Observer-based nonlinear control of power system using sliding mode control strategy

Electric Power Systems Research 84 (2012) 135– 143 Contents lists available at SciVerse ScienceDirect Electric Power Systems Research jou rn al h om epage: www.elsev ier .co Observ ste control M. Ouass a Department o ées, B.P b Department o P. 765, a r t i c l Article history: Received 13 N Received in re Accepted 28 O Available onlin Keywords: Sliding mode c Lyapunov met Damper curren Power system Transient stab stabi team tage a ions. ode o back dynam the ystem tran 1. Introdu Successful operation of power system depends largely on the ability to provide reliable and uninterrupted service. The reliabil- ity of the power supply implies much more than merely being available. Id frequency a as power ch vital operat performanc transfer to t by an adequ The hig together w require can important n pendent of to the appli sient stabili on feedback these works formance th these studie machine. In to control t ∗ Correspon E-mail add itati 7th order model of the synchronous machine. Feedback lineariza- tion is recently enhanced by using robust control designs such as H∝ control and L2 disturbance attenuation [11,12]. Several modern control approaches including methods based on the passivity prin- 0378-7796/$ – doi:10.1016/j. eally, the loads must be fed at constant voltage and t all times. However, small or large disturbances such anges or short circuits may transpire. One of the most ion demands is maintaining good stability and transient e of the terminal voltage, rotor speed and the power he network [1,2]. This requirement should be achieved ate control of the system. h complexity and nonlinearity of power systems, ith their almost continuously time varying nature, didate controllers to be able to take into account the onlinearities of the power system model and to be inde- the equilibrium point. Much attention has been given cation of nonlinear control techniques to solve the tran- zation problem [3,4]. Most of these controllers are based linearization technique [5–9]. The main objective in was to enhance the system stability and damping per- rough excitation control. The nonlinear model used in s was a reduced third order model of the synchronous Ref. [10], the feedback linearization technique was used he rotor angle as well as the terminal voltage, using ding author. Tel.: +21 2678 03 50 59; fax: +21 2524 66 80 12. resses: [email protected], [email protected] (M. Ouassaid). ciple [13,14], fuzzy logic and neural networks [15,16], backstepping design [17,18], have been used to design continuous nonlinear control algorithms which overcome the known limitations of tra- ditional linear controllers: Automatic Voltage Regulator (AVR) and the Power System Stabilizer (PSS) [19–22]. New modern control techniques will continue to fascinate researchers looking for further improvements in high performance power system stability. The sliding mode control approach has been recognized as one of the efficient tools to design robust controllers for complex high-order nonlinear dynamic plants oper- ating under various uncertainty conditions. The major advantage of sliding mode is the low sensitivity to plant parameter variations and disturbances which relaxes the necessity of exact modelling [23]. In this paper, a sliding mode controller has been constructed based on a time-varying sliding surface to control the rotor speed and terminal voltage, simultaneously, in order to enhance the tran- sient stability and to ensure good post-fault voltage regulation for power system. It is based on a detailed 9th order model of a system which consists of a steam turbine and SMIB and takes into account the stator dynamics as well as the damper winding effects and practical limitation on controls. However damper currents