a simple fuzzy modeling of permanent magnet synchronous generator

7/28/2019 A Simple Fuzzy Modeling of Permanent Magnet Synchronous Generator

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Faculty of Electrical Engineering

Universiti Teknologi Malaysia

VOL. 11, NO. 1, 2009, 38-43

ELEKTRIKAhttp://fke.utm.my/elektrika

38

A Simple Fuzzy Modeling of Permanent Magnet

Synchronous Generator

N. Selvaganesan1*

and R. Saraswathy Ramya2

1Faculty in Avionics, Indian Institute of Space science and Technology(IIST), Department of Space, Govt. of India,Thiruvananthapuram 695022, India.

2Application Development Group, Collabera Solutions Pvt. Ltd, Bangalore-560095, India.

*Corresponding author: [email protected], Tel: +91 471 2563484, Fax: +91 471 2564806

Abstract:This paper presents the design methodology of fuzzy based modeling of permanent magnet synchronous generatorsystem. An analytical model of the system is presented in the nonlinear state space form based on fundamental physical laws.The fuzzy based modeling scheme for Permanent Magnet Synchronous Generator is developed using the general Takagi-Sugeno fuzzy model. The feasibility of the proposed scheme for Permanent Magnet Synchronous Generator is demonstrated

using MATLAB for load disturbances and the obtained responses are compared with mathematical model responses.

Keywords: T-S Fuzzy model, permanent magnet synchronous motor, rotor reference frame, fuzzy system.

1.INTRODUCTION

Permanent Magnet Synchronous Generators (PMSGs) are

receiving significant attention from industries for the lasttwo decades. They have numerous advantages over themachines which are conventionally used. Currentresearch in the design of the PMSG indicates that it hashigh torque to current ratio, large power to weight ratio,

high efficiency, high power factor and robustness [1].Currently, there is much interest in using brushless

electronically commutated servo machines in highperformance electromechanical systems and the

application of neodymium-iron-boron (Nd2Fe14B) andsamarium cobalt (Sm1Co5 and Sm2Co17) rare-earthmagnets results in high torque and power density,efficiency and controllability, versatility and flexibility,simplicity and ruggedness, reliability and cost, weight-to-torque and weight-to-power ratios, better starting

capabilities [2]-[5].Building models of reality is a central topic in many

disciplines of engineering and science. Models can be

used for simulations, analysis of the system's behaviorand for a better understanding of the underlying physicalmechanisms in the system. In control engineering, amodel of the plant can be used to design a feedbackcontroller or to predict the future plant behavior in orderto calculate optimal control actions. The mathematical

modeling of PMSG is discussed by researchers in variousliteratures [1]-[6].

Recent advances in the theory of fuzzy modeling and anumber of successful real-world applications showthat fuzzy models can be efficiently applied to complexnonlinear systems intractable with standard linear

methods. The idea of fuzzy modeling was first proposed

by Zadeh [7] and has subsequently been pursued by manyothers for more than two decades. The fuzzy modeling forcomplex processes is regarded as one of the key problemsin fuzzy systems research [8] [9].

In the field of fuzzy modeling, the Takagi - Sugeno (T-S) fuzzy model [10]-[15] has been used to approximateaccurately the dynamics of complex systems. Besides thecapability of modeling nonlinear systems, there are other

properties that make fuzzy models interesting not onlytheoretically but also for the industrial practice. Few

researches [16]-[17] have attempted in making a fuzzy

model for permanent magnet synchronous motor drive.But the fuzzy modeling of PMSG is largely unexplored.This paper focuses the attention on T-S fuzzy modeling

of PMSG drive. The proposed fuzzy model for PMSG issimulated with load disturbances and the responses are

compared with the mathematical model responses.The rest of the paper is described in four sections.

Section 2 explains the mathematical model of PMSGdrive. Section 3 deals with proposed T-S fuzzy model forPMSG drive. Section 4 describes the comparison betweenthe mathematical model and the fuzzy model under loaddisturbances. Finally the conclusion is narrated in section5.

