A nonlinear voltage controller based on interval type 2 fuzzy logic control system for multimachine power systems

Electrical Power and Energy Systems 45 (2013) 456–467

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

A nonlinear voltage controller based on interval type 2 fuzzy logic control system for multimachine power systems A. Abbadi ⇑, L. Nezli, D. Boukhetala Laboratoire de Commande des Processus, Département d’Automatique, Ecole Nationale Polytechnique, El Harrach, Algeria

a r t i c l e

i n f o

Article history: Received 26 March 2012 Received in revised form 15 August 2012 Accepted 26 September 2012 Available online 7 November 2012 Keywords: Power system Type-2 fuzzy logic control system (T2 FLCS) Lyapunov stability Pole placement Linear matrix inequalities (LMIs)

a b s t r a c t In this paper, we propose an interval type-2 fuzzy controller that has the ability to enhance the transient stability and achieve voltage regulation simultaneously for multimachine power systems. The design of this controller involves the direct feedback linearization (DFL) technique. The DFL compensated system model is transformed into an equivalent type 1 T–S fuzzy model using linearly independent functions. This paper highlights the mathematical foundations for analyzing the stability and facilitating the design of stabilizing controllers of the interval type 2 Takagi–Sugeno fuzzy control systems (IT2 T–S FLCSs). Sufficient conditions for designing the interval type-2 fuzzy controller with meeting quadratic D stability constraints are obtained by using Linear Matrix Inequalities (LMIs). Based on only local measurements, the designed controller guarantees transient stability, voltage regulation and satisfies desired transient responses. The proposed controller is applied to two-generator infinite bus power system. Simulation results illustrate the performance of the developed approach regardless of the system operating conditions. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Power system stability has been recognized as an important problem since the electricity has been used in everyday life. Although power system stability may be broadly defined according to different conditions, two major issues are frequently considered; the first is the problem of transient stability and the second is the voltage regulation. In order to obtain high quality transient stability and voltage regulation, many researches has been established and numerous papers are published [1–10]. Generator excitation systems play a fundamental role in power system control. Indeed, the basic function of the excitation system is to supply and automatically adjust the field current of the synchronous generator to regulate the terminal voltage. The power system stabilizer (PSS) provides the supplementary signal through the excitation automatic voltage regulator (AVR) loop which dampens the power oscillations. The common feature of AVR/PSS controllers is that they are typically based on models established by approximate linearization of the nonlinear equations of a power system at certain operating point [2,3]. But, in case of large disturbances, these linearized control techniques are not effective since the modern power systems are highly complex and nonlinear and their operating conditions can vary over a wide range. Hence, ⇑ Corresponding author. Address: Electrical and Computer Engineering Department, LREA, University of Medea, Algeria. Tel.: +213 778359368; fax: +213 25581253. E-mail address: [email protected] (A. Abbadi). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.09.020

attention has been focused on the application of nonlinear controllers, which are independent of the equilibrium point and take into account the important nonlinearities of the power system model. The application of nonlinear control techniques to solve the transient stabilization problem has been given much attention [5–8]. Most of these nonlinear excitation controllers are based on feedback linearization technique [5–7]. The drawback of the existing nonlinear excitation controller designs is that the post-fault voltage varies considerably from the pre-fault one; this is mainly due the heavy inherent nonlinear characteristics of the terminal voltage which prevents us from using it as feedback variable. In the last decades, nonlinear control approaches based on the Takagi–Sugeno fuzzy model solved this voltage regulation problem by including terminal voltage as feedback variable [9,10]. Takagi– Sugeno fuzzy logic control systems (T–S FLCSs) have been mostly considered as one of the best suitable tools for modeling and control of nonlinear systems. Based on the Lyapunov method, some significant stability analysis results have provided sufficient stability conditions in Linear Matrix Inequalities (LMIs) terms to ensure the overall system stability. The controller is synthesized via the parallel distributed compensation (PDC) technique [11–13]. However, based on precise type-1 (T1) fuzzy sets, the type-1 T–S FLCSs have the common problem that they cannot fully handle the linguistic and numerical uncertainties with an unknown, uncertain and perturbed nonlinear dynamical system. Recently, the applications of type-2 fuzzy logic controllers to uncertain control processes have received considerable attention

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[14]. The type-2 fuzzy logic controller based on the T2-fuzzy sets can deal with both the linguistic and the numerical uncertainty effectively; these fuzzy sets include a third dimension and footprint of uncertainty. Thus, the type-2 fuzzy logic controller can obviously outperform its T1 counterpart under the situation of high uncertainty. One of the first studies on type-2 fuzzy logic controller that have been employed for solving the rotor angle stability problems is reported in [15]. There, for both small and large disturbance, oscillations are damped by using a power system stabilizer (PSS). With this controller, considerable efforts have been devoted to damp frequency oscillations in the power systems, and less attention has been paid to the voltage quality and to the excursions of generator rotor angles. In this paper, motivated by exploiting the advantages of the interval type-2 fuzzy controller, we have investigated the transient stability enhancement and the voltage regulation of multimachine power systems in the framework of the interval type-2 Takagi–Sugeno fuzzy logic control system (IT2 T–S FLCS). The aim of this work is to present and discuss the stability of the IT2 T–S FLCS through the Lyapunov stability theory; where a set of LMI-based stability conditions are derived to guarantee the system stability and facilitate the control synthesis. Moreover the desired transient specification behaviors of the closed loop system, which is commonly expressed in terms of transient responses [16], can be guaranteed by constraining the closed-loop poles to lie in a suitable sub-region of the left-half of the complex plane [17]. IT2 T–S FLCS theory is advantageous in that it is able to handle nonlinearity smoothly, which is an attractive method to overcome the heavy nonlinear characteristics of the terminal voltage in power system. In this paper, The DFL technique has been used to linearize and decouple a nonlinear n machine power system to n independent DFL compensated models. These compensated models are described by type 1 T–S models. The proposed IT2 fuzzy controller is simulated on a two-generator infinite bus power system. The simulation results exhibit the effectiveness of the designed controller, in the sense that both voltage regulation and system stability enhancement can be achieved in the presence of variation of operation points, fault location, network parameters and step change of the mechanical input power. The layout of the paper is as follows. In Section 2, the background materials concerning type 1 T–S fuzzy model, IT2 fuzzy controller and IT2 T–S FLCS are introduced. Sufficient conditions for stability and pole location requirements formulated in terms of LMIs are represented in Section 3. In Section 4, the T–S fuzzy model of the power system is described. The nonlinear voltage controller derived from the proposed methodology and based on constructed type 1 T–S Fuzzy model is designed in Section 5. This control scheme is implemented in a two-machine infinite bus power system and simulation results are provided to demonstrate the performance of the proposed controller in Section 6. Finally, conclusions are drawn in Section 7.

