Design of power system stabilizers using two level fuzzy and adaptive neuro-fuzzy inference systems

Electrical Power and Energy Systems 35 (2012) 47–56

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

Design of power system stabilizers using two level fuzzy and adaptive neuro-fuzzy inference systems S.M. Radaideh a,⇑, I.M. Nejdawi a, M.H. Mushtaha b a b

Electrical Engineering Department, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, Jordan Electrical Engineering Department, Jordan University of Science and Technology, P.O. Box 630176, Irbid 22300, Jordan

a r t i c l e

i n f o

Article history: Received 30 January 2008 Received in revised form 14 August 2011 Accepted 20 August 2011 Available online 5 November 2011 Keywords: Power system stabilizer Conventional controller Fuzzy systems Adaptive fuzzy technique

a b s t r a c t This paper deals with a design of two level power system stabilizer. Former one is conventional, while the latter level is designed using one of the following two methods: fuzzy inference system (FIS) and adaptive neuro-fuzzy inference system (ANFIS). The main function of the conventional level is to stabilize unstable or poorly stable systems, while the second – which is designed using (FIS) or (ANFIS) – improves the total response in order to achieve required results. This technique is applied on a three machine nine bus power system. As expected, the Adaptive neuro-fuzzy inference system (ANFIS) damps out the low frequency oscillations in the better manner than fuzzy inference system (FIS). Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Power systems are complex and nonlinear. They produce a low frequency electromechanical oscillations because of many factors as operating conditions. Electromechanical oscillations between interconnected synchronous generators are phenomena inherent to power systems. To overcome this problem, or in other words, to damp out these oscillations, power system stabilizers (PSSs) are widely used in power stations. Many studies are executed in designing stabilizers. Early researches deal with lead-lag compensation as a conventional type [1,2]. The main idea in this type is concentrating in shifting unstable and poorly stable poles deeper to the left side plane. This method depends on controlling phase margins and gain margins [3]. In the recent years, several topics are produced which depend on modern control theory. Optimal control theory was used in [4,5] to optimize the parameters of power system stabilizers and automatic voltage regulator (AVR), while fuzzy power system stabilizers (FPSSs) with multi-inputs have a great deal of study as in [6,7]. Robust H1 power system stabilizers are introduced in [1,8]. In this paper, a novel approach for designing PSS is introduced. We present a two level power system stabilizer that can improve damping of oscillation modes. Former level introduces pole placement of unstable poles to the left side plane. Insufficient modes

⇑ Corresponding author. E-mail addresses: [email protected] (S.M. Radaideh), [email protected] (I.M. Nejdawi), [email protected] (M.H. Mushtaha). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.08.022

damping forced us to add another level that can compensate the control signal u(t). The rest of the paper is arranged as follow: mathematical model of a power system is illustrated in Section 2. Two level fuzzy system is introduced in Section 3. Section 4 shows a two level adaptive fuzzy power system stabilizer, while Section 5 presents simulation results. Comments and conclusions are drawn in Section 6. 2. Mathematical model of power system Each synchronous machine is described by a nonlinear fourthorder model [1,9].

x_ i ¼

T mi  T ei  Di ðxi  1Þ Mi

d_ i ¼ xb ðxi  1Þ E_ 0qi ¼

Efdi  ðvdi  v0di Þidi  E0qi T 0doi

K ai ðU i  V i þ V refi Þ  Efdi E_ fdi ¼ T ai

ð1:aÞ ð1:bÞ ð1:cÞ

ð1:dÞ

where xi is the rotor speed of the ith machine, di is the rotor angle of the ith machine, E0qi is the internal (quadrature) voltage of the ith machine, Efdi is the field voltage of the ith machine, Tm is the mechanical torque, Te is the electrical torque, D is the damping

