Week 5 Lecture EEE 576
Week 5 Lecture EEE 576 EEE 576
Popular in Power System Dynamics
Popular in Electrical Engineering
This 69 page Class Notes was uploaded by Shammya Saha on Sunday March 6, 2016. The Class Notes belongs to EEE 576 at Arizona State University taught by Dr. Vittal in Fall 2015. Since its upload, it has received 93 views. For similar materials see Power System Dynamics in Electrical Engineering at Arizona State University.
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Date Created: 03/06/16
EEE 576 Power System Dynamics Week 5 Class Note Professor : Dr. Vijay Vittal How Does Excitation System effect Stability? • Increases “steady-state” stability limit. Precautions: • Voltage regulator gain requirement at no-load is different from that needed for good performance under loading conditions. • Growing oscillations in large interconnected systems are further aggravated by regulator gain, so regulator gain has to be set accurately. • Excitation control is also complicated by differing control requirements at different instants of time - Transient - Long Term Hence , It is important to determine this limit i.e., to find the exciter design and control parameters that can provide desirable performance at a competitive cost. Transient Stability Considerations In this case machine is subjected to a large impact for a short time which then could result in a significant reduction of terminal voltage and hence an ability to transfer synchronizing power. V V P t D sin x For thi siuation the m ostbe nefiialattri utesof t ev olage regulatora r speed and h ih ceilng voltageto hold V tatth enee ded lv el W hen the faulti rem oved d ueto switching x will change and exc iati nc hange isagain required. 2-Machine System E 1 2 P sin x1 x 2 1 2 x 1 x 12u E 1 E hav2certain values to holdV at1.0 pu. t If the powerfactor is unity E & E 1ave th2 same magnitude and the phasor diagramisgiven by If E 1 E ar2 held constantat thesevalues, power tran sferredto infinite bus variessinusoidially with andhasa max at 90 Assume E & 1 subj2ctedtoperfectregulatoraction. Aim of thisis toholdV at1.tpu and PF 1.0 2 j2 2 j2 E 11 jI 1 I e E 21 jI 1 I e E E 2 2 1 I cos 2 or E E 1 1 2 2 1 2 cos 1 P 2 sin 2cos 2 2 sin 2cos 2 2 cos 2 tan 2 In this analysis we assume regulatorsa1t o2 E & E -Ideal regulation hasnostability limit -Operates in region where 90 -Such a physical system not realizablesince there isalways a lag in excitation response Hence considering a more practical scenario where E 21.0pu constant & let PF vary. Wehave the phasor diagram shown here. PF constrained by 2 1 2 1 2 , & 1 2 E 1 jI 1 I sin jI cos E e j1 1 1 j2 E 21 jI 1 I sin jI cos E e 2 E 21 I 2sin 1 2 2 2 2 2 2 2 1 I sin I cos 1 1 2I sin I sin I cos 1 2 2 1 2I sin I 1 2I sin I 0 I 2sin 2sin 2 1sin 1 Hence E sin 2sin sin 2sin 2 sin1 tan 1 2 2cos 2 Once i2established, wecanfix,I, and 1 The relationship between these variables isnonlinear. P V Itcos P sin 2 P occursat 90 max 2 E 1 2 j1 2.235 (26.6) I 1.414 45 116.6 maygo beyond 90 toachievemax powerandrequires E 1 2pu These examplesdepicteffectof excitationunderideal conditions.These conditions will not berealizedin practice. Important question: Ability of system to maintain synchronism during and following disturbance. Important factor: Machine behavior and power network dynamic interactions. Assumptions: Power supplied by prime movers does not change in the period of interest. Effect of excitation largely related to ability