course if we were so what you would get is A inverse g this g dash sorry d x k. So if you could take out this inverse beforehand and keep then at every step all you have to do is just multiply this A inverse which have already computed and stored with this g dash x k. So this is what you do at every step of course there is a small issue here that explicitly computing the inverse and storing it is not a good idea when you have got very large systems because this A matrix even if A is a sparse matrix A inverse is likely to be a full matrix. So if you have got a network consisting of thousands of nodes you will find that A is a large matrix but which is also sparse normally the admittance matrices of the network are in fact sparse. So this A matrix when you take out the inverse is not necessarily sparse it will be usually a full matrix. So storing this A inverse again becomes a big issue. So instead what we do is we store the LU factors of A you would have done in a previous course on power system computation that solving an equation can be done by computing the LU factors of this A matrix and then doing backward and forward substitution. Now it turns out that you can by appropriately ordering your variables y get reasonably sparse LU factors. So the thing is that instead of storing A inverse you store the LU factors of A and then you just do backward and forward substitution. So this of course requires you to know a little bit about linear system solution. So I request you to look at any book on matrix computations which will invariably talk of how to solve linear equations using this LU factors and backward and forward substitution. Then what I am trying to tell you will become clear. The fact remains that if A is constant you can store the LU factors and if A is a small matrix you can store the inverse as well beforehand and at every time step in your of your numerical integration all you have to do is actually do some simple calculations in order to get y k from x k. You do not have to again at every step compute A a nu. So this is the advantage in case you can get the algebraic equations of your system in this form. This will not be true for example if x d double dash is not equal to x q double dash unfortunately or if you have got non-linear loads that is loads which cannot be represented as a simple linear relationship between the current and the voltage. So this is something which you should keep in mind. So your algebraic equation formulation can be done in the way I mentioned and then you do a partitioned solution you solve the algebraic equation to get y the variables y then use both x and y to get the new values of x by some numerical integration technique. So since we have removed the stiffness by modeling simplifications you can use explicit methods. It is a good idea to use explicit methods which are quite accurate like rangekutta fourth order. Please do not Euler method can be used but only in an academic sense you can just do it for the first time using Euler method but then you will have to constrain your time step to be quite small and you would get of quite an inaccurate solution otherwise. So what your trans instability program does is basically solve these differential algebraic equations. The algebraic equations arise because you neglected the network or stator transients and of course transients associated with any other fast acting device in the system. So what that fast acting device in the system is depends on what study you have you may have things like HVDC links and also other power electronic devices which you may decide to model by algebraic equations instead of differential equations because the transients associated with them are quite fast. Now somebody of course may ask that let us not neglect the fast transients retain all the differential equations can we do the can we actually study the system by considering both the fast and slow transients the answer is yes. So if you want to capture both fast and slow transients you will not make these modeling simplifications you would instead use a more detailed model and then use a variable time step implicit method like backward Euler. So this is some discussion which we had done in around the ninth and tenth lecture of this course in case you want to capture both fast and slow transients it is a good idea to use a variable step implicit method. But since you have neglected fast transients and made the modeling simple simpler we can in this case use explicit method. Implicit methods remember are more complicated because they will require you to solve non-linear algebraic equations in every time step in case you have got a non-linear system. So this is the differential equations in a synchronous machine are non-linear and you will find that using implicit method is not I mean if you want to do implicit method use implicit method very honestly then you would have to solve algebraic equations using iterative methods at every time step within every time step. So that becomes real pain. So I would suggest that you think of using a partitioned explicit method at least the first time you try to program a transient stability simulation. Now there are situations where you want to study slow the alternative which I mention may be appropriate for example if you want to study relative angular motion which is around at 0.1 to 1 hertz or 2 hertz. But you also may be interested in longer term dynamics in that case you have to you can model a system by using a transient stability like assumptions. But remember that once the fast transient fast transient in this case of the swings and the relative angular movement have died down you can think of increasing the time step. So you can actually use a transient stability program for long term transient behavior as well. But remember if you are studying long