Advanced Mateials Reseach Online: 214-6-18 ISSN: 1662-8985, Vols. 96-961, pp 1146-115 doi:1.428/www.scientific.net/amr.96-961.1146 214 Tans Tech Publications, Switzeland A Double Exponential Function Fitting Algoithm fo Optimize Paamete of µh Cuve Jiao Du 1,2,a, Meng Song 1, Nannan Hu 1, Kunnan Cao 1,Wentao Huang 1,3, Peng Xu 1,2, Yiyang Li 1,2 1 Yunnan Electic Powe Reseach Institute, Kunming City, Yunnan Povince, P.R. China, 65217 2 School of Electical Engineeing, Kunming Univesity of Science and Technology, Kunming, P.R. China, 655 3 School of Electical Engineeing, Univesity of Chongqing, Chongqing, P.R. China, 444 a 921562872@qq.com Keywods:electic powe system secuity; ion-coe satuation; double expontential function fitting algoithm Abstact.Satuable magnetic cicuit is a common phenomenon in powe system. It can lead to wavefom distotion, coefficient of self inductance decease and measuement eo incease. So establishing a mathematical model fo satuable magnetic cicuit and evealing the mathematical elationships in satuable phenomenon. It is helpful to eliable and safe opeation of powe system. This pape base on double expontential function fitting algoithm and simplex algoithm to establish the function elationship between elative pemeability and magnetic field intensity. Applying fo the model,we can obtain the wavefom of magnetic flux in tansfome coe and tansfome voltage wavefom. The eseach has cetain significance in the abstact and foegound in the application fo the study of satuable phenomenon. Intoduction BH cuve is the key to studying satuable phenomenon. Geneally speaking,the BH cuve can be detemined in laboatoy. Theefoe thee is no simple fomula fo this kind of cuve. But if you look at the mathematics of the poblem, BH cuve can be gotten accoding to the polynomial fitting, cubic spline intepolation and so on. The data of cuve fitting would be in eo. Because calculation of seconday voltage tansfomes and magnetic flux of ion coe have many pocedues, calculation eo has the tend of becoming bigge. Theefoe, in this pape a new method of fitting µ H ( µ : elative pemeability) cuve is poposed. Applying fo the model, we can educe the amount of calculation and lowe the calculation eo. Model Building It is well known that expeimental BH cuve of silicon steel sheet,as shown in figue(1). Fig. 1. BH cuve of silicon steel sheet. The elationship between elative pemeability and magnetic field intensity, expessed by the simple equation(1). µ cuve shown in figiue(2). H All ights eseved. No pat of contents of this pape may be epoduced o tansmitted in any fom o by any means without the witten pemission of Tans Tech Publications, www.ttp.net. (ID: 13.23.136.75, Pennsylvania State Univesity, Univesity Pak, USA-1/5/16,6:58:28)
Advanced Mateials Reseach Vols. 96-961 1147 B µ = (1) H µ In equation(1), the µ is pemeability of vacuum, µ is elative pemeability, B is magnetic induction intensity,h is magnetic field stength. Fig. 2. µ H cuve of silicon steel sheet. Now, we utilized the cuve-fitting to establish µ H fun-ction. µ H cuve look like a impulse function. So double expontential function fitting algoithm can expess µ H cuve. The expession can be given as equation(2). In equation(2). A,α, β is the fitting coefficient. αx βx y = A( e e ) (2) The basic ideas concening cuve fitting is that find the optimum solution of A,α, β, and make diffeence between the data of the cuve fitting and the expeimental data as small as possible. The eo function can be given as equation(3). αi e (,, ) = [ ( )] 2 e βi E A α β µ A e e = min (3) In odo to find the optimum solution of A,α, β, we need solving the patial deivatives of E function. Then make patial deivatives of E function equal to zeo,it can be shown as equation(4). E( A, α, β ) A E( A, α, β ) α E( A, α, β ) β = = = (4) The simplex method is used to solve the system of algebaic equation(4). Since ou pices ae 1 closely calculated, A = 2.3584, α =.59429, β =.1769. µ H cuve as shown in figue(3).