are not available for measurement. Consequently, an observer of damper currents is proposed. see front matter © 2011 Elsevier B.V. All rights reserved. epsr.2011.10.014 er-based nonlinear control of power sy strategy aida,∗, M. Maaroufib, M. Cherkaouib f Industrial Engineering, Caddi AAyad University, Ecole Nationale des Sciences Appliqu f Electrical Engineering, Mohammed V University, Ecole Mohammadia d’Ingénieurs, B. e i n f o ovember 2009 vised form 27 February 2011 ctober 2011 e 25 November 2011 ontrol hods ts observer ility a b s t r a c t This paper presents a new transient chronous power generator driven by s high performance for the terminal vol and a wide range of operating condit technique. First, a nonlinear sliding m structed. Second, the stabilizing feed which takes into account the stator shown to be asymptotically stable in Machine-Infinite-Bus (SMIB) power s combined observer–controller for the ction the exc m/locate /epsr m using sliding mode . 63, Sidi Bouzid, Safi, Morocco Agdal, Rabat, Morocco lization with voltage regulation analysis approach of a syn- turbine and connected to an infinite bus. The aim is to obtain nd the rotor speed simultaneously under a large sudden fault The methodology adopted is based on sliding mode control bserver for the synchronous machine damper currents is con- laws for the complete ninth order model of a power system, ics as well as the damper effects, are developed. They are context of Lyapunov theory. Simulation results, for a single- , are given to demonstrate the effectiveness of the proposed sient stabilization and voltage regulation. © 2011 Elsevier B.V. All rights reserved. on and the turbine’s servo-motor input. It is based on a 136 M. Ouassaid et al. / Electric Power Systems Research 84 (2012) 135– 143 The rest of this paper is organized as follows. In Section 2, the dynamic equations of the system under study are presented. A new nonlinear observer for damper winding currents is devel- oped in Section 3. In Section 4, the nonlinear excitation voltage and rotor s number of n based nonl in Section 6 2. Mathem The syst Fig. 1. It co bine and co synchronou ematical m winding an plant, whic as follows: Electrical e x˙1 = a11x1 + a16 x˙2 = a21x1 + a26 x˙3 = a31x1 + a36 x˙4 = a41x1 + a46 x˙5 = a51x1 + a56 Mechanica x˙6 = a61x6 x˙7 = ωR(x6 Turbine dy x˙8 = a81x8 Turbine va x˙9 = a91x9 where x = variables, of control Appendix The mac nents vd an vt = (v2d + v with vd = c11x1 + + c16 c vq = c21x1x6 + c22x2x6 + c23x3 + c24x4x6 + c25x5 + c26 sin(−x7 + �) (12) where cij are coefficients which depend on the coefficients aij, on the bus Le. T ilabl ts id d cur wer the will elop g cu cont ctric] = ] = [a1 [a4[ a1 a2 a31[ a1 a2 a34 struc S as[ xˆ1 xˆ2 xˆ3 n, an] = xˆ1, xˆ ing g t obs] = xˆ4 an tract] = x˜4 a x5. peed controllers are derived. Section 5 deals with a umerical simulations results of the proposed observer- inear controller. Finally, conclusions are mentioned . atical model of power system studied em to be controlled, studied in this work, is shown in nsists of synchronous generator driven by steam tur- nnected to an infinite bus via a transmission line. The s generator is described by a 7th order nonlinear math- odel which comprises three stator windings, one field d two damper windings. The mathematical model of the h is presented in some details in [10,24] can be written quations: + a12x2 + a13x3x6 + a14x4 + a15x6x5 cos(−x7 + �) + b1ufd (1) + a22x2 + a23x3x6 + a24x4 + a25x6x5 cos(−x7 + �) + b2ufd (2) x6 + a32x2x6 + a33x3 + a34x4x6 + a35x5 sin(−x7 + �) (3) + a42x2 + a43x3x6 + a44x4 + a45x6x5 cos(−x7 + �) + b3ufd (4) x6 + a52x2x6 + a53x3 + a54x4x6 + a55x5 sin(−x7 + �) (5) l equations: + a62 ( x8 x6 ) − a62Te (6) − 1) (7) namics [25]: + a82x9 (8) lve control [25]: + a92x6 + b4ug (9) [id, ifd, iq, ikd, ikq, ω, ı, Pm, Xe] T