Rangsit University Mueang Pathum Thani, Pathum Thani, Bangkok 12000, Thailand

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1 Internatonal Journal of Innovatve Computng, Informaton and Control ICIC Internatonal c 08 ISSN Volume 4, Number, February 08 pp. 6 NONLINEAR DISTURBANCE OBSERVER-BASED BACKSTEPPING CONTROL FOR A DUAL ECITATION AND STEAM-VALVING SYSTEM OF SYNCHRONOUS GENERATORS WITH ETERNAL DISTURBANCES Adrak Kanchanaharutha and Ekkacha Mujjalnvmut Department of Electrcal Engneerng Rangst Unversty Mueang Pathum Than, Pathum Than, Bangkok 000, Thaland adrak@rsu.ac.th Department of Electrcal Engneerng Faculty of Engneerng Kng Mongkut s Unversty of Technology Thonbur Pracha Utht Road, Bang Mod, Bangkok 040, Thaland ekkacha.muj@kmutt.ac.th Receved June 07; revsed October 07 Abstract. Ths paper presents a nonlnear stablzng state feedback control for a dualexcted and steam-valvng system of synchronous generators va a backsteppng strategy combned wth a nonlnear dsturbance desgn. The dsturbance observer s used to estmate unavodably external dsturbances. The resultng controller s employed to stablze the system stablty and reject undesred external dsturbances. In order to demonstrate the effectveness of the developed desgn, numercal smulaton results are provded to llustrate that the presented control can mprove dynamc performances, rapdly suppress system oscllatons of the overall closed-loop dynamcs, and despte havng nevtably external dsturbances, performs better than two conventonal nonlnear control technques: a backsteppng desgn and an ntegral backsteppng desgn. Keywords: Dual exctaton and steam-valvng system, Backsteppng control, Nonlnear dsturbance observer, Integral backsteppng control. Introducton. It s well known that mprovng power system stablty has attracted a lot of researcher s attenton due to power systems wth rapd ncrease of the sze and complexty. Addtonally, t becomes a key ssue of the operaton of modern power systems. Thus, there are currently a lot of effects to fnd hgh-performance stablzng controllers whch are able to mtgate the results from many severe contngences such as voltage collapse, slandng faults, and loss of synchronsm. The exctaton control of synchronous generators s an effectve way for transent stablty enhancement of power systems []. Further, a varety of exctaton control strateges have been proposed [-5]. Besdes from the exctaton control, a steam-valvng control scheme [6-8] s also a promsng and effectve way capable of stablzng the overall closed-loop dynamcs of synchronous generators and effectvely accomplshng the control performances. However, t was found that the power system stablty and operaton were greatly mproved wth the help of a combnaton of generaton exctaton control wth steam-valvng control. Ths coordnaton has receved a great attenton n the power engneerng communty [9-]. Ths strategy can offer an opportunty to ncrease a degree of freedom for achevng further desred control performances.

