Robust controller for synchronous generator with local load via VSC

Electrical Power and Energy Systems 29 (2007) 348–359 www.elsevier.com/locate/ijepes

Robust controller for synchronous generator with local load via VSC J. Cabrera-Va´zquez a, Alexander G. Loukianov a

b,*

, Jose´ M. Can˜edo b, Vadim I. Utkin

c

Universidad de Guadalajara, Centro Universitario de Ciencias Exactas e Ingenierı´as, Departamento de Electro´nica, Av. Revolucio´n No. 1500, Mo´dulo ‘‘O’’, Apdo. Postal 44840, Guadalajara Jalisco, Mexico b Centro de Investigacio´n y de Estudios Avanzados del IPN, Apdo. Postal 31-438, Plaza La Luna, C. P. 44550, Guadalajara, Jalisco, Mexico c Department of Electrical Engineering, The Ohio-State University, Columbus, OH 43210-1272, United States

Abstract The objective of this paper is to design a nonlinear observer-based excitation controller for power system comprising a single synchronous generator connected to an infinite bus with local load. The controller proposed is based on the using first singular perturbation systems concepts and then Sliding Mode Control technique combining with Block Control Principle. To reduce ‘‘chattering’’ a nonlinear observer with estimation of the mechanical torque and rotor fluxes is designed. This combined approach enables to compensate the inherent nonlinearities of the generator and to reject external disturbances.  2006 Elsevier Ltd. All rights reserved. Keywords: Synchronous generator; Singular perturbations; Sliding mode; Nonlinear observer; Lyapunov stability

1. Introduction Design of robust stabilizing feedback controllers for power systems with modeled and unmodeled uncertainties remains one of the important problems in control theory. A fruitful and relatively simple approach is based on the use the concept of Variable Structure Systems with sliding mode [1]. This scheme provides, first and foremost, performance robustness, and secondly decomposition and simplicity of the control design procedure. It is known that the model of power system is highly nonlinear and therefore, during last decade several modern control approaches including adaptive linear control [2–4], methods based on the passivity principle [5–9], fuzzy logic and neural networks [10–15], control based on the direct Lyapunov method [16,17], feedback linearization technique [18–23] have been used to design continuous nonlinear control algorithms which overcome the known limitations of traditional linear controllers [24–29]. Basically, these con*

Corresponding author. E-mail addresses: [email protected] (J. Cabrera-Va´zquez), [email protected] (A.G. Loukianov), [email protected] (J.M. Can˜edo), [email protected] (V.I. Utkin). 0142-0615/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2006.09.001

trol algorithms based on a reduced-order power system model, are sensitivities to the plant parameter variations and external disturbances, do not taken in consideration practical limitation on the control and the effects of the exciter system and load dynamics. Additionally, these control schemes do not consider the generator voltage regulation and the plant state estimation problems. The purpose of this investigation is to design the excitation control for the single machine infinite bus system (SMIB) with a local load (see Fig. 1). In this paper, we shall resort to sliding mode [1] and block control [30] techniques combined with singular perturbation methods, to obtain a simpler controller, computationally law demanding, that takes into account structural constraints. The main feature of the proposed control is robustness to disturbances and plant parameter variations. The advantage of singular perturbation method in the control application is the possibility of dealing with reduced order model instead of the fullorder system. A switching function is formed using block control linearization technique [30] and neglecting the fast dynamics of the exciter system. But discontinuities in the sliding mode control can excite the unmodeled exciter dynamics, leading to oscillations in the state vector. This phenomenon is known as ‘‘chattering’’ [1,31]. To prevent

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

349

these oscillations a nonlinear observer is designed and the ideal sliding mode in the auxiliary observer loop is produced.

where is = (id, iq)T, ir = (if, ig, ikd, ikq)T, Vs = (Vd, Vq)T, Vr = (Vf, 0, 0, 0)T, id and iq are the direct-axis and quadrature-axis stator currents; if is the field current; ikd, ikq and ig are the direct-axis and quadrature-axis damper windings currents; x is the angular velocity; Vd and Vq are the direct-axis and quadrature-axis terminal voltages; Vf is the excitation voltage; Rs and Rf are the stator and field resistances; Rg, Rkd and Rkq are the damper windings resistances; Ld and Lq are the direct-axis and quadrature-axis self-inductances; Lf is the rotor self-inductance; Lkd and Lkq are the direct-axis and quadrature-axis damper windings self-inductances; and Lmd and Lmq are the direct-axis and quadrature-axis magnetizing inductances. The torque Te can be expressed in terms of the currents as follows:

2. Plant model

T e ¼ ðLq  Ld Þid iq þ Lmd iq ðif þ ikd Þ  Lmq id ðig þ ikq Þ

Fig. 1. Single machine-infinite bus system, SMIB with local load.