Bae et al. in [19] presented a method for constructing T–S fuzzy model using the sum of products of linearly independent functions from nonlinear systems. To this end the following nonlinear system is considered

x_ ¼ FðzðtÞÞgðtÞ

ð1Þ

T

T

T

T

where g (t) = [x (t)u (t)] , the matrix function F(z(t)) is represented as

2

f11 ðzðtÞÞ f12 ðzðtÞÞ    f1ðnþmÞ ðzðtÞÞ

3

7 6 6 f21 ðzðtÞÞ f22 ðzðtÞÞ    f2ðnþmÞ ðzðtÞÞ 7 7 6 FðzðtÞÞ ¼ 6 7 .. .. .. .. 7 6 . . . . 5 4 fn1 ðzðtÞÞ fn2 ðzðtÞÞ    fnðnþmÞ ðzðtÞÞ

ð2Þ

where fij(z(t)) is the (i, j) element of the F(z(t)) matrix. The nonlinear system (1) can be rewritten as follows

"

# w X _ fi ðzðtÞÞF i gðtÞ xðtÞ ¼ F0 þ

ð3Þ

i¼1

where

fi ðzðtÞÞ ¼

v Y l g iij ðzðtÞÞ;

8i ¼ 1; 2; . . . w

ð4Þ

j¼1

where ‘v’ is the least number of linearly independent functions gj (z(t)). When we express fi (z(t)) to the form of Eq. (4), lij is ‘1’ if fi (z(t)) has the term of gj(z(t)), otherwise lij is ‘0’. It is also possible to rewrite the nonlinear system (1) to the form of Eq. (3). Substituting Eq. (4) into Eq. (3) gives

" _ xðtÞ ¼ F0 þ

w Y v X

l g jij ðzðtÞÞF i

#

gðtÞ

ð5Þ

i¼1 j¼1

and Eq. (5) is equivalent to

" _ xðtÞ ¼ F0 þ

# w Y v X 1 X l hjk ðzðtÞÞg jkij F i gðtÞ

ð6Þ

i¼1 j¼1 k¼0

where

hj0 ðzðtÞÞ ¼

g j1  g j ðzðtÞÞ ; g j1  g j0

g j0 ¼ minfg j ðzÞg; z

hj1 ðzðtÞÞ ¼

g j ðzðtÞÞ  g j0 g j1  g j0

ð7Þ

g j1 ¼ maxfg j ðzÞg z

for all j = 1, 2, . . ., v. To verify that Eqs. (5) and (6) are equivalent, we need to show that the following expressions are true l

g jij ðzðtÞÞ ¼

1 X

l

hjk ðzðtÞÞg jkij

ð8Þ

k¼0 1 X hjk ðzðtÞÞ ¼ 1

ð9Þ

k¼0

2. T1 T–S fuzzy model, IT2 fuzzy controller and IT2 T–S FLCS An IT2 T–S FLCS is referred to a closed-loop system consisting of a type1 T–S fuzzy model and an IT2 fuzzy controller connected in a closed loop [18]. 2.1. Type 1 T–S fuzzy model It should be pointed out, that almost all nonlinear dynamical systems can be represented by type 1 Takagi–Sugeno fuzzy models to high degree of precision. In fact, it is proved that type 1 Takagi– Sugeno fuzzy models are universal approximations of any smooth nonlinear system [11], [18] and [19].

for all j = 1, 2, . . ., v. Eq. (9) is shown from Eq. (6). And Eq. (7) is derived from l

g jij ¼ ½hj0 ðzðtÞÞg j0 þ hj1 ðzðtÞÞg j1 lij ( hj0 ðzðtÞÞg j0 þ hj1 ðzðtÞÞg j1 ; lij ¼ 1 lij gj ¼ 1ð¼ hj0 ðzðtÞÞg j0 þ hj1 ðzðtÞÞg j1 Þ; lij ¼ 0 l

l

ð10Þ

l

g jij ¼ hj0 ðzðtÞÞg j0ij þ hj1 ðzðtÞÞg j1ij using the T–S fuzzy model representation, Eq. (6) is rewritten as

_ xðtÞ ¼

r X hi ðzðtÞÞ½Ai xðtÞ þ Bi uðtÞ i¼1

ð11Þ

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A. Abbadi et al. / Electrical Power and Energy Systems 45 (2013) 456–467

where

r¼2

v

hi ðzðtÞÞ ¼

v Y hkik ðzðtÞÞ

ð12Þ

k¼1

½Ai Bi  ¼ F 0 þ

q Y v X

l g kijkk F j

j¼1 k¼1

for all i = 1, 2, . . ., r and ik is a kth bit of the binary number of ‘‘i  1’’ which is represented by binary number having v bits. From Eqs. (6) and (12), we have

hi ðzðtÞÞ P 0; i ¼ 1; 2; . . . ; r r X hi ðzðtÞÞ ¼ 1

ð13Þ

i¼1

for all t. It has been shown that, the so called T1 T–S fuzzy model proposed by Bae et al. in [19] offers an effective way to represent nonlinear dynamic systems in terms of a set of fuzzy IF–Then rules whose consequent parts represent locally linear input–output relations that characterize a general nonlinear system. A typical rule of this fuzzy model, where the antecedents are type-1 fuzzy sets, and the consequent is a crisp number (a linear dynamical system), is of the following form: Plant rule i:

IF z1 ðtÞis M i1

and . . . zp ðtÞ is M ip

_ THEN xðtÞ ¼ Ai xðtÞ þ Bi uðtÞ;

i ¼ 1; 2; . . . r

ð14Þ

Here z(t) = {z1(t), . . . , zp(t)} are known as premise variables, i.e., the nonlinear terms appeared in the system equations. These premise variables are usually functions of the state variables. Also, Mij is the fuzzy set, r is the number of model rules, Ai and Bi are the system and input matrices, respectively. Also, x(t) is the system state vector; and u(t) is the input vector. The overall system dynamics is described as follows

Pr _ xðtÞ ¼

i¼1 qi ðzðtÞÞðAi xðtÞ þ Bi uðtÞÞ Pr i¼1 qi ðzðtÞÞ

ð15Þ

q ðzðtÞÞ hi ðzðtÞÞ ¼ Pr i i¼1 qi ðzðtÞÞ for all t. The term Mij .

with qi ðzðtÞÞ ¼

Mij ðzj ðtÞÞ

M ij ðzj ðtÞÞ

ei IF z1 ðtÞ is M 1

ei and . . . zp ðtÞ is M p

THEN uðtÞ ¼ K i xðtÞ;

ð17Þ

i ¼ 1; 2; . . . r

e i represents the IT2 fuzzy set of premise variable j in rule i, where M j and Ki is linear state feedback gain for ith subsystem. r is the number of rules.  i ðzðtÞÞ and Lower and upper firing strengths of the ith rule, w wi(z(t)), are given by

 e i ðz1 ðtÞÞH    Hl  e i ðzp ðtÞÞ  i ðzðtÞÞ ¼ l w M1

Mp

ð18Þ

wi ðzðtÞÞ ¼ l e i ðz1 ðtÞÞH    Hl e i ðzp ðtÞÞ M1

where

Mp

l Me i and l Me i represent the jth (j = 1, . . . , p) lower and upper j j

membership functions of rule i and ‘‘w’’ is a t-norm operator. The overall fuzzy controller proposed by Biglarbegian et al. in [18] is as follows

Pr uðtÞ ¼ m

i¼1 wi ðzðtÞÞK i xðtÞ Pr i¼1 wi ðzðtÞÞ

Pr n



i¼1 wi ðzðtÞÞK i xðtÞ Pr  i¼1 wi ðzðtÞÞ

ð19Þ

where m and n are design parameters that weight the sharing of lower and upper firing levels of each fired rule.

where p Y

several assumptions about the membership functions. This approach is applicable only in specific situations. Moreover, no systematic method is introduced to identify the membership function parameters required [18]. In an attempt to address the stability of the IT2 T–S FLCS, a new inference mechanism is introduced by Biglarbegian et al. in [18]. The new inference engine has the advantages that the closed mathematical form derived can be easily used for control design, and the conditions necessary to guarantee the asymptotic stability of the IT2 T–S FLCS that use this inference engine can be easily converted into convex optimization problems in terms of linear matrix inequalities (LMIs). In this section, an interval type-2 fuzzy controller is proposed to stabilize the nonlinear plant represented by the T1 T–S fuzzy model (15). The rule of the interval type-2 fuzzy controller is of the following format: Control rule i:

ð16Þ

j¼1

is the grade of the membership of zj(t) in

2.2. IT2 fuzzy controller In this paper, our objective is to reformulate the design of a stabilizing IT2 fuzzy controller through the Lyapunov stability theory as convex optimization problem. The major operation in the existing IT2 fuzzy logic system (FLS) is type reduction which reduces the T2 FLS output to a type-1 fuzzy set so that the defuzzification can be used to obtain a crisp output. In the interval type-2 fuzzy controller used so far, there are two ways to perform type-reduction: using the iterative Karnik–Mendel (KM) procedure to calculate the type-reduced fuzzy sets or using the Wu–Mendel uncertainty bounds method to approximate the type-reduced set [14]. These approaches do not provide any generalized methodology to help guarantee the stability of the control system. Designing stabilizing controllers is accomplished through simulations or by imposing ad hoc assumptions to derive conditions for the stability of closed-loop control systems. Recently, Lam and Seneviratne in [20] investigated stability analysis of the IT2 T–S FLCS based on

2.3. IT2 TS FLCS To obtain stability conditions for the IT2 T–S FLCS using rigorous mathematical analyses, closed-form equations are required. An IT2 T–S FLCS is formed by connecting the T1 T–S fuzzy model (15) and the IT2 fuzzy controller (19) in a closed loop. The closed loop of the fuzzy control system can be expressed as

Pr i;j;l¼1 dijl ðzðtÞÞG ijl _ xðtÞ xðtÞ ¼ Pr i;j;l¼1 dijl ðzðtÞÞ

ð20Þ

where

 l ðzðtÞÞ dijl ðzðtÞÞ ¼ qi ðzðtÞÞwj ðzðtÞÞw

ð21Þ

Gijl ¼ Ai  mBi K j  nBi K l It is straightforward to show that r X

Gijl ¼

i;j;l¼1

Pr

i;j;l¼1 G ijl

r r X r r X r X r X X X Giii þ Gijj þ Gijl i¼1

i–j j¼1

can be expressed as

ð22Þ

i¼1 j–l l¼1

Note that the system represented by Eq. (20) can be rewritten as:

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A. Abbadi et al. / Electrical Power and Energy Systems 45 (2013) 456–467

Pr diii ðzðtÞÞGiii _ xðtÞ xðtÞ ¼ Pi¼1 r i;j;l¼1 dijl ðzðtÞÞ P P   2 ri
This theorem is reduced to the well known Lyapunov stability theorem for linear systems when s = 1. Applying theorem 1, the following sufficient condition for the fuzzy control system (24) is provided.

