MATLAB SYMBOLIC COMPUTATION FOR THE STEADY STATE MODELING OF SYMMETRICALLY LOADED SELF EXCITED INDUCTION GENERATOR. Gurung K., Freere P.

Transcription

1 TB BO OPUTTON O THE TED TTE ODENG O ET ODED E ETED NDUTON GENETO Gurung K., reere P. Department of Electrical and Electronics Engineering Kathmandu Uniersity, P.O.Box: 650, Kathmandu, Nepal orresponding uthor E-ail: krishna@ku.edu.np BTT This paper presents the use of atlab symbolic computation technique to model and simulate self excited induction generator. n this technique, the computer itself carries out both the tedious job of deriing the complex coefficients of the polynomial equations and soling them. Hence the modeling and programming becomes ery simple yet ersatile. Good agreement between the results obtained from the conentional method and that obtained using symbolic computation alidates the effectieness of this new technique. Key words: ymbolic computation, induction generator, self excitation. NTODUTON elf excited induction generators (EG) hae become ery popular in micro and pico hydro systems of Nepal. This is mainly because they are robust, easily aailable and inexpensie. They require little maintenance and hence are ery much suitable for remote area applications. Both the frequency and magnetizing retance of EG ary with load in order to maintain an ext balance of tie and retie power ross the air gap. Hence, it is a crucial step in the steady state analysis of a EG to determine the per unit frequency and the magnetizing retance m for gien mhine parameters, speed, excitation capitance and load impedance []. The balance of tie and retie power ross the air gap can be realized by equating the real and imaginary terms of total admittances, connected ross the terminal representing air gap, respectiely to zero. The usual prtice is to derie the complex coefficients of the non-linear equations manually and sole them using numerical methods. The mathematical manipulations required are tedious, time consuming and liable to human error. t requires tremendous human effort for curate programming and debugging. The model lks flexibility as the coefficients are alid only for a gien circuit configuration. nclusion of the core loss resistance or load inductance will increase the order of the equations. The aboe shortcomings can be oercome by the use of symbolic computation technique in atlab. The symbolic computation technique allows one to sole the EG goerning equations without haing to derie the complex coefficients of polynomial equations manually. single command sole can be used to sole multiple equations and the user does not need to bother about the numerical methods inoled. This makes the modeling and simulation of EG ery simple yet ersatile. Good agreement between the results obtained

2 from the conentional method and that obtained using symbolic computation alidates the effectieness of this new technique. teady state modeling of self excited induction generator ost of the steady state models of EG deeloped by different researchers are based on per phase equialent circuit. These models use the following two basic methods; i) oop impedance method and ii) Nodal admittance method. The steady state model based on nodal admittance method and used in [] is presented here. This model makes assumptions that the load is purely resistie, core loss component is neglected and the mhine parameters (except for magnetizing retance) remain constant. igure. Per phase equialent circuit of EG igure. aboe shows the per phase equialent circuit of EG, where the different symbols represent: s,, p.u. stator, rotor and load resistances respectiely s,,, p.u stator, rotor, magnetizing and excitation retances respectiely s,,,, p.u stator, rotor, magnetizing, load and excitation admittances respectiely. p.u.frequency p.u. rotor speed s,, p.u. stator, rotor and load currents respectiely Eg, Vt p.u. air gap and load terminal oltage respectiely The total current at node a in the aboe figure can be written as E ( ) 0..()

3 Where ( ) ( / ) ( / ) ( / ).() Under self-excitation E 0, therefore sum of total admittance connected ross the air gap must be zero i.e. 0..() s the admittances are complex quantities, the real and imaginary parts of equation can be equated to zero. Therefore, eal( ) 0 mag( ) 0..(4)..(5) or gien alue of shaft speed, generator parameters, excitation capitance and load impedance, solution of equation 4 gies the p.u. output frequency. The corresponding alue of magnetizing retance can then be found from equation 5 using the alue of obtained from 4. fter determining the alues of and, the air gap oltage Eg can be determined using the experimentally obtained magnetization cure, which relates Eg/ and. Now, different quantities can be calculated using the following relations describing figure. E g / E g / V t..(6) P in P out

4 onentional solution methods n the conentional methods, complex coefficients of equations 4 and 5 are manually deried and then soled using numerical methods. n the aboe example, equations 4 and 5 can be simplified to equations 6 and 7 respectiely (7) )] /( [..(8) where (9) ( ) [ ] bc ( ) bc bc bc Disadantages of conentional solution methods lthough the conentional methods are effectie in simulating the EG performance, they hae common disadantages as correctly pointed out by T.. han in [] and []. They can be listed out as:. ll the coefficients of the non-linear equations or a higher order polynomial need to be deried manually. The mathematical manipulations are tedious, time-consuming and prone to human errors.. The expressions for the coefficients are ery long and complicated, which require tremendous human effort for curate programming and debugging.

