Adaptive Fuzzy Sliding Mode Power System Stabilizer · PDF fileInternational Journal of Automation and Computing 10(4), August 2013, 281-287 DOI: 10.1007/s11633-013-0722-0 Adaptive

International Journal of Automation and Computing 10(4), August 2013, 281-287

DOI: 10.1007/s11633-013-0722-0

Adaptive Fuzzy Sliding Mode Power System Stabilizer

Using Nussbaum Gain

Emira Nechadi1 Mohamed Naguib Harmas1 Najib Essounbouli2 Abdelaziz Hamzaoui21Electrical Engineering Department, Ferhat Abbas University of Setif 1, Algeria

2Centre of Research for Science and Information Technology and Communication Laboratory, Champagne Ardennes University, France

Abstract: Power system stability is enhanced through a novel stabilizer developed around an adaptive fuzzy sliding mode approachwhich applies the Nussbaum gain to a nonlinear model of a single-machine infinite-bus (SMIB) and multi-machine power systemstabilizer subjected to a three phase fault. The Nussbaum gain is used to avoid the positive sign constraint and the problem ofcontrollability of the system. A comparative simulation study is presented to evaluate the achieved performance.

Keywords: Multi-machine power system stabilizer, adaptive fuzzy, sliding mode, Nussbaum gain, Lyapounov stability.

1 Introduction

Power systems are complex nonlinear systems that oftenexhibit low frequency oscillations due to insufficient damp-ing caused by adverse operating conditions which can leadto a devastating loss of synchronism[1]. Power system sta-bilizers are used to suppress these oscillations and improvethe overall stability[1−4]. The computation of the fixed pa-rameters of these stabilizers is usually based on the lin-earized model of the power system around a nominal op-erating point[5−7]. Operating conditions often change as aresult of load variations and/or major disturbances. Thesechanges affect power system dynamic behavior which re-quires adjustment of stabilizer parameters. Keeping thelatter at fixed values will greatly degrade power systemperformance[7]. Conventional stabilizers, using lead–lagcompensators have been based on linearized power systemmodel to damp oscillations. Disturbances, varying load-ing conditions and therefore frequently changing operatingpoint were not taken into consideration[8−11]. However, alot of researches about the design of power system stabi-lizers have been conducted, using a wide range of strate-gies, such as sliding controller[12], adaptive controller[13−15],and adaptive fuzzy controllers[16, 17]. A comparison of someapproaches to designing power system stabilizers was pre-sented in [18]. One of these possible methods is the ap-plication of adaptive fuzzy sliding controller. Remarkableresearch effort has been done in the last decade to put for-ward intelligent fuzzy logic based power system stabilizer(PSS) as well as optimality in adapting to changing operat-ing conditions as in [19−21]. However, these linear modelbased control strategies often fail to provide satisfactory re-sults over a wide range of operating conditions. Moreover,during severe disturbances, PSS action may actually causethe generator under its control to lose synchronism in anattempt to control its excitation field.

For the last few years, optimization techniques for aconventional[22−23] and dual PSS[24−26] have also been ap-plied using different algorithms such as particle swarm op-timization (PSO), genetic algorithms (GA), chaotic opti-

Manuscript received December 5, 2012; revised April 16, 2013

mization algorithm (COA) and neuro-fuzzy system (NFS).Contribution made in this paper consists in a new adap-

tive fuzzy sliding mode power system stabilizer using aNussbaum gain. Stability of the overall system is guar-anteed via Lyapunov synthesis.

In the following sections of this paper, a nonlinear powersystem model is presented first, followed by the develop-ment of an adaptive fuzzy sliding mode controller usingNussbaum gain and the addressed stability issue. In Sec-tion 3, the study of simulation results for different operatingconditions on single-machine and multi-machine power sys-tem is described.

2 Power system model

The power system model considered in this paper is anonlinear model representing a synchronous machine con-nected to an infinite bus via a double circuit transmissionline. Fig. 1 shows the power system schematic diagram in-cluding turbine, transformer, automatic voltage regulatorand PSS[4].

Fig. 1 Single-machine infinite-bus (SMIB) power system

A nonlinear representation of the power system during atransient period after a major disturbance has occurred inthe system[17, 27, 28]:

{x1 = x2

x2 = f (x) + g (x) u(1)

where x =[

x1 x2

]T

=[

∆ω ∆PM

]T

∈ R2 is the

282 International Journal of Automation and Computing 10(4), August 2013

state vector, ∆ω is the speed deviation, ∆P = Pm − Pe

is the accelerating power, M is inertia moment coefficientof the synchronous machine, u ∈ R is the input, f (x) andg (x) are nonlinear functions, and g (x) 6= 0 in the control-lability region (see the appendix).

