 Welcome to class 15 on topics in power electronics and distributed generation. We have been looking at methods of detecting unintentional islands and these are essentially what we call anti-alending algorithm. We looked at passive anti-alending methods and in the last class we looked at the relationship between power mismatch in between the D G and the load and the voltage level after the disconnection of your upstream switch. So, it could be a upstream breaker. So, what would be the voltage and the idea is if the voltage goes beyond some particular bound a o voltage level or a under voltage level then you would declare that it is a unacceptable situation it is a situation of unintentional island. So, today we will look at the relationship between the reactive power mismatch and the frequency deviation that would be seen on the in the island. So, this is essentially a passive anti-alending method assuming the same R L C or load model and the D G model as we have earlier been discussing. So, again the assumptions are that the L and C draws equal reactive power at 50 hertz which means that your Q L is equal to Q C and in this condition essentially your L and C is under resonance at 50 hertz and you can actually write an expression for the quality factor you are assuming that the quality factor is large. So, Q L divided by P and Q C divided by P is the quality factor and you are assuming that the quality factor is larger than 1 greater than or equal to 1 and you are also assuming that your D G unit is operating at unity power factor which means that Q D G is 0 under nominal conditions. So, the next the question is now if you have a situation where your L and C is different from what was nominal then what would be your new reactive power mismatch and the corresponding change in frequency L comma C Q F is Q L divided by P D G is operating at unity power factor. If actual reactive components L prime comma C prime then your actual resonant frequency is F prime is 1 by 2 pi square root of L plus delta L and C plus delta C where this is L prime and this is C prime. So, you can write an expression now for the deviation in frequency and your F is 1 by square root of L C. So, you can writing the previous expression for F prime and you know what F is you can simplify it you will get square root of L C divided by and you can then take F prime to be a maximum level or F prime to be a minimum level and then you could consider a threshold F min minus F by F is less than this could be further simplified and you have minus 1 over here and minus 1 over here and you could simplify it as F min and then you could take the reciprocal and then you get F by F max square less than or equal to. So, here we have neglected the product term delta L delta C. So, let us call this expression 1 and we also know for the for your ideal load case you have Q equal to Q load is 0 your delta Q is essentially what comes in from your grid side in the model that you are having and this is Q load minus Q dG and for the case where now you have deviation from the nominal value this turns out to be equal to V square 1 by 2 pi F into L plus delta L. So, this can be written as and we know that Q L is equal to Q C is the quality factor times P. So, this can be written as delta Q can be written as Q L by 1 plus delta L by L. So, because Q L and Q C is quality factor times P you can write it as delta Q by P is equal to Q F into 1 by delta L by L and this 1 plus delta L by L can be simplified at taking an approximation of 1 plus 1 plus x 1 by 1 plus x. So, this is equal to Q F. So, you have now a relationship between delta Q P Q F and the difference the variations in the load and if you look at what we had we had an expression for variation in the load and your F min and F max. So, we could substitute this in the in the in the expression that we just have to write a relationship as 1 minus F by F min. So, similar to what we did for the case of voltage where we had a nominal voltage and then we looked at what would be the power mismatch to determine what would be the resulting voltage amplitude after opening of the upstream breaker. Here we can look at what is the reactive power mismatch assuming the load is a RLC load to see then what would be the frequency deviation when you have a reactive power mismatch or the reactive power that is coming in from the grid. So, we will again look at a few examples of what would be the frequency deviation we can expect for some change in reactive some delta reactive power that is coming in from the from the substation or any upstream opening breaker. So, we look at an example D G operating with power output equal to 1 per unit and it is operating at UPF unity power factor and the load is also considered 1 per unit. So, power wise it is perfectly matched and we will assume that on the feeder the original power factor on the feeder was 0.707 lag 0.71 lag essentially what it means is that your reactive power being consumed by the loads on the feeder is 1 by 1 divided by the power factor it is about 1 per unit. Then we will assume that this particular feeder because of this large reactive power that is being drawn has power factor correction capacitors that are added on the feeder to try and make it unity power factor. So, essentially it means you have added a capacitor capacitance drawing war of 1 per unit. So, that you have essentially unity power factor coming in from your upstream substation breaker then your quality factor is Q L by P load is 1 and then we will assume that say for some variation in the value of the installed capacitance you see a actual power factor of 0.99 lag rather than unity power factor you are seeing 0.99 lag at the upstream breaker. So, essentially this could mean that you are having a C prime which is slightly smaller or essentially a inductive load which is larger which means your L prime is again something smaller than the L that you originally thought. Then you can now calculate your delta