5 . The model lks flexibility as the coefficients are alid only for a gien circuit configuration. or example, inclusion of the core-loss resistance or the addition of compensation capitie retance will change the order of the equations. any researchers hae proposed different techniques to tkle these problems recently [-5]. Howeer they still require some degree of manual manipulation of the mathematical equations before these techniques are applied. This paper proposes a noel technique using TB symbolic computation, which eliminates all the aboe problems. ymbolic computation in atlab TB can compute on symbolic ariables just as on constants. This exempts one from the tedious job of manual manipulation of the complex equations to obtain the final two equations 7 and 8. t can also sole seeral simultaneous equations hence the user does not need to use numerical methods to sole the complex equations 7 and 8 obtained after the manipulation. ince TB performs both the jobs, steady state modeling and simulation of EG becomes ery simple and effectie [6]. ew simple examples below illustrate the symbolic computation that can be done in TB (ersion 6.5 or aboe). ommand eturns syms a b creates two symbolic ariables a & b. x (a b)^ x (a b) x expand(x) x a ab b y (a - b)^ y (a - b) y expand(y) y a - ab b z x^ y^ z (a ab b ) (a - ab b ) z simple (z) z a 4 a b b 4 Equation can be written down in the similar fashion. imilarly, the following commands can be used to sole the Z for a and b if the Z happens to be a complex quantity which is true for equation. ommand eturns - Equates real terms of Z to zero. zreal real(z) - assigns the equation name as zreal. zimag imag(z) - Equates imaginary terms of Z to zero. - ssigns the equation name as zimag.

6 [a,b] sole( zreal, zimag) oles zreal and zimag for a and b a double(a) eturns the numeric alues of a and b b double(b) n this way equation can be soled directly using atlab ymbolic omputation technique. This exempts one from haing to manually derie equations 7 and 8 and using numerical methods to sole them. EUT ND DUON n order to erify the alidity of the new technique the generator equialent circuit used in [] was simulated using atlab symbolic computation and the results were compared with the one obtained from the conentional method. The EG used for this purpose is a -phase, 4- pole, 60 Hz, kw, 80V,.7, -connected squirrel cage induction mhine whose per phase equialent circuit parameters in pu are: nd the magnetization cure is represented mathematically as E g ; 0 <. (0) < The oltage regulation cure of the EG for different loading at constant speed (rated) from both the methods are plotted in the figure below. 00 EG Voltage egulation cure Terminal oltage() symbolic computation onentional method with core loss oad current() igure. Voltage regulation cure of EG rom the figure, it can be seen that the results obtained using the symbolic computation technique is in good agreement with that obtained from the conentional method. t also demonstrates the effect of adding a core loss component to the EG equialent circuit. While the effect of core loss can be significant for more curate analysis, it cannot be included in the conentional method without significantly increasing the complexity in the mathematical manipulation required. On the other hand it can be included in the symbolic computation technique with much ease.

7 ONUON rom the aboe results it can be concluded that the atlab ymbolic computation technique is ery effectie for the simulation of EG. This technique has the adantage that there is no need to manually derie the complex coefficients of the polynomial equations. t also exempts one from using complex numerical methods to sole the polynomial equations. odeling becomes ery simple yet ersatile. ore loss and other component can be included easily. The programming and debugging become ery easy. EEENE. han T.., 995. nalysis of elf-excited nduction Generators Using an teratie ethod. EEE transtion on Energy onersion, 0(), nagreh.n. 00. Tehing the elf Excited nduction Generator using TB, nternational Journal of Electrical Engineering Education, 40() han T.., 994. teady tate nalysis of elf-excited nduction Generators, EEE transtion on Energy onersion, 9(, andhu K.., 00. teratie odel for the nalysis of elf-excited nduction Generators, Electric Power omponents and ystems,, Nigim K.., alama..., Kazerani. 00. oling polynomial algebraic equations of the stand alone induction generator. nternational Journal of Electrical Engineering Education, 40(), Gurung K., reere P., 006. Three phase self excited induction generator with a single phase load... esearch Thesis, Kathmandu Uniersity.