ð23Þ

without loss of generality, Eq. (20) can be expressed as follows:

Prðr2 þ1Þ=2 v s ðtÞH s s¼1 _ xðtÞ xðtÞ ¼ P rðr 2 þ1Þ=2 v s ðtÞ s¼1

ð24Þ

where

Hs ¼

Gijj þGjii 2

; s ¼ i þ jðj1Þ þr 2

Giii ; 8 > 2d ; > > ijl <

v s ¼ > 2dijj ; > > :

diii ;

GTiii P þ PGiii < 0; i ¼ 1; 2; . . . r  T   Gijj þ Gjii Gijj þ Gjii PþP 6 0; 2 2  T   Gijl þ Gilj Gijl þ Gilj PþP 6 0; 2 2

i < j; s:t: hi \ hj – /

8i; j < l; s:t: hi \ hj \ hl – /

  s ¼ i þ r j  1 þ ðl1Þðl2Þ and j < l 2

8 G þG ijl ilj > ; > > 2 < > > > :

Theorem 2. The equilibrium of the IT2 T–S fuzzy control system described by Eq. (24) is globally asymptotically stable if there exists a common positive definite matrix P such that

2 ðr1Þ

2

ð30Þ The control objective is to determine the gain matrix Fi for the IT2 T–S FLCS such as the IT2 fuzzy controller (19) is able to drive the system states towards the origin, i.e., x(t) ? 0 as time t ? 1 using the stability conditions of Theorem 2.

i < j;

and j ¼ l;

2

s ¼ i þ iði1Þ þ r ðr1Þ and i ¼ j ¼ l 2 2   s ¼ i þ r j  1 þ ðl1Þðl2Þ and j < l 2 s ¼ i þ jðj1Þ þr 2

2 ðr1Þ

s ¼ i þ iði1Þ þr 2

2 ðr1Þ

2 2

and j ¼ l;

3.2. Pole placement

i
and i ¼ j ¼ l ð25Þ

3. Design requirements and LMI formulation In this study, to determine the fuzzy state feedback controller, the following design requirements are considered:  Stabilization: Design a controller such that the closed-loop IT2 T–S FLCS is asymptotically stable, i.e.

limxðtÞ ¼ 0

ð26Þ

t!1

for all initial condition x(0).

In the synthesis of control system, meeting some desired performances should be considered in addition to stability. Generally, stability condition (Theorem 1 and 2) does not directly deal with the transient responses of the closed-loop system [16]. In contrast, it is well known that the transient response of the dynamic system is closely related to the locations of poles. The desired damping can be achieved by constraining the closed loop poles to lie in a prescribed region. Motivated by Chilali and Gahinet’s theorem [17], we consider a disk LMI region Dp,r where the lower bound on both exponential decay rate and damping ratio of the closed-loop response are imposed; which is the most commonly used in practical controller design. The disk LMI region Dp,r considered is centered at (p, 0) with radius r > 0. The characteristic function is given by:

fDp;r ðzÞ ¼

 Pole placement: Design a controller such that the closed-loop eigen values of the IT2 T–S FLCS are located in a prescribed sub-region (D) in the left half of the complex plane to prevent too fast controller dynamics and achieve desired transient behavior, i.e.

rðAi  mBi K j  nBi K l Þ  D

ð27Þ

for all initial condition x(0). 3.1. Stabilization



r pþz p þ z r

 ð31Þ

As shown in Fig. 1, if k = 1xn ± j xd is a complex pole lying in Dp,r with damping ratio 1, undamped natural frequency xn and damped pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi natural frequency xd, then 1 > 1  ðr2 =p2 Þ; xn < p þ r, and jxdj < r Theorem 3. The IT2 T–S FLCS (24) is Dp,r-stable (all the complex poles lying in LMI region Dp,r) if and only if there exists a positive symmetric matrix Q such that

rQ

pQ þ ðAi  mBi K j  nBi K l ÞQ

pQ þ Q ðAi  mBi K j  nBi K l ÞT

rQ

The basic stability condition, proposed by Tanaka and Sugeno, for ensuring stability of Eq. (24) is given as follows.

s ¼ 1; 2; . . . ; rðr 2 þ 1Þ=2

ð28Þ

In other words, the fuzzy system (20) is quadratically stabilizable via the fuzzy control law (19) if there exists a common positive definite matrix P such that

ðAi  mBi K j  nBi K l ÞT P þ PðAi  mBi K j  nBi K l Þ < 0; ¼ 1; 2; . . . ; r

<0 ð32Þ

Theorem 1 [11]. The equilibrium of a fuzzy control system described by Eq. (24) is asymptotically stable in the large if there exists a common positive-definite matrix P such that

H Ts P þ PH s < 0;

!

i; j; l ð29Þ Fig. 1. Circular region (D) for pole location.

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A. Abbadi et al. / Electrical Power and Energy Systems 45 (2013) 456–467

The proof and more details of this theorem can be found in [17].



Eqk ðtÞ ¼ E0qk ðtÞ þ xdk  x0dk Idk ðtÞ

3.3. Stable fuzzy control with pole-placement constraints

Efk ðtÞ ¼ kck ufk ðtÞ n X Idk ðtÞ ¼ E0qj ðtÞðGkj sin dkj ðtÞ  Bkj cos dkj ðtÞÞ

The problem of designing IT2 T–S FLCS that guarantees stability and provides desired transient response can be solved by using above LMI constraints (30) and (32) (Theorem 2 and Theorem 3). These design requirements can be recast as an LMI feasibility problem by multiplying the inequalities (30) and (32) on the left and right by P1. Two new variables Q = P1 and Yi = KiQ are defined. Solving these LMI constraints directly leads to the following LMI formulation of the objectives fuzzy state feedback synthesis problems [16]. Theorem 4. The IT2 T–S FLCS (24) is Dp,r-stablizable if and only if there exists a positive symmetric matrix Q and Yisuch that the following LMI condition is satisfied.

pQ þ QATi  ðm þ nÞY Ti BTi

pQ þ Ai Q  ðm þ nÞBi Y

i

rQ

ð34Þ

It guarantees a global stability with a desired transient behavior. Remark 1. It should be pointed out that the pole placement is fundamentally related to the cases for which ‘‘i = j = l ’’ and not necessary for the other cases, it suffices to locate the poles of only dominant term in the prescribed LMI region since Theorem 3 imposes additional constraints in our problem.

