 Thank you. I, I wish to start by apologizing for not being there, but thanks for arranging this room webinar so allowing me to talk to you. So, since there was enough information about me during the introduction. I'll just gonna move on to the talk, and the talk will be mainly on for location in power systems. So before I start, I'd like to acknowledge the sponsors of this work. I mean, mainly it was jointly sponsored by Department of Energy and National Science Foundation. And we did the work on this for school we are see that we are part of current. And I'd like to acknowledge the contributions of students when you think and Arthur Morko. So with that, we move on to the outline so I, I'd like to even though probably you all know what the full location problem is I'd like to just briefly describe it. And, and then talk about the formulation of that's proposed solution. And then I'll talk a bit about the implementation of that solution and some of the issues, and then show some examples using typical power systems under faults. And then summarize the contributions and tight answers if you have any questions after that. So, so the fault location problem is is an age old problem. It's not a new problem and it's essentially was down to catching and locating short circuits occurring in power systems. And these short circuits occur due to unexpected, but most of the time natural causes like lightning striking a line or a tree branch falling on a conductor. Or sometimes vegetation growing and reaching out to the to the height of the overhead lines and touching them, things like that so they cannot be always avoided. And when they happen, they trip the circuit breakers and take out the entire transmission lines out of service. And as a result, you know, they need to be dealt with located and repaired and so forth. So the problem is some of these lines are as I'm showing here in this diagram it's essentially the power system. Systems can be thought of as a graph where the nodes are the substations and and the branches are either the transmission lines or the transformers. And some of these lines may be really long in the order of hundreds of kilometers. So therefore if there's a fault, and even if the fault is cleared by the relays. It's important to know the exact location of the fault so that the crew can can go and attend the fault and the terrain may be rough, especially in rough weather in the middle of snowstorms and this and that. And so, so it's important to locate the fault as close as possible. So, so it's actually a two step process. The first thing is to identify the line which is faulted. And then once we know that then we would like to also know exactly where the fault is along that line. So we need to know the distance from one of the terminals of the line to the fault point. So I'm showing on this diagram or circles with the letter P, and they represent phasor measurement units. You are probably familiar with this but just to clarify phasor measurement units are these devices. Sample use high frequency sampling a sample voltages or currents. And, and they time stamp these samples using GPS. No matter where they are, they are synchronized with respect to each other to the GPS system and using these samples, they do a FFT and produce a phasor representing the voltage at that point where the measurement is taken. And they provide these phasor measurements, quite high rate, typically anywhere from 30 to 60 times a second. So they can provide these phases quite accurately and in a synchronized fashion. So, so the systems power systems are equipped with these PMUs. But not necessarily at all the buses at all the substations as shown here they may be located at a certain percentage of the substations in the system. Okay, so, so again the motivation even though I went through this, obviously public safety is one, we need to locate the fault fast, and also we wish to minimize the service interruption, because anytime you have a fault. And some of the customers may suffer because of that. And also, there is a cost associated with the reduced transmission capacity, and sometimes associated with that reduced security margin. This is related to the stability of the system either voltage or the angle stability, maybe reduce the margin for that. So, as I said this is a very old problem. It has been studied in the past. And by many people, and the solutions offered can be sort of categorized under three categories. Obviously, I mean you can probably group them in different ways but this is one way. You can look at it. One is those techniques which use the power frequency in other words you filter out the higher frequencies and you try to capture the 60 hertz signals. And there, you're having the voltage and the current captured, you try to find the effective impedance, looking into the line and there's a fault this impedance certainly becomes much smaller. And based on that, you try to estimate the location of the fault. There's also this category of methods which rely on traveling ways because anytime there's a fault, the fault generates or initiates a traveling way, a signal travels both ways. And if you have a way of capturing the wave front of this traveling way, and since these are synchronized sensors capturing it, you can capture the arrival times in a synchronized fashion. And based on that, you can triangulate and try to determine the