2 A. KANCHANAHARUTHAI AND E. MUJJALINVIMUT So far, the combnaton above has concentrated on the coordnaton of sngle-excted and steam-valvng control. Ths means that the d-axs feld voltage s regarded as a constant throughout consderaton; nevertheless the only q-axs feld voltage s utlzed to acheve the desred control objectves. Thus, a coordnaton of d-axs and q-axs feld voltages, called dual-exctaton s a more effectve method than the coordnaton of sngle-excted and steam-valvng control. Also, ths method s a promsng dea of ncreasng greater flexblty for the system stablty enhancement. Roughly speakng, ths scheme can be regarded as an ncluson of the addtonal degree of freedoms and offers an opportunty to determne the effectve control on account of both d-axs and q-axs feld wndngs whch are separately desgned. To the best knowledge of the authors, there s less attenton devoted to the coordnaton of dual-excted and steam-valvng control of synchronous generators [-4]. Wth the help of a coordnated passvaton scheme [], a nonlnear control was presented to demonstrate hat dynamc performances were consderably mproved superor to the feedback lnearzng scheme. A desgn procedure based on an mmerson and nvarance method for a nonlnear feedback stablzng control law was report n [3,4]. Ths scheme has provded an opportunty to acheve power angle stablty along wth frequency and voltage regulaton, and to ensure that the closed-loop system dynamcs are transently and asymptotcally stable. In those works, the resultng controller performed better than a coordnate passvaton controller [] n terms of the rapd dampng of oscllatons n all tme responses followng small or large dsturbance. However, even f the I&I control desgn has presented the hgh-performance stablzng controller and was applcable for many types of practcal systems, t had several dsadvantages such as no systematc way to select the mappng, a target dynamcal system, and an energy functon, respectvely. It has been found that n practce, most engneerng systems have often dsturbances capable of degradng nevtably the desred control performances of the closed-loop dynamcs. The dsturbances consdered nclude external dsturbances, parametrc uncertantes and other unknown nonlnear terms. Therefore, the desred control desgn method needs to nclude the dsturbance dynamcs to reject the effects of the abovementoned dsturbances. Recently, a dsturbance observer method s an approach for compensatng the result from external dsturbances and msmatched dsturbances/uncertantes. Ths method has been wdely accepted n compensatng the effects of dsturbances. The dsturbance observer s utlzed to estmate dsturbances appearng n the system. There are currently the development of dsturbance observer desgn combned wth most popular nonlnear control methods such as backsteppng method [5] and sldng mode method [6], as presented n [6-0]. Based on the abovementoned references, dsturbance observer-based control s a promsng method capable of rejectng external dsturbances and mprovng robustness aganst uncertantes [6] smultaneously. It also provdes an effectve way to handle external dsturbances and system uncertantes. Addtonally, dsturbance observer desgn method can be further extended to several problems n control system socetes, such as adaptve control [], fnte-tme control [], and trackng control [3]. Further, ths method can be successfully appled for numerous knds of real engneerng systems such as flght control systems [6], permanent magnet synchronous motors [6], arbreathng hypersonc vehcle systems [7], power systems [8], and electrohydrolc actuator systems [3]. Those ndcate mportant applcaton potentals of the dsturbance observer-based control method to deal wth the effect of unavodably external dsturbances. However, even though the control desgn methods presented n [-4] have good control performances, external dsturbances and uncertantes have not been taken nto account before. Dsturbances and uncertantes arsng n the system nevtably may lead to undesred control performances, and eventually make the system unstable.

3 NONLINEAR DISTURBANCE OBSERVER-BASED BACKSTEPPING CONTROL 3 Ths paper stll contnues ths lne of nvestgaton, but a systematc procedure to synthesze a nonlnear feedback stablzng control law on the bass of a backsteppng control [5] combned wth the dsturbance observer desgn [6] s developed to cope wth the effects of external dsturbances. Therefore, the prmary contrbutons of ths work le n that: The use of a nonlnear dsturbance observer-based backsteppng control strategy to stablze the system n the presence of external dsturbances has not been nvestgated before; The overall closed-loop system s nput-to-state n spte of havng external dsturbances; In comparson wth a backsteppng control and an ntegral backsteppng control, the developed control law offers better dynamc performances and a satsfactory dsturbance rejecton ablty. The rest of ths paper s organzed as follows. A dynamc model of a dual-excted and steam-valvng system of synchronous generators s brefly presented, and some sgnfcant lemmas together wth the problem statement are gven n Secton. Controller desgn and stablty analyss are developed n Secton 3 whle smulaton results are stated n Secton 4. Fnally, n Secton 5, a concluson s gven.. Power System Model Descrpton and Prelmnares... Power system models. In ths subsecton, a dynamc model of a synchronous generator wth dual-excted and steam-valvng controller can be obtaned as follows: δ = ω ω s + d t, ω = M + d t, P m = P m P me P m E q V dσ sn δ E d V qσ cos δ dσ qσ V dσ sn δ Dω ω s qσ T HΣ dσ Ė q = dσ T d0 qσ Ė d = qσ T q0 + C H T HΣ u G + d 3 t, E q + dσ dσ dσ T V d0 cos δ + u fd + T d d0 4 t, E d qσ qσ qσ T V q0 sn δ + u fq T q0 + d 5 t, where δ s the power angle of the generator, ω denotes the relatve speed of the generator, D 0 s a dampng constant, and E q and E d are the q-axs and d-axs nternal transent voltages, respectvely. d and q are the d-axs and q-axs transent reactances, respectvely. P e s the electrcal power delvered by the generator to the voltage at the nfnte bus V, ω s s the synchronous machne speed, ω s = πf, H represents the per unt nertal constant, f s the system frequency and M = H/ω s. dσ = d + T + L s the reactance consstng of the drect axs transent reactance of SG, the reactance of the transformer, and the reactance of the transmsson lne L. Smlarly, dσ = d + T + L s dentcal to dσ except that d denotes the drect axs reactance of SG. qσ and qσ denote the q-axs reactances smlar to d-axs reactance. T d0 and T q0 are the d-axs and q-axs transent open-crcut tme constants. u fd and u fq are the q-axs and d-axs feld voltage control nputs to be desgned, respectvely. P me s the ntal value of mechancal power, and C H s the assgned coeffcent of hgh-pressure cylnder. T HΣ s the equvalent tme constant of steam valve control systems. u G s the steam-valvng control nput

