Papers of Control Systems

8/7/2019 Papers of Control Systems

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2/35Department of Electrical Engineering, JNTUCEA 185


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1. Speed Control of Induction Motor using Optimal Fuzzy Gains

scheduling

Abstract

In this paper, a optimal fuzzy gain scheduling of PIcontroller is adopted to speed control of an inductionmotor. First, a designed fuzzy gain scheduling of PIcontroller is investigated, in which fuzzy rules areutilized on-line to adapt the PI controller parametersbased on the error and its first time derivative.However, the major disadvantage of the fuzzy logiccontrol is the lack of design techniques, for thispurpose we propose an optimization technique of thefuzzy logic adapter parameters using geneticalgorithm. The effectiveness of the complete proposed

control scheme is verified by numerical simulation.The numerical validation results of the proposedscheme have presented good performances comparedto the fuzzy controller which have parameters chosenby the human operator.

Keywords: fuzzy logic, genetic algorithm, PIcontroller, adaptation, optimization, and vectorcontrol.

1. Introduction

The control of the AC electric machines

known a considerable development and apossibility of the real time implantationapplications. It is widely recognized that theinduction motor is going to be the main actuatorfor industrial purposes. Indeed, as compared tothe DC machine, it provides a betterpower/mass ratio, a simpler maintenance andrelatively lower cost. However, it istraditionally for a long time, used in industrialapplications that do not require highperformances, this because its control is a morecomplex problem, its high nonlinearity and itshigh coupled structure. Furthermore, the motor

parameters are time-varying during the normaloperation and most of the state variables are notmeasurable. Since Blashke and Hasse havedeveloped the new technique known as vectorcontrol, the use of the induction machinebecomes more and more frequent. This controlstrategy can provide the same performance asachieved from a separately excited DC

machine, and is proven to be well adapted to all type ofelectrical drives associated with induction machines.

The most widely used controller in the industrialapplications is the PID-type controllers because of theirsimple structures and good performances in a wide rangeof operating conditions. In control by fuzzy logic, thelinguistic description of human expertise in controlling aprocess is represented as fuzzy rules or relations. Thecontrollers based on fuzzy logic (FLC) can be consideredas non-linear PID controller where their parameters aredetermined on-line based on the error and its derivative.However, this standard FL controller cannot react tochange in operating conditions. The FL controller needs

more information to compensate nonlinearities when theoperation conditions change. When the number of thefuzzy logic inputs is increased, the dimension of the rulebase increases too. Thus, the maintenance of the rule baseis more time-consuming. Another Disadvantage of the FLcontrollers is the lack of systematic, effective and usefuldesign methods, which can use a priori knowledge of theplant dynamics. In this paper, to overcome disadvantages of PID controllers and FLC, a combinationbetween them together. PID parameters controller can betuned on-line by an adaptive mechanism based on a fuzzylogic for induction machine speed control. However, themajor drawback of fuzzy control is the lack of design

technique. Most of the fuzzy rules are human knowledgeoriented and hence rules will deviate from person toperson in spite of the same performance of the system.The selection of suitable fuzzy rules, memberfunctions and their definitions along the universe ofdiscourse always involve a painstaking trial-and errorprocess. GA most known and is most largely employed inthe technique of global research with a capacity to exploreand exploit a given operation space using measurement of the available performance. Recently ofmany applications combining the fuzzy concepts and GAappeared, particularly, the use of GA for the fuzzy logicsystems control design. Thus approaches are cagenetic-fuzzy system. In this paper a techniqueoptimize the parameters of fuzzy adapter of PI controlleris discussed; the controller resulting from combination is known on the name: adaptive FLC-PI-GAin order to apply it to the speed control of the inductionmachine.

A fuzzy gain scheduling of conventional PI controller isinvestigated, in which the fuzzy logic system is used on-line to

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generate the PI controller parameters. Then, anoptimal fuzzy gain scheduling of PI controller isbeing designed in which an optimization techniqueusing genetic algorithm is developed to optimize thefuzzy logic controller. Finally, the combinedproposed controller was applied for induction motorspeed control through a numerical simulation.

2. Indirect field-oriented control of

induction motor

The dynamic model of three-phase, Y-connectedinduction motor can be expressed in thed-qsynchronously rotating frame as:

(1)

Where is the coefficient of dispersion and is givenby:

(2)

Ls,Lr,Lm stator, rotor and mutual Inductances;Rs , Rr stator and rotor resistances; e , r electrical and rotor angularFrequency;sl slip frequency ( e r) ;r rotor time constant ( Lr/ Rr) ;

p

pole pairs

The main objective of the vector control of inductionmotors is, as in DC machines, to independently control thetorque and the flux; this is done by using a d-q rotatingreference frame synchronously with the rotor flux spacevector. In ideally field-oriented control, the rotor fluxlinkage axis is forced to align with the d-axes, and itfollows that

=0 (3)

(4)

Applying theresult of (3) and (4), namely field oriented control, the

torque equation become analogous to the machine and can be described as follows:

(5)

And the slip frequency can be given as follow:

(6)

Consequently, the dynamic equations (1) yield:

(7)

The decoupling control method with compensation is to chooseinverter output voltages such that:

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3.The speed control of the IM by an

adaptive controller FLC-pi

In this paper PID parameters controller can beadjusted by an adaptive mechanism based on a fuzzy

inference (adaptive FLC-PI).

3.1. Fuzzy gain scheduling of PI controller

Gain scheduling means a technique where PIcontroller parameters ( k p and ki gains) are tunedduring control of the system in a predefined way. Itenlarges the operation area of linear controller (PI) toperform well also with a nonlinear system .Thediagram of this technique is illustrated in fig. 1. Thefuzzy inference mechanism adjusts the PI parametersand generates new parameters during process control,so that the FLC adapts the PI parameters to operatingconditions based on the error and its first timedifference.

Fig. 1 PI control system with fuzzy gain adapter.

3.2. Description of the fuzzy scheduler

The parameters of the PI controller used inthe direct chain k p and ki are normalized into therange between zero and one by using the followinglinear transformations :

The inputs of the fuzzy adapter are: The errore and

the derivative of error e , the outputs are : thenormalized value of the proportional action ( kp )and the normalized value of the integral action ( ki).The problem of selecting the suitable fuzzy controllerrules remain relying on expert knowledge and try anderror tuning methods. The parameters k pand ki aredetermined by a set of fuzzy rules of the form:

Ife is Ai, and e is Bi, then ' k p is Ci, and ' ki is Di.(12)Where Ai, Bi ,Ci and Di are fuzzy sets on

corresponding supporting sets.The fuzzy rules in (12) may be extracted

operators expertise or based on the step response of the

process.. By using the membership functions, we havethe following conditions

The fuzzy outputs k p and kican be calculated by thecentre of area defuzzification as:

Fig 2 shows the block diagram of the indirect fieldoriented control by an adaptive controller FLCPI.

4. Speed control of IM with an optimal fuzzy gain

scheduler of pi controller

4.1. Genetic Algorithms

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GAs are parallel and global search techniqueswhich take the concepts from evolution theory andnatural genetics to evolve solutions to problems. The

basic idea is to maintain a population of chromosomes(representing candidate solutions to

the concrete problem being solved) that evolves overtime through a process of competition and controlledvariation. GAs is theoretically and empiricallyproven to provide robust search in complex spaces,giving a valid approach to problem requiring efficientand effective searching. Evaluation of individual fitness; Formation of gene pool (intermediate population); Recombination and mutation.

4.2. Design of fuzzy-genetic system

4.2.1. Membership parameters optimization

GA is applied to modify the membershipfunctions. When modifying the membershipfunctions, these functions are parameterized with oneto four coefficients (Fig. 3), and each of thesecoefficients will constitute a gene of the chromosomefor the GA.

Fig. 3 Some parameterized membership functions

4.2.2. Fuzzy rule base optimization

GA is used to modify the decision table of anFLC, which is applied to control a system with two

input

(trial-and error) and one input (command action)variables. A chromosome is formed from the decisiontable by going row-wise and coding each outputfuzzy set as an integer in 0, 1 n, where n is thenumber of membership functions defined for theoutput variable of the FLC. Value 0 indicates thatthere is no output, and value kindicates that theoutput fuzzy set has the k-th membership.

4.2.3. Optimization algorithm using GA of the

Fuzzy adapter

The application of the GA in the optimizationprocess of the FL controllers can be formulated asfollows:1. Start with an initial population of solutions thatconstitutes the first generation (P (0)).2. Evaluate P (0):a) Take each chromosome (KB) from the population andintroduce it into the FLC,b) Apply the FLC to the controlled system for anadequate evaluation period,c) Evaluate the behavior of the controlled system byproducing a performance index to the KB.3. While the termination condition is not met, do

a) Create a new generation (P(t+1)) by applying theevolution operators (selection, crossover and mutation)to the individuals in P(t),b) Evaluate P (t+1)

c) t = t+1.4. End.The mechanism of this optimization procedure can berepresented in fig. 4.

