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  STEADY-STATE PERFORMANCE AND DYNAMIC STABILITY OF A SELF- :EXCITED INDUCTION GENERATOR EEDING AN INDUCTION MOTOR Sung-Chun Kuo Li Wang Department of Electncal Engineenng National Cheng Kung University Tainan, Taiwan 70101, R. 0 C. Abstract: This paper presents a novel scheme on steady-state performance of an isolated three-phase self-excited induction generator (SEIG) upplying a loaded induction motor. An approach based on d-q axis induction-machine model is employed to derive steadystate equations of the studied SEIG. Eigenvalue analyses based on synchronous reference frame are employed to determine the critical operating conditions and dynamic stability of the studied machines. The required minimum excitation capacitance of the SEIG, the maximum torque of the induction motor load, the combined maximum operating efficiency, etc. of the studied system can b: easily investigated. Experimental results obtained from a laboratory I. 1 kW induction machine driven by a dc motor and a 300 W induction motor with a dc generator as its shaft load are also performed to confirm the feasibility of the proposed method. Keywords: self-excited induction generator, steady-state performance, dynamic stability, induction motor load. NOMENCLATURE per-phase resistance, inductance, and reactance r per-phase magnetizing reactance and inductance magnetizing current voltage and current angular speeds of reference fiame and rotor per-phase excitation capacitance differentiation operator with respect to time t induction generator and motor quantities d-axis and q-axis quantities stator and rotor quantities magnetization quantities base quantities I. INTRODUCTION It is well known that an extemally driven induction machine can maintain self excitation when an appropriate value of a capacitor bank is appropriately connected across the twminals of the induction machine [l]. Such induction machine is called a self-excited induction generator (SEIG). The primary advantages of a SEIG over conventional synchronous generator are brushless construction with squirrel-cage rotor, reduced size, without DC supply for excit;3tion, reduced maintenance cost, and better transient characteristics. In recent years, SEIGs have received increased attention and they have been widely employed as suitable isolated power sources in small hydroelectric and wind energy applications [2-4]. According to the analyzed results of available references, the studies of an SEIG feeding static loads have been reported. Performances of an SEIG feeding a dynamic load such as an induction motor need to be further investigated. Shridhar, et al. [ ] reported a method to predict the steady- state behavior of a SEIG feeding an induction motor. The per-phase equivalent circuit model was obtained from a steady-state condition and it can not be used for analyzing SEIG’s transient characteristics. The author employed a novel eigenvalue sensitivity technique to analyze parallel operated SEIG feeding an induction motor load [6] In this paper, the authors propose an eigenavlue analysis based on synchronous reference frame to determine dynamic stability of a SEIG feeding a loaded induction motor. Experimental results obtained from a laboratory 1.1 kW induction generator supplying a 300 W induction machine load are also employed to validate the proposed method. 11. MACHINE MODELS Fig. I One-line diagram of the studied SEE feeding an IM load Fig. 1 shows the one-line diagram of a self-excited induction generator SEIG) feeding an induction motor load. The excitation capacitor C provides the required reactive power for the SEIG. Fig. 2  shows the d-q axis equivalent- circuit model of a three-phase symmetrical induction machine with arbitrary reference frame. The excitation capacitance C is connected to the induction machine’s stator terminals. The voltage equations of Fig. 2  can be written as below [7]. 0-7803-5935-6/00/ 10.00 c) 2000 IEEE 211  PMg < g -Nig 4g'PL.g The d-q axis equivalent-circuit model of the induction motor load is similar to Fig. 2  except that the currents in Fig. 2  are in the opposite direction. Connecting the induction motor load to the common bus, the voltage equations can be written as below. c' I- I I I+- I +I I I -I Fig 2 D-q xis equivalent circuit of an nduction generator. The voltage-current equations of the capacitor bank can be expressed as below. 3) The shaft torque and the rotor speed of the SEIG are related by thr: following equation. where T, is mechanical input torque and H, is inertia constant of the studied SEIG. The prime mover of the SEIG is simulated by a shunt DC motor whose speed-torque characteristic is given by 5) where VDC, RA, , and are the DC voltage, armature resistance, machine constant, and field flux of the DC motor, respectively. The no-load speed of the prime is given by vDC go Ka -- The torque equation of the induction motor load is given by where T, is the load torque. If the q-axis is aligned with the stator terminal voltage phasor by setting vdp 0 and p(vdl) = 0 in 1)- 3), the studied machine can be referred to as the synchronous reference frame. The system synchronous frequency is given by I2O 10 7 70 \. i X: experimental result I I I 1 I 0.5 IO 1.5 2.0 25 30 Im A) Fig. 3 Magnetization curve between X, and I, of the studied SEIG Due to nonlinear behavior of the magnetizing reactance X, in the studied SEIG, previous authors intended to approximate the air-gap voltage V, versus X, by a piecewise linear equation [4] or to fit the air-gap flux linkage versus 1/M using a polynomial method [SI. The nonlinear characteristic relating X, SZ) versus I, A) shown in Fig. 3  is determined by experimental tests. Such nonlinear relationship can be fitted with continuous function as: X = a[arctan(pI, - ) + 91 1 I 9) where magnetizing current I, is defined by: I The coefficient a s employed to make the estimated X, to match the measured reactance with respect to the measured maximum I,. The coefficients p and y decide the maximum value and the initial value of the magnetization reactance, respectively. The coefficient 9 makes the flux linkages to be zero when I, is equal to zero. The maximum value and initial value of magnetization reactance can be derived from (9) by using the limit theorem. Combining 1)- 7) and replacing arbitrary angular speed o with synchronous angular speed a,, the complete system consists of 11 nonlinear differential equations. The major advantage of using synchronous reference frame is that all variables are presented as dc quantities. Under steady state 0-7803-5935-6/00