 Based on the understanding of how linear systems behave during transient conditions, we were able to understand a simple linear circuit consisting of capacitors and inductors in the previous lecture. In fact, there was one capacitor and two inductors in that circuit. We could take out a time response and very importantly, we could take out a few characteristics of the system. In fact, the characteristics of the system depend on the model parameters in addition to the circuit itself. Now, one of the most important things which came out of that study was that there is a combination of fast and slow transients. Now, this is a very fundamental you know kind of modeling issue which we shall encounter in our studies of power systems. So, I would recommend that a great deal of attention be paid on these the modeling principles as far as systems in which there are a mixture of fast and slow transients. A mixture of fast and slow transients also bring into play several effects when we try to numerically simulate these systems that is when you are trying to do understand the system by numerically integrating them. A stiff system or a stiff system in which there are fast and slow transients coming into the picture requires some specific tools numerical tools to solve. We shall see that we shall be able to make some modeling simplifications based on the fact that there is a clear time scale difference between the nature of transients which are seen in certain circuits. Now, so this lecture we will try to understand stiff systems and multi time scale modeling. We will do that of course by considering again the same example which we discussed in the previous class. If you find time by the end of the lecture we will also move on to numerical integration methods just a brief overview. Now, if you recall what we did in the previous class we consider this particular circuit it was consisting of a voltage source which is connected to a circuit consisting of 2 inductors a capacitor. The values are of importance here you had a resistance of 0.1 ohm, 10 millihenry inductor, 1 henry there is a fat inductor and a 100 microfarad capacitor and we try to analyze the transients associated with this. Now, if you look at it from mathematical point of view it boils down to trying to solve this particular circuit this particular system it is a we write down the state space equations for this system. The states being the current to the inductor 10 millihenry the current to the 1 henry inductor and the voltage across the capacitor. This of course is a system we derived last time and we try to find out what was the response if the initial conditions are these I mean this is the initial conditions before the circuit is energized by an input. Now, what we saw was the response of course is given by this formula e raise to a t into the initial conditions plus this convolution into integral eventually the response of I L 1 I L 2 and V C turns out to be this which was evaluated effectively required us to evaluate this where P is the Eigen vector matrix and it required us to get lambda 1 lambda 2 and lambda 3 and the Eigen vector matrix. Now, this was got using psi lab you can also use mat lab for getting the Eigen values of the matrix A. Now, if you recall what A is this now the most notable feature here when we did this study was lambda 1 was approximately equal to this lambda 2 was this there was a conject complex conjugate pair and you had a real Eigen value which also had a negative part. So, it was this is stable system. Now, these were the Eigen values and the most important feature which struck us then was that the magnitude of this Eigen value these this pair of Eigen values and this one is vastly different this is a very small Eigen value this is a very large magnitude Eigen value. So, one thing which we had mentioned sometime a large magnitude of Eigen value is associated with very fast rates of change. So, that is one important thing. So, we have got some part of the system is going to some part of the response is going to move very fast and some of it very slowly. Another thing which we saw when we looked again like the at the Eigen vectors the Eigen vectors were approximated by this. This is of course, an approximation we did some rounding of these were very small values. So, we have done this rounded of rounded it off to 0 this was practically point this is practically 0.1 this minus j 0.1 and so on. So, the approximate Eigen vectors were these and the most important thing we saw was in the second state in the Eigen vector corresponding to the second state and the first and second Eigen value these are corresponding to the first and second Eigen value. In fact, the complex pair the second state has no practically 0 observability these were of course, rounded off, but we saw that these were almost 0. So, you will not observe terms corresponding to e raise to lambda 1 and lambda 2 t in the response. In fact, lambda 1 and lambda 2 t were complex numbers. So, the first and second Eigen values in fact, point out towards oscillatory response and the final response of course, was evaluated to be this. So, what you notice here of course, here you have got this particular oscillatory response this comes out because of the complex pair of Eigen values conjugate pair of Eigen values you also have this particular mode this is practically not observable in the variable i l 2. So, this is where we stopped last time. Now, one of the important things which we can see here is there is of course, this is a very this particular transient is associated with a high rate of change you look at the real part as well as the imaginary part rather the frequency as well as the rate of decay are quite high. So, if you look at just we will redraw this circuit. So, what we see is this in i l 1 is this and i l 