Transients in Power Systems
Transients in Power Systems ECE 524
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This 15 page Class Notes was uploaded by Fredy Okuneva on Thursday October 22, 2015. The Class Notes belongs to ECE 524 at University of Idaho taught by Staff in Fall. Since its upload, it has received 36 views. For similar materials see /class/227736/ece-524-university-of-idaho in ELECTRICAL AND COMPUTER ENGINEERING at University of Idaho.
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Date Created: 10/22/15
61 39OU NOISSEIS SWELLSAS EBAOd NI SiNEJISNVEIi vzs 333 m Univergityofldaha p a Jig i Lav 9 ECE524 Session 18 Page 1321 Transients in Power Systems Spring 2008 General solution of wave equation a Bit m 5 0 D Alembert 8X2 K812 0 Solution F g x 3E Where K LC or CL 0 Then TIR Vii C v which is the propagation velocity 0 Also we can nd v L CM WUMi W L d1 g m U 1 Egg Spring 2008 o ThenF gxvt Distributed Parameter Line Model ECE524 Session 18 Page 1421 Transients in Power Systems Spring 2008 Propagation Velocity 0 Notice that for overhead lines L 2 H f lnj f Hm I I x 39 g Q And C gaff Fm r Ur 0 Using air as a dielectric 8 LL l f g 9 Then LC Huang K 57 080 t3 0 Therefore lv xLC ittoeo lc where c is the speed oflight c When the ground return is included then 8r gt 1 and u gt 1 and this does not hold a The same is true with cables as well 0 We also de ne the characteristic impedance as g 26 o For overhead lines 26 ranges from 3009 to 5009 o For underground cables ZC ranges from 30 to 80 Distributed Parameter Line Model Spring 2008 52 W m Universityofidahe ho Eat m Universityofldaho m Universityofldaho m m 3 217 39OU NOISSES SIAEIJSAS HEIMOd NI SiNElISNVHJ 1725 EDEI Session 41 Page 118 Spring 2008 ECE524 Transients in Power Systems Numerical Oscillations in EMTPLike Programs 1 Causes of Numerical Oscillations The Electromagnetic transients program and its variants all use the the trapezoidal rule numer ical integration method The trapezoidal rule is a second order numerical integration technique that is simple to implement astable numerically stable in stiff systems and fast However it is also susceptible to numerical oscillations when differentiating step changes in voltage or current The program user should be aware of the potential for these oscillations when simulating circuits and systems using EMTP The most common events leading to numerical oscillations are a step change in current through and inductor or a step change in voltage across a capacitor The cause if this problem can be seen by looking at the differential equations for the inductor and the capacitor r i Inductor Voltage 1 L L g 1 dt at do CL paC LtOT39 Current t C H t 2 One can think of the numerical oscillations as resulting from forcing an in nite 171 It across and inductor or an infinite dvdt on a capacitor One possible solution is to produce a more accurate circuit model by including parasitic capacitances internal resistances in a capacitor 11 Mathematical Representation The problem can be represented by modeling the equation ll it 3 using the trapezoidal rule We can integrate both sides of the equation resulting in the following e V ewe t 1t tm mlt 1175 At 4 Next we rewrite the equation using the trapezoidal rule Remember we are finding an approxi mation of the area of the trapezoid bounded by 3 t and t At ya 3105 Ar 9 5ltmlttgt are Am 5 1 17 was 933 msi Session 41 Page 218 Spring 2008 ECE524 Transients in Power Systems Now we can solve this equation for 1t which could be either the current through a capacitor or the voltage across an inductor 2 we mltt m mom ye At 6 Start out with rt At 0 We will now have y undergo a step change from yt At 0 to 10 The function will stay at 10 for the rest of the time period of interest Ve would expect yt and xt to behave as shown in Figure 1 Note that we would expect to be an impulse Dirac Delta Function at at 1 z t and zero everywhere else no L0 Time t At t At Figure 1 Plots showing yt and expected It for example case Plug the know values into the equation for 361 resulting in 2 1 t 1lt gt At 7 Now step ahead and nd xt At 2 2 2 i m Ar N Ain 0 0 At 8 if we continue ahead another time step we nd that 2 A I 9 tt 2 t N tan Universityofldaho Luz 31w ECE524 Transients in Power Systems Session 41 Page 618 Spring 2008 122 Capacitor Voltage The second example is a single phase system with a sinusoidal voltage source supplying a pi circuit A switch