4 4 A. KANCHANAHARUTHAI AND E. MUJJALINVIMUT to be desgned. dj t, j =,, 3, 4, 5 are external dsturbances and system parameter varatons. For convenence, let us defne new state varables as follows: x = δ δ e, x = ω ω s, x 3 = P m P me, x 4 = E qv snx + δ e E qev sn δ e + msn x dσ + δ e sn δ e, x 5 = E d V cosx + δ e E d V cos δ e, qσ where m = dσ qσ. dσ qσ Subsequently, after dfferentatng the state varables, we have the dynamc model of the dual exctaton and steam-valvng system of synchronous generators wth lumped dsturbances can be expressed as an affne nonlnear system as follows: ẋ = fx + gxux + dt, 3 where fx = gx = ux = f x f x f 3 x f 4 x f 5 x = g 3 x g 4 x g 53 x C H T HΣ u G u fd T d0 u fq T q0 x M x 3 Dx x 4 x 5 P me x 3 T HΣ aq E q + b q cosx + δ e V snx + δ e dσ a d E d b d snx + δ e V cosx + δ e, dt = qσ, = 0 V snx + δ e 0 dσ V cosx + δ e 0 0 qσ d t d t d t d t d 3 t d 3 t = V snx + δ e, d 4 t dσ T d d0 4 t d 5 t d 5 t V cosx + δ e qσ T q0, 4 where a q = dσ, a dσ T d0 d = qσ qσ T q0 operaton s defned n the set D = {x S R R R R 0 < x < π, b q = dσ dσ dσ T d0 operatng equlbrum s denoted by x e = [0, 0, 0, 0, 0] T. V, b d = qσ qσ V qσ T q0. The regon of }. The open loop

5 NONLINEAR DISTURBANCE OBSERVER-BASED BACKSTEPPING CONTROL 5 For the sake of smplcty, the power system consderng 3 and 4 can be expressed as follows. ẋ = d t, ẋ = M x 3 Dx x 4 x 5 + d t, ẋ 3 = f 3 x + g 3 x C H T HΣ u G + d 3 t, ẋ 4 = f 4 x + g 4 x u fd T d0 ẋ 5 = f 5 x + g 53 x u fq T q0 + d 4 t, + d 5 t. Assumpton.. The external dsturbances d j t, j =,, 3, 4, 5 are bounded. Addtonally, the frst dervatves of the dsturbances above are also bounded. Remark.. The assumpton above s often employed n the dsturbance estmator based control desgn technque for real engneerng applcatons [6], such as arbreathng hypersonc vehcle systems [7], power systems [8], and electrohydrolc actuator systems [3]. In partcular, the dervatve of the dsturbance wll appear n the estmaton error equaton 0. Ths assumpton s necessary to be used to analyze the overall closed-loop system stablty as gven n Secton Prelmnares. In ths subsecton, some mportant lemmas are mentoned as follows for convenence of the reader. Consder the followng system ẋ = ft, x, u, x R n, u R m. 6 Defnton.. [4] A contnuous functon α : [0, a [0, + belongs to class K f t s strctly ncreasng and α0 = 0. It belongs to class K f a = + and ar + as r +. Lemma.. [4] Let V : [0, R n R be a contnuously dfferentable functon such that α x V t, x α x V t + V x ft, x, u W 3x, x ρ u > 0, for all t, x, u [0, R n R m, where α and α are class K functons, ρ s a class K functon, and W 3 x s a contnuous postve defnte functon on R n. Then, system 6 s nput-to-state stable ISS. Lemma.. [4] Consder the followng system 6. satsfed system ẋ = ft, x, u s globally nput-to-state stable. lm t + u = 0. 5 If the followng condtons are then the states of the system 5 wll asymptotcally converge to zero, that s, lm t + x = 0. Remark.. It s observed that the second subsystem of the system 5 depends upon the state varables x 3, x 4 and x 5. Therefore, the dynamc equatons consdered are not the strct-feedback form. The backsteppng desgn presented n ths work needs to be extended for the nonlnear control desgn for non-strct-feedback form.

