REDUCED-ORDER models of synchronous machines

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1 A Library of Secon-Orer Moel for Synchronou Machine Olaoluwapo Ajala Stuent Member IEEE Alejanro Domínguez-García Member IEEE Peter Sauer Life Fellow IEEE an Daniel Liberzon Fellow IEEE Abtract Thi paper preent a library of econ-orer reuce moel for ynchronou machine that can be utilize in controller eign an power ytem analyi ta. The reuce moel are evelope through the ytematic reuction of a nineteenth-orer moel uing ingular perturbation techniue an the ynamic tate of the reulting moel are the power angle an angular pee. Uing two tet cae the reuce moel are valiate by comparing their voltage freuency an phae profile with that of the high-orer moel an that of the ocalle claical moel. Inex Term Synchronou machine euce-orer moeling Singular perturbation analyi. I. INTODUCTION EDUCED-ODE moel of ynchronou machine fin application in power ytem analyi control eign an eucation with each application reuiring implifie moel that capture ynamical phenomena relevant to the intene ue. Thi ha le to the proliferation of ynchronou machine moel in the literature [ 3 4] with varying egree of complexity computational cot an tate-pace imenion. One of uch moel i the o-calle claical moel avocate in [5] an [6] a econ-orer ynamic moel that capture ynamic of the machine phae an angular pee. Analytically the claical moel i the implet ynchronou machine moel but it ha certain limitation that retrict it application to firt wing tability analyi i.e. tability analyi for the firt 0 econ [7 8 9]. A a reult if we conier that a power ytem may be table in the firt wing but untable in ubeuent wing it i clear that the claical moel though imple i unreliable for analyi an control eign ta extening beyon a one econ time interval. For example the eign of a generator ynchronization metho reuire a moel that capture ynamic of the generator phae freuency an voltage magnitue over the entire ynchronization perio. A econ-orer moel uch a the claical moel houl uffice but the firt wing tability contraint coul mae it inapplicable if the ynchronization perio i greater than one econ. There i therefore a nee to evelop moel that poe the implicity of the claical moel but alo the temporal breath that it lac. The main contribution of thi paper i the evelopment of econ-orer ynchronou machine moel that when compare to the claical moel have the ame tate-pace imenion are ignificantly more accurate over a long time interval The author are with the Department of Electrical an Computer Engineering Univerity of Illinoi at Urbana-Champaign Urbana IL 680 USA. {ooajala alean pauer liberzon}@illinois.edu. an are ueful for a broaer range of application. Uing ingular perturbation analyi a our main tool [ 0 3] the econ-orer moel preente in thi paper are erive by ientifying the fatet ynamic tate in a high-orer moel eveloping approximate manifol euation for them which are algebraic euation an replacing the ifferential euation for thee tate with the algebraic counterpart. Our approach to eveloping the propoe machine moel i bae on evelopment in [ 3 4] where zero-orer an firt-orer approximation of manifol for fat ynamic tate are ue to evelop reuce-orer moel. In [3 4] the ue of integral manifol for moel orer reuction i introuce with ome application preente an in [] the techniue i ue to evelop the two-axi moel the one-axi moel an the claical moel. The remainer of the paper i organize a follow. In Section II we preent the high-orer moel an the claical moel of a ynchronou machine. In Section III we evelop a library of econ-orer moel from the high-orer moel uing ingular perturbation analyi. Finally in Section I we valiate the econ-orer moel evelope uing numerical example an in Section we comment on implication of the preente reult. II. PELIMINAIES We begin thi ection by preenting a high-orer moel of the ynchronou machine aopte in thi wor. In aition the time-cale propertie of the moel are icue. Afterwar we introuce the o-calle claical moel an ecribe how it can be evelope from the high-orer moel. A. High-Orer Synchronou Machine Moel The high-orer ynchronou machine moel we ecribe in thi ection i bae on the evelopment in [ 3]. The component inclue in the moel are: i three amper wining ii a woun-rotor ynchronou machine iii an IEEE type DCA excitation ytem [5] an iv a Woowar ieel governor DEGO [6] couple to a ieel engine which act a the prime mover. Next we provie mathematical expreion that ecribe the ynamic behavior of thee component. [Note that the moel i preente utilizing the 0 tranformation with all parameter an variable cale an normalize uing the per-unit ytem]. Aumption. The ynchronou machine i connecte to an electrical networ bu through a hort tranmiion line.