Fig. 4 Evolutionary learning of an FLC

The fuzzy adapter consists of two inputs (error and itsderivative) and two outputs ( kp) and ( ki), where eachinput has seven membership functions. These subsets arelabeled by linguistic terms such as: Zero (Z), Negative

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(N)... etc. GA is used to search the appropriateparameters values and to modify the decisions tableof the FLC, where the chromosome is formed fromthe decision table and to code each membershipfunction by a integer number from 0 to 2, number 2indicates the number of membership function definedfor the two outputs. So, we can present the equivalentcode by: Small (S): 1, Big (B): 2 and No output: 0. InGA, we only need to select some suitable parameters,such as generations, population size, crossover rate,mutation rate, and coding length of chromosome,then the searching algorithm will search out aparameter set to satisfy the designer's specification orthe system requirement. In this paper, GA will beincluded in the design of fuzzy gains tuner of the PIcontroller. The parameters for the GA simulation areset as follows: The parameters for the GA simulationare set as follows:(1) Initial population size: 30;(2) Maximum number of generation: 100;(3) Crossover: Uniform crossover with probability0.8;(4) Mutation probability: 0.01.In this paper, the performance is measured using thefollowing criteria.(5) Minimum integral of squared which is given asfollows:

Fig. 5 shows the tuning scheme of PI controlleradapted by a fuzzy system where their parameters are

optimized by the genetic algorithm.

4.2.4. Results of optimization procedure

The results obtained for the parameters optimizationof the membership functions are represented in fig. 5to fig. 7.

Fig. 5 Membership Fig. 6 Membership functionsof functions e e

The resulting rule bases from the optimization procedureare shown in table I and II. In the tables for example, thefirst rule for the output kpand kis:If

eis A11 And

e is A21 So

kp

is B12 and

ki

is B21

Where B12 is the second fuzzy set of thConsequent ( kp) and B21 is the first fuzzy set of thesecond consequent ( ki). We can re-write this rule as:Ife is NB And e is NB So kp B and ki

is S.

Fig. 7 Membership functions of kp and kiTable I: rule bases of the output kp

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5. Simulation results

To prove the rightness and effectiveness ofproposed control scheme, we apply the designedcontroller to the control of the induction motor. Itmainly consists of an induction motor, a ramp

comparison current-controlled pulse width modulated(PWM) inverter, a slip angular speed estimator, aninverse park, an outer speed feedback control loop anda fuzzy gain scheduling of PI controller or fuzzy gainscheduling of PI controller optimized by GA for thespeed control. Fig.8 to Fig.10 show the varioussimulation results.

Fig. 8 The IM rotor speed control with adaptive FLC-PI optimized by GA for two different Rr.

Fig. 9 The IM rotor speed control with adaptiveFLC-PI optimized by GA for two different J.

Fig. 10 Simulated results comparison of adaptive PI usingfuzzy inference and adaptive PI using fuzzy inferenceoptimized by GA of IM

6. Conclusion

In this work, we proposed a method ofcombination between the fuzzy controller aconventional PI controller in order to overcomedisadvantages of PI controllers and FLC, this combination

gave us an adaptive PI controller which presatisfactory performances (no overshoot, minimal risetime, best disturbance rejection). The major drawback ofthe fuzzy controller is the insufficient analytical designtechnique (choice of the rules, the membership functionsand the scaling factors). That we chose with the use of thegenetic algorithm for the optimization of this controller inorder to control IM speed. In the system, GA is used todesign an adaptive PI controller using fuzzy controllerwith optimal parameters. The optimal fuzzy gscheduling of PI controller is used to achieve robustperformance against parameter variations and externaldisturbances. Simulation results have shown that proposed optimal controller is robust with regard parameter variations and external load disturbance (noovershoot, minimal rise time, best disturbance rejection).Finally, the proposed controller provides drive robustnessimprovement and assures global stability.

Appendix:

References:

[1] R. D. Lorenz and D. B. Lawson: A SimplifiedApproach to Continuous On-Line Tuning of Field-Oriented Induction Machine Drives, IEEE Trans. On

Industry application, Vol/26, Issue/3, May/June (1990).

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2. FAULT DETECTION FOR POWER TRANSFORMERS

USING NEURAL NETWORKS

NAGIREDDY RAVI1 N. Yadaiah2MIEEE1 Mini Hydel Stations, Pochampad, Andhra Pradesh Power Generation Corporation Limited, POCHAMPAD 500

082, A. P., INDIA, E-mail:[email protected] Dept. of Electrical and Electronics Engineering, College of Engineering ( Autonomous), Jawaharlal Nehru

Technological University, ANANTAPUR- 515 002, A. P., INDIA, E-mail:[email protected]

Abstract- This paper presents the methodology forincipient fault detection in Power transformers usingartificial neural networks. An artificial neuralnetwork is used for identifying the practicaltransformer faults. The conventional Dissolved GasAnalysis method to detect incipient faults has beenimproved using artificial neural networks and iscompared with Rogers ratio method with availablesamples of field information.

Key words: Incipient faults, Power Transformers,Neural Networks.

I. INTRODUCTION

The transformer is an important componentin power system network. The failures of thetransformer hinder the proper operation of electricalsystem network and its reliable operation is animportant factor. Any fault in the transformer sidecauses power outages and blackouts, thereby hampersthe power quality. Power quality is one of the mainconcerns of the utilities and inferior power qualitycauses many problems such as malfunctions,instabilities, short lifetime and so on, hence tomaintain power quality is most important. Thereplacement of power transformer is very expensiveand time consuming and therefore it is essential todetect incipient faults as early as possible, whichenable rectification of fault with minimuminterruption of service.

The faults that occur within transformerbroadly classified into two types, one is an internalincipient faults and other is an internal short circuitfaults. The majority of incipient faults occurring inpower transformer give evidence very early in theirdevelopment stages through the transformer oil gasanalysis. The transformer may function externally, it may cause serious problem with someinsulation deterioration. The fault detectiontechniques such as Dissolved Gas Analysis (DGA)[1-2] and partial discharge analysis [3] are currentlyused for detection of incipient faults. In this paper, amethod based on DGA has been improved usingArtificial Neural Networks (ANN) to identify thefaults. It is an off-line technique and requires DGAreport. In addition to this, it is high in cost and timeconsuming. The performance of the ANN basedmethod has been compared with the existing methodsuch as Rogers ratio by using the field information ofDGA samples from Andhra Pradesh (IndDistribution, Transmission, Generation Companies.

II. PROBLEM FORMULATION

In this paper, the problem of identificationof internal incipient faults in power transformer hasbeen considered. Currently, DGA based methods are

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being used, but these cannot provide accurate results.The uncertainty in the process of sampling which isdue to improper collection of oil samples, transport ofoil samples to laboratory. The errors in measuringand analyzing in the laboratory due to leaks in thevacuum system during gas extraction andcontamination of columns used in gaschromatograph. The involvement of group discussionby experts is necessary to achieve a conclusion and tomake decision for disassembly. Due to this timeconsuming and tedious procedure, automatic methods

described below have proposed for identification ofincipient faults in the power transformers.

An ANN method based DGA has been usedto overcome the limitations of existing methods. Toimplement the proposed method, the DGA data of oilsamples for different transformers in Andhra Pradesh

(India) Generation, Transmission and DistributionCompanies, which contains no fault and fasamples, has been collected. This data has been usedto train the ANN and the performance of this methodhas been compared with conventional method.

III. ARTIFICIAL NEURAL NETWORKS

Artficial neural networks composed ofsimple elements operating in parallel [7]. Theseelements are inspired by biological nervous system.An artififial neuron with activation function is shownin Fig.1.

Fig .1 Artificial neuron with activation function

The NET is weighted input of neuron and OUT is the

output of neuron. An activation function is appliedfor each neuron in the network. There are differenttypes af actvation functions such as linear, piecewise,hardlimiter, unipolar sigmoid, bipolar sigmoid etc.

IV. FAULT DIAGNOSIS USING DGA

DGA is an earliest tool for the detection ofincipient faults in transformers. Hydrocarbon oils areused as insulating fluids in transformers because oftheir high dielectric strength, heat transfer propertiesand chemical stability. Insulating material willdecompose under the stresses of thermal and

electrical. These generate gaseous decompositionproducts, which dissolve in mineral oil. The natureand the amount of the individual component gasesextracted from the oil may be indicative of the typeand degree of abnormality. The gases thataregenerated and dissolved in oil of the transformer are

shown in Table I. The ratios of these gases are usedfor interpretation of faults.

TABLE IDISSOLVED GASES IN OIL.

Formula of the gas Name of the gasC2H2 AcetyleneC2H4 EthyleneCH4 MethaneH2 Hydrogen

C2H6 Ethane

There are two commonly used conventionalratio analysis methods namely Doernenburg ratiosmethod and Rogers ratios method [1]. These tworatio analysis methods have been used for DGA todetect the incipient faults in power transformer. TheDGA data provides information about the conditionof the transformer and advance warning developing faults.

A. Doernenburg Ratios Method

This method utilizes gas ratios listed inTable II and suggests the existence of three types offault types such as (i) Thermal fault, (ii) Corona and(iii) Arcing [1]. This table is formulated

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dissolved gases of transformer oil. The limitation ofthis method is that, it can detect only three faults andis very complex.