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S.M. Radaideh et al. / Electrical Power and Energy Systems 35 (2012) 47–56

torque, xd is the direct reactance, x0d is the sub-transient reactance, Ui is the supplementary control signal, M is the inertia constant, and TA is the time constant. The dynamic behavior of the system described in (1) can be represented in a set of n first order non-linear differential equations [1]:

x_ ¼ fðx; uÞ

ð2:aÞ

y ¼ gðx; uÞ

ð2:bÞ

where x is the state vector with n state variable x, u the vector of inputs to the system, and y is the vector of outputs. To investigate the small-signal stability at one operating point, Eq. (2) needs to be linearized. Assume x0 is the initial state vector at the current operating point and u0 is the corresponding input vector. Because the perturbation is considered small, the nonlinear function f can be expressed in terms of Taylor series expansion. By using only the first order terms, the approximation for the ith state variable xi leads to the following equations with m number of inputs:

x_ i ¼ x_ i0 þ Dx_ i x_ i ¼ fi ðx0 ; u0 Þ þ

ð3:aÞ

Δδ j

K1 ii K1 ij +

ΔTm

K2 ij

− − − −

Δ E′q j Δ E′qi

ΔTm − ΔTe

Δω i

1 Mi . s

Δδ i [rad ]

2π f s

Δδ j

K 2 ii

K 5 ii

K 4 ii − K 3ii ′ K3ii 1 + sTdoi

−

−+ Δ Efd −

− K Ai 1 + sTAi

ΔVt − u + +

+

K 5 ij −

K6 ij

K 6 ii 1 K3 ij

u

+

Δ E′q j

K 4 ii

Δ E′q j Fig. 1. Transfer function block diagram of a linearized model of multimachine power system [10].

3. Two level fuzzy system stabilizer

@fi @fi @fi Dx1 þ    þ Dxn þ Du1 þ    @x1 @xn @u1

Dx_ ¼ ADx þ BDu

ð5:aÞ

In this section, the proposed controller is divided into two levels. The former is a conventional level, while the latter is designed using a fuzzy theory [11]. Pole placement technique is used in designing a conventional controller. The main function of this level is to stabilize unstable or poorly stable plants. while the task of the second level is to produce a control signal that can achieve needed response. Fig. 2 shows the structure of the two level fuzzy controller. Note: Sensor in the feedback path in Fig. 2 has unity gain. It converts the speed deviation to an equivalent voltage signal that can be combined with the reference signal DVref_i.

Dy ¼ CDx þ DDu

ð5:bÞ

3.1. Conventional controller

ð3:bÞ

where x_ i0 ¼ fi ðx0 ; u0 Þ. So, we obtain

Dx_ i ¼

@fi @fi @fi @fi Dx1 þ    þ Dxn þ Du1 þ    þ Dum @x1 @xn @u1 @um

ð4Þ

Thus, the linearized system can be written as:

where A is the state matrix, B the control matrix, C the output matrix, and D is the feed forward matrix. From the stability viewpoint, the state matrix A is the most important. This matrix can be represented as:

2 @f

1

@x1

6 6 A ¼ 6 ... 4

@fn @x1



@f1 @xn

.. . 

@fn @xn

3

7 .. 7 . 7 5

ð6Þ

The system dynamics can be expressed by a set of linear differential equations in the small-perturbation variables Dxi, Ddi, DE0qi , DEfd as following [10].

_i¼ Dx

Di K 1i K 2i 0 1 Dx i  Ddi  DEqi þ DTmi Mi Mi Mi Mi

Dd_ i ¼ 2pf  xi DE_ 0qi ¼ 

ð7:aÞ ð7:bÞ

K 4i 1 1 Ddi  0  E0 þ  DEfdi T doi  K 3i qi T 0doi T 0doi

ð7:cÞ

K Ai  K 5i K Ai  K 6i 1 _ K Ai  Dd  DEqi  DEfdi þ uEi T Ai T Ai T Ai T Ai

ð7:dÞ

DE_ fdi ¼ 

where K-constants are called Phillips–Hefferon constants. Fig. 1 illustrates the transfer function of a multi-machine power system for small signal stability analysis.