term behavior you have to make appropriate modeling you know you have to model a system appropriately. For example if you are doing a long term simulation you may have to model turbine boiler and other controls in more detail. If it is a very long term simulation you may have to model things like tap changing, tap changing, transformers and so on. You may have to model things like you know the action of over over excitation limiters in excitation system controllers and so on which are relatively slower acting kind of devices. So remember that there is a distinction made between different kinds of programs required to study different kinds of phenomena. But if you use variable time step implicit methods you can use a common model and use variable time steps to capture both the fast and slow transients. So some people some of some programs are in fact nowadays written in that way. Of course there is one important point which you should remember that suppose I have I have modeled a system in which I have which I have modeled fast transients. Fast transients are modeled and I would like to use a variable time step implicit method. Now in case your system which is modeling fast transients has got periodic periodic switchings like for example in a HVDC system. You have got you know if you have got what is known as a 12 pulse thyristor bridge. You will have switchings occurring every 12th of a cycle and if you want to capture every switching in that case if even if you use a variable time step implicit method you will not be able to increase your time steps because you are you will be forced to capture every switching. Every time there is a switching there is a kind of an event which you have to capture and that occurs every 12th of a second or 12th of a cycle for a 12 pulse converter. In that case it would not be a good idea to use a complete model of the system with no simplifications for the study of both fast and slow transients. It is better to use a program which models switchings and everything to study fast transients have a separate program and have a simplified model which uses larger time steps and possibly explicit methods and removes the stiffness out of the system use that for studying other phenomena. So, you have two separate programs that is why in this course I have talked of a special program called a transient stability program otherwise you may say that you model the whole system in detail and use variable time step in implicit methods. Do not make a distinction between fast and slow transient but given the example which I gave you like things like h v d c links etcetera where you are constrained to keep the time steps small you will never be able to increase it because you are capturing every switching. In that case it does not make sense to have one overarching kind of program which models everything because you will not be able to increase the time steps because you are capturing every switching event. So, if you are you cannot so there is no point in modeling switchings in case your interest is in long term dynamics. So, it is a good idea to make a separate program for long term dynamics a separate program for transient stability and a separate program for studying very fast transients like switching in power electronic devices. So, this is one thing which is a general principle which you should should guide you when you are developing programs. Now, let us now move on we have discussed one stability tool called a transient stability tool we can use this to simulate larger system considering consisting of many machines and so on. So, we move on now to studying a new phenomena we for some time we took a diversion we studied a analysis tool power system analysis ability analysis tool. Now, we move on and try to study a phenomena called voltage stability right in the beginning where it is a kind of half discussed it. Now, when I introduce you to the topic of voltage stability a very popular example is this single source connected to a single load. So, they have got a voltage source a sinusoidal voltage source which is connected to a load R through an impedance which is in fact, the source impedance X and if you look at the characteristics of the system for a variable R it is quite interesting. If you look at the voltage power and voltage characteristic rather this is not the power and voltage characteristic in fact, it is the power versus resistance characteristic I am sorry this is resistance and power characteristic. So, you have got resistance in the X axis and power dissipated in the load or the load power on the Y axis. So, it is a variation of load power with load resistance we assume E is equal to 1 and X is equal to 0.2 and now we plot the steady state power as a function of the resistance you will find that as resistance is reduced from infinity that is open circuit conditions you will find that the power starts increasing. So, if I reduce the resistance power increases, but there is a point in fact, it is the maximum power point at which you will find that any reduction in the resistance actually causes a reduction in the power. This is because a reduction in resistance eventually causes the voltage drop to be so significant that power starts dropping, but this occurs only after R is equal to X which is the maximum power point. Now, what is the significance of this is that if I go on reducing the load resistance your power increases, but after a point it decreases. Now, the reason why I show you this graph is that it is certainly not true that if you reduce resistance power will increase always if your R is at is beyond this point you will find it actually reducing R decreases the power. So, this is an interesting point which you should remember, but of course, this is not expected to occur. For example, unless R is near about equal to the