ADAPTIVE TYPE-2 FUZZY CONTROLLER FOR LOAD FREQUENCY CONTROL OF AN INTERCONNECTED HYDRO-THERMAL SYSTE

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014

ADAPTIVE TYPE-2 FUZZY CONTROLLER FOR LOAD FREQUENCY CONTROL OF AN INTERCONNECTED HYDRO-THERMAL SYSTEM INCLUDING SMES UNITS Dr. R.Vijaya Santhi1 and Dr. K.R.Sudha2 1

Assistant Professor,Department of Electrical Engineering, Andhra University, India 2 Professor,Department of Electrical Engineering, Andhra University, India

ABSTRACT This present paper includes the study Load Frequency Control (LFC) of power systems with several nonlinearities like Generation Rate Constraint(GRC) and Boiler Dynamics (BD) including Superconducting Magnetic Energy Storage (SMES) units using Type-2 Fuzzy System (T2FS) controllers . Here, Load frequency control problem is dealt with a three – area interconnected system of Thermal-Thermal-Hydal power system by observing the effects and variations of dynamic responses employing conventional controller, Type-1 fuzzy controller and T2FS controller considering incremental increase of step pertubations by 10% in the load. The salient advantage of this controller is its high insensitivity to large load changes and plant parameter variations even in the presence of non-linearities. As the non-linearities were considered in the system, the conventional and classical Fuzzy controllers does not provide adequate control performance with the consideration of above nonlinearities. To overcome this drawback T2FS Controller has been employed in the system. Therefore, the efficacy of the proposed T2FS controller is found to be better than that of conventional controller and Type-1 Fuzzy controller in cosidreration with overshoot, settling time and robustness.

KEYWORDS Load Frequency Control(LFC), Type-2(T2) Fuzzy Controller, Generation Rate Constraint (GRC), Boiler Dynamics(BD), Superconducting magnetic energy storage (SMES).

1. INTRODUCTION Inorder to maintain system frequency and inter-area oscillations within limits, Load Frequency Control (LFC) plays a vital role in large scale electric power systems. Both area frequency and tie-line power interchange varies with variation in power load demand. The motives of load frequency control (LFC)[1][2] are to minimize the transient deviations in theses variables and to ensure their steady state errors to be zeros. When dealing with the LFC [3] problem of power systems, certain unexpected pertubations, parametric uncertainties and the model uncertainties of the power system leads for the designing of controller. In large interconnected power system , DOI : 10.5121/ijfls.2014.4102

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014

generation of power is done by thermal, hydro, nuclear and gas power units.Usually, nuclear units are kept at base load close to their maximum output owing to their high efficiency with no participation in system Automatic Generation Control (AGC)[4]. Since these type of plants produces a very small percentage of total system generation, so such plants donot play an significant role in AGC of a large power system. In order to meet peak demands, Gas plants are used. Thus the natural choice for LFC falls on either thermal or hydro units.

In past, the area of LFC constrained to interconnected thermal systems and relatively lesser attention has been focussed to the LFC of interconnected hydro-thermal system [5] involving thermal and hydro subsystem of widely different characteristics. Concordia and Kirchmayer [6] have studied the AGC of a hydro-thermal system considering non-reheat type thermal system neglecting generation rate constraints and boiler dynamics. Since frequency has become a common factor, a change in active power demand at one point is reflected throughout the system,.

Mostly in the load frequency control studies, the boiler system effects and the governor dead band effects are neglected. But for the realistic analysis of system performance, these should be incorporated as they have considerable effects on the amplitude and settling time of oscillations.

From the past literature, under continuous-discrete mode with classical controllers, Nanda, Kothari and Satsangi [7] are the first to present comprehensive analysis of LFC of an interconnected hydrothermal system.

In the past decades, fuzzy logic controllers (FLCs) have been successfully developed for analysis and control of nonlinear systems [8][9]. The fuzzy reasoning approach is motivated by its ability to handle imperfect information,especially uncertainties in available knowledge. Stimulated by the success of FLCs, Talaq [10], Yesil and Chang[11] proposed different adaptive fuzzy scheduling schemes for conventional PI andor PID controllers. These methods provide good performances but the system transient responses are relatively oscillatory.