As mentioned earlier, the main task of the first level concerns in shifting unstable or poorly stable poles deeper to the left half plane. The technique used in achieving our goal is called pole placement using state feedback. Fig. 3 illustrates how the states are fed back using gain constants. Specific selection of the feedback gains allows to place poles any where in the plane. This can be achieved using Ackermann’s formula [12]:

 1 K ¼ ½ 0 0 . . . 0 1   B A  B . . . An1  B  aC ðAÞ

ð8Þ

where aC(s) is the desired characteristic equation, and n is the order of the plant.

Supplementary Control Signal

Δω i

Synchronous Generator Conventional Stabilizer Fuzzy Logic Stabilizer

Sensor

@fi Dum þ @um

+ ΔV ref _ i

Fig. 2. Two level fuzzy system stabilizer.

S.M. Radaideh et al. / Electrical Power and Energy Systems 35 (2012) 47–56

49

Fig. 3. State feedback system.

3.2. Fuzzy logic power system stabilizers Due to the high order of the plant transfer function, there is no closed form that can determine the best location of poles that can minimize overshoot and settling time. Because of this, fuzzy logic controller is used to enhance the control signal u(t). This can improve the overall system response. Fig. 4 shows the fuzzy logic controller terminals. 4. Two level adaptive fuzzy system stabilizer The main problem with the pure fuzzy system is that its inputs and outputs are fuzzy sets (that is, words in natural languages), whereas in engineering systems the inputs and outputs are realvalued numbers. To solve this problem, Takagi, Sugeno, and Kang suggested another fuzzy system whose inputs and outputs are real-valued variables [11]. Fig. 5 illustrates the operation of Sugeno rule. 4.1. Basic structure of ANFIS [13,14] The main consequents of the Takagi–Sugeno–Kang (TSK) fuzzy rules are linear combinations of their preconditions. Generally, for a multi-input fuzzy system, the TSK rules are in the following forms:

 In layer 1, every node is an input node that just passes external signals to the following layer.  In layer 2, every node is an adaptive one with a node function defined by:

ljAi ðxi Þ

ð10Þ

where xi is the input to the ith-node and Aji is the linguistic label (large, small, etc.) associated with this node. The output of this node specifies the degree to which the given xi satisfies the quantifier Aji . In this study, the function ljAi ðxi Þ is the bell-shaped function with a maximum value equal to 1 and a minimum value equal to 0, with j the following parameters faji ; bi ; cji g:

0

x  cj lAj ðxi Þ ¼ @1 þ i j i i ai

IF x1 is ¼

aj0

þ

AND x2 is

aj1 x1

þ  þ

Aj2

. . . AND xn is

Ajn

THEN y ¼ fj

ajn xn

ð9Þ

Fig. 6 shows the basic ANFIS structure of simple two fuzzy rules. ANFIS is composed of six functional layers, as shown in the previous figure.

ð11Þ

Parameters in this layer are referred to as premise parameter set S1. They can be trained using the ANFIS hybrid – back propagation and least square estimate – learning algorithm. While in layer 3, every node is a fixed – not adaptive – node which multiplies the incoming signals. This layer implements the fuzzy AND operator. The outputs of these nodes are given by:

wj ¼ lAj ðx1 Þ  lAj ðx2 Þ j ¼ 1; 2 1

Aj1

!2bj 11 i A

ð12Þ

2

Nodes in layer 4 are fixed nodes and calculate the ratio of the firing strength of the jth rule to the sum of all firing strengths of the rules:

j ¼ w

wj 2 P wj

ð13Þ

j¼1

In layer 5, every node j in this layer is an adaptive node and has the following output:

 j ðpj x1 þ qj x2 þ r j Þ  j fj ¼ w w

Fig. 4. Terminals of fuzzy system stabilizer.

