Reaching phase free adaptive fuzzy synergetic power system stabilizer

Electrical Power and Energy Systems 77 (2016) 43–49

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Reaching phase free adaptive fuzzy synergetic power system stabilizer Z. Bouchama a,c,⇑, N. Essounbouli b, M.N. Harmas c, A. Hamzaoui b, K. Saoudi d a

Sciences and Technology Department, Bordj Bou Arreridj University, Algeria CReSTIC Laboratory, University of Reims, Champagne Ardennes, France c QUERE Laboratory, Department of Electrical Engineering, Ferhat Abbas University, Sétif1, Algeria d Department of Electrical Engineering, Bouira University, Algeria b

a r t i c l e

i n f o

Article history: Received 22 December 2014 Received in revised form 3 October 2015 Accepted 10 November 2015 Available online xxxx Keywords: Power system stabilizer Synergetic control Reaching phase Fuzzy logic system Indirect adaptive Multi-machine power system

a b s t r a c t In this paper, an adaptive fuzzy power system stabilizer is developed based on robust synergetic control theory and terminal attractor techniques. The main contribution consists in making the dynamic system insensitive to parameters variation. This aim is achieved using a new synergetic controller design such that power system states start, evolve and remain on a designer chosen attractor toward the equilibrium point therefore avoiding transient mode. Rendering the design more robust, fuzzy logic systems are used to approximate the unknown power system dynamic functions without calling upon usual model linearization and simplifications. Based on an indirect adaptive scheme and Lyapunov theory, adaptation laws are developed to make the controller handle parameters variations due to the different operating conditions occurring on the power system and to guarantee stability. The performance of the proposed stabilizer is evaluated for a single machine infinite bus system and for a multi machine power system under different type of disturbances. Simulation results show the effectiveness and robustness of the proposed stabilizer in damping power system oscillations under various disturbances and better overall performance than classical PSS and some other types of power stabilizers. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction In many situations power system stabilizers (PSS) are used to generate supplementary control signals for the excitation system in order to damp low-frequency oscillations caused by load disturbances or short-circuit faults. [1,2]. So called conventional power system stabilizers (CPSS) have been widely used relying on leadlag phase compensator [3,4] whose parameters were calculated using power system model linearized around a given operating point regardless of a normally wide range of operating conditions. Thus adequate oscillations damping couldn’t be achieved in a permanently changing load condition. Power systems are nonlinear systems, with configurations and parameters that fluctuate with time that which require a fully nonlinear model and an adaptive control scheme for a practical operating environment. Therefore, an adaptive PSS which considers the nonlinear nature of the plant and adapts its parameters to changes in the environment is necessary and is addressed in this paper. Many papers have addressed this issue using nonlinear approaches such as variable structure technique [5], with meta-heuristic methods [6–8], neural network ⇑ Corresponding author at: Sciences and Technology Department, Bordj Bou Arreridj University, Algeria. E-mail address: [email protected] (Z. Bouchama). http://dx.doi.org/10.1016/j.ijepes.2015.11.017 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

based PSS [9] and fuzzy adaptive schemes [10,11]. In most of these approaches parameters were not optimized and implementation not easy to realize due to use of non-measurable variables in the control law as in [12]. However, a new nonlinear adaptive fuzzy approach based on synergetic control theory (SCT) has been developed for nonlinear power system stabilizers [13] (AFSPSS) to overcome above mentioned problems. Similar to sliding mode approach but without its devastating chattering drawback, synergetic control provides a continuous control law driving system states to a predesigned attractor and on to the operating equilibrium point. The main advantage of this approach is that once system states reach the attractor, the system dynamics remain insensitive to a class of parameter variation and disturbances. However, the robustness of synergetic control law is not guaranteed during transient motion (reaching phase equivalent) when system states trajectories have not attained the attractor. A new approach to avoid this disadvantage inherent to both sliding mode and synergetic control, an adaptive fuzzy synergetic control law is proposed thus eliminating the reaching phase. In the present paper, the main contribution consists in determining a new mathematical synergetic attractor model, such that system states start motion on this attractor, reducing transient dwelling time, on to the equilibrium point. Relyingon fuzzy sets