Q that is coming in from the grid is 1 by 0.99 in this case you are having 0.14 per unit lag essentially is the reactive power coming in from the grid and the question is what would be the resulting frequency change that you would see when a upstream breaker opens. So, you can use a relationship that we have just derived you have 1 minus f by f max P is 1 per unit quality factor is 1. So, you have f by f max is 0.92 or your f max equal to. So, you can see that with 0.99 power factor essentially you see frequency deviation of almost 4 hertz 3.9 hertz. So, you could say if you had set a over frequency relay at 50.5 hertz that would immediately detect that islanding has occurred and disconnect your DG in response to the feeder frequency rising to a higher value. So, similarly we could look at what would happen say instead of the power factor being 0.99 lag what would be the situation if your power factor was 0.99 lead that would correspond to a situation where you have slightly lower reactive load or a slightly larger value of capacitance in the power factor correction in capacitor bank. Then you would get delta Q again is minus 0.14 per unit. So, when your upstream breaker opens S 1 opens in this case the frequency would drop and you have 1 minus. So, your frequency would drop to about 46.8 hertz and. So, in this case say if you had a under frequency relays may be it is set to open at 48 hertz. So, seeing 46.8 hertz it would disconnect the DG and say that there is a situation abnormal situation. So, the DG has to be a disconnected. So, if the DG is operated at unity power factor reactive power compensation is not perfect then over and under frequency relays. So, if you again assuming that your load model is a RLC load model and you are assuming that your DG is operating at unity power factor. In some situations it might actually be beneficial to operate the DG under non unity power factor because you may have reactive power elements in your facility and you may want to operate it. So, that you do local compensation rather than always operate it at unity power factor. So, you have to realize that these are under assumptions that these thresholds are valid. And again you may say that assuming that all the loads on the feeder can be approximated as a RLC load is not always valid. But again the reason why people look at RLC load as a model for a situation of unintentional island is so that you can do a test that is repeatable. In fact, more common load might be a induction machine or a machine load with or a inductance voltage behind a inductance type of load. And if the machine has a large inertia it can actually sustain an island for quite a while depending on how large that inertia is. But there is nothing like a standard induction machine with standard value of core loss or standard value of friction, windage. These are parasitic elements in a induction machine, but you could accurately specify your resistance to be of 2 percent tolerance, your inductance to be of some percent tolerance. And you could actually do tests in a repeatable manner irrespective of whether it is being done in one location or the other. So, assuming the simplified RLC model for the feeder allows you to conduct tests in a repeatable manner. Again the other assumptions we are assuming that the DG is being controlled in a with constant P, constant Q. You might have situations where there might be some other methods for controlling the DG. You might control your rather P and Q, you might control your direct and quadrature axis of currents. So, depending on how your controllers your thresholds can actually also differ. One needs to keep in mind that here we are assuming a single DG with load and assuming that each of each of this element and where you are switching a upstream opening as upstream breaker. And we are looking at the power mismatch and the reactive power mismatch. And in an actual feeder the loads are constantly going up and down your DGs may vary in a sparse set point. So, statistically you might have a duration of time where your power generated by the DG might have a small mismatch with the power being consumed by the load as the penetration of DGs on the feeder goes up. So, the chances that your delta P and delta Q becomes really small becomes higher once your DG penetration goes up. So, there is always a chance of unintentional island for which one needs to be careful. As we have seen there are actually implications of continuing sustained unintentional island. So, we can then look at see that these passive methods would work well under low penetration of DG on the feeder. So, initially when there are just a few photovoltaic panels on the roof and you are having large loads on the feeder then there would be no problem. And another situation that we have just seen is that so it works well. Another result that we saw from the examples that we did is that it would work well if your grid voltage amplitude and frequency range is maintained in a tight range. So, if you look at a standard such as IEEE 1547 which is a standard for interconnecting distributed energy resources with the grid they would specify say V max of 110 percent V min of 85 percent then you would say F max of F norm nominal plus 0.1 hertz and F min what is suggested as that if the voltage amplitude or the frequency goes out of this particular range you disconnect in 100 milliseconds. So, you have a instantaneous disconnection if your voltage and frequency got goes out of this range. And if your grid is holding your nominal values in a very tight range then such a tight window for voltage and frequency would be acceptable. Say in the Indian grid this would mean that you may not be able to connect the DG with the grid at all. So, just directly following the standards would not work in all situations you will have to look at what is realistic in a particular local scenario. And then see whether that particular method would work or not in for detection