6 6 A. KANCHANAHARUTHAI AND E. MUJJALINVIMUT Problem statement: The control objectve of ths paper s to solve the problem of the stablzaton of the system 5 wth the external dsturbances d, whch can be formulated as follows: wth the help of the nonlnear dsturbance observer-based backsteppng control technque [7], to desgn a stablzng state feedback controller ux and dsturbance estmaton ˆd as follows: u = ϕ ˆd = φ x, ˆd x, u, ˆd 7 such that the overall closed-loop systems 5 and 7 are nput-to-state stable, where ˆd s the estmate of d. For the developed desgn procedure n the next secton, a combnaton of the backsteppng strategy and dsturbance observe desgn wll be presented to obtan a composte nonlnear controller 7. In comparson wth the conventonal backsteppng method, the proposed approach wll ntroduce the dsturbance estmaton terms nto vrtual control varables. These terms are also used for compensatng the external dsturbances at each step, and the estmaton error dynamcs are ncluded for the closed-loop stablty analyss. 3. Controller Desgn and Stablty Analyss. In ths secton, we am at dervng the control laws for stablzng the dual-exctaton and steam-valvng system of synchronous generators. Our proposed desgn process can be dvded nto three subsectons. The frst subsecton conssts of desgnng the nonlnear dsturbance observer to onlne dentfy the unknown, but bounded, dsturbances. The second subsecton presents a desgn procedure combnng the backsteppng control law wth the dsturbance estmator ntroduced nto the vrtual control laws n each desgn step. Based on Lyapunov stablty arguments, the overall closed-loop system n the presence of external dsturbance s nvestgated n the fnal subsecton. Also, the resultng controller can acheve stablty and performance specfcatons. 3.. Nonlnear dsturbance observer desgn. In accordance wth [6-8] n order to guarantee the control performance of the system 5, the nonlnear dsturbance observer can be desgned as ˆd = λ x j p j, j =,, 3, 4, 5, ṗ = f x + ˆd, ṗ = f x + ˆd, ṗ 3 = f 3 x + g 3 x C H T HΣ u G + ˆd 3, ṗ 4 = f 4 x + g 4 x u fd T d0 ṗ 5 = f 5 x + g 53 x u fq T q0 + ˆd 4, + ˆd 5, where λ j > 0 s a desgn parameter. Thus, based on 8 the dsturbance estmaton dynamcs can be expressed n the followng form: ˆd j = λ ẋ j ṗ j = λ d j ˆd j, j =,, 3, 4,

7 NONLINEAR DISTURBANCE OBSERVER-BASED BACKSTEPPING CONTROL 7 Let us defne the dsturbance estmaton error as e j = d j ˆd j, and we have the estmaton error dynamcs as follows. ė j = λ j e j + d j Backsteppng desgn. Accordng to the concept reported n [8], the stablzaton problem for the system 5 s solved by desgnng a backsteppng control. The desgn procedures are developed step by step as follows. Step : Consderng, the frst subsystem 5, a Lyapunov functon s selected as V = z + e, where z = x. Then the tme dervatve of V along the system trajectores becomes V = z d + e λ e + d = λ e + z z d + e d = λ e + z x + z x x + z d + e d. From, t s seen that x s regarded as the vrtual control varable wth the dsturbance estmate ˆd as follows. x = k + z 4ϵ ˆd, 3 where k > 0 and ϵ > 0. After substtutng 3 nto, we have V = k + z λ 4ϵ e + z e + z z + e d = k + z λ 4ϵ e + z + ϵ e + z z + e d 4ϵ = k z λ ϵ e + z z + e d, 4 where z = x x. Step : Let us defne the Lyapunov functon of Step as V = V + z + e. 5 Then the tme dervatve of V along the system trajectores s as follows: V = k z λ ϵ e + z z + e d + e d +z M x 3 Dx x 4 x 5 + d x z d x ˆd λ e = k z λ ϵ e λ e + z z + M z x 3 M z x 4 M z x x 5 z x z [ ] +z x 3 x 3 Dx x 4 x M 4 x 5 x 5 + d x d z z x ˆd λ e + e d + e d. 6

