 So, welcome to lecture 25. So, in this lecture what we will do is, we will understand how do we apply finite element method to analyze insulation and design insulation in an effective way in a high voltage equipment and here we will be talking mostly for a power transformer with you know high voltages involved on primary and secondary sides of the transformer. So, before going into finite element method, let us see some basics of high voltage insulation design. So, what are the types of breakdown mechanisms? Tribes of breakdown mechanisms are jump or bulk oil breakdown, creepage breakdown, corona and partial discharge. I will briefly describe this in further slides and then we are also going to see two withstand theories because finite element method will give you the stress distributions inside the problem domain, but we need to also compute withstand and that withstand or strength also can be calculated by using finite element analysis. So, we will go further now. Now, let us understand this jump or bulk oil breakdown and creepage breakdown. Now, as you can see here, this is our standard configuration of a high voltage lead in the vicinity of a ground plane. If we go on increasing this voltage applied to this high voltage lead, at some point when the electric field intensity exceeds certain value, then there could be a direct flash work from this high voltage lead to ground. So, that is called as bulk oil breakdown or jump breakdown. In creepage breakdown, if the same high voltage lead for some reason, if it has to be supported from this ground plane by using a insulating piece as shown here, rectangular piece, then what we observe here is this equipotential contours are cutting this surface of this insulating piece and there is a tangential electric field intensity all along this surface and the corresponding what is known as creepage stress. And thing to note here is the creepage strength is much lower than the bulk oil strength. That is why even though the creepage stress is lower, since the creepage strength is also lower that is why it is important to you know find out what is the creepage stress and compare with corresponding creepage strength and how do we calculate that stress and strength we will see in you know further slides. Now, let us go further. Now, the third you know is a partial breakdown mechanism is corona. Now, these both these mechanisms corona and partial discharge, they are partial breakdown mechanisms because only locally there is some breakdown happening. Now, this corona discharge is a well known phenomena in case of high voltage transmission lines. So, when you know particularly monsoon season when moisture is there, the air molecules around this high voltage conductor get ionized and then there is a corresponding energy loss and associated with energy loss you also have glow discharge as well as sound you can hear. So, you know this corona discharge and corona phenomenon needs to be you know mitigated and there are you know established ways to you know mitigate these corona effects. Coming to partial discharge, suppose you have a bulk solid insulation and inside that you have void of say either air or oil then since the dialectical constant of air or oil void would be less as compared to the dialectical constant of the solid insulation and since electric field intensity is inversely proportional to the dialectical constant the permittivity. Then what is going to happen? Higher stress is going to come across this void and there could be a local breakdown and discharge. Now, one may you know wonder then what is so great about this let there be local discharge since complete breakdown is not happening. But if you allow this kind of local discharges for a longer period of time then you know slowly and steadily insulation is going to get deteriorated and eventually there could be a complete flash over one day. So, that is why again partial discharge activities inside any insulation should be minimized and you know this corona and partial discharge phenomena can be also analyzed by using finite element analysis. Now, let us go further. So, now, let us understand how do we actually you know design bulk you know oil duct that means you know instead of having complete one oil duct between low voltage and high voltage findings of a transformer what is generally done is this entire oil duct is split into small oil ducts. So, these are individual small oil ducts and these are the solid insulating barriers which basically divide this whole oil duct into smaller ducts. Now, what why this is done because subdivision of oil ducts increases kV per mm withstand stress. So, suppose what does it mean if suppose you have 10 mm oil duct and if suppose it withstands 10 kV then you know 20 mm will not withstand 20 kV it may withstand may be 19 kV that is the meaning of you know increase in kV per mm withstand stress for lower ducts. Now, these barriers obviously would basically arrest propagation of discharge streamers between these two electrodes, but one has to remember that barriers should be as seen as mechanically possible otherwise there will be increased stress in oil because one may think since the insulation strength or withstand for solid insulation is much greater than that of oil then why not go on increasing the thickness of these barriers. So, that overall withstand is you know input significantly, but even though that may happen what is going to happen is the oil ducts smaller oil ducts in that case will get stressed to higher stress levels and that could be a problem. Now, how that happens let us you know analyze one simple case of a oil solid composite system wherein both these thicknesses of these two insulations