2 Damper wining moel: Let Φ t an E t enote the flux linage of two amper wining aligne with the uarature axi -axi of the ynchronou machine let Φ t an E t enote the flux linage of a amper wining an a fiel wining repectively aligne with the irect axi -axi of the ynchronou machine an let I an I enote the -axi an -axi component of the tator output current repectively. Then the amper wining ynamic can be ecribe a follow: τ Φ = Φ X X I E τ Φ = Φ X X I E t τ Ė = E X X I X X X X Φ X X I E where X enote the machine leaage reactance X enote the machine tator reactance X an X enote machine tranient reactance an X enote the machine ub-tranient reactance an τ = ω X 0 X m τ = X X mx f an τ = X X m are time contant with X X X f X enoting leaage reactance X m an X m enoting mutual reactance an enoting wining reitance. Stator wining an networ moel: Let Φ t an Φ t enote the -axi an -axi component of flux linage for the tator wining repectively let Φ e t = X e I an Φ e t = Xe I enote the -axi an -axi component of flux linage for the electrical line repectively let ω t enote the machine angular pee in electrical raian per econ an let δ t enote the power angle of the ynchronou machine in electrical raian. At the electrical networ bu let l an δ l enote the voltage magnitue in per unit an the voltage phae relative to a reference frame rotating at the nominal freuency in electrical raian repectively. Let l := l co δ δ l l := l in δ δ l Φ t := Φ t Φ e t Φ t := Φ X m X f t Φe t. Then the tator wining an networ ynamic are ecribe by: δ = ω t Φ = ω t Φ = ω t ω Φe 0 ω Φe 0 Φ l e I Φ l e I = e I ω t Φ e l = e I ω t Φ e l ω 0 Φ = X e I X X X X Φ X X X X E Φ = X e I X X X X Φ X X X X E t 3 where X e := X X e an X e := X X e X e enote the per-phae line reactance X enote a machine ub-tranient reactance e enote the per-phae line reitance enote the per-phae tator reitance an enote the nominal freuency in electrical raian per econ. 3 Excitation ytem moel: Let E f t enote the output voltage of the machine excitation ytem let U f t enote the exciter control input let Ū f t enote the rate feebac variable of the voltage regulator an let :=. Aumption. The effect of magnetic aturation on the machine excitation ytem are negligible. Then the ynamic of the machine excitation ytem can be ecribe a follow: τ Ė = E X X I X X X X Φ X X I E E f τ f Ė f = E f U f τ u U f = U f K u Ū f K u K u E f K u r Ū f = Ūf K u E f where r enote the reference voltage magnitue τ = X f f τ f = L f K g = f K g = LtLm t Ku = Nt L m N t t X enote the machine tator reactance τ u enote the amplifier time contant K u enote the amplifier gain X f enote the fiel wining reactance f enote the fiel wining reitance L f enote the unaturate fiel inuctance K g enote the lope of the unaturate portion of the exciter aturation curve f enote the exciter circuit reitance L t an L m enote erie an magnetizing inuctance of the tabilizing tranformer which i ue to tabilize the excitation ytem through voltage feebac [] repectively t enote the erie reitance of a tabilizing tranformer an Nt N t enote the turn ratio of the tabilizing tranformer. 4 Prime mover an pee governor moel: Let T m t enote the mechanical torue output of the machine. For the pee governor ytem let P a enote the output of it actuator with P a = P a an let P b enote the output of it electric control box with P b = P b. Let P u = P a τ 4 P a enote the valve poition of the ieel engine which act a the prime mover. Then the pee control ytem of the ynchronou machine can be expree a follow: M ω = T m Φ ti Φ ti D 0 ω τ m T m = T m P u τ a P a = Pa κ P b τ 3 P b P a τ 5 τ 6 τ P b = P b P c P u ω τ D 0 P b 4 5