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Z. Bouchama et al. / Electrical Power and Energy Systems 77 (2016) 43–49

universal approximation properties, we develop a reaching phase free indirect adaptive fuzzy synergetic power system stabilizer. Basics in synergetic control design Synergetic control design procedure basics can be introduced as follows [14–16]:let’s consider an nth order nonlinear dynamic system described by (1):

x_ ¼ f ðx; u; tÞ

ð1Þ

where x represents the system state vector, u the control input vector and f a nonlinear function. Synthesis of a synergetic controller begins with a choice of a function of the system state variables called the macro-variable (2).

w ¼ wðx; tÞ

ð2Þ

The control objective is to force the system to operate on the manifold w = 0. The designer can select the characteristics of the macro-variable according to performances and control specifications (overshoot, control signal limits, etc. . .). Then a desired dynamic evolution of the macro-variable is chosen such as (3):

T w_ þ w ¼ 0; T > 0

ð3Þ

T is a positive constant imposing the designer chosen speed convergence to the desired manifold. Differentiating the macro-variable (2) along (1) leads to (4):

dw x_ w_ ¼ dx

ð4Þ

Combining Eqs. (1), (3) and (4), leads to (5):

T

dw f ðx; u; tÞ þ w ¼ 0 dx

ð5Þ

Solving for the control law u, leads to (6):

u ¼ uðx; wðx; tÞ; T; tÞ

ð6Þ

Thus the continuous control law, not causing chattering as in the sliding mode control approach, depends on designer chosen macro-variable w and constant T imposing therefore desired dynamics. In this process, system control design needs no model linearization but rather relies on a complete nonlinear system. The new constraint (3) as in sliding mode techniques reduces system order by one while enabling control designer to achieve global stability and parameter insensitivity. Adequate selection of the macro-variable ensures the performance and the stability sought [12–16].

f(x) and g(x) are nonlinear functions and g(x) – 0 in the controllable region. The synergetic synthesis of the power system stabilizer begins by defining a macro-variable given in (8).

w ¼ kx1 þ x2

ð8Þ

w_ ¼ K T x þ f ðx1 ; x2 Þ þ gðx1 ; x2 Þu

ð9Þ

Without loss of generality, let us chose K T ¼ ½0 ka in which k is a designer chosen positive constant. Combining Eqs. (9) and (5) yields

1 f ðx1 ; x2 Þ þ gðx1 ; x2 Þu ¼  w  K T x T

The synergetic control law is then obtained and is given by (11):

u¼


 1 1 f ðx1 ; x2 Þ  w  K T x gðx1 ; x2 Þ T

w ¼ kx1 þ x2 ¼ 0

ð12Þ

Eq. (12) represents a straight line (the manifold) through the origin with slope 1=k. As shown in Fig. 1, the dynamic motion of power system state variables under synergetic control consists of transient phase during which system states move toward the manifold and then move along it to the origin. However, during transient mode the power system is sensitive to parameters variation, unlike when the system states are on the manifold (w ¼ 0) and thus insensitive to parameters variation. To enhance robustness and avoid the transient mode, we present a reaching phase free synergetic controller leading to more robust performance. Reaching phase free indirect adaptive power system design In this new approach, we will reformulate the macro-variable dynamics (8) by defining a new macro-variable given in (13).

W ¼ wðx; tÞ  zðx0 ; tÞ

ð13Þ

where w(x, t) the macro-variable defined in (8) and zðx0 ; tÞ a term added to the macro-variable, defined by (14) [20]

zðx0 ; tÞ ¼

i 2 hp  atanðtÞ wðx0 Þ p 2

The time derivative of Eq. (13) can be rewritten as:

The main goal of PSS is to damp power oscillations that occur upon perturbations such as sudden change of loads or in the event of short-circuit occurrence. These oscillations hamper power flow drastically and may cause inability to meet power demand or even loss of synchronism that may eventually lead, in the worst case, to blackouts. Let x1 = Dx designate speed deviation and x2 = Pm  Pe accelerating power system. It is possible to represent the system with the following nonlinear equations [13,17–19]:

ð7Þ

where a = 1/(2H) and H is the per unit machine inertia constant,

x = [x1, x2]T e R2 is a measurable system state vector and, u represents the control law to be designed, i.e. the PSS output.

ð11Þ

Notice that the control law (11) forces state trajectories to satisfy (3). According to this equation, the trajectories converge to manifold w ¼ 0 then move along it to the origin. This means that after a transient duration the following equation is satisfied.

