Electric Power Systems Research

Electric Power Systems Research 84 (22 35 43 Contents lists available at SciVerse ScienceDirect Electric Power Systems Research jou rn al h om epage: www.elsevier.com/locate/epsr Observer-base nonlinear control of power system using sliing moe control strategy M. Ouassai a,, M. Maaroufi b, M. Cherkaoui b a Department of Inustrial Engineering, Cai AAya University, Ecole Nationale es Sciences Appliquées, B.P. 63, Sii Bouzi, Safi, Morocco b Department of Electrical Engineering, Mohamme V University, Ecole Mohammaia Ingénieurs, B.P. 765, Agal, Rabat, Morocco a r t i c l e i n f o Article history: Receive 3 November 29 Receive in revise form 27 February 2 Accepte 28 October 2 Available online 25 November 2 Keywors: Sliing moe control Lyapunov methos Damper currents observer Power system Transient stability a b s t r a c t This paper presents a new transient stabilization with voltage regulation analysis approach of a synchronous power generator riven by steam turbine an connecte to an infinite bus. The aim is to obtain high performance for the terminal voltage an the rotor spee simultaneously uner a large suen fault an a wie range of operating conitions. The methoology aopte is base on sliing moe control technique. First, a nonlinear sliing moe observer for the synchronous machine amper currents is constructe. Secon, the stabilizing feeback laws for the complete ninth orer moel of a power system, which takes into account the stator ynamics as well as the amper effects, are evelope. They are shown to be asymptotically stable in the context of Lyapunov theory. Simulation results, for a single- Machine-Infinite-Bus (SMIB power system, are given to emonstrate the effectiveness of the propose combine observer controller for the transient stabilization an voltage regulation. 2 Elsevier B.V. All rights reserve.. Introuction Successful operation of power system epens largely on the ability to provie reliable an uninterrupte service. The reliability of the power supply implies much more than merely being available. Ieally, the loas must be fe at constant voltage an frequency at all times. However, small or large isturbances such as power changes or short circuits may transpire. One of the most vital operation emans is maintaining goo stability an transient performance of the terminal voltage, rotor spee an the power transfer to the network [,2. This requirement shoul be achieve by an aequate control of the system. The high complexity an nonlinearity of power systems, together with their almost continuously time varying nature, require caniate controllers to be able to take into account the important nonlinearities of the power system moel an to be inepenent of the equilibrium point. Much attention has been given to the application of nonlinear control techniques to solve the transient stabilization problem [3,4. Most of these controllers are base on feeback linearization technique [5 9. The main objective in these works was to enhance the system stability an amping performance through excitation control. The nonlinear moel use in these stuies was a reuce thir orer moel of the synchronous machine. In Ref. [, the feeback linearization technique was use to control the rotor angle as well as the terminal voltage, using Corresponing author. Tel.: +2 2678 3 5 59; fax: +2 2524 66 8 2. E-mail aresses: ouassai@yahoo.fr, ouassai@emi.ac.ma (M. Ouassai. the excitation an the turbine s servo-motor input. It is base on a 7th orer moel of the synchronous machine. Feeback linearization is recently enhance by using robust control esigns such as H control an L 2 isturbance attenuation [,2. Several moern control approaches incluing methos base on the passivity principle [3,4, fuzzy logic an neural networks [5,6, backstepping esign [7,8, have been use to esign continuous nonlinear control algorithms which overcome the known limitations of traitional linear controllers: Automatic Voltage Regulator (AVR an the Power System Stabilizer (PSS [9 22. New moern control techniques will continue to fascinate researchers looking for further improvements in high performance power system stability. The sliing moe control approach has been recognize as one of the efficient tools to esign robust controllers for complex high-orer nonlinear ynamic plants operating uner various uncertainty conitions. The major avantage of sliing moe is the low sensitivity to plant parameter variations an isturbances which relaxes the necessity of exact moelling [23. In this paper, a sliing moe controller has been constructe base on a time-varying sliing surface to control the rotor spee an terminal voltage, simultaneously, in orer to enhance the transient stability an to ensure goo post-fault voltage regulation for power system. It is base on a etaile 9th orer moel of a system which consists of a steam turbine an SMIB an takes into account the stator ynamics as well as the amper wining effects an practical limitation on controls. However amper currents are not available for measurement. Consequently, an observer of amper currents is propose. 378-7796/$ see front matter 2 Elsevier B.V. All rights reserve. oi:.6/j.epsr.2..4