is the vector of state ufd the excitation control input, ug the input power system. The parameters aij and bi are described in A. hine terminal voltage is calculated from Park compo- d vq as follows [10,24]: 2 q) 1/2 (10) c12x2 + c13x3x6 + c14x4 + c15x5x6 os(−x7 + �) + c17ufd (11) infinite Re and Ava curren vq, fiel the po section and ikq 3. Dev windin For the ele d dt [ x1 x2 x3 d dt [ x4 x5 where H1(t) = H2(t) = F11 = F12 = To con surface S(t) = The d dt [ xˆ1 xˆ2 xˆ3 where switch curren d dt [ xˆ4 xˆ5 where Sub d dt [ e1 e2 e3 where x4 and phase voltage V∞ and the transmission line parameters hey are described in Appendix A. e states for synchronous generator are the stator phase and iq, voltages at the terminals of the machine vd and rent ifd. It is also assumed that the angular speed ω and angle ı are available for measurement [26]. In the next development of an observer of the damper currents ikd be presented. ment of a sliding mode observer for the damper rrents inuous time systems, the state space representation of al dynamics of the power system model (1)–(5) is: F11 [ x1 x2 x3 ] + F12 [ x4 x5 ] + [ b1 b2 0 ] ufd + H1(t) (13) F21 [ x1 x2 x3 ] + F22 [ x4 x5 ] + [ b3 0 ] ufd + H2(t) (14) 6 cos(−x7 + �), a26 cos(−x7 + �), a36 sin(−x7 + �)] T , 6 cos(−x7 + �), a56 sin(−x7 + �)] T 1 a12 a13x6 1 a22 a23x6 x6 a32x6 a33 ] , F21 = [ a41 a42 a43x6 a51x6 a52x6 a53 ] , 4 a15x6 4 a25x6 x6 a35 ] , F22 = [ a44 a45x6 a54x6 a55 ] t the sliding mode observer, let define the switching follows: − x1 − x2 − x3 ] ≡ [ e1 e2 e3 ] = 0 (15) observer for (13) is constructed as: F11 [ xˆ1 xˆ2 xˆ3 ] + F12 [ xˆ4 xˆ5 ] + [ b1 b2 0 ] ufd + H1(t) + K [ sgn(xˆ1 − x1) sgn(xˆ2 − x2) sgn(xˆ3 − x3) ] (16) 2 and xˆ3 are the observed values of id, ifd and iq, K is the ain, and sgn is the sign function. Moreover, the damper erver is given from (14) as: F21 [ xˆ1 xˆ2 xˆ3 ] + F22 [ xˆ4 xˆ5 ] + [ b3 0 ] ufd + H2(t) (17) d xˆ5 are the observed values of ikd and ikq. ing (13) from (16), the error dynamics can be written: F11 [ e1 e2 e3 ] + F12 [ x˜4 x˜5 ] + K [ sgn e1 sgn e2 sgn e3 ] (18) nd x˜5 are the estimation errors of the damper currents M. Ouassaid et al. / Electric Power Systems Research 84 (2012) 135– 143 137 ration The swit K = min × {−a11 −a22 −a33 where � is a Theorem 1 if the switch Proof. The the stability and field cu posed slidin Vobs = 1 2 ST� where � is Therefor V˙obs = ST� = G1 where G1 = a11e21 G2 = a21e1e G3 = a31ωe Using the d negatives. T mode condi stability of Accordin parametric The swit condition o into the slid d dt [ e1 e2 e3 ] = eve ad to n fun latio ς1 ς1 is ign o ller rmin dyn e de ts are 1 vt ( + vd vt +c14 Fig. 1. Control system configu ching gain is designed as: |e1| − (a12e2 + a13ωe3 + a14x˜4 + a15ωx˜5)sgn e1 |e2| − (a21e1 + a23ωe3 + a24x˜4 + a25ωx˜5) sgn e2 |e3| − (a31ωe1 + a32ωe2 + a34ωx˜4 + a35x˜5) sgn e3 } − � (19) positive small value. . The globally asymptotic stability of (18) is guaranteed, ing gain is given by (19). stability of the overall structure is guaranteed through of the direct axis and quadrature axis currents x1, x2, rrent x3 observer. The Lyapunov function for the pro- g mode damper current is chosen as: S (20) an identity positive matrix. e, the derivative of the Lyapunov function is S˙ = [ e1 e2 e3 ]T � ( F11 [ e1 e2 e3 ] + F12 [ x˜4 x˜5 ] + K [ sgn e1 sgn e2 sgn e3 ]) + G2 + G3 (21) How may le the sig in simu S(t) |S(t)| + where 4. Des contro 4.1. Te The the tim curren dvt dt = + a12e1e2 + a13ωe1e3 + a14e1x˜4 + a15ωe1x˜5 + K |e1| 2 + a22e22 + a23ωe2e3 + a24e2x˜4 + a25ωe2x˜5 + K |e2| 1e3 + a32ωe2e3 + a33e23 + a34ωe3x˜4 + a35e3x˜5 + K |e3| esigned switching gain in (19), both G1, G2 and G3 are herefore, V˙obs is a negative definite, and the sliding tion is satisfied [27]. Furthermore the global asymptotic the observer is guaranteed. g to (19) by a