 Welcome to class 40 in topics in power electronics and distributed generation. We have been looking at damping of resonances in LCL filters and one way to provide damping is passive damping and in passive damping what is done is to add a resistor or a network with a resistor inert to the LCL filter and the objective of the passive damping is to minimize the quality factor of the filter at resonance and do that with minimum power dissipation. So, we looked at the progression of possible passive damping schemes starting with a simple resistive damping and then a split capacitor resistive damping and a split capacitor RL damping and as one goes with higher complexity it is possible to reduce the power dissipation but with the higher complexity you have more components it can be more expensive. So, there is a tradeoff in where what could be applicable. To model such a LCL filter one can make use of circuit analysis and one could then make state space models of the equivalent circuit of the power converter grid connected power converter with a dammed LCL filter. The state space models can be used to derive transfer functions of between different parameters say one transfer function may be the capacitor voltage to the inverter output voltage then you could look at what the quality factor is and then we derived an expression for a approximate expression for the quality factor using parameters the parameter k and with certain assumptions. We also can make use of the state space model of the LCL filter with the damping to evaluate power loss and the power loss is evaluated at the two major frequencies of excitation which is the fundamental frequency and the frequency at which the major ripple occurs which is the switching frequency. So, now you that one can have ways to quantify the quality factor and the and the power loss one can then look at how to go about designing the passive damping network. So, the first step that we looked at was we took an example of a 40 kVA 250 volt 3 phase 4 wire inverter with a LCL filter and as a starting point we had the grid side inductor and the inverter side inductor to be equal to L by 2 and we saw that that was a good choice for keeping the grid current ripple to a minimum given a constant value of L 1 plus L 2. Then the next question was to how to split the capacitor between C 1 and C D and C D is taken as a C times C 1 with C 1 plus C D being equal to C and this C is essentially the C selected in the LCL design procedure. So, if you take a value of AC equal to 0 so what is plotted over here is AC versus the quality factor the minimum possible quality factor and AC equal to 0 would correspond to a situation where C D equal to 0 and AC tending to large values would correspond to a situation where C D equal to C and as C D becomes larger and larger we can see that the quality factor Q f comes down but again as AC increases the power dissipation increases and you can see that for a range of switching frequencies the power dissipation is increasing with AC and it is also increasing as the switching frequency reduces. So, if you select a value of AC greater than 1 the knee of this particular curve is close to AC being equal to 1. So, if you select AC being equal to 1 you will get a quality factor of less than 3 and with minimum with a small reduction in quality factor further as AC is further increased. So, selection of a AC in this particular range might be a reasonable starting point and taking AC equal to 1 might be a suitable design. Also one can look at with AC equal to 1 what is the value of R D that can be selected and a suitable choice for R D is square root of L by C and then with the state space model one can plot the poles of this particular filter transfer function and one can see that as R D is increased you have one pole at the origin you have another pole which moves to the left you have a complex conjugate pole player and the maximum damping is obtained when R D is equal to square root of L by C. So, selection of R D is equal to square root of L by C and C 1 is equal to C D we have a suitable design for a starting design for a SCR split capacitor resistive passive damping for a LCL filter. So, the next question is how to select the damping inductor if you want to go for a SCRL damped network. So, the question is what should LD be? So, address this particular question one can go back and take a look at the expression for the quality factor and for the power loss and if we saw that the expression for the quality factor is approximate expression is twice square root of 1 minus K times omega fundamental by omega resonance the whole square plus 1. So, you can see that to minimize this particular function if you select K to be equal to omega R by omega f u then this term within the square would go away and you would get a quality factor close to 2 and in this particular design we have the resonant frequency close to 1 kilo Hertz and the fundamental frequency is 50 Hertz. So, if you plot this particular expression the minimum happens for K close to 20 and what is plotted is the quality factor versus K and if you look at then the power loss versus K the power loss consists of two terms one is because of the fundamental power loss which is essentially this blue curve over here and you have the power loss due to the switching ripple which is the red curve over here the black curve is the total of the two. You can see that the power loss versus K graph has a knee close to around K is equal to 5. So, selecting a value of may be K greater than 5 would one can have minimal further reduction in power loss. So, if you take a value of K may be from 10 to larger value of K there is no significant reduction in power loss as the value of K is increased. And you can see that for this particular design if you select the K of 20 you would get a power loss of about 0.5 of around 0.06 percent for K is equal to 20. So, the previous expression for the quality factor versus K was based on an approximation as we saw we could also determine the exact quality factor numerically. So, if you plot the exact numerical quality factor versus K we again see that this is a curve with a minimum and the value of the minimum is again close to a value of 2, but the location of where the minimum occurs has shifted from a value of close to 20 to a value of close to 10. So, there is a factor of 2 between the approximate and the exact quality factor minimum point based on a exact numerical analysis of where you would have the minimum value of K. So, you could then make use of this particular