2 is this. Now, in i l 2 we are hardly seeing any oscillatory component that is because of course, I mean intuitively this being a very large inductor it does not happy with large rates of change. So, you will find that you know this large inductor prevents a large rate of fast rate of change of the current. So, that is why in i l 2 one is not surprised to see that the faster transient is not visible. So, in fact, if you look at the response of i l 2 you will find that it starts from 0 of course, if it grows to 10 amperes. So, it starts with 0 and grows to 10 amperes you can see this as t tends to infinity the value of i 2 becomes 10. This is roughly 4 times the time constant associated with this Eigen value. So, Eigen value is approximately 0.1 the magnitude of the Eigen value. So, the time constant associated with this is roughly 10 seconds. So, around 40 seconds you will see that this builds up to hear about the final value. So, this is around 40 seconds you hardly see any oscillatory response in this. Now, what about i l 1 if you look at i l 1 the response is primarily consisting of this component and a small component which is oscillatory a decaying oscillatory waveform. So, what you have got is a decaying oscillation and then again this settles down to 10 amperes in 40 around 40 seconds. So, this is the response of i l 2 and this is the response and of course, v c is interesting you look at how v c looks like it consists of this as well as this. So, what it has here is exponentially decaying component as well as a decaying oscillatory component of high frequency. So, what you are likely to find is something of this kind roughly around 40 seconds it will become a very small value and in the beginning there is an oscillatory component. Now, when you are studying this circuit a thing which will get suggested automatically is can we simplify the analysis of this circuit. For example, if you look at the state space equation of the system it appears that the response for i l 2 the slowly varying variable will be preserved even if I make certain approximations in this. Just look at what I am trying to say what I am saying is as far as the response of i l 2 goes this is the one Henry inductor goes I can obtain it simply by trying to study this circuit. In fact, what is the response of this circuit it is exactly what we got for i l 2 time constant is of course, 10 seconds the lambda for this circuit is 0.1 minus 0.1. So, what we you know very a simple simplification of the circuit does not seem to affect the response of i l 2. So, it appears that we can actually make a simplification in fact, looking at it a bit formally if I if I said d i l 1 in this I artificially set it to be 0 I mean set this value here to be 0 the rest of the things remain exactly the same. So, what I have done is just set these two things to 0 in that case what we have here is only one differential equation corresponding to i l 2 and i l 1 and v c can in fact, be written in terms of i l 2. So, what we have done is you have got now d i l 2 by d t is equal to v c this is a differential equation and i l 1 is nothing, but i l 2 from this last equation. Since, this has been set to 0 you have 10000 i l 1 is equal to minus 10 plus sorry 10000 i l 1 minus 10000 i l 2 is equal to 0 which actually gives you i l 1 is equal to i l 2 and from this we will get v c is equal to what what we have here is minus 10 times i l 1 minus 100 v c plus 100 is equal to 0. So, what we have from that is we will get is v c is equal to 100 which is nothing, but v c is equal to 1 minus 0.1 i l 2 because i l 1 is equal to i l 2. So, this is what v c is. So, it is algebraically. So, i l 1 and v c are algebraically related to i l 2. So, what we have is basically a state space equation d i l 2 by d t is equal to v c which is nothing, but 1 minus 0.1 i l 2. So, this is my one and only differential equation if I make this approximation. So, if I make this approximation you have reduced your system size to this and if you want to get v c and i l 1 you have to use this algebraic equation and this algebraic equation. So, what we have if you look at this particular system if you look at the response of this particular system we have this is the same or practically the same responses before. What about i l 1 i l 1 also follows the same response because i l 1 is equal to i l 2 and v c is 1 minus 0.1 i l 2 and v 2. So, if you look at the response of v c it is simply this is 1. So, you will have this 10 is not to scale like this. So, this is v c. Now, if you look at we had drawn these responses before if you consider the whole system in its full glory you have got this if you make the approximation which I made you will get this. So, the basic issue which I would like to point out here is that if you have got a system please note carefully if you have got a system in which there are fast and slow transients and your interest is in the slow transients then you can get the slow transient behavior roughly correctly you know you will get it almost correct if you assume the rate of change of the variables which you think are associated with the fast transients is set to 0. So, what I have done is that I have artificially set the derivative in this equation to 0 and made i l 1 and i l v c algebraically related to i l 2. So, you can make a general statement of this kind. So, if you have got some fast variables then as far as the behavior the slow behavior of the system can be obtained simply by replacing this algebraic equation rather this differential equation by an algebraic equation that is I am making x 1 and x 2 x 2 algebraically related to x 1. So, in such a case I will be able to capture the slow transient without making too many errors. So, the basic point is first thing you need is this is a clear separation of fast and slow