The main motive of this paper is to determine the Load Frequency Control and inter-area tie power control problem for a wide area power system with following certain uncertainities. From the literature, many authors have proposed fuzzy logic based controllers to power systems [12] inorder to take care of these uncertainties. This fuzzy logic, also called as Type-1 fuzzy, can further be modified to Type-2 fuzzy by giving grading to the membership functions which are themselves fuzzy. Or in other words, in Type-2 fuzzy sets, at each value of the variable the membership is a function but not just a point value. Therefore, a Type-2 fuzzy set can be visualized as a three dimensional. The advantage of the third dimension gives an extra degree of freedom for handling uncertainties. Taking this feature into consideration, a robust decentralized control scheme is designed using Type-2 Fuzzy logic [13][14][15]. The proposed controller is simulated for a three area power system in the presence of Generation Rate Constraint (GRC) and Boiler Dynamics (BD)[16] including Superconducting Magnetic Energy Storage(SMES) units was compared with conventional PI controller and Type-1 Fuzzycontroller. Results of simulation show that the T2 fuzzy controllers guarantee the robust performance .

2. POWER SYSTEM MODELLING AND PROBLEM FORMULATION: Usually, tie line power are used to interconnect control areas for a large scale power system.

However, for the design of LFC a simplified and linearized model is usually used. The detailed power system modeling of three area system containing two reheat steam turbines and one hydro -turbine tied together through power lines including Superconducting Magnetic Energy Storage 14

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(SMES) units with Generation Rate Constraint (GRC) and Boiler Dynamics (BD) for load frequency control is investigated in this study as shown in Fig.1 with Area Control Error(ACE) and its derivative are given as the inputs to the controllers [17]. Three areas have been installed with SMES1, SMES2 and SMES3 inorder to stabilize frequency oscillations. The interconnected power system model is shown in Fig-3. The Parameters of the three areas is given in Appendix.

Modelling of Speed Governors and turbines are discussed in [18]. Power generation can be changed only to a specified maximum rate in a power system having steam plants. the generation rate for the steam plants can be restricted, by adding limiters to the governors. The Generation Rate Constraint (GRC) value for thermal units of 3%/min is considered. To prevent the excessive control action, two limiters, bounded by ± 0.0005 within the automatic generation controller are used. By adding limiters to the turbines GRCs for all the areas are taken into consideration. Fig-2 shows the model to represent the boiler dynamics. Representations for combustion controls are also incorporated. This model is used inorder to study the responses of coal fired units with poorly tuned combustions controls and with well tuned controls.The limiter of -0.01 ≤ ∆PSMi, i=1, 2 ≤ 0.01 [puMW] based on a system MW base is equipped for each SMES unit. “Parameters values of SMES1, SMES2 and SMES3 are set at Ksm1 = Ksm2 = Ksm3= 0.12 and Tsm1 = Tsm 2 =Tsm3= 0.03 sec[19]”.

3.TYPE 2 (T2) FUZZY LOGIC CONTROLLERS: Zadeh [20] introduced type-2 fuzzy sets. The fuzzification of a type-1 fuzzy set gives the Type-2 sets. To describe the membership function by numbers, type-1 fuzzy sets requires the developer, in the discrete case, or by a function, where continuous membership function is given by the fuzzy . So, `non-fuzzy' (or crisp) representation is given by the fuzziness of a system which employs fuzzy sets . A fuzzy system that uses Type-2 fuzzy sets and/or fuzzy logic and inference is called a Type-2 (T2) fuzzy system. Infact, a Type-1 (T1) fuzzy system can be defined as the system that employs ordinary fuzzy sets, logic, and inference. In order to solve many practical problems, T1 fuzzy systems, especially fuzzy logic controllers and fuzzy models are modelled. As per Mendel,“A Type-1 fuzzy set (T1 FS) has a grade of membership that is crisp, whereas a Type - 2 fuzzy set (T2 FS) has a grade of membership that is fuzzy, so T2 FS are ‘fuzzy-fuzzy’ sets”. To represent the fuzzy membership of fuzzy sets footprint of uncertainty (FOU) is employed, which is a 2-D representation, with the uncertainty about the right end point of the right side of the membership function and with the uncertainty about the left end point of the left side of the membership function. The type-1 fuzzy sets, which represents uncertainty by numbers in the range [0, 1] can be handled by the general framework of fuzzy reasoning .