factor of 2 and look at evaluate designs now at different switching frequencies. So, here what is shown is the point of minimum quality the quality factor versus K for power converter designs where the switching frequency is selected from close to 10 kilohertz to 2.25 kilohertz. So, for a wide range of switching frequency you go through the procedure of defining the LCL filter with SCRL damping and you look at where the exact quality factor is minimum and that happens at half omega r by omega u. So, the point half omega r by omega u coincides with the minimum value of the quality factor in the quality factor versus K expression for this range of practical range of switching frequency selections in a power converter. So, again if you then look at what would be the effect of this reduction of K by a factor of 2 in terms of the power loss one can again plot the power loss versus K. So, what is plotted here is the power loss and per unit versus K for again for a range of switching frequencies from 9.75 kilohertz to 2.25 kilohertz and all the cases you can see that for the selection of 0.5 omega r by omega f u the points of power loss lie beyond the knee of this particular curve. So, even for a low switching frequency such as 2.25 kilohertz you are having a power loss of less than 0.6 percent for the selection given selection of K. One could also see look at the effect of this particular selection of K is equal to half omega r by omega fundamental in terms of the location of the pole 0 plots using our state space model. So, from the state space model we could look at say for example, the transfer function of capacitor voltage to your inverter voltage and see where the pole locations and say the 0 locations how they vary as the value of L is varied. You will take a nominal value of L D to be given by this particular expression and that would correspond to L D being equal to 2 R D by the resonant frequency and if you then vary L D from one tenth of this particular nominal value to twice the nominal value you get the locus of the poles and zeros. So, you start with say 0.1 L D and you get one trajectory of this pole 0 plot you have a complex conjugate pair and you can see that for low values of L D, L D nominal you have very poor damping and again if the value of L D nominal becomes large again you have poor damping. Similarly, you have another complex conjugate pole pair which takes a trajectory such as this. So, you have this would be 0.1 L D and you have another pair of complex zeros moving from L D is 0.1 L D nominal to 2 L D having this particular trajectory. So, if you look at the point at which L D is equal to this particular value then at that particular point one can see that there is one complex pole and a 0 at the origin you have a pair of complex conjugate poles and you have a 0 and this would be correspond to the point where you get the maximum damping again in this S plane as you get closer to the j omega axis your damping reduces and as you get closer to the real axis in the left half S plane your damping is actually improved. So, this would correspond to the point of having the best damping and corresponding selection for the given value of L D where K is given by half omega resonance by the fundamental frequency. Another factor in a filter design is to look at how the filter would work for a variety of grid conditions in particular the grid condition the grid may be strong. So, this would imply that your impedance of the grid is close to 0 or the grid may be weak would this would imply that your z grid is large and if you think in terms of the model the grid as inductance reactance plus a resistance then you could think about lumping the grid impedance along with the value L 2 and think of L 2 prime to be equal to L 2 plus L grid. One could also think about the resistance term in the grid interconnection, but the resistance term would only provide additional damping to your filter. So, we would consider the situation where essentially z grid primarily consists of the inductive term. So, if you look at the range where L grid is varied from 0 to L 2 itself essentially what it means is L 2 prime is varied from the range of L 2 to twice L 2 and one can look at the quality factor as this value of L grid is varied over this particular range. And you can see that you start off with a number which is closer to the quality factor which is a number closer to 2.25 and at 2 twice L 2 you end up with a quality factor which is less than 2.26. So, there is not a major variation in the quality factor as the grid impedance is varied. So, this particular design could work over a fair range reasonable range of grid impedance. So, with this we could then summarize the procedure that we took for the filter design. So, one thing we could start off with is to identify what is your pass band and what is your stop band. Of course, the fundamental frequency would be in the pass band plus potentially some of the harmonic frequencies if one is looking at control at those harmonic frequencies. The switching frequency would be in your stop band and we could use that to determine what your resonant frequency is for the filter. And if you take too low a value of resonant frequency you would end up with large values of L or C or both. The next step is to identify a constraint on the value of L minimum and this is again we saw this is from the ripple constraint the amount of ripple that is being allowed to inject to the grid. A second constraint minimum value of inductance is from your maximum reactive power that is drawn by the capacitor. So, you are saying L 1 and L 2 you could then find what is your L min which would be the maximum of L 1 min or L 2 min. You also have a maximum value of the inductance that you would use in the filter and this is based on your DC bus voltage because if you have too large a filter inductance then the voltage drop across the filter inductor would add to the grid voltage and you would need higher DC bus voltage to be able to operate the power converter in a linear modulation range. So, you get a range of particular value of the inductance and then in this particular range what one could see what value of inductance leads to the minimum power loss in your LCR filter and using that you could then decide the value of L 1 and L 2. Another condition that can be derived is L 1 is equal to L 2 is equal to L by 2 would be a good design from the point of view of keeping the cost of the inductor low by minimizing the ripple injection into the grid by for a given constant value of L 1 