trans there is a mixture of fast and slow transients in the response as is seen in our system here. So, this is a very very important you know modeling principle or modeling simplification which we are making. Now, this we in fact this kind of modeling simplification we seem to making all the time without knowing it. For example, you just take a simple case of a a bulb and connect it to the plug and you this is a bulb this is a bulb and the basic point is I want to study how this bulb lights up. Now, when you switch this on the bulb practically lights up instantaneously it is only a resistive kind of load. But of course, a purist may say well their parasitic inductances and capacitance associated with this wire which connects the bulb to the plug plug point. So, what you somebody may say is well if you really want to study how this bulb is going to light up you ought to model all the inductances the distributed inductances and capacitance as in a transmission line. Now, the point is you really do not need to do that because the transients associated with this die down very soon. So, that is why we can practically treat this wire as if it is directly a simple resistance less connection to this point resistance and zero dynamics associated with this. Of course, if you are interested in fact, if one really are looking at time scales of a few nanoseconds or a micro seconds you may actually see some transients evolving even in this simple circuit. But usually we may not be interested in those fast transients. So, in such a case remember you can make an approximation you can actually neglect the dynamics associated with that fast transients by simply taking the state equations and setting the right hand side where we write the d by d t is equal to 0 in those states which we think are associated with the fast transients. Of course, now this seems to be a kind of a chicken and egg story because to know which elements or which states in a circuit are associated with slow or fast transients itself requires you to do an analysis of the system. So, you have to do a full blown analysis and then find out which states are associated with the fast transients and then you can make this modeling simplification. But actually engineers do not do that in most situations it is fairly obvious to an engineer what are the things which decay fast and which other thing which are the transients which really move very fast and which are the transients which move very slowly and which are the states associated with them even at a rough way. So, even through engineering intuition one should be able to find this out. For example, in this particular circuit it is quite clear that the large inductance the magnetizing inductance in that particular circuit 1 Henry is much larger compared to the leakage that is the 0.1 Henry inductance and 100 micro farad capacitor likely it looks a large value. So, I could have guessed that I could have probably made this assumption. So, if I were an engineer I would not have waited I would have just made this assumption and in so far as the slow transients are concerned I would have made very little error even if I had just taken this circuit. So, if I wanted to represent my slow transient correctly it was adequate to have a circuit of this kind. Now, what we really have done is there was an inductor here, but it was a very small value. So, we assume that it goes into steady state right away it kind of is algebraically related to the current here and we also have a capacitor here which you have basically said it is open circuited. So, we have basically got this particular circuit which is marked in red by making the d by d t's associated with these states equal to 0. But remember if you do not want if you want to make this approximation without actually doing the analysis of the circuit then you have to rely on engineering intuition. Now, you can also make a converse kind of analysis suppose you have got a system which has got a mixture of slow and fast transients and you are interested in what happens just after the transient has occurred you do not want to we are not really interested in the fast rather the slow variations or even the final steady state, but you are interested in what happens just a few maybe just for short while after the transient has occurred. For example, in this circuit remember that your V c varies like this this is around 1 and this is around 40 seconds. Now, if my aim is to study this fast transient here so of course, they could be situations where you are interested in the fast transients. So, if you are interested in the fast transients can you guess what has to be done well one thing is you need to know the states which are associated with the fast transients again this is an intuitive thing how do you know which states are associated with the fast transients you need to really look at something more than what we have studied till now I will just mention that in a few moments from now, but right now let us try to do it intuitively. So, if I want to study the fast transients probably if I assume we will just first write it down these are the if we assume that this particular state remains where it is during this extremely short time period you know that this is a very fast transient so it will decay very fast if it is stable. So, I rather I should say that I want to see the transient in a very small window in that window I L 2 does not change at all. So, what I will do is I will rewrite my circuit in this fashion I will just rewrite it on another page d I L 1 by d t d I L 2 by d t and d V c by d t is equal to what I will do is this is the original state space equation what I will do is I will set this equal to 0 this also as 0. So, what it means is that I L 2 d I L 2 is equal to 0 the rest of the things of course, remain the