2. THE PMSG SYSTEM MODEL

The dynamic characteristics of a PMSG can be modeledbased on the d-q axis and the differential equations thatdescribe the circuitry and torsional-mechanical dynamics[1]-[5]. Consider the generation system when the

permanent magnet DC motor is used as prime moverwhose differential equations are as follows.

a a aa r a

a a a

di r k 1= i - + u

dt L L L(1)

rarm m ma qs rm

kd 3P B

= i - i - dt J 4J J (2)

rmrm

d=

dt(3)


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N. SELVAGANESAN, R. SARASWATHY RAMYA / ELEKTRIKA, 11(1), 2009, 38-43

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where, ra, La, ia, ka are armature resistance, armatureinductance, armature current , armature constantrespectively. r, rm and rm are angular velocity,

mechanical angular velocity and mechanical angulardisplacement respectively. P is number of poles. m isflux linkages. Bm and J are moment of inertia and viscous

friction co-efficient respectively. rqsi is the q-axis stator

current component. The transient dynamics of generatormust be integrated with prime mover equations. In therotor reference frame, mathematical model of three phasePMSG [1] with resistive load can be expressed as follows

rqs r rs L m

qs rm ds rm

ls lsm m

di r +R P P=- i + - i

3 3dt 2 2L + L L + L

2 2

(4)

rr rds s Lds qs rm

ls m

di r +R P=- i + i

3dt 2L + L

2

(5)

Where, rdsi is d-axis stator current component RL and rs are

load resistance and stator resistance respectively. Lls and

mL are leakage and magnetizing inductancerespectively. The q-axis voltage and d-axis voltage are

given [1] as follows.

r rqs s ss qs r ss ds r m

du r L i L i

dt

= + +

(6)

r rds s ss ds r ss qs

du r L i L i

dt

= + +

(7)

Where, uds, uqs are q- axis stator voltage component, d-

axis stator voltage component respectively.ls m

3L + L

2is

Lss which is the self inductance of stator winding. Bymeans of Parks transformation the generated threecurrent and voltages are obtained.

3. T-S FUZZY MODEL

Fuzzy modeling is derived based on T-S fuzzy model. Itis derived by utilizing the concept of Parallel DistributedCompensation. The design procedure is conceptuallysimple and natural. It begins with representing a non-

linear plant based on T-S fuzzy model. The fuzzy modelproposed by Takagi and Sugeno is described by fuzzy

IF-THEN rules, which represent local linear input-outputrelations of a nonlinear system. The main feature of a T-Sfuzzy model is to express the local dynamics of eachfuzzy implication (rule) by a linear system model. Theoverall fuzzy model of the system is achieved by fuzzyblending of the linear system models. Many nonlinear

dynamic systems can be represented by T-S fuzzy modeland are proved as universals approximators [13]-[15].

Consider a general system as follows:

sx(t)=f(x(t))

y(t)=h(x(t)) (8)

Where, x Rn is the state vector; sx(t) denotes x(t).

and x(t+1) in continuous-time and discrete-time systems

respectively; y Rm is the systems output; f() and h()

are nonlinear functions with appropriate dimensions. A

fuzzy representation of (8) is composed of the following

rules:

IF z1(t) is F1i and .. and zg(t) is Fgi THEN

sx(t) = Aix(t)+bi

y(t)= Cix(t), i = 1, 2, ., r (9)

Where, z1(t) ~ zg(t) are the premise variables whichconsists of states of the system; Fji (j = 1, 2, , g) arefuzzy sets, r is the number of fuzzy rules, Ai and Ci the

are system and output matrices with appropriatedimensions and bi R

n denotes the constant bias term,

which is generated by the exact fuzzy modelingprocedure.