for example, setting of relays detection of unintentional islands etcetera. We will look at another method of detection of unintentional island and that is based on looking at the direction of power flow. So, if you look at a situation where you have normal loads connected to the distribution system the power that is being drawn by at your point of common coupling would be from the grid into your load to your facility. And when you have distributed generation so connected at a load then now you have the possibility that power might be sent in either it might be drawn into the facility if your DG power is less than the facility loads or it might be sent back out into the grid if there is excess power available at the DG. Suppose the DG is controlled so that power is always drawn into the facility and is not so if you control the power flow to be only in the direction into the facility then you could make use of that to detect a situation of unintentional island. So, this is by sizing the DG to be less than your load in your facility or controlling your adjusting your DG operating power such that it does not export power at your point of common coupling then you could actually detect a situation of unintentional island. So, the reason why this is this would be the case is that say if you have a open upstream breaker and assuming that say all other loads on the feeder goes to 0 say if your L feeder your P feeder goes to 0 then to maintain the unintentional island with the DG you need to actually send power back out at your point of common coupling even at 0 load you will have some losses core losses in your interconnection transformer etcetera. So, you could always detect a situation where your power is flowing out from a facility to maintain your voltage on the feeder to be some finite positive value and if you are if you detect such a situation where the power is going out from the facility into the grid that means that some upstream breaker the grid has disconnected and the facility is now supporting the feeder and that can be used to detect a situation of unintentional island based on the power flow direction. But you can see that you now have restrictions on the sizing of your DG and always ensuring that your power in the DG is less than what is being consumed on the load. So, if you look at this particular case it puts restrictions on how you operate your DG which may not be actually be acceptable say suppose your DG is a photovoltaic system or a wind turbine you may want to operate it. So, that you are harvesting the maximum energy possible and having such restrictions would not be economically viable. So, all methods of anti islanding detection have some drawbacks you might have some restrictions you might have non detection zones etcetera which might lead to constraints on how you operate the system. So, now we could look at what are considered active anti islanding methods and essentially active anti islanding methods interact with the DG control and essentially you could have situations where you might periodically change the operating point of the DG and because you are doing that in a periodic manner you might have power or reactive power mismatch which causes detection that you have you are in an unintentional island or you might look at inject some harmonics and when your grid is connected with the your RLC load model you can you know that the grid maintains the voltage. So, as soon as you are disconnected from the grid there is nothing to maintain the voltage. So, if you are injecting harmonics you would see a corresponding harmonic distortion at your point of common coupling. Similarly, you can have unbalance which is detected you can detect changes in power factor also you could say try to shift your voltage operating point or your frequency operating point in a active manner where your DG controls itself is trying to actually shift the operating point of the DG and you can see that these methods are actually modifying the control of the DG in a active manner to detect an island. Some of these methods such as trying to change the operating point etcetera would be statistical and probabilistic in the sense that you might be able it might work with a single DG, but when you have multiple DGs one might try to take the operating point up and the second DG might bring the operating point down and it might cancel out overall. So, some of these things might work on a individual basis, but not in a large collective group. So, you will have to look at the implications of such situations also if you look at injection of harmonics and looking at monitoring unbalance etcetera these affect power quality. So, you have possibilities of power quality imbalance instabilities in the grade etcetera which can affect the operation. So, we will look at what is the underlying methods of some of these active and anti-aligning schemes especially the ones that try to shift your frequency and voltage operating point. Essentially what you are trying to do is we saw in the conditions that are required to operate your island as a sustained island. There are two conditions one is the real and reactive power between your DG and the load should match and the second thing was that the operating point should be stable. So, if the operating point is not stable then your potentially a voltage would exponentially drift away from whatever is the point it was at the when the upstream breaker disconnected or the frequency might drift away and you can make use of the fact that you can detect this drift in frequency and voltage to detect a situation of unintentional island. So, essentially the active anti-aligning methods so we will first look at how potentially the voltage amplitude can be shifted. So, if you remember based on our RLC model for the feeder and the DG we had an expression for your delta P by P. So, this was a relationship that we had and we also had delta R by R. So, if you look at it on a small signal basis you