3 3 where τ τ 3 τ 4 τ 5 an τ 6 enote time contant of the control ytem τ a = τ5τ6 τ 5τ 6 κ enote a controller gain for the actuator P c enote the power change etting of the machine M enote the inertia of the machine D0 enote the friction an winage amping coefficient of the machine τ m enote the time contant of the engine an D 0 = D with D enoting the roop coefficient. [Note that for alient pole machine X = X o that E t = 0 an for rounrotor machine X = X ]. B. High-Orer Moel Time-Scale Propertie The following obervation are bae on tanar parameter value obtaine from ynchronou machine moel in [ 3 6] an an eigenvalue analyi of thee moel. Obervation. The ynamic of Φ Φ E Φ Φ Φ e Φ e E E f U f Ū f T m P u P a P b P a an P b are much fater than thoe of ω an δ. Obervation. For ɛ = 0. enoting a contant the parameter τ τ τ f τ u τ m τ a τ τ τ 5 τ 6 τ 5τ 6 τ 5τ 6 κ D are O ɛ. Bae on thee obervation the nineteenth-orer machine moel ecribe by 5 can be expree compactly a: ẋt = f xt zt ɛ x0 = x 0 ɛżt = g xt zt ɛ z0 = z 0 6 [ ] [ where xt = δ ω an zt = Φ Φ E Φ Φ Φ e Φ e E E f U f Ū f T m P u P a P b P a P b ]. In the remainer of thi paper we refer to the element of zt a the fat tate an element of xt a the low tate. Other obervation which will prove ueful in Section III-B an III-C are: Obervation 3. The ynamic of Φ Φ Φ e an Φ e much fater than thoe of Φ Φ E an E. are Obervation 4. The ynamic of Φ an Φ are much fater than thoe of E an E. C. Claical Moel The claical moel of a ynchronou machine i a econorer moel whoe formulation i bae on the following aumption [7]: i the machine can be moele a a contant magnitue voltage ource with a erie reactance ii the mechanical rotor angle of the machine can be repreente by the angle of the voltage ource iii amping can be neglecte an iv the machine mechanical power input i contant. Thu the claical moel can be obtaine from the high-orer moel by etting τ = 0 τ = 0 = 0 ω t = Conier a poitive contant ɛ where ɛ < an a function fɛ efine on ome ubet of the real number. We write fɛ = O ɛ i if an only if there exit a poitive real number uch that: fɛ ɛ i a ɛ 0. = 0 e = 0 X = X τ = τ = τ m = to give: δ = ω M ω = T m 0 E 0 X e l in δ δ l D 0 ω E where E 0 = 0 E 0 e an X := X X e enote contant. III. A LIBAY OF SECOND-ODE MODELS In thi ection a library of two-imenional ynamic moel for ynchronou machine are evelope from the high-orer moel preente in Section II-A. By utilizing the time-cale propertie ecribe in Section II-B an ingular perturbation analyi the nineteenth-orer machine moel i reuce to the elemental moel the ampe moel an the emi-ampe moel. The moel are bae on the following implifying aumption: Aumption 3. The angular pee of the machine ω t i ufficiently cloe to the nominal pee of the machine o that = O ɛ. ω t A. The Elemental Moel The elemental moel i formulate by replacing the ifferential euation for the fat tate with algebraic counterpart calle zero-orer approximate manifol. The manifol are evelope by etting τ τ an all O ɛ parameter in 6 to zero to give: Φ e 0 t = Xe I Φ e 0 t = Xe I E 0t = X X I Φ 0t = X X I Φ 0t = X X I E f0 t Φ 0 t = e I l in Φ 0 t = e I l co K u r δ t δ l δ t δ l E f0 t = E 0t = X X I E f0 t U f0 t = E f0 t Ū f0 t = K u E f0 t Ū f0 t = K u E f0 t P u0 t = P c D 0 ω t T m0 t = P u0 t alo P a0 = 0 P a0 = 0 P b0 = 0 an P b0 = 0 where 0 ubcript enote a zero-orer approximation an e := e X e := X X e X e := X X e. 7 8