are 10 mm and dielectric constants are relative permittivities are 2.2 and 4.4 of oil and solid respectively will get simply E 1 D 1 is equal to E 1 D 1 plus E 2 D 2 is equal to 100 kV because total voltage difference is 100 kV. So, now since E is inversely proportional to permittivity E 1 will be equal to twice E 2 right and then if you substitute in place of E 1 twice E 2 and then we will get E 2 as 3.3 kV per mm and then E 1 is 6.6 double of you know E 2. Now, if we actually find equivalent oil gap it can be found out like this. So, total voltage difference upon oil duct plus solid insulation thickness into ratio of two permittivities solid oil permittivity divided by solid insulation permittivity. So, then you get this as 100 by 15 and that is 6.6 kV as obtained earlier. So, thing to note is here this equivalent oil gap is now 15 mm. So, although you have solid barriers if we have to analyze this in terms of one equivalent oil gap it will be 15 mm. Now and if you increase this number of barriers or their thickness what is going to happen? This equivalent oil gap is going to further reduce and it will stress these oil ducts and since oil has lower dielectric withstand that could lead to some problems. So, that is the reason that barrier should be as thin as mechanically possible from mechanical strain point of view otherwise there will be increased stress in oil. Now, let us understand how to design individual duct for that we will use cumulative stress and withstand theory. So, what is well known is that smaller the stress gap or length or any insulation higher is its strength. So, that is clear from this these are the withstand curves blue ones you can see that if the gap is small withstand is higher and as the gap length increases the withstand becomes lower and that is the same reason that I mentioned on the previous slide it is always better to have smaller you know smaller and smaller ducts. So, that their strength is higher and when we say stress and strength remember the unit is KV per mm. So, now have you understood how the withstand behaves as a function of gap length now let us understand how to calculate cumulative stress. Now, let us take one gap suppose this is from here to here this is one gap here like for example, if suppose this is this gap and let us assume that maximum electric field intensity is on one of the ends in this case I am assuming on the left end and on the right end here it is E min for that duct and now let us divide this gap into equally spaced nodes 1, 2, 3 up to N and now let the voltages be V0, V1, V2, V3 and the distance between adjacent nodes 1 mm. So, now E0 and En we already know which is E max and E min. So, E1 is given by V0 minus V1 upon 1 mm so that is why you get KV per mm whereas here to we have to note that all the voltages are in kilo volts. So, similarly E2 instead of now having 1 mm we will take 2 mm because we are talking of cumulative stress. So, cumulative means we have to increase the gap. So, E2 will be V0 minus V2 upon 2 mm, E3 will be V0 minus V3 upon 3 mm. So, this is the cumulative stress. So, now all these cumulative stress values now they can be plotted as given by this graph and cumulative withstand will be something like this and at every point then we can calculate the margin that is the margin between withstand and the stress and this margin should be good enough. Now, see in case of insulation design the margin has to be much higher as compared to you know other design aspects because insulation breakdown is statistical phenomena and you know repeatability is not as good as compared to other engineering field phenomena. So, that is why you need to have good amount of margin between strength and stress. The value of margin at various place in the insulation should be more or less equal that means insulation everywhere is optimally utilized. You should not have a case wherein at some point in the insulation you have very less margin somewhere else you have ample margin. So, that is not good insulation design. Of course, this is a challenging design aspect and it may not be possible to equalize stresses everywhere in the insulation but our effort should be in that direction. So, this is how you know you can design bulk you know oil insulation by dividing into sub ducts and then designing individual ducts by using this cumulative stress and withstand theory. Now, let us understand this creepage and how we can use finite element method to you know analyze creepage phenomena and increase the withstand. Now, again this is suppose is one high voltage electrode and let us say this is another high voltage electrode and let us assume that this is at ground potential then the equipotential counters would be as shown. And now suppose you want to increase the withstand of this overall you know insulation gap and then you can do it by putting some insulating you know piece of inverted L shape. But then what we are doing although we may have increased the overall breakdown strength of this gap but what is happening is along this surface of this insulating piece you are having significant amount of tangential electric field intensity and corresponding creepage stress. So, now we can see here these equipotential counters are crossing and there will be surface electric field intensity and the corresponding creepage stress. And as I mentioned earlier since creepage strength is smaller than the bulk insulation strength then that is right that is why you need to basically watch creepage stress closely. How do we basically increase the creepage withstand? What we could do is we can shape this insulating piece barrier here so that the insulating piece is almost along the equipotential