IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 15, NO. 1, MARCH 2000 61 Optimization-Based Steady State Analysis of Three Phase Self-Excited Induction Generator Abdulrahman L. Alolah, Senior Member, IEEE, and Majeed A. Alkanhal, Member, IEEE Abstract—This paper presents a steady state analysis of three phase self-excited induction generator. The problem is formulated as a multidimensional optimization problem. A constrained optimizer is used to minimize a cost function of the total impedance or admittance of the circuit of the generator to obtain the frequency and other performance of the machine. Classic gradient optimizers are used for this purpose as “fmin” and “constr” routines built in basic Matlab. Unlike the other reported methods of analysis, the same equation is to be solved irrespective of the unknown parameters. The analysis presented is validated by experimental results. Index Terms—Induction generators, Matlab, steady state analysis. Fig. 1. Equivalent circuit of three phase self-excited induction generator under R–L load. NOMENCLATURE F; u C; Xc p.u frequency and speed, respectively; Per phase value of excitation capacitance ( F) and its p.u reactance (at base frequency), respectively; Rs ; Rr ; RL p.u stator, rotor, and load resistances, respectively; Xs ; Xr ; XL p.u stator, rotor leakage, and load reactances at base frequency, respectively; p.u saturated and unsaturated magnetizing reacXm ; X0 tances at base frequency, respectively; Eg ; V0 Air gap and terminal voltages, respectively; Ic ; Il ; Is p.u. per phase excitation capacitance, load, and stator currents, respectively; Vb ; Ib ; Zb Base voltage, current, and impedance, respectively; fb ; Nb Base frequency and speed in Hz and rpm, respectively. has several advantages such as low unit and running cost, free from current collecting problems, ruggedness and self protection against large over loads and short circuit faults. The frequency and value of the voltage generated by these generators are highly dependent on speed, excitation capacitance and load –. This paper deals with a new approach for the analysis of a three phase self-excited induction generator. This problem is formulated as a numerical optimization problem where no detailed derivation of analytical equations is needed. Gradient optimizers such as those built in the mathematical software Matlab are utilized to solve the total impedance or admittance equations of the circuit of the generator to obtain the frequency and other performance of the machine. Unlike the other reported methods of analysis, the same equation is to be solved irrespective of the unknown parameters. The analysis presented is validated by experimental results. II. ANALYSIS I. INTRODUCTION HEN THE rotor of an induction machine is driven at a suitable speed, the rotor residual magnetism induces a small emf in the stator windings. This emf can be made to build up if a stator current at a leading power factor is supplied by a suitable means. This can be achieved by connecting a capacitor bank of a sufficient value, across the stator terminal . In this case, the armature reaction flux assists the original flux causing the emf to continue to build up until an equilibrium is reached due to the magnetic saturation of the machine. Under these conditions the machine is a self-excited generator. This machine W Manuscript received July 27, 1998; revised March 31, 1999. The authors are with the Department of Electrical Engineering, College of Engineering, King Saud University, P.O. Box 800, Riyadh 1142 1, Saudi Arabia. Publisher Item Identifier S 0885-8969(00)02213-0. A. Loop-Impedance Method Fig. 1 shows the equivalent circuit of three phase self excited generator under R–L load. In this circuit all the parameters are assumed to be constants and are independent of saturation except the magnetizing reactance Xm . Core loss and effect of harmonics in the machine are neglected . From Fig. 1 and under steady state: Is Zt = 0 where Zt = Zs + [ZL ==Zc] + [ZM ==Zr ] Zs = RS =F + jXS ; ZL = RL=F + jXL Zc = 0JXc =F 2 ; Zm = jXm Zr = Rr =(F 0 u) + jXr : 0885–8969/00$10.00 © 2000 IEEE (1) 62 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 15, NO. 1, MARCH 2000 Since Is 6= 0, it implies that Zt = 0, or t) = 0 (2) t ) = 0: (3) real (Z imag (Z Equations (2) and (3) can be solved simultaneously for two unknowns. These two unknowns may be (F and Xm ), (F and Xc ), (F and Z L ), or (F and u). According to the two unknowns different forms of (2) and (3) are to be solved. These two equations can be solved by Newton–Raphson method . Another approach is by rearranging the two equations as two high degree polynomials in F and the other unknown , . B. Nodal-Admittance Method By the same way, the total admittance (Yt ) of the circuit of Fig. 1 is also equal to