plus L 2. So, once you have selected your resonant frequency and your L then that would give you the design for your capacitor then for a split capacitor based damping design you could select C 1 is equal to C D is equal to C by 2 and R D is equal to square root of L by C this would give you a low value of Q factor and total power loss in a SCR damped LCL filter. So, if you want to then go to a split capacitor R L damping one could select a K to be equal to half omega s by omega fundamental, omega fundamental being a major excitation which causes losses in this damping branch and the corresponding value of L D would be equal to R D by K omega fundamental. So, at the end of the design you want to go back and validate what your attenuation is the different performance constraints are being met if they are not being met for example, if the attenuation is on the lower side you want to slightly improve that attenuation you could increase the value of L and go back and then evaluate the value new value of C and go through the design procedure. So, at this particular point we would also like to see how good this particular procedure is. So, what is shown over here is the damping factor K versus the quality factor for a SCRL damped LCL filter. So, what is shown in red is the analytical value of the quality factor versus K and what is shown in the blue curve is the lab measured quality factor. So, you can see that the shape of the curve the value at which the minimum occurred is similar for the analytical and the lab measurements. In the lab the quality factor is reduced a little bit further because in practical filter you would have additional stray resistance in your windings in capacitors etcetera which provides a slightly higher level of damping. You could also then look at the measurements of the analytical and the measured power loss for the different values of K and you can see that the power loss at the fundamental frequency and the power analytical power loss measured matches each other closely and similarly for the power loss at the switching ripple varies closely with the variation of K. So, the overall model can actually match the measurements that you would make in a physical design. So, this procedure can be used in a realistic manner. So, at this particular point we have actually looked at range of things and factors in this course we have looked at distribution systems we have looked at distribution systems from the point of view of what the power converter can expect when it is interfaced to a distribution system in terms of the distribution system protection, the time frames in which the protection systems act, the type of voltage distributions that one can expect along the feeder etcetera and see whether the power converter can actually operate in that particular context connected to a system rather than assume the grid to be a constant value of voltage source we can actually refine it further by including some of the aspects of the distribution system along with the model of the distributed generation inverter. The second part that we looked at is how the power electronic system can be evaluated in terms of the life cycle cost and when you consider the life cycle cost there are factors such as efficiency, reliability, aging that comes into play when you are evaluating the cost and that can be useful in the design of the power converter. So, you have the upfront cost, the efficiency related factors and reliability factors. So, with this we then looked at the main components of the inverter. So, the major components in a power converter would be the power semiconductor devices, your DC bus for a voltage source inverter this corresponds to the DC bus capacitor and we have also looked at the output filter. There are many other hardware components that go into a power converter such as circuit breakers, contactors, surge arrestors, precharging circuit etcetera which also can be looked at a closely. You also have to think about issues like how the enclosure is built, how cabling and bus bar connections are made within the power converter, what are the cooling loops, what is the air exchange mechanism or if there is a water cooling, what is a where how the water cooling loop is built and the associated equipment. You also have a important aspect which is related to the control of the converter and in terms of control there are two aspects to it. One is the hardware related aspects which might involve circuit boards for protection, you might have the gate drive, how you feed the control power, the voltage and the current sensing, the controller often in today the controller of high power converters will be digital controllers and you also then have the firmware or the algorithm related aspects and some of the firmware or algorithm related aspects would be, so one would be the phase lock loop or unit vector generation. So, important task is to find out where the what the frequency, the amplitude, the phase of your fundamental component is a three phase system, you might want to know what where the angle of the positive sequence fundamental voltages, you also have current control, you have voltage control often you need to may need to control your DC bus voltage, you also have to look at what the broader real and reactive power control, how that is being achieved in some applications, harmonic control might be important if you are in particular if you are looking at active filtering or trying to damp out harmonics etcetera, you are we just looked at design of the passive damping, it is also possible to consider active damping of resonances, also a major aspect of the power control is how to deal with the distribution grid, we saw that the contact the grid distribution protection can occur in a longer time frame and the challenge is to have a right through of the power converter when there is grid disturbance and the disturbances may be longer term, voltage regulation, frequency shifts etcetera or it may be shorter term, sag, swells, outages for shorter durations and a variety of the situations can power converter right through the disturbance and help reenergize the system when the grid recovers, you also have issues like active island, anti islanding and many such control related issues, so in this course we are not looking at the details of the control related issues, starting point for looking at some of the control issues would be a course on electric drives that would be available on NPTEL, so at this point we will wrap up this course and I hope you have found it useful, thank you.