Using the singleton fuzzifier, product fuzzy inferenceand weighted average defuzzifier [14], the final output ofthe fuzzy systems is inferred as follows:

sx(t)=r

i i i

i = 1

(z(t))(A x(t)+b )

y(t)=r

i i

i = 1

(z(t))(C x(t)) (10)

Where z(t) = [z1(t) z2(t) . zg(t)]T, and

i

i r

i

i = 1

(z(t)) (z(t))=

(z(t))

withg

i ji j

j = 1

(z(t))= F (z (t))

note thatr

ii = 1

(z(t)) = 1 for all t, where i (z(t)) 0

for i = 1, 2, , r.

Considering equations (4) and (5), the fuzzymembership functions are specified appropriately inpremise parts, associated entries of matrices Ai, Ci and biin the consequence parts are represented by a TS fuzzymodel. From the investigation of a large class of

continuous-time and discrete-time systems, it is foundthat nonlinear terms have a common variable or dependonly on one variable. If we take this as the premisevariable of fuzzy rules, a simple fuzzy dynamic modelcan be obtained.

From the equations (4) and (5) X1 and X2 are the statevariables of quadrature-axis current and direct-axiscurrent respectively and the speed is chosen as the

premise variable. The S1 and S2 denotes the maximum andthe minimum values of speed [0 - 444] rad/sec. The

discretised equations of (4) and (5) are written as follows

1

(11)

1

(12)

where t is sampling time and is chosen as 0.0001sec forsimulation purposes. The system matrices (Ai), input

matrices (bi), output matrices (Ci) for the given fuzzymodel is based on maximum and minimum values (i=1,


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2) of the premise variables is developed and they are asfollows.

s L1

ss

1

s L1

ss

R +R- -S

LA =

R +RS -

L

ands L

2

ss

2

s L2

ss

R +R- -S

LA =

R +RS -

L

m 1

ss1

SLb =

0

and m 2

ss2

S

Lb =

0

[ ]1C = 1 0 and [ ]2C = 0 1

Fuzzy sets chosen for the model are r 11

1 2

-SF =

S -Sand

r 22

2 1

-SF =

S -S

Which is obtained from the expression

S1F1+ S2F2 = r (13)

Final output of the fuzzy model is computed usingparallel distributed compensation and is obtained as

21

1221

FF

FYFYYY

+

+= (14)

From the fuzzy modelingrqsi and

rdsi are determined.

Based onrqsi and

rdsi values uqs and uds are calculated. By

Parks transformation, the 2-axis quantity is converted in

3-axis quantity. Hence, three phase currents and voltagesare obtained.

4. SIMULATION RESULTSThe specification of the PMSG drive is given in

Appendix [I]. The fuzzy modeling for PMSG is simulated

using MATLAB. The q-axis current, d-axis current, q-

axis voltage, d-axis voltage, three phase currents, three

phase voltages are obtained and compared with

mathematical model simulation results.

4.1 Without load disturbance

The mathematical model and the fuzzy model are

evaluated by superimposing their responses in the same

graph for the load resistance of 100. The iqs, zoomed

view ofrqsi ,

rdsi , uqs, zoomed view of uqs, uds, zoomed

view of uds, three phase currents and three phase voltages

are shown in the Figure 1 to Figure 9 respectively. A load

of 100 is applied to mathematically modeled PMSG.

The results such asrqsi = 0.425A,

rdsi = 0.035A, uq =

42.96V, ud =0.336V, three phase currents of magnitude

0.425 A, three phase voltages of magnitude 42.95V are

obtained. Similarly for fuzzy model, the results such as

rqsi = 0.425A,

rdsi = 0.033A, uq = 42.96V, ud =0.336V,

three phase currents of magnitude 0.425A and three phase

voltages of magnitude 42.92V are also obtained.