Uncertainity cannot be determined with its exact value, because of its complexity and rather type-1 fuzzy sets gives much senser than using crisp sets [21]. So, it is difficult to measure an uncertain membership function . To overcome this difficulty, we require another type of fuzzy sets, those which has ability to handle these uncertainties. Those type of fuzzy sets are called type-2 fuzzy sets. As the type-2 fuzzy logic has better capability to cope up with linguistic uncertainities , type-2 is a good replacement for type-1 fuzzy system..

Infact, the Type 1- fuzzy and Type-2 fuzzy sets operation are similar, but while using with interval fuzzy system; by limiting the FOU, fuzzy operator is being done as two T1 membership functions, UMF and LMF inorder to produce firing strength which is shown in Fig - 4.

Defuzzification is a mapping process from fuzzy logic control action to a non-fuzzy (crisp) control action. Defuzzification on an interval Type2 fuzzy logic system using centroid method is 15

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014

shown in Fig -4. In Type-2 fuzzy set, at each value of primary variable the membership is a function and it is not just a point value; the secondary membership function whose domain, i.e., the primary membership is in the interval [0,1], then their range, the secondary grades may also be in the interval [0,1]. Since, the foot of membership functions is not a single point but designed over an interval, therefore Type -2 fuzzy logic controller can also be refered as Interval Type-2 fuzzy logic controller. Interval type2 fuzzy logic operation is shown in Fig. 4.The Interval Type-2 membership functions and operators are designed and are employed in the IT2FLS toolbox. An Inference FS is a rule base system that uses fuzzy logic, instead of Boolean logic that is utilized in data analysis. Its basic structure includes four components (Fig - 5): Fuzzification: Translates inputs (real values) to fuzzy values.

Inference System: To obtain a fuzzy output, fuzzy reasoning mechanism is applied.

Type Defuzzificator/Reductor: To transduces one output to precise values, defuzzificator is employed; the type reductor converts a Type 2 Fuzzy Set into a Type- 1 Fuzzy Set. Knowledge Base:It contains data base which consists of set of fuzzy rules, and a membership functions set. The two normalized input variables, ∆ ACE and ∆ AC E , are first fuzzified by two interval T2 fuzzy sets (Fig -6), namely “positive” and “negative” represented by  P (∆ ACE ) and

 N (∆ ACE ) respectively. The primary memberships are generated by blurring the trapezoidal T1 fuzzy sets 1  P (∆ ACE ) ,  N (∆ ACE ) ,  P ( ∆ AC E ) , and  N ( ∆ AC E ) . The interval T2 fuzzy sets secondary membership functions are all constant.

The definitions of the T1 fuzzy sets are as follows:

(−∞,− L1 ] 0   p (∆ACE ) =  ( L1 + ∆( ACE )) / 2 L1 [− L1 , L1 ]  1 [ L1 , ∞) 

-----(1)

After shifting the membership functions of the T1 fuzzy sets upward and downward by θ1 ∈ [0, 0.5] for  P (∆ ACE ) and  N (∆ ACE ) along the membership axes, the boundary membership functions of the primary memberships of the interval T2 fuzzy sets[22][13] (i.e.),  P L ( ∆ ACE ) ,

 PU (∆ ACE ) ,  NL (∆ ACE ) ,  NU (∆ ACE ) ). These boundary membership functions form the shaded bands in Fig -6 which are called footprints of uncertainty (FOU). The design parameters θ1and θ 2are used to control the degree of uncertainty of the interval T2 fuzzy sets.

Inorder to realize the AND operations in the rules, Zadeh fuzzy logic AND operator (i.e., min( )) is used.