ð42Þ

Q ek ðtÞ ¼ E0qk Idk ðtÞ

ð43Þ

Eqk ðtÞ ¼ xadk Ifk ðtÞ

ð44Þ

V tdk ðtÞ ¼ x0dk ðtÞIqk ðtÞ

ð45Þ

V tqk ðtÞ ¼ E0qk ðtÞ  x0dk Idk ðtÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi V tk ðtÞ ¼ V 2tqk ðtÞ þ V 2tdk ðtÞ

ð46Þ ð47Þ

  1 1 v fk ðtÞ  T 0dok E0qk I_qk þ Pmk þ ððxdk kck Iqk ðtÞ kck ð48Þ

and by differentiating Eq. (42), the nonlinear generator model (35)– (37) has been directly transformed to a system whose closed-loop dynamics are linear

Dd_ k ðtÞ ¼ Dxk ðtÞ D x _ k ðtÞ ¼  k Dxk ðtÞ  0 DPek ðtÞ Dx 2Hk 2Hk 1 1 DP_ ek ðtÞ ¼  0 DP ek ðtÞ þ 0 v fk ðtÞ T dok T dok where

ð49Þ ð50Þ ð51Þ

vfk(t) is the feedback control law

v fk ðtÞ ¼ kd Ddk ðtÞ  kx ðtÞDxk ðtÞ  kP k

k

ek

DPek ðtÞ

4.1. Dynamic model of power system In the used model, the multimachine power system is reduced into a network with generator nodes only. For the design of the excitation controller the classical third-order single-axis dynamic generator model is used, whereas other differential equations that represent dynamics with very short time constants have been neglected. In general, for a power system with n-generator, the dynamic model of the kth generator can be written as follows [21,22]: Mechanical equations

v fk ðtÞ ¼ kv

ð36Þ

with

k

DV k ðtÞ  kxk ðtÞDxk ðtÞ  kPek DPek ðtÞ

DV k ðtÞ ¼ V tk ðtÞ  V tk0 ð37Þ

ð53Þ

The design objective is to improve the voltage regulation and transient stability performance of the multimachine power system, subjected to severe disturbances. The DFL nonlinear controller (48) and (52) can achieve effective transient stability enhancement since that the equilibrium points were well defined for dk(t), xk(t) and Pek(t) (respectively dk0, x0 and Pmk). The output feedback law does not consider voltage deviation, due to which the terminal voltage regulation is not guaranteed and this performance is unacceptable in practice operations. To solve this problem, the feedback control law has been proposed as [21]

ð35Þ

Generator electrical dynamics

ð52Þ

and where

Ddk ðtÞ ¼ dk ðtÞ  dk0 Dxk ðtÞ ¼ xk  x0 DPek ðtÞ ¼ P ek ðtÞ  Pmk

4. T–S fuzzy model of power system

Electrical equations:

ð41Þ

Pek ðtÞ ¼ E0qk ðtÞIqk ðtÞ

 x0dk ÞIdk ðtÞÞ

Given a solution (Q, Yi), the fuzzy state feedback gains are obtained as

1 E_ 0qk ðtÞ ¼ 0 ðEfk ðtÞ  Eqk ðtÞÞ T d0k

n X E0qj ðtÞðBkj sin dkj ðtÞ þ Gkj cos dkj ðtÞÞ j¼1

ufk ðtÞ ¼

ð33Þ

d_ k ðtÞ ¼ xk ðtÞ  x0 D x x_ k ðtÞ ¼  k ðxk ðtÞ  x0 Þ þ 0 ðPmk  Pek ðtÞÞ 2Hk 2Hk

Iqk ðtÞ ¼

According to the model described above, it can be found that the synchronous generator is nonlinear through the excitation loop. To cancel these nonlinearities, the DFL technique is applied, indeed by employing the DFL compensating law [21,23]

<0

K i ¼ Y i Q 1

ð40Þ

j¼1

4.2. Feedback linearization compensation design

Ai Q þ QATi  ðm þ nÞY Ti BTi  ðm þ nÞBi Y i < 0    T Ai þ Aj Ai þ Aj ðm þ nÞ T T ðm þ nÞ Q þQ Y j Bi  Bi Y j  2 2 2 2 ðm þ nÞ T T ðm þ nÞ  Y i Bj  Bj Y i 6 0 2 2 ðm þ nÞ T T ðm þ nÞ ðm þ nÞ T T Ai Q þ QATi  Y j Bi  Bi Y j  Y l Bi 2 2 2 ðm þ nÞ Bi Y l 6 0  2 ! rQ

ð38Þ ð39Þ

ð54Þ

ð55Þ

The system stability enhancement can be guaranteed by the feedback control of xk(t) and Pek(t) and the voltage regulation can be expected by the feedback control of Vtk(t). The terminal voltage can be represented as [21]

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8" #2 < n X V tk ðtÞ ¼ E0qk  x0dk E0qj ðtÞðGkj sin dkj ðtÞ  Bkj cos dkj ðtÞÞ : j¼1 " x0dk

þ

#2 91=2 n X

= 0 Eqj ðtÞ Bkj sin dkj ðtÞ þ Gkj cos dkj ðtÞ ; j¼1

  V tk ðtÞ ¼ f dk ðtÞ; E0qk ðtÞ; dj ðtÞ; E0qj ðtÞ ðj – kÞ

tions), the sum of products of linearly independent functions method will be applied to construct the type 1 T–S fuzzy model from the nonlinear system (62). The nonlinear system (62) can be rewritten as follows:

ð56Þ

x_ k ðtÞ ¼ F k0 ð57Þ

Assuming that, while considering the kth generator, are constant and neglect the effect of xj (where j 2 {1, 2, . . . , k  1, k + 1, . . . , n}). The dynamic equation for terminal voltage variation can be written as

V_ tk ðtÞ  fk1 Dxk ðtÞ þ fk2 P_ ek ðtÞ

xk ðtÞ ¼ ½DV tk ðtÞ; Dxk ðtÞ; DP ek ðtÞT

gk ðtÞ ¼ ½DV tk ðtÞ; Dxk ðtÞ; DPek ðtÞ; v fk ðtÞ 2



i

h 1 þ x0dk Bkk E02 qk ðtÞBkk  Q ek ðtÞV tqk ðtÞ

x0dk



x0dk Bkk

1þ V tk ðtÞ

V tk ðtÞIqk ðtÞ

Pek ðtÞ

F k2

1 þ x0dk Bkk V tqk ðtÞ  V tk ðtÞIqk ðtÞ

ð60Þ

where fk1(t) and fk2(t) are highly nonlinear functions. Employing the DFL compensating law (48), and substituting Eq. (42) in Eq. (58) we get from Eq. (58)

fk ðtÞ fk ðtÞ DV_ tk ðtÞ  fk1 ðtÞDxk ðtÞ  20 DPek ðtÞ þ 20 v fk ðtÞ T dok T dok

ð61Þ

The nonlinear generator model (35)–(37) is now represented by the following DFL compensated system model

