Chapter Eight system modeling, balanced and unbalanced fault analysis. symmetrical components are used extensively for fault study. In these calculations the positive negative and zero sequence impedance networks are given by the manufacturer in some way shape or form either through drawings or listings or whatever, but they're usually supplied in some manner. So that the system engineers and technicians can do the calculations required to put the system into service. Each of the sequence networks are then connected together in various ways depending on the fault or the characteristic of the fault and the voltages and The analysis is usually carried out after the fact. All analysis in engineering start with a formulation of appropriate models a model and in power system analysis we almost invariably mean mathematical models is a set of equations or relations, which appropriately describe the interaction between the different quantities of the system.

Hence, we use these models to visually and mathematically symbolize the concept of symmetrical components. In most all power systems, the main network components of interests are transmission lines or feeder lines, transformers and synchronous machines. Since we have developed the tools for symmetrical components We now can work with a simplified single line diagrams for these networks. So, we need to know what the single line diagrams of these network components are. The generator in general may be represented by a stark connected equivalent with possibility of neutral to Earth resistance as shown here, together with a three phase diagrams for the positive sequence, negative sequence and the zero sequence circuits. The neutral path is not shown in the positive and negative sequence circuits as the neutral current is always zero for balance conditions also by design, the generator generates a balanced voltage supply and hence only the positive sequence network will have a supply voltage.

Remember that the three sequence vector groups when added will equal the actual generated supply voltages and currents. Since the three component networks are balanced networks, they may be represented by a single line diagram. In a fault calculation, the component currents can now be simply calculated and their subsequent unbalanced phase circuits from there. When a fault is applied to a power system supplied by a synchronous generator, the initial current supplied by the generator will start at a larger value and over a period of several cycles, it will decrease from its initial value to a steady state value. The initial value of this current is called subtracts one current or the initial symmetrical rms current, subtracts one current decreases rapidly during the first few cycles after the fault is initiated, but its value is defined as the maximum value that occurs at fault at the fault inception. After a few cycles of sub transient current, the current will continue to decrease for several cycles, but at a slower rate, this current is called the transient current.

Although like the sub transit and current it is continually changing. The transient current is defined as a maximum value which occurs after a few cycles of sub transient current. After several cycles of transient current, the current will reach a final steady state value This is called a steady state current of the synchronous machine. The reason why the current supplied by the synchronous generator is changing after a fault is because the increased current through the armature of the generator creates a flux that counteracts the flux produced by the rotor. This results in a reduced flux through the armature and therefore, a reduced generated voltage. However, because the decrease in flux takes time, the generator voltage will be initially higher and decrease over time.

We account for the changes in the generator voltage in our modeling by using different values of reactance in series with the internal generator voltage, we use three values of reactance to model the generator During the period of the fault, the sub transient reactance x d double primed is used during the initial few cycles, the transition reacts and reaction transit reactance x d prime is used for the period following the initial few cycles until the steady state value is reached. The synchronous reactance XD is used for the steady state period. The reactance of the of a synchronous motor are the same as a synchronous generator. If the line to a synchronous motor develops a three phase fault, the motor will no longer receive electrical energy from the system but its field remains energized and the inertia of its rotor and connected load will keep the rotor churning for some time.

The motor is Then acting like a generator and contributes current to the fault. So, during fault conditions, motors are often treated as generators. Transformers tend to be a very confusing area for working in network components. But what we want to do is we want to reduce these complex items to more simpler, easier to handle models and Transformers can appear in many forms and they can either be single phase Transformers that are connected together to form a three phase bank or they could be a three phase bank all by themselves. That complexity that the introduced into the system are due to the fact that first of all, A turns ratio that you have to deal with. Secondly is their various configurations and how they're connected, whether they're connected delta y, y, delta, delta zigzag or y, y, they all have the connections have to be taken into consideration as well.

So the first thing we can do in dealing with network components is we can understand the fact that we're dealing with balanced systems. In other words, the positive negative and zero sequence systems are balanced by nature. So first of all, we can get rid of the fact that they're a three phase configuration. Not that we can ignore it, we will have to return to it later. But for the present time, let's assume that we're only dealing with a per phase quantity so the inside of a transformer or the model of the transformer would look like this. This would represent either the positive or the negative sequence impedance of the transformer it is the per phase positive or negative sequence impedances.

The items that are listed here cover all of the modeling items that are required in a transformer. There's the primary and the secondary winding resistance, there's the magnet magnetizing reactance, eddy current and core losses and all the rest of them you can see them all listed there. The first thing we can do to add to the analysis and make it simpler, is we can convert all of these impedances and values to per unit values by converting them to per unit value. We essentially can remove the turns ratio of the transformer. Because normal fault and load currents are very much larger than the magnetizing current we can omit the magnetizing branch from our model. The leakage impedance on one side of the transformer can be referred to the other side of the transformer by multiplying by the square of the turns ratio or if we're working in per unit, we can simply add the primary and secondary impedances.