location of the fault. This requires high frequency sampling because you need to capture this very fast traveling wave, which means that you probably cannot use the conventional instrument transformers like the current transformers and power transformers, including cut-offs around, you know, 5000, 5 kilohertz range. So you probably need to use one of those, you know, optical instrument transformers, optical CTs or PPs. So that is some additional complication here for implementing these methods. And then there's the whole group of methods which rely on AI and machine learning and so forth. And they try to use either, you know, the history, the recorded history or simulated fault scenarios to train. And then hopefully get a successful prediction when they actually measure the voltage and the current during the fault. So the method or the approach I will be talking about is probably can be classified as in the first category, but with a different sort of tilt in the sense that we are assuming that the system is not observable. In other words, we don't have enough PMUs or measuring instruments or sensors that will allow us to capture what's happening at the terminals of the fault of the line by these PMUs. So, so we assume that the phaser measurement units are installed only at a very small percentage of the system buses which represent the substations. And we also assume that we know the network topology and line models and so forth pre-fault before the fault occurred. So under this assumption, the question is can, can a fault on any line be located, and its fault type can be identified because we are also interested in identifying what kind of fault it was, whether it was three phase to ground fault or phase to phase fault, a phase to phase to ground fault, single phase to ground fault and so forth. So, so this is the question we wish to answer. So, so we start out to formulate this problem we start out by looking at the pre-fault network. Let's just write the network equations, admittance matrix times the bus voltages equating to the net injected currents. And here I'm showing just any, any branch in the line connecting buses K and M. And the corresponding rows columns in admittance matrix are highlighted there, and the pre-fault voltage multiplied by that and gives you the pre-fault net injected current vector. So, so then if we consider, if you consider a fault along that line KM line at the point F, which is an unknown point obviously, and consider a fault current and unknown fault current IF. That can be sort of incorporated into the network equations by augmenting the equations by one row column adding this fictitious bus F where this unknown fault is occurring. And we have this equation, and here we are also doing a difference equation instead of just writing the equation after the fault, we are writing the equation in such a way that we subtract the post fault network equations from the pre-fault network equations. So as a result, if you look at the equation here, you will see that we are calling this vector now delta V indicating that this is the difference between the post fault values minus the pre-fault values at each of those buses. And assuming that this fault occurs all of a sudden, so we approximate the changes in the net injected currents at every bus with zero because typically loads and generation, they don't really change that fast. In the instantaneous, the only change we expect of course is this fault current, which is the reason why the system is changed. So, given this equation, we can apply a reduction, a chron reduction, and get this equation. The interesting thing is, when we do that, we get our original admittance matrix back without any changes, and then of course here we have the changed values, the difference values in the bus voltages. And on the right hand side, because of the sparse connections of this fictitious point F to K and M, we will only see changes brought to entries in rho K and rho M in the current injection vector. So as a result, we have this picture where the right hand side current vector is empty completely other than those two entries corresponding to the terminals of the faulted branch. And then the voltage solution is essentially the changes at each and every bus voltage in the system. So, at this point, you can also think about it like this, we have the actual system with the faulted line, the fault current injection somewhere along the line, unknown point along the line. And the voltages have changed as a result of this fault by delta D, delta VK, delta M amount at each end, as well as everywhere else in the system. And then we can replace this picture by a picture where there is no fault so the line is not faulted, but we have these fictitious equivalent injected currents representing exactly the same operating condition created by the fault. So these two are equivalent and we tend to now use the bottom equivalent picture for the reasons we will see. So, so not only that we get this nice equation where y bus remains the same and there are only two non zeros on the right hand side vector. But also, these values, even though we don't know them, these are unknown values, but if we, we know what these values are that that those values can actually be used to determine the distance from the two terminals from terminal M and from terminal K. You can actually calculate the percentage length to