4 4 Alo the output voltage i ecribe by: = e I X e I l co δ t δ l = e I X e I l in δ t δ l an the output current i ecribe by I = e Ku r e X e X e X e l in δ t δ l X e e X e X e l co δ t δ l X e e X e I = e e l co δ t δ l X e e X e X e Ku r X e e X e l in δ t δ l X e e X e. Subtituting the zero-orer approximate manifol formulate in thi ection into 5 the elemental moel i given by: δ = ω M ω = P r D 0 ω e I C r l C C r r l co δ δ l C Cx l in δ δ l C x r l in δ δ l where C r C Cx an C x are contant with C r = e X C e = Ku X Cx = e X C e x = e X X X e X e X e e X e an P r = P c D 0 D 0 = D 0 D 0 an I = I I. The ynamic circuit of the elemental moel i epicte in Fig.. For the pecial cae where an e are O ɛ we et e = 0 from where it follow that C r = 0 an C x =. j t r - j t C e j X X I e X e I ji e j t jx e jx j t j e e l l j t e Fig. : Dynamic circuit of ynchronou machine elemental moel. B. The Dampe Moel The ampe moel i formulate by replacing an with firt-orer approximate manifol an replacing the ifferential euation for other fat tate with zero-orer approximate manifol. By uing a firt-orer approximation for the amper wining manifol the effect of amper wining on the machine repone are capture by the reulting reuce moel. The following implifying aumption i employe: Aumption 4. The per-phae line reitance e i O ɛ. Starting with the tate oberve to have the fatet ynamic Φ t Φ t Φ e t an Φ e t we formulate the following zero-orer approximation by etting = 0 e = 0 0 an ω t = : Φ 0 t = l in δ t δ l Φ 0 t = l co δ t δ l Φ e 0 t = l in δ t δ l Φ e 0 t = l co from where it follow that: I = X X X X X e E t I = δ t δ l X X = 0 Φ X X X e t l in δ t δ l an X e X X X Φ X X X e t X E X X X e t l co δ t δ l X e. Next for the ubeuent fatet tate Φ t an Φ t which are amper wining tate we erive a firt-orer approximation of it manifol. Manifol for Φ t an Φ t can be expree a power erie in τ an τ repectively to give: Φ t = Φ 0t τ Φ t τ Φ t Φ t = Φ 0t τ Φ t τ Φ t from where it follow that firt-orer approximation are given by: Φ t Φ 0t τ Φ t Φ t Φ 0t τ Φ t. Expreion for Φ 0t Φ t Φ 0t an Φ t are erive uing the following tep: Subtitute into to give: τ τ Φ0t τ Φ t = Φ 0t t τ Φ t X X I E t Φ0t τ Φ t = Φ 0t t τ Φ t X X I E t. 3 Uing the zero-orer approximation in 0 ubtitute expreion for I an I into 3 an euate the 0 τ an τ 0 term to give: Φ 0t = Xe X e Φ 0t = Xe X e where X e := X X e. E t X X l in δ t δ l X e E t X X l co X e δ t δ l

5 5 Alo euate the τ an τ term to give: X e X e Φ t = 3 τ X e in δ t δ l X e X e Φ t = 3 τ X e X e E t X X l X e co δ t δ l X e X X X e E t X X l X e X X l l X e X e X e E f t τ X e l where = l co δ t δ l δ t δ l l in δ t δ l l = l co δ t δ l l in δ t δ l δ t δ l an X e := X X e. Next for the amper wining tate oberve to have the lower ynamic E t we erive a firt-orer approximation of it manifol. A manifol for E t can be expree a a power erie in τ to give: E t = E 0t τ E t τ E t 4 from where it follow that a firt-orer approximation i given by: E t E 0t τ E t. 5 Expreion for E 0t an E t can be erive uing the following tep: Subtitute an 4 into to give: τ E 0t τ E t = t E 0t τ E t X X I X X X τ Φ t. X X X e X 6 Uing the zero-orer approximation in 0 ubtitute the expreion