proper selection of �, the influence of uncertainties of the SMIB can be much reduced. ching gain must large enough to satisfy the reaching f sliding mode. Hence the estimation error is confined ing hyerplane:[ e1 e2 e3 ] = 0 (22) = c1 where f (x) = vq vt d d +c14 Furthermor age and its e1 = vt − vret And its erro de1 dt = c17 v v . r, if the switching gain is too large, the chattering noise estimation errors. To avoid the chattering phenomena, ction is replaced by the following continuous function n: a positive constant. � f sliding mode terminal voltage and speed al voltage control law amic of the terminal voltage (23), is obtained through rivative of (10) using (11) and (12) where the damper replaced by the observer (17): vd dvd dt + vq dvq dt ) = vq vt dvq dt + c17 vd vt dufd dt[ c11 dx1 dt + c12 dx2 dt + c13x6 dx3 dt + c13x3 dx6 dt dxˆ4 dt + c15x6 dxˆ5 dt + c15xˆ5 dx6 dt + c16 dx7 dt sin(−x7 + �) ] vd dufd 7 vt dt + f (x) (23) vq t + vd vt [ c11 dx1 dt + c12 dx2 dt + c13x6 dx3 dt + c13x3 dx6 dt dxˆ4 dt + c15xˆ5 dx6 dt + c15x6 dxˆ5 dt + c16 dx7 dt sin(−x7 + �) ] e, we define the tracking error between terminal volt- reference as: f (24) r dynamic is derived, using (23), as follows: d t dufd dt + f (x) (25) 138 M. Ouassaid et al. / Electric Power Systems Research 84 (2012) 135– 143 According to the (24), the proposed time-varying sliding surface is defined by: S1 = K1e1(t) (26) where K1 is design a co law. The co u(t) = ueq(t where ueq( system’s be switching i maintains t parameter input is obt following c From th S˙1 = K1c17 v v Therefore, t ueq(t) = − c By choosing un(t) = −˛1 where ˛1 is (29) and (30 u(t) = dufd dt Using the p mode contr 4.2. Sliding We now tive. The s proposed co Step 1: T e2 = x6 − ω where ωref selected as: S2 = K2e2(t where K2 is the sliding dS2 dt = K2 ( The x8 can To ensure th nonlinear c x∗8eq = x6 a62 ( The nonline x∗8n = −˛2 x a where ˛2 is Stator and Field fdu fdu dˆi iˆ di qi ig. 2. n, th ed as 6 62 (a6 a fau cal p , it be ing a (37), , it ha 6 62 (a6 Tˆe is be slidin K2(− T˜e = p 2: S biliz d as: − x8 ilize 3e3(t K3 is iven K3 ( a81x8 + a82x9 − dx∗8 dt ) (42) onsider the steam valve opening x9 as a second virtual con- e equivalent control x∗9eq is obtained as the solution of the m S˙3(t) = 0. 1 a82 ( dx∗8 dt − a81x8 ) (43) he stabilizing function of the steam valve opening is obtained 1 82 ( dx∗8 dt − a81x8 − ˛3 sgn(e3) ) (44) a positive constant feedback gain. The next step is to ntrol input which satisfies the sliding mode existence ntrol input is chosen to have the structure: ) + un(t) (27) t) is an equivalent control-input that determines the havior on the sliding surface and un(t) is a non-linear nput, which drives the state to the sliding surface and he state on the sliding surface in the presence of the variations and disturbances. The equivalent control- ained from the invariance condition and is given by the ondition [23] S1 = 0 and S˙1 = 0 ⇒ u(t) = ueq(t) e above equation: d t dufd dt + K1f (x) = 0 (28) he equivalent control-input is given as: vt 17vd f (x) (29) the nonlinear switching input un(t) as follows: vt c17vd sgn(e1) (30) a positive constant. The control input is given from (27), ) as follows: = − vt c17vd (f (x) + ˛1 sgn(e1)) (31) roposed control law (31), the reachability of sliding ol of (25) is guaranteed. mode rotor speed controller focused our attention to the rotor speed tracking objec- liding mode-based rotor speed control methodology nsists of three steps: he rotor speed error is: ref (32) = 1 p.u. is the desired trajectory. The sliding surface is ) (33) a positive constant. From (32) and (6), the derivative of surface (33) can be given as: a61x6 + a62 x8 x6 − a62Te ) (34) be viewed as a virtual control in the above equation. e Lyapunov stability criteria i.e. S˙2S2 ≺ 0 we define the ontrol input x∗8eq as: a62Te − a61x6) (35) ar switching input x∗8n can be chosen as follows: 6 62 sgn(e2) (36) a positive constant. fdi F The obtain x∗8 = x a When electri Hence design In occurs xˆ∗8 = x a where should speed dS2 dt = where Ste the sta define e3 = x∗8 To stab as: S3 = K where (8) is g dS3 dt = If we c trol, th proble x∗9eq = Thus, t as: x∗9 = a Currents Observer Eq. (16) Damper Currents Observer Eq. (17) + - q fˆdi kˆdi kˆqi Block diagram of the sliding mode damper currents observer. e stabilizing function of the mechanical power is : 2Te − a61x6 − ˛2 sgn(e2)) (37) lt occurs, large currents and torques are produced. This erturbation may destabilize the operating conditions. comes necessary to account for these uncertainties by higher performance controller. as electromagnetic load Te is unknown, when fault s to be estimated adaptively. Thus, let us define: 2Tˆe − a61x6 − ˛2 sgn(e2)) (38) the estimated value of the electromagnetic load which determined later. Substituting (38) in (34), the rotor g surface dynamics becomes: ˛2 sgn(e2) − a62T˜e) (39) Te − Tˆe is the estimation error of electromagnetic load. ince the mechanical power x8 is not our control input, ing error between x8 and its desired trajectory x∗8 is (40) the mechanical power x8, the sliding surface is selected ) (41) a positive constant. The derivative of S3 using (40) and as: M. Ouassaid et al. / Electric Power Systems Research 84 (2012) 135– 143 139 1 AVR+PSS controllers 9 1.006 1.008 AVR+PSS controllers 3 PSS con ed contr dden where ˛3 is valve openi dS3 dt = −˛3K Step 3: I error is defi e4 = x9 − x∗9 By defining S4 is found dS4 dt = K4 ( To satisfy t ugeq(t) is giv ugeq = 1 b4 ( Next, the fo ug = 1 b4 ( d 4.3. Stabilit Theorem 3 with stabiliz machine infi of the outpu respectively Co 8 9 10 11 12 13 14 15 160 0.2 0.4 0.6 0.8 Time (s) Te rm in al v ol ta ge (p .u. ) Proposed controller 8 0.994 0.996 0.998 1 1.002 1.004 R ot or s pe ed (p .u. ) 8 9 10 11 12 130 35 40 45 50 55 60 Time (s) R ot or a ng le (d eg ree ) AVR+ Propos Fig. 3. Simulated result of the proposed controller under large su a positive constant. Substituting (44) in (42), the steam Proof. ng sliding surface dynamics becomes: 3 sgn(e3) (45) n order to go one step ahead the steam valve opening ned as: (46) a new sliding surface S4(t) = K4e4(t) the derivative of by time differentiation of (46) and using (9): a91x9 + a92x6 + b4ug − dx∗9 dt ) (47) he reaching condition S˙4S4 ≺ 0, the equivalent control en as: dx∗9 dt − a91x9 − a92x6 ) (48) llowing choice of feedback control is made: x∗9 dt − a91x9 − a92x6 − ˛4 sgn(e4) ) (49) y analysis . The dynamic sliding mode control laws (31) and (49) ing functions (38) and (44) when applied to the single nite power system, guarantee the asymptotic convergence ts vt and x6 = ω to their desired values vtref and ωref = 1, . Vcon = 12S 2 1 Using (2 derived as f V˙con = S˙1S1 + K1 + K4 Substitutin V˙con = −˛1 − ˛4 − ˛2 To make the adaptive law dT˜e dt = �a62 Thus: V˙con = −˛1 = − 10 11 12 13 14 15 16 Time (s) Proposed controller 14 15 16 trollers oller fault and operating point Pm = 0,6 p.u. nsider a positive definite Lyapunov function: + 1S2 + 1S2 + 1S2 + 1 T˜2 (50) 2 2 2 3 2 4 2� e 8), (39), (45) and (47), the derivative of (50) can be ollows: + S˙2S2 + S˙3S3 + S˙4S4 + T˜e 1 � dT˜e dt = K1c17 vd vt dufd dt f (x) + K2(−˛2sgn(e2) − a62T˜e) − ˛3K3 sgn(e3)( a91x9 + a92x6 + b4ug − dx∗9 dt ) + T˜e 1 � dT˜e dt (51) g the control laws (31) and (49) in (51) produces: K21 e1 sgn(e1) − ˛2K22 e2 sgn(e2) − ˛3K23 e3 sgn(e3) K24 e4 sgn(e3) − K22a62T˜ee2 + T˜e 1 � dT˜e dt = −˛1K21 |e1| K22 |e2| − ˛3K23 |e3| − ˛4K24 |e4| + ( 1 � dT˜e dt − K22a62e2 ) T˜e (52) time derivative of Vcon strictly negative, we choose the as: K22 e2 (53) K21 |e1| − ˛2K22 |e2| − ˛3K23 |e3| − ˛4K24 |e4| 4∑ i=1 ˛iK 2 i |ei| < 0 (54) 140 M. Ouassaid et al. / Electric Power Systems Research 84 (2012) 135– 143 8 9 10 11 12 13 14 15 160 0.2 0.4 0.6 0.8 1 Time (s) Te rm in al v ol ta ge (p .u. ) AVR+PSS controllers Proposed controller 8 9 10 11 12 13 14 15 160.