ð14Þ

where {pj, qj, rj} is referred to as the consequent parameter set S2. They can also be trained using ANFIS learning algorithms.

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S.M. Radaideh et al. / Electrical Power and Energy Systems 35 (2012) 47–56

Fig. 5. Sugeno rule operation.

Fig. 6. Architecture of ANFIS system.

Fig. 7. Three machine nine bus power system.

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S.M. Radaideh et al. / Electrical Power and Energy Systems 35 (2012) 47–56

Degree of membership

1

NE

ZO

PS

PMPB

0.8 0.6 0.4 0.2 0 -0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

input1 Fig. 8a. Input membership functions.

ZO

Degree of membership

1

PS

PM

PB

0.8 0.6 0.4 0.2 0 -0.04

-0.02

0

0.02

0.04

output1 Fig. 8b. Output membership functions.

Fig. 9. Two level stabilizer.

0.06

0.08

0.1

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S.M. Radaideh et al. / Electrical Power and Energy Systems 35 (2012) 47–56

signals go forward until layer 5 and the consequent parameter vector {pj, qj, rj} is identified by means of the least squares estimate (LSE). This step can be summarized in solving the over constrained simultaneous linear equations shown in Eq. (16).

Table 1 Operating conditions at nominal conditions. Nominal conditions P (pu)

Q (pu)

Generator G1 G2 G3

0.716 1.630 0.850

0.270 0.066 0.109

Load A B C

1.250 0.900 1.000

0.500 0.300 0.350

2

 ð1Þ x 1

6 ð2Þ 6x 6 1 6 . 6 . 4 .

2

r1

3

ð16Þ

h  i ðkÞ ðkÞ ðkÞ are the kth training pair, with k = 1, 2, . . ., p, x1 ; x2 ; d

  ðkÞ ðkÞ x1 ; x2 .

Operating point (pu)

Eigenvalues

P1 = 0.716 P2 = 1.630 P3 = 0.850

89.908 ± j0.7409 0.009 ± j3.427 4.6208 ± j13.9012 2.7809 ± j9.2724

Q1 = 0.270 Q2 = 0.066 Q3 = 0.109

72.4625 27.3572 16.0872 20.3437

The single node in the layer 6 is a fixed node that computes the overall output as the summation of all incoming signals. Eq. (15) illustrates what is meant.

 j fj w

.. .

3

2 ð1Þ 3 6p 7 d 6 17 7 7 6 ð2Þ 7 ð2Þ 7 6  ð2Þ  x x 7 6 6 d 7 q1 2 2 7 7¼6 7 76 7 6 .. .. 7 6 r 2 7 4 ... 7 5 5 6 . . 6 7 ðpÞ 4 p2 5 ðpÞ ðpÞ d  2 x2  x q2 ð1Þ  ð1Þ  x 2 x2

 ðkÞ  ðkÞ x 1 and x2 are the fourth layer outputs associated with the input

Table 2 Open loop eigenvalues.

X

 1ð2Þ xð2Þ x 1

ðpÞ  ðpÞ  ðpÞ x x 1 1 x1

where

y¼

 1ð1Þ xð1Þ x 1

ð15Þ

j

4.2. ANFIS design and training The ANFIS hybrid learning algorithm is composed of a forward pass and a backward pass. In the forward pass, keeping constant the available values of the premise parameters set S1, functional