The block diagram of a conventional lead-lag power sys-tem stabilizer is shown in Fig. 2, in which a conventionalsingle-input is presented. Parameters and details can befound in [22].

The two inputs of an IEEE power system stabilizer, un-like the conventional single-input PSS, is shown in Fig. 3,in which a dual-input PSS3B is presented. Parameters anddetails can be found in [24].

Fig. 2 Conventional power system stabilizer

Fig. 3 Dual-input power system stabilizer

3 Adaptive fuzzy sliding mode powersystem stabilizer

Consider a single-input-single-output (SISO) nonlinearsystem described by (1), with the sliding surface as

S = x2 + βx1 (2)

where β is a positive constant.The time derivative of the sliding surface (2) is given by

S = f (x) + βx2 + g (x) u. (3)

Assuring the existing condition (4)

SS < 0. (4)

Hence, the control law is

u = g−1 (x) (−f (x)− βx2 − ksgn (S)) . (5)

Control law (5) allows ensuring the system stabilizationand robustness, but it has the drawback in the computationof k, which is not a straightforward task. In a more realisticcase where f (x) and g (x) are unknown, they are replacedby their fuzzy estimations.

In previous researches on the indirect adaptive fuzzymethod, the controller with g−1 (x) can be singular becauseit cannot be guaranteed that g (x) is not equal to zero at anymoment, where g (x) denotes the approximation of g (x). Inthis paper, a Nussbaum gain will be incorporated into thecontrol law in order to estimate function g−1 (x)[29].

Definition 1[30]. A function is called a Nussbaum-typefunction if it has the following properties:

limy→∞

sup1

y

∫ y

0

N (ς) dς = +∞ (6)

limy→∞

inf1

y

∫ y

0

N (ς) dς = −∞. (7)

Throughout this paper, the even Nussbaum function:

N (ς) = cos(π

2

)ςeς2 (8)

is employed, and ς is a variable to be determined later.Lemma 1[30]. Let V (·) and ς (·) be smooth functions de-

fined on [0, tf ), with V (t) > 0, ∀t ∈ [0, tf ), and N (·) be aneven Nussbaum-type function. If the following inequalityholds:

V (t) 6 c0 + e−c1t

∫ t

0

g′ (x (τ))N (ς) ςec1τdτ+

e−c1t

∫ t

0

ςec1τdτ , ∀t ∈ [0, tf ) (9)

where c0 represents some suitable constant, c1 is a positiveconstant, and g′ (x (τ)) is a time-varying parameter whichtakes values in the unknown closed interval I =

[l−, l+

],

with 0 /∈ I,then V (t) , ς (t) and∫ t

0g′ (x (τ))N (ς) ςdτ must

be bounded on [0, tf ).The proof of this lemma can be found in [30].Based on the universal approximation theorem, the un-

known function f (x) and constant k can be approximatedby (10) and (11), respectively:

f (x, θf ) = ξT (x) θf (10)

k (x, θk) = ξT (x) θk (11)

where θ = [θ1, θ2, · · · , θm]T is the parameter vector andξ = [ξ1, ξ2, · · · , ξm]T is the vector of fuzzy basis functions.

The approximation error is given by

δf = f (x)− ξT (x) θ∗f (12)

δk = k − ξT (x) θ∗k (13)

in which θ∗f and θ∗k are the optimal parameter values. Wedefine

θf = θf − θ∗f (14)

θk = θk − θ∗k. (15)

Theorem 1. For the nonlinear system (1), if we choosethe following control law

u = N (ς)(−f (x)− βx2 − ksgn (S)

)(16)

with

ς = S(θT

f ξ (x) + βx2

)(17)

and the following adaptation laws:

θf = γ1Sξ (x)− γ1θf (18)

θk = γ2g (x) N (ς) |S| ξ (x)− γ2θk (19)

E. Nechadi et al. / Adaptive Fuzzy Sliding Mode Power System Stabilizer · · · 283

such that |δfS| 6 ε, γ1, γ2 are positive constants andg′ (x) = −g (x), then the stability of the closed loop sys-tem can be guaranteed.