for I an Φ t into 6 an euate the 0 τ term to give: E 0t = X X l in δ t δ l N l D where N = τ τ X e Xe X X X X X X X X D = τ X e X e X X X X. τ X e X e Alo euate the τ term to give: E t = N D l O τ X e 3 X X X X an where N = τ D = X e D Finally for other tate oberve to have fat ynamic E E f U f Ū f T m P u P a P b P a P b we erive the following zero-orer approximation by etting τ an all O ɛ parameter except τ an τ to zero: E f0 t = E 0t = Xe K u r X e E f0 t N l D X X l co X e U f0 t = E f0 t Ū f0 t = K u E f0 t P u0 t = P c D 0 ω t T m0 t = P u0 t δ t δ l 7 alo P a0 = P a0 = P b0 = P b0 = 0 where N = τ τ X e X e X X X X X X an D = τ X e X e X X τ X e X e X X X X. Subtituting the firt-orer an zero-orer approximate manifol in 5 0 an 7 into 5 an etting O τ term to zero the ampe moel for a non-alient pole machine i given by: δ = ω M ω = P r D 0 ω C x l in δ δ l C X e r l in δ δ l C l co δ δ l δ δ l C l in δ δ l δ δ l C C l l in δ δ l 8 where C C x C an C are contant C = Ku C x = X X C = C C C C X e X e with C = τ X X X e C = X X C D = τ X e Xe X X X e τ X e X X C = an C = C C C C C with C = τ X X X e C = X X C D =

6 6 τ X e X X C = τ Xe Xe.The ynamic circuit of the ampe moel i epicte in Fig. δ t δ l δ t δ l l with = l co l in δ t δ l l l in X X X e = l co δ t δ l δ t δ l δ t δ l. Note that for alient pole machine C = 0 wherea for roun-rotor machine C x = 0. j C X C e e l j t r - e l j t X X I X C e I ji e e j t jx jx j e j t e - - l l j t e Fig. : Dynamic circuit of ynchronou machine ampe moel. C. The Semi-Dampe Moel Thi moel i only applicable to roun-rotor machine. Employing Aumption 4 the emi-ampe moel i evelope by replacing with a firt-orer approximate manifol an replacing the ifferential euation for other fat tate with zero-orer approximate manifol. Starting with fatet tate Φ t Φ t Φ e t Φ e t Φ t an Φ t we evelop the following zero-orer approximation by etting = 0 e = 0 = 0 ω t = τ = 0 an τ = 0: Φ 0 t = l in δ t δ l Φ 0 t = l co δ t δ l Φ e 0 t = l in δ t δ l Φ e 0 t = l co δ t δ l Φ 0t = X X I E t Φ 0t = X X I E t from where it follow that: I = l in δ t δ l 9 X e l co δ t δ l. X e E t I X e = E X e t Next we erive a firt-orer approximate manifol for E t having the form E t E 0t τ E t. Subtituting the expreion for Φ 0t in 9 an the power erie expanion in 4 into it follow that: τ E t 0t τ E t = E 0t τ E t X e X e X X l in δ t δ l. X e Euating the τ 0 term in 0 we have that: E 0t = X X l in X e 0 δ t δ l an euating the τ term we have that: E t = Xe X X l X e l where = l co δ t δ l δ t δ l l in δ t δ l. Finally for other tate oberve to have fat ynamic i.e. E E f U f Ū f T m P u P a P b P a an P b the following zero-orer approximation erive by etting τ τ an all O ɛ parameter except τ to zero are ue: K u r E f0 t = E 0t = Xe X e U f0 t = E f0 t E f0 t X X X e Ū f0 t = K u E f0 t P u0 t = P c D 0 ω t T m0 t = P u0 t l co δ t δ l 3 with P a0 = P a0 = P b0 = P b0 = 0. Subtituting the zero-orer approximate manifol in 3 an the firt-orer approximate manifol E 0t τ E t into 5 the emi-ampe moel i given by: δ = ω M ω = P r D 0 ω C l l in δ δ l C l co δ δ l δ δ l C X e r l in δ δ l 4 where C an C are contant with C = K u an C = τ X X. The ynamic circuit X e of the emi-ampe moel i epicte in Fig. 3 l with = l co δ t δ l δ t δ l l in δ t δ l. j t r - e l j t ' I ji e C e j X C e e j t jx jx j t j e e - - l l j t e Fig. 3: Dynamic circuit of ynchronou machine emi-ampe moel.