the fault by using these two expressions and these are straightforward to derive from the prong reduction process. So if we knew these virtual or sort of equivalent injected currents, then we could have easily found the location of the fault using these expressions. So, so then the question is, how does it work? Does it really work before we try to formulate a solution? I'd like to show you an example just to show you how this thing works. So this is a very simple, small example, but it's interesting because this is a distribution system model and it contains not only three phase branches, but it also contains two phase, single phase branches. So it's a mixed three phase system having some branches, three phase, some branches, two phase, some branches, single phase. And similarly, the laws may be single phase, three phase or two phase. So it's a mixed system of lines in a distribution system. In this system, we simulate a fault along this branch or line section connecting bus two to bus five, and we do it in such a way that the fault occurs exactly one fourth of the line length from bus two. So, so by doing that, and if you simulate the fault, and we actually, we actually check how these virtual or equivalent injections looks like. And in this case, by the way, the fault is a phase to phase phase a phase B to ground fault. So this is the type of fault we are simulating along this line. So for this case, if you look at the results, you will see that we see all zeros as estimated as the entries in the current vector, except for the two ends, and the two ends only the two phases are significant. Because those are the phases which are folded like phase C doesn't show up as a virtual injection. And not only that if you look at the ratio here you see that it's one to three indicating that one part is three and the other part is one, which is exactly how we actually calculate based on these injections, the location of the fault the percentage starting from one end, going to the other. So this is just an example to illustrate or to verify this formulation. And now, if, obviously, this is a very simple approach, if we had a chance to measure the pre fault and the post fault voltages at all buses, that means if we have access to delta V. And it would have been a trivial process, we could have just measured these multiplied by the matrix, and that would have given us the two non zero entries in delta I, and from there we could just move forward and located the fault. However, unfortunately, we don't have PMUs at every bus, or we don't have enough PMUs to make the system fully observable so that we have access to those values in the voltage vector delta V. So, so what we can do is we can take the inverse of the admittance matrix, which we call Z, Z bus here, and rewrite this equation in the reverse order, and then assume that only a portion of the voltage buses are have PMUs, so we only have access to a small percentage of these voltage differences, measurements, and the rest of them we don't know. So that will partition the equation like this, and, and then we can look at it, essentially like this we are carving out a portion of Z bus, based on the available set of voltage measurements, and we are having this equation, which is a sort of rectangular coefficient matrix times delta I equals delta Dr, which is a, which is a, an undetermined set of equations. So that that that gives us this picture. In this picture, obviously, we are stuck, and we need to figure out a way to solve this equation. And the nice thing is that even though this is an undetermined set of equation, we already know that the solution is a sparse vector, there are only a few non zeros in it. It's full of zeros. So we exploit that. And obviously, this is an old problems, nothing new, and it has been looked at earlier. So for this kind of undetermined set of equations, where you have a smaller number of equations than the number of unknowns. But you know that the unknown vector contains only few non zeros, even though you don't know where they are, which ones are those you know that there are a few. So, so this kind of sparse problems are dealt with earlier. And interestingly, of course, the group who dealt with it and came up with this nice solution is actually from there from your institution and statistics. I'm Shirani and his co workers, they have worked on this problem and produced some nice tools to use for these types of problems. And one of them is based on this last so the known as the least absolute thing which and selection operator, and using that optimization solvents optimization problem. They can actually find a solution to this under determined but sparse set of equations. So, so in our case, the sparse set of equations are given like this, unfortunately, this is a complex phaser equation. These are complex matrix, these are phasers these are complex numbers so we split them and turn the equation to a much larger dimension but real problem like this, the real and imaginary parts of the currents and so forth. It's a simple conversion and then apply the last so approach and and solve it this way. So, so it seems like this is a nice solution, and that will give us some usable results. There are a few caveats. So, one of them is that, as you know, I mean the systems power systems are protected by fast relaying action that are released, and they respond to fault currents, and