Design of synergetic power system stabilizer

x_ 1 ¼ ax2 x_ 2 ¼ f ðx1 ; x2 Þ þ gðx1 ; x2 Þu

ð10Þ

Fig. 1. Geometric interpretation of control law in the phase plane.

ð14Þ

Z. Bouchama et al. / Electrical Power and Energy Systems 77 (2016) 43–49

where

45

lF li ðxi Þ is the membership function of the linguistic variable

xi, and yl is the point in R at which lGl achieves its maximum value

lGl ðyl Þ ¼ 1).

(assuming

Introducing the concept of fuzzy basis function vector n(x), (18) can be rewritten as:

yðxÞ ¼ hT nðxÞ

ð19Þ

where hl = [h1...hM]T and n(x) = [n1(n). . .nM(x)]T are the fuzzy basis functions defined as:

Qn

nl ðxÞ ¼ P Fig. 2. Single-machine infinite-bus power system.




 2 1 wðx0 Þ w_ ¼ K T x þ f ðx1 ; x2 Þ þ gðx1 ; x2 Þu  2 p 1þt

ð15Þ

The Eq. (13) is used for the design of a synergetic power system stabilizer which assures asymptotic stability of the system, in which w = 0 is guaranteed at the beginning of the system operation. Moreover the proposed macro-variable eliminates the transient phase and replaces the disadvantages mentioned previously by two contributions [20]: (i) System sensitivity and fragile robustness during the transient phase, are handled for the latter has been removed. (ii) Reduction of convergence time as the reaching phase has been eliminated Using Eqs. (5) and (15), the synergetic control law is then obtained as:

u¼



  1 2 1 1 wðx0 Þ  f ðx1 ; x2 Þ  W  K T x 2 gðx1 ; x2 Þ p t þ 1 T

ð16Þ

If f and g are known, one can easily construct the synergetic control law (16). However, power system parameters are not well known and imprecise; therefore it is difficult to implement the control law (16). Therefore an adaptive fuzzy synergetic controller using fuzzy logic system is proposed to circumvent these problems.

i¼1

M l¼1

lF li ðxi Þ

Q n

i¼1



lF li ðxi Þ

It is to be noted that the approximation error issue has been addressed in great details in [21,22] where the Stone-Weierstrass theorem is used to prove that fuzzy systems can approximate any continuous real function on a compact set to any arbitrary accuracy while fuzzy rules are derived based on experts’ recommendations. Therefore the new control law is rewritten as:

uc ¼



 1 2 1 ^f ðx=h Þ  1 W  K T x wðx Þ  0 f T g^ðx=hg Þ p 1 þ t2

Consider the control problem of the nonlinear system (7), if the ^ are used and the parameters vector hf and control action uc, ^f and g hg adjusted by the adaptive law where h_ f ¼ r1 w n and

h_ g ¼ r 2 w nðxÞ uc , the closed loop system signals will be bounded and tracking error will converge to zero asymptotically. Proof. Define the optimal parameters of fuzzy systems

 i h ^hf ¼ arg minh 2Z sup n ^f ðx=hf Þ  f ðxÞ x2R f f

ð22Þ

 i h ^hg ¼ arg minh 2Z sup n ^f ðx=hg Þ  f ðxÞ x2R g f

ð23Þ

where Zf and Zg are constraint sets for hf and hg, respectively. h Defining the minimum approximation error as:

e ¼ f ðx1 ; x2 Þ  ^f ðx=^hf Þ þ ðgðx1 ; x2 Þ  g^ðx=^hf ÞÞuc

In a practical real case where f and g are unknown, they are replaced by their fuzzy estimates ^f ðx=hf Þ ¼ hTf nðxÞ and g^ðx=hg Þ ¼

Then, we have:

Rð1Þ : IF x1 is F l1 and . . . and xn is F ln THEN y is Gl

ð17Þ

A fuzzy logic system performs a mapping from U = U1  . . .  Un  Rn to R, where x = (x1, . . ., xn)T e U and y e R represent input and output of the fuzzy logic system, respec-

þ

2

p




 1 wðx0 Þ  g^ðx=hg Þuc 1 þ t2

1 ¼ e þ ð^hTf  hTf Þ:nðxÞ þ ð^hTg  hTg Þ:nðxÞ  W T

fuzzy implication, F l1  . . .  F ln ! Gl , which is a fuzzy set defined in the product space U  R. Using singleton fuzzification, product inference, and centreaverage defuzzification, the output of the fuzzy system is obtained as: [21,22].