The complete mathematical description of a single machine infinite-bus system that will be considered includes the mechanical and electrical dynamics of the three phase synchronous machine (including both the field and damper windings) with exciter system and external networks with load constraints. The mechanical dynamics are described by the swing equations given by [32] dd ¼ x  xs dt 2H dx ¼ Tm  Te xs dt

ð1Þ ð2Þ

where d is the power angle of generator; x is the angular velocity; xs is the rated synchronous speed, H is the inertia constant; Tm is the mechanical torque applied to shaft; and Te is the electrical torque. After Park’s transformation, the electrical dynamics which include both the rotor and stator windings, using the currents as the state variables, can be expressed as follows: " #     dis Vs is dt L di ¼ GðxÞ þ ð3Þ r ir Vr dt 3 2 0 Lmd 0 Lmd 0 Ld 60 Lq 0 Lmq 0 Lmq 7 7 6 7 6 6 Lmd 0 Lf 0 Lmd 0 7 7; L ¼6 60 Lmq 0 Lg 0 Lmq 7 7 6 7 6 4 Lmd 0 Lmd 0 Lkd 0 5 2

0 Rs

6 xLd 6 6 60 G ¼6 60 6 6 40 0

Lmq 0 xLq 0

Lmq 0 xLmq

Lkq 0

xLmq

Rs 0

xLmd Rf

0 0

xLmq 0

0 0

0 0

0 0

Rg 0

0 Rkd

0 0

0

0

0

0

Rkq

3 7 7 7 7 7 7 7 7 5

ð4Þ

The exciter system model is given by sf V_ f ¼ V f þ bf u

ð5Þ

where u is the control input, and sf is the time constant. The mechanical input torque Tm is assumed to be constant. Thus, T_ m ¼ 0

ð6Þ

The equilibrium equations for the external networks of the synchronous machine connected to an infinite bus can be written for the generator dynamics as #" # "R #    "1 e 0 x ied VdV1 d ied Le Le d ¼ ð7Þ  VqV1 dt ieq ieq 0 L1e x RLee q and for the load dynamics as         iLd Vd iq þ ieq d iLd ¼  RL  xLL : LL dt iLq Vq iLq id þ ied 

ð8Þ

The relations in terminals of the generator are given by       id  ied Vd iq þ iLq 1 RL ¼ ðI  Adq Þ þ xLL Vq iq  iLq id þ ied " 1#    )   Vd ied f7 h2 1 þLL þ LL  Ax V f  Adq 1 V iLq f8 0 q ð9Þ

where ie = (ied, ieq)T is the current in the transmission line, iL = (iLd, iLq)T is the current in the loads, kr = (kf, kg, kkd, kkq)T is the rotor flux, V1 is the value of the infinite-bus voltage; Le and Re are the transmission line resistance and inductance, respectively, LL and RL are the load resistance and inductance, respectively, f7 ¼ a71 sinðdÞ þ a73 kf þ a75 kkd  a77 id þ xða74 kg  a76 kkq þ a78 iq Þ; f8 ¼ a81 cosðdÞ þ a84 kg þ a83 kkq  a88 iq þ xða83 kf  a85 kkd þ a87 id Þ;

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

350

" Adq ¼

LL a71  LLLe

0 " Re L

L

Ax ¼

Le

LL x

LL x

#

0 LL a81  LLLe # 

Re LL Le

;

I¼

"L

L

;

A1 dq

1

0

0

1



¼

Le

0

0 LL Le

#

where x = (x1, . . ., x10)T, x1 = d, x2 = x, x3 = kf,

;

T

T

T

where

The transformation (10) reduces system (3) to the following form: " #     dis is Vs dt ðxÞ þ B ð11Þ ¼ A e e dkr kr Vr dt with Ae(x) = TL1G(x)T1 and Be = TL1. The torque Te can be expressed now in terms of the currents and fluxes as T e ¼ a23 kf iq þ a24 kg id  a25 kkd iq þ a26 kkq id þ a28 id iq

ð12Þ

where a2,i, i = 1, . . ., 8 are positive constant parameters depended on the generator parameters. The state variables as well as the parameters of the model (1)–(12) are expressed in per unit. Combining Eqs. (1)–(12) and using relationship (10), the complete model of the SMIB is represented in the nonlinear state-space form:

sf z_ ¼ z þ bf u x_ 4 ¼ A44 x4 þ A45 x5 x_ 5 ¼ f 5 ðxÞ þ b5 z x_ 6 ¼ f 6 ðx2 ; x6 ; V d ; V q Þ

T

x5 ¼ ðid ; iq Þ ¼ ðx7 ; x8 Þ ;

;

where ai,j, i = 7,8,j = 1, . . ., 7 are positive constant parameters depended on the power system parameters. Analyzing relations between the fluxes and currents shows that the sensitivity of the fluxes with respect to parameter variations is lower than that of the currents. Therefore, it is more suitable the representation of the electrical dynamics in terms of the stator current is and rotor flux kr. Such model can be obtained from (3) using the following transformation between the fluxes and currents:     is is ¼T ð10Þ kr ir

x_ 1 ¼ x2  xs x_ 2 ¼ f2 ðx4 ; x5 Þ þ b2 ðx5 Þx3 þ d 2 T m x_ 3 ¼ f3 ðx4 ; x5 Þ þ b3 z