8 8 A. KANCHANAHARUTHAI AND E. MUJJALINVIMUT From 6, t can be observed that x 3, x 4 and x 5 are consdered as the vrtual control varables wth the dsturbance estmates ˆd and ˆd as follows. x 3 = M [ k + + ĉ z z + Dx 3 4ϵ M ˆd + x ], x 4 = x 7 3, x 5 = x 3, where k > 0, ϵ > 0 and ĉ = x 4ϵ z + x ˆd λ. After substtutng 7 nto 6, we obtan V = k z λ e k + + ĉ z λ e + 4ϵ M z x 3 x 3 M z x 4 x 4 x M z x 5 x 5 + z e z + x z ˆd λ e + e d + e d. 8 Based on Young nequalty, t can be straghtforwardly computed of the terms n 8 as e z z + ϵ e 9 4ϵ x z + x z ˆd λ e 4ϵ x z + x ˆd λ zϵ = ĉ z + ϵ e. 0 Substtutng 9 and 0 nto 8 and then defnng z = x x, = 3, 4, 5, we get V k z k z λ ϵ e λ ϵ e + z M z 3 z 4 z 5. Step 3: we select a Lyapunov functon as follows: V 3 = V + z + e. After takng dervatves of both sdes of, one has V 3 = V + z ż + e ė = k z k z λ ϵ e λ ϵ e + z M z 3 z 4 z 5 + [z ẋ x ż x ż x z z ˆd λ e x ˆd λ e λ e + e d ]. 3 Substtutng ẋ, = 3, 4, 5 from 5, ż, ż and x nto 3 yelds V 3 k z k z λ ϵ e λ ϵ e λ e e d [ z +z 3 M + f 3x + g 3 x C H u G + d 3 x 3 T HΣ ] + x 3 x d x z x ˆd λ e [ +z 4 z 3 x 3 ˆd λ e + x 3 ˆd λ e x 3 f x + d z z M + f 4x + g 4 x u fd + d T d0 4 x 4 x 4 f x + d z

9 NONLINEAR DISTURBANCE OBSERVER-BASED BACKSTEPPING CONTROL 9 ] + x 4 x d x z x ˆd λ e [ +z 5 z M + f 5x + g 53 x u fq z 4 x 4 ˆd λ e + x 4 ˆd λ e + d T q0 5 x 5 x 5 f x + d z ] + x 5 x d x x z x ˆd λ e z 4 5 ˆd λ e + x 4 ˆd λ e. 4 From 4, n order to acheve the desred control performance, we choose the control law as follows: C H u G = [ z T HΣ g 3 x M f 3x ˆd 3 + x 3 + x 3 f x + x k 3 + ] + ĉ 3 + ĉ 3 + ĉ 33 + ĉ 34 z 3 4ϵ 3 u fd = [ z T d0 g 4 x M f 4x ˆd 4 + x 4 + x 4 f x + x k 4 + ] 5 + ĉ 4 + ĉ 4 + ĉ 43 + ĉ 44 z 4 4ϵ 4 u fq = [ z T q0 g 53 x M f 5x ˆd 5 + x 5 + x 5 f x + x k 5 + ] + ĉ 5 + ĉ 5 + ĉ 53 + ĉ 54 z 5, 4ϵ 5 where ĉ = 4ϵ x z ĉ 4 = [ x 4ϵ x ], = 3, 4, 5. + x ˆd λ, ĉ = 4ϵ x z + x ˆd λ, ĉ3 = 4ϵ [ x x z z + x ˆd λ ], Remark 3.. In accordance wth the results presented n [8], some auxlary terms ĉ, ĉ,..., ĉ 4 are ntroduced nto the vrtual state varables 7 and the fnal controller 5 to deal wth the crossng terms arsng from the effect of dsturbances, compensaton errors, and system states. On the other hand, these auxlary terms are not ncluded n the conventonal backsteppng method, thereby leadng to unsatsfactory control performances. Substtutng the presented control law 5 nto 4, we have V 3 = k z k z λ ϵ e λ ϵ e λ e e d [ + z e x e x e + x x e x e z z z z x + k + ] + ĉ + ĉ + ĉ 3 + ĉ 4 z 4ϵ [ ] x z ˆd λ e + x ˆd λ e + x x z ˆd λ e. 6