This is now a more simple model for the network components. The positive negative and sequence impedances may be represented like this for the positive and negative sequence components of the transformer. For the time being, let's leave the zero sequence impedance of the transformer model and we will return to it shortly but it is a little bit more complex in that it relates to the actual connections of the transformer. There are a couple of things to keep in mind in dealing with three phase transformers are three single phase transformer banks and how they are connected First of all, there could be a phase shift from the primary to the secondary depending on these connections. And secondly, the zero sequence current may or may not be transferred from one side of the transformer to the other. The first of these arises from the fact that Y delta or delta y transformer connections produce phase shifts from the primary to the secondary.

Depending on these connections, this phase shift may be either plus or minus 30 degrees from the primary to the secondary for positive seats. voltages and currents. It is straightforward then to show that the negative sequence shifts are in the opposite direction from the positive That is to say, if the connections advance the positive sequence across the transformer, then it will retard the negative sequence. And conversely, if the connections retard the positive sequence across the transformers, then it will advance the negative sequence components. This does not turn out to affect the set setting up of the sequence networks, but it does affect the reconstruction of the phase, currents and voltages. For example, let's look it up delta y connected transformer.

The current through the top right side windings of this transformer bank we designate needed is I subscript capital a current through the top left side winding we have designated and can hardly see it there but it is I subscript smally. The phase line current which we call line a is current flowing from the line into the transformer bank at that position is actually a sum of the current flowing into the top left hand side of the bank but it also is subtracts the current flowing in the middle bank which is IB So, the current flowing into that node showing there is I A minus IB that current flow In the left side of the transformer bank, top left side of the transformer bank ay ay ay ay ay ay ay ay ay. is made up of. The symmetrical components. I A naught plus a one plus ay ay ay ay ay two.

Similarly, the current flowing in the middle bank, on the left hand side is IB and that is also made up of components IB zero plus IB one plus IB two. So if we look at the current i A minus I B, it is made up of the sum, or the differences actually of the two right hand side of those equations, which is the symmetrical components for ay ay ay ay ay b. If we group the components together, so we have the zero sequence component components grouped together in brackets, the positive sequence component grouped together in brackets and the negative To sequence components grouped together in brackets, it would look like this. We know that the zero sequence currents are in phase and equal to each other. Hence, that term of the equation ay ay ay ay ay, not minus IB naught is equal to zero, because those two items is a zero and B zero or equal, you subtract one from the other and you get zero.

So what does that mean? That means we're only left with positive and negative sequence components. And what happens to the zero sequence occurrences actually, it they are circulating in the Delta side of the transformer, and they never get transferred over to the prop the positive side of the transformer. Now we can simplify the equation To the have IA minus IB is all equal to only the sum of the positive and negative sequence components. And that is all equal to i subscript capital A, which is the right side of the transformer. Notice that we don't have to worry about the turns ratio of the transformer because we're working in per unit values.

Plotting these, these phasers, the A, the current ay ay one might look like this. Then the IB one is 120 degrees, shifted from ay ay ay ay ay one, because they are balanced and they are 120 degrees apart. Then if we take minus v one and put it on the head of the arrow of ay ay ay ay ay ay one, essentially The two vectors together. The result and gives us ay ay ay ay minus i sorry i a one minus IB one which forms and I saw Sally's triangle, which is our familiar, one two root three ratio of the sides and introduces a 30 degree angle at the base of the saucer these triangle. So what that tells us is that the current ay ay ay ay ay ay one minus IB one, is equal to, root three, ay ay ay ay ay ay one.

At 30 degrees. Which is, ay ay ay ay ay one minus IB one is a line current A, which is also equal to root three high capital subscript de at 30 degrees. eyeline is now is the Delta side the current into the Delta side of the transformer. So this is really a significant fact of a characteristic of a Delta Wye transformer in that it introduces a phase shift into the currents flowing into one side and out the other and it also eliminates the zero sequence components going through the transformer. The second important feature of Transformers arises because ungrounded y connections are open circuited to zero sequences at their terminals. A delta connected winding on the other hand will provide a short circuit to zero sequence currents induced from the Y connected wire.

Thus zero sequence networks of Transformers may take on one of several forms. The winding configurations of a transformer will determine whether or not zero sequence currents can be transformed between the windings. We have seen that the there is no zero sequence current transformed across a Delta Wye transformer in our previous slide. They basically circulate around the Delta side of the transformer. Because zero sequence current do not add up to zero at the neutral point, they cannot flow in a neutral without a neutral conductor or a ground connection. If the neutral has a neutral conductor or if it is grounded, the zero sequence current from the phases will add together to equal three times the zero sequence current of any one of the phases.