zero, or t) = 0 (4) t ) = 0: (5) real (Y imag ( y Equation (4) can be first solved for F then (5) may be solved for Xm or Xc . If the two unknowns are F and u, (5) is first solved for F then (4) may be solved for u. According to the two unknowns and like (2) and (3), different forms of (4) and (5) are to be solved. C. Proposed Method The above approaches of analysis of the self-excited induction generator require lengthy and tedious algebraic derivations and consequently a lot of time is wasted. With the rapid developments in computer hardware and software, new, fast, and user friendly program packages are now available. As a result, many problems of engineering that used to cost long analysis will find more efficient numerical-based solutions. In this paper, the problem of analysis of three phase self-excited induction generator is solved numerically utilizing new computer facilities. Instead of step by step analytical derivation of several equations, a global optimizer is used to find the required parameters of the generator. Two schemes have been developed. First, a scheme of two sequential minimizations is used. A single variable minimizer is used to find the zero of (4) to find the frequency (F ). Then, the same minimizer is used to find the zero of (5) to yield the value of Xc or Xm . A more efficient and reliable scheme, which uses several-variable constrained minimization, solves the problem directly. This scheme simultaneously finds the values of F and Xc or Xm , that minimize the total impedance of (1). Several compilers and mathematical software packages have the required gradient optimizers of the above solution integrated in their libraries. The authors of this paper developed an integrated program routines to solve the proposed problem using Matlab. In this respect “fmin” and “constr” functions of the basic Matlab can be used to solve the total impedance or admittance equations of the circuit of the generator to obtain the frequency and the other unknown of the machine circuit. A list of the Matlab programs developed for the proposed schemes are given in Table I. Once these values are obtained, the circuit of Fig. 1 is to be solved with the help of the magnetization curve to yield the performance of the generator . III. RESULTS AND DISCUSSIONS There are three external elements of self-excited induction generator that can be controlled; these are the speed, excitation capacitance, and load. Changing any one of these elements will change Xm and F . In order to confirm the feasibility, reliability and accuracy of the proposed two methods, the variations of F and Xm under different conditions have been computed by the programs of Table I. Both of the two methods yielded the same results. The computations were carried out on the three phase induction machine with the data given in the Appendix. Figs. 2 and 3 show the variations of Xm and F versus motor speed “u” and excitation capacitance, respectively. The variations are shown under different loads. As can be seen from these figures Xm changes in a convex manner. Above Xm = X0 the machine is not saturated and consequently will not operate. The range of operation is maximum at no load and decreases as the load impedance decreases. Regarding the frequency it can be noted that it increases as the speed increases and decreases as the capacitance increases. From the figures it can be realized that, at the same speed and excitation capacitance, the load variation has no significant effect on the frequency. Fig. 4 shows the variation of Xm and F versus load impedance “ZL ” at fixed speed and excitation capacitance. From this figure it can be noted that at high load impedance Xm is nearly constant and less than X0 . As the impedance decreases Xm increases until it exceeds X0 when the excitation capacitance is less than the required value. It can be noted that at the same load impedance, speed and excitation capacitance, the frequency increases as the power factor decreases. The difference decreases as the load impedance increases. The above results and findings are in complete agreement with that of the other methods of analysis of three phase self-excited induction generator –. IV. EXPERIMENTALVERIFICATION To verify the validity of the above analysis, the induction solutions integrated in their libraries. The authors of this generator under study was tested experimentally under paper developed an integrated program routines to solve different conditions. No load test as a motor under variable voltage yielded the magnetization curve of Figs. 5–7 show some computed results and their counterpart experimental measurements. Fig. 6 shows the variation of the open circuit voltage versus the rotor speed at fixed excitation capacitance. Fig. 7 shows the terminal voltage variation versus load current under a load of unity power factor and fixed excitation capacitance and speed. The experimental results of Figs. 6 and 7 are quoted from . Good correlation between the computed and experimental results can be seen from these two figures. ALOLAH AND ALKANHAL: OPTIMIZATION-BASED STEADY STATE ANALYSIS OF THREE PHASE SELF-EXCITED INDUCTION GENERATOR TABLE I MATLAB PROGRAMS 63 64 IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 15, NO. 1, MARCH 2000 = 25 Fig. 2. Variation of Xm and F versus speed for different loads C F,: No load, : ZL = 3.0 p.u, p.f = 1.0, : ZL = 1.5 p.u, p.f = 0.8 —: Xm , - - -:F . 