T

x4 ¼ ðkg ; kkd ; kkq Þ ¼ ðx4 ; x5 ; x6 Þ ;

ð13aÞ ð13bÞ

T

x6 ¼ ðied ; ieq Þ ¼ ðx9 ; x10 Þ ; z¼Vf f3 ðxÞ ¼ a33 x3 þ a35 x5  a37 x7 f2 ðxÞ ¼ a24 x7 x4  a25 x8 x5 þ a26 x7 x6 þ a28 x7 x8 ; b2 ðx5 Þ ¼ a23 x8 ;

b5 ¼ ð0; b7 ÞT ;

d 2 ¼ xs =2H ;

T

f 5 ðxÞ ¼ ðf7 ðxÞ; f8 ðxÞÞ ; f 6 ¼ ðf9 ðx2 ; x6 ; V d ; V q Þ; f10 ðx2 ; x6 ; V d ; V q ÞÞT ; f7 ¼ a71 sin x1 þ a73 x3 þ a75 x5  a77 x7 þ x2 ða74 x4  a76 x6 þ a78 x8 Þ; f8 ¼ a81 cos x1 þ a84 x4 þ a86 x6  a88 x8 þ x2 ða83 x3 þ a85 x5  a87 x7 Þ; 1 Re x9 þ x2 x10 ; f9 ¼ ðV d  V 1 d Þ Le Le 1 Re x10  x2 x9 ; f10 ¼ ðV q  V 1 q Þ Le Le 2 3 2 a46 a44 0 0 6 7 6 A44 ¼ 4 0 a55 0 5; A45 ¼ 4 a57 a64 0 0 a66

a48

3

7 0 5; a68

where aij (i = 2, . . ., 8; j = 1, . . ., 9), b3 and b7 are positive constant parameters depending on Rs, Rf, Rg, Rkd, Rkq, Re, RL, Ld, Lq, Lkd, Lkq, Lmd, Lmq, Le, LL and V1. It is assumed that the power angle x1, the terminal voltage Vg, the speed x2 and the stator currents x7 and x8 are available for measurement, and that the control input u(t) should remain in: juðtÞj 6 U m

ð14Þ

with Um > 0. 3. Ideal sliding mode The control objectives are first to stabilize the rotor angle x1 and make the speed x2 be equal to the rated synchronous speed xs. Secondly, the voltage regulation is important as well. To simplify the control algorithm, we set sf = 0 in (13d) that results in z ¼ bf u

ð15Þ

Substituting (15) in (13a)–(13f) and (13g), yields ð16aÞ ð16bÞ

ð13eÞ

x_ 1 ¼ x2  xs x_ 2 ¼ f2 ðx4 ; x5 Þ þ b2 ðx5 Þx3 þ d 2 T m x_ 3 ¼ f3 ðx4 ; x5 Þ þ b3 u x_ 4 ¼ A44 x4 þ A45 x5 x_ 5 ¼ f 5 ðxÞ þ b5 u

ð13fÞ ð13gÞ

x_ 6 ¼ f 6 ðx2 ; x6 ; V d ; V q Þ where b3 ¼ b3 bf and b5 ¼ b5 bf .

ð16fÞ

ð13cÞ ð13dÞ

ð16cÞ ð16dÞ ð16eÞ

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

3.1. Speed control design The subsystem (16a), (16b) and (16c) has the Block Controllable form [30]. Therefore, this subsystem can be linearized using block control technique by the following transformation: z1 ¼ x1 :¼ u1 ðxÞ z2 ¼ x2  xs :¼ u2 ðxÞ   f2 ðx4 ; x5 Þþ 1 rx ¼ x3  :¼ u3 ðx; T m Þ b2 ðx5 Þ þd 2 T m  k z ðx2  xs Þ ð17Þ where kz > 0. The transformed subsystem (16a), (16b) and (16c) then is represented in new variables z1, z2 and rx as z_ 1 ¼ z2 z_ 2 ¼ k z z2 þ b2 ðx5 Þrx r_ x ¼ fx ðx; T m Þ þ  bx ðxÞu

ð18Þ ð19Þ

where 3 3 3 fx ¼ ou ðf2 þ b2 x3 þ d 2 T m Þ þ ou f þ ou ðA44 x4  A45 x5 Þ þ ox2 ox3 3 ox4  ou3 b 7    f , b ¼ b  ða x þ a x  a x x 3 24 4 26 6 28 8 Þ; and bx ðtÞ is a ox5 5 a23 x8 positive function of time. A discontinuous control law is proposed as u ¼ U m signðrx Þ;

U m > 0:

ð20Þ

Sliding mode stability can be analyzed using the following candidate of Lyapunov function: 1 V x ¼ r2x : ð21Þ 2 Find the time derivative of (21) along the trajectories of system (19) with control (20): V_ x ¼ fx ðx; T m Þrx  U m  bx ðxÞsignðrx Þrx x ðxÞ  fx ðx; T m Þ: 6 jrx j½U m b