If ∆ACE is P and ∆AC E is P, then U is N For an interval T2 fuzzy interface, the firing set becomes a firing interval [RL,RU]=[min( ∆ACE PL, ∆AC E PL,U NL), min( ∆ACE PU, ∆AC E PU,UNU)] The rules are shown in Table-1.

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4. SIMULATION RESULTS To illustrate robust performance of the proposed Type-2 Fuzzy controller we have chosen different cases:

Case I(a),Case I(b) & CaseI(c): Step increase in demand of the first area ∆ PD1: In this case, step increase in demand of the first area ∆ PD1 is applied. The frequency deviation of the first area, Δf1, the frequency deviation of the second area, Δf 2, the frequency deviation of the third area, Δf3, and inter area tie-power signals of the closed-loop system are shown in Fig -7,Fig -8.

Step increase in demand of the second area ∆ PD2 is applied. The frequency deviation of the first area, Δf1, the frequency deviation of the second area, Δf 2, the frequency deviation of the third area, Δf3, and inter area tie-power signals of the closed-loop system are shown in Fig -9, Fig -10.

Step increase in demand of the third area ∆ PD3 is applied. The frequency deviation of the first area, Δf1, the frequency deviation of the second area, Δf 2, the frequency deviation of the third area, Δf3, and inter area tie-power signals of the closed-loop system are shown in Fig -11, Fig 12.Using proposed method, the frequency deviations and inter area tie-power quickly driven back to zero and controller using T2 fuzzy controller has the best performance in control and damping of frequency and tie-power in all responses when compared with conventional PI and Type-1 Fuzzy controller [12].

Case II: Step increase in demand of the first area ∆ PD1 , second area ∆ PD2 and third area ∆ PD3 is applied. This is the condition, for which perturbation is given in all the three areas. In this case, a step increase in demand of the first area ∆ PD1 , the second area ∆ PD2 and third area ∆ PD3 is applied. The frequency deviation of the first area Δf 1 , the frequency deviation of the second area Δf2, the frequency deviation of the third area Δf 3 is shown in Fig -13,fig-14. The frequency deviations and inter area tie-power quickly driven back to zero by employing proposed controller.

Type- 2 fuzzy controller has the best performance in control and damping of frequency and tiepower in all responses when compared with conventional PI and Type-1 Fuzzy controller[12].

The robust performance for the above cases is shown numerically at a particular operating condition is listed in Table-2. In this study, settling time, overshoot and undershoot are calculated for 10% band of the step load change in each area and in all three areas and simulation results for 10% band of step load change for the operating point shown in Appendix. Upon examination of Table-2, reveals that the performance of the proposed Type-2 Fuzzy controller is better than conventional PI and Type-1 Fuzzy controller.

5.CONCLUSIONS From the Table-2, the power system results are shown with the variation of 10% load. Under Hydro-thermal-thermal combination, the proposed Type-2 Fuzzy control gives a better dynamic performance and also reduces the oscillations of frequency deviation and the tie line power..

Simulation results proves that the proposed controller guarantees the robust stability performance like frequency tracking and disturbance attenuation under a wide range of parameter uncertainty and area load conditions. The results shows that under large parametric uncertainty, the proposed type-2 fuzzy controller provided decentralized stability of the overall system. To demonstrate performance robustness of proposed method, the Settling Time , Maximum Overshoot , and 17

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014

Undershoot are being considered.. It gave an appreciable performance as compared to conventional PI controller and Type I Fuzzy controller for the given operating condition.