36 M. Ouassai et al. / Electric Power Systems Research 84 (22 35 43 The rest of this paper is organize as follows. In Section 2, the ynamic equations of the system uner stuy are presente. A new nonlinear observer for amper wining currents is evelope in Section 3. In Section 4, the nonlinear excitation voltage an rotor spee controllers are erive. Section 5 eals with a number of numerical simulations results of the propose observerbase nonlinear controller. Finally, conclusions are mentione in Section 6. 2. Mathematical moel of power system stuie The system to be controlle, stuie in this work, is shown in Fig.. It consists of synchronous generator riven by steam turbine an connecte to an infinite bus via a transmission line. The synchronous generator is escribe by a 7th orer nonlinear mathematical moel which comprises three stator winings, one fiel wining an two amper winings. The mathematical moel of the plant, which is presente in some etails in [,24 can be written as follows: Electrical equations: ẋ = a x + a 2 x 2 + a 3 x 3 x 6 + a 4 x 4 + a 5 x 6 x 5 + a 6 cos( x 7 + + b u f ( ẋ 2 = a 2 x + a 22 x 2 + a 23 x 3 x 6 + a 24 x 4 + a 25 x 6 x 5 + a 26 cos( x 7 + + b 2 u f (2 ẋ 3 = a 3 x x 6 + a 32 x 2 x 6 + a 33 x 3 + a 34 x 4 x 6 + a 35 x 5 + a 36 sin( x 7 + (3 ẋ 4 = a 4 x + a 42 x 2 + a 43 x 3 x 6 + a 44 x 4 + a 45 x 6 x 5 + a 46 cos( x 7 + + b 3 u f (4 ẋ 5 = a 5 x x 6 + a 52 x 2 x 6 + a 53 x 3 + a 54 x 4 x 6 + a 55 x 5 + a 56 sin( x 7 + (5 Mechanical equations: ẋ 6 = a 6 x 6 + a 62 ( x8 x 6 a 62 T e (6 ẋ 7 = ω R (x 6 (7 Turbine ynamics [25: ẋ 8 = a 8 x 8 + a 82 x 9 (8 Turbine valve control [25: ẋ 9 = a 9 x 9 + a 92 x 6 + b 4 u g (9 where x = [i, i f, i q, i k, i kq, ω, ı, P m, X e T is the vector of state variables, u f the excitation control input, u g the input power of control system. The parameters a ij an b i are escribe in Appenix A. The machine terminal voltage is calculate from Park components v an v q as follows [,24: v t = (v 2 + v 2 q /2 ( with v = c x + c 2 x 2 + c 3 x 3 x 6 + c 4 x 4 + c 5 x 5 x 6 + c 6 cos( x 7 + + c 7 u f ( v q = c 2 x x 6 + c 22 x 2 x 6 + c 23 x 3 + c 24 x 4 x 6 + c 25 x 5 + c 26 sin( x 7 + (2 where c ij are coefficients which epen on the coefficients a ij, on the infinite bus phase voltage V an the transmission line parameters R e an L e. They are escribe in Appenix A. Available states for synchronous generator are the stator phase currents i an i q, voltages at the terminals of the machine v an v q, fiel current i f. It is also assume that the angular spee ω an the power angle ı are available for measurement [26. In the next section the evelopment of an observer of the amper currents i k an i kq will be presente. 3. Development of a sliing moe observer for the amper wining currents For continuous time systems, the state space representation of the electrical ynamics of the power system moel ( (5 is: x x b x x 2 = F x 2 + 4 F2 + b t x 2 u f + H (t (3 x 3 x 5 3 x x 4 x = F t x 2 x 2 + 4 b F22 + 3 u 5 x x 5 f + H 2 (t (4 3 where H (t = [a 6 cos( x 7 +, a 26 cos( x 7 +, a 36 sin( x 7 + T, H 2 (t = [a 46 cos( x 7 +, a 56 sin( x 7 + T [ a a 2 a 3 x 6 [ a F = a 2 a 22 a 23 x 6, F 2 = 4 a 42 a 43 x 6, a a 3 x 6 a 32 x 6 a 5 x 6 a 52 x 6 a 53 33 [ a4 a 5 x 6 [ F 2 =, F 22 = a 24 a 25 x 6 a 34 x 6 a 35 a 44 a 45 x 6 To construct the sliing moe observer, let efine the switching surface S as follows: ˆx x e S(t = ˆx 2 x 2 ˆx 3 x 3 e 2 = (5 Then, an observer for (3 is constructe as: [ ˆx b ˆx + 4 F2 + u t ˆx f 5 ˆx 2 ˆx 3 = F [ ˆx ˆx 2 ˆx 3 + H (t + K [ sgn(ˆx x sgn(ˆx 2 x 2 sgn(ˆx 3 x 3 b 2 (6 where ˆx, ˆx 2 an ˆx 3 are the observe values of i, i f an i q, K is the switching gain, an sgn is the sign function. Moreover, the amper current observer is given from (4 as: t [ [ ˆx ˆx 4 = F ˆx 2 ˆx 2 5 ˆx 3 [ ˆx + 4 F22 + ˆx 5 [ b 3 u f + H 2 (t (7 where ˆx 4 an ˆx 5 are the observe values of i k an i kq. Subtracting (3 from (6, the error ynamics can be written: [ e sgn e x + 4 F2 + K (8 t x 5 e 2 = F [ e e 2 sgn e 2 sgn where x 4 an x 5 are the estimation errors of the amper currents x 4 an x 5.