based on the setting of these they can, you know, clear the fault, depending on how fast they operate it can take one or two cycles or more, but they don't certainly allow the fault to remain and to reach the post fault steady state value. And this is precisely what we are using because we need the post fault steady state value in order to use those pre fault and post fault steady state difference equations. So therefore, we wanted to make sure that we actually have access to those values and turns out that if you capture the voltage transients. And you can, because the PM use internally they have access to high frequency sampling of signals. So if you have access to those voltage transients during the initial few cycles, they can be used to predict the post fault state voltage phases. And one way and probably not the only way but the one way to do that is using chronic analysis. So here, throwing an example of that so this is the fault duration and then here default is cleared. So it doesn't allow this to continue and settle in a post fault steady state, which would have been like this if you allowed it to settle. And this is precisely what we want to use in the formulation so therefore what we need is to be able to use this window here when the fault is still on and there's a transient going on use that transient data to capture it, and to use that with some manipulation using the pro analysis and estimate the state long term steady state behavior of the signal. And indeed, you can do it. And here I'm showing one example of that here there's the fault. And the fault is cleared right there. And from then on, if you use the front analysis and do the prediction. The yellow signal is 100 times magnified version of the prediction error, and the blue and the red lines show the estimated and the measured signals, which are fairly, fairly close. And then of course you can predict in the end the steady state post fault steady state value based on this prediction. So that's, that's one issue that to be resolved. And based on that, we have looked at an example here this is a an example system with 118 sort of substations models and the three phase system for each and every one of these are three phase lines, the loads are three phase and so forth. And did test several different scenarios here. For instance, in this scenario we are looking at the three phase to ground fault occurring at 20% of the line. This line connecting bus 50 and 57 and looking at the currents, estimated by last hope, you notice that they are not strictly zero, there are certainly large ones that appear here. And those actually correspond to indeed bus 50 and bus 57 phases ABC, as shown here, but in order to sort of filter them out, we are using a threshold as you can see we put a threshold here, and we sort of throw out anything below the threshold, and just consider the ones about to find these virtual currents. And based on those and based on the expressions I showed earlier, using the ratios, we can actually estimate the publication fairly closely you know it's supposed to be 20% and 80% and the numbers we get for different phases for this case. You can also do a different kind of fault where there's fault between two phases face to face fault, and it's happening at 5% of the line length of line 101 to 102. Here as you can see since this is a face to face fault the number of non zeros we expect in the current vector are smaller. In this case, only two on each end 101 and 102. Again, we are using the threshold here. And, and we get the results which are again approximate but close this is 4.8 instead of five. This is 3.9 instead of five and this is 95 20 instead of 95 and this is 96 10 instead of 95%. Now, we were thinking about how to improve this a bit further. And then we said okay, we are identifying the faulted line based on this last algorithm. Why put all the burden on last so why not, you know, help it to further improve the accuracy of destination. Essentially what we do is once we look at once we lock in on the faulted line, then we know that those are the faulted line corresponding columns in Z are so you can pull those and create an ordinary problem in this case because now we have every no which line is the faulted line. So using this kind of approach by just, you know, pulling the columns corresponding to the identified line. We can indeed, if you look at this case for instance, the ones that we looked at earlier. In this case, we ran into some non zeros which are not supposed to be there by using lasso, even though the rest were correct. And then doing the oil less, we can clean it out and then get these results as shown here. And, and then and then similarly, we can do it for another case shown here between 50 and 57. Again, there was an incorrect value appearing here, and we were able to clean it up by using the oil less approach here. So, I guess with that, I would like to conclude this is a way to locate faults. If you have a limited number of synchronized voltage phasor measurements in a given power grid. You can improve this by using some sort of a prediction algorithm, based on something like pro need to take the transient sample recordings and determining the possible steady state values that way, because those are not going to be captured in use since the relays are not going to allow you to keep the fault that long and active and live in the