10 0 A. KANCHANAHARUTHAI AND E. MUJJALINVIMUT It s observed that some terms of the last two lnes n 6 can be changed nto the followng nequaltes: x z + x z ˆd λ e x + x 4ϵ z ˆd λ z + ϵ e = ĉ z + ϵ e, 7 x z + x z ˆd λ e x + x 4ϵ z ˆd λ z + ϵ e = ĉ z + ϵ e, 8 [ ] x z x + x z z ˆd λ e [ ] x x + x 4ϵ z z ˆd λ z + ϵ e = ĉ 3 z + ϵ e, 9 x z e x z + ϵ e = ĉ 4 z + ϵ e x 4ϵ x. 30 After substtutng the nequaltes 7-30 and then combnng those nequaltes wth 6, we have V 3 k j zj + e j d j λ 8ϵ e λ 4ϵ e λ ϵ e 3 j= In the followng subsecton, we analyze the stablty of the closed-loop system wth the control law 5 based on the control desgn of ths subsecton Stablty analyss. In ths subsecton, the overall closed-loop stablty of the system 5 wth the proposed control law 5 and the error estmaton dynamcs 0 are analyzed wthn the framework of Lyapunov theory. Before the closed-loop stablty s carred out, the closed-loop system dynamcs can be expressed as follows: ż ż ż 3 = z k + 4ϵ z + e = z + M z 3 z 4 z 5 = M z k + + ĉ 4ϵ k 3 + 4ϵ 3 + ĉ 3 + ĉ 3 + ĉ 33 + ĉ 34 z + e x e x z ˆd λ e z 3 + e 3 x 3 e x 3 e z z ż 4 ż 5 + x 3 z x = M z + x 4 z x = M z e + x 3 x z z ˆd λ e x 3 ˆd λ e x 3 ˆd λ e k ĉ 4 + ĉ 4 + ĉ 43 + ĉ 44 4ϵ 4 e + x 4 x z z ˆd λ e x 4 ˆd λ e x 4 ˆd λ e k ĉ 5 + ĉ 5 + ĉ 53 + ĉ 54 4ϵ 5 z 4 + e 4 x 4 z e x 4 z e z 5 + e 5 x 5 z e x 5 z e 3 ė j + x 5 x e + x 5 x z z z ˆd λ e x 5 ˆd λ e x 5 ˆd λ e, = λ j e j + d j, j =,, 3, 4, 5. Therefore, we can summarze the control desgn n the followng theorem. Theorem 3.. Under Assumpton., the nonlnear dsturbance observer-based backsteppng controller 5 can guarantee that the overall closed-loop system consstng of the

11 NONLINEAR DISTURBANCE OBSERVER-BASED BACKSTEPPING CONTROL system and the dsturbance observer error dynamcs 3 wth the developed controller s nput-to-state stable. Proof: To demonstrate the closed-loop stablty of the presented control strategy, let us defne the followng Lyapunov functon for the closed-loop dynamcs 3. V 3 = z j + e j. 33 j= After computng the tme dervatve of the Lyapunov functon canddate 33, the closed-loop system can be expressed as V 3 k j zj + e j d j λ 8ϵ e λ 4ϵ e λ ϵ e. 34 j= After selectng λ = a 0 + 8ϵ, λ = a 0 + 4ϵ, λ = a 0 + ϵ, = 3, 4, 5, a 0j > 0, j =,, 3, 4, 5, we obtan V 3 k j zj a 0j e j + e j d j k j zj a 0 e + e d, 35 j= j= j= [ where e = [e, e, e 3, e 4, e 5 ] T, d = d, d, d 3, d 4, d ] T 5, a0 = mn{a 0, a 0,..., a 05 }. Besdes, we rewrte the nequalty 35 as V 3 k j zj θa 0 e θa 0 e + e d, 36 j= where 0 < θ <. Provded that e d a 0 θ, we obtan V 3 5 j= k jz j θa 0 e 0. In accordance wth Lemma., t can be concluded that the overall closed-loop dynamcs 3 are nput-to-state stable. Ths completes the proof. Remark 3.. Accordng to the result of Theorem and Lemma 9. n [4], t can be observed that under Assumpton., the dsturbance estmaton error wll approach to any specfed level va a sutable selecton of the observer gan λ. Assumpton 3.. The dsturbances satsfy the condton of lm t + d j t = 0, j =,, 3, 4, 5. Theorem 3.. Under Assumptons. and 3., the closed-loop dynamcs 3 under the control law 5 and the dsturbance estmaton 8 wll asymptotcally converge to zero. Proof: It s seen from the closed-loop system 3 that d j can be regarded as an nput of the system. After combnng Theorem 3. wth Assumpton 3., t follows from Lemma. that all trajectores of z j and e j of the closed-loop dynamcs converge to zero. Ths means that z j 0 and e j 0 as t +. Ths completes the proof. 4. Smulaton Results. In ths secton, smulaton results are gven to ndcate the effectveness of the developed strategy. The proposed controller s evaluated va smulatons of a sngle-machne nfnte bus SMIB power system consstng of dual-excted and steam-valvng control as shown n Fgure [3]. The performance of the proposed control scheme s evaluated and verfed n MATLAB envronment. The physcal parameters pu., the controller parameters, and ntal condtons used for ths power system model are the same as those used n [3] as follows: j=