same. So, what we have is I L 2 see if I said d I L 2 is equal to 0 it implies that I L 2 is equal to as a function of time is nothing but I L 2 0. So, what I am going to do is make this approximation. So, when I am interested in what happens for a short while after a particular disturbance or the initial period after a transient is occurred or some disturbances occurred then I can make this assumption if I know that this variable is associated with slow transients only. So, if I know that this is what I can do. So, in such a case you will have d I L 1 by d t d V c by d t you know what I L 2 is because I have assumed it to be 0. So, you will have simply the state space equations as this is of course, 0 because I have assumed that I L 2 does not change in the short period which we have. So, this is the kind of a converse kind of approximation which we can make. Now, this particular circuit or this particular set of equations in fact, describes if this is set to 0 then in fact it describes this. So, in the fast period when we can say that this the current in this does not move at all. So, we have for all practical purposes disconnected this the current through this stays at 0 it does not change at all in that case you have got this fast dynamical equivalent of the circuit and the Eigen values of this will have to be found out. In fact, if you do a study of this system the Eigen values you will get are or I will just write down the directly the response of this circuit it will turn out to be I L 1 is equal to 0.5 e raise to minus 5 t sin 999.9 that is almost 1000 and v c of t this is the final time response for this circuit. I am just writing it down directly you can take out the Eigen values and Eigen vectors using psi lab cos plus 0.005 sin. So, this is our response. So, what you see is that I L v c is this if you look at try actually find out what v c is if you evaluate this is this whereas, what we had got using the complete system was something like this. So, actually only in the initial portion this gives the correct result. So, if you are only interested in the initial behavior the very in a very short time period if you are interested in the behavior you can assume that the slow state or the state associated with the slow variable is just frozen at its pre disturbance value. So, that is an important modeling point which you should appreciate. Now well one of the simple reasons why we would need to do this modeling approximation is that if you have a system consisting of fast and slow transients you can actually reduce the size of the system you want to study. In fact, the differential as we have in this of course, simple example we did not get much of an improvement as far as the size of the system is concerned. In the we managed to get rid of two variables two state variables and were left with just one state variable d i l 2 by d t when we wanted to understand the slow transients. And if you wanted to understand the fast transients you could assume that state associated with the slow transient i l 2 was frozen at its pre disturbance value. So, in this particular example there seems to be no very great advantage computation or otherwise of making this modeling simplification. But when you come to large systems you know consisting of very many diverse components it does make sense to make these modeling simplifications. So, one important thing you should note which is true of any engineering modeling is you do not have to model the system in its full glory you do not have to start from Maxwell's equation and model everything in terms of PDEs and so on you know you can make some approximations get simplified models you can even neglect the transients d by d t is associated with some elements which you know from engineering judgment you know they are associated only with the slow transient. So, you can actually make this kind of time scale distinction and the modeling can be appropriate even in a power system since we have many diverse components a huge variation in time scales. So, you can shift your attention to this particular slide. So, I have drawn a kind of a transient and control spectrum. So, if you just look at the kind of transients you are likely to study in fact you will be studying some of these transients in this particular course. At the fastest transients the fastest transients in a power system are those associated with traveling wave phenomena on the transmission lines. In fact, in such a case you will have to model transmission lines in a great amount of detail you know you may have to model them as PDEs partial differential equations. But of course, when you are studying lightning and switching transients you can assume some things like the speed of the synchronous generators etcetera not going to vary much in 0.01 seconds or 0.0001 seconds. So, you can make modeling simplifications. So, if you are only interested in looking at lightning and switching transients you can assume that the slow transients are frozen at the predisturbance values. As you move towards the right in the spectrum after lightning and switching transients you have slower network transients as well as some torsional transients associated with the shaft of turbines and generators. Even at a slower time scale say from half a second to around 10 seconds we have what are known as relative angle dynamics. The dynamics associated with machines staying in synchronism. So, if you give a for example, a push to a synchronous machine which is connected to another synchronous machine you are likely to excite oscillations of around 1 hertz or so. And also if you give a very large disturbance you may have loss of stability. All these phenomena take place in a period of around half a second to around 10 seconds. So, all these things are visible in that if you take a snapshot in