ð22Þ

Under the following condition: U m > juxeq ðx; T m Þj

ð23Þ

with equivalent control uxeq [1] calculated from r_ x ¼ 0 (19) as uxeq ðx; T m Þ ¼ ð bx ðxÞÞ1 fx ðx; T m Þ

ð24Þ

the value of V_ x (22) is negative. Therefore, the state will reach the manifold rx = 0 after finite time interval [0,ts] [1]. Once this is achieved, the sliding motion on rx = 0 is governed by the following second-order system: z_ 1 ¼ z2 z_ 2 ¼ k z z2

ð25Þ

with desired eigenvalue kz, that corresponds to the linearized mechanical dynamics. If kz > 0 then limt!1z2(t) = 0 and x1 ðtÞ ! x11 ¼ const x2 ðtÞ ! x21 ¼ xs

as t ! 1

ð26Þ

351

where x11 is the steady-state value of angle x1 corresponded to the value of mechanical torque Tm. Additionally, from rx(t) = 0 for t P ts it follows: x3 ðtÞ ¼ x3d ðx4 ðtÞ; x5 ðtÞ; T m Þ; 1 x3d ¼ ½f2 ðx4 ; x5 Þ þ d 2 T m  b2 ðx5 Þ

ð27Þ

The basic property of the control for sliding mode is that the control provides the subspace {n = (x1,x2,x3)T = (x11 ,xs,x3d)T,g 2 R7} invariant where g = (x4,x5,x6)T. The dynamics of vector g on this subspace are the zero dynamics. To obtain these dynamics, it is necessary to substitute the equivalent value of the control on the invariant subspace, uxeq ðx11 ; xs ; x31 ; g; T m Þ (24) and the values x1 = x11, x2 = x21 = xs (26) and x3 = x3d (27) in (16d), (16e) and (16f), that yields g_ ¼ f g ðx11 ; xs ; x3d ; g; V d ; V q ; Þ þ bg uxeq ðx11 ; xs ; x3d ; g; T m Þ

ð28Þ

where fg = (f4(g), f5(x11,xs,x3d,g), f6(xs,g,Vd,Vq))T, T f4(g) = A44x4 + A45x5, bg ¼ ð0; 0; 0; b7 ; 0; 0; 0Þ and b7 ¼ b7 bz . Dividing the right part of this system into linear and nonlinear parts, then the zero dynamics can be represented as _ g¼A g gþgg ðx11 ;xs ;x3d ;g;T m ;V d ;V q Þ 2 a46 0 a48 0 a44 0 60 a55 0 a57 0 0 6 6 6 a64 0 a 0 a 0 66 68 6 6 whereAg ¼6 xs a74 a75 xs a76 a77 xs a78 0 6 6 a84 xs a85 a86 xs a87 a88 0 6 6 0 0 0 0 a99 40 2

0 0

0

0

0

0

0 0 0 0 0 xs

3 7 7 7 7 7 7 7; 7 7 7 7 5

xs a10;10 3

7 6 a x ðg;T Þ m 7 6 53 3d 7 6 7 60 7 6 6 a cosx þ b ux ðx ;x ;x ;g;T Þ 7 gg ¼6 71 11 7 eq 11 s 3d m 7: 7 6 7 6 a81 sinx11 7 6 7 6 d ðV V 1 Þ 5 4 9 d d d 9 ðV q V 1 Þ q It can be shown that the matrix Ag is Hurwitz, and the nonlinear term gg(Æ) is bounded. Therefore, a solution of Eq. (28) is ultimately bounded [33]. 3.2. Voltage regulator design The second objective of the excitation control is to regulate the voltage Vg at the terminals of the synchronous generator, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V g ¼ jVs j ¼ V 2d þ V 2q : ð29Þ

352

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

To derive the voltage dynamics, we use the balanced equation for the external net (9). This equation can be represented as       Vd hd ðxÞ bd ð30Þ ¼ z; z ¼ bf u þ hq ðxÞ Vq 0 where   LL hd ¼ ða73 x3 þ a74 x2 x4 þ a75 x5 þ a76 x2 x6 Þ L11     RL Le þ LL a77 Le xs LL Le þ LL x2 a78 Le þ x7 þ x8 Le L11 Le L11   LL Re  RL Le LL V 1 d þ ; x9 þ Le L11 Le L11   LL ða87 x2 x3 þ a84 x4 þ a83 x2 x5 þ a86 x6 Þ hq ¼ L22     xLL Le þ LL a85 Le x2 RL Le þ LL a88 Le þ x7 þ x8 Le L22 Le L22   LL V 1 RL Le  LL Re q þ ; x10 þ Le L22 Le L22   1 L11 ¼ 1  LL a71  ; Le   1 L L b7 bf : and bd ¼ L22 ¼ 1  LL a81  Le L11 It is considered that bd  0 since bd is very small. Then, the pattern of the voltage Vg can be obtained. From (29), (30), (16a)–(16e) and (16f) we have dV g ¼ fv ðx; T m Þ þ bv ðxÞu ð31Þ dt where   LL V d a73 V q a87 x2  þ b3 bv ¼ V g L11 L22   LL V d ðRL Le þ LL a77 Le Þ V q x2 ðLL Le þ LL a85 Le Þ þ þ b7 bf ; Le L11 Le L22 Vg  ohq d þ V fv ¼ V1g V d oh x_ and bv (t) is the positive function q ox ox for t P 0. With the purpose of regulating the voltage Vg, a discontinuous control is defined as u ¼ U m signðrv Þ;