In the backwards pass, the error rates propagate backward and  1ðkÞ and x  ðkÞ the premise parameters S1, x 2 are updated by the gradient descent technique. For more details of this technique, refer to Ref. [15]. The only user specified information is the number of membership functions in the input–output as training information. Choosing the correct number of membership functions is a fundamental question often raised in these applications. There are many techniques applied to find the suitable number of membership functions, but in this paper, the number of membership functions for input variable is determined by trial and error for simplicity. In this paper, only one ANFIS structure is used to estimate the supplementary control signal – direct input of the synchronous generator – of a three-machine nine-bus power system. The input layer for ANFIS has one input which is a combination between reference input voltage DVref_i and the voltage signal DU that is equivalent to the speed deviation Dxi as will be shown in Fig. 9 with changing the fuzzy logic level with adaptive fuzzy one. The training procedure is achieved based on learning technique. Tuning of the parameters of the membership functions in adaptive neuro-fuzzy logic controller is achieved with back propagation (BP) technique using training data set. A training set of 50 input output pattern representing the control signal will be used to train the ANFIS

Fig. 10. Training data.

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S.M. Radaideh et al. / Electrical Power and Energy Systems 35 (2012) 47–56

in1mf1

in1mf2

Degree of membership

1

in1mf3

in1mf4

in1mf5

0.8 0.6 0.4 0.2 0 -0.2

-0.15

-0.1

-0.05

0

0.05

0.1

input1 Fig. 11. Input membership functions of ANFIS structure.

Fig. 12. Speed deviation under nominal loading conditions (TLFPSS vs. TLAFPSS).

structure. This number of the training data was chosen due to our experts, and it was found to be enough to hold large number of operating points. Using these data, hybrid algorithm is applied to adapt premise and consequent parameters.

Fig. 13. Speed deviation when (P1 = 0.362 pu, Q1 = 0.162 pu) (TLFPSS vs. TLAFPSS).

5. Simulation results Three machine nine bus power system [9] shown in Fig. 7 is used to illustrate the effectiveness of the proposed stabilizers. Details of the system data will be given in Appendix A.

Table 1 shows the operating conditions of the three generators at nominal loading conditions. Using a linearized fourth-order model of each machine in Eq. (7), open loop eigenvalues can be obtained. They are found in Table 2.

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S.M. Radaideh et al. / Electrical Power and Energy Systems 35 (2012) 47–56

Participation factors with coupling factors method proved that generators G2 and G3 are the optimum locations [16] of PSS to damp out the modes of oscillations. To design the first level, the following set of pole pairs are chosen to achieve pole placement using state feedback. Note that these poles are chosen in order to achieve specific amount of overshoot, and settling time.

ð89:908  j0:7409Þ; ð1:5  j3:5Þ; ð1:9  j12:5Þ; ð2:3  j9:5Þ; ð72:4625Þ; ð27:35Þ; ð16:0872Þ; and ð20:3437Þ 5.1. Fuzzy logic level The non-conventional level is firstly composed of fuzzy logic controller. This stabilizer has one input and one output as illustrated in Fig. 4. The following five statements are the rules used in a fuzzy controller:     

IF IF IF IF IF

(Input (Input (Input (Input (Input

is is is is is

NEG) THEN (Output is ZO). ZO) THEN (Output is PS). PS) THEN (Output is PB). PM) THEN (Output is PM). PB) THEN (Output is PB).

5.2. Selection of membership function One may ask, how to determine the membership function? How to choose one from these alternatives? Generally, there are two approaches to select a membership function. The first approach is to use the knowledge of human experts. Usually, this approach can only give a rough formula of the membership function. Not only this, but also fine tuning is required. While in the second approach, various sensors (observations) are used to collect data to determine a suitable membership function. In this paper, we depend on a combination between two approaches. We depend on the knowledge of the importance of the control signal. We also observed it and saw its values after dividing it to three parts – small, medium, and large. After this, we decided that Gaussian type membership function may be suitable to satisfy our requirements. This shape of membership functions is chosen because of the natural distribution of the data included in it.

Fig. 14. Speed deviation when (P3 = 1.1 pu, Q3 = 0.15 pu) (TLFPSS vs. TLAFPSS).