8 f ðtÞ f ðtÞ > DV_ tk ðtÞ ¼ fk1 ðtÞDxk ðtÞ  Tk20 DPek ðtÞ þ Tk20 v fk ðtÞ > > dok dok < Dk x0 _ k ðtÞ ¼  2H Dx Dxk ðtÞ  2H DPek ðtÞ k k > > > _ : DPek ðtÞ ¼  T 01 DPek ðtÞ þ T 01 v fk ðtÞ dok

3 0 0 0 0 6 7 Dk x0 0  2H  2H 0 7; ¼6 k k 4 5 0 0  T 01 T 01 dok dok 2 3 0 0  T 01 T 01 dok dok 6 7 ¼ 40 0 0 0 5 0 0

ð59Þ



dok

0

ð65Þ 2

0 1 0 0

6 F k1 ¼ 4 0 0

3

7 0 0 0 5; 0 0 0

ð66Þ

0

As the number of linearly independent functions is 2 (fk1(t) and fk2(t) (Eqs. (59), (60) respectively)) and for each function fki(t) (i = 1, 2) two triangular fuzzy sets are assigned, then four fuzzy rules are formulated. The type 1 T–S fuzzy model of the nonlinear system (62) is such as:

x_ k ðtÞ ¼

4 X hki ðzðtÞÞ½Aki xk ðtÞ þ Bki uk ðtÞ

ð67Þ

i¼1

where

hk1 ¼ M k10 M k20 ; hk2 ¼ M k10 M k21 ; M kj0 ðzðtÞÞ ¼

ð62Þ

ð64Þ T

F k0

fk1  

ð63Þ

where

ð58Þ

with



# 2 X þ fkj ðzðtÞÞF kj gk ðtÞ j¼1

E0qj (t)

fk2

"

fkj1  fkj ðzðtÞÞ fkj1  fkj0

;

hk3 ¼ M k11 M k20 ; hk4 ¼ M k11 M k21

Mkj1 ðzðtÞÞ ¼

fkj ðzðtÞÞ  fkj0 fkj1  fkj0

½Ak1 Bk1  ¼ F k0 þ fk10 F k1 þ fk20 F k2 ;

½Ak2 Bk2  ¼ F k0 þ fk10 F k1 þ fk21 F k2

½Ak3 Bk3  ¼ F k0 þ fk11 F k1 þ fk20 F k2 ;

½Ak4 Bk4  ¼ F k0 þ fk11 F k1 þ fk21 F k2 ð68Þ

4.3. Equivalent T–S fuzzy model of the power system Since the dynamic equation of terminal voltage variation is a nonlinear function (fk1(t) and fk2(t) are linearly independent func-

The nonlinear functions fk1(t) and fk2(t) are dependent on the operating conditions but bounded with a certain operating region. As in the reference [21], the following bounds of fk1(t) and fk2(t) are considered as

Fig. 2. A two-machine infinite bus power system.

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Table 1 System parameters.

xd (p.u.) x0d (p.u.) xT (p.u.) xad (p.u.) T 0do (s) H (s) D (p.u.) kc (p.u.) x12 (p.u.) x13 (p.u.) x23 (p.u.) x0 (rad/s)

5. The nonlinear voltage controller design Generator # 1

Generator # 2

1.863 0.257 0.129 1.712 6.9 4.0 5.0 1.0

2.36 0.319 0.11 1.712 7.96 5.1 3 1.0 0.55 0.53 0.6 314.159

Based on the type-1 T–S fuzzy model (67), four rules fuzzy controller of the following format are adopted:

ei IF f k1 ðtÞ is M k1

THEN uki ðtÞ ¼ K ki xk ðtÞ;

0:266 6 f12 6 3:794

2:832 6 f21 6 0:233;

0:241 6 f22 6 3:670

i ¼ 1; 2; . . . 4

ð69Þ

e i and M e i are triangular interval type-2 fuzzy sets of IFwhere M k1 k2 part. Referring to Eq. (19), the IT2 fuzzy controller is defined as follows:

P4 uk ðtÞ ¼ mk

3:526 6 f11 6 0:259;

ei and f k2 ðtÞ is M k2

i¼1 wki K ki xk ðtÞ P4 i¼1 wki

P4  nk



i¼1 wki K ki xk ðtÞ P4  i¼1 wki

ð70Þ

The problem of designing IT2 fuzzy controller that guarantees voltage regulation, transient stability enhancement and provides desired transient response can be solved by using LMI constraint (33) (Theorem 4). By solving LMI feasibility problem (33), we can

Fig. 3. Power system responses for Case 1, fault location k = 0.1. Temporary fault + Permanent fault.

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obtain the state feedback gains which guarantee global stability and desired transient behavior by constraining the closed-loop poles to lie within the disk Dpk ;rk . The stability region Dpk ;rk is a disk of center (pk, 0) and radius rk. The LMI synthesis is performed for a set of values, (pk, rk) = (17.5, 16). The LMI region has the following characteristic function:

 fDpk ;rk ðzÞ ¼

16

17:5 þ z

17:5 þ z

16



K 22 ¼ ½4283:8  2237:5 6297:9 K 24 ¼ ½3023:6  1417:9 4175:6

6. Simulations results

(1) Generator #1: K 11 ¼ ½41:29 20:73  74:49;

K 12 ¼ ½40:94 22:18  73:80

K 13 ¼ ½32:88 17:36  52:47;

K 14 ¼ ½29:91 14:45  51:15

n1 ¼ 5:0445

K 21 ¼ ½4047:8  1958:3 5886:1; K 23 ¼ ½3367:2  1708:1 4349:9; m2 ¼ 6:9149; n2 ¼ 6:8937

ð71Þ

The lower bound of damping ratio imposed by this disk region is 0.4051. The controller tuning parameters mk and nk were designed based on the best power system responses. The state feedback gains and the controller tuning parameters obtained are as follows:

m1 ¼ 6:8452;

(2) Generator #2

In this section, the performances of the closed-loop power system are exhibited. The IT2 T–S FLCS designed above was simulated on a two generator infinite bus power system which is shown in Fig. 2. The generator and the transmission line parameters are listed in Table 1 [6]. To demonstrate the effectiveness of the proposed IT2 fuzzy controller, different sets of faults sequences, fault locations, change in transmission line parameters and step change in mechanical input power are considered. The symmetrical three phase short circuit fault considered occurs on one of transmission lines between the generator #1 and the generator #2. The fault location is indexed by a constant k which is the fraction of the line to the left of the fault.