rv ¼ V g  V ref



r_ x r_ v



 ¼

   fx ðx; T m Þ bx ðxÞ þ u f v ðx; T m ; V_ ref Þ bv ðxÞ

There is only one input for the control, that is, the excitement of the synchronous machine. Therefore, to achieve two objectives, a combined discontinuous control is chosen of the form u ¼ U m signðrÞ  rx for jz2 j > e r¼ rv for jz2 j 6 e

 and

e¼

ð35Þ e1

for

jrv j > e3

e2

for

jrv j 6 e3 ð36Þ

with e1 > 0, e2 > 0, e3 > 0 and e2 < e1. It is assumed that the mechanical dynamics are slower than the electric ones and they take advantage the resources of the control, first to stabilize the angle x1 and speed x2 and then to regulate the generator voltage Vg. For U m > max juaeq ðx; T m Þj; a

a ¼ x; v

the state vector first converges to the manifold rx = 0 in a finite time and then the sliding mode motion, described by the linear system (25), ultimately confines to the vicinity jz2j < e1 (Fig. 2). Thereafter, the control algorithm regulates the voltage Vg. After the voltage error convergences so that jrvj 6 e3 is satisfied, the action of the controller decreases to the narrow part of the layer for the speed error z2, from e1 to e2 (e1 > e2). When there is jz2j 6 e2, the control (32) ensures the finite time convergence of rv to zero. The vicinity jz2j 6 e2 can be made arbitrarily small by choosing e2 small enough. In the limit, as e2 ! 0 we have jz2j ! 0. Note, due to the synchronism property of the machine, in steady state x(1) = xss we have uxeq ðxss ; T m Þ ¼ uveq ðxss ; T m Þ that means, one excitation control input achieves two control objectives.

ð32Þ

where Vref is a reference signal. Then r_ v ¼ f v ðx; T m Þ þ bv ðxÞu

ð33Þ  _ where f v ¼ fv ðx; T m Þ  V ref . Similar to the speed controller case, it can be shown that for U m > juveq ðx; T m Þj;

 uveq ¼ b1 v ðxÞf v ðx; T m Þ

ð34Þ

the control error rv(t) vanishes at one finite time. 3.3. Speed and voltage combined controller Using (19) and (33) the equation of the motion projection of the system on the subspace rx and rv can be written as

ð37Þ

Fig. 2. Voltage error rv and speed error z2.

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

4. Real sliding mode The key feature of the designed sliding controller is its robustness to uncertainties. During the reaching time period, the task of forcing the trajectories toward the sliding manifold rx = 0 and rv = 0 and maintaining them, is achieved by the control (35), provided the plant parameters variations satisfy the inequality (37). However, the proposed controller provides the desired performance of the closed-loop system only in the ideal case, i.e. in the absence of unmodeled dynamics (13d), sf = 0. 4.1. Chattering problem In the case sf50, the switching control leads to finite frequency oscillations in the state vector, ‘‘chattering’’ [1,31], since discontinuities in the control excite the unmodeled dynamics (13d). In this case we have r_ x ¼ fx ðx; T m Þ þ bx ðxÞz 1 bf z_ ¼  z þ u sf sf

ð38Þ

where bx ¼ b3  a23b7x8 ða24 x4 þ a26 x6  a28 x8 Þ and the derivative V_ x (22) depends on the variable z only, but not on the control input u: V_ x ¼ fx ðx; T m Þrx þ bx ðxÞzrx :

ð39Þ

In order to examine the system behavior, assume steadystate conditions with zðtÞ ¼ bf uðtÞ ¼ bf U max : The step response of the exciter system (38) for the first switch at ts from u(t) = Umax to u(t) = Umax at rx = 0 is given by  tts zðtÞ ¼ bf U max 1  2e sf : ð40Þ For some initial time interval Dt = t  ts variable z(t) < bf u(t) = bfUmax and V_ x > 0 in (39) results for the case fx(x,Tm)rx > 0. It is only after the decay of the exponential term in (40), i.e. after some delay period D t(sf), that j zðtÞ j>j b1 x ðxðtÞÞfx ðxðtÞ; T m Þ j is established once more and V_ x < 0 indicates convergence to sliding manifold rx = 0. During this delay period the trajectory has a deviation from ideal motion. Similar derivations hold for the next switch from u(t) = Umax to u(t) = Umax. Repetition of this process creates the ‘‘zig-zag’’ motion or chattering. Chattering results in low control accuracy and high heat losses in electrical power circuits. This phenomenon has been considered as serious obstacles for applications of sliding mode control in many papers and discussions. A recent study and practical experience showed that chattering caused by unmodeled dynamics may be eliminated in systems with asymptotic observers. This idea was proposed by Bondarev et al. in [31]. In spite of the presence of unmodeled dynamics, ideal sliding mode is possible. It is described by a singularly perturbed differential equation

