7/30/2019 Excitation Requirements of Three Phase Self-excited Induction Generator Under Single Phase Loading With Minimum Unbalance

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EXCITATION REQUIREM ENTSOF THREE PHASE SEL F-EXCITED INDUCTI ONGENERAT OR UNDER SINGL E PHASE L OADING WTHMI NIMUM UNBALANCEA.I. Alolah Majeed A. AlkanhalEE Dept., K ing Saud University, P.O.Box 800,Riyadh 11421, Saudi Arabia

Absfract: This paper presents an optimization method todetermine the excitation requirements of three phase self-excitedinduction generator under single phase mode of operation.Asingle phase load is connGcted to the generator through twoexcitation capacitors. The values of these capacitorsare chosento ensure minimum self-excitation of the machine in addition tominimization of the unbalaqce between the stator voltages. Theproblem is formulated as a multidimensional optimizationproblem. A sequential genetic (GA)/gradient optimizer is used tominimize a cost function of the summation oftheequivalentimpedance of the generator plus the ratio of the negative to thepositive sequence voltages, to obtain the valuesof the frequencyand the excitation reactances. These values are then used todetermine the other performance of the machine. A classicgradient solver initialized by a simpleGA isused to solve for theunknown parametersin the formulated optimization problem.Kqwordr : : nduction generators, Excitation requirements,Singlephasemode, unbalance f a c to r , Selfexcitation.

Listof Symbolsp.u phasor value of voltageandcurrent, respectivelypuN valueof V and f , respectivelyp.u impedanceand admittance, respectivelyp.u resistanceandreactance,respectivelyIZ and I 1 respectivelyp.u saturatedand unsaturatedmagnetizingreactancesat basefrequency, respectiveyAir gapvoltagesBasevoltage,current and impedance, respcctivclyBasefnquency andspeedinHz andrpm. respectivelyp.u frequencyand speed, respectivelyIL90"and IL120". respectivelyunbalancefactor I VJVp

Subscriptspositive,negativeandzerosequences,respectivelystatorphasesrotor and stator, respectiveyload

I. Introductionphase self-excited induction generators arebecoming more and more popular due to variousadvantages over the synchronous generators. Squirrel cageinduction generators are cheap, robust, freefrom currentcollecting problems, self protected against large overloads and short circuit faultsand can be operatedtroublefree for many years [1,2]. Therefore, self-excitedinduction generators are considered well suited forgenerating electricity from nonconventional energysources and for supplying electrical energy in remote andrural areas [3-51. Nearly all the research work has beencarried out on three phase self-excited induction

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generators are directed towards the analysis of themachine under three phase balanced conditions [1-61.Most of the loads in remote and rural areas aresinglephase loads. Recently, attempts have been done to analyzethree phase self-excited induction generator under singlephase loading by using just one excitation capacitor [7,8].The machine under this case is unbalanced due tothevoltage and current negative sequence components.Unbalanced operation leads to voltage stresses and overheat in the machine. However, three phaseself-excitedinduction generator can be operated under minimumunbalanced condition even with single phase load byusing two excitation capacitors.Thi s paper deals with the analysis of athree phase self-excited induction generator under single phase load usingtwo excitation capacitors. The values of these elementsare chosen to ensure self-excitation of the machine and tominimize the unbalance between the stator voltages. Theanalysis of this system iscarried out using the principle ofsymmetrical components. The resulted main equations areformulated in a multidimensional optimization problemand solved by anumerical optimization technique, whereno detailed derivation of analytical equations is needed.AGA initialized-Gradientoptimizer is utilized to minimize acost function of the summation of the equivalentimpedance of the generator plus the unbalance factor, toobtain the frequency and the value of the excitationreactances. Once these values are known, otherperformanceof the machine can be easily obtained.

11.AnalysisSingle phase load can be supplied froma three phase self-excited induction generator as shown in Fig.1. As thegenerator frequency varies with the load, speed andexcitation Fapacitors, the circuit is referred to the basefrequency. Voltage and curtent equations of the systemunder study of Fig. 1 are:vu+v* +v, =0VL - vu=0v, +v, =0Vb 4 v, =0I L - I u+ ,- I ,= 0I h +I , - ,- I , =0V, =- j I IX l /FJV,=- I , xz/FJAccording to the three phase symmetrical components,stator voltages and currents are:

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(9)whereg stands for voltage orcurrent.