connecting the source to the pi circuit closes at 16667 msec causing a step changing in the capacitor voltage on the capacitor near the source Figure 5 shows EMTP simulation results The rst plot shows the current through the capacitor connected to the switch Notice the large amplitude of the oscillations in this case The second plot zooms in on the oscillations themselves and the third plot shows that the voltage across the capacitor is not impacted The EMTP data le is provided below aw x104 I 0 2 quotI l I l L I I 0002 0004 0006 0008 001 0012 0014 0016 0018 002 p A lca l N l 1 l I l l 1 l I 0017 0017 0017 0017 0017 00171 00171 00171 00171 00171 00171 10 X 15 1 l I l l l I I 10 a I 5 0 4L 0 l l l l 1 0002 0004 0006 0008 001 0012 0014 0016 0018 002 Time sec 4 l Va1V Figure 5 Capacitor example BEGIN NEW DATA CASE C EMTP Datafile showing numerical oscillations when changing the C voltage across a capacitor C C Miscellaneous data C DeltaTlt TMaxlt X0pt C0ptlt EpsilnltTolMatlt T8tart 10E5 002 C IDutltIPlotltIDoublltKSSOutlt Max0utlt IPunlt MemSavlt 1Catlt NEnergltIPrSup 111 1 1 C C Circuit data C Bus1gtBu52gtBu53 gtBus4gtlt Rlt Llt C 67139 2h ECE524 Transients in Power Systems Session 41 Page 1018 Spring 2008 it it Figure 8 Snubber circuit provides a bypass path for the switch the numerical oscillations are avoided and a more accurate model of the power electronic circuit is used However some circuit topologies done need snubbers diode recti ers for example in addi tion switches with large safe operating areas SOA such as IGBT s and MOSFET s don t need snubbers In this cases the program user can add numerical snubbers These numerical snubbers can also be added for conventional switches as well although they have less of a basis in reality If a capacitor alone is used a capacitance value of roughly 1 2 nF will suf ce If a RC snubber is used as shown in Figure 8 the time constant for the RC snubber must still be greater than the simulation time step to avoid problems The resistance value should be chosen so the RC time constant is a minimum of 2 3 times the simulation time step At Performance of the RC snubber will vary with the circuit and the user may need to vary R and C values for best performance The inductor current example from earlier was simulated with a RC snubber across the switch as shown in Figure 9 The first plot shows the resulting voltage Notice that there are no oscillations The second plot zooms in on the voltage waveform Note that the voltage sees a decaying exponential from overdamped RLC response The third plot shows the snubber current Notice also that the time step used in the data le below is smaller to capture the LC resonance BEGIN NEW DATA CASE C EMTP Datacase showing numerical oscillations resulting from opening C a switch in series with an inductor C C Add a snubber across the switch C C C Miscellaneous data C DeltaTlt TMaxlt X0ptlt COptlt EpsilnltTolMatlt TStart 4E6 003 C IOutlt IPlotltquotIDoubllt KSSOutltMax0utlt IPunltMemSavlt ICatlt NEnerglt IPrSup 11 1 1 5 2h ECE524 Session 41 Page 1118 Transients in Power Systems Spring 2008 x16 1 r e mww V bus V c I l i 0005 001 0015 002 0025 003 l snubber A m ac ca Figure 9 Inductor current example with snubber added C C Circuit data C Bus1 gtBu52 gtBus3 gtBus4gtlt Rlt Llt C GENl B081 20 1 BUS2 25 C Snubber circuit RC 3DELTAT GENl 8081 1500 010 1 BLANK ends circuit data C C Switch data C Bus gtBus gtlt Tcloselt Topenlt Ie 0 BUSl BU32 1 1E 3 BLANK ends switch data C C Source data C Bus gtltIltAmplitudeltFrequencylt TOlPhiOlt OPhi0 lt Tstartlt Tstop 14GEN1 10000E3 60 0 1 111 BLANK ends Source data C C Output Request Data C Bus gtBus gtBusgtBusgtBus gtBus gtBus gtBus gtBusgtBus gtBusgtBusgtBus gt BUSl BUSZ GENl BLANK ends output requests BLANK ends plot request BEGIN NEW DATA CASE BLANK ends all cases 23 Reducing the Time step In some cases reducing the simulation time step can also eliminate numerical oscillations How ever7 this is not a general purpose solution It depends on the presence of resistances in the circuit preferably in parallel with inductances or in series with the capacitances In many cases the required time step is also far too small for practical simulation Also since the amplitude of the oscillation varies with lAt the smaller time step may also make the oscillations larger of St P 395 Um 7W W Universityofldaho Cox 1 meow a mw ma a w ww m acukohm Ohm mg gmr db 1 4 mam Scum 1W 07 QM skunk