B. Rogers Ratios MethodThis method has been proposed by Rogersand utilizes gas ratios listed in Table III and it revealsfive different types of faults such as (i) Lowtemperature thermal fault, (ii) Medium temperaturethermal fault (iii) High temperature thermal fault, (iv)Partial discharges and (v) High Energy arcing. It isone method most commonly used for diagnosis of thetransformer incipient faults.

TABLE II

GAS RATIOS FOR DORENENBERG RATIOS

METHOD

Ratio1 (R1) CH4/H2

Ratio2 (R2) C2H2/C2H4Ratio3 (R3) C2H2/CH4Ratio4 (R4) C2H6/ C2H2

TABLE III

GAS RATIOS FOR ROGERS RATIOS

METHOD

Ratio1 (R1) CH4/H2Ratio2 (R2) C2H2/C2H4Ratio3 (R3) C2H4/ C2H6

The code generation of each Rogers ratiosfor classification is described in Table IV and it wasdefined by Rogers [4]. The coding of each fault typewith respect to their characteristics is given in TableV.

TABLE IV

CLASSIFICATION TABLE BY ROGERS

C2H2/ C2H4 CH4/H2 C2H4/C2H6

Range ofGas

Ratios0 1 0 3.0

TABLE V : TYPES OF FAULTS

Characteristic Fault Type Code ofcharacteristicgases

No Fault 0 0 0Low temperature fault (7000C) 0 2 2Low energy partial discharges 0 1 0High energy partial discharges 1 1 0Low energy discharges 1-2 0 1-2High energy discharges 1 0 2

C. ANN based fault detection

An Artificial Neural Network is good toolfor the applications like pattern classification, faultdetection; function approximation due to inherent

characteristics of learning, generalization, faultolerance etc. In this paper it has been used for off-line fault detection of power transformers incipientfault with knowledge of DGA of transformer oilsamples. The problem of fault detection is a potentialapplication of ANN [5-6].A multi-layer feed-forwardartificial neural network has been proposed identify the fault and a generalized delta rule hasbeen used for training of network. Implementationinvolves the generating of training data, and design ofANN structure, training of network and testing. Inthis application, gas ratios become the inputs forneural network and types of faults becomeoutputs of the network. To start with, a three layernetwork has been considered, in which size of inputlayer is equal to number of gas ratios and size ofoutput layer is equal to the number of different faultsand the nodes in the hidden layer are selected basedon experience and a schematic diagram of a threelayer network is shown in Fig. 2.

Fig .2 Multi-layer feed-forward Neural Network.

D. Algorithm

Step 1: The data of oil samples of dissolved gasanalysis of various normal and faulty transformers

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are collected. The data collected is used for trainingthe neural network.Step 2: The faulty type samples have been confirmedby actual faults.Step 3: The ANNs are established on these samplesusing multi-layer back-propagation.Step 4: The inputs are the gas ratios of the DGA testreport , the outputs are the fault types for the neuralnetwork.Step 5: The trainining of the network is carried outwith all training pairs from the training set byapplying the input vector to the network input.Step 6: Calucate the output of the networkStep 7: Calculate the error between the networkoutput and the desired output i.e, the target vectorfrom the traing set.Step 8:Adjust the weights of the network forminimizing the error.Step 9: Repeat steps 1 through 4 for each vector inthe training set until the error for the entire set isacceptably low.Step 10: The trained network is tested using test data.

D. Simulation Results

In order to illustrate these methods such as Rogersratios method and ANN based method, experimentalresults of DGA for various power transformers havebeen collected from the Andhra Pradesh PowerDistribution, Transmission, and Generationcompanies. The input vector consists of three gasratios such as (i) C2H2/C2H4, (ii) CH4/H2, and (iii)C2H4/C2H6 and the output vector involves fiveelements (i) No fault, (ii) Low Temperature fault

(1500C - 3000C), (iii) Medium Temperature fault(3000C 7000C), (iv) High temperature fault(>7000C) and (v) Low energy partial discharge. Thecollected samples are listed in Table VI withrespective fault type and these samples are used totrain ANN. The fault type samples were confirmedby the actual faults in the internal examination of thetransformer.

TABLE VI : COMPOSITION OF THE TRAINING

DATA

FaultCode

Fault Type TrainingSamples

0 No fault (NF) 4001 Low temperature fault(150oC-300oC) (LTF)

25

2 Medium temperature fault(300 0C-7000C) (MTF)

14

3 High temperature fault(>7000C) (HTF)

20

4 Low partial energydischarge (LPD)

13

The ANNs were trained with 472 trainingsamples. The three gas ratios are considered as inputsand five nodes in output layer to identifdifferent types of faults, in which the nodes are set to

1 for respective fault and 0 for no fault. Aftertraining, the network is tested with typical samples and the results are summarized in Table VII.The performance of both the methods is comparedand the results show that the ANN method detectsmore accurately than Rogers ratio, method as shownin Table VII.

Table VII : MPARISON OF ANN METHOD

WITH ROGERS RATIO METHOD

Sa

m

ple

no

C2

H2

C2H4

CH4

H2 C2H6

ANN

Method

RogersRati

omethod

ActualFault

1 1.09

125

25.08

27.21

169.1

NF NF NF

2 0 0.98

4.86

21.54

113

NF NF NF

3 1.98

0.67

3.12

11.82

120.8

NF UNIDENTIFIABLE

NF

4 0 145.6

5.92

69.05

23.09

NF UNIDENTIFIABLE

NF

5 0.23

2.67

74.59

24.28

74 LTF

LTF LTF

6 3.86

8.74

76.93

34.88

52.89

LTF

UNIDENTIFIABLE

LTF

7 1 166

67.25

11.6

96.23

MTF

MTF

MTF

8 1.8

456.2

90.32

54.17

2.19

HTF

HTF HTF

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9 0.34

60.32

39.31

45.42

7.92

HTF

UNIDENTIFIABLE

HTF

10 3.

97

15

4.5

9 29

7

21

1.67

LP

D

LPD LP

D

V. CONCLUSIONS

An ANN method has been developed forfault detection of power transformers. Theperformance of ANN method has been tested andcompared with Rogers ratios method and found that,ANN detects more accurately.

ACKNOWLEDGMENT

The authors would like to thank thecomments from the reviewers. Also would like to

thank the authorities of JNTU College ofEngineering, Anantapur, India and Power GenerationCorporation Limited, Hyderabad, India for providingall infrastructural facilities.

REFERENCES

[1] IEEE guide for the Interpretation of GasesGenerated in Oil-Immersed Transformers,ANSI/IEEE std.C57.104, 1991.

[2] C.E. Lin, J.M. Ling and C. L. Huang, An expertsystem for transformer fault diagnosis andmaintenance using dissolved gas analysis IEEETrans. on Power Delivery, Vol.8, No.1, 1993, pp231-238.

[3] IEEE trial-use guide for the detection of AcousticEmissions from partial discharges in oil-immersed Power Transformers, IEEE std.C57.127-2000.

[4] Y. C. Huang and Hong-Tzer Yang, Developing aNew transformer fault Diagnosis System throughEvolutionary fuzzy logic, IEEE Trans. onPower Delivery, Vol. 12, No.2, 1997, pp 761-767.

[5] R. P. Lippmann, An Introduction to Computingwith Neural Nets, IEEE ASSP magazine, Vol.4, No.2, 1987, pp 4-22.

[6] N. Yadaiah, L.Sivakumar, B.L.Deekshatulu andV.Sri Hari Rao, "Neural Network Architecturesfor Describing Nonlinear Input-Output

Relations", Electronic Modeling, Vol. 24, No. 3,2002, pp 48-61.

[7] N. Yadaiah, L.Sivakumar, Nagireddy Ra"Design of Load Frequency Control in PowerSystems Using Neural Networks",ElectricEnergy Systems Management Proc. Indian

Scenario, Roorkee, Sep 4-5, 1998, pp 134 - 138

[8] MATLAB Documentation Wavelet Tool Box,Ver. 7, The Mathworks Inc., Natick, MA.

N. Yadaiah received B.E. in Electrical andElectronics Engineering from College ofEngineering, Osmania University, Hyderabad, India,in 1988. M.Tech. in Control systems from IITKharagpur, India in 1991 and Ph.D. in Electrical andElectronics Engineering from Jawaharlal NehruTechnological University, Hyderabad, India in 2000.He received Young Scientist Fellowship (YSF) ofAndhra Pradesh State Council for Science Technology, in 1999. He is currently a Professor inElectrical and Electronics Engineering at JawaharlalNehru Technological University. He is Fellow ofInstitution of Engineers (India), Institution Electronics & Telecommunications Engineers

(India), Member of IEEE, Life Member of IndianSociety of Technical Education, Life Member ofSystems Society of India. His research includes inthe areas of Artificial Neural Networks, Fuzzy Logic,System identification, fault identification, AdaptiveControl, Nonlinear Systems and process control.

N. Ravi received B.E. in Electrical & ElectronicsEngineering from College of Engineering, JawaharlalNehru Technological University Hyderabad, India, in1998 and M.Tech. in Power Electronics from thesame College in 2006. He is currently working as an

Assistant Divisional Engineer in Andhra PradeshPower Generation Corporation Ltd., Andhra Pradesh,Hyderabad, India.