Fig.1 Connectionofsingle phase load to three phase self-excitedinduction generator with twoexcitation capacitorsAs can be realized from eqns. (1) and(9), zero sequencevoltage and consequently current are zero in the aboveconnection.Moreover and as shown in Fig.2, sequence voltages andcurrents are related as follows:Ip=VJZ,' VYpI" =V,/Z,= v,Y"wherez,=z*+m/izq z,=zs+, / /z ,

Z= j XmZ,= R, / (F+u)+XZs =RJF+ j X,2, =R, / (F-U)+XSolving equations (I)-( 11) yields:

I ,= Y,[ (l -a)Yp - j a F 2 /X 2 ]

where D, =XIX, f l p$. Y,) +j XI+X)From eqns. (12) and (13), the unbalance factor isgivenbv: )I( J s I= xlx,Y p+-{ax2-x1}.-~~x1x2Y,+Q{x2-ax1)( 6Also the input admittance of the machineas seen acrossthe load is:

Fig2Positive sequence equivalent circuitof thegeneratorIII.ProblemFormulation

From Fig.1 and under self excitation:I LZ,=0 (17)whereZ =( ZL+Z,), Z,=I / U, and ZL= L/F +XLSinceIL O , t implies that Z, =0.Also to ensure minimum unbalance, the unbalance factor(U)must be minimum.The above problem is formulated as an optimizationproblem according to the following formula:Minimize(@ whereQ is given by :wherea isa weighting parameter (0 c a cl).The goal is to find the correct values for the variablesF ,X, and X2 that will attain the minimum valueof Qforevery load and speed. To ensure minimum excitationrequirements, the value of X, is taken to be equal to theunsaturated value (X,) 31.The proposed method to solve for thethree unknowns inthe above formula is a sequential numerical optimizer.This optimizer consists of two parts. First, a simplestandard genetic algorithm (GA) is used to bring thevalues of the unknowns close to the region of the(optimality) solution. Then aclassical constrained solverwil l take over tomodify and finetunethe valuesof theseunknowns. This sequential optimizer takes advantageofthe capability of GA to locate the region of the optimalsolution without being trapped in a local minimum. Inaddition, it avoids theGA slow convergence. The gradientsolver will swiftly converge to the correct solution valuessince it is properly initialized by the GA close to thesolution region.

Q=a Z I +( l - a)U (18)

W.ResultsThe data of the machine under study are given in theAppendix [7]. Fig.3 shows the variation of the value of thetwo reactances ofthe excitation capacitors against the loadimpedance, for different values of lagging power factor.The reactance across the lagging phase isless than that ofthe leading phase. Both reactances increase as the loadimpedance increases: The variation of the unbalance0-7803-5935-6/00/$10.00 c) 2000 IEEE 25 8

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factor Uagainst the load impedance, if the values ofX ,and X, of Fig.3 are used to excite the generator, is shownin Fig.4. The factor increases as the load impedanceincreases. The factor also increases as the power factordecreases. I t s maximum value does not exceed7% at 0.5load power factor.

0. 00

-0.522

-0.70 pf = 1.0 A pfs0.8 v pfz0.51 I I I

\\ \

-0.90 pf=l.O A pf=0.8 V pfsO.5I I I 1

-1.8ipfs1.0 A pf=0.8 v pfz0.5I-2.6 I 1 I I i

0 8 ZL , .u 16(b)Fig.3 Variation of the excitation reactances with the load impedance atdifferent power factors (pf) , (a)X, and (b) X,

V. C onclusionsThis paper presents an optimization method to determinethe excitation requirements of three phase self-excitedinduction generator under single phase mode of operation.A single phase load is connected to the generator throughtwo excitation capacitors. The valuesof these capacitorsare chosen to ensure minimum self-excitation of themachine in addition to minimization of the unbalancebetween the stator voltages. A sequential genetic(GA)/gradient optimizer is used to minimize a costfunction of the summationof the equivalent impedanceofthe generator plus the unbalance factor. From thisoptimization the valuesof the frequency and the excitationreactances are obtained. These values are then used todetermine the other performance of the machine. Theresults indicate that (i) the excitation requirementsincreases with the decrement in load impedance, (ii) boththe excitation requirements and the unbalance factor

increase as the load power factor decreases, and (iii) themaximum value of the unbaIance factor does not exceed7% at0.5 load power factor (lagging).

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