Now consider the Lyapunov function candidate:



l¼1 y i¼1 lF li ðxi Þ  yðxÞ ¼ P Q M n l¼1 i¼1 lF l ðxi Þ i

ð18Þ

ð25Þ

^g  hg Þ Let /f ¼ ð^ hf  hf Þ and /g ¼ ðh Therefore we may rewrite (25) as:

1 w_ ¼ /Tf nðxÞ þ /Tf nðxÞuc  W þ e T

Q n l

ð24Þ

_ ¼ K T x þ f ðx1 ; x2 Þ þ gðx1 ; x2 Þuc þ g^ðx=hg Þuc W

tively.F li and Gl are labels of fuzzy sets in Ui and R, respectively, where l = 1, 2, . . ., M. Each fuzzy IF-THEN rule of (17) defines a

PM

ð21Þ

Theorem [21]:

Fuzzy logic systems

hTg nðxÞ. A brief recall of fuzzy logic systems is given in what follows. A basic configuration of fuzzy logic systems [21] consists of a collection of fuzzy IF-THEN rules:

ð20Þ

V¼


 1 1 1 W2 þ /Tf /f þ /Tg /g 2 r1 r2

ð26Þ

ð27Þ

where r1 and r2 are positive constants that will be used as learning rates in the adaptation procedure. Time derivative of V is obtained as:

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Z. Bouchama et al. / Electrical Power and Energy Systems 77 (2016) 43–49

Fig. 3. System response to three phase fault for case 1.

_ þ 1 /T /_ f þ 1 /T /_ g V_ ¼ WW r1 f r2 g
 1 1 1 ¼ W /Tf :nðxÞ þ /Tf :nðxÞuc  W þ e þ /Tf /_ f þ /Tg /_ g T r1 r2  1   1  1 6 /Tf r 1 WnðxÞ þ /_ f þ /Tg r 2 WnðxÞuc þ /_ g þ we  jWj r1 r2 T where /_ f ¼ h_ f and /_ g ¼ h_ g .

Substituting h_ f and h_ g into (28), leads to

ð28Þ

1 V_ 6 We  W2 T

ð29Þ

Based on the universal approximation theorem, term We is very

small, such that V_ 6 0, and such that all closed-loop system signals are bounded.

Fig. 4. System response to 0.1 p.u. step in reference voltage for case 2.

Fig. 5. System response to 0.2 pu step in mechanical torque for case 3.

Z. Bouchama et al. / Electrical Power and Energy Systems 77 (2016) 43–49

In order to complete the proof we need to show that the tracking error converges to zero asymptotically, we need to prove that: lim t?1|W(t)| = 0. Inequality (29) leads to:

1 1 1 T V_ 6 jWjjej  jWj2 6  jWj2  ðjWj  TjejÞ2 þ jej2 T 2T 2T 2 having:

1 2T

ð30Þ

ðjWj  TjejÞ2 P 0 empowers rewriting (30) as follows:

1 T V_ 6  jWj2 þ jej2 2T 2

ð31Þ

Integrating both sides of (31), then we have:

Z

t

0

jWðsÞj2 ds 6 2T½jVð0Þj þ jVðtÞj þ T 2

Z 0

t

jej2 ds

ð32Þ

Defining constants: a and b as: a ¼ 2T½jVð0Þj þ jVð1Þj and b = T2, therefore, (32) can be further simplified as:

Z 0

t

jWðsÞj2 ds 6 a þ b

Z 0

t

jeðsÞj2 ds

ð33Þ

If e e L2, then we have W e L2. From (33), we know that W is _ 2 L1 , bounded and every term in (25) is bounded. Hence, W; W _ 2 L1 , we conclude using Barbalat lemma [23], if W e L2 \ L1 and W

that: lim t?1|W(t)| = 0, the system is stable and the error will asymptotically converge to zero. Simulation results

In this study, we will investigate the performance of the proposed power system stabilizer as it is applied to both single machine infinite-bus and multi-machine power systems. The success of the proposed PSS, with the single-machine infinite-bus case, motivates us to test its capability on a multi-machine model.