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3. Design of Fuzzy Controller for a Robot arm: A systematic

ApproachB.Krishna Madhuri

Abstract

Most fuzzy controllers have been treated andused as black-box controllers in the sense that theiranalytical structures are unknown. Knowing theexplicit structure information will enable one toinsightfully understand how a fuzzy control works. Inthe present paper a novel technique is presented forderiving input out put relations for the fuzzy

controller that uses Zadeh AND operator forsymmetrical triangular input fuzzy sets. The roboticarm control with and without the gravitational effectis simulated to know the efficacy of the fuzzycontroller using symmetrical input membershipfunctions.

Index Terms-- Fuzzy controller, triangular fuzzysets, robot arm, gravitational torque.

I. INTRODUCTION

Fuzzy controllers are constructed generally via

heuristic approaches as opposed to the mathematicalapproaches exclusively used in conventional control.The fuzzy controllers have been treated and used asblack box controllers. With out analytical structureinformation, precise and effective mathematicalanalysis and design are very difficult to achieve.Availability of the structure information may lead to

less trail and error effort and produce better controlperformance.

In literature analytical structures are derivedmainly for fuzzy controllers using Zadeh Aoperator [1]-[5]. Revealing the analytical structure ofa fuzzy controller that uses Zadeh AND operator isfar more difficult even for triangular input fuzzy sets,which are simplest fuzzy sets because this operator

requires the comparison of membership functions.

Recently that is extended to derive therelationship between input space divisions neededdue to the use of Zadeh AND operator and inputfuzzy sets [6][7].

In this paper an analytical structure forsymmetrical class of input fuzzy sets is developed.The robotic arm control is simulated to know theefficacy of the fuzzy controller.

II. DEVELOPMENT OF A GENERALTECHNIQUE

A Mamdani type fuzzy controller, whichemploys two input variables, is taken. The two inputvariables are e (n) and r (n) and system output is y(n). The scaled error and change of error of y (n) are

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E(n) = Ke e(n) = Ke (SP(n)-y(n))

R(n) = Kr r(n) = Kr (e(n)-e(n-1))

Where Ke and Kr are scaling factors and

SP (n) is output command signal. E (n) and R (n) arefuzzified by triangular fuzzy sets. The membership

functions for E (n) and R (n) are noted by E i, E i+1, R j and R j+1respectively.

The rules are:

Rule 1: IF E (n) is Ei+1 AND R (n) is RJ+1THEN u(n) is H1

Rule 2: IF E (n) is Ei+1 AND R (n) is RjTHEN u (n)is H2

Rule 3: IF E (n) is Ei AND R (n) is Rj+1THEN u (n)is H3

Rule 4: IF E (n) is Ei AND R (n) is RjTHEN

u (n) is H4

Zadeh fuzzy logic AND operator is used to realizethe AND operations in the rules.

The popular centroid defuzzifier is employed whichyields

4321

44332211uu

hhhhKu(n)KU(n)+++

+++== .

where Ku is the scaling factor and U (n) is theoutput of fuzzy controller.

III. ANALYTICAL STRUCTURE FOR

TRIANGULAR MEMBERSHIP FUNCTIONS:

Mathematical definitions of the fuzzy sets are

+

=+

][ 1 5 ,1 ,

1 51 5 ,[,3 0

1 5E ( n )

1 5,(0 ,

1iE

+

=+

][ 1 2 ,1

1 21 2 ,[,2 4

1 2R ( n )

1,(0 ,

1jR

Over all input space divisions of all the four rules dueto Zedah AND operator are shown in Fig. 1.

Fig. 1. Over all input space division for all the four

rules

For each region a unique analytical in erelationship can be obtained for each fuzzy rulebetween two membership functions being ANDed.

Table I: Rule Firing

Analytical structure of the fuzzy controller isobtained by putting these results into the defuzzifier,

and it is tabulated in Table II

IV. CASE STUDY

A system representing the complete modal forRobot arm control is shown in Fig. 2. with fuzzy PDcontroller. The disturbance Tgris due to gravity.Simulated results are shown in Fig. 3 to Fig. 5.

Fig. 2 Block diagram for the control of robot armusing fuzzy controller

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Table II Analytical Structure

Fig. 3 Step response without disturbance

A. Effect of gravitational torque:

Introducing the gravity torque Tgr = 21 sin for thesystem considered in fig.4., there are two cases to beanalyzed.

Fig. 4 Response with disturbance

B. Performance Analysis:

From the graphs it can be observed that the risetimeand settling time for the fuzzy controller usingsymmetrical input fuzzy sets are less compared torise time and settling time for the conventcontroller. I.e. fuzzy controller gives the

response.C. When arm is moving down:

The effectiveness of controller when the arm ismoving down is shown in Fig. 5. It is observed thatfuzzy controller response compared to conventionalcontroller is better in this case also.

Fig. 5. Response when arm is moving down

V. Conclusions

In this paper analytical structure of fuzzycontroller using symmetric triangular fuzzy sets isdeveloped. And it is applied to control the roboticarm. Results are compared with the conventionalcontroller. Finally from the simulated results it can beconcluded that the system using fuzzy controller isfast responding irrespective of the presence disturbance torques.

V. REFERENCES

[1] H. Ying, W. Siler, and J.J. Buckley,control theory: A Non-linear case,Automatica,vol.26, pp.513-520, 1990.[2] A. E. Hajjaji and A. Rachid, Explicit formulasfor fuzzy controller, Fuzzy Sets Syst., vol. 62, pp.135-141, 1994.[3] W. Li, Design of a hybrid fuzzy proportional plus conventional integral-derivativecontroller, IEEE Trans. Syst., vol. 6, no. 4, pp. 449-463, Nov. 1998.

[4] G.K. Mann, B.G. Hu, and R. G. Analysis of direct action fuzzy PID contrstructures IEEE Trans. Syst., Man, Cybern, B.Cybern., vol. 29, pp.371-388, 1999.[5] Hao Ying, A General Technique for DerivingAnalytical Structure of Fuzzy Controllers UsiArbitrary Trapezoidal Input Fuzzy Sets and Zadeh

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AND Operator. Automatica, Vol .39, pp. 1171-1184, 2003.[6] K.A. Gopala Rao, K.R.Sudha, AnalyticalStructure of Three Input Fuzzy PID Power SystemStabilizer with Decoupled Rules, WSEAS Trans. onCIRCUITS and SYSTEMS, pp.965-970, June 2004.

[7] K.A. Gopala Rao, K.R.Sudha, AnalyticalStructure of PID Fuzzy Logic Power System

Stabilizer Proceedings of IASTED Modelingsimulation, and Optimization MSO2004 August 2004

VI. ACKNOWLEDGEMENTS

I thank my guide Dr. K.A. Gopala Rao, Professor andHead, Department of Electrical Engineering f

introducing me to this area of research.

4. Load Frequency Control Using Fuzzy Gain Scheduling

of PI Controller

A.Sreenath1, Y.R.Atre2.1P.G.Student, Electrical Engineering dept, Walchand College of Engg, Sangli.

2Electrical Engineering dept, Walchand College of Engg, Sangli (M.H-416415).

E-Mail Address: [email protected]@gmail.com

Abstract

In this paper, a fuzzy gain scheduledproportional and integral (FGPI) controller wasdeveloped to regulate and to improve the frequencydeviation in a two-area electrical interconnectedpower system. Also, a conventional proportional andintegral (PI), and a fuzzy logic (FL), controllers wereused to control the same power system for theperformance comparison. Two performance criteriawere utilized for the comparison. First, settling timesand overshoots of the frequency deviation werecompared. Later, the absolute error integral analysismethod was calculated to compare all the controllers.

The Simulation results show that the FGPI controllerdeveloped in this study performs better than the othercontrollers with respect to the settling time andovershoot, and absolute error integral of thefrequency deviation.

Keywords-Two area power system; Load-Frequencycontrol; Fuzzy logic controller

1. Introduction.

Large scale power systems are normally composed ofcontrol areas or regions representing coherent groupsof generators. The various areas are interconnectedthrough tielines. The tie-lines are utilized contractual energy exchange between areas aprovide inter-area support in case of abnoconditions. Area load changes and abnormconditions, such as outages of generation, lead tomismatches in frequency and scheduled powinterchanges between areas. These mismatches haveto be corrected through supplementary control.

Load Frequency Control (LFC) of interconnectedsystems is defined as the regulation of power outputof generators within a prescribed area, in response tochange in system frequency, tie-line loading, or therelation of these to each other; so as to maintain

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scheduled system frequency and/or establishedinterchange with other areas within predeterminedlimits [1].