353

with solutions close to those of the ideal system. An asymptotic observer serves as a bypass for the high-frequency component; therefore the unmodeled dynamics are not excited. Preservation of sliding stage modes in systems with asymptotic observers enabled successful applications of discontinuous control. 4.2. Observer- based solution To prevent chattering and estimate unmeasured excitation flux x3, rotor flux x4 = (x4,x5,x6)T, and mechanical torque Tm, a nonlinear observer is proposed as ^x_ 2 ¼ f^ 2 ð^ x4 ; x5 ðtÞÞ þ d 2 Tb m þ l1 ðx2  ^x2 Þ _ Tb m ¼ l2 ðx2  ^x2 Þ ^x_ 3 ¼ f^ 3 ð^ x4 ; x5 ðtÞÞ þ b3 u

ð41bÞ

^_ 4 ¼ A44 x ^4 þ A45 x5 ðtÞ x

ð41dÞ

ð41aÞ

ð41cÞ

^4 ¼ ð^x4 ; ^x5 ; ^x6 ÞT and Tb m are the estimated variwhere ^x3 ; x ables, l1 and l2 are observer gains, f^ 2 ð^ x4 ; x5 ðtÞÞ ¼ a23 x8 ðtÞ^x3 þ a24 x7 ðtÞ^x4  a25 x8 ðtÞ^x5 þ a26 x7 ðtÞ^x6 þ a28 x7 ðtÞ x8 ðtÞ and f^ 3 ð^ x4 ; x5 ðtÞÞ ¼ a33^x3 þ a35^x5  a37 x7 ðtÞ. Considering the stator current x5(t) = is(t) in (41a)– (41c) and (41d) as a known (measured) function of time, the dynamics of the estimation errors can be obtained by subtracting the Eqs. (16b)–(16d), (6), (41a)–(41c) and (41d) as the following linear system with time varying parameters:      e_ 1 A11 A12 ðtÞ e1 ¼ ð42Þ e_ 2 0 A22 e2 where e1 = (e2, em)T, e2 = (e3, e4, e5, e6)T, i = 2, . . ., 6, em ¼ T m  Tb m ,   l1 a29 A11 ¼ ; l2 0   a23 ðtÞ a24 ðtÞ a25 ðtÞ a26 ðtÞ ; A12 ðtÞ ¼ 0 0 0 0 2 3 a33 0 a35 0 60 a44 0 a46 7 6 7 A22 ¼ 6 7; 4 a53 5 0 a55 0 0

a64

0

ei ¼ xi  ^xi ,

a66

a23 ðtÞ ¼ a23 x8 ðtÞ; a24 ðtÞ ¼ a24 x7 ðtÞ and a25 ðtÞ ¼ a25 x8 ðtÞ; a26 ðtÞ ¼ a26 x7 ðtÞ: It is easy to see that the spectrum of the system (42) consists of (1) the eigenvalues of the block A11, which can be assigned by appropriatechoice of the observer gains 

0 dm ; ½ 1 0  is obserl1 and l2, since the pair 0 0 vable, and

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

354

Table 1 Calculated mathematical model parameters and their variations

a23 a24 a25 a26 a28 am a33 a35 a37 a44 a46

Nominal

Increment

0.48514 0.13789 0.46667 0.800 0.020 0.00003763 0.71222 0.64556 0.11067 2.776 2.579

– – – – – – +0.11745 +0.12911 +0.02213 – –

a48 a53 a55 a57 a64 a66 a68 a71 a73 a75 a77

(2) the eigenvalues of the matrix A22: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 k3;4 ¼  ða33 þ a55 Þ  ða33  a55 Þ2 þ 4a35 a53 ; 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 2 ða44  a66 Þ þ 4a46 a64 k5;6 ¼  ða44 þ a66 Þ  2 2 which are real and negative.

Nominal

Increment

0.322 30.321 33.333 5.000 9.849 14.286 7.143 711.30 26.046 28.759 42.203

– – – – – – – – 0.521 0.118 0.020

Increment

0.26017 1.5008 1.0377 685.44 13.630 20.133 46.799 0.84848 0.96364 0.88207 345.08

– – – – – – – – – – –

^4 ; x5 ; Tb m Þj; U m > j^uxeq ðx1 ; x2 ; ^x3 ; x 1 ^4 ; x5 ; Tb m Þ ^uxeq ¼ ð^bx ð^ x4 ; x5 ÞÞ f^ x ðx1 ; x2 ; ^x3 ; x