12 A. KANCHANAHARUTHAI AND E. MUJJALINVIMUT The parameters of dual-exctaton and steam-valvng system and transmsson lne: ω s = πf rad/s, D = 5, H = 5, f = 60 Hz, T d0 = 0, T q0 = 4, V = 0, ω = ω s, q =.6, q = 0.38, d =.6, d = 0.3, T HΣ = 0.4, T = 0.3, L = 0.7. The controller parameters of the proposed controller are ϵ j = 0, k j = 0, λ j = 50, j =,,..., 5. Intal condtons δ e = rad, P me =.749, E qe =.0703, E de = 0.5, ˆd j0 = 0, j =,, 3, 4, 5. Fgure. A sngle lne dagram of SMIB model wth SMES Addtonally, the external dsturbances d j, j =,, 3, 4, 5 actng on the underlyng system are assumed to be: d t = 0, 0 t 0, 0.5 snt, 0 t < cost, 0 t < 5 d t =, 5 t < 0, d 3 t =, 5 t < snte t, 0 t coste t, 0 t snt, 0 t < 5 0. cost, 0 t < 5 d 4 t =, 5 t < 0, d 5 t =.5, 5 t < snte 3t, 0 t coste t, 0 t 0 The controller parameters are set as ϵ j = 0, k j = 0, λ j = 50, j =,,..., 5. The tme doman smulatons are carred out to nvestgate the system stablty enhancement and the dynamc performance of the desgned controller, as gven n 5, n the system n the presence of external dsturbances. The control performance of the proposed controller nonlnear dsturbance observer-based backsteppng controller s compared wth that of the followng nonlnear controllers. A backsteppng controller [5] s provded by C H u G = [ k 3 z 3 z T HΣ g 3 x M f 3x + ] 3 Dẋ Mk ż + ż + k ẋ, u fd = [ k T d0 4 z 4 + z g 4 x M f 4x ] 3 Dẋ Mk ż + ż + k ẋ, u fq = [ k T q0 5 z 5 + z g 53 x M f 5x ] 3 Dẋ Mk ż + ż + k ẋ, wth z j = x j x j, j =,, 3, 4, 5, x = 0, x = k z, x 3 = 3 Dx Mk z + z + k x, x 4 = x 5 = x 3, ż = c z + z, ż = f x + c x. The controller parameters of ths scheme are set as c j = 0, j =,,..., 5. 37

13 NONLINEAR DISTURBANCE OBSERVER-BASED BACKSTEPPING CONTROL 3 An ntegral backsteppng controller s gven by C H u G = [ k 3 z 3 z T HΣ g 3 x M f 3x + Dẋ Mk ż + ż 3 u fd T d0 u fq T q0 + k ẋ + βẋ + βẋ ], = [ k 4 z 4 + z g 4 x M f 4x Dẋ Mk ż + ż 3 + k ẋ + βẋ + βẋ ], [ = g 53 x k 5 z 5 + z M f 5x Dẋ Mk ż + ż 3 + k ẋ + βẋ + βẋ ], 38 where z j = x j x j, j =,, 3, 4, 5, x = β t x 0 τdτ, β > 0, x = k z βx, x 3 = Dx 3 Mk z + z + k βx + βx, x 4 = x 5 = x 3, ż = c z +z, ż = f x + c βx + βx. The controller parameters of ths scheme are set as c j = 0, j =,,..., 5, β =. The smulaton results are presented and dscussed as follows. Tme trajectores of the power angle, frequency, mechancal power together wth d-axs and q-axs transent nternal voltages under three controllers are presented n Fgures and 3, respectvely. Also, n order to llustrate the effectveness of dsturbance observer, Fgure 4 shows tme hstores of external dsturbances and dsturbance estmaton. From these fgures, t can be observed that the tme responses can acheve the control objectves n the presence Fgure. Controller performance power angles δ rad., frequency ω ω s rad/s. and mechancal nput P m pu. Sold: The proposed control, Dashed: Backsteppng control, Dotted: Integral backsteppng control