APPENDIX The typical values of parameters of Hydro-thermal-thermal system for nominal operating condition are as follows[13][5] K p1 = K p2 = Kp3= 120 , Tg1 = Tg2 = 0.2 , T12 =T23=T31= 0.0707 , Tw = 1 f = 60 hz

Tp1 = Tp2 = Tp3= 10 , Kr1 = Kr2 = 0.333, Tr1 = Tr2 =10 R1 = R2 = R3= 2.4, B1 = B2 = B3= 0.425 , Tt1 = Tt2 = 0.3 a12 =a23=a31= -1 Kd = 4 Kp = 1 Ki = 5

Boiler Dynamics data: K1= 0.85, Kib= 0.03,

K2= 0.095, Tib= 26,

K3= 0.92, Trb= 69

Cb= 200,

Td= 0 ,

Tf= 10,

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014 [12] Shayeghi H, Jalili A, Shayanfar H.A, (2005)“Fuzzy PI Type Controller for Load Frequency Control Problem in Interconnected Power System”, 9th World Multi Conf. on Systemic Cybernetics and information, Orlando, Florida, U.S.A., July 10-13, pp. 24-29.

[13] Sudha K.R, Vijaya Santhi R,(2011) “Robust decentralized load frequency control of interconnected power system with Generation Rate Constraint using Type-2 fuzzy approach”, Electrical Power and Energy Systems, Vol. 33, pp. 699–707.

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SMES 1

SMES 2 Reheat Thermal Plant Area 2

Reheat Thermal Plant Area 1 Tie line

Load Disturbance

Hydal Plant Area 3

Load Disturbance

SMES 3

Load Disturbance

Fig-1: Three - Area Interconnected Power System including SMES units

Fig- 2: Boiler dynamics

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Fig-3:Block Diagram of Three Area Interconnected system

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Fig- 4: Membership Function and Interval Type-2 Fuzzy Reasoning

Fuzzy Rule Bases

output U(n)

Defuzzification

inputs

E(n) R(n)

Fuzzy Inference

Fuzzification

Type-Reducer

Fig- 5:The structure of the T2 fuzzy PI controller

 ( ACE ) NL

N

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PL

P

PU

θ2+0.5

θ20.5 -L2-P2

- L2 -L2+P2

Universe of Discourse

L2-P2 L2

L2+P2

Fig- 6: Membership functions of the Interval T2 fuzzy sets

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Fig -10: ∆Ptie12, ∆Ptie23, ∆Ptie31with step increase in second area ∆PD2 with GRC and SMES Units 26

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014 x 10

1 .5

-4

C o n ve n t i o n a l P I c o n t r o l l e r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P ro p o s e d T y p e -2 F u z z y c o n t ro lle r

1

0 .5

d e lt a f 1

0

-0 . 5

-1

-1 . 5

-2

-2 . 5

-3

1 .5

0

x 10

1

2

3

4

5 t im e , s e c s

6

7

8

9

10

-4

C o n ve n t i o n a l P I c o n t r o l l e r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P ro p o s e d T y p e -2 F u z z y c o n t ro lle r

1

0 .5

d e lt a f 2

0

-0 . 5

-1

-1 . 5

-2

-2 . 5

-3

4

0

x 10

1

2

3

4

5 t im e , s e c s

6

7

8

9

10

-4

C o n ve n t io n a l P I c o n t ro lle r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P ro p o s e d T y p e -2 F u z z y c o n t ro lle r 2

d e lt a f 3

0

-2

-4

-6

-8

0

1

2

3

4

5 t im e , s e c s

6

7

8

9

10

Fig -11: ∆f1,∆f2,∆f3 with step increase in third area ∆PD3 with GRC, BD including SMES Units

27

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014 x 10

3

-6

C o n ve n t i o n a l P I c o n t r o l l e r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P ro p o s e d T y p e -2 F u z z y c o n t ro lle r 2

d e lt a P t ie 1 2

1

0

-1

-2

-3

0

2

x 10

10

4

6

8

10 t im e , s e c s

12

14

16

18

20

-5

C o n ve n t i o n a l P I c o n t r o l l e r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P rp o s e d T y p e -2 F u z z y c o n t ro lle r

8

6

d e lt a P t ie 2 3

4

2

0

-2

-4

-6

-8

0

8

1

x 10

2

3

4

5 t im e , s e c s

6

7

8

9

10

-5

6 C o n ve n t i o n a l P I c o n t r o l l e r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P ro p o s e d T y p e -2 F u z z y c o n t ro lle r