this period. In fact, if you are studying slow relative angle dynamics you do not model a transmission line by you know the partial differential equations. In fact, you do not even you can even neglect the d by d t is associated with the transmission line inductances and capacitances. At a even slower scale are the prime over dynamics and flow slow frequency and load changes. These are slower much slower. In fact, although it is true that I can cause a load change by simply going and switching off a bulb. On an aggregate sense you know if you look at a substation a extra high voltage substation or the power system as a whole the overall power you know power being consumed at a given time does not vary very dramatically. I mean you can almost predict how the load changes. In fact, you know at unearthly hours like around 1 or 2 am the load is very low. And after 10 o clock in the morning it starts rising. There is a kind of peak at around 10 30 or 11 and maybe at after 8 o clock in the night there is another big peak that is the highest peak in the system. This is the typical kind of loading scenario. So, these of course things these load changes can be considered as very slow changes. They they cannot even call them as disturbance unless you have a sudden load trip of say you know 100 megawatt in in a say 20 gigawatt system. So, this kind of thing can be considered as disturbance, but generally load changes slowly and it is not really changing at a very fast rate. Similarly, you can even you know these are the inherent you know elements in the system the power elements in the system. There are elements which we put into the power system. These are usually control systems designed by us. For example, protection systems are a kind of a special controls which are put in a power system which kind of isolate faulty equipment. So, the equipment protection can be in the range of around 1 cycle to around 1 second. So, this is typically the equipment range of equipment protection the transit rather the time at which equipment protection acts. Power electronic controls on the other hand also rather they also are very fast. I mean you are trying to control for example, a rectifier in a HPDC station that also is a very fast kind of moving system and the controls associated with them are also fast. You are trying to control the firing of the electronic valve. So, basically you find that the control systems there also are fairly fast. You also have other control systems like for example, the excitation in a synchronous machine is also controlled by what is known as automatic voltage regulator that also you know you can consider as acting in the range of 0.1 to around 1 or 2 seconds. So, that is the typical you can say this response and deliberately not giving too much precision into what I am trying to say. When we come to that topic later in this course when you model AVRs automatic voltage regulators excitation systems you will come to know about the physical elements concerned with these systems. You also have system protection schemes like you know I mentioned that if a system loses synchronism you have got synchronous machines losing synchronism with other synchronous machines. You may wish to do islanding that is controlled system separation of the system or you can do under frequency load shedding and so on under different circumstances. For example, under frequency load shedding is done when there is a sudden very gross kind of load generation imbalance. So, you may these are all called system protection schemes. Prime mover controllers and governors they act between the range the actuators associated with these things require around 1 to 100 seconds and of course, manual control is much much slower of the order of minutes. You do have continuous monitoring of the system and manual control actions, but usually manual control actions are associated with changing the set points of various automatic regulators in the system as well as in some you know unusual cases you may even want to you know actuate your protection systems manually. So, manual controls of course, are much much slower and you know they can take a longer longer time. Now of course, one of the important points which we have come across is that a very important thing. In fact, the chicken and egg story you know when do you really say that a particular state is associated with a particular Eigen value or a mode. See the basic point which we really tried to tell you today was that if you have got a slow transients find out the you know state variables which are associated with the slow transients and the fast transients. Then in the fast transients you set d by d t equal to 0 I mean set the right hand side rather the left hand side of your state equation corresponding to the derivatives of those fast variables equal to 0. So, I have converted your algebraic equations rather the differential equations into algebraic equations, eliminate them if you can and just work with the slow subsystem with the x 2 the faster variables simply algebraically related to your slower variables. So, this is basically what we have learnt, but the whole point is how do you know which state is associated with you know the faster and slower transients. I told you you have to do it intuitively, but that is not a very you know comforting answer for students who are learning and engineering subject for the first time. So, let us just quickly look at the Eigen vectors associated perhaps they will give us a clue. So, if you look at the Eigen vectors associated with the slow and fast transients remember lambda 1 and lambda 2 lambda 1 is minus 5 plus a plus or was it minus it was plus j lambda 2 was minus 5 minus j and lambda 3 was 0.1 minus 0.1 what you see here is in the second state variable you do not observe lambda 1 and