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 1.01 Time (s) R ot or s pe ed (p .u. ) AVR+PSS controllers Propsed controller 1 65 70 75 80 le (d eg ree ) contro control dden From the ab guaranteed Remark. control com a smooth sl dufd dt = − v c1 x∗8 = x6 a62 ( a Fig. 5. Trackin Ref. [10]). 8 9 10 11 12 13 40 45 50 55 60 Time (s) R ot or a ng AVR+PSS Proposed Fig. 4. Simulated result of the proposed controller under large su ove analysis, it is evident that the reaching condition is . � In order to eliminate the chattering, the discontinuous ponents in (31), (38), (44) and (49) can be replaced by iding mode component to yield: t 7vd ( f (x) + ˛1 S1(t) |S1(t)| + �2 ) 62Te − a61x6 − ˛2 S2(t) |S2(t)| + �3 ) x∗9 = 1 a82 ( ug = 1 b4 ( d where �i � boundary l tem traject reduced sig 4 6 8 10 12 14 0 0.2 0.4 0.6 0.8 1 Time (s) Te rm in al v ol ta ge (p .u. ) Proposed control FLT Ref. [10] Backstepping Ref. [17] 4 0.99 0.995 1 1.005 1.01 R ot or s pe ed (p .u. ) g performance comparison of the proposed observer based controller and nonlinear con 4 15 16 llers ler fault and operating point Pm = 0,9 p.u. dx∗8 dt − a81x8 − ˛3 S3(t) |S3(t)| + �4 ) x∗9 dt − a91x9 − a92x6 − ˛4 S4(t) |S4(t)| + �5 ) 0 is a small constant. This modification creates a small ayer around the switching surface in which the sys- ory remains. Therefore, the chattering problem can be nificantly [23]. 6 8 10 12 14 Time (s) Proposed control FLT Ref. [10] Backstepping Ref. [17] trollers (backstepping Ref. [17] and feedback linearization technique M. Ouassaid et al. / Electric Power Systems Research 84 (2012) 135– 143 141 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 Time (s) Te rm in al v ol ta ge (p .u. ) +20% from the nominal value −20% from the nominal value Nominal value 2 3 4 5 6 7 8 9 10 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 Time (s) R ot or s pe ed (p .u. ) +20% from the nominal value −20% from the nominal value Nominal value Fig. 6. Dynamic tracking performance of the proposed control scheme under parameter perturbations. 1 .) 1.006 1.008 1.01 Fig. 7. Perform linearization t 5. Simulat In order and, hence linear cont Power Syst significant Fig. 8. Perform linearization t 0.4 0.6 0.8 Te rm in al v ol ta ge (p .u 0.994 0.996 0.998 1 1.002 1.004 R ot or s pe ed (p .u. ) 0 2 4 6 8 10 12 0 0.2 Time (s) Proposed control FLT Ref. [10] Backstepping Ref. [17] 0 0.99 0.992 ance comparison, under −20% parameter perturbations, of the proposed control sche echnique Ref. [10]). ion results and discussion to show the validity of the mathematical analysis , to evaluate the performance of the designed non- rol scheme, simulation works are carried out for the em under severe disturbance conditions which cause deviation in generator loading. The performance of the nonline model of S kinds of n limiters, etc are: max|vfd| = 0 2 4 6 8 10 12 0 0.2 0.4 0.6 0.8 1 Time (s) Te rm in al v ol ta ge (p .u. ) Proposed control FLT Ref. [10] Backstepping Ref. [17] 0 0.99 0.992 0.994 0.996 0.998 1 1.002 1.004 1.006 1.008 1.01 R ot or s pe ed (p .u. ) ance comparison, under +20% parameter perturbations, of the proposed control sche echnique Ref. [10]). 