source reactance. So, if normally of course, this is not occur usually a source impedance is much smaller than the load resistance that you will put. So, if you put a heater in your room, so if you have it is winter and you want to increase the temperature of your room you connect a heater into your room and you connect another heater in parallel to it we expect that the total power output will increase. So, you take a heater connected to your plug point you will find that the heater dissipates some amount of power which is the heating to heat the room. You take another heater put it in parallel with that that heater also lights up and you know you will find that the room becomes warmer quickly, but if the source impedance is very high, if the source impedance is very high which is normally not the case in that case putting another heater would cause such a voltage drop that eventually load power decreases. So, this does not occur normally and so you know this something which you normally do not encounter in our daily lives. This is another way of looking at this graph this is the plot of the load voltage on the y axis with load power. So, this is what really I wanted to say if I decrease my resistance I will find that the load power is increasing, but beyond a point decrease in r causes a reduction in the load power and interesting the voltage drops very significantly beyond this point. So, if you in fact take this resistance below this point this amount of load power you will find that the voltage starts dropping so substantially that the load power in fact starts decreasing when you decrease r remember that load power is v square by r voltage across the load divided square divided by r. Now, normally this would not be a problem in the sense that we do not go on decreasing load the way I showed you rather you do not decrease the resistance the way I showed you. In fact you in fact do have devices which implicitly do this however for example, consider what you see on your screen you have got the same system except that this resistance is fed wire transformer which has got a turns ratio 1 is to a. So, if a is equal to 1 we are back to the old system because you can refer the resistance on to the primary and that is simply r itself. So, if you have got of course, a tap ratio of a that is if you have got normally 1 is to 1 transformer, but of normally you have a tap ratio of a then the resistance gets reflected on to the primary side as r by a square. So, the thing is that suppose I have got a scheme or please pay attention suppose I have got a scheme a load or transformer in which I can change the tap setting and therefore, I can change a so that I try to regulate the voltage at the load. So, what I do is that in case I find this voltage to be low I increase this a the tap a if I increase this tap a the voltage which appears here would be a into v. So, if I find voltage dropping here or is low here I can change the tap ratio and ensure that you get whatever voltage or rather the voltage the nominal voltage here at the load, but from a system point of view for the source the source starts seeing a resistance which is r by a square. So, if a is greater than 1 this resistance is low. So, I can have a scheme think of the scheme of regulating load power. So, I am a very selfish load in the sense that if the voltage decreases I still want to draw the same amount of power. So, what will I do is voltage decreases on the primary side of the transformer I increase the tap ratio a and reduce effectively reduce the resistance which is seen by the source and draw the same roughly the same amount of power. So, the moment I see voltage decreasing I effectively reduce my resistance. So, this is what I would call as a dynamic load and a quite a selfish load at that who tries to keep his load power constant by changing the tap ratio. Now, this can sometimes lead to a runaway situation where you may have instability picture this the voltage here drops the voltage on the primary side of the transformer drops and I as a selfish load change the tap a I increase the tap a. So, that in spite of the fact that the voltage dropped here I get the nominal voltage v here as a result I will draw the same amount of power, but because I have changed the tap ratio the system is seeing a lower resistance and therefore, the voltage may decrease further. Now, this may become a runaway process in case this x is large normally reduction in a r will cause a reduction in r effectively will cause load power to increase the amount of power dissipated in the load will increase, but of course, if this r starts approaching x in that case a decrease in r may cause in the fact load power also to decrease. So, you may have a runaway situation in the sense that I change the tap a that decreases the effective resistance seen by the source that reduces the voltage further the drop in voltage further drop in voltage results in again increasing a that reduces the resistance seen by the source again that further reduces the voltage and it becomes a runaway process and a voltage collapses. Of course, it would not really collapse because a real tap changing transformer would have limits, but the point is that your taps would go on increasing and a voltage would go on decreasing to a point at which to a very low voltage. So, this can happen I mean this can happen of course, if the source impedance is large normally the normal source impedance is not very large. So, it is certainly not true under normal situations that if you deduce your load resistance tippy means of course, reasonable if for example, you have got one bulb on in your home if you switch on another bulb it does not cause a voltage collapse it is unlikely to happen, but suppose a source impedance does increase how can source impedance increase for example, a