Fig. 4. Power system responses for Case 1, fault location k = 0.95. Temporary fault + Permanent fault.

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The simulations are performed for the following cases. Case 1. Temporary fault + Permanent fault The performance of the proposed controller is tested under the following fault sequence:  Stage 1: The system is in a pre-fault steady state.  Stage 2: A fault occurs at t = 1 s.  Stage 3: The fault is removed by opening the breakers of the faulted line at t = 1.15 s.  Stage 4: The transmission lines are restored at t = 2.5 s;  Stage 5: Another fault occurs at t = 3 s;  Stage 6: The fault is removed by opening the breakers of the faulted line at t = 3.15 s  Stage 7: The system is in a post-fault state. The operating points considered are:

d10 ¼ 52:72 ; 

d20 ¼ 54:48 ;

Pm10 ¼ 0:95 p:u:;

V t10 ¼ 1:00 p:u:

Pm20 ¼ 0:95 p:u:;

V t20 ¼ 1:02 p:u:

The fault locations considered are k = 0.1 and k = 0.95. The corresponding closed loop system responses are shown in Figs. 3 and 4.

Case 2. Permanent fault + Step increase of the mechanical input power The transient stability enhancement and the voltage regulation are tested under the fault sequence described below:  Stage 1: The system is in a pre-fault steady state.  Stage 2: A fault occurs at t = 1 s.  Stage 3: The fault is removed by opening the breakers of the faulted line at t = 1.15 s.  Stage 4: A mechanical input power of the generator #1 has a 30% step increase at t = 1.5 s.  Stage 5: The system is in a post-fault state. The following operating points are considered:

d10 ¼ 46:00 ;

Pm10 ¼ 0:87 p:u:;

V t10 ¼ 1:02 p:u:

d20 ¼ 44:69 ;

Pm20 ¼ 0:86 p:u:;

V t20 ¼ 1:10 p:u:

The fault location is k = 0.01. The corresponding closed loop system responses are shown in Fig. 5. Case 3. Permanent fault + Change in transmission line parameters

Fig. 5. Power system responses for Case 2: Permanent fault + Step increase of the mechanical input power.

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The transmission line parameters are changed x12 = x13 = x23 = 0.7. The operating points considered are: 

d10 ¼ 52:58 ; d20 ¼ 55:33 ;

Pm10 ¼ 0:80 p:u:; Pm20 ¼ 0:85 p:u:;

to

V t10 ¼ 1:01 p:u: V t20 ¼ 1:012 p:u:

The fault location is k = 0.02. The corresponding closed loop system responses are shown in Fig. 6. The fault sequence considered is as follows:  Stage 1: The system is in a pre-fault steady state.  Stage 2: A fault occurs at t = 1 s.  Stage 3: The fault is removed by opening the breakers of the faulted line at t = 1.15 s.  Stage 4: The system is in a post-fault state. In order to evaluate the performance of the proposed controller more accurately, the physical limit of the excitation voltage and the saturation effect of the synchronous generator are also considered in the simulation. The considered physical limits of the excitation voltage in the simulation are:

3 6 kck ufk 6 6;

k ¼ 1; 2

ð72Þ

when the saturation effect of the synchronous generator is taken into account, Eq. (37) can be rewritten as in [21]:

 1  E_ 0qk ðtÞ ¼ 0 Efk ðtÞ  ðxdk  x0dk ÞIdk ðtÞ  kfk E0qk ðtÞ T d0k

ð73Þ

where

kfk ¼ 1 þ

nk 1 bk  0 Eqk ðtÞ ak

ð74Þ

The saturation parameters are chosen as [6]

a1 ¼ 0:950; a2 ¼ 0:935;

b1 ¼ 0:051; b2 ¼ 0:064;

n1 ¼ 8:727; n2 ¼ 10:878;

ð75Þ

In the simulation, three kinds of controllers, the IT2 fuzzy (IT2 F) controller proposed in this paper, the type-1 fuzzy (T1 F) controller presented in [10] and the robust excitation (Rob Exc) controller described in [6] are tested separately. To make a fair comparison between the IT2 fuzzy controller proposed and the T1 fuzzy controller applied in [10], the

Fig. 6. Power system responses for Case 3: Permanent fault + Change in transmission line parameters.

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Table 2 Values of performance indices under different conditions. Controller

Scenario

Comparator indices of transient stability (deviation of power angle (D@))

Comparator indices of voltage regulation (deviation of voltage (DV))