have delta P by P is minus delta R by R. So, let us call this one now in the DG controls if we add a term delta P by P norm. So, essentially what you have is your power in the DG is from your MPPT algorithm or something which drives your power command and you are adding a additional term to it where you are looking at deviations from the nominal voltage and adding a term K times P norm by V norm. So, essentially this term corresponds to essentially the expression 2 and this would go say if you have a synchronous machine it can go to the exciter of the synchronous machine or it can go to the in phase current current command in a inverter. So, based on that you could actually adjust your actual power that is being put out by the DG to the governor. So, essentially this determines what would be your actual power and if you look at this particular expression where delta P by P norm is K times delta V by V norm and if you look at the expression for the RLC load model the physical RLC load model you can see that the polarity of the sign between delta P and the voltage and the resistance is such that you could essentially based on K have emulated a positive a negative resistance in a dynamical system essentially a positive resistance gives you damping. So, a negative resistance would give you something which is exponentially diverging. So, you can introduce an instability by controlling the term K by tuning the term K to actually cause your voltage to diverge rather than damp down in a dynamical system. So, essentially what your dynamical resistance that is being emulated can be negative and so if you look at your RLC load model. So, essentially what you could have is your dynamically emulating a resistance and this resistance along with your actual load resistance let us call this as R emulated. So, R emulated in parallel with R would be your effective resistance in your feeder with after anti after islanding and if that resistance becomes effectively negative you would have a situation where dynamically you have a system which would instead of damp down it would essentially become unstable. You could also see what this physically means. Suppose you have a situation where in response to the disconnection of the grid if your voltage falls the possibility of the voltage falling could be because your power that was being injected is lesser than what is required and this particular term would then further reduce the power which means that the resulting voltage would fall further down and this becomes a positive loop and eventually the entire feeder voltage would collapse. If the voltage rises on disconnection then this term would inject additional power causing a further rise in voltage essentially causing your final voltage to just go outside your acceptable range. So, we could in a similar manner where we looked at the voltage and then altered the frequency we could do a similar thing with by measuring the power the frequency you could alter the reactive power and in response to the change in frequency and the reactive power you could adjust the operation of the d g and with the objective of taking the frequency now outside the nominal range. We will start looking at this problem we may be able to probably wrap it up in the next class. So, this is essentially what you do when you are having a active frequency shift. So, we saw in our RLC load model your q of the load is V square by 2 pi f l and the nominal value is was 0 and now we will look at what would be the situation if your frequency is something which shifted to f plus delta f and what would be the net reactive power that is being drawn by the load. So, you can write your delta q load the original value of 0 with the change in frequency you would have delta q load equals and taking the 1 plus delta f by f term to the numerator this is approximately equal to 1 minus delta f by f and we saw again the assumption that we had was that q l and q c are resonant at 50 hertz. So, you could write this as your delta q delta q that is coming from your grid due to a change in frequency would be. So, if you plot say frequency versus your q load and plotted close to your nominal range essentially you would have a curve with a negative slope and your load is resonant at 50 hertz. So, at 50 hertz your q is 0 and if your frequency drops by some delta f then essentially you would require some additional delta q or if your frequency rises then essentially the load would act more capacitive because it is a parallel RLC model that we are assuming for the load. So, then we could look at what would happen if there is a l and c variation around nominal and essentially what will do is will look at the change in the operating reactive power of the d g to be similar to what would happen when there is a nominal l and c variation that is happening on this particular island. So, if you have so this is the nominal curve and suppose you had your l and c to be say l minus delta l c minus delta c. Essentially what would happen over there is that your effective power factor correction capacitance is less or l minus delta l means that your reactive power that the load is drawing is more. So, you are adding a delta q load. So, you could think of it as corresponding to a new higher frequency at which it would settle. So, this would say let us call it as some f max and say this would be your plus delta q load. Similarly, you could look at a situation where l and c parameters where l plus delta l or c plus delta c then you could think of it as essentially in the parallel RLC load your reactive power would drop or it would settle in at a lower frequency f max. So, we will look at the change in operating point of your d g to emulate a equivalent change in your l and c l and c values and use that to see what would be the change required to actually cause your frequency to just be driven out to be driven to a very large value or to a very small value. We will do this in the next class. Thank you.