Many investigations have been reported in thepast pertaining to load frequency control of a multi-area interconnected power system. In the literature,some control strategies have been proposed based onclassical linear control theory .However, because ofthe inherent characteristics of the changing loads, theoperating point of a power system changescontinuously during a daily cycle. Thus, a fixedcontroller may no longer be suitable in all operatingconditions. There are some authors who have appliedvariable structure control [3] to make the controllerinsensitive to system parameters change. However,this method requires information on the system stateswhich are very difficult to know completely. In viewof this, a new area load frequency controller based onfuzzy gain scheduling of PI controller is proposed inthis paper. Gain scheduling is a technique commonlyused in designing controller for non-linear systems.Its main advantage is that controller parameters canbe changed very quickly in response to changes in thesystem dynamics because no parameter estimation isrequired. Besides being an effective method tocompensate for non-linear and other predictablevariations in the system dynamics, it is also simplerto implement than automatic tuning or adaptation.However, the transient response can be unstablebecause of abruptness in system parameters. Besides,it is impossible to obtain accurate linear timeinvariant models at variable operating points. Somefuzzy gain scheduling of PI controllers have been

proposed to solve such problems in power systems[4] and [5] that developed different fuzzy rules forthe proportional and integral gains separately. Fuzzylogic control presents a good tool to deal withcomplicated, non-linear and indefinite and time-variant systems [6]. In this paper, the rules for thegains are chosen to be identical in order to improvethe system performance. The comparison of theproposed FGPI, the conventional PI controllers, andthe fuzzy logic controller suggests that the overshootsand settling time with the proposed FGPI controllerare better than the rest.

2. Two area power system.An interconnected power system can be

considered as being divided into control areas whichare connected by tie lines. In each control area, allgenerators are assumed to form a coherent group. Thepower system is subjected to local variations ofrandom magnitude and duration. Hence, it is required

to control the deviations of frequency and tie-linepower of each control area.

An uncontrolled two-area interconnected powersystem is shown in Figure 1 where, fis the systemfrequency (Hz), iR is regulation constant (Hz/per

unit), gTis speed governor time constant (sec), tT

is turbine time constant (sec) and pT is powersystem time constant (sec).

The overall system can be modelled as a multi-variable system in form of

)()()( tdLtuBtxAx ++= , (1)

Where A is the system matrix, B and L are input anddisturbance distribution matrices, x(t), u(t) and d(t)are state, control and load changes disturbancevectors respectively.

X(t)=[f1 Pg1 Pd1 Ptie12 f2 Pg2 Pd2 ]T

[ ]21)( uutu = T

[ ]21)( dd PPtd = T,

where denotes deviation from the nominal values.

1u and 2u are the control outputs inFigure1. The system output, which depends on areacontrol error (ACE) shown as

)()(

)()(

2

1

2

1

txCA

A

ty

tyty (2)

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iiitiei fbPACE += , ,(3)

Where bi is the frequency bias constant, if is the

frequency deviation and itieP, is the change intie-line power for areai and Cis the output matrix [4].

Fig.1: Two Area Interconnected System

area model parameters of Fig.1 are defined in theappendix.

3. Fuzzy logic in power systems.

Fuzzy set theory and fuzzy logic establishthe rules of a nonlinear mapping [6]. The use of fuzzysets provides a basis for a systematic way for theapplication of uncertain and indefinite models [4].Fuzzy control is based on a logical system calledfuzzy logic is much closer in spirit to human thinkingand natural language than classical logical systems[5,6]. Nowadays fuzzy logic is used in almost allsectors of industry and science. One of them is load-frequency control [2]. The main goal of load-frequency control in interconnected power systems isto protect the balance between production andconsumption. Because of the complexity and multi-variable conditions of the power system,conventional control methods may not givesatisfactory solutions.

The fuzzy controller for the single input, singleoutput type of systems is shown in Fig. 2 [3].

In this figure, Kp and Ki are the proportional andintegral gains, respectively. The fuzzy controllerinput can be the derivative of e together with thesignal E. The fuzzy controller block is formed byfuzzification of E, the inference mechanism and

defuzzification. Therefore, Y is a crisp value, and u isa control signal for the system.

Fig.2. The simple fuzzy controller

4. Fuzzy gain scheduled PI controller.

Gain scheduling is an effective way of controllingsystems whose dynamics change non-linearly withoperating conditions [4]. It is normally used when therelationship between the system dynamics aoperating conditions are known, and for which asingle linear time-invariant model is insufficient. Inthis paper, we use this technique to schedule theparameters of the PI controller according to changeof the new area control error ACE, and ACE, asdepicted in Fig. 3.

Fig.3. The scheme of fuzzy gain scheduling.

By taking ACE as the system output, the control

vectors for the conventional PI and I controllers,respectively can be given in the following forms:ui = -KPACEi- Ki (ACEi)dt

= - KP(Ptie,i+bifi) - Ki(Ptie,i+bifi)dt

Fuzzy logic shows experience and preference throughmembership functions, which have different shapesdepending on the experience of system experts. Sameinference mechanism is realized by seven rules forthe two FGPI and the FL controllers. The appropriaterules used in the study are given in Table 1.

Fig.5. Membership functions for FL Controller of

(a) ACE, (b) ACE, (c) Kp, Ki

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Fig.6. Membership functions for FGPI Controller

of (a) ACE, (b) ACE, (c) Kp, Ki

Membership functions shapes of the errorand derivative error and the gains are chosen to beidentical with triangular function for both fuzzy logiccontrollers. However, their horizontal axis ranges aretaken different values because of optimizing thesecontrollers. The membership function sets of FL forACE, ACE, Kp and Ki are shown in Fig. 5, whilethe ones for FGPI controller are shown in Fig.6.Defuzzification has also been performed by thecenter of gravity method in all studies.5. Simulation study.Simulations were performed using the conventionalPI, Fuzzy Logic (FL) and the proposed FGPIcontrollers applied to a two-area interconnectedelectrical power system. The same system parametersgiven in Tables 2 were used in all controllers for acomparison.

Two performance criteria were selected in thesimulation. The frequency deviation graphs were firstplotted with Matlab 7.0-Simulink software. Here,settling times and overshoots of the frequencydeviation of the controllers were compared againsteach other. The comparison results are provided in

Table 2 and 3.

Fig 7. a, b, c, d shows the responses for frequencydeviation of area1 (f1)p.u (Pd1=0.01 p.u.).

f11

Time(sec)Fig a. Without Controller

Fig 8. e,f shows the responses for Change inmechanical power in area1(Pm1).(ii)Change inmechanical power in area2(Pm2).Change in Tielinepower (Ptie).

f11

Controller

Frequency Deviation in area 1(f1)

Settlingtime(sec) (for

5% band

of the stepchange)

MaximumOvershoot

(HZ)

FGPI 3.2 -0.013

FLC 6.2 -0.022

ConventionalPI

4.9 -0.024

209

Time(sec)

Fig b. With PI Controller

f1

Time(sec)Fig c. With Fuzzy Logic Controller

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Time(sec)Fig d. With FGPI Controller

Time(sec)

Fig e. Without Controller

Table 1

Table 2

System performances for all controllers onsettling times and overshoots for frequency deviationof area1.

System performances for all controllers withsteady state error for frequency deviation of area1.

6.Conclusion.

In this paper, a new fuzzy gain scheduling ofPI controller was investigated for automatic load-frequency control of a two-area interconnectelectrical power system. In the simulations, horizontal ranges of membership functions of the FLand the two FGPI controllers were taken differentlyin order to decrease the oscillations of frequencydeviation in all areas. The proposed controller is verysimple and easy to implement, since it does notrequire any information about the system parameters.According to the experimental results, it performs

significantly better than other controllers in settling time and absolute error integral while itperforms closer in the overshoot magnitude. conclusion, the proposed fuzzy gain scheduling PIcontroller is recommended to generate good qualityand reliable electric energy.

Appendix.

Two-area power system parameters:Tg=0.08 R1=2.4 R2=2.4 Tp=20 Tt=0.3 12=1

Controller

Frequency Deviation inarea 1 (f1)

Steady state error(ess)

FGPI -0.000067

FLC -0.00383

Conventional PI -0.00136

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References.

[1]. Demiroren A, Yesil E.Automatic generationcontrol with fuzzy logic controllers in the power systemincluding SMES units. Electr Power Energy Syst 2004;26:291305.

[2]. C am E, Kocaarslan I. Load frequency control in twoarea power systems using fuzzy logic controller. EnergyConversion Manage 2005; 46:23343.

[3].Meliopoulos APS, Cokkinides GJ, BakirtzisAG.Load-frequency control service in a deregulatedenvironment. Decision Support Syst 1999; 24:24350.

[4]. Talaq J, Al-Basri F. Adaptive fuzzy gain schedulingfor load frequency control. IEEE Trans Power syst 1999;

14(1):14550.[5]. Chang CS, Fu W. Area load-frequency control usingfuzzy gain scheduling of PI controllers. Electr Power systRes 1997; 42:14552.

[6]. Tang KS, Man KF, Chen G, Kwong S. An optimalfuzzy PID controller. IEEE Trans Ind Electron 2001;48(4):75765.[7].CC. Lee, Fuzzy logic in control systems: Fuzzy logic

controller, parts I and II.IEEE Trans. Sqst., ManCybern... 20 (2) (1990).

[8]. El-Sherbiny MK, El-Saady G, Yousef AM.Efficientfuzzy logic load-frequency controller. Energy ConversMgmt 2002; 43:185363.