ð43Þ

The parameters of A12(t) and its derivatives are bounded. Therefore, the linear system (42) with time varying parameters is asymptotically stable. The resulting estimates ^xi , i = 2, . . . ,6 and Tb m are employed in the control law (20), in particularly forming the estimate of switching function rx (17) as 1 ½f2 ð^ x4 ; x5 Þ þ d 2 Tb m  k z ðx2  xs Þ : b2 ðx5 Þ ^4 ; x5 ; Tb m Þ ^ 3 ðx2 ; ^x3 ; x ¼u

a74 a76 a78 a81 a84 a86 a88 a83 a85 a87 b7

Nominal

^x ¼ 0 sliding mode is established after finite time, and r holds exactly thereafter. Due to the observer (41a)–(41c) ^x ¼ 0 will and (41d) stability, the sliding mode motion on r coincide after some transient time period in the observer, with the motion on rx = 0, which is described by the linear reduced-order system (25). 2 X1 [rad.]

1.8 1.6 1.4

^x ¼ ^x3  r

1.2

ð44Þ

that allows definition of an ideal sliding mode controller for the observer loop as

1 0.8 0.6 0.4

rx Þ u ¼ U m signð^

ð45Þ

0

^4 ; x5 ; T^ m Þ þ ^ ^_ x ¼ f^ x ðx1 ; x2 ; ^x3 ; x bx ð^ x4 ; x5 Þu ð46Þ r u3 u3 ^ u3 ^4  where f^ x ¼ o^ ðf2 þ b 2x3 þ d 2 T m Þ þ o^ ðA44 x f 3 þ o^ ox2 o^x3 o^ x4  o^ u3 o^ u3 b7 ^  A45 x5 Þ þ ox5 f 5 þ l2 ðx2  ^x2 Þ,bx ¼ b3  a23 x8 ða24^x4 þ a26^x6  ob Tm ^ a28 x8 Þ and bx ðtÞ it is a positive function of the time. Stability of the auxiliary observer loop is examined via a similar Lyapunov function as in (21) 1 2 ^ Vb x ¼ r 2 x

0.2 Time [sec.]

Then

4

6

8

10

12

14

16

18

20

2 Vg [p.u.]

1.8 1.6 1.4 1.2

ð47Þ

1

Substitution of (46) under control (45) into time derivative of (47) results in _ ^4 ; x5 ; Tb m Þ^ Vb x ¼ f^ x ðx1 ; x2 ; ^x3 ; x rx  U m ^ x4 ; x5 Þsignð^ rx Þ^ rx bx ð^

0.4

Hence, under similar assumption as in (23)

2

Fig. 3a. Angle x1.

0.8

^4 ; x5 ; Tb m Þ P j^ rx j½U m ^ x4 ; x5 Þ  f^ x ðx1 ; x2 ; ^x3 ; x bx ð^

0

0.6

0.2 Time [sec.]

0

0

2

4

6

8

10

12

14

Fig. 3b. Terminal voltage Vg.

16

18

20

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359 385

355

-0.45 X2, X2est, (p.u.)

X4, X4est [p.u.]

-0.5

-0.55

380

-0.6

-0.65 375

-0.7

-0.75

Time, (sec.)

370 0

2

4

6

8

10

12

14

16

18

20

Time, [sec.]

-0.8

0

Fig. 3c. Speed x2 and its estimate ^x2 .

2

4

6

8

10

12

14

16

18

20

Fig. 3f. q-axis(1) damper winding flux x4 and its estimate ^x4 .

1.5

1 X5, X5est [p.u.]

Tm, Tmest (p.u.)

0.95 0.9

Tm

1

0.85 0.8 Tmest

0.5

0.75 0.7 0.65

Time, (sec.)

0

0

2

4

6

8

10

12

14

16

18

20

Time, [sec.]

0.6

Fig. 3d. Mechanical torque Tm and its estimate Tb m .

0

2

4

6

8

10

12

14

16

18

20

Fig. 3g. d-axis damper winding flux x5 and its estimate ^x5 . 1.4 X3, X3est [p.u.]

1.3

-0.45

1.2

-0.5

X6, X6est [p.u.]

1.1

-0.55

1 -0.6

0.9 -0.65

0.8 -0.7

0.7 0.6

-0.75 Time, [sec.]

0.5

0

2

4

6

8

10

12

14

16

18

Time, [sec.]

20

Fig. 3e. Excitation flux x3 and its estimate ^x3 .

-0.8

0

2

4

6

8

10

12

14

16

18

20

Fig. 3h. q-axis(2) damper winding flux x6 and its estimate ^x6 .

5. Simulation results The performance of the proposed sliding mode observer-based controller was tested on the complete power system (see Fig. 1).

The parameters of the synchronous machine, transmission and exciter systems and load, all in p.u., except where indicated, are Rs = 0.003, Rf = 0.021, Rg = 0.725, Rkd = 10.714, Rkq = 8.929, Re = 0.05, Ld = 1.81,

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

356

1

2

Tm, Tmest (p.u.)

X1, [rad.]