Table 3 Results of two level fuzzy power system stabilizer (TLFPSS) under various operating points. Operating point

Two level fuzzy power system stabilizer (TLFPSS) Generator # 2

Nominal P1 = 0.362 pu Q1 = 0.162 pu P3 = 1.1 pu Q3 = 0.15 pu

Generator # 3

Max. peak

Min. peak

Settling time

Max. peak

Min. peak

Settling time

2.05  103 2.3  103

2  103 1.85  103

3.3 6.8

2..01  103 2.28  103

2  103 1.9  103

3.15 6.7

1.3  103

1.2  103

5.3

1.15  103

1.1  103

5

Table 4 Results of two level adaptive fuzzy stabilizer (TLAFPSS) under various operating points. Operating point

Two level adaptive fuzzy power system stabilizer (TLAFPSS) Generator # 2

Nominal P1 = 0.362 pu Q1 = 0.162 pu P3 = 1.1 pu Q3 = 0.15 pu

Generator # 3

Max. peak

Min. peak

Settling time

Max. peak

Min. peak

Settling time

6.5  104 8  104

6.45  104 6.5  104

3.1 6.3

6.45  104 7.6  104

6.5  104 6.5  104

3.1 6.2

4.5  104

4.4  104

5.2

3.8  104

3.6  104

5

S.M. Radaideh et al. / Electrical Power and Energy Systems 35 (2012) 47–56 Table A1 Generator and exiter data [1]. Parameter

Generator

M xd xq x0d D T 0d0 Ka Ta

G1

G2

G3

47.280 0.1460 0.0969 0.0608 0.0000 8.9600 200.00 0.0100

12.800 0.8950 0.8645 0.1198 0.0000 6.0000 200.00 0.0100

6.0200 1.3125 1.2578 0.1813 0.0000 5.8900 200.00 0.0100

Table A2 Line data (in pu) [1]. Line

R

X

B/2

1–4 2–7 3–9 4–5 4–6 5–7 6–9 7–8 8–9

0.0000 0.0000 0.0000 0.0100 0.0170 0.0320 0.0390 0.0085 0.0119

0.0276 0.0625 0.0586 0.0850 0.0920 0.1610 0.1700 0.0720 0.1008

0.0000 0.0000 0.0000 0.0880 0.0790 0.1530 0.1790 0.0745 0.1045

Table A3 Bus data (nominal loading condition) [1]. Type

P

Q

V

1 2 3 4 5 6 7 8 9

Slack PV PV PQ PQ PQ PQ PQ PQ

– 1.63 0.85 0.00 1.25 0.90 0.00 1.00 0.00

– – – 0.00 0.50 0.30 0.00 0.35 0.00

1:040\0 1.025 1.025 – – – – – –

Table A4 Voltages and currents data [9].

xd  x0d xq  x0d

Generator 1

some observations that were got from the case of two level fuzzy power system stabilizer (TLFPSS). Fig. 11 illustrates the membership functions of the input signal of an adaptive fuzzy logic controller. It is noted that input membership functions in Fig. 8a is similar in shape (Gaussian membership functions) as that produced from ANFIS structure. The main difference between two figures is parameters of the membership function. This is expected due to hybrid algorithm that adjust ANFIS parameters (see Fig. 8b). The speed deviations of G2, and G3 under nominal loading condition when using two level fuzzy power system stabilizers (TLFPSS) and two level adaptive fuzzy power system stabilizers (TLAFPSS) are shown in Fig. 12a and b respectively. Fig. 13a and b illustrates the speed deviations of generators G2 and G3, respectively, when changing the loading conditions of generator G1 (P1 = 0.362 pu and Q1 = 0.162 pu). Fig. 14a and b shows the speed deviations of generators G2 and G3 when changing the loading conditions of generator G3 (P3 = 1.1 pu and Q3 = 0.15 pu), respectively. It is shown in the previous figures that two level adaptive fuzzy power system stabilizer (TLAFPSS) introduced an amazing amount of damping oscillation modes rather than two level fuzzy power system (TLFPSS). The following two tables quantify the results obtained when using two level fuzzy and adaptive fuzzy power system stabilizer under various operating points. Tables 3 and 4 illustrate numerically the difference between the results obtained in the two methods. 6. Comments and conclusions