Proof. Choose the Lyapunov function candidate to be

V =1

2STS +

1

2γ1θT

f θf +1

2γ2θT

k θk. (20)

Therefore,

V = STS +1

γ1θT

f θf +1

γ2θT

k θk =

ST (f (x) + g (x) u + βx2) +1

γ1θT

f θf +1

γ2θT

k θk =

ST((

g′ (x) N (ς)+1) (

θTf ξ (x)+βx2

)− θT

f ξ (x)+δf

)−

kg (x) N (ς) sgn (S))

+1

γ1θT

f θf +1

γ2θT

k θk =

[g′ (x) N (ς)+1

]ς − θT

f Sξ (x)+δfS− k∗g (x) N (ς)

|S| STS−

θTk g (x) N (ς) |S| ξ (x) +

1

γ1θT

f θf +1

γ2θT

k θk. (21)

Using (18) and (19), we can obtain

V = −k∗g (x) N (ς)

|S| STS − θTf θf−

θTk θk + δfS +

(g′ (x) N (ς) + 1

)ς . (22)

The following inequalities are valid

−θTf θf 6 −1

2θT

f θf +1

2

∥∥θ∗f∥∥2

(23)

−θTk θk 6 −1

2θT

k θk +1

2‖θ∗k‖2 . (24)

And we can rewrite V as

V 6 −k∗g (x) N (ς)

|S| STS − 1

2θT

f θf − 1

2θT

k θk +1

2

∥∥θ∗f∥∥2

+

1

2‖θ∗k‖2 + ‖ε‖+

(g′ (x) N (ς) + 1

)ς . (25)

Let α = min{

λmin

(2k∗g(x)N(ς)

|s|

), λmin (γ1) , λmin (γ2)

}

and β = 12

∥∥θ∗f∥∥2

+ 12‖θ∗k‖2 + ‖ε‖. Then, (25) becomes

V 6 −αV + β +(g′ (x) N (ς) + 1

)ς . (26)

Multiplying both sides of (26) by eαt, we can obtain

d

dt

(V (t) eαt) 6 βeαt + eαt [

g′ (x) N (ς) + 1]ς . (27)

After integrating (27) over [0, tf ] , it follows that

0 6 V (t) 6 β

α+

[V (0)− β

α

]e−αt+

e−αt

∫ t

0

[g′ (x) N (ς) + 1

]eατ ςdτ . (28)

Noting that 0 < e−αt < 1 and βαe−αt > 0, we have[

V (0)− βα

]e−αt 6 V (0). Then the above equality becomes

0 6 V (t) 6 η + e−αt

∫ t

0

[g′ (x) N (ς) + 1

]eατ ςdτ (29)

where η = βα

+ V (0).According to Lemma 1, it can be concluded from

(32) that V (t) and∫ t

0g′ (x) N (ς) ςdτ are bounded,

i.e., S, θf , θk, ς (t) and∫ t

0[g′ (x) N (ς) + 1] eατ ςdτ are also

bounded. ¤

Fig. 4 Speed deviations in nominal, heavy loading and light loading cases

284 International Journal of Automation and Computing 10(4), August 2013

4 Simulation

The soundness of the proposed PSS was tested, and theperformance as well as robustness tests were conducted andcompared with a conventional stabilizer and a dual-inputpower system stabilizer through simulations. Good tran-sient behavior with the proposed control under severe oper-ating conditions were illustrated by the following case stud-ies. The speed variation and accelerating power are chosenas the power system control variables. Five fuzzy sets foreach input are sufficient for the PSS to be designed.

4.1 Simulation cases for an SMIB

To assess the performance of the proposed controller,simulations were carried out for different operating condi-tions.

A three-phase fault test is applied to an infinite bus as inFig. 1, lasting 60ms before being cleared. When the powersystem is strongly perturbed, the proposed stabilizer re-acts rapidly and prevents an eventual loss of synchronism.Therefore, it enables the system to reach a stable operatingpoint very quickly.

The simulation results for nominal load, heavy load andlight load are shown in Fig. 4. It is clear that the proposedPSS (PPSS) exhibits superior performance to the conven-tional PSS (CPSS) and dual-input PSS (DPSS) power sys-tem stabilizers. The simulation results shown in Fig. 4 in-dicates a good transient behavior of the proposed PSS.

4.2 Simulation of multi-machine powersystem

To evaluate the performance of the proposed control, weperformed simulation for multi-machine power system asin Fig. 5 with the aim to compare the performance of the

proposed PSS with the conventional PSS and dual-inputPSSs.

In order to validate stability enhancement due to the pro-posed stabilizer, a three-phase fault test is applied to bus 7(Fig. 5) with a duration of 60ms before it is cleared. Threeoperating conditions were investigated: Cases of nominalload, light load and heavy load. The fault is cleared andthe controller helps the system to reach a stable operat-ing point very quickly. As shown in Fig. 6, the proposedapproach shows better control performance than the con-ventional PSS and dual PSSs in terms of settling time anddamping effect.

Fig. 5 Multi-machine power system

To further evaluate the robustness of the proposed stabi-lizer, the power system is subjected to heavy load operation(Fig. 7) and light load operation (Fig. 8). The results showagain the clear oscillations damping of the proposed con-troller in multi-machine power system, as compared withits conventional counterpart, but we have a small deterio-ration in performance.