[9].Chaturvedi DK, Satsangi PS, Kalra PK. Loadfrequency control: A generalised neural networkapproach. Electr Power Energy Systems 1999;21:40515.

[10]. Wang Y, Zhou R, Wen C. Robust load-frequencycontroller design for power systems. IEE Proc Control1993;140(1):1116.

5. Optimal Power Flow by Enhanced Genetic Algorithm

V.B.VIKRAM.D UMME SALMA, ASSISTA

AbstractThis paper presents an enhanced geneticalgorithm (EGA) for the solution of the optimalpower flow (OPF) with both continuous and discretecontrol variables. The continuous control variablesmodeled are unit active power outputs andgenerator-bus voltage magnitudes, while the discreteones are transformer-tap settings and switchable

shunt devices. A number of functional operatingconstraints, such as branch flow limits,load busvoltage magnitude limits, and generator reactivecapabilities, are included as penalties in the GAfitness function (FF). Numerical results on ieee testsystem are presented and comparision of results

with normal genetic algorithm and enhanced geneticalgorithmare shown.

NOMENCLATURE

Bus voltage angle vector.

UL Load (PQ) bus voltage magnitude vector.

PG Unit active power output vector.

UG Generation (PV) bus voltage magnitude

vector.

t Transformer tap settings vector.

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bsh Bus shunt admittance vector.

X System state vector.

U System control vector.

A hat above vectors and denotes that the entry

corresponding to the slack bus is missing. For

simplicity of notation, it is assumed that there is

only one generating unit connected on a bus.

I. INTRODUCTION

Since its introduction as network constrainedeconomic dispatchby Carpentier and its definitionas optimal power flow(OPF) by Dommel andTinney , the OPF problem has been the subject ofintensive research. The OPF optimizes a powersystem operating objective function (such as theoperating cost of thermal resources) while satisfyinga set of system operating constraints, includingconstraints dictated by the electric network. OPF hasbeen widely used in power system operation andplanning . After the electricity sector restructuring,OPF has been used to assess the spatial variation ofelectricity prices and as a congestion managementand pricing tool .In its most general formulation, theOPF is a nonlinear, nonconvex, large-scale, staticoptimization problem with both continuous anddiscrete control variables. Even in the absence of

nonconvex unit operating cost functions, unitprohibited operating zones, and discrete controlvariables, the OPF problem is no convex due o theexistence of the nonlinear (AC) power flow equalityconstraints. The presence of discrete controlvariables, such as switchable shunt devices,transformer tap positions, and phase shifters, furthercomplicates the problem solution.

The literature on OPF is vast, and [1] presentsthe major contributions in this area. Mathematicalprogramming approaches, such as nonlinear

programming (NLP), quadratic programming (QP)[2], [3], and linear programming (LP) [4],[5], havebeen used for the solution of the OPF problem.Some methods, instead of solving the originalproblem, solve the problems KarushKuhnTucker(KKT) optimality conditions. For equality-constrained optimization problems, the KKTconditions are a set of nonlinear equations, which

can be solved using a Newton-type algorithm. InNewton OPF [6],the inequality constraints are addedas quadratic penalty terms to the problem objective,multiplied by appropriate penalty multipliers.Interior point (IP) methods, convert the inequalityconstraints to equalities by the introduction nonnegative slack variables. A logarithmic barrierfunction of the slack variables is then added to theobjective function, multiplied by a barrierparameter, which is gradually reduced to zeduring the solution process. The unlimited pointalgorithm uses a transformation of the slack anddual variables of the inequality constraints whichconverts the OPF problem KKT conditions to a setof nonlinear equations, thus avoiding the heuristicrules for barrier parameter reduction required by IPmeth ods.

OPF programs based on mathematicalprogramming approaches are used daily to solve

very large OPF problems.However, they are notguaranteed to converge to the global optimum of thegeneral nonconvex OPF problem, although thereexists some empirical evidence on the uniqueness ofthe OPF solution within the domain of interest Toavoid the prohibitive computational requirements ofmixed-integer programming, discrete controlvariables are initially treated as continuous, andpost-processing discretization logic is subsequentlyapplied Whereas the effects of discretization onload tap changing transformers are small and usuallynegligible, the rounding of switchable shunt devicesmay lead to voltage infeasibility, especially whenthe discrete VAR steps are large, and requiresspecial logic.The handling of nonconvex OPFobjective functions, as well as the unit prohibitedoperating zones, also present problems tomathematical programming OPF approaches.

Recent attempts to overcome the limitationsof the mathematical programming approaches andgenetic algorithms (GAs) [7], [8].

In [7], a simple genetic algorithm (SGA) isused for OPF solution. The control variablmodeled are generator active power outputs andvoltages, shunt devices, and transformer tapsBranch flow, reactive generation, and voltagemagnitude constraints are treated as quadraticpenalty terms in the GAfitness function (FF).To

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keep the Ga chromosome size small, only a 4-bitchromosome area is used for the encoding of eachcontrol variable. A sequential GA solution schemeis employed to achieve acceptable control variableresolution. Test results on the IEEE 30-bus system,comprising 25 control variables, are presented.

In [8], a GA is used to solve the optimalpower dispatch problem for a multinode auctionmarket. The GA maximizes the total participantswelfare, subject to network flow and transportlimitation constraints. The nodal real and reactivepower injections that clear the market are selected asthe problem control variables. A GA with twoadvanced operators, namely, elitism and hillclimbing, is used. A 10-bit chromosome area isdevoted to each control variable. Test results on a17-node, 34-control variable system are presented.

This paper presents an enhanced geneticalgorithm (EGA) for the solution of the OPF. Thecontrol variables and constraints included in theOPF and the penalty method treatment of thefunctional operating constraints are similar to theones in [7] with the following improvements:switchable shunt devices and transformer taps aremodeled as discrete control variables. Variablebinary string length is used for different types ofcontrol variables, so as to achieve the desired

resolution for each type of control variable, withoutunnecessarily increasing the size of the GAchromosome. In addition to the basic geneticoperators of the SGA used in [7] and the advancedones used in [8], problem-specific operators,inspired by the nature of the OPF problem, havebeen incorporated in our EGA. With theincorporation of the problem-specific operators, theGA can solve larger OPF problems. Test results onIEEE 30-buses system with 24 control variablesdemonstrates the improvement achieved with the aidof problem-specific operators.

II. OPTIMAL POWER FLOW PROBLEM

FORMULATION

The OPF problem can be formulated as mathematical optimization problem as follows:

Min(x,u) (1)

S.t.g(x,u) = 0 (2)

h(x,u) 0 (3)

u U

where

x=[ ULT ] T .

u=[PGT UGT tT bshT ] (6)

The equality constraints (2) are the nonlinear powerflow equations. The inequality constraints (3) are

the functional operating constraints, such as

Branch flow limits (MVA, MW or A)

load bus voltage magnitude limits;

Generator reactive capabilities;

Slack bus active power output limits.

Constraints (4) define the feasibility region of theproblem

Control variables such as

Unit active power output limits;

Generation bus voltage magnitude limits;

Transformer-tap setting limits (discrete values);

Bus shunt admittance limits (continuous discrete

Control).

III. GENETIC ALGORITHMS

GAs are general purpose optimizationalgorithms based on the mechanics of natuselection and genetics. They operate on strin

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structures (chromosomes), typically a concatenatedlist of binary digits representing a coding of thecontrol parameters (phenotype) of a given problem.Chromosomes themselves are composed ofgenes.The real value of a control parameter, encoded in agene, is called an allele..

Gas are an attractive alternative to otheroptimization methods because of their robustness.There are three major differences between GAs andconventional optimization algorithms. First, GAsoperate on the encoded string of the problemparameters rather than the actual parameters of theproblem. Each string can be thought of as achromosome that completely describes onecandidate solution to the problem. Second, GAsuse a population of points rather than a single pointin their search. This allows the GA to exploreseveral areas of the search space simultaneously,reducing the probability of finding local optima.Third, GAs do not require any prior knowledge,space limitations, or special properties of thefunction to be optimized, such as convexity,. Theyonly require the evaluation of smoothness, the so-called fitness function (FF) to assign a quality valueto every solution produced.

Assuming an initial random population producedand evaluated, genetic evolution takes place by

means of three basic genetic operators:

1) Parent selection;

2) Crossover;

3) mutation.

Parent selectionis a simple procedure whereby twochromosomes are selected from the parentpopulation based on their fitness value. Solutionswith high fitness values have a high probability

Of contributing new offspring to the nextgeneration. The selection rule used in our approachis a simple roulette-wheel selection [9]. Crossoverisan extremely important operator for the GA.It isresponsible for the structure recombination(information exchange between mating

chromosomes) and the convergence speed of the GAand is usually applied with high probability (0.60.9). The chromosomes of the two parents selectedare combined to form new chromosomes that inheritsegments of information stored in parentchromosomes.

Fig. 1. Simple genetic algorithm (SGA).

Until now, many crossover schemes, such as single

point,multipoint, or uniform crossover have been

proposed in the literature. Uniform crossover [9] as

been used in our implementation. While crossover is

the main genetic operator exploiting the information

included in the current generation, it does

produce new information. Mutation is the operator

responsible for the injection of new information.