M. Ouassai et al. / Electric Power Systems Research 84 (22 35 43 37 Fig.. Control system configuration. The switching gain is esigne as: K = min { a e (a 2 e 2 + a 3 ω + a 4 x 4 + a 5 ω x 5 sgn e a 22 e 2 (a 2 e + a 23 ω + a 24 x 4 + a 25 ω x 5 sgn e 2 a 33 (a 3 ωe + a 32 ωe 2 + a 34 ω x 4 + a 35 x 5 sgn where is a positive small value. } (9 Theorem. The globally asymptotic stability of (8 is guarantee, if the switching gain is given by (9. Proof. The stability of the overall structure is guarantee through the stability of the irect axis an quarature axis currents x, x 2, an fiel current x 3 observer. The Lyapunov function for the propose sliing moe amper current is chosen as: V obs = 2 ST S (2 where is an ientity positive matrix. Therefore, the erivative of the Lyapunov function is [ T ( [ e e sgn e V obs = S T x Ṡ = e 2 F e 2 + 4 F2 + K sgn e x 2 e 5 3 sgn = G + G 2 + G 3 (2 where G = a e 2 + a 2 e e 2 + a 3 ωe + a 4 e x 4 + a 5 ωe x 5 + K e G 2 = a 2 e e 2 + a 22 e 2 2 + a 23 ωe 2 + a 24 e 2 x 4 + a 25 ωe 2 x 5 + K e 2 G 3 = a 3 ωe + a 32 ωe 2 + a 33 e 2 3 + a 34 ω x 4 + a 35 x 5 + K Using the esigne switching gain in (9, both G, G 2 an G 3 are negatives. Therefore, V obs is a negative efinite, an the sliing moe conition is satisfie [27. Furthermore the global asymptotic stability of the observer is guarantee. Accoring to (9 by a proper selection of, the influence of parametric uncertainties of the SMIB can be much reuce. The switching gain must large enough to satisfy the reaching conition of sliing moe. Hence the estimation error is confine into the sliing hyerplane: e e t e 2 = e 2 = (22 However, if the switching gain is too large, the chattering noise may lea to estimation errors. To avoi the chattering phenomena, the sign function is replace by the following continuous function in simulation: S(t S(t + ς where ς is a positive constant. 4. Design of sliing moe terminal voltage an spee controller 4.. Terminal voltage control law The ynamic of the terminal voltage (23, is obtaine through the time erivative of ( using ( an (2 where the amper currents are replace by the observer (7: v t t where = v (v v t t + v q v q = v q v q t v t t + v c u f 7 v t t + v [ x c v t t + x 2 c 2 t + x 3 c 3 x 6 t + x 6 c 3 x 3 t +c 4 ˆx 4 t + c 5 x 6 ˆx 5 t + c 5 ˆx 5 x 6 t + c 6 x 7 t sin( x 7 + v = c u f 7 + f (x (23 v t t f (x = v q v q v t t + v [ x c v t t + x 2 c 2 t + x 3 c 3 x 6 t + x 6 c 3 x 3 t +c 4 ˆx 4 t + c 5 ˆx 5 x 6 t + c 5 x 6 ˆx 5 t + c 6 x 7 t sin( x 7 + Furthermore, we efine the tracking error between terminal voltage an its reference as: e = v t v ref t (24 An its error ynamic is erive, using (23, as follows: e t v = c u f 7 + f (x (25 v t t

38 M. Ouassai et al. / Electric Power Systems Research 84 (22 35 43 Accoring to the (24, the propose time-varying sliing surface is efine by: i iˆ S = K e (t (26 where K is a positive constant feeback gain. The next step is to esign a control input which satisfies the sliing moe existence law. The control input is chosen to have the structure: i q i f u f - + Stator an Fiel Currents Observer Eq. (6 iˆq iˆf u(t = u eq (t + u n (t (27 where u eq (t is an equivalent control-input that etermines the system s behavior on the sliing surface an u n (t is a non-linear switching input, which rives the state to