Numerical Methods. (2) Here, P e (δ(t)) is given by the power flow equations:

Transcription

1 .0 Problem set-u Numerical Methods We reviously showed that we may write the swig equatio for a sigle machie as a air of first-order differetial equatios, accordig to: Re H P m P ( ( t e ( ( Here, P e (δ(t is give by the ower flow equatios: P ei E i G ii j ji E E i j Y ij cos( ij i j (3 where E i, E j are the iteral bus voltage magitudes havig agles of δ i ad δ j, resectively, ad Y ij =G ij +jb ij =Y ij /_θ ij is the elemet i the i th row, j th colum of the aroriate Y- bus. Deedig o the articular time iterval we are itegratig, the aroriate Y-bus will be Y, Y, or Y 3, corresodig to re-fault, fault-o, ad ost-fault coditios, resectively, as described i the otes called Multimachie.

2 he solutio to ( ad ( is foud by itegratig both sides of each equatio, resultig i t t Re ( t ( d Pm Pe ( ( d H 0 0 t ( t (0 ( d ( d 0 t 0 (4 (5 Note that δ aears i the itegrad of (4, ad that ω aears i the itegrad of (5. But these fuctios are what we are tryig to comute. Ad so we observe that i geeral, we will ot be able to obtai ay closed form exressio for δ ad ω, ad so we caot solve for them i closed form. hus, our oly choice is to resort to umerical itegratio. More comactly, we may defie x =ω ad x =δ, so that (4 ad (5 may be writte as x x f f ( x ( x, x, x x f ( x (6a

3 At this oit, we eed to recall that, if we wat to rereset additioal odes beyod the iteral machie odes (i.e., if we wat to rereset ay load buses, the differetial-algebraic system (DAS of the ower system electromechaical ositive-sequece timedomai simulatio roblem has aother set of equatios to deal with; the algebraic set, which is a fuctio of the algebraic variables ad the states. hus, our real roblem is stated as (6a lus the algebraic set, i.e., x = f (x, y 0 = g(x, y (6b where x rereset the state variables, y reresets the algebraic variables, ad 0=g(x, y reresets the algebraic equatios. Oe advatage we do have i seekig to solve (6b is that we do kow x(0, i.e., we kow the iitial agles ad seeds δ(0 ad ω(0, so it is a iitial value roblem (δ(0 3

4 comes from the solutio of the ower flow equatios, ad ω(0=0. Figure 0a, take from a very famous aer o this subject [], shows the so-called iterface roblem betwee the differetial ad algebraic equatios of the DAS. his roblem refers to the eed to solve both the ODEs of the system (o the left of Fig. 0a as well as the algebraic equatios of the system (o the right of Fig. 0a. Fig. 0a I geeral, there are two classes of methods for addressig the iterface roblem ad thus solvig the ODEs ad the algebraic equatios: Alteratig (or artitioed aroach: Here, we 4

5 . solve the differetial equatios usig values of the algebraic variables from the iitial ower flow solutio;. solve the algebraic equatios usig agles from the differetial equatios; 3. solve the differetial equatios usig values of algebraic variables from ste ; retur to ste. he alteratig method alies umerical itegratio to oly the discretized ODEs (ad ot to the origial algebraic equatios. For this reaso, the alteratig method may use either exlicit itegratio (which itegrates oly ad so caot iclude the algebraic equatios or imlicit itegratio (which itegrates by solvig oliear algebraic equatios ad so ca iclude the algebraic equatios. Direct solutio method: Here, we combie the discretized ODEs (which are algebraic equatios ad the origial algebraic equatios ito a sigle set of algebraic equatios. his sigle set of algebraic equatios are oliear, ad so a oliear solver such as Newto-Rahso ca be used to solve them. Because the direct solutio method solves together the 5