4

d e lt a P t ie 3 1

2

0

-2

-4

-6

-8

-1 0

0

2

4

6

8

10 t im e , s e c s

12

14

16

18

20

Fig -12: ∆Ptie12, ∆Ptie23, ∆Ptie31with step increase in third area ∆PD3 with GRC, BD including SMES Units 28

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014 x 10

2

-4

C o n ve n t i o n a l P I c o n t r o l l e r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P ro p o s e d T y p e -2 F u z z y a p p ro a c h

1

0

d e lt a f 1

-1

-2

-3

-4

-5

-6

0

1

x 10

2

2

3

4

5 t im e , s e c s

6

7

8

9

10

-4

C o n ve n t i o n a l P I c o n t r o l l e r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P ro p o s e d T y p e -2 F u z z y c o n t ro lle r

1

0

d e lt a f 2

-1

-2

-3

-4

-5

-6

3

0

x 10

1

2

3

4

5 t im e , s e c s

6

7

8

9

10

-4

2

C o n ve n t i o n a l P I c o n t r o l l e r T y p e - 1 F u z z y ( S h a y e g i 's ) c o n t r o l l e r P ro p o s e d T y p e -2 F u z z y c o n t ro lle r

1

0

d e lt a f 3

-1

-2

-3

-4

-5

-6

-7

0

2

4

6

8

10 t im e , s e c s

12

14

16

18

20

Fig -13: ∆f1,∆f2,∆f3 with step increase in first area ∆PD1, second area ∆PD2 and third area ∆PD3 with GRC, BD including SMES Units 29

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014 x 10

3.5

-6

C o n ve n t io n a l P I c o n t ro lle r Ty p e -1 F u z z y (S h a y e g i's )c o n t ro lle r P ro p o s e d Ty p e -2 F u z z y c o n t ro lle r

3 2.5 2

d e lt a P t e 1 2

1.5 1 0.5 0 -0 . 5 -1 -1 . 5

0

x 10

6

2

4

6

8

10 tm e,s ec s

12

14

16

18

20

-5

C onventional P I c ontroller Ty pe-1 F uz z y (S hay egi's )c ontroller P ropos ed Ty pe-2 F uz z y c ontroller

5 4

delta P tie23

3 2 1 0 -1 -2 -3

0

3

x 10

2

4

6

8

10 tim e,s ec s

12

14

16

18

20

-5

2 1

delta P tie31

0 -1 Conventional P I c ontroller Ty pe-1 F uz z y (S hay egi's )c ontroller P ropos ed Ty pe-2 F uz z y c ontroller

-2 -3 -4 -5 -6 -7

0

2

4

6

8

10 tim e,s ec s

12

14

16

18

20

Fig -14: ∆Ptie12, ∆Ptie23, ∆Ptie31with step increase in demand of first area ∆PD1, second area ∆PD2 and third area ∆PD3 with GRC, BD including SMES Units 30

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014 ∆ACE

(∆ AC E )

N

Z

P

N

P

N

N

Z

N

P

P

P

N

N

N

Table-1: Control rules forT1 and T2 Fuzzy controller

Case -I(a)

∆f1

∆f2

∆f3

∆Ptie12

∆Ptie23

∆Ptie31

Case -I(b)

∆f1

∆f2

∆f3

∆Ptie12

∆Ptie23

Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy

Settling Time secs >10 10 5.2 >10 >10 6.1 >10 >10 8.8 >10 >10 5.9 >10 >10 6.39 >10 >10 8.6 >10 >10 8 >10 >10 5.3 >10 >10 8.9 >10 >10 5.6 >10 >10

Maximum Overshoot 1.9x10-4 2.0x10-4 1.8x10-4 0.57x10-4 0.52x10-4 0.187x10-4 1.45x10-4 1.4x10-4 0.74x10-4 4.1x10-5 3.78x10-5 2.33x10-5 7.03x10-5 6.71x10-5 6.41x10-5 2.13x10-5 2.06x10-5 1.23x10-5 0.57x10-4 0.48x10-4 0.16x10-4 1.8x10-4 2.0x10-4 1.8x10-4 1.45x10-4 1.47x10-4 0.75x10-4 7.15x10-5 6.9x10-5 6.47x10-5 6.1x10-5 5.9x10-5

