Chapter nine. Another example of three phase normalization. In this example, we have three generators that are connected together through three Transformers in three lines, the generator ratings, our G one is 200, MVA at 20 kV, with a synchronous reactance of 15%. And we're using synchronous reactance because we're talking about steady state operating conditions at this time. So we're not going to talk about synchronous reactance anymore, we'll just call it the reactance of the internal reactance of the generator generator to has a 300 amp VA at 18 kV with an internal impedance which is reactance at 20% on its own base, generator three is 300. Va at 20 kV with an internal impedance of 20% on its own base transformer T one is 300 MVA at with a ratio of 220 kV 222 kV and it is connected y to delta with the Y on the high side Delta on the low side and it has an internal impedance of 10% on its own base transformer T two is a little bit different because it is not a transformer bank, but three individual Transformers connected together in a y two Delta configuration with the Y side on the high and a Delta on the low.

Its ratio is their individual ratings are 100, MVA and the ratios of the industry Transformers 130 kV 225 kV and when we do connect them together they will be in a y two Delta configuration. These Transformers each have an internal impedance of 10% on it their own base transformer t three is 300 MVA and its transformer ratio is 220 kV to 22 kV, it is connected y to y and it has a internal impedance of 10% on its own bass lines, l one and l two each have are rated at 75 ohms actual impedance in line three is 50 ohms actual impedance all impedances here, if you haven't gathered yet are all assumed to be reactance is or lagging, type impedances. And our task will be to draw the reactance diagram indicating per unit values, choosing generator three circuit as a common base. In other words, we'll use this MVA and its voltages as our, our base quantities.

Before starting a problem, we will have to take a closer look at T two and the unusual way that it is presented to the problem. Although that is not really unusual because it's quite often in the system that we use individual Transformers to form a three phase bank. However, the ratings were given as a single phase rating, and those have to be changed to three phase bank ratings. In order to do that we have to come up with line to line voltages and three phase MVA ratings for the entire world. Bank. Let's look at how these individual transformers are connected.

We have three 100 MVA Transformers that are connected in this configuration y to delta and the individual coil to coil voltage ratings are 130 kV 225 kV, which means the high side or the high voltage side is connected why and that hundred and 30 kV is really a line to neutral rating on the transformer and that has to be changed to a line to line rating. On the secondary side. It's connected delta. So the line to line voltage is 25 kV, so we don't have to do anything with that. The individual Transformers have an an internal impedance of 10% which is based on the manufacturers rating of 100 We'll have to use that for calculation of our internal impedance when we get to the three phase bank. Now in converting the high side voltage to a line to line rating, we just use a phase relationship that we often use and that is the line the line is root three times the line to neutral, which is root three times 130 giving us 225 kV line to line voltage high side rating.

The low side as we said earlier, I said there was just still not a problem because it is a line to line rating of 25. The three phase VA rating for the bank now is three times the individual single phase rating, which now is 300, MVA. We can now regard transformer to from a three phase rating and the three phase rating of the T two bank is now 300 MVA, its ratio is 225 kV to 25 kV and it has an internal impedance of 10% on its own base we can now start the process of per unit analysis and the first step is specifying the MVA base and the base voltages. In this particular case, we are told at the beginning that we are to choose d3 as the base for our analysis. So, if we look at generator three or G three, the MVA base for that generator was quoted as 300 MVA, so that will be our s base and the voltage quoted by that generator Or when we first looked at it was 20 kV.

So our V base is 20 kV. Now I've subscripted three VB three and you'll see why in a few minutes. But we have to start because we've specified those as our basis, we have to start at G three. Now, remember that the voltage basis throughout the system are determined by the transformer line to line voltage ratios. So if we look at the transformer key three, its ratio is 222 22 kV. So it is a 10 to one ratio.

So, T three low voltage base is 20 kV. That's what we are told. And if we go through the voltage ratios, the lines voltages will have to be 200 kV the lines are connected together. So, they all have to be at the same voltage at any one time. So, the line voltages line one base line two base line three base, they are all equal to 200 kV. In the case of our transformer number two, the voltage ratio for that transformer is 225 kV to 2025 kV or nine to one.

So, the low voltage key to low voltage base is 200 kV times the nine to one ratio the inverse the nine to one ratio and we find that the low voltage key to Lv base is 22.2 to kV I leaves one other voltage to be considered and that is the voltage at the G one area and the voltage ratio of the transformer T one again is 10 to one. And if we look at that ratio it, it's going to be, we're going to take the 200 volts and divide by 10 and that will give us a T one low voltage base of 20 kV. Now let's have a look at how we divide these voltages up in a little bit more detail. Again, I'm repeating myself because it is important that these voltage zones are determined by the line to line transformer ratios. So looking at our starting point, which is the base three, which is our Voltage base for our G three generator connected on the low side of the transformer, everything connected inside that blue box has a V base of 20 kV because of the turns ratio sorry because of the voltage ratio of the transformer, the high side voltage is going to be 200 kV.

So, the voltage base of everything connected inside that green box is 200 kV. Similarly, on the low side of T two, everything connected inside that orange box has to use v base two which is 22.22 kV And lastly, generator g one which is connected to the low voltage side of transformer one. Everything inside that red box has to use Voltage base one, which is 20 kV. proceeding to step three of the analysis process using per unit values, we need to determine the impedance base impedance bases of the various zones in our circuit that we're analyzing here. We know that the impedance space is given by the square of the voltage base over the S base. We also can use kV a base and MVA base.