1.8

0.9

1.6

0.8

1.4

-20% Lmd

0.7

1.2

Tm

0.6

1 0.5

0.8

Tmest 0.6

0.4

0.4

0.3

0.2

- 20 % Lmd

0.2

Time, [sec]

0 0

5

10

15

20

25

30

35

40

0.1

Time, (sec.)

Fig. 4a. Angle x1.

0

0

5

10

15

20

25

30

35

40

Fig. 4d. Mechanical torque Tm and its estimate Tb m . 1.2 Vg, [p.u.]

1.4 X3, X3est, [p.u.]

1 1.3

0.8

1.2

-20% Lmd

1.1

0.6

1

0.4 0.9

0.2

-20% Lmd

0.8

0

0.7 Time, [sec]

-0.2

0.6

0

5

10

15

20

25

30

35

40 Time, [sec]

0.5

Fig. 4b. Terminal voltage Vg.

0

5

10

15

20

25

30

35

40

Fig. 4e. Excitation flux x3 and its estimate ^x3 .

378

-0.45

X2, X2est [p.u.]

X4, X4est, [p.u.]

377.5

-0.5

377

-0.55

376.5

-0.6 -20% Lmd

376

-0.65 -20% Lmd

375.5

-0.7

375

-0.75 Time, [sec]

374.5

0

5

10

15

20

25

30

35

Time, [sec]

40

-0.8 0

5

10

15

20

25

30

35

40

Fig. 4c. Speed x2 and its estimate ^x2 .

Fig. 4f. q-axis(1) damper winding flux x4 and its estimate ^x4 .

Lq = 1.76, Lkd = 1.831, Lkq = 1.735, Lmd = 1.66, Lmq = 1.61, Le = 0.3, H = 3.525 s, bf = 1, sf = 0.015 s, dref = 1.3314, xs = 377 rad s1, Tm = 0.9463 and V1 = 1.

For these parameters we obtain the parameters of mathematical model (13a)–(13f) and (13g) presented in Table 1, considered as nominal parameters. After Lmd experienced

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

357

-0.45

1 X5, X5est, [p.u.]

X6, X6est, [p.u.]

0.95

-0.5

0.9 -0.55

0.85 -0.6

0.8 -20% Lmd

-0.65

0.75

-20% Lmd

-0.7

0.7

-0.75

0.65 0.6

Time, [sec.]

Time, [sec.]

0

5

10

15

20

25

30

35

40

-0.8 0

5

10

15

20

25

30

35

40

Fig. 4g. d-axis damper winding flux x5 and its estimate ^x5 .

Fig. 4h. q-axis(2) damper winding flux x6 and its estimate ^x6 .

an increment of 20%, some parameters of model (13a)–(13f) and (13g), namely, a33, a35, a37, a73, a75 and a77 are modified and their corresponding increments are presented in the

same Table 1. The controller gain was adjusted to kz = 7 and the observer gains were chosen as l1 = 200 and l2 = 187, resulting in the eigenvalues k1 = k2 = 100. The

Fig. 5. Comparison of regulators.

358

J. Cabrera-Va´zquez et al. / Electrical Power and Energy Systems 29 (2007) 348–359

remaining observer eigenvalues were calculated using (43) as k3 = 0.123, k4 = 33.922, k5 = 0.883 and k6 = 16.179. The mechanical torque disturbance is presented in the second 10. Also a short circuit in the 30th s with a release time of 150 ms. Figs. 3a–3g and 3h show the behavior of the variables of states when a short circuit is caused in the 6th s with a release time of 150 ms. The system SMIB undergoes a step change in the torque at the 10th s. The nominal value 0.902 p.u. goes down 0.5 p.u. The dynamics related to the weight of the coils have not been modeled for this system. Figs. 4a–4g and 4h show the behavior of the state variables, and also, besides the two previous perturbations, an internal perturbation in introduced with Lmd decreasing 50% from the nominal value of 1.66 p.u. This variation takes place in the interval from 20 to 30 s. The system has no dynamics modeled for the one reeled of the excitatory one. Fig. 5 shows the comparison that became between a system of conventional control (line dashed) with respect to the proposed control (line solid). 6. Conclusions A nonlinear controller based on the combination of the sliding mode control, block control linearization and singular perturbations techniques has been proposed. A model, which takes into account all of the interactions between the electrical and mechanical dynamics and load constraints, has been described. A nonlinear observer for the estimation of the excitation and rotor fluxes and the mechanical torque, has been designed. This new controller has been tested through simulation under three very important perturbations in the power systems: • variation of the mechanical torque; • a large fault (a 150 ms short circuit); • variation of parameters. The results of the simulation show that with the proposed observer-based sliding mode controller, we can eliminate many of the problems encountered with other types of controllers, such as eliminating oscillations and high variations after internal and external perturbations to the system. This controller takes into account all of the electrical and mechanical dynamics. This controller is much more robust, simple to design, and uses much less CPU power than other alternatives. Acknowledgements The authors thank the support of CONACYT Mexico, Grant 46069Y. References [1] Utkin VI, Guldner J, Shi J. Sliding mode control in electromechanical system. London, UK: Taylor and Francis; 1999.

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