Figure 1. q-axis current

Figure 2. Zoomed view of q-axis current

Figure 3. d-axis current

Figure 4. q-axis voltage

Fuzzy model

Mathematical model

Mathematical model

Fuzzy model


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Figure 5. Zoomed view of q-axis current

Figure 6. d-axis voltage

Figure 7. Zoomed view of d-axis voltage

Figure 8. Three phase current

Figure 9. Three phase voltage

4.2 With load disturbance

The PMSG drive is subjected to load disturbance by

varying the load resistance from 100 to 50 at 1 secand 50 to 10 at 2sec. Figure 10 to Figure 15 shows the

rqsi ,

rqsi , uqs, uds, three phase currents and three phase

voltages under changes in load. Initially a load of 100 isapplied to PMSG and the responses are observed. A

change in load of 50 and 10 is applied at 1 sec and 2sec respectively and its corresponding mathematicalmodel and fuzzy model responses are observed.

In mathematical model, when the load is 100 ,rqsi =

0.425A,rdsi = 0.035A, uq = 42.96V, ud =0.336V,

three phase currents of magnitude 0.425A and three phasevoltages of magnitude 42.96V are obtained. With the

change in load to 50 at 1 secrqsi = 0.83A,

rdsi =

0.13A, uq = 42.835V, ud = 0.66V, three phase currents ofmagnitude 0.8374 A and three phase voltages ofmagnitude 42.83V are obtained. The change in load of 10 is introduced at 2 sec and its corresponding variations

inrqsi = 2.6A,

rdsi = 1.96A, uq = 41.18V, ud = 2.04V,

three phase currents of magnitude 3.108A and three phase

voltages of magnitude 41.24V are also obtained.When the fuzzy modeled PMSG is subjected to 100

load,rqsi = 0.425A,

rdsi = 0.035A, uq = 42.75V, ud

=0.32V, three phase currents of magnitude 0.425A andthree phase voltages of magnitude 42.75V are obtained.

The change in load to 50 is introduced at 1sec and thecorresponding variations of

rqsi = 0.825A,

rdsi = 0.144A,

uq = 42.825V, ud = 0.656V, three phase currents ofmagnitude 0.8374A and three phase voltages ofmagnitude 42.842V are obtained. With the change in load

to 10 at 2 secrqsi = 2.40A,

rdsi = 2A, uq = 41V, ud =

1.875V, three phase currents of magnitude 3.108A, threephase voltages of magnitude 41.24V are obtained.

From the results it is observed that for a small

variation inrqsi ,

rdsi , uq and ud, the net three phase

currents and voltages are not affected much. Also it is

found that, when PMSG drive is subjected to loaddisturbance the responses of fuzzy modeled PMSG tracksthe mathematical model responses accurately.

Fuzzy model

Mathematical model

Mathematical model

Fuzzy model


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42

Figure 10. q-axis (100 to 50 at 1sec and 50to 10 2sec)

Figure 11. d-axis current (100 to 50 at 1sec and 50

to 10 at 2sec)

Figure 12. q- axis voltage (100 to 50 at 1sec and 50

to 10 at 2sec)

Figure 13. d-axis voltage (100 to 50 at 1sec and 50to 10 at 2sec)

Figure 14. Three phase current (100 to 50 at 1sec and

50 to 10 at 2sec)

Figure 15. Three phase voltage (100 to 50 at 1sec and 50to 10 at 2sec)

5. CONCLUSION

The design methodology of fuzzy based modeling forPMSG drive is presented. The Takagi-Sugeno basedsingleton fuzzy model is developed and simulated for

load disturbances and its responses are compared withmathematical model responses. From the simulation it is

observed that both the responses are matching closely.

REFERENCES

[1] Sergey Edward Lyshevski, ElectromechanicalSystems, Electric Machines, and AppliedMechatronics, CRC Press, FL2001.

[2] Kai Zhang, Hossein M. Kojabadi and Peter Z.Wang,Liuchen Chang, "Modeling of a Converter-Connected Six-Phase Permanent Magnet

Synchronous Generator", IEEE PEDS 2005, pp.1096-1100.