2 4 6 8 10 12 Time (s) Proposed control FLT Ref. [10] Backstepping Ref. [17] me and nonlinear controllers (backstepping Ref. [17] and feedback ar controller was tested on the complete 9th order MIB power system (202 MVA, 13,7 kV), including all onlinearities such as exciter ceilings, control signal . and speed regulator. The physical limits of the plant 10 p.u., and 0 ≤ Xe(t) ≤ 1 2 4 6 8 10 12 Time (s) Proposed control FLT Ref. [10] Backstepping Ref. [17] me and nonlinear controllers (backstepping Ref. [17] and feedback 142 M. Ouassaid et al. / Electric Power Systems Research 84 (2012) 135– 143 Table 1 Parameters of the transmission line in p.u. Parameter Value Le , inductance of the transmission line 0.4 Re , resistance of the transmission line 0.02 Table 2 Parameters of the power synchronous generator in p.u. Parameter Value Rs , stator resistance 1.096 × 10−3 Rfd , field resistance 7.42 × 10−4 Rkd , direct damper winding resistance 13.1 × 10−3 Rkq , quadrature damper winding resistance 54 × 10−3 Ld , direct self-inductance 1.700 Lq , quadrature self-inductances 1.640 Lfd , rotor self inductance 1.650 Lkd , direct damper winding self inductance 1.605 Lkq , quadrature damper winding self inductance 1.526 Lmd , direct m Lmq , quadrat V∝ , infinite b D, damping H, inertia co Table 3 Parameters of Parameter Tt , time cons Kt , gain of th R regulation Tg , time con Kg , gain of th The system ulation of t implement tions have b values used 5.1. Observ First, we mance of th given in Fig rotor angle considered paper is a closer to th barkers of with those As it can be rately track the differen 5.2. Comparison of dynamic performances for proposed controller and non linear controllers In order to prove the robustness of the proposed controller, the results are compared with two non linear controllers: (i) Feed- back Linearization Technique (FLT) Ref. [10] and (ii) Backstepping Ref. [17]. The operating point considered is Pm = 0.725 p.u. The fault occurs closer to the generator bus. The simulation results are presented in Fig. 5. It can be seen that both dynamics of the ter- minal voltage and the rotor speed settle to their prefault values very quickly with proposed observer based controller. It is obvious that with the derived high control accuracy and stability can be achieved. 5.3. Robustness to parameters uncertainties The variation of system parameters is considered for robust- valuation of the proposed observer-based controller. The of t ased The in F the p ted a anc llers l sch nd p anc clus his p evel syst onou nt st agnetizing inductance 1.550 ure magnetizing inductance 1.490 us voltage 1 constant 0 nstant 2.37 s the steam turbine and speed governor. Value tant of the turbine 0.35 s e turbine 1 constant of the system 0.05 stant of the speed governor 0.2 s e speed governor 1 configuration is presented as shown in Fig. 1. The sim- he proposed sliding mode damper currents observer is ed based on the scheme shown in Fig. 2. Digital simula- een carried out using Matlab–Simulink. The parameter in the ensuing simulation are given in the Appendix B. ness e values H incre tively. shown tion of simula perform contro contro eters a perform 6. Con In t been d power synchr transie er based controller performance evaluation verify the stability and asymptotic tracking perfor- e proposed control systems. The simulated results are s. 3 and 4. It is shown terminal voltage, rotor speed and of the power system, respectively. The operating points are Pm = 0.6 p.u. and 0.9 p.u. The fault considered in this symmetrical three-phase short circuit, which occurs e generator bus, at t = 10 s and removed by opening the the faulted line at t = 10.1 s. The results are compared of the linear IEEE type 1 AVR + PSS and speed regulator. seen, the proposed controller can quickly and accu- the desired terminal voltage and rotor speed despite t operating points. of the gene The slid ear observe voltage and control law both the ob Lyapunov s Simulati linear cont and voltage observer–c opposed to and nonline that the des a great robu he transmission line (Le, Re) and the inertia constant by +20% and −20% from their original values, respec- responses of the terminal voltage and rotor speed are ig. 6. In addition to the abrupt and permanent varia- ower system parameters a