1 3 Fig. 3. Variation of Xm and F versus C for different loads u = 1.0 p.u, : No load, : ZL = 3.0 p.u, p.f = 1.0, : ZL = 1.5 p.u, p.f = 0.8 —:Xm , - - -:F . Fig. 5. Magnetization curve of the machine under study. Fig. 6. V0 versus load current under fixed C and u. Fig. 7. V0 versus u under no load and C 1 3 Fig. 4. Variation of Xm and F versus the load. V. CONCLUSIONS In this paper, an optimization-based steady state analysis of three phase self-excited induction generator has been introduced. The problem is solved numerically utilizing new computer facilities. Instead of step by step analytical derivation of several equations, a global optimizer is used to find the required parameters of the generator. Two methods have been = 20 F. proposed. First, a scheme of two sequential minimizations is used. A single variable minimizer is used to find the value of the frequency (F ) then, the same minimizer is used to find the value of the other unknown (Xc or Xm ). A more efficient and reliable scheme which uses several-variable constrained minimization solves the problem directly. This scheme simultaneously finds the values of Xc or Xm , and F that minimize the total impedance of the generator. The authors of this paper developed a series of integrated program routines to solve the ALOLAH AND ALKANHAL: OPTIMIZATION-BASED STEADY STATE ANALYSIS OF THREE PHASE SELF-EXCITED INDUCTION GENERATOR proposed problem using Matlab. In this respect “fmin” and “constr” functions of the basic Matlab were used to solve the total impedance or admittance equations of the circuit of the generator to obtain the frequency and other performance of the machine. Experimental test results confirm the feasibility and accuracy of the proposed methods. APPENDIX The rating of the machine under study is 0.75 kW. The base values are: Vb =220 V, Ib = 2.31 A, fb = 60 Hz and Nb = 1800 rpm. The experimental tests the following p.u. parameters: Rs = 0.111, Rr = 0.132, Xs = Xr = 0:157, X0 = 2:64. The magnetization curve of Fig. 5 can be approximated by a polynomial of degree 3 as follows: Eg =F = X i ki (Xm ) where: 02 92318 3 = 00 418359 k0 = 2:5954 k1 k2 = 1:8711 k = ; : 65  N. Malik and S. Haque, “Steady state analysis and performance of an isolated self-excited induction generator,” IEEE Trans., vol. EC 1, no. 3, pp. 134–139, 1986.  A. Al-Jabri and A. Alolah, “Limits on the performance of three-phase self-excited induction generators,” IEEE Trans., vol. EC 5, no. 2, pp. 350–356, 1990. , “Capacitance requirement for isolated self-excited induction gen erator,” IEE Proc., pt. B, vol. 137, no. 3, pp. 155–159, 1990.  N. Malik and A. Al-Bahrani, “Influence of the terminal capacitor on the performance characteristics of a self-excited induction generator,” in IEE Proc., vol. 137, 1990, pp. 168–173.  A. S. Al-Khalaf, “Comparison between performance of isolated selfexcited induction and reluctance generators,” M.Sc thesis, King Saud University, Riyadh, Saudi Arabia, Mar. 1989. Abdulrahman L. Alolah (S’85–AM ’86–M ’87–SM ’96) was born in 1957. He received the B.Sc. degree in electrical engineering from King Saud University, Riyadh, Saudi Arabia, in 1979 and the Ph.D. degree in electrical and electronic engineering from the University of Bradford, England, in 1986. In May 1986, Dr. Alolah joined King Saud University as an Assistant Professor. He was promoted to Associate Professor and then to Professor in May 1990 and November 1994, respectively. His current research interests include electrical machines and power electronics. : REFERENCES  C. Wagner, “Self-excitation of induction motors,” AIEE Trans., vol. 58, pp. 47–51, 1939.  L. Quazene and G. McPherson, “Analysis of the isolated induction generator,” IEEE Trans., vol. PAS 102, no. 8, pp. 2793–2798, 1983. Mageed A. Alkanhal (S’88–M’94) received the B.Sc. and the M.Sc degrees in electrical engineering from King Saud University, Riyadh, Saudi Arabia, in 1984 and 1986, respectively. He received the Ph.D. degree from Syracuse University, Syracuse, NY, in 1994, in electrical engineering. His fields of research interest are electromagnetics, telecommunications, and modem analytical and numerical methods as applied to electrical engineering problems.