Bus

Quantity

55

This paper has developed a two level power system stabilizer (PSS) design to enhance the damping of oscillation modes. The first level in the proposed stabilizer is achieved with pole placement using state feedback, while the second level is composed using one of the following techniques: fuzzy logic theory and adaptive fuzzy technique. Participation factor method was used to determine the optimum sites of PSS in the system. The results show that using (TLAFPSS) can damp out a low frequency oscillations more efficiently than (TLFPSS). Appendix A

Generator 2

Generator 2

E0q0

0.0852 0.0361 1.0558

0.7760 0.7447 0.7882

1.1312 1.0765 0.7679

E0d0 Iq0 Id0 Vq0 Vd0 d0

0.0419 0.6780 0.2872 1.0392 0.0412 2.2717°

0.6940 0.9320 1.2902 0.6336 0.8057 61.0975°

0.6668 0.6194 0.5615 0.6661 0.7791 54.1431°

Fig. 9 illustrates a two level fuzzy stabilizer with the plant. 5.3. Adaptive fuzzy level Two level adaptive fuzzy logic stabilizer will be used in this section to control the oscillation modes of ‘‘Three-machine nine-bus power system’’. Fig. 9 shows the structure of two level adaptive fuzzy logic stabilizer when applied to the generator – plant by changing the fuzzy logic controller with the ANFIS structure as a non-conventional level. Fig. 10 illustrates the shape of the training data used in adjustment the adaptable coefficients. These data are obtained from

Power system data (Tables A1–A4). References [1] Hardiansyah, Furuye Seizo, Irisawa Juichi. A robust H1 power system stabilizer design using reduced-order models. Electr Power Energy Syst 2006;28:21–8. [2] Tse CT, Tso SK. Refinement of conventional PSS design in multimachine system by modal analysis. IEEE Trans Power Syst 1993;8(2). [3] Ogata Katsuhiko. Modern control engineering. 3rd ed. Prentice-Hall International, Inc.; 1997. [4] Fleming RJ, Sun Jun. An optimal multivariable stabilizer for a multimachine plant. IEEE Trans Energy Convers 1990;5(1). [5] El-Zonkoly AM. Optimal tuning of power systems stabilizers and AVR gains using particle swarm optimization. Expert Syst Appl 2005. [6] Lakshmi P, Abdullah Khan M. Design of a robust power system stabilizer using Fuzzy logic for a multimachine power system. Electr Power Syst Res 1998;47(March):39–46. [7] Lie Tjing T, Sharaf AM. An adaptive fuzzy logic power system stabilizer. Electr Power Syst Res 1996;38(May):75–81. [8] Asgharian R. A robust H1 power system stabilizer with no adverse effect on shaft torsional modes. IEEE Trans Energy Convers 1994;9(3). [9] Anderson PM, Fouad AA. Power systems control and stability. Iowa State University Press; 1977. [10] Andreoiu Adrian. Genetic algorithm based tuning and economic worth as ancillary services. PhD thesis, Department of Electric Power Engineering, Chalmers University of Technology, Göteborg, Sweden; April 2004. [11] Wang Li-Xin. A course in fuzzy systems and control. Prentice-Hall, Inc.; 1997. [12] Phillips Charles L, Troy Nagle H. Digital control system analysis and design. 3rd ed. Prentice-Hall International, Inc.; 1995.

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[15] Jang Jyh Shing Roger. ANFIS: adaptive-network-based fuzzy inference system. Man Cybernet 1993;23(3):665–85. [16] Hsu YY, Chen CL. Identification of optimum location for stabilizer applications using participation factors. IEE Proc Part C 1987;134(3):238–44.