With a small probability, random bits of

offspring chromosomes flip from 0 to 1 and vice

versa and giveconstraints . new characteristics that

do not exist in the parent population . In our

approach, the mutation operator is applied with a

relatively small Probability (0.0001-0.001) to every

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The FF evaluation and genetic evolution takepart in an iterative procedure, which ends when amaximum number of generations is reached, asshown in Fig. 1. When applying GAs to solve aparticular optimization problem (OPF in our case),two main issues must be addressed:

1) the encoding, i.e., how the problem physicaldecision variables are translated to a GAchromosome and its inverse operator, decoding ;

2) the definition of the FF to be maximized by theGA (the GA FF is formed by an appropriatetransformation of the initial problem objectivefunction augmented by penalty terms that penalize

the violation of the problem)

Fig. 2. GA chromosome structure.

IV. GENETIC ALGORITHM SOLUTION TO

OPTIMAL POWER FLOW

A. Encoding

The chromosome is formed as shown in Fig. 2.There are four chromosome regions (one for eachset of control variables),namely, 1) PG ;2)UG ; 3)tand 4) bsh . Encoding is performed using differentgene-lengths for each set of control variables,depending on the desired accuracy. The decoding of

a chromosome to the problem physical variables isperformed as follows:

1) continuous controls taking values in theinterval [uimin,uimax]

ui = uimin +( uimax _uimin).K/(2Nui -1) (7)

2) discrete controls taking values u i 1, ui 2.., ui m

ui = ui m with m=int[/(2Nui).K+1.5]

and log2M Nui log2M +1

(8)

where is the decimal number to which the binary

number in a gene is decoded and is the gene length

(number of bits) used for encoding control variable .

B. Fitness Function (FF)

GAs are usually designed so as to maximize theFF, which is a measure of the quality of eachcandidate solution. The objective of the OP

problem is to minimize the total operating cost (1).

Therefore, a transformation is needed to convertthe cost objective of the OPF problem toappropriate FF to be maximized by the GA. TheOPF functional operating constraints(3) are includedin the GA solution by augmenting the GA FF byappropriate penalty terms for each violatedfunctional constraint. Constraints on the controlvariables (4) are automatically satisfied by tselected GA encoding/decoding scheme (7) and (8).

Therefore, the GA FF is formed as follows:

NG NG

FF= A/( Fi( PGi) + wj.Penj) (9)i=1

Pen j=| hj(x,u) |. H (hj(x,u)) (10)where

FF fitness function;

A constant;

Fi( PGi) fuel cost function of unit (in our case, a

quadratic function);

wj weighting factor of functional operating215


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constraint;

Penj penalty function for functional operating

Constraint;j(x,u) violation ofjthfunctional operating

constraint, if positive;

H(.) Heaviside (step) function;

NG number of units;

NC number of functional operationconstraints.

Given a candidate solution to the problem,represented by a chromosome, the FF is computedas follows.

Step 1) Decode the chromosome to determine the

actual control variables, , using (7) and (8).

The computed control vector satisfies, by

design, constraints (4).

Step 2) Solve the power flow (2) to compute the

state vector,

Step3) Determine the violated functional constraints

(3) and compute associated penalty functions

Step 4) Compute the FF using (9). & (10)

In Step 2, a simple fast decoupled load flow (FDLF)is used with no PV-PQ bus-type switching, sincegenerator reactive capabilities are incorporated inthe functional operating constraints and no localcontrol adjustments, such as tap and switchable

shunts , since the settings of these controls are

determined by the GA. Therefore, only a few loadflow iterations are required for convergence. TheFDLF and matrices are formed andactorized only once in the beginningthe effect ofthe changes of shunt admittances on the matrix is

neglected. In case that, due to the random (yetwithin limits) initial selection of the contrvariables, the load flow does not converge within apredefined number of iterations (set to 8), largepenalty terms, proportional to the maximum

power mismatch, are added to the FF.

VII. C. ADVANCEDAND PROBLEM-SPECIFIC GENETIC

A. Operators

One of the most important issues in the geneticevolution is the effective rearrangement of tgenotype information. In the SGA crossover is themain genetic operator responsible for theexploitation of information while mutation bringsnew nonexistent bit structures. It is widerecognized that the SGA scheme is capable oflocating the neighborhood of the optimal or near-

optimal solutions, but, in general, requires a largenumber of generations to converge. This problembecomes more intense for large-scale optimizationproblems with difficult search spaces and lengthychromosomes, where the possibility for the SGA toget trapped in local optima increases and convergence speed of the SGA decreases. At thispoint, a suitable combination of the basadvanced,and problem-specific genetic operatorsmust be introduced in order to enhance performance of the GA. Advanced

Fig. 3. Gene swap operator.

and problem-specific genetic operators usuallycombine local search techniques and expertisederived from the nature of the problem.

A set of advanced and problem-specificgenetic operators has been added to the SGA in

order to increase its convergence speed and improvethe quality of solutions. Our interest was focused onconstructing simple yet powerful enhanced genetic

operators that effectively explore the problem searchspace. The advanced features included in our GAimplementation are as follows.

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1) Fitness Scaling: In order to avoid earlydomination of extraordinary strings and toencourage a healthy competition amongequals, a scaling of the fitness of thepopulation is necessary [9]. In ourapproach, the fitness is scaled by a linear

transformation.2) Elitism: Elitism ensures that the best solutionfound thus far is never lost when movingfrom one generation to another.The bestsolution of each generation replaces arandomly selected chromosome in the newgeneration [10].

2) Hill Climbing:In order to increase the GAsearch speed at smooth areas of the search

space a hill-climbing operator is introduced,which perturbs a randomly selected controlvariable. The modified chromosome is accepted ifthere is an increase in FF value;otherwise, the oldchromosome remains unchanged

This operator is applied only to the bestchromosome (elite) of every generation [8].Inaddition to the above advanced features, which arecalled advanced despite their wide use in mostrecent GA implementations to distinguish betweenthe SGA and our EGA, operators specific to theOPF problem have been added.

All problem-specific operators introducerandom modification to all chromosomes of a newgeneration. If the modified chromosome proves tohave better fitness, it replaces the original one in thenew population. Otherwise, the originalchromosome is retained in the new population. Allproblem-specific operators are applied with aprobability of 0.2. The following problem-specificoperators have been used.

1) Gene Swap Operator (GSO):This operator

randomly selects two genes in a chromosome andswaps their values, as shown in Fig. 3. This operatorswaps the active power output of two units, thevoltage magnitude of two generation buses, etc.Swapping among different types of control variablesis not allowed.

Fig. 4. Gene copy operator.

Fig. 5. Gene inverse operator.

Fig. 6. Gene max-min operator.

2)Gene Copy Operator (GCO):This operatorrandomly selects one gene in a chromosome andwith equal probability copies its value to predecessor or the successor gene of the samecontrol type, as shown in Fig. 4. This operator

has been introduced in order to force consecutive

controls (e.g., identical units on the same bus) tooperate at the same output level.

3) Gene Inverse Operator (GIO):This operatoracts like a sophisticated mutation operator. randomly selects one gene in a chromosome andinverses its bit-values from one to zero and viceversa, as shown in Fig. 5. The GIO searches for bit-structures of improved performance, exploitsnewareas of the search space far away from the

current solution, and retains the diversity of thepopulation.

4) Gene Max-Min Operator (GMMO):TheGMMO tries to identify binding control variableupper/lower limit constraints.It selects a randomgene in a chromosome and, with the s

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probability (0.5), fills its area with 1 s or 0 s, asshown in Fig. 6.

D. Enhanced Genetic Algorithm (EGA)

In the EGA, shown in Fig. 7, after the application of

the basic genetic operators (parent selection,crossover, and mutation) the advanced and problem-specific operators are applied to produce the newgeneration. All chromosomes in the initialpopulation are created at random (every bit in thechromosome has equal probability of beingswitched ON or OFF). Due to the decoding processselected [(7) and (8)], the corresponding controlvariables of the initial population satisfy theirupperlower bound or discrete value constraints (4).

However, the initial population candidate

solutionsmay not satisfy the functional operatingconstraints(3) or even the load flow constraints (2)since the random,within limits, selection of thecontrol variables may lead to load flow divergence(as already discussed in Section I V-B). Populationstatistics computed for the new generation includemaximum, minimum, and average fitness values andthe 90% percentile.

Fig. 7. Enhanced genetic algorithm (EGA).