to help generator maintain output power in the period of interest. Main FactorsAffecting Performance 1.Impact of disturbance – type, location, and duration 2.Transmission system capability to maintain strong synchronizing forces during the transient caused by the disturbance 3.Turbine – generator parameters a)Synchronous machine parameters -Inertia constant - xd -' do b) Abilityof excitation system to hold the flux levelof the synchronous machine and increaseoutput powerduring the transient c)Fastacting protectionsystem Effect of excitation on small signal stability and long term stability: While high gain fast acting exciters greatly benefit transient stability following large impacts they are not necessarily beneficial in damping oscillations following a large disturbance. This kind of analysis involves the use of a linear system model. Addressing the probleb using a one machine infinite bussystem.Thesubscript isomitted. Te K 1 K E2 q (1) K K K E q 31 K ' sE FD 3 4 1 K ' s (2) 3 do 3 do Vt K 5 K E6 q (3) js T m e (4) ' do Direct axisO.C.timeconstant Constants K 1 K d6pend onsystem parameters& initial conditions.This material covered in chapter 6. To thegeneratormodela simple modelof a regulator- excitationsystemisadded E FD K E V 1 s (5) t E K ERegulatorgain -Exciter-regulatortimeconstant E E K 3 K E V K 3 4 q 1K 3 sdo 1 sE t 1K 3 do K K3 E K3K 4 1K ' s 1 s K 5 K E6 q 1K ' s 3 do E 3 do K 3 K6 E K 3 K5 E K K3 4 E q 1 K 3' do1 s E 1 K 3 sdo s E 1 K 3 sdo E K 3 K5 E K 3 14 s E q 1 K ' s 1 s K K K 1 K ' s 1 s K K K 3 do E 3 6 E 3 do E 3 6 E T K K K 3 K5 E K 3 K4s 3 4 E e 1 2 1 K ' s 1 s K K K 3 do E 3 6 E K K K s 1 K 5 E 2 3 4 E E K 4 E K 1 K ' s s 1 1 1K K 3 6 E 3 do E E K 3' do K 3 do E K K s 1 5 E K 2 4 E K 4 E 1 ' 1 K K K (6) do s s 1 1 3 6 E E K 3' do K 3' do E From(4)forT 0 m 2 s R T e R T e (7) J 2H Combining (6)and (7)and rearranging, the following characteristicequation isobtained 4 3 2 s s s s 0 (8) 1 1 E K 3'do 1 K K K 3 6 E K 1 R K 3' do E 2H R K 1 K 1 K 2 4 2H E K 3' do 'do R K 1 K K 3 6 E K 2 4 K 5 E 1 2H K 3' do E 'do E K 4 Applying Routh'sCriterion s 1 s 0 2 s a 1 2 1 s b1b 2 0 s c 1 a1 1 a2 1 0 b 1 a b 0 1 a1 1 2 2 1 c1 b 1a2b 2 1 1 No.of signchanges in the firstcolumn (1,,a 1b 1c1) No.of rootswithpositiverealparts Forstability,a ,b1,a1d c 01 1. 1 1 0 E K 3' do Both aEd' 0do 'do 1 E K 3 K 3sanimpedancefactorand isnot negativeunless thereis ' excessiveseriescapitance.Even then do islargeenough E tosatisfyinequality. 2.a 1 0 ' R K2K 3 4 E 1 K E do E K 6 2H K '3do E K 3'do E K Eslimited to valuesgreater thansome negative numbers. This constraint iseasilysatisfiedin the physicalsystem. 3.b 0 1 R K K2 4 1 K 4 6K ' 5 do K 1 2 K E 2H ' do E K 4 do E The terms in parenthesesarepositivefor loadconditions. This inequalityplacesa limiton the maximum valueof K Eor stableoperation. 4.c 0 1 K K 2 4 1 K K 3 E K 1 6K K 2 5 This condition puts a lowerlimiton the valueof K E Effect on Electrical Torque From(3.13) wecan computetheelectricaltorqueasa function of angularfrequency T K K K e K 1 2 3 4 2 2 2 1 jK 3 do 1 K '3 do The realcomponentin theaboveequationis thesynchronizing torquecomponentwhich is reduced by thedemagnetizing effectof armaturereaction. At verylow frequencies T KsK K1K 2 3 4 1.07551.2578* .3072*1.7124 0.4138 In an unregulated machine positive damping is introduced by the armature reaction and is given by the imaginary part of the Te equation. From(3.40) we note that Te K 2 14 s EK K K 2 5 E K 1 1 K6K E s ' do E sdo E K3 K 3 Theeffect of the term K2K 4 s Essmall compared to K K 2 5 E T K K K e K 1 2 5 E 1 2 E K K 6 E ' dosE s'do K 3 3 The aboveequationcan beseparatedinto itsrealcomponent givingT asd imaginarycomponent givingT ata partidularfrequency 1 2 K 2 K5 E K K 6 E ' do E T K 3 s 1 2 2 1 K K6 E ' do E ' do E K 3 K 3 K 2 K5'E do E K 3 T d 2 2 1 K K ' ' E K 3 6 E do E do K 3 Tdwill have thesamesignas K .Th5squantity can benegativeat someoperatingpoints.In thiscase theregulationreducesinternal damping. At verylowfrequencies T K K 2 5 KE 3 s 1 1K K3K 6 E 1.2578x -.0409x.3072x1 1.0755- 1.0892 1.3072x.4971x1 Which is higher than the valueobtainedfor theunregulated machine. Hence,theregulatorimprovesthesynchronizing forcesatlow frequencies,andreducesinherentdamping when K is ne5ative. Wearenormallyinterestedin the performanceof excitationsystems with moderateor high response.Forsuchsystemswecanmakethe following observations basedon theanalysisdone. With K positive,the effect of the regulator is tointroduce 5 a negative synchronizing torque and positivedamping torque component. The constant K is 5ositivefor low valuesof externalsystem reactanceand lowgenerator output. The reductionin K due 5o regulator actioninsuch casesis usuallynot of much concernbecause K isso high. 1 With K n5gative the regulator actionintroduces a positive synchronizing torque component and a negative damping torque component. This effect is more pronounced as the exciter responseincreases. For high valuesof external system reactanceand high generator outputs K i5 negative.Such situations arecommonly encountered in practice.Forsuch casesa high responseexciter is beneficialin increasingsynchronizing torque.However,in doing soitintroduces negative damping. We thus haveconflicting requirements with regard to exciter response. Root – Locus Analysis of a Regulated M1achine connected to an infinite bus In this caseweconsidera more detailedrepresentation of the exciter.The limiter and saturation function areomitted. G F is the ratefeedbacksignal. V sstabilizingsignal. The linearizedmachine equations areused to simulate the effect of damper windings and damping torques.A damping torque component - D isadded. To study the effect of the different feedback loops, we manipulate the block diagram so that all feedback loops originate at the same take off point. Common tak eoff pointV ,and tte feedback loopswewant to study arethe regulator and ratefeedbackG s F K K 2Hs Ds K K K K N s 3 6 1 R R 2 3 5 1 K ' s 2Hs Ds K K K K 3 do 1 R R 2 3 4 We willuse a specificexample to conduct thelinearanalysis. The machine data is the same as that used inchapters4,5,&6 and the loadingconditioninexample6.7is used. The exciter data are K A400 A 0.05 K E 0.17 E 0.95 K R1.0 R 0.0 2H 4.745 D 2.0pu ' 5.95 do The constants K 1K in 6u at the specifiedoperating point are K 11.4479 K 3 0.3072 K 5 0.0294 K 21.3174 K 41.8052 K 6 0.5257 We willexamine the rootlocusplotand time responseof V for a t stepchange inV .For REF a)G s 0 F sK F b)G sF 1 s F K F 0.04, 1.F5 Root locus without G (sF Root locus without G (sF Root locus with G (F) Root locus with G (F) Thesystem isclearly unstablefor this value of gain without the ratefeedback With G F s 0thesystem response asdiscussed in Ex7.7 is dominated by two pairsof complex rootsnear the imaginary axis.The pair that causes instability is determined by thefield winding and exciter parameters. Root-Locus Poles and Zeros Condition Zeros Poles (a) KF= 0 -0.2110 + j10.4525 -0.2732 -0.2110 - j10.4525 -20.0000 0.1789 -0.3502 +j10.7275 -0.3502 -j10.7275 (b) KF= 0.04 -0.4034 +j10.6930 -20.0000 -0.4034 -j10.6930 -0.3502 +j10.7275 -0.2110 +j10.4525 -0.3502 -j10.7275 -0.2110 -j10.4525 -0.2110 +j10.4525 -1.1972 + j0.8324 -0.2110 -j10.4525 -1.1972 - j0.8324 -1.0000 -0.2732 0.1789 Approximate System Representation • Dynamic performance is dominated by two pairs of complex roots that are particularly significant at low frequencies. • In this frequency range, damping is inherently low and stabilizing signals are needed to improve damping. As a result we develop an approximate model for the excitation system that is valid at low frequencies. The approximate system is composed of two subsystems – 1. Exciter field effects 2. Inertial effects K K2K 3 E G s 1 K K3K 6 E x K ' s K ' 1 E 3 do 3 do E s 2 1 K K K 1 K K K 3 6 E 3 6 E K 3 K6EIn allcasesof interest K 2 K 6 G sx E K '3 do 'do E 2 1 K K K s K K s 3 6 E 6 E K 2 E ' do E (*) 2 E K 3 do K K6 E s K ' s 3 do E 'do E K K 2 E 'do E 2 2 s 2 xx x x undamped natural freq. xdamping ratio K K6 E E K '3 do x ' x 2 K ' do E x 3 do E Wearemainly interestedin the system frequency of oscillatio n ascompared to .xhe damping is usxallysmalland the systemis poorly damped. The function G s xan bedetermined by measurement.Monitor terminal voltage whileinjecting a sinusoidalinput at the voltage regulator summing junction. The resulting amplitude and phase plotcan be used to identify G s .x Wecanalso estimateG s bycxlculation for a given operatingcondition.This procedurehassome drawbacks. • Gains and time constants may not be known precisely. •The model based on K -K1co6stants is load dependent and only applies to a one machine infinite bus system •Later I will provide a method for multimachine systems Estimate of G s x This approachassumes that the generator under study is connected to an equivalentinfinite bus of voltageV throug a transmissionlineof impedance Z R jX e e e This isassumed to be the The venin equivalentimpedanceseen at the generator terminals. FromY BUS GetY ii Ze 1 Yii The equivalentinfinite bus voltageV is calulatedby subtracting thedrop I Z fioe thegeneratorterminal voltageV , iiereI is tiegeneratorcurrent. • Please check example 8.6 and example 8.7 from the Text Book “” Power System Control and Stability , Anderson Supplementary Stabilizing Signals Wesaw earlier th at 1 2 K 2 K5 E K 3 K K6 E ' do E T d 2 2 1 K K6 E ' do E ' do E K 3 K 3 The voltageregulator introduces a damping torque component proportional to K 5 Under heavyloadingconditions K can be negative.Long 5 termstabilityisa concernunder such conditions. In Ex 8.7 we note that the excitation system introduces a large phase lag at low system frequencies just above the natural frequency of the excitation system. Thus the voltage regulator introduces negative damping. To counter this effect and to improve system damping, artificial means of producing torques in phase with speed are introduced. These are called “supplementary stabilizing signals” and the networks used to generate these signals are known as power system stabilizer networks. Stabilizing signals are introduced in excitation systems at the summing junction where the ref. voltage and the signal produced from the terminal voltage are added to obtain the error signal For any generator in the system the unit performance can be conveniently characterized by the block diagram shown below. Since the purpose of aP SS i t introduce a dam ping torque com po nent,a l gi alsin altou sefor controllng gene r tr excitaton isthe speed deviation . If te excitertransferfunc ton G EXs,and t ege nerat r transferfunction betw een E FD and Te we r pure gains,a directfeedb ack of w ould resul i a dam ping torque com po nent. However,in practiceboth thegeneratorand theexciter (dependingon thetype)exhibit frequencydependentgain andphasecharacteri stics. HencethePSS transferfunctionG s shosld have appropriatephasecompensation circuitstocompensate for thephaselag betweenexciterinput andelectricaltorque. In te idealc ase,w ih t ep hasec haracteriticof G ssb einga n exac ti verseo fthe exciterand gene r tr phase charac tristc t be com pe nsat d,the PSS w o ul resultin ap uredam p ig torque atallosc ilatig frequencies. T heg eneratorm od elassum ed neglects Am o risseur t sim pliy t erep r senttion. Damper windings havea significant effect on generator phase characteristicand should beconsideredin establishing the parameters of the PSS. Let us examine the principleof PSS operationbyconsidering a specificsystem. K 11.591, K 21.5, K 3 0.333, H 3.0 K ' 1.91, K 0.12, K 0.3, G s K 200 3 do 5 6 EX A Since iR verysmallin comparison to K ' wewil3nedoect its effect in examining PSS performance. E que to PSS isgiven by K K3 A E q K E6Vq s 1 K 3 sdo E q K 3 A 0.333200 66.66 V s 1 K 3 sdo K K K 3 A 6 11.91s .33200.3 1.91s 21 Examine the PSS phasecompensation required to producedamping torqueat a frequencyof 10rad/s, s j j10 Eq 66.66 Vs 21 j19.1 T T due to PSS K E due to PSS pss e 2 q at10rad/s T PSS K 66.66 1.5x66.66 3.522 42.3 V 2 21 j19.1 21 j19.1 s If TPSShas to bein phase with , he sinalshould beprocessed through a phase-lead network so that thesignalis advancedby 42.3 0 at a frequency 10rad/s. The amount of damping introduced dependson the gain of the PSS transfer function at that frequency. TPSS Gainof PSS at 10rad/s 3.522 TPSS Damping torque coefficient Gainof PSS at 10rad/s 3.522 Net TPSS due to AVR and PSS T PSS Gainof PSS at 10rad/s 3.522 AVR E K 3 K G Exs K K E q 1 K ' s 4 1 s 5 6 q 3 do R K K 1 s K G s E q 3 4 R 5 Ex s K 3' do R K ' 3 do R 1 K K3G s6 Ex Change in airgap torque due to E q T eE K E2 q q 0.60.333K K 0.012s E q 2 5 A 0.0382s 1.93s10.1K A 11.1 j0.18 T eEq at 10rad/s 17.18 j19.3 0.2804 0.3255 j Damping torque component due to E q T D q 0.3255 j s 0 T 0.3255 0 D q With 10rad/sthe damping torque coefficient T D q 0.3255602 Torque 10 12.27pu pu speedchange T PSS 12.27 Gainof PSS at 10rad/s 3.522 With a gain of 12.27 3.48 the PSS produces just 3.522 enough damping tocompensatefor negativedampingdue to theexciter.As PSS gain isincreasedamountof damping increases. The PSS is describedby G s K 0 0 1T s11T s 3 s 1T s01T s 12T s 4 Wecanin generalsplit these upinto thefollowing blocks Phase compensation Gain Washout sT 1sT 1 sT K 0 1 3 0 1 s0 1 sT2 1 sT 4 The phase compensation blocks provide the appropriate phase-lead characteristics to compensate for the phase lag between the exciter input and the generator electrical torque. The signal washout blockservesasa high -passfilteh wit time constant0 high enough to allowsignalsassociated withoscillatnsin to passunchanged. Without it,steady R changes inspeedwouldmodify terminal voltage. isin the 0 range of 1-20s. We willuse a simplifiedrepresentation used earlier toshow the effect of the PSS. The simplifiedmodel has paramete n,n defined earlier. undamped
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