the sliing surface an maintains the state on the sliing surface in the presence of the parameter variations an isturbances. The equivalent controlinput is obtaine from the invariance conition an is given by the following conition [23 S = an Ṡ = u(t = u eq (t From the above equation: v Ṡ = K c u f 7 + K v f (x = (28 t t Therefore, the equivalent control-input is given as: u eq (t = v t f (x (29 c 7 v By choosing the nonlinear switching input u n (t as follows: v t u n (t = sgn(e c 7 v (3 where is a positive constant. The control input is given from (27, (29 an (3 as follows: u(t = u f = v t (f (x + sgn(e t c 7 v (3 Using the propose control law (3, the reachability of sliing moe control of (25 is guarantee. 4.2. Sliing moe rotor spee controller We now focuse our attention to the rotor spee tracking objective. The sliing moe-base rotor spee control methoology propose consists of three steps: Step : The rotor spee error is: e 2 = x 6 ω ref (32 where ω ref = p.u. is the esire trajectory. The sliing surface is selecte as: S 2 = K 2 e 2 (t (33 where K 2 is a positive constant. From (32 an (6, the erivative of the sliing surface (33 can be given as: S 2 t = K 2 ( a 6 x 6 + a 62 x 8 x 6 a 62 T e (34 The x 8 can be viewe as a virtual control in the above equation. To ensure the Lyapunov stability criteria i.e. Ṡ 2 S 2 we efine the nonlinear control input x 8eq as: x 8eq = x 6 (a 62 T e a 6 x 6 (35 a 62 The nonlinear switching input x 8n can be chosen as follows: x 8n = x 6 2 sgn(e 2 (36 a 62 where 2 is a positive constant. u f Damper Currents Observer Eq. (7 Fig. 2. Block iagram of the sliing moe amper currents observer. Then, the stabilizing function of the mechanical power is obtaine as: x 8 = x 6 (a 62 T e a 6 x 6 2 sgn(e 2 (37 a 62 When a fault occurs, large currents an torques are prouce. This electrical perturbation may estabilize the operating conitions. Hence, it becomes necessary to account for these uncertainties by esigning a higher performance controller. In (37, as electromagnetic loa T e is unknown, when fault occurs, it has to be estimate aaptively. Thus, let us efine: ˆx 8 = x 6 (a 62 ˆT e a 6 x 6 2 sgn(e 2 (38 a 62 where ˆT e is the estimate value of the electromagnetic loa which shoul be etermine later. Substituting (38 in (34, the rotor spee sliing surface ynamics becomes: S 2 = K 2 ( 2 sgn(e 2 a 62 T e (39 t where T e = T e ˆT e is the estimation error of electromagnetic loa. Step 2: Since the mechanical power x 8 is not our control input, the stabilizing error between x 8 an its esire trajectory x 8 is efine as: = x 8 x 8 (4 To stabilize the mechanical power x 8, the sliing surface is selecte as: S 3 = K 3 (t (4 where K 3 is a positive constant. The erivative of S 3 using (4 an (8 is given as: S 3 = K 3 (a 8 x 8 + a 82 x 9 x 8 (42 t t If we consier the steam valve opening x 9 as a secon virtual control, the equivalent control x 9eq is obtaine as the solution of the problem Ṡ 3 (t =. ( x 9eq = x 8 a 82 t a 8 x 8 (43 Thus, the stabilizing function of the steam valve opening is obtaine as: ( x 9 = x 8 a 82 t a 8 x 8 3 sgn( (44 iˆk iˆkq