47

Application to the single-machine infinite bus model A nonlinear power system model consisting of a single machine connected to an infinite bus (SMIB) represented in Fig. 2 is selected to assess performance and effectiveness of the proposed controllers. Performance obtained with the proposed reaching phase free adaptive fuzzy power system stabilizer (RPFSPSS) is compared to those obtained using a conventional (CPSS) [1], using a fuzzy power system stabilizer (FPSS) [17] and using an adaptive fuzzy synergetic power system stabilizer (AFSPSS) [13], under different operating conditions. The nonlinear simulation of the power system model is carried out under the following severe fault cases: Case 1: With the generator operating at nominal condition specified by an active power of 0.9 pu and reactive power 0.3 p. u., and a three-phase fault to ground on the transmission line occurring at t = 0.2 s with 0.06 s duration. Simulation of the system responses are presented in Fig. 3 with different stabilizers. The proposed stabilizer effectively exhibits and confirms superior performance in improving the damping of oscillations compared to the AFSPSS, FPSS and CPSS. Case 2: A 0.1 p.u. step increase in reference voltage was applied at t = 0.1 s while the generator is operating at an active power of 0.7 p.u. with 0.7 power factor. As illustrated in Fig. 4 the system responses with AFPSS and AFSPSS have a better performance than the CPSS whereas responses obtained with RPFSPSS show rapid elimination of unwanted oscillations thus enabling better power flow. Case 3: the result shown in Fig. 5. was simulated with a 0.2 p.u. disturbance in mechanical torque occurring at t = 0.5 s, while the system is operating with an active power of 0.85 p.u. with 0.9 leading power factor. As observed, the stabilizers CPSS, AFPSS and AFSPSS show satisfactory performance. It is worth mentioning that the oscillations were damped much faster with the proposed PSS.

Fig. 6. Multi machine power system.

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Z. Bouchama et al. / Electrical Power and Energy Systems 77 (2016) 43–49

Fig. 7. Speeds deviations response in nominal operating point.

Fig. 8. Speed deviation response for heavy operating point.

Fig. 9. Speeds deviations response in light operating point.

Z. Bouchama et al. / Electrical Power and Energy Systems 77 (2016) 43–49 Table 1 Loading operating conditions for the system (in p.u). Nominal P

Heavy

Light

Q

P

Q

P

Q

Generator G1 1.25 G2 0.9 G3 1.0

0.50 0.30 0.35

2.0 1.80 1.50

0.80 0.60 0.60

0.65 0.45 0.50

0.55 0.35 0.25

Load A B C

0.50 0.30 0.35

2.0 1.80 1.50

0.80 0.80 0.60

0.65 0.45 0.50

0.55 0.35 0.25

1.25 0.9 1.0

Application to the multi-machine model In this study, a three-machine nine-bus power system shown in Fig. 6 is considered. Details of the system data are given [1] To assess the effectiveness and robustness of the proposed method over a wide range of loading conditions, three different cases designated as nominal, lightly and heavily loading are considered. The generator and system loading levels at these cases are given in Table 1. In this scenario, the performance of the proposed controller under transient conditions is verified by applying a 6-cycle threephase fault at t = 0.2 s, on bus 7 at the end of line 5–7 [1]. The fault is cleared by permanent tripping of the faulted line. The speed deviations of the generators G1, G2 and G3 under the nominal, lightly and heavily loading conditions are shown in Figs. 7–9. To evaluate the performance and effectiveness of the proposed controller, system response obtained with the proposed stabilizer is compared to those obtained using conventional PSS, fuzzy power system stabilizer and using an adaptive fuzzy synergetic power system stabilizer. It is evident from the results in Fig. 3 that damping of the low frequency oscillations with both stabilizers: conventional PSS and FPSS require more time and show more oscillations before the speed deviation response is stabilized. The adaptive fuzzy synergetic PSS improves the damping of oscillations despite changes in operating conditions. However, superior performance is clearly obtained with the proposed controller. The latter provides significantly better damping enhancement of power system oscillations, while one can easily observe overshoot and settling time reduction. Conclusion A new robust adaptive power system stabilizer based on synergetic control and fuzzy systems, has been proposed in this paper with the following main contribution: – enhancement in robustness is evident through a convergence time reduction by way of transient mode removal. – elimination of chattering through the use of a continuous control law. – ease in implementation for not relying on non-measurable variables.