6 discretized ODEs ad the algebraic equatios, it caot use exlicit itegratio; rather, it must use imlicit itegratio. Figure 0b illustrates the various desig decisios associated with buildig a time-domai simulatio software alicatio. he to attribute (hardware is whether oe wats to arallelize the comutatios or ot. he secod layer is whether oe uses alteratig or direct solutio methods. he third layer is whether oe uses exlicit or imlicit itegratio methods (we will discuss this further i these otes. he fourth layer deicts the tye of oliear solver chose. he fifth layer shows that a liear equatio solver is also eeded, because most oliear solvers require oe. Figure 0b 6

7 owards the ed of these otes (Sec 5., , we retur to the toic of solvig ODEs vs. algebraic equatios, while exlaiig how to treat the etwork whe retaiig odes beyod those reresetig iteral machie odes. Regardless of whether we use the alteratig aroach (with either exlicit or imlicit itegratio or the direct solutio aroach (with imlicit itegratio, we must solve ODEs of the roblem osed i (6b (reeated here for coveiece: x = f (x, y 0 = g(x, y (6b Ad so let s look at a simler iitial value ODE roblem i oe-dimesio (i.e., a sigle state variable: So we wat to solve x f ( x( t, x(0 x0 (7 t x ( d x( t f ( x( d 0 t 0 (8 7

8 If we are goig to try a umerical solutio, i.e., oe usig comuters, the we must deal with discrete time. I discrete time, (8 becomes: x( k k 0 f ( x( k d k f ( x( k d 0 x( k k f k ( x( k d (9 From (9, we ca write: x( k x( k k f k ( x( k d (0 Equatio (0 says the followig: If we wat to kow x at the ext ste, k, we eed to kow x at the last ste, k-, ad we eed to kow the itegral i (0. his itegral is givig us the chage i x from the last ste to the ext, i.e., x k f k ( x( k d ( 8

9 Sice we do kow x(0, we ca say that we kow x at the last ste. hus, solvig our roblem requires oly the ability to comute the itegral i (. here are may methods that will allow us to comute the itegral i (. We will review oly a few of them. You should read Aedix Sectio B. i your text as backgroud material for this toic..0 Euler method Cosider lottig our fuctio f(x(t as a fuctio of t. What we wat to do, based o (, is to obtai the area uder the curve of f(x(t from t=k- to t=k, as illustrated i Fig.. 9

10 f(x(k f(x(t f(x(k- k- k Fig. Here is a roosed aroach: Assume that f is costat throughout the iterval at f(k-. his assumtio is clearly ot good if k is large, but it might be reasoable if k is made small eough. his aroach aroximates the itegral as the area shaded by the vertical lies i Fig.. Oe observes that it misses the area shaded by the horizotal lies i Fig.. 0

11 f(x(k f(x(k- f(x(t error k- k Fig. Aalytically, (0 becomes x( k x( k f ( x( k ( where the last term is x. his aroach is also called the forward rule because we assume a value for f ad hold it costat as we move forward i time.. Alterative develomet of forward rule We may also develo the forward rule i aother way.

12 Assume that we kow x(k- ad that is chose small eough so that x(k is close to x(k-. he by aylor series, x( k x( k x( k x x tk tk x ( tk 3! x tk... (3 where O( is the remaider of the aylor series, ad its argumet idicates that the lowest ower of reset i the remaider is. If is small eough, O( is egligible ad x( k x( k x t (4 k where the last term is x. his is illustrated i Fig. 3.