counters. So, now since the equipotential counters are along the you know surface the tangential electric field is reduced considerably and you know stress is correspondingly reduced and this way we can increase the you know quality and reliability of the insulation. Now let us go further how do we actually you know assess the creepest you know stress whether it is below certain you know allowed value. Again we will you know use the same theory cumulative stress and withstand theory. Now here what we will do here if you see on the vertical you know portion here E max will be somewhere here E min the tangential component E min will be somewhere here minimum. So, then starting from E max to E min you can again calculate the cumulative stress and the corresponding cumulative withstand is given by this you know curve. Now you know equation for cumulative withstand is available in the literature I have given some references in these slides. So, you can refer that similarly on this horizontal you know surface E max will be somewhere here E min will be somewhere here. So, again from starting from here to here you can calculate and plot cumulative stress and then you already have this cumulative withstand and again you can compare margin between withstand and stress at various points and ensure that you have certain minimum margin available at any portion. Now let us go to another case study wherein we design lead to ground insulation. Now this is again high voltage lead and ground and we have you know some insulation on this lead right this is paper insulation. So, now here we will calculate stress by using finite element method and maximum stress will be at this point in oil and it will not be generally here in the solid because electric field is inversely proportional to dielectric constant. So, typically the electric field here will be higher in the oil at this surface at this interface in the oil the electric field intensity will be higher as compared to at this point in the solid in the paper insulation. So, the finite element method will give you the maximum stress at this point right. So, which will be typically at the minimum distance between the you know insulated lead to ground that straight line. So, how do we calculate then the withstand? withstand can be calculated by using stressed oil volume. Now what is stressed oil volume is shown here. Stressed oil volume is generally calculated by first computing the area between E max and 90 percent of E max counter. On this counter here the stress values will be 90 percent of E max and this shaded area will give you stressed oil area between 100 percent stress and 90 percent stress values. If you multiply this area by the lead length which will be in the direction perpendicular to this plane right. So, that lead length multiplied by this stressed area will give you stressed oil volume and strength will be some function of stressed oil volume. What is that equation you can you know refer these references here and then E max calculated by FEM should be considerably less than the strength calculated by this you know relation. So, this is how you can actually you know design you know high voltage lead to ground you know gap. Now let us understand how do we do coding in FEM for calculating stress oil volume and you know corresponding strength. So, now we know electric field intensity is given by E is minus gradient of V and V in FEM formulation is N 1 V 1 plus N 2 V 2 plus N 3 V 3 N 1 N 2 N 3 are the shape functions of you know 3 nodes of the element under consideration. So, now substituting V here then you will get this expression remembering again that you know V 1 V 2 V 3 are not you know functions of x and y that is why they are taken out and del operator acts only on N 1 N 2 and N 3 which are functions of x and y as you know I have given for ready reference the expression for N 1. So, now del N i is given by P i A x hat plus Q i A y hat upon 2 delta where P i is y 2 minus y 3 and sorry P 1 if we are taking because this is N 1. So, P 1 will be y 2 minus y 3 and Q 1 will be x 3 minus x 2. So, this also we have seen earlier. So, then substituting all these you know expressions here in E then we get expression for E as this right and then the magnitude of E will be simply square root of E x square plus E y square for the element under consideration. Going further now let us see how to calculate electric field and stress-toil volume. So, first what we will do is we set up a for loop going from 1 to the total number of elements then by this command this also we have seen earlier when we were studying some other codes by this command we get global node numbers of the element under consideration into this nodes you know matrix which is a column matrix of you know having 3 entries. So, global node numbers get assigned to 3 entries of this matrix node and then by these 2 commands again x c and y c they are they are having 3 entries they are also matrices and they are having 3 entries. So, those 3 entries will be x and y coordinates of the corresponding 3 global node numbers right and then we are initializing this p and q to 0 and then p 1 p 2 p 3 q 1 q 2 q 3 as I mentioned on the previous slide and we have seen this earlier number of times and then area of the element under consideration will be simply 0.5 into p 2 q 3 minus p 3 q 2 refer l 18 for this and then e x and e y based on the previous slide we can write you know this and of course here also it is evident e x will be this divided by 2 delta with a minus sign similarly e y will be this bracketed term divided by 2 delta with a minus sign. So, these are the 2 expressions here and then the resultant electric field intensity of the element under