49

Simulation studies were carried out for single and multimachine power systems for severe different operating conditions showing the effectiveness of the proposed design stabilizer. Furthermore a comparative study between a classical PSS, Fuzzy PSS, adaptive fuzzy PSS and the proposed alternative has shown better performance of the latter over its counterparts providing faster damping of low frequency oscillations therefore improving greatly power system stability. References [1] Anderson PM, Fouad AA. Power system control and stability. New York: IEEE Press; 1993. [2] Kundur P. Power system control and stability. McGraw-Hill Inc.; 1994. [3] Talaq J. Optimal power system stabilizers for multi machine systems. Electr Power Energy Sys 2012;43:793–803. [4] Sambariy DK, Prasad R. Robust tuning of power system stabilizer for small signal stability enhancement using metaheuristic bat algorithm. Electr Power Energy Sys 2014;61:229–38. [5] Cao Y, Jiang L, Cheng S, Chen D, Malik OP, Hope GS. A nonlinear variable structure stabilizer for power system stability. IEEE Trans Energy Conversion 1994;9:489–95. [6] Alkhatib H, Duveau Jean. Dynamic genetic algorithms for robust design of multimachine power system stabilizers. Electr Power Energy Sys 2013;45:242–51. [7] Mostafa HE, El-Sharkawy MA, Emary AA, Yassin K. Design and allocation of power system stabilizers using the particle swarmoptimization technique for an interconnected power system. ElectrPowerEnergy Sys 2012;34:57–65. [8] Ali ES. Optimization of power system stabilizers using BAT search algorithm. ElectrPower Energy Sys 2014;61:683–90. [9] Segal R, Kothari ML, Madnani S. Radial basis function (RBF) network adaptive power system stabilizer. IEEE Trans Power Systems 2000;15:722–7. [10] Ghasemi A, Shayeghi H, Alkhatib H. Robust design of multimachine power system stabilizers using fuzzy gravitational search algorithm. ElectrPowerEnergy Sys 2013;51:190–200. [11] Keumarsi V, Simab M, Shahgholian G. An integrated approach for optimal placement and tuning of powersystem stabilizer in multi-machine systems. ElectrPowerEnergy Sys 2014;63:132–9. [12] Jiang Z. Design of a nonlinear power system stabilizer using synergetic control theory. Elect. Power Syst Res 2009;79(6):855–62. [13] Bouchama Z, Harmas MN. Optimal robust adaptive fuzzy synergetic power system stabilizer design. Electr Power Syst Res 2012;83(1):170–5. [14] Santi E, Monti A, Donghong Li, Proddutur K, Dougal RA. Synergetic control for DC-DC boost converter: implementation options. IEEE Trans Ind Appl 2003;39 (6):1803–13. [15] Kondratiev I, Santi E, Dougal R, Veselov G. Synergetic control for m-parallel connected DC-DC buck converters, PESC, 30th Annual IEEE, vol. 1, 2004. p. 182–8. [16] Jiang Z, Dougal RA. Synergetic control of power converters for pulse current charging of advanced batteries from a fuel cell power source. IEEE Trans Power Electron 2004;19(4):1140–50. [17] Hosseinzadeh N, Kalam A. An indirect adaptive fuzzy logic power system stabiliser. Electr Power Energy Syst 2002;24(10):837–42. [18] Nechadi E, Harmas MN, Hamzaoui A, Essounbouli N. A new robust adaptive fuzzy sliding mode power system stabilizer. ElectrPowerEnergy Sys 2012;42:1–7. [19] Saoudi K, Harmas MN. Enhanced design of an indirect adaptive fuzzy sliding mode power system stabilizer for multi-machine power systems. ElectrPowerEnergy Sys 2014;54:425–31. [20] Al-khazraji, Essounbouli N, Hamzaoui A, Nollet F, Zaytoon J. Type-2 fuzzyslidingmodecontrolwithoutreachingphase for nonlinearsystem. Eng Appl Artif Intel 2011;24:23–38. [21] Wang LX. Stable adaptive fuzzy control of nonlinear systems. IEEE Trans Fuzzy Sys 1993;1(2):146–55. [22] Kosko B. Fuzzy systems are universal Approximators. In: Proc IEEE Int Conf Fuzzy Systems, San Diego, Ca., 1992. p. 1143–62. [23] Hou M, Duan G, Guo M. New versions of Barbalat’s lemma with applications. J Control Theory Appl 2010;8(4):547–54.