13 x(k x(k- x(t error k- k x f (x Fig. 3 Recallig that, (4 becomes x( k x( k f ( x( k (5 which is the same as (, our forward rule. From both Figs. ad 3, we ca observe that the forward method will icur some error, ad this error will icrease with larger values of. 3

14 Oe major roblem with this method is that if is too large, the error will roagate from oe time ste to aother. his meas that error icurred at oe time ste k will add to the error icurred at later time stes. Solutio methods where this ca hae are said to be umerically ustable. However, occurrece of this heomeo does deed o our choice of. I will rovide you with a way to cosider this issue.. Numerical stability of forward rule he followig develomet may be hard to follow if you have ot had a course i discrete-time systems ad cotrol. ISU offers such as course as EE 576. he text by Frakli & Powell is a well-kow text i this area. I will state some Facts, without roof, that we eed i order to cosider umerical stability of itegratio schemes. his will give you a framework to cosider this issue. If you wat more deth, take EE 576, or read the Frakli & Powell book, or read a similar oe. 4

15 Fact : A trasfer fuctio i Lalace, H(s corresodig to cotiuous time imulse resose h(t, may be obtaied for a dyamical system. Fact : he eigevalues of the system are the oles of the trasfer fuctio, i.e., the roots of the deomiator of H(s. Fact 3: he system is stable if all oles are i the left-hadlae. Fact 4: We may discretize H(s i a umber of differet ways, ad the articular way of discretizatio will secify the maig betwee the Lalace variable s (used i the trasform H(s of the cotiuous-time fuctio h(t to the z-domai variable z (used i the trasform H(z of the discrete-time fuctio h(k. Secifically, use of the forward itegratio rule corresods to fidig a z- domai trasfer fuctio H ( z H ( s z s (6 where H(z is the z-trasform of the system s discrete time imulse resose (see Frakli & Powell,

16 Fact 5: Whereas the cotiuous-time system is stable if all oles of H(s are i the left-half-lae, the discrete-time system is stable if all oles of H(z are withi the uit circle. his is because z/(z-γ corresods to a discrete-time ole at γ, which has a z-trasform of: z z k u(k (7a where u(k is the uit ste-fuctio such that 0, k < 0 u(k = {, k 0 (7b From (7a, we see that if γ >, the time-domai term will go to ifiity as time icreases. his would idicate ustable behavior. Fact 6: From Fact 4, a ole of H(s, call it s, mas to the z- lae accordig to: s z z s (8 Coclusio: he stability of the discrete-time system (as idicated by whether all z are withi uit circle deeds o 6

17 he stability of the cotiuous time system through the locatio of cotiuous time system eigevalues, s, ad he time ste. Let s look at how a eigevalue s mas to the z-lae through the fuctio (8. s-lae j*im(s=jω z =+s z-lae j*im(z s σ=re(s jω σ - SABLE Re(z SABLE - Fig. 4 From Fact 5, we recall that if all oles z are withi the uit circle, H(z is stable. So this imlies that for stability, the oles must ma to the z-lae so that 7

18 z (9 But the forward rule corresods to a maig of z (0 s Substitutio of (0 ito (9 yields: But s ( s j ( Substitutio of ( ito ( results i Or j (3 j (4 akig the magitude of the comlex umber withi (4 (5 Squarig both sides results i 8

19 (6 Observe, i (6, that if σ >0 (a right-half-lae ole!, the (5 will be violated. Sice (5 must be satisfied for the discrete-time-system to be stable, we see that a ustable cotiuous time system ecessarily imlies a ustable discrete time system. his is a good thig, because we would ot be hay if our itegratio method could stabilize a ustable simulatio. Ad so we are o loger iterested i ustable cotiuous-time systems sice we kow our itegratio method will also show them to be ustable, as desired. What we are iterested i are the stable cotiuous time systems. Is it ossible for our itegratio method to cause them to be ustable? Ad so we will assume ow that σ <0 (imlyig a stable cotiuous time system. Exadig (6, we get (7 Subtractig off from both sides, ad factorig out a, 9

20 0 (8 Now will always be ositive (it is ste size. herefore for (8 to be true, it must be the case that Solvig for results i 0 (9 (30 Equatio (30 must be satisfied i order to be sure that the forward itegratio method does ot create umerical istability. But there are may eigevalues! Which oe to choose? he aswer is to choose the oe with the smallest ratio corresodig to the right-had-side of (30. his meas we wat the eigevalues with small σ ad large ω. For these kid of eigevalues, it will be the case that σ << ω. herefore, (30 collases to 0