system. Now finally, once the faulted branches identified, you can further improve on the accuracy of the estimation by doing one last, you know, these squares estimation, and that will provide you with better estimates for the fault location. The main advantage of this, of course, will disappear as we have as we install more and more numbers of PM use in the system. So, this is sort of an intermediate solution until that happens. On the other hand, if this works quite well. And you might save on investing on extra PM use if you can do the job with fewer PM use, instead of installing a PM at every bus, then maybe it might be a good solution to save some money some funds. Investing in PM use. I have included some publications at the bottom here, which are related to the presentation to present it work. And then I have my URL here in case you need to contact me for any further questions other than those you already have now. So I'm ready to answer any questions if you have it. Thank you. Okay, shall I go ahead and start answering the questions from the chat. Yes. Yes, so there is a question let me repeat the question. How well does this method work on identifying and locating high impedance faults. Yeah, so high impedance faults probably will not be handled by this because of of their. I mean, the way they they present themselves. They contain a lot of jitter a lot of high frequency signals and typically those faults. Since they don't produce sufficiently high currents. They may go under the radar in this, in this particular approach because the approach sort of relies on on the fact that the current is significant and and so are the equivalent currents that that we use to represent it. So more than likely it will not be able to catch high impedance faults. Okay, so there's another question to ask related to the distribution system. So transmission is easy with the amount of data available have you smart meter voltage event measurements used. I think we haven't we this this these are the results that I'm showing are all simulated results using the systems that we use. We have not used an actual distribution system for this purpose but we have used the. The distribution subcommittee system that there are a number of systems that I can be distribution subcommittee put together as typical distribution systems. One has 123 buses three phase. We use those systems to test this, but not using actual fault data. So, yeah, so there's another question about how you choose the threshold for less of fault current results. And that was just by trial and error by looking at, of course, since we are simulating results we know which, which equivalent currents are supposed to be there, and comparing those values for the equivalent currents and the other parents, we came up with an appropriate threshold for a given system. And turns out that once for a given system you set this threshold for different faults for different scenarios that threshold pretty much holds true, as long as you are using the same system. So that may be system dependent configuration topology dependent, but that was mainly the reason why we wanted to find an improvement. And that's the reason why we went to that OLS augmentation, because this threshold may be a bit sort of subjective, this choice of that threshold. I don't see any others. I have one question. Yes. Yeah. In the event of a cycle. Someone tries to manipulate a new. How can I distinguish it is a cycle or a fourth. Yes. Well, yeah, I think I don't know actually we haven't looked at that, but the thing is, unless the cyber attack is sort of persistent, because it should coincide with the fault otherwise. This is there's no way to find to determine the fault ahead of time, the full time. This should be a persistent attack to the, to the PMU. And maybe there is a way to identify persistent attacks in general. But no, we haven't looked at what will happen in that case. The other question I have is, if you know the three parts of our gene. What was the minimum number of PMU data or where should they be placed. Is there a placement problem? Because you are trying to, you're trying to identify reports with minimal number of new mission. So, yeah, I guess this is the PMU placement problem. You are referring to, I mean, there are, there are some results. Some of the last few papers showing how to choose the right locations for these sensors, so that, for instance, you make each and every branch identifiable, because this impedance bus matrix is actually notoriously conditioned. And, and, in fact, I'm not showing all the details but we are actually using a QR decomposition. So we're not directly using Z bus but we are using a QR decomposition. So, so there are some, some tools, mathematical tools to determine the locations where you should have those sensors, those PMUs, in order for you to be able to identify faults on each and every branch in the system. So, we have done some of it in the first students going new fans pieces that are some examples of that, showing how you can choose locations for PMUs so that you do have that guarantee, but unfortunately in the power systems area. This is very much topology dependent so it's there is no generic way to say like on only need a certain percentage of buses to be covered in order for this to happen for each system it turns out to be a different number, different location. Any more questions.