Fig. 6 Speed deviations in nominal case

E. Nechadi et al. / Adaptive Fuzzy Sliding Mode Power System Stabilizer · · · 285

Fig. 7 Speed deviations in heavy loading of the system

Fig. 8 Speed deviations in light loading of the system

In this study, we used different operating points todemonstrate the effectiveness of the proposed control in os-cillations damping after the occurrence of large disturbanceon a power system by providing better transient responseand stronger robustness than other stabilizers. The adap-tive fuzzy sliding mode using Nussbaum gain scheme per-mits to take account of severe load variation and varyingoperating conditions.

5 Conclusions

Based on the adaptive fuzzy sliding mode controller andthe Nussbaum gain, we introduced a new power system sta-bilizer that enhances damping and improves transient dy-namics of a single-machine infinite-bus and multi-machinepower system stabilizers. Different load conditions as wellas severe perturbations were used to evaluate the proposed

286 International Journal of Automation and Computing 10(4), August 2013

power system stabilizer effectiveness in rapidly reducingoscillations that could lead to loss of synchronism if nottreated. Simulation results exhibited its superior perfor-mance over classical PSSs. Real power system remains tobe thoroughly investigated under the proposed stabilizer.

Appendix

Parameters of single (SMIB) operating conditions:Nominal load: P = 0.9/unit, Q = 0.3/unitHeavy load: P = 1.60/unit, Q = 0.6/unitLight load: P = 0.45/unit, Q = 0.35/unitMulti-machine power system parameters:G1:Nominal load: P = 0.72/unit, Q = 0.27/unit.Heavy load: P = 2.21/unit, Q = 1.09/unit.Light load: P = 0.36/unit, Q = 0.16/unit.G2:Nominal load: P = 1.63/unit, Q = 0.07/unit.Heavy load: P = 1.92/unit, Q = 0.56/unit.Light load: P = 0.80/unit, Q = −0.11/unit.G3:Nominal load: P = 0.85/unit, Q = −0.11/unit.Heavy load: P = 1.28/unit, Q = 0.36/unit.Lightload: P = 0.45/unit, Q = −0.20/unit.Nonlinear functions:

f (x) =xe + xd

T ′d0 (xe + x′d)(∆P − Pm) +

V 2

2T ′d0 (xe + x′d)2

((xe + xd) (x′d − xq)

(xe + xq)+ (xd − x′d)

)sin (2δ)+

(Q + V 2

(sin2 (δ)

xe + xq

+cos2 (δ)

xe + x′d+

(x′d − xq) cos (2δ)

(xe + xq) (xe + x′d)

))×

ωsm∆ω +KAV sin (δ)

T ′d0 (xe + x′d)(Vref − Vt)

g (x) =KAV sin (δ)

T ′d0 (xe + x′d).

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Emira Nechadi received her bachelor degree in control sys-tems, master degree in control systems, and Ph. D. degree in

control systems, all from the University of Setif, Algeria. in2002, 2004, and 2013, respectively.

Her research interests include sliding mode control, adaptivefuzzy control, fuzzy systems, synergetic control and power sys-tems.

E-mail: [email protected] (Corresponding author)

Mohamed Naguib Harmas receivedthe Ph.D. degree in control systems fromFerhat Abbas University of Setif 1, andM. Sc. degree in electrical engineering andelectronics from California State Univer-sity of Sacramento, Sacramento, USA. Heis currently a professor at Setif University,Algeria.

His research interests include robust non-linear control, power system and power

electronics control, and fuzzy synergetic design.E-mail: [email protected]

Najib Essounbouli received his bache-lor degree in electrical from the Universityof Sciences and Technology of Marrakech(FSTG) Morocco, and DEA. and Ph.D.degrees both in electrical engineering fromReims University of Champagne Ardennes,in 2000 and 2004 respectively. From 2005to 2010, he was an assistant professorwith University Institute of Troyes, ReimsChampagne Ardennes University. Since

2010, he has been a professor with the same institute.His research interests include fuzzy logic control, robust adap-

tive control, renewable energy and control drive.E-mail: [email protected]

Abdelaziz Hamzaoui received hisbachelor degree in electrical engineeringfrom the Polytechnic School of Algiers(ENPA), Algeria, in 1982, and DEA. andPh.D. degrees both in electrical engineer-ing from Reims University of ChampagneArdennes, in 1989 and 1992, respectively.He is currently a professor and the direc-tor of the Technology Institute of Troyes,Reims Champagne Ardennes University.

His research interests include intelligent control, fuzzy control,and robust adaptive control.

E-mail: [email protected]