21 (3 hese eigevalues are tyically the fast modes, usually associated with the excitatio system. hese eigevalues are closely related to the time costats of the cotrol systems. You ca test out the above theory with a stability rogram that uses the forward itegratio rule by decreasig the time costat of the excitatio system for oe of your geerators, keeig the time ste fixed. At some oit, you will see istability. he begi decreasig your time ste, ad the you will see it stabilize! he cost of decreasig the time ste is that it icreases the comutatio time. here are other ways of imrovig umerical erformace of a itegratio method. We will look at two of these.. Reduce the error: we will look at two methods of doig this.. Use a imlicit itegratio method: we will look at two methods of doig this.

22 3.0 Reducig the error wo algorithms that imrove o the forward (Euler method are redictor-corrector ad Ruge-Kutta. We will look at both briefly. 3. Predictor-corrector method his method is called the modified Euler i your text. he idea here is that we will take a ste to comute x(k (a redictor ad the we will use that calculatio to recalculate that same ste (the corrector. Ste : Predict x(k usig Euler to get x(k: I (3, the exressio uder f is x-dot of k-. x ( k x( k f ( x( k x ( k (3 Ste : Use x (k to obtai a corrected value x c (k: a. Get a corrected derivative f c as the average of the derivatives at x(k- ad x (k: f c f ( x( k f ( x ( k (33

23 b. he aly the forward rule: x c ( k x( k x( k f c f ( x( k f ( x ( k (34 his method is illustrated i Fig. 5. x t k 4 Ad you get this oe. Average these two sloes x (k x(k- x(t 3 k- k Fig. 5 Uderstadig the method is facilitated by observig the sequece of oits i Fig. 5. he derivative f(x(k- is obtaied at oit. he redicted oit x (k is obtaied at oit. he derivative of the redicted oit f(x (k is 3

24 obtaied at oit 3. he fial oit, oit 4, is obtaied from averagig the two obtaied derivatives f(x(k- ad f(x (k. Oe ca observe ituitively from Fig. 5 that the error will be reduced. his method ca be show to be equivalet to cosiderig u to the secod derivative term i the aylor series, therefore the error is of O( 3. his is a sigificat imrovemet over the Euler method, but it is still a umerically ustable algorithm. I other words, for the redictor-corrector method, for a give miimum eigevalue, ca be larger tha it ca be i the Euler method, but it is still true that the algorithm may be ustable if is too large. 3. Ruge-Kutta method (roouced Ru-gah Kut-tah his algorithm was develoed i 895, ad it also alies the idea of averagig, similar to redictor-corrector, but i a slightly differet way. here are differet R-K algorithms of differet order. We will oly study oe of them, the 4 th order R-K. 4

25 he 4 th order R-K method requires, at each successive time ste, comutig 4 differet icremets x j, as follows: x K Icremet x j f ( x( k Derivative used Start-oit derivative oly x K f x ( k K First iterior derivative x x 3 K K 3 f f x ( k 4 4 K x( k K 3 Secod iterior derivative Arox. edoit derivative Note the followig about the K i s.. K i is always used to comute K i+.. Each K i is ot a derivative but rather a icremet i the itegratio variable, i.e., xi Ki xi ( k x( k (35 5

26 Ay of the K i s could be used to obtai the ew value x(k. 3. Use of K to obtai the ew value x(k is equivalet to the Euler method. 4. he derivatives are comuted at four differet locatios i the iterval: he begiig of the time ste x(k- he first iterior oit x(k-+k / he secod iterior oit x(k-+k / he aroximate ed-oit oit x(k-+k 3 Figure 6 below illustrates the sequece of calculatios, which ca be uderstood by followig the sigle arrows from oit to oit to oit 3, ad the the double arrows from oit 3 to oit 4 to oit 5, ad the the trile arrows from oit 5 to oit 6 to oit 7 to oit 8. 6