M. Ouassai et al. / Electric Power Systems Research 84 (22 35 43 39.8.6 Terminal voltage (p.u..8.6.4 Propose controller Rotor spee (p.u..4.2.998.2.996 Propose controller 8 9 2 3 4 5 6 Time (s 6.994 8 9 2 3 4 5 6 Time (s 55 Rotor angle (egree 5 45 4 Propose controller 35 3 8 9 2 3 4 5 6 Time (s Fig. 3. Simulate result of the propose controller uner large suen fault an operating point P m =,6 p.u. where 3 is a positive constant. Substituting (44 in (42, the steam valve opening sliing surface ynamics becomes: S 3 = 3K 3 sgn( (45 t Step 3: In orer to go one step ahea the steam valve opening error is efine as: e 4 = x 9 x 9 (46 By efining a new sliing surface S 4 (t = K 4 e 4 (t the erivative of S 4 is foun by time ifferentiation of (46 an using (9: S 4 = K 4 (a 9 x 9 + a 92 x 6 + b 4 u g x 9 (47 t t To satisfy the reaching conition Ṡ 4 S 4, the equivalent control u geq (t is given as: ( u geq = x 9 b 4 t a 9 x 9 a 92 x 6 (48 Next, the following choice of feeback control is mae: ( u g = x 9 b 4 t a 9 x 9 a 92 x 6 4 sgn(e 4 4.3. Stability analysis (49 Theorem 3. The ynamic sliing moe control laws (3 an (49 with stabilizing functions (38 an (44 when applie to the single machine infinite power system, guarantee the asymptotic convergence of the outputs v t an x 6 = ω to their esire values v tref an ω ref =, respectively. Proof. Consier a positive efinite Lyapunov function: V con = 2 S2 + 2 S2 2 + 2 S2 3 + 2 S2 4 + 2 T e 2 (5 Using (28, (39, (45 an (47, the erivative of (5 can be erive as follows: T e v V con = Ṡ S + Ṡ 2 S 2 + Ṡ 3 S 3 + Ṡ 4 S 4 + T e = K c u f 7 t v t t + K f (x + K 2 ( 2sgn(e 2 a 62 T e 3K 3 sgn( + K 4 (a 9 x 9 + a 92 x 6 + b 4 u g x 9 T e + T e t t (5 Substituting the control laws (3 an (49 in (5 prouces: V con = K 2 e sgn(e 2K 2 2 e 2 sgn(e 2 3K 2 3 sgn( 4K 2 4 e 4 sgn( K 2 2 a 62 T T e e e 2 + T e = K 2 t e ( 2K 2 2 e 2 3K 2 3 4K 2 4 e T e 4 + t K 2 2 a 62e 2 T e (52 To make the time erivative of V con strictly negative, we choose the aaptive law as: T e = a 62 K 2 t 2 e 2 (53 Thus: V con = K 2 e 2K 2 2 e 2 3K 2 3 4K 2 4 e 4 = 4 ik 2 i e i < (54 i=

4 M. Ouassai et al. / Electric Power Systems Research 84 (22 35 43..8.6 Terminal voltage (p.u..8.6.4.2 Propose controller Rotor spee (p.u..4.2.998.996.994.992 Propse controller 8 9 2 3 4 5 6 Time (s 8.99 8 9 2 3 4 5 6 Time (s 75 Rotor angle (egree 7 65 6 55 5 Propose controller 45 4 8 9 2 3 4 5 6 Time (s Fig. 4. Simulate result of the propose controller uner large suen fault an operating point P m =,9 p.u. From the above analysis, it is evient that the reaching conition is guarantee. Remark. In orer to eliminate the chattering, the iscontinuous control components in (3, (38, (44 an (49 can be replace by a smooth sliing moe component to yiel: u f t = v ( t f (x + c 7 v S (t S (t + 2 x 8 = x ( 6 S 2 (t a 62 T e a 6 x 6 2 a 62 S 2 (t + 3 ( x 9 = x 8 a 82 t S 3 (t a 8 x 8 3 S 3 (t + 4 ( u g = x 9 b 4 t S 4 (t a 9 x 9 a 92 x 6 4 S 4 (t + 5 where i is a small constant. This moification creates a small bounary layer aroun the switching surface in which the system trajectory remains. Therefore, the chattering problem can be reuce significantly [23.. Propose control Backstepping Ref. [7 FLT Ref. [ Terminal voltage (p.u..8.6.4.2 Propose control Backstepping Ref. [7 FLT Ref. [ 4 6 8 2 4 Time (s Rotor spee (p.u..5.995.99 4 6 8 2 4 Time (s Fig. 5. Tracking performance comparison of the propose observer base controller an nonlinear controllers (backstepping Ref. [7 an feeback linearization technique Ref. [.