lambda 2 there is very little observability. So, perhaps it is practically 0. So, if such a situation occurs you can say maybe you can say that i l 2 is not associated with the fast transients, but there is a pitfall in just looking at these Eigen vectors to come to this conclusion. Let us look at the converse thing which are the variables associated with the slow transients. So, you will directly say i l 2 was associated with the slow transients, but look at the components corresponding to i l 2 corresponding to lambda 3. The Eigen vector components corresponding to the 3 states and the Eigen value minus 0.1 this is a slow transient you will see that. In fact, i l 1 also is associated with it you will observe this slow transient not only in i l 1, but you will also observe it in v c. So, just looking at these Eigen vector components of p you will not tell you that i l 2 is associated with the slow transient. So, what you need to do here really is look at both p and p inverse if you look at both p and p inverse. So, we will do this computation using psi lab. So, I will just write down the A matrix first the A matrix was minus 10 0 minus 100 0 0 1. So, this is A. So, the Eigen values of A are obtained from this command this is what we got. So, this is the Eigen value 0.1 we rounded of it to 0.1 minus 0.1 this is my approximately minus 5 plus or minus j 1005 j j or i i is i or j r square root of minus 1. Now, if you want to get the right Eigen values right Eigen vectors you have to use this command and p gives you the right Eigen vectors. In fact, you see that this is roughly j into 0.1 these are very small values this is an extremely small value. So, we have approximated it to be 0. In fact, if you look at what I wrote here on the sheet these are just approximations of what you have seen there. So, just rounded it off you know. So, this is approximately 0 and this is what we get is p. So, p is I will just rewrite it p is now what is p inverse p inverse is what we called as q. So, if you look at q which is nothing but p inverse this is roughly I will just write it down roughly. So, it is roughly I will just write it down while you can note it down also. So, it is minus j 5 plus j 5 approximately and this is roughly 0.015 and this is j 5 minus j 5 it is not a surprise that this Eigen vector this is the complex conjugate of this. So, minus 1.40 this is a complex conjugate 1.043 I encourage you to download psi lab and just try out these example try out this particular example yourself this is almost 0. So, what we will do is do an element by element multiplication of this kind we will multiply this by this this by this this by this and we will multiply this by this this by this this by this and so on. So, what we are doing effectively is a kind of a element by element multiplication of p and p inverse transpose. So, what we will do is use this command p dot star inverse of p dot transpose this is a transpose. So, what we will get in such a case. So, what we have done is p dot star into p inverse transpose this dot star implies a element by element multiplication. So, this is given by 0.5, 0.5 roughly 0, roughly 0, roughly 0, 0.99. So, this is almost 1 and we have got 0.5, 0.5 and 0. So, if I compute p I compute p inverse and I do the computation p dot star that is a element by element multiplication of p and p inverse transpose this is what I get. What we see here is this of course corresponding to state 1, state 2 and state 3 this is Eigen value 1, Eigen value 2 and Eigen value 3 this is I L 1, I L 2 and we see what we see is that this particular thing is called a participation matrix, participation matrix and it gives you the participation of a state in a mode. For example, I L 1 is associated with the complex pair of modes, V c is also associated with the complex pair of modes and I L 2 is associated fully it is 1 it is a normalized measure is completely associated with lambda 3. So, actually this particular matrix p dot star p inverse transpose this is the element by element multiplication of these two matrices actually gives you an interesting matrix called the participation matrix and this participation matrix in fact, gives you correctly the association of various states to various modes. So, if the participation of state number 3 in Eigen value Eigen value 3 is 0 it means that this state is not participating in that particular Eigen value. So, if you look at this participation matrix what you see is I L 1 participates in the complex mode, complex pair of mode Eigen values and I L 2 participates in this. In fact, remember that I L 3 is the slow mode and lambda 1 and lambda 2 correspond to the fast transient. So, I L 1 and we see correspond to the fast transients and I L 2 corresponds to the slow transient. So, we can actually if you take out this participation matrix we will we are able to tell which states are associated with which modes. Of course, there may be situations in which many states are associated with a certain mode like for example, I L 1 and we see are both associated with the fast transient. So, you have to consider them together whenever you do fast modeling or fast transient modeling. Fast transient modeling means the slow transients are assumed to be frozen in the predisturbance state. So, this is a very important concept called participation. We will I will give you a reference for this in the next class again when we recap this particular part of the subject again. In the next class we shall move on to the next part of this course that is numerical integration techniques for you know numerical integration techniques for dynamical systems. Numerical integration techniques will be required in systems which are too complicated to handle by linearized analysis of the kind I have shown you today. So, with this we will end today's lecture.