These in the zeros just cancel out in this case, but it allows us to go directly from our ratings into that equation. So we can just plug in the KV a base or a KV, a values and the MVA values and they'll work out as well. So in the Case of the impedance base for zone one which is read it is the KV base one squared over the MVA of base one and that is 20 squared over 300 which gives 1.33 ohms. Similarly, for the green zone, the MVA base is the same as 300. The KV base is now 200. So it's 200 squared over 300 to give us 133.33 ohms.

And for the Blue Zone or said base three, we have 20 kV squared all over 300 again comes up with the answer of 1.33 ohms. And finally, the base for the orange zone or the low voltage side of key to the MVA base. is still the same, it is 300. But decay The base is now to 22.22 squared over 300 gives us 1.65 ohms. We can now proceed to calculate the per unit values for the impedances of the system and we use it we can use the standard equation where we divide the actual value by the base value in order to come up with the with the per unit value. In the case where the manufacturer has specified the per unit quantity on its own base, the we have to use the base conversion equation to calculate the manufacturer change the manufacturers ratings into our new rating.

Using our new voltage basis and our new, our new VA basis, just for reference purposes, I've listed the manufacturer's specifications here in the, in this yellow box. So as we go through the calculations, you can refer to those and and see where where the where the numbers are coming from. First of all the lines one and two, they were specified as 75 ohms. So we use the first equation where we put the actual value over the base value and that's j 75 over 133.33. That gives us a value of j 0.5625 per unit. line two was given to us as 50 ohms.

So we just divide j 50. By 130 3.33 to come up with J 0375 per unit. In the case of generator one, the manufacturer specified that as being 200 MVA at 20 kV, with an internal impedance of 15% on its own base, so we have to come up with a new value, the specified value was 15% or point one five, so we need to convert that using our new base values. And the the new base values for the voltages are the same as the old ones 20 kV. So in the equation 20 squared over 20 squared would just cancel out to be one. So we're really need to only consider the ratios of the of the new per unit base s base to the old s base, which is 300 over 200 and gives us a an answer of J 0.25 per unit g two.

In this case, the MVA quantities are the same. They are both 300. So that term in the equation would cancel out to one and it's just the ratios of the manufacturers base bolt base voltage which is 18 k v squared over the new base voltage which is 22.22 squared. And that would give us an answer or a new question. per unit value of j 0.1312 per unit and G three is very easy because the both the Old and the New voltages and MVA are the same. And it was already that's our that was actually our starting point.

So, the impedance of G three is exactly what it was stated j 0.2 per unit Transformers present an interesting situation in calculating per unit impedance in in, in most systems, the, the the manufacturer has specified the impedance of transformers and they've also specified the voltage basis in using our impedance conversion formula that That changes the impedance from the manufacturer specification to the new, the new impedance that we're looking for on our new basis. You have to consider the new voltage base and the old voltage base in a transformer. One of the windings, if it's a two winding transformer is in one of the voltage bases the other. winding is in the in another voltage base. So really, you might think you have two per unit values for a transformer. But in actual fact you don't so and keep in mind that we've already looked at impedances.

In regard to transformer if you're working Transformers if you're working in per unit values, the turns ratio is disappear and as long as you're talking about per unit and pins and even per unit and payments on the low side is exactly the same as the per unit payments on the high side. And that should work out with our formula. So let's just check it out here. We're gonna look at transformer one transformer one, the manufacturer has specified the impedance as 10% on its own base, which is J 0.10 per unit, we want to convert that to using our new base values, the old s base values are identical 303 hundred so the next term of that equation will cancel out it'll be one the new voltages on the high side of the transformer is 200, the new voltage basis 200 kV, so, we now put the manufacturers specification for voltage base on the numerator, which is 220 kV 220 kV squared over 200 kV squared times the original specified per unit impedance gives us j 0.121.

That's j 0.1 to one. Now let's consider the low side of that transformer the low side of the transformer. The first part of the equation is identical to the previous one that we laid out. The specified impedance from the manufacturer is the same, the spaces are the same, so we can just move on to the voltages in this case, we're on the low side of the transformer, which is the manufacturers specification of a base on the low side was 22 kV. Our specification in our in our new in our new base value is 20 kV, so you have 22 kV squared over 20 kV square multiplied by the manufacturers. specification for per unit impedance comes out to J 0.1 to one, which is identical to our previous calculation on the high side.

So, this should not be a surprise to us because we already know that impedances if after specified in per unit values are the same on either side of the transformer. If you take which we consider transformer three, you can see that the numbers are the same as the transformer one, so the answers are going to be the same and doesn't matter which side of the transformer you're on. The values for per unit are going to be the same. So, that leaves us with considering transformer two and the manufacturer specification for that transformer was point one per unit j per point one per unit. The s basis again are the same 300 over 300. So they don't change anything significantly.

The voltage specified by the manufacturer was 25 kV, our new base voltage is 22.22 kV. So in the equation we have 25 squared over 22.22 squared, and the answer comes out to J 0.1266 per unit. If we now look at the high side of the transformer, the voltage ratios of the new base and the old base workout to 225 squared over 200 squared, and if you do the math, it comes out to the same answer which should not be a surprise to us. J 0.1266 per unit Now that we have converted all of the components to per unit values, we can draw our per unit equivalent circuit and it would look like this. Now, we have to remember though that Transformers T two and T one r, y delta transformers, the primary being Delta connected, the secondary being y connected. So there is a phase shift in going across the transformer.

Depending on the phase shift depends on which direction you're going in. If you are going from the secondary to the primary, the secondary lags the primary by 30 degrees. So in going from the secondary to the primary, you have to add 30 degrees. If you're going from the primary to the secondary, you would have to subtract 30 degrees. The other transformer that's in our circuit is a yy connected transformer. So there is no phase shift to worry about in going from the primary to the secondary or vice versa.

So this ends the chapter