M. Ouassai et al. / Electric Power Systems Research 84 (22 35 43 4.8.2 +2% from the nominal value.6 +2% from the nominal value Terminal voltage (p.u..8.6.4 Nominal value 2% from the nominal value Rotor spee (p.u..4.2.998.996 2% from the nominal value Nominal value.2.994 2 3 4 5 6 7 8 9 Time (s.992 2 3 4 5 6 7 8 9 Time (s Fig. 6. Dynamic tracking performance of the propose control scheme uner parameter perturbations.. Terminal voltage (p.u..8.6.4.2 Propose control Backstepping Ref. [7 FLT Ref. [ 2 4 6 8 2 Time (s Rotor spee (p.u..8.6.4.2.998.996.994 Propose control.992 Backstepping Ref. [7 FLT Ref. [.99 2 4 6 8 2 Time (s Fig. 7. Performance comparison, uner 2% parameter perturbations, of the propose control scheme an nonlinear controllers (backstepping Ref. [7 an feeback linearization technique Ref. [. 5. Simulation results an iscussion In orer to show the valiity of the mathematical analysis an, hence, to evaluate the performance of the esigne nonlinear control scheme, simulation works are carrie out for the Power System uner severe isturbance conitions which cause significant eviation in generator loaing. The performance of the nonlinear controller was teste on the complete 9th orer moel of SMIB power system (22 MVA, 3,7 kv, incluing all kins of nonlinearities such as exciter ceilings, control signal limiters, etc. an spee regulator. The physical limits of the plant are: max v f = p.u., an X e (t..8 Terminal voltage (p.u..8.6.4.2 Propose control Backstepping Ref. [7 FLT Ref. [ 2 4 6 8 2 Time (s Rotor spee (p.u..6.4.2.998.996.994 Propose control.992 Backstepping Ref. [7 FLT Ref. [.99 2 4 6 8 2 Time (s Fig. 8. Performance comparison, uner +2% parameter perturbations, of the propose control scheme an nonlinear controllers (backstepping Ref. [7 an feeback linearization technique Ref. [.

42 M. Ouassai et al. / Electric Power Systems Research 84 (22 35 43 Table Parameters of the transmission line in p.u. Parameter Value L e, inuctance of the transmission line.4 R e, resistance of the transmission line.2 Table 2 Parameters of the power synchronous generator in p.u. Parameter Value R s, stator resistance.96 3 R f, fiel resistance 7.42 4 R k, irect amper wining resistance 3. 3 R kq, quarature amper wining resistance 54 3 L, irect self-inuctance.7 L q, quarature self-inuctances.64 L f, rotor self inuctance.65 L k, irect amper wining self inuctance.65 L kq, quarature amper wining self inuctance.526 L m, irect magnetizing inuctance.55 L mq, quarature magnetizing inuctance.49 V, infinite bus voltage D, amping constant H, inertia constant 2.37 s Tabl Parameters of the steam turbine an spee governor. Parameter Value T t, time constant of the turbine.35 s K t, gain of the turbine R regulation constant of the system.5 T g, time constant of the spee governor.2 s K g, gain of the spee governor The system configuration is presente as shown in Fig.. The simulation of the propose sliing moe amper currents observer is implemente base on the scheme shown in Fig. 2. Digital simulations have been carrie out using Matlab Simulink. The parameter values use in the ensuing simulation are given in the Appenix B. 5.. Observer base controller performance evaluation First, we verify the stability an asymptotic tracking performance of the propose control systems. The simulate results are given in Figs. 3 an 4. It is shown terminal voltage, rotor spee an rotor angle of the power system, respectively. The operating points consiere are P m =.6 p.u. an.9 p.u. The fault consiere in this paper is a symmetrical three-phase short circuit, which occurs closer to the generator bus, at t = s an remove by opening the barkers of the faulte line at t =. s. The results are compare with those of the linear IEEE type AVR + PSS an spee regulator. As it can be seen, the propose controller can quickly an accurately track the esire terminal voltage an rotor spee espite the ifferent operating points. 