three-phase short-circuit is t the terminal of the generator. Figs. 7 and 8 show the es of the combined observer controller and the other Ref. [10] and Ref. [17]. It can be seen that the designed eme is able to deal with the uncertainties of param- reserve the global stability of the system with good es in transient and steady states. ion aper, a new nonlinear observer–controller scheme has oped and applied to the single machine infinite bus em, based on the complete 7th order model of the s generator. The aim of the study is to achieve both ability enhancement and good postfault performance rator terminal voltage. ing mode strategy was adopted to develop a nonlin- r of damper winding currents and nonlinear terminal rotor speed controller. The detailed derivation for the s has been provided. Globally exponentially stable of server and control laws has been proven by applying tability theory. on results have confirmed that the observer-based non- roller can effectively improve the transient stability regulation under large sudden fault. The combined ontroller scheme demonstrates consistent superiority a system with linear controllers (IEEE type 1 AVR + PSS) ar controllers. It can be seen from the simulation study igned sliding mode observer based-controller possesses stness to deal with parameter uncertainties. M. Ouassaid et al. / Electric Power Systems Research 84 (2012) 135– 143 143 Appendix A. The coefficients of the mathematical model a11 = −(Rs + Re)(LfdLkd − L2md)ωRD−1d , a12 = −Rfd(LmqLkd − L2md)ωRD−1d , a13 = (Lq + Le)(LmdLkd − L2md)ωRD−1d a15 = −Lmq(LfdLkd − L2md)ωRD−1d , a14 = Rkd((Ld + Le)Lmd − L2md)ωRD−1d , a16 = −V∞((Ld + Le)Lkd − L2md)ωRD−1d b1 = (LmdLkd − L2md)ωRD−1d , a21 = −(Rs + Re)(LmdLkd − L2md)ωRD−1d , a22 = −Rfd((Ld + Le)Lkd − L2md)ωRD−1d a23 = (Lq + Le)(LmdLkd − L2md)ωRD−1d , a24 = Rkd((Ld + Le)Lmd − L2md)ωRD−1d , a25 = −Lmq(LmdLkd − L2md)ωRD−1d a26 = −V∞(LmdLkd − L2md)ωRD−1d , b2 = ((Ld + Lfd)Lkd − L2md)ωRD−1d , a31 = −(Ld + Le)LkqωRD−1q a32 = LmdLkqωRD−1q , a33 = −(Rs + Re)LkqωRD−1q , a34 = LmdLkqωRD−1q , a35 = −Lmq.RkqωRD−1q , a36 = V∞LkqωRD−1q a41 = −(Rs + Re)(LfdLmd − L2md)ωRD−1d , a42 = Rfd((Ld + Le)Lmd − L2md)ωRD−1d , a45 = −Lmd(Lmq.Lfd − L2md)ωRD−1d a43 = (Lq + Le)(LmdLd − L2md)ωRD−1d , a44 = −Rkd((Ld + Le)Lfd − L2md)ωRD−1d , a46 = −V∞(Lmd.Lfd + L2md)ωRD−1d b3 = ((Ld + Le)Lmd − L2md)ωRD−1d , a51 = −(Ld + Le)LmqωRD−1q , a52 = LmdLmqωRD−1q , a53 = −(Rs + Re)LmqωRD−1q a54 = LmdLmqωRD−1q , a55 = −Rkq(Lq + Le)ωRD−1q , a56 = −V∞LmqωRD−1d , a61 = −D(2H) −1, a62 = (2H)−1 a81 = −(Tm = Kg c12 = a12Le 16 = c21 = Le + a , c2 here we ha Dd = (Ld + Appendix B [24,25] (Tab References [1] Y. 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Wang, Nonlinear decentralized control of large scale power systems, ca 36 (2000) 1275–1289. ello, Measurement of synchronous machine rotor angle from analy- o sequence harmonic components of machine terminal voltage, IEEE ons on Power Delivery 9 (1994) 1770–1777. n, Sliding mode control design principles and applications to drives, IEEE Transactions on Industrial Electronics 40 (1993) Observer-based nonlinear control of power system using sliding mode control strategy 1 Introduction 2 Mathematical model of power system studied 3 Development of a sliding mode observer for the damper winding currents 4 Design of sliding mode terminal voltage and speed controller 4.1 Terminal voltage control law 4.2 Sliding mode rotor speed controller 4.3 Stability analysis 5 Simulation results and discussion 5.1 Observer based controller performance evaluation 5.2 Comparison of dynamic performances for proposed controller and non linear controllers 5.3 Robustness to parameters uncertainties 6 Conclusion Appendix A The coefficients of the mathematical model Appendix B The parameters of the system are as follows [24,25] (Tables 1–3) References