line trip occurs there are several lines which are feeding you know you normally have several parallel lines in a transmission network and one of the line strips then x will increase, but even so the source impedance is usually much much lower than the load resistance. So, you will never have a situation happening if you just a triple load a triple transmission line it usually will not result in x being so large because there are other parallel paths. So, typically this will not happen even after weakening a transmission network, but this can happen in case this source E itself has a source impedance. Now, think of this situation this source E is usually a generator this is your transmission network the transmission network source impedance is normally much smaller than whatever load you know whatever rated load or typical load you will put on this network. So, even if this actually this transmission line is made out of two lines even if you trip on transmission line normally you will not have a situation where reduction in R results and reduction in load power this typically does not occur, but one more phenomena which you should remember which can occur is relating to this source this source is usually an equivalent of generator a generator. Now, a generator is not a not strictly speaking a stiff voltage source, but it can be made like a stiff voltage source you can keep its magnitude at its terminal constant because we can change the field voltage which is applied to a synchronous generator. So, typically a synchronous generator although it has got extremely poor voltage regulation you can still make it a stiff voltage source because we can increase the field voltage every time the loading on the synchronous generator changes. So, if the loading increases you increase the field voltage and something which we have discussed in the modeling of a static excitation system is that typically there is a very large margin there is a large range or large ceiling voltages provided in an excitation system, but even so if you starts loading a synchronous machine you will find that the field voltage will go on increasing, but beyond the point will not be able to do that because the field voltage would have hit its limit or the field current would have hit a limit after which the field will start getting hot or rather heated up beyond the rating of the machine. So, even although a synchronous generator field can be controlled so as to regulate the terminal voltage and therefore, I have practically a constant e at the terminals there is a limit. So, if for example, you have got a transmission line a two transmission line you trip this transmission line and now the reactive power loading on the machine increases in that case you may have encountered a situation where this synchronous machine hits its field voltage limit in trying to trying to regulate the voltage it hits its limit in such a situation you no longer have a well regulated source here, but instead you will have something like this the full force of the synchronous reactance of a synchronous machine will come in series with the transmission line reactance and remember that this impedance can be quite large a synchronous reactance of a synchronous machine may be 2 per unit on its own base near about 2 per unit on its own base. So, this can be really very very large and under this situation you may encounter a very large source impedance effectively a large source impedance and thereby come into a situation where reducing load reducing the resistance actually reduces the load and therefore, you may have come going to a voltage reduction spiral if you have got a controlled selfish load which reduces or increases a which increases a in case the voltage drops. So, this is the reason why you may have under certain circumstances a pure voltage collapse scenario though rare it can occur in case you have got a weak transmission network coupled with some of the synchronous machine hitting the limits you can of course, have other situations fast voltage collapse occurring for example, if you have got induction machines connected to very weak lines. So, if there is a sudden disturbance which causes the slip of a induction machine to deviate like you have got a fault or you connect another induction machine in parallel and voltage dips you can have induction machine stalling. So, this is also example of faster voltage collapse where induction machine draws more current because its slip is a slip increases that causes a further drop which further you know causes a larger current to be drawn by the induction machine because its slip deviation has become large and you will find a complete voltage collapse and all induction machines at a bus stall this is also an example of voltage instability. So, we will try to simulate this example in the next class. So, next in the next class we will have a simulation of a system like this and we will try to see what happens and try to you know simulate this situation and show that the tap changing action of a transformer in order to maintain regulate voltage at the load bus in order that the load kind of regulates the power. So, if you regulate the voltage at a load bus effectively regulating the power. So, you are becoming a selfish load. So, you are drawing the same amount of power in spite of the fact that the voltage is reduced. In fact, the system would be happy if you were a responsive load in the sense that voltage drop you draw lesser power, but in case of tap changing transformers you have got a control which tries to override this voltage dependence and tries to draw the same amount of power in spite of a reduced voltage. So, we will take an example and show you that under special circumstances this can lead to a voltage collapse. So, this is something we will do in the next lecture.