5.2. Comparison of ynamic performances for propose controller an non linear controllers In orer to prove the robustness of the propose controller, the results are compare with two non linear controllers: (i Feeback Linearization Technique (FLT Ref. [ an (ii Backstepping Ref. [7. The operating point consiere is P m =.725 p.u. The fault occurs closer to the generator bus. The simulation results are presente in Fig. 5. It can be seen that both ynamics of the terminal voltage an the rotor spee settle to their prefault values very quickly with propose observer base controller. It is obvious that with the erive high control accuracy an stability can be achieve. 5.3. Robustness to parameters uncertainties The variation of system parameters is consiere for robustness evaluation of the propose observer-base controller. The values of the transmission line (L e, R e an the inertia constant H increase by +2% an 2% from their original values, respectively. The responses of the terminal voltage an rotor spee are shown in Fig. 6. In aition to the abrupt an permanent variation of the power system parameters a three-phase short-circuit is simulate at the terminal of the generator. Figs. 7 an 8 show the performances of the combine observer controller an the other controllers Ref. [ an Ref. [7. It can be seen that the esigne control scheme is able to eal with the uncertainties of parameters an preserve the global stability of the system with goo performances in transient an steay states. 6. Conclusion In this paper, a new nonlinear observer controller scheme has been evelope an applie to the single machine infinite bus power system, base on the complete 7th orer moel of the synchronous generator. The aim of the stuy is to achieve both transient stability enhancement an goo postfault performance of the generator terminal voltage. The sliing moe strategy was aopte to evelop a nonlinear observer of amper wining currents an nonlinear terminal voltage an rotor spee controller. The etaile erivation for the control laws has been provie. Globally exponentially stable of both the observer an control laws has been proven by applying Lyapunov stability theory. Simulation results have confirme that the observer-base nonlinear controller can effectively improve the transient stability an voltage regulation uner large suen fault. The combine observer controller scheme emonstrates consistent superiority oppose to a system with linear controllers (IEEE type AVR + PSS an nonlinear controllers. It can be seen from the simulation stuy that the esigne sliing moe observer base-controller possesses a great robustness to eal with parameter uncertainties.

M. Ouassai et al. / Electric Power Systems Research 84 (22 35 43 43 Appenix A. The coefficients of the mathematical moel a = (R s + R e (L f L k L 2 m ω RD, a 2 = R f (L mq L k L 2 m ω RD, a 3 = (L q + L e (L m L k L 2 m ω RD a 5 = L mq (L f L k L 2 m ω RD, a 4 = R k ((L + L e L m L 2 m ω RD, a 6 = V ((L + L e L k L 2 m ω RD b = (L m L k L 2 m ω RD, a 2 = (R s + R e (L m L k L 2 m ω RD, a 22 = R f ((L + L e L k L 2 m ω RD a 23 = (L q + L e (L m L k L 2 m ω RD, a 24 = R k ((L + L e L m L 2 m ω RD, a 25 = L mq (L m L k L 2 m ω RD a 26 = V (L m L k L 2 m ω RD, b 2 = ((L + L f L k L 2 m ω RD, a 3 = (L + L e L kq ω R Dq a 32 = L m L kq ω R Dq, a 33 = (R s + R e L kq ω R Dq, a 34 = L m L kq ω R Dq, a 35 = L mq.r kq ω R Dq, a 36 = V L kq ω R Dq a 4 = (R s + R e (L f L m L 2 m ω RD, a 42 = R f ((L + L e L m L 2 m ω RD, a 45 = L m (L mq.l f L 2 m ω RD a 43 = (L q + L e (L m L L 2 m ω RD, a 44 = R k ((L + L e L f L 2 m ω RD, a 46 = V (L m.l f + L 2 m ω RD b 3 = ((L + L e L m L 2 m ω RD, a 5 = (L + L e L mq ω R Dq, a 52 = L m L mq ω R Dq, a 53 = (R s + R e L mq ω R Dq a 54 = L m L mq ω R Dq, a 55 = R kq (L q + L e ω R Dq, a 56 = V L mq ω R D, a 6 = D(2H, a 62 = (2H a 8 = (T m, a 82 = K m (T m, a 9 = (T g, a 92 = K g (T g Rω R, b 4 = K g (T g, c = R e + a L e ω R c 2 = a 2 L e ω R, c 3 = L e (a 3 ω R, c 4 = a 4 L e ω R, c 5 = a 5 L e ω R, c 6 = V + a 6 L e ω R, c 7 = b L e ω R c 2 = L e + a 3 L e ω R, c 22 = a 32 L e ω R, c 23 = a 33 L e ω R + R e, c 24 = a 34 L e ω R, c 25 = a 35 L e ω R, c 26 = V + a 36 L e ω R here we have enote D = (L + L e L f L k L 2 m (L + L f + L k + 2L 3 m, D q = (L q + L e L kq Lmq 2 Appenix B. The parameters of the system are as follows [24,25 (Tables 3 References [ Y. Guo, D.J. Hill, Y. Wang, Global transient stability an voltage regulation for power systems, IEEE Transactions on Power Systems 6 (2 678 688. [2 C. Zhu, R. Zhou, Y. Wang, A new nonlinear voltage controller for power systems, International Journal of Electrical Power an Energy Systems 9 (996 9 27. [3 Z. Xi, G. Feng, D. Cheng, Q. Lu, Nonlinear ecentralize saturate controller esign for power systems, IEEE Transactions on Control Systems Technology (23 59 58. [4 W. Lin, T. Shen, Robust passivity an feeback esign for minimum-phase nonlinear systems with structural un certainty, Automatica 35 (999 35 48. [5 A. Isiori, Nonlinear Control System, Springer Verlag, Berlin, 989. [6 Y. Wang, D.J. Hill, R.H. Mileton, L. Gao, Transient stability enhancement an voltage regulation of power systems, IEEE Transactions on Power Systems 2 (993 62 628. [7 E. De Tuglie, S. Marcello Iannone, F. Torelli, Feeback-linearization an feeback feeforwar ecentralize control for multimachine power system, Electric Power Systems Research 78 (28 382 39. [8 C.A. King, J.W. Chapman, M.D. Ilic, Feeback linearizing excitation control on a full-scale power system moel, IEEE Transactions on Power Systems 2 (994 2. [9 S. Jain, F. Khorrami, B. Faranesh, Aaptive nonlinear excitation control of power system with unknown interconnections, IEEE Transactions on Control Systems Technology 4 (994 436 447. [ O. Akhrif, F.A. Okou, L.A. Dessaint, R. Champagne, Application of mulitivariable feeback linearization scheme for rotor angle stability an voltage regulation of power systems, IEEE Transactions on Power Systems 4 (999 62 628. [ L. Yan-Hong, L. Chun-Wen, W. Yu-Zhen, Decentralize excitation control of multi-machine multi-loa power systems using Hamiltonian function metho, Acta Automatica Sinica 35 (29 92 925. [2 T. Shen, S. Mei, Q. Lu, W. Hu, K. Tamura, Aaptive nonlinear excitation control with L2 isturbance attenuation for power systems, Automatica 39 (23 8 9. [3 A.Y. Pogromsky, A.L. Frakov, D.J. Hill, Passivity base amping of power system oscillations, in: Proceeings of the IEEE CDC, Kobe, Japan, 996. [4 R. Ortega, A. Stankovic, P.A. Stefanov, Passivation approach to power systems stabilization, in: Proceeings of the IFAC NOLCOS, Enshee, Netherlans, 998. [5 P. Hoang, K. Tomsovic, Design an analysis of an aaptive fuzzy power system stabilizer, in: Proceeings of the IEEE PES Winter Meeting, 996. [6 J. Wook Park, G. Ronal Harley, K. Ganesh Venayagamoorthy, Decentralize optimal neuro-controllers for generation an transmission evices in an electric power network, Engineering Applications of Artificial Intelligence 8 (25 37 46. [7 M. Ouassai, A. Nejmi, M. Cherkaoui, M. Maaroufi, A nonlinear backstepping controller for power systems terminal voltage an rotor spee controls, International Review of Automatic Control 3 (28 355 363. [8 A. Karimi, A. Feliachi, Decentralize aaptive backstepping control of electric power systems, International Electric Power Systems Research 78 (28 484 493. [9 P. Kunur, Power System Stability an Control, MacGraw-Hill, 994. [2 M.S. Ghazizaeh, F.M. Hughs, A generator transfer function regulator for improve excitation control, IEEE Transactions on Power Systems 3 (998 437 44. [2 Ail., A. Ghanakly, A.M. Farhou, A parametrically optimize self tuning regulator for power system stabilizers, IEEE Transactions on Power Systems 7 (992 245 25. [22 Q. Zhao, J. Jiang, Robust controller esign for generator excitation systems, IEEE Transactions on Energy Conversion (995 2 27. [23 V.I. Utkin, J. Gulner, J. Shi, Sliing Moe Control in Electromechanical Systems, Taylor an Francis, Lonon, 999. [24 P.M. Anerson, A.A. Foua, Power System Control an Stability, IEEE Press, 994. [25 D.J. Hill, Y. Wang, Nonlinear ecentralize control of large scale power systems, Automatica 36 (2 275 289. [26 F.P. e Mello, Measurement of synchronous machine rotor angle from analysis of zero sequence harmonic components of machine terminal voltage, IEEE Transactions on Power Delivery 9 (994 77 777. [27 V.I. Utkin, Sliing moe control esign principles an applications to electric rives, IEEE Transactions on Inustrial Electronics 4 (993 23 36.