27 Fig. 6 7

28 Oce each of the icremets K -K 4 are comuted, the the fial icremet is obtaied by takig a weighted average of the four icremets, where the middle icremets are weighted heaviest, accordig to (36. x 6 K K K 3 K 4 (36 he middle icremets (K ad K 3 are weighted heaviest as they are comuted based o sloes that will be more reresetative of the sloe i the iterval tha the begiig (K ad fial (K 4 icremets. he R-K method ca be show to be equivalet to cosiderig u to the fourth derivative term i the aylor series, therefore the error is O( 5. Although this is a sigificat imrovemet over the Euler or the P/C method, R-K is also a umerically ustable algorithm therefore the stability domai, although elarged relative to the uit circle of the Euler, is bouded, as show i the aedix. 8

29 4.0 wo geeral tyes of itegratio methods We have so-far exlored the Euler method, the P/C method, ad the R-K method of umerical itegratio. hese are i a class of itegratio methods called exlicit methods. hey are so-called because they evaluate x(k exlicitly as a fuctio of values at revious stes ad their derivatives. All exlicit methods are umerically ustable, i.e., they have a bouded stability domai. Aother class of itegratio methods that does ot have this roblem is called imlicit methods. We will study two of these: Backwards rule ad raezoidal rule. Imlicit methods require a value of the fuctio x(k at a future time ste. o get this, we eed to erform a extra ste. Imlicit methods are good for stiff roblems, where exlicit methods must utilize small ste-sizes to work. Stiff roblems are ofte characterized by large differeces betwee the real arts of system eigevalues. 9

30 5.0 Backwards rule Recall that our itegratio roblem is characterized by (0, reeated here for coveiece. x( k x( k k f k ( x( k d (0 where we have the rst term (because we kow the iitial value ad eed to evaluate the d term, which corresods to the area uder the curve of f vs. time, as illustrated i Fig., reeated here for coveiece. f(x(k f(x(t f(x(k- k- k Fig. 30

31 I the forward (Euler method, we assumed f is costat throughout the iterval at f(k-. his was simle, ad coveiet, sice we already kew f(k-. Now we will agai assume f is costat throughout the iterval. his time, however, we will assume that it is costat at f(k istead of f(k-, as illustrated i Fig. 8. f(x(k f(x(t error f(x(k- k- k Fig. 8 he, (0 becomes x( k x( k f ( x( k (37a 3

32 Of course, (37a is still a aroximatio, but this time it icludes area as idicated i Fig. 8 (as oosed to the error of the Euler method as idicated by Fig.. But ote a roblem with this method: we do ot kow x(k!!!! If we do ot kow x(k, how do we evaluate f(x(k? he aswer to this questio lies i observig that (37a is ot a differetial equatio, but rather it is a algebraic equatio, ad there is oly oe ukow, x(k. herefore we may solve it! Of course, we will eed to accout for the fact that there is o reaso to assume f is liear ad i fact, most of the time, f is oliear. herefore we will eed to use a oliear algebraic solver to solve it. he Newto-Rahso is oe such solver with which we are familiar. o aly it, let s rewrite (37a as x(k x(k f(x(k = 0 (37b ad the defie: F ( x( k x( k x( k f ( x( k 0 (38 3

33 Aly aylor series exasio u to the first derivative, yieldig F( x( k x( k F( x( k F( x( k x( k 0 (39 he derivative F (x(k is the derivative of F with resect to x, ot t. Solvig (39 for x(k results i F( x( k F( x( k x( k (40 I Newto-Rahso, we must guess a iitial solutio, ad the we use (40 to iterate. I the multi-dimesioal case, (40 becomes J ( x( k F( x( k x( k (4 where J is the Jacobia give by See Kudur, for more o this. 33