14 4 A. KANCHANAHARUTHAI AND E. MUJJALINVIMUT Fgure 3. Controller performance the q-axs transent nternal voltage E q pu. and the d-axs transent nternal voltage E d pu. Sold: The proposed control, Dashed: Backsteppng control, Dotted: Integral backsteppng control Fgure 4. External dsturbances and dsturbance estmaton

15 NONLINEAR DISTURBANCE OBSERVER-BASED BACKSTEPPING CONTROL 5 of external dsturbances. It s obvously seen that the proposed controller offers better dynamc performances, such as a shorter settlng tme, a short rse tme, a faster convergence rate, and no steady-state error. It s capable of accomplshng a satsfactory dsturbance rejecton ablty despte havng external dsturbances. From Fgures and 3, one can observe that under the proposed control, the adverse effects caused by the dsturbances are removed from the system after a short perod. On the other hand, the backsteppng control and the ntegral backsteppng control clearly exhbt undesred control performances such as unsatsfactory overshoots, slowly suppressng system oscllatons, and nonzero steady-state error. Thus, the proposed controller offers superorty over backsteppng and ntegral backsteppng controller snce the dsturbance observer desgn, employed to estmate the external dsturbance, ntroduces the dsturbance estmaton to the vrtual control laws n each step and the fnal controller 5. Fgure 4 llustrates tme hstores of dsturbances and ther estmatons usng the proposed observer capable of effectvely estmatng and compensatng the external dsturbances successfully. Apart from ths, the dsturbance estmator of the developed strategy quckly approaches to the external dsturbances wth very fast convergence rates and no oscllatons as shown n Fgure 4. From the smulaton results mentoned prevously, t s evdent that as the presented backsteppng method combned wth the dsturbance observer scheme s appled to the SMIB power system wth external dsturbances, the advantages over both conventonal backsteppng and ntegral backsteppng methods are as follows. The proposed control law s effectvely desgned to stablze the system n the presence of undesred dsturbances. The developed control strategy can make the overall closed-loop dynamcs converge more quckly to a desred equlbrum pont. In partcular, t obvously performs well and has consderably effectve dsturbance rejecton ablty. It offers obvously superor transent performances llustrated by the rapdly suppressng system oscllatons n all tme trajectores n spte of havng external dsturbances. The process of desgnng the desred control law ncludes some auxlary terms nto the vrtual control laws and the fnal controller. These terms can counteract the crossng terms arsng from dsturbances, compensaton errors, and system states. In contrast, these terms are not ncluded n the conventonal backsteppng method and the ntegral backsteppng method. Thus, both backsteppng methods provde unsatsfactory control performances. 5. Concluson. In ths paper, the nonlnear dsturbance observer-based backsteppng control strategy has been proposed for a dual exctaton and steam-valvng system of synchronous generators n the presence of external dsturbances. The developed approach s utlzed to offer a satsfactory dsturbance rejecton performance and acheve mproved dynamc performances. To valdate the proposed scheme, a dynamc model of the dualexcted and steam-valvng system of synchronous generators s used to evaluate through a smulaton envronment. The smulaton results have demonstrated that the developed control method provdes an mproved transent performance and performs satsfactorly n rejectng external dsturbances better than backsteppng and ntegral backsteppng methods. The comparatve results wth other two controllers confrm the effectveness of the proposed controller capable of rapdly suppressng system oscllatons n the closedloop system dynamcs and rejectng unavodably external dsturbances. REFERENCES [] P. Kundur, Power System Stablty and Control, McGraw-Hll, 994.

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