V. TEST RESULTS

In this section, the proposed EGA solution of theOPF is evaluated using A. IEEE 30-Bus System

It has 41-branch system . It has a total of 24 controlvariables as follows: five unit active power outputs,six generator-bus voltage magnitudes, four

transformer-tap settings, and nine bus shuntadmittances. The input data for IEEE 30System are given in appendix A.The gene lengthfor unit power outputs is 12 bits and for generatorvoltage magnitudes is 8 bits. They are both treatedas continuous controls. The transformer-tap settings

can take 17 discrete values (each one is encodedusing 5 bits): the lower and Fig. 8. FF comparisonfor IEEE 30-bus system. upper limits are 0.9 p.u.and 1.1 p.u., respectively, and the step size is 0.0125p.u. The bus shunt admittances can take six discretevalues (each one is encoded using 3 bits): the lowerand upper limits are 0.0 p.u. and 0.05 respectively, and the step is 0.01 p.u. (on systemMVA basis). The GA population size is taken equalto 60, the maximum number of generations is 200,and crossover and mutation are applied with initial

probability 0.9 and 0.001, respectively. Two sets of20 test runs were performed; the first (SGA) withonly the basic GA operators and the second (EGA)with all operators, including advanced and problem-specific operators. The FF evolution of the best ofthese runs is shown in

Fig. 8. FF comparison for IEEE 30-bus

system

The best and worst solutions of the second set of 20runs (EGA) are shown in Table I. The operating

costs of the best and worst solutions are 802.73$/hand 802.34 $/h, respectively, The differencesbetween the values of the control variables in thebest and worst solutions are significant. Toperating cost of all EGA-OPF solutions is slightly

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less as shown in Table below

VII. CONCLUSIONS

A GA solution to the OPF problem has beenpresented and applied to small and medium sizepower systems. The main advantage of the GAsolution to the OPF problem is its modeling

flexibility: nonconvex unit cost functions, prohibitedunit operating zones, discrete control variables, andcomplex, nonlinear constraints can be easilymodeled. Another advantage is that it can be easily

coded to work on parallel computers. The maindisadvantage of GAs is that they are stochastic

algorithms and the solution they provide to the OPFproblem is not guaranteed to be the optimum.Another disadvantage is that the execution time andthe quality of the provided solution deteriorate withthe increase of the chromosome length, i.e., the OPFproblem size. The applicability of the GA solutionto large-scale OPF problems of systems with severalthousands of nodes, utilizing the strength of parallelcomputers, has yet to be demonstrated.

REFERENCES

[1] J. A. Momoh, M. E. El-Hawary, and R. Adapa,A review of selected optimal power flow literatureto 1993, IEEE Trans. Power Syst., pt. I and II, vol.14, pp. 96111, Feb. 1999.

[2] G. F. Reid and L. Hasdorf, Economic dispatch

using quadratic programming, IEEE Trans. PowerApparat. Syst., vol. PAS-92, pp. 20152023,

[4] R. C. Burchett, H. H. Happ, and K. A. Wirgau,Large-scale optimal power flow, IEEE Trans.Power Apparat. Syst., vol. PAS-101, pp. 37223732, Oct. 1982.

[5] B. Stott and E. Hobson, Power system securitycontrol calculation using linear programming,IEEE Trans. Power Apparat. Syst., pt. I and II, vol.PAS-97, Sept./Oct. 1978.

[6] R. Mota-Palomino and V. H. Quintana, Apenalty function- method for solving power systemconstrained economic operation problems, IEEETrans. Power Apparat. Syst., vol.,June 1984.

[7] D. I. Sun, B. Ashley, B. Brewer, A. Hughes,andW. F. Tinney, Optimal power flow by Newtonapproach, IEEE Trans. Power Apparat. Syst., vol.PAS-103, pp. 28642880, 1984..

[8] L. L. Lai, J. T. Ma, R. Yokoyama, and M. Zhao,

Improved genetic algorithms for optimal powerflow under both normal and contingent operationstates, Elec. Power Energy Syst., no. 5,1997.

[9] D. E. Goldberg, Genetic Algorithms in Search:Addison-Wesley, 1989. crossover landscape, inProc. 3rd Int. Conf. Genetic Algorithms, 1989,

[10] L. Davis, Handbook of Genetic Algorithms. NewYork: Van Nostrand, 1991.

[3] T. Numnonda and U. D. Annakkage, Optimalpower dispatch in multinode electricity market

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using genetic algorithm, Elec. Power Syst. Res.,

vol. 49, pp. 211220, 1999.

APPENDIX A

IEEE 30 BUS SYSTEM

No.of buses : 30

No.of lines :41

No.of generators : 6

Bus data:

Bus Type Pgen Qgen Pload Qload Vspecified Qmin Qmax Yshunt

1 Slack 1.3848

-0.0279

0 0 1.05 0 00

2 P-V 0.5756

0.0247 0.217

0.127

1.0338 -0.2 0.60

3 P-Q 0 0 0.02

4

0.01

2

1 0 0

0

4 P-Q 0 0 0.076

0.016

1 0 00

5 P-V 0.245 0.2257 0.94 0.19 1.0058 - 0.62 0

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6 2 0.15 5

6 P-Q 0 0 0 0.3 1 0 0 0

7 P-Q 0 0 0.228

0 1 0 00

8 P-V 0.35 0.3484 0.3 0.2 1.023 -0.15

0.50

9 P-Q 0 0 0 0 1 0 0 0

10 P-Q 0 0 0.058

0.75 1 0 00.19

11 P-V 0.1793

0.3078 0 0 1.0913 -0.1 0.40

12 P-Q 0 0 0.112

0.016

1 0 00

13 P-V 0.1691

0.3783 0 0.025

1.0883 -0.15

0.450

14 P-Q 0 0 0.062

0.018

1 0 00

15 P-Q 0 0 0.082

0.058

1 0 00

16 P-Q 0 0 0.035

0.009

1 0 00

17 P-Q 0 0 0.09 0.034 1 0 0 0

18 P-Q 0 0 0.032

0.007

1 0 00

19 P-Q 0 0 0.095

0.112

1 0 00

20 P-Q 0 0 0.022

0 1 0 00

21 P-Q 0 0 0.17

5

0.01

6

1 0 0

022 P-Q 0 0 0 0.06

71 0 0

0

23 P-Q 0 0 0.032

0 1 0 00

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24 P-Q 0 0 0.087

0.023

1 0 00.04

25 P-Q 0 0 0 0 1 0 0 0

26 P-Q 0 0 0.03

5

0 1 0 0

027 P-Q 0 0 0 0.00

91 0 0

0

28 P-Q 0 0 0 0.019

1 0 00

29 P-Q 0 0 0.024

1 0 00

30 P-Q 0 0 0.106

1 0 00

Pgmax and Pgmin for generators :

Generator

bus no

Pgmin Pgmax

1 0.5 2.0

2 0.2 0.8

5 0.15 0.5

8 0.1 0.3511 0.1 0.3

3 0.2 0.8

a, b, c constants for generators :

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Generator

No

a b c

1 0 2 0.00375

2 0 1.75 0.0175

3 0 1 0.0625

4 0 3.25 0.002075

5 0 3 0.025

6 0 3 0.025

Line Data :

Line From bus To bus R X Half line

charging

Suscepta

nce

Tap ratio Max

Power (pu)

1 1 2 0.0192 0.0575 0.0264 1 1.3

2 1 3 0.0452 0.1852 0.0204 1 1.3

3 2 4 0.057 0.1737 0.0184 1 0.65

4 3 4 0.0132 0.0379 0.0042 1 1.3

5 2 5 0.0472 0.1983 0.0209 1 1.3

6 2 6 0.0581 0.1763 0.0187 1 0.65

7 4 6 0.0119 0.0414 0.0045 1 0.9

8 5 7 0.046 0.116 0.0102 1 0.7

9 6 7 0.0267 0.082 0.0085 1 1.3

10 6 8 0.012 0.042 0.0045 1 0.32

11 6 9 0 0.208 0 1.0155 0.65

12 6 10 0 0.556 0 0.9629 0.32

13 9 11 0 0.208 0 1 0.65

14 9 10 0 0.11 0 1 0.65

15 4 12 0 0.2560 0 1.0129 0.65

16 12 13 0 0.1400 0 1 0.65

17 12 14 0.1231 0.2559 0 1 0.32

18 12 15 0.0662 0.1304 0 1 0.32

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19 12 16 0.0945 0.1987 0 1 0.32

20 14 15 0.2210 0.1997 0 1 0.16

21 16 17 0.0824 0.1923 0 1 0.16

22 15 18 0.1070 0.2185 0 1 0.16

23 18 19 0.0639 0.1292 0 1 0.16

24 19 29 0.0340 0.0680 0 1 0.32

25 10 20 0.0936 0.2090 0 1 0.32

26 10 17 0.0324 0.0845 0 1 0.32

27 10 21 0.0348 0.0749 0 1 0.32

28 10 22 0.0727 0.1499 0 1 0.32

29 21 22 0.0116 0.0236 0 1 0.32

30 15 23 0.1000 0.2020 0 1 0.16

31 22 24 0.1150 0.1790 0 1 0.16

32 23 24 0.1320 0.2700 0 1 0.16

33 24 25 0.1885 0.3292 0 1 0.16

34 25 26 0.2544 0.3800 0 1 0.16

35 25 27 0.1093 0.2087 0 1 0.16

36 27 28 0 0.3960 0 0.9581 0.65

37 27 29 0.2198 0.4153 0 1 0.16

38 27 30 0.3202 0.6027 0 1 0.16

39 29 30 0.2399 0.4533 0 1 0.16

40 8 28 0.0636 0.2000 0.0214 1 0.32

41 6 28 0.0169 0.0599 0.0650 1 0.32

224

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