7 7 I. NUMEICAL ALIDATION In thi ection imulation reult comparing the high-orer moel the claical moel the elemental moel the emiampe moel an the ampe moel of a roun-rotor ynchronou machine are preente. We conier a two-bu power ytem with a ynchronou machine connecte to a contant power loa through a hort electrical tranmiion line. See Fig. 4 for a one-line iagram an Table II for the ytem parameter. where: j t t t e e t arctan t jx e j l t t l e P jq l l Fig. 4: One line iagram of a power ytem with a ynchronou machine connecte to a contant power loa through a hort tranmiion line. A. Cae Thi cae i ue to highlight the high-fielity of the econorer moel in comparion to the claical moel an we conier the ytem repone to an increae in real power eman by the loa. A roun-rotor ynchronou machine i coniere an a table euilibrium point for the high-orer moel i choen a the common initial conition for all the moel. The real power eman by the loa i increae from 0.05 [pu] to 0.5 [pu] at time t = 30 an the reference voltage magnitue r i change at time t = 30 to eep the bu voltage magnitue at l = [pu]. Numerical reult are epicte in Fig. 5. Figure 5 how that the elemental moel the emi-ampe moel an the ampe moel have an overall better accuracy than the claical moel an that after one econ the error of the claical moel repone increae exponentially. B. Cae Thi cae i ue to compare the fielity of the elemental moel the emi-ampe moel an the ampe moel. The machine whoe parameter are ecribe in Table II i employe. The real power eman by the loa i increae from 0.05 [pu] to 0.5 [pu] at time t = 30 from 0.5 [pu] to 0.35 [pu] at time t = 530 from 0.35 [pu] to 0.3 [pu] at time t = 3030 an from 0.3 [pu] to 0.5 [pu] at time t = For each loa change the reference voltage magnitue r i change to eep the bu voltage magnitue at l = [pu]. The root mean uare error of the moel relative to the high-orer moel are outline in Table I an numerical reult are preente in Fig. 6. TABLE I: oot Mean Suare Error MSE ω δ ampe moel.9093[rpm] [pu] [eg] emi-ampe moel.9093[rpm] [pu] [eg] elemental moel [rpm] [pu] [eg] The MSE reult from Cae how that although the elemental moel the emi-ampe moel an the ampe moel match in accuracy for machine voltage magnitue repone the ampe moel an the emi-ampe moel have a higher accuracy for machine angular freuency repone. TABLE II: Sytem parameter for a alient pole ynchronou machine Damper wining Stator wining IEEE DCA exciter DEGO pee governor Tranmiion line parameter τ τ τ X X X X X X τ τ f τ u value [] 0.04 [] 3.63 [] [pu] [pu].7997 [pu] 0.9 [pu] [ra/] [pu] 0.4 [pu] 0.3 [pu] 5.04 [] 0 8 [] 0.00 [] 0 [].7997 [pu] [pu] X K u 00 K u 0 [] τ 0 4 [] τ 0 [] τ [] τ [] τ [] τ [] τ m [] κ 0 P r 0 [pu] M 0.88 [ ] D [/ra] D [/ra] e X e. CONCLUDING EMAKS [pu] [pu] In thi paper we introuce a library of econ-orer ynchronou machine moel compriing of the elemental moel the ampe moel an the emi-ampe moel. We alo howe how thee moel an the o-calle claical moel can be obtaine from a high-orer machine moel. While the claical moel i obtaine by ientifying mall an large parameter in the high-orer moel an etting them to zero an infinity repectively the library of econorer moel are obtaine by ientifying fat an low tate in the high-orer moel an replacing ifferential euation for the fat tate with algebraic counterpart referre to a