At the neutral point or three I zero and then flow through the neutral conductor or ground to complete the path. The following are some of the different transformer winding configurations and their effect on the zero sequence currents. They are the models that have to be used for the zero sequence network if the transformer is content connected in this fashion. For example, a y two y connected ungrounded transformer would have this as the model for the zero sequence network. If we ground one side only, this is how the model of the zero sequence network would look. And if both sides are grounded, we would have this as our connection In four, zero sequence network, this is the Delta two y, which we've already seen and gone through, this would be the Delta two y with the Y side grounded.

This would be the Delta two Delta unground ungrounded, of course transformer, zero sequence network. Now let's look at how we model transmission lines. Here's an example how a transmission line would be represented on an impedance diagram. In a balanced three phase system, the per phase impedance or self impedance, shown here is said subscript s of each phase of the line is the same. We can calculate the single phase current, but we must take into account the voltage drop across the mutual impedance costs By the currents in the other two phases, we can then write ohms law for the each phase of the line. In this case we're picking the a phase and voltage a phase.

That's the line to neutral voltage and the voltage in the a phase is equal to the self impedance of the a phase line times the line current which is, ay ay ay ay ay, plus the mutual impedance of the line times IB plus the mutual impedance times I see. Now we're assuming that the line is fully transposed, so that the mutual impedance is the same for B on to A and C on a as well. So we only have one term for a mutual impedance and it said sub m. We can rewrite this equation in color The terms of Zed m because it's the same, and we would have VA is equal to Zed s times IA, plus Zed m times the quantity i b plus IC. We know that in a balanced system that the currents have to add to zero, because of purchase current law, then ay ay ay ay ay plus IB plus IC is equal to zero or we can say that IB plus IC is equal to minus ay ay ay ay ay ay.

We can rewrite the previous equation now and substitute minus ay ay ay ay ay, for the term IB plus IC, and where we end up with an equation where VA is equal to Zed s times IA minus Zed m times i. And again, we can collect the like terms on the right side of the equation which is, the current ay ay ay ay ay ay ay. We can now divide both sides of the equation by ay ay ay ay ay. and because VRI is an impedance the impedance of the line, the collective impedance can be written as Zed as minus said M. And this, in fact is the positive sequence impedance for a transmission line, and we'll designate it as Zed subscript a one is equal to Zed s minus set M. And in actual fact that negative sequence impedance, because it is a passive network, there's no active elements in it is the same.

The negative sequence impedance is the same as a positive sequence impedance, so we can write a term Zed a two which is the negative sequence impedance is given. By the same quantity as a positive sequence impedance which is as minus said M for the zero sequence impedance because the zero sequence currents are all equal and in essentially the same direction, we can say that I a zero is equal to IB zero is equal to IC zero, then we can rewrite that equation that we started out with. And this time we're using our zero sequence quantities, we end up with an equation, which after we boil it down, works out to the actual zero sequence impedance is equal to v a zero, all over i a zero which is equal to the quantity Have the Zed s plus two Zed M. And this in in fact is our zero sequence impedance.

The zero sequence impedance is always larger than the positive and or the negative sequence impedance because we are adding two times a mutual impedance to the self impedance instead of subtracting the mutual impedance from the self impedance as a final summation just to make sense out of all the terminology that we've been introducing here, dealing with the sequential opponent components as well as actual impedances and sequential impedances. Remember that in the fact that we're dealing with impedances They don't change when you're shifting from one phase to the next phase, the voltages and currents could change depending on the impedance, but the impedance in each leg of the balanced sequential component is the same, but it's acted on by the positive negative and zero sequence voltages. So, in terms of the positive sequence impedance for example, Zed subscript a one is equal to the quantity Zed s minus said AM, which is also equal to the B phase positive sequence impedances should which is also equal to the C phase positive sequence impedance.

So, we a lot of times just refer to the sequential impedance the the positive sequence impedance for the line being Add one which is equal to set S minus Zed M. Similarly, the negative sequence impedance in spite of the fact that it's equal to the positive impedance impedance, in the case of a line, they are also equal in each one of the phases. So a lot of times it's referred to as the negative sequence for the line or Zed l two. And the same thing can be said for the zero sequence impedance, each one of the phases has the same zero sequence impedance, which is equal to Zed s plus two Zed M. And a lot of times is just referred to as the zero sequence impedance for the line. Keep in mind that the in the case of the line itself, lying if it's transposed, and and designed properly The impedances of the line itself are all equal and each one is equal to the sum of the sequential impedances for the lines.

So, sometimes the actual impedance of the lines that l is not sometimes but it is made up of Zed l one plus L two plus said l not or l l zero. And the same thing can be said for the for the B phase as well as the C phase. This ends chapter eight