ITAE

IAE

ITAE

IAE

IT2 F controller

Case Case Case Case

1: k = 0.1 1: k = 0.95 2 3

1297.9182 1783.3941 11073.6634 552.9339

325.1802 468.7699 1180.873 202.5082

667.6194 905.3896 417.366 281.208

88.2114 257.1107 136.9393 112.7018

T1 F controller

Case Case Case Case

1: k = 0.1 1: k = 0.95 2 3

1370.8132 1950.2244 11202.9241 629.4725

325.4119 487.3378 1193.2678 208.3982

716.485 1005.6814 547.7617 336.2093

189.1578 268.293 149.7778 117.1817

Rob Exc controller

Case Case Case Case

1: k = 0.1 1: k = 0.95 2 3

1964.6366 2681.509 2994.7013 1084.691

415.0785 592.6864 478.654 269.6603

1307.0291 1625.8936 6945.6249 763.9051

260.6438 337.545 682.2143 168.5205

equivalent T–S fuzzy model of the power system (67) is adopted in both systems: the IT2 T–S FLCS and the T1 T–S FLCS. In addition, the poles of both controllers are confined in the same prescribed region. Figs. 3–6 show the system performances when subjected to different faults. It can be concluded from the simulation results that: 1. The robust excitation controller can damp out the oscillations at the designed operating points in the presence of different fault location. However the voltage regulation is unsatisfactory essentially when there is a step increase of the mechanical input power (Fig. 5). 2. With the T1 fuzzy controller, the transient stability can be improved in the presence of fault location variations, operating point variations. The T1 fuzzy controller is robust with regard to uncertain network parameters and can achieve satisfactory post-fault voltage level. 3. The IT2 fuzzy controller can enhance the transient stability and damp out oscillations effectively regardless of the system operating points and fault locations and, simultaneously, achieve good post-fault voltage regulation. The robustness with regard to uncertain network parameters is guaranteed by this controller. To compare the performance of controllers more accurately, some performance indices based on the system performance characteristics are considered. Scalar integral performance indices have proved to be the most meaningful and convenient measures of dynamic performance [24]. Two performance indices, i.e., the integral of absolute error (IAE) and the integral of time multiplied by absolute error (ITAE) defined below are used to make a quantitative comparison in this study. Since it is not practicable to integrate up to infinity, the convention is to choose a value of T sufficiently large so that DZi for t > T is negligible. We used T = 20 s.

IAE ¼

Z

! 2 X jDZ i j dt

T

0

ITAE ¼

i¼1

Z 0

T

t

ð76Þ

!

2 X jDZ i j dt

ð77Þ

i¼1

As the objectives of the proposed controller are the ability to enhance transient stability and achieve voltage regulation, deviation of power angle and voltage levels are considered in the determination of the performance indices. It should be noted that the lowest values of these indices reflect the best system response in terms of characteristics in the time domain.

Numerical results of performance indices for all operating conditions are given in Table 2. As can be seen in Table 2, the IT2 fuzzy controller significantly outperforms the T1 fuzzy and the robust excitation controllers in the voltage regulation; since our proposed control scheme provides lesser error values in terms of voltage deviation in all cases. Compared to its T1 counterpart and to the robust excitation controller, the IT2 fuzzy controller presents excellent system stability enhancement, regardless of different condition of operation points, fault locations and network parameters. 7. Conclusion In this paper, the key requirements of an analytical methodology that guarantees a stable IT2 fuzzy controller design with pre-specified transient performance are given. Indeed, based on Lyapunov stability theory, LMI-based stability conditions are obtained to guarantee the stability of the IT2 fuzzy control system and synthesize the IT2 fuzzy controller under pole placement constraints. The addressed control tasks are to reformulate transient stability, voltage regulation and desired transient performance as convex optimization problems with linear matrix inequality (LMI). Solving these LMI constraints provides the fuzzy statefeedback controller such as the proposed IT2 fuzzy controller meets the required objectives. The effectiveness of the controller is demonstrated on a twogenerator infinite bus example system. It was shown that a welltuned IT2 fuzzy controller guarantees the overall stability of the multimachine power systems and presents robustness with regard to uncertain network parameters. The simulations illustrate that the transient stability is greatly enhanced and the post-fault voltage level is well achieved, regardless of operation points, fault locations, uncertain network parameters and change in mechanical input power. Another important issue of the proposed method is that the described control scheme in this paper could be viewed as a starting point in the design of other controllers like optimal interval type-2 Takagi–Sugeno fuzzy logic controller, robust interval type-2 Takagi–Sugeno fuzzy logic controller or it could be considered as a basis for future controller conception with predefined performance specifications (decay rate conditions, constrains on control input and output, and disturbance rejection). Appendix A dk(t) the angle of the kth generator, in radian;xk(t) the relative speed of the kth generator, in rad/s; Pmk the mechanical input

A. Abbadi et al. / Electrical Power and Energy Systems 45 (2013) 456–467

power, in p.u.; Pek(t) the electrical power, in p.u.; x0 the synchronous machine speed, in rad/s, x0 = 2pf0; Dk the per unit damper constant; Hk the inertia constant, in sec. E0qk ðtÞ the transient EMF in quadrature axis of the kth generator, in p.u.; Efk(t) the equivalent EMF in the excitation coil, in p.u.; T 0dok the direct axis transient open circuit time constant, in second; Eqk the EMF in quadrature axis of the kth generator, in p.u.; Vtk the generator terminal voltage, in p.u.; xdk the direct axis reactance of the kth generator, in p.u.; x0dk the direct axis transient reactance of the kth generator, in p.u.; Idk the direct axis current, in p.u.; Iqk the quadrature axis current, in p.u.; kck the gain of the excitation amplifier, in p.u.; ufk the input of the SCR amplifier of the kth generator; xadk the mutual reactance between the excitation coil and the stator coil of the kth generator; Ykj = Gkj + jBkj the kth row and jth column element of nodal admittance matrix, in p.u.; Qek the reactive power, in p.u.; Ifk the excitation current. References [1] Colbia-Vega A, De León-Morales J, Fridman L, Salas-Peña O, Mata-Jiménez MT. Robust excitation control design using sliding-mode technique for multimachine power systems. Electr Power Syst Res 2008;78:1627–34. [2] Dysko A, Leithead WE, O’Reilly J. Enhanced power system stability by coordinated PSS design. IEEE Trans Power Syst 2010;25(1):413–22. [3] Kundur P, Klein M, Rogers GJ, Zywno MS. Application of power system stabilizers for enhancement of overall system stability. IEEE Trans Power Syst 1989;4:614–26. [4] Guo Y, Hill DJ, Wang Y. Global transient stability and voltage regulation for power systems. IEEE Trans Power Syst 2001;16(4):678–88. [5] Wang Y, Guo G, Hill D J. Robust decentralized nonlinear controller design for multimachine power systems. Automatica 1997;33(9):1725–33. [6] Guo Y, Hill DJ, Wang Y. Nonlinear decentralized control of large-scale power systems. Automatica 2000;36(9):1275–89. [7] Tan YL, Wang Y. Transient stabilization using adaptive excitation and dynamic brake control. Control Eng Pract 1997;5(3):337–46. [8] Jalili M, Yazdanpanah MJ. Transient stability enhancement of power systems via optimal nonlinear state feedback control. Electr Eng 2006;89(2):149–56.

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