8 Time econ ω rpm high-orer moel ampe moel emi-ampe moel elemental moel claical moel Time Minute per-unit Time econ high-orer moel ampe moel emi-ampe moel elemental moel claical moel Time Minute Time econ δ egree high-orer moel ampe moel emi-ampe moel elemental moel claical moel Time Minute Fig. 5: Cae numerical reult: machine angular freuency voltage magnitue an phae ω rpm Time econ high-orer moel ampe moel emi-ampe moel elemental moel Time Minute per-unit Time econ high-orer moel ampe moel emi-ampe moel elemental moel Time Minute 0 04 δ egree high-orer moel ampe moel emi-ampe moel elemental moel Time econ Time Minute Fig. 6: Cae numerical reult: machine angular freuency voltage magnitue an phae. approximate manifol zero-orer or firt-orer. The library of econ-orer moel were valiate by comparing their repone to thoe of a high-orer moel an the claical moel for given tet cae. EFEENCES [] P. Kunur N. J. Balu an M. G. Lauby Power ytem tability an control. McGraw-Hill 994. [] P. Sauer an A. Pai Power Sytem Dynamic an Stability. Stipe Publihing L.L.C [3] P. Kraue O. Waynczu S. Suhoff an S. Peare Analyi of Electric Machinery an Drive Sytem er. IEEE Pre Serie on Power Engineering. Wiley 03. [4] L. Wang J. Jatevich an H. W. Dommel eexamination of ynchronou machine moeling techniue for electromagnetic tranient imulation IEEE Tranaction on Power Sytem vol. no. 3 pp. 30 Aug [5] S. Crary Power Sytem Stability: Tranient tability er. General Electric erie. John Wiley 947. [6] E. Kimbar Power Sytem Stability. ol. 3. Synchronou Machine. Wiley 956. [7] A. Pai Energy Function Analyi for Power Sytem Stability er. Power Electronic an Power Sytem. Springer Boton MA 989. [8] P. M. Aneron an A. A. Foua Power ytem control an tability er. IEEE Pre power engineering erie. IEEE Pre 003. [9] S. Y. Calian an P. Tabuaa Ue an abue of the wing euation moel in 05 54th IEEE Conference on Deciion an Control CDC Dec 05 pp [0] P. Kootović H. K. Khalil an J. O eilly Singular Perturbation Metho in Control: Analyi an Deign er. Claic in Applie Mathematic. Society for Inutrial an Applie Mathematic 986. [] H. K. Khalil Nonlinear Sytem. Pearon Eucation Limite 03. [] J. H. Chow Time-Scale Moeling of Dynamic Networ with Application to Power Sytem B. A.. an T. M. E. Springer 98. [3] P. W. Sauer S. Ahme-Zai an P.. Kootovic An integral manifol approach to reuce orer ynamic moeling of ynchronou machine IEEE Tranaction on Power Sytem vol. 3 no. pp. 7 3 Feb [4] P.. Kootovic an P. W. Sauer Integral manifol a a tool for reuce-orer moeling of nonlinear ytem: A ynchronou machine cae tuy IEEE Tranaction on Circuit an Sytem vol. 36 no. 3 pp Mar [5] IEEE ecommene Practice for Excitation Sytem Moel for Power Sytem Stability Stuie IEEE St eviion of IEEE St pp. 07 Aug. 06. [6] PowerWorl corporation. 07 Woowar ieel governor moel. [Online]. Available: { powerworl.com/webhelp/content/tranientmoel HTML/Governor%0DEGO.htm?tocpath=Tranient% 0Stability%0A-On%0TS%7CTranient% 0Moel%7CGenerator%7CGovernor%7C 4} [7] A. Foua an. ittal Power Sytem Tranient Stability Analyi Uing the Tranient Energy Function Metho. Prentice Hall 99.