[3] Aleksandr Nagorny and Narajan V. Dravid, Highspeed Permanent magnet synchronous

motor/generator design for flywheel applications,National research council associateship award,

Cleveland, Ohio, 2005.[4] Sergey Edward Lyshevski and Alberico Menozzi,

Control of Permanent magnet brushless DC motor,Proceedings of the American Control Conference,Arlington, VA June 25-27, 2001,pp. 2150-2155.

[5] B. Dehkordi, A. M. Gole and T. L MaguirePermanent Magnet Synchronous Machine Model

for Real- Time Simulation, InternationalConference on Power Systems Transients (IPST05),Montreal, Canada, June 19-23, 2005

Mathematical model

Fuzzy model

Fuzzy model

Mathematical model

Mathematical model

Fuzzy model


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[6] Kazuo Tanaka and Hua O. Wang, Fuzzy ControlSystem Design and Analysis: A linear MatrixInequality Approach, John Wiley & sons, 2001

[7] P. C. Sen, "Electric Motor Drives and Control Past,Present, and Future", IEEE Transactions onIndustrial Electronics, vo1.37, No.6, pp.562-575,1990.

[8] R. Krishnan, "Selection Criteria for Servo MotorDrives", IEEE Transactions on IndustryApplications, Vo1.M-23, No.2, pp. 270-275, 1987.

[9] G. R. Slemon, Electric Machines and Drives,Addison-Wesley Publication Company, pp. 503511,1992.

[10]L. A. Zadeh, Outline of a new approach to theanalysis of complex systems and decision processes,IEEE Trans. SMC, SMC3, pp. 28-44, 1973.

[11]M. Sugeno and T. Yasukawa, A fuzzy-logic-basedapproach to qualitative modeling.IEEE Trans. Fuzzy

Systems, FS-1, pp. 7 - 3 1, , l993.[12]Zhu Wenbiao and Sun Zengqi, Fuzzy modeling for

complex process, IEEE Proc. of the 3rd

World

Congress on Intelligent Control and Automation, pp1271-2175, 2000.

[13]T. Takagi and M. Sugeno, Fuzzy Identification ofSystems and Its Applications to Modeling andControl, IEEE SMC, vol.15, no.1, pp.116-132,

1985.[14]Kuang-Yow Lian, Chian-Song Chiu, Tung-Sheng

Chiang, and Peter Liu Secure Communications ofChaotic Systems with Robust Performance via FuzzyObserver-Based Design, IEEE Transactions onfuzzy systems, Vo1.9, No.1, pp. 212-221, 2001.

[15]V. Natarajan, P. Kanagasabapathy, N. Selvaganesanand P. Natarajan, Fuzzy observer design with n-shift

multiple key for cryptography based on 3-D hyperchaotic oscillator,Iranian Journal of Fuzzy Systems,Vol. 3, No. 2 pp. 21-32, 2006.

[16]Zhang Bo, Li Zhong and Mao Zong Yuan, A typeof fuzzy modeling of the chaotic system ofPermanent Magnet Synchronous MotorIEEE Conf.,pp 880-883.

[17]Jacek Kabzifski, Fuzzy modeling of disturbancetorques/forces in rotational/linear interior permanent

magnet synchronous motors, Proc. On EPE-2005,pp 1-10.

Appendix I

PMSG Specification

No. of phases 3

No. of poles 4

Rated Voltage (V) 43 V

Rated Current 6.9 ARated Power 135 W

Stator Resistance (rs) 0.5

Self Inductance (Lss) 0.01 H

Flux Linkage (m) 0.069 N-m-A-1

Viscous Friction co-

efficient (Bm)

0.0006 N-m-s-rad-1

Moment of Inertia (J) 0.0075 Kg-m2

Rated Speed () 399 rad-s-1

Model H234-G Kollmorgen

a simple fuzzy modeling of permanent magnet synchronous generator

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