Undershoot -1.7x10-4 -1.6x10-4 -0.9x10-4 -1.8x10-4 -1.7x10-4 -1.4x10-4 -2.4x10-4 -2.3x10-4 -1.7x10-4 -7.1x10-5 -6.7x10-5 -6.43x10-5 -6.18x10-5 -5.85x10-5 -3.5x10-5 -1.82x10-5 -1.78x10-5 -0.66x10-5 -1.8x10-4 -1.7x10-4 -1.3x10-4 -1.6x10-4 -1.6x10-4 -0.9x10-4 -2.46x10-4 -2.3x10-4 -1.7x10-4 -3.9x10-5 -3.6x10-5 -2.17x10-5 -7.04x10-5 -6.8x10-5 31

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014

∆Ptie31

Case –I(c)

∆f1

∆f2

∆f3

∆Ptie12

∆Ptie23

∆Ptie31

Case –2

∆f1

∆f2

∆f3

∆Ptie12

∆Ptie23

∆Ptie31

Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy Conventional PI Type-1 Fuzzy Type-2 Fuzzy

6.7 >10 >10 7.58 >10 >10 7.35 >10 >10 7.12 >10 >10 6.3 ----13.66 >10 >10 7.8 >10 >10 6.66 >10 10 9.4 ----8.6 >10 >10 6.7 ----18.78 >10 >10 7.9 10 9.6 7.13

3.52x10-5 1.7x10-5 1.8x10-5 0.7x10-5 1.3x10-4 1.3x10-4 0.81x10-4 1.26x10-4 1.38x10-4 0.71x10-4 2.7x10-4 2.6x10-4 2.0x10-4 1.3x10-6 2.13x10-6 1.58x10-6 9.43x10-5 9.43x10-5 7.9x10-5 6.11x10-5 5.7x10-5 3.43x10-5 1.3x10-4 1.2x10-4 0.53x10-4 1.32x10-4 1.26x10-4 0.5x10-4 1.8x10-4 1.7x10-4 0.9x10-4 2.58x10-6 3.43x10-6 0.46x10-6 5.53x10-5 5.9x10-5 3.6x10-5 1.65x10-5 2.1x10-5 1.0x10-5

-6.4x10-5 -2.2x10-5 -2.25x10-5 -1.28x10-5 -2.5x10-4 -2.49x10-4 -1.8x10-4 -2.5x10-4 -2.5x10-4 -1.7x10-4 -2.4x10-4 -2.1x10-4 -0.98x10-4 ---0.33x10-6 -2.7x10-6 -6.2x10-5 -5.9x10-5 -3.3x10-5 -9.4x10-5 -9.4x10-5 -7.8x10-5 -----0.1x10-4 -------0.26x10-4 -0.26x10-4 -0.05x10-4 -----1.36x10-6 -1.9x10-5 -2.3x10-5 -0.95x10-5 -5.6x10-5 -6.29x10-5 -3.6x10-5

Table -2: The numerical analysis

32

International Journal of Fuzzy Logic Systems (IJFLS) Vol.4, No.1, January 2014

Authors Dr. R.Vijaya Santhi received her B.Tech. degree in Electrical and Electronics Engineering from S.V.H Engineering College, Machilipatnam, Nagarjuna University in 2003.She did her M.Tech in Power systems, from JNTU Kakinada in 2008. awarded her Doctorate in Electrical Engineering in 2014 by Andhra University.Presently, she is working as Assistant Professor in the Department of Electrical Engineering, Andhra University, Visakhapatnam, India.

Dr.K.R.Sudha received her B.E. degree in Electrical and Electronics Engineering from GITAM; Andhra University 1991.She did her M.E in Power Systems 1994. She was awarded her Doctorate in Electrical Engineering in 2006 by Andhra University. During 1994-2006, she worked with GITAM Engineering College and presently she is working as Professor and Head in the Department of Electrical Engineering, AUCE(W), Andhra University, Visakhapatnam, India.

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