1148 Themal, Powe and Electical Engineeing III Fig. 3. µ H cuve fitting. Fom the µ H cuve (figue 3), the fitting data accods with the expeiment data basically. Fitting function can be shown as equation(5). 1.6 H.11H µ = 2.3584 ( e e ) (5) The function elation between exciting cuent and magnetic field stength in tansfome can be shown as equation(6). H NI e = (6) l In equation(6). I e is exciting cuent, N is tuns,l is loop length. Eq.(6) substituted in Eq(5), The function elation between elative pemeability and exciting cuent can be expess as equation(7). NIe.6 NIe.11 1 l l µ = 2.3584 ( e e ) (7) Model Application Using the model(7), the magnetic flux and induced electomotive foce in tansfome coe can be calculated. Model of powe tansfome as shown in the figue(4). Coil(1) is pimay winding. The paametes as shown in the Table Ⅰ. Fig. 4. Model of powe tansfome.
Advanced Mateials Reseach Vols. 96-961 1149 Table 1. Tansfome Paamete The capacity / fequency of tansfome : 15MVA/ 5Hz Loss esistance / emanence flux of ion coe:1.85e+6ohm/ T 2 Loop length / sectional aea of ion coe: 2m/ 1 m Coil esistance R1 Coil leakage inductance L1 Nominal voltage peak Coil(1) 1.1111 ohm Coil(2) 1.1111 ohm Nominal cuent peak.11789h 5kV 3A 5.11789H 23kV 5A 3 Tuns of coil Accoding to Faaday law of electomagnetic induction, Induced electomotive foce at pimay winding can be expessed as equation(8). dh S e = µ µ N( ) (8) dt Suppose exciting cuent equals ated cuent. Using the model(7), the ion coe magnetic fluxtime function pictue as shown in the figue(5). 3 2 1 B(T) -1-2 -3.1.2.3.4.5.6.7.8.9.1 Fig. 5. Ion coe magnetic flux-time function pictue. When ion coe is not satuated,the magnetic flux-time function is a sine function. So using the model(8), the induced electomotive foce at seconday winding is a sine wave as shown in the figue(6). 25 2 15 1 5 e(kv) -5-1 -15-2 -25.1.2.3.4.5.6.7.8.9.1 Fig. 6. Induced electomotive foce -time function pictue. Suppose exciting cuent is geate than ated cuent. Using the model(7), the ion coe magnetic flux-time function pictue as shown in the figue(7). 25 2 15 1 5 B(T) -5-1 -15-2 -25.1.2.3.4.5.6.7.8.9.1 Fig. 7. Ion coe magnetic flux-time function pictue. Because of coe satuation, magnetic induction intensity in the ion coe is a flat wave. Using the model(8), the induction electomotive foce at seconday winding is a peaked wave. It shown in figue(8).
115 Themal, Powe and Electical Engineeing III 1 8 6 4 2 e(kv) -2-4 -6-8 -1.1.2.3.4.5.6.7.8.9.1 Fig. 8. Ion coe induction electomotive foce -time function pictue. Conclusion Detailed calculations shows that it the esult of the model(7) is in accodance with the theoetical analysis. So the model is feasible and available. These eseach and pactices ae useful efeences to us. Refeences [1] QianSheng Hu,MinQiang Hu, Electomechanics, Beijing:China Electic Powe Pess,29. [2] William H.Hayt, John A.Buck, Engineeing Electo- magnetics, XiAn: XiAn JiaoTong Univesity Pess,29. [3] DeXin Xie,ShiYou Yang, Numeical Analysis of Engineeing Electomagnetic Field, Beijing:China Machine Pess,28. [4] Qian Li,TianBi Tu,XiaoYu Wang, An Impove Method fo the Fitting of Excitation Cuve of Electomagnetic Voltage Tansfome, Tansfome, vol.35,pp.28-3,mach.1998 [5] YanPeng Hao,GuoLi Wang,YanMing Li,Heng,Xie, A Double Exponential Function Fitting Algoithm fo Impulse Paamete Evaluation, High-Voltage Technology,vol.26,pp.31-35, June.2..
Themal, Powe and Electical Engineeing III 1.428/www.scientific.net/AMR.96-961 A Double Exponential Function Fitting Algoithm fo Optimize Paamete of μh Cuve 1.428/www.scientific.net/AMR.96-961.1146