consideration will be simply square root of e x square plus e y square right e x e y e net for all elements will get stored in these matrices when we run this for loop. Now, we go further and understand how do we calculate stress-toil volume again we you know run a for loop and then we check whether the sub domain number is 1. Now, this why 1 because in the you know FEM analysis that is done here number 1 is assigned to oil and number 2 is assigned to paper insulation on the you know lead right. So, if the sub domain number of that element is 1 right then you basically take into this matrix the corresponding net electric field intensity of that element and store it. Right and then and you also note down the element number biasing this you know vector column vector a ELE underscore oil because this will be useful when finally we will have to you know calculate a total stressed area and volume. Now, actually we will calculate maximum of all the you know e oils that we have got here in this you know column vector suppose there are 100 elements in oil finite elements in oil we are finding out which element has the maximum you know e max maximum electric field intensity. So, that will be given by e 100 will be maximum of the values that are stored in this you know column vector column matrix and e 90 will be 90 percent of that right. So, now what we have to do we have to find out those elements in which the electric field intensity is between 90 percent and 100 percent values. So, that can be you know simply obtained by using this you know set of statements. And then you know you can if it is in between 90 and 100 percent of e max then we basically note down you know that element number because remember this here we are already we have already stored you know element numbers in this you know matrix ELE oil matrix which is again a column vector. So, now so we note down all those elements in oil which are having stress value between 90 percent and 100 percent of e max. And then we calculate stress oil volume as sum of areas of all those elements all those elements which have stress between 90 and 100 percent right. And then this total stressed area between 90 percent and 100 percent stress values is multiplied by length in z direction perpendicular to this plane of you know board which is in this case it is you know length is 5 meters. So, when we do that we get the stress oil volume and strength is a function of stress oil volume the electric stress we have obtained from finite element analysis. And then basically we compare you know using this stress oil volume we calculate the strength and then we compare this against this. So, here you can see electric stress is much lower than strength. So, it is a good insulation design. Now, we come to the last topic of this lecture wherein we will see how do we use design of experiment technique for you know you know designing a good you know insulation system. So, here again we take the same configuration insulated lead to ground. Now, we want to optimize this insulation system. So, what are the variables? One is the bare diameter, second is the insulation thickness and third is the gap. Now, the question arises what should be the optimum combination of this A and B, A, B and C or a designer of this system would generally know what are the values of A, B, C he can vary you know in certain range which will depend on established design and manufacturing practices as well as it depends on you know availability of you know material in standard sizes. So, now here we first note down the total you know range of these 3 factors A, B, C. So, A is you know allowed to vary from A to 12 mm, B is allowed to vary from 2 to 4 mm and gap is allowed to vary from 20 to 25 mm. So, now we want to find out what will be the optimum combination of this. So, now instead of doing you know number of simulations with various combinations you can do a set of experiments set of 9 experiments as per this design of experiments technique and then we have 9 combinations of these 3 factors. So, for example, for first combination all these factors are at level 1. We get E max values from FEM analysis as given here. Now, what we do? We do analysis of means. Now, these are in these 3 by 3 matrix all these values are means. So, what is for example, this mean A is at level 1 for 3 experiments. These are the 3 experiments wherein A is at level 1. So, that mean value will be average of these 3 values. So, this is that average. So, similarly, then you can calculate you know means for other cases. Now, you can see here for the considered you know levels of the factors, we need to find out which is the most influential factor deciding you know the performance figure which in this case is the electric field intensity maximum electric field intensity. Now, from this table itself you can see that the means of B are having maximum you know range it is varying from 5.7 to 7.1 that means B is more influential in the considered ranges of 3 factors and that is evident from this percentage effect calculation. Now, it is evident here also that when B is very you are you know the change is much you know more as compared to the other 2 factors. So, what does this all tell you is that you can very suitably B so that you get maximum advantage in terms of getting better performance figure. So, you know this way you can of course I have just told you some basic things about this design of experiments technique you can actually refer these references and then understand more about the power and potential of this technique for not only insulation design but for design of any other engineering aspect. Thank you. So, we will conclude here lecture 25. Thank you.