34 J F x F x F x F x x xx( k (4 Of course, (4 is solved usig a liear solver such as LUdecomositio, based o J ( x( k x( k F( x( k (43 We address three issues regardig this method: liear solvers, etwork solutios, ad umerical istability. 5. Liear solvers [] A imortat observatio about imlicit methods is that (43 is just a roblem i the form of Ax=b ad therefore simly requires a liear solver to obtai the aswer. Although solutio to simultaeous liear equatios is a very old toic, because imlicit methods devote over 90% of their comutatio to this roblem, it is still a very imortat toic for roblems that require high comutatioal seed. 34

35 Researchers have ut much effort ito writig comutatioal libraries for erformig solutio of liear equatios. he eole who write these libraries kow much more about solvig liear equatios tha you do. Never ivert the matrix; Always use the libraries; Never write your ow method. here are a umber of stadard, ortable solver libraries available, icludig: BLAS (Basic liear algebra subrograms: May liear algebra ackages icludig LAPACK, ScaLAPACK ad PESc, are built o to of BLAS. Most suercomuter vedors have versios of BLAS that are highly tued for their latforms. ALAS (Automatically ued Liear Algebra Software ackage is a versio of BLAS that, uo istallatio, tests ad times a variety of aroaches to each routie ad selects the versio that rus fastest. ALAS is substatially faster tha the geeric versio of BLAS. 35

36 LAPACK (Liear Algebra PACKage solves dese or secial case sarse systems of equatios deedig o matrix roerties such as: o Precisio: sigle, double o Data tye: real, comlex o Shae: diagoal, bidiagoal, tridiagoal, baded, triagular, traezoidal, Heseberg, geeral dese o Proerties: orthogoal, ositive defiite, Hermetia (comlex, symmetric, geeral. LAPACK is built o to of BLAS, which meas it ca beefit from ALAS. LAPACK is a library that you ca dowload for free from ScaLAPACK is the distributed arallel (MPI versio of LAPACK. It cotais oly a subset of the LAPACK routies. ScaLAPACK is also available from PESc (Portable, Extesible oolkit for Scietific Comutatio is a solver library for sarse matrices that uses distributed arallelism (MPI. It is desiged for geeral sarse matrices with o secial roerties, but it also works well for sarse matrices with simle 36

37 roerties like badig & symmetry. It has a simler Alicatio Programmig Iterface tha ScaLAPACK. Whe choosig a solver, ick a versio that s tued for the latform you re ruig o, ad use the iformatio that you have about your system to select the oe that will be most efficiet. You will have to do some research ad discuss with eole to gai a level of kowledge to eable you to most effectively make this selectio. Figures 9 ad 0 illustrate the relatio betwee imlicit itegratio methods, oliear solvers, & liear solvers. Fig. 9 37

38 Fig Network solutios his sectio recas iformatio give o. 3-5 of these otes. I our roblem formulatio, because we elimiate all odes excet machie iteral odes, we are able to kow all ode voltage magitudes ad thus write the etire roblem (swig dyamics ad etwork equatios as a sigle set of state equatios where the oly ukows are the states (agles ad seed. 38

39 However, if we rereset load as somethig other tha costat imedace, the we may wat to retai idetity of these buses. But we will ot kow the bus voltage magitude (or the agle, yet, because there is o swig dyamics at this bus, there will be o state equatio for it. We will see later that the solutio to this roblem is to write a set of algebraic equatios for the etwork. hese equatios, essetially the Y-bus relatio, are the solved together with the state equatios of the swig dyamics. I the exlicit itegratio methods, the oly way to do this is through what is called a artitioed aroach where the state equatios are first solved ad the the etwork equatios are solved searately. he reaso why this is the oly way to do this is because the two sets of equatios (ODEs ad algebraic equatios must use differet methods for solutios. Oe roblem ecoutered by the artitioed aroach is referred to as the iterface error, where the solutio to the state equatio is ot quite cosistet with the solutio to the etwork equatios. 39

40 he artitioed aroach may be used with imlicit itegratio methods also. I this case, imlicit methods will also ecouter the roblem of iterface error. I additio, imlicit itegratio methods eable a secod aroach called the direct solutio (or simultaeous aroach, where the etwork equatios are embedded i (43 ad solved at the same time as the state equatios. his very coveiet method elimiates iterface error. 5.3 Numerical stability for backwards rule he followig develomet is very similar to that give i Sectio.. Agai, additioal readig may be foud i the the text by Frakli & Powell. I will agai state facts, ad i fact Facts -3 ad 5 are exactly as before. his aalysis differs oly i Facts 4 ad 6. Fact : A trasfer fuctio i Lalace, H(s corresodig to cotiuous time imulse resose h(t, may be obtaied for a dyamical system. 40

41 Fact : he eigevalues of the system are the oles of the trasfer fuctio, i.e., the roots of the deomiator of H(s. Fact 3: he system is stable if all oles are i the left-hadlae. Fact 4: We may discretize H(s i a umber of differet ways. Whereas use of the forward itegratio rule corresods to fidig a z-domai trasfer fuctio H ( z H ( s z s (6 where H(z is the z-trasform of the system s discrete time imulse resose (see Frakli & Powell, , use of the backwards itegratio rule corresods to fidig a z-domai trasfer fuctio H ( z H ( s z s z (44 Fact 5: he discrete-time system is stable if all oles of H(z are withi the uit circle. his is because z/(z-γ 4

42 corresods to a discrete-time ole, which has a z- trasform of: z z k u(k (7 From (7, we see that if γ >, the time-domai term will go to ifiity as time icreases. Fact 6: From Fact 4, whereas a ole of H(s, call it s, mas to the z-lae i the forward rule accordig to: s z z s (8 i the backwards rule, a ole of H(s, s, mas to the z- lae accordig to: s z z z s (45 Coclusio: For the forward rule, the stability of the discrete-time system deeds o 4

43 he stability of the cotiuous time system through the cotiuous time system eigevalues, s, ad he time ste. For the forward rule, we cocluded that the deedece o was that had to satisfy (30 he questio we must aswer ow is: For the backwards rule, (a does stability of the discrete-time also deed o, ad (b if so, i what way? Let s take the same aroach by lookig at how a eigevalue s mas to the z-lae through the fuctio (45. Our real questio here is: what are the coditios o to force all left-had-lae eigevalues of H(S to ma ito the uit circle of the z-lae? 43

44 s-lae j*im(s=jω z =/(-s z-lae j*im(z s σ=re(s x σ jω - SABLE Re(z SABLE - Fig. From Fact 5, we recall that if all oles z are withi the uit circle, H(z is stable. So this imlies that for stability, the oles must ma to the z-lae so that z (9 But the backwards rule corresods to a maig of z s (46 44

45 45 rick: Add ad subtract ½ to the right-had-side: s z (47 Gettig commo deomiator for the term i brackets: s s s s z ( (48 Now substitute s =σ +jω to obtai j j z (49 Now collect real ad imagiary terms to get j j z ( ( (50 Now we will use this rule: If for three vectors a, b, ad c, a=b+c, the a < b + c. Alicatio of this rule to (5 results i

46 46 ( ( ( ( z (5 Defie A as the square-root term i the umerator of (5; B as the square-root term i the deomiator, i.e., ( ( ( ( B A (5 he (5 becomes: B A z (53 Now cosider oe eigevalue havig σ P <0, corresodig to a stable ole of the cotiuous time system. he (5 idicates that B>A>0. he 0<A/B<. he the right-had-side of (53 must be i [0,, i.e., B A z (54

47 Ad so the discrete-time-system must be stable. he imlicatio is that umerical itegratio usig the backwards rule of a stable cotiuous-time-system will result i a stable discrete-time-system, ideedet of choice of. Likewise, we ca cosider the case of havig oe eigevalue havig σ P >0, corresodig to a stable ole of the cotiuous time system. But i this case, it will ot be good eough to simly show that we must evaluate z ca be greater tha ; we will eed to show that z > so that we ca be certai a ustable cotiuous time system results i a ustable discrete time system. Doig so results i z ( (55 Here we ca see that, for σ P >0, if σ P <, the the umerator is greater tha the deomiator, ad z >. he coditio σ P < is easy to satisfy i ractice, sice ustable eigevalues tyically have very small real arts. 47

48 Equatio (55 may also be used to verify our earlier coclusio: for σ P <0, the deomiator is always greater tha the umerator, ad z <. 6.0 raezoidal rule he traezoidal rule is a imlicit method that aroximates the area uder the curve of the secod term below x( k x( k k f k ( x( k d (0 usig the formula for the area of a traezoid, as: A ( a b (56 a b Fig. 48

49 Figure 3 illustrates the alicatio of the traezoidal method. f(x(k f(x(t error f(x(k- k- k Fig. 3 herefore (0 becomes x( k x( k f ( x( k f ( x( k (57 49

50 Discretizatio usig raezoidal itegratio corresods to fidig a z-domai trasfer fuctio of H ( z H ( s s z z (58 (usti s method, or the biliear trasformatio. his corresods to the maig z s s / / (59 as idicated i Fig. 4. z =(+s //(- s / s-lae z-lae j*im(s=jω j*im(z s σ=re(s x jω σ - SABLE Re(z SABLE - Fig. 4 50

51 5 We may agai show that the traezoidal itegratio aroach is umerically stable. Aedix: Stability Domai Aalysis of Itegratio Methods Assume that the geeral form of a differetial equatio is, ( x t f x. We liearize f i its eighborhood as follows., (, ( ( (, ( x t x t x f x x t f t t x t f x ( he high order items ca be omitted. Let, ( t x x f, ad we ca get x t x t f t t x t f x x, ( (, ( ( If 0, we ca do trasformatio x t x t f t t x t f x x, ( (, (, hus ( ca be trasformed to x x (3 Exressio (3 is usually called test equatio (see the defiitio as follows. For a set of differetial equatios, i are the egevalues of Jacobia matrix. Now we aly a umerical method to exressio (3, for examle Forward Euler method. x z R x h R x h x h x x ( ( ( (4 where h z. Assume that there is disturbace o x, ad the resultig disturbace o x is.

52 he, R( z If we wat, we just eed R( z. Defiitio 3 : he fuctio R(z the umerical solutio after oe ste for is called the stability fuctio of the method. It ca be iterreted as x x, with x 0, z h, the famous Dahlquist test equatio. he set S z C ; R( z is called the stability domai of the method. I order to aalyze the umerical stability of -order Exlicit Ruge-Kutta(ERK method, we just eed to calculate their stability domai. Coclusio 3 : If the Exlicit Ruge-Kutta is of order, the z z R( z z O( z!! he stability domai of first to fourth order Exlicit Ruge-Kutta methods is show i the figure below. 5

53 Stability domai of -order Exlicit Ruge-Kutta Similarly, we ca fid stability domai of some imlicit methods, as geeralized i below table. Numerical Method Formula Exressio Stability Fuctio Backward Euler x x hf x ( raezoidal rule x x f ( x f ( x h R( z z z / R( z z / Imlicit Midoit rule x x hf ( x x z / R( z z / 53

54 For ower system domai simulatio, raezoidal rule is usually used. It ca be see from its stability fuctio that the stability domai of raezoidal rule is the whole left comlex domai. [] B. Stott, Power system dyamic resose calculatios, Proceedigs of the IEEE, Vol. 67, No., Feb., 979. [] 007 resetatio slides from Paul Gray, Uiversity of Norther Iowa, Hery Neema, Uiversity of Oklahoma, Charlie Peck, Earlham College. [3] E. Hairer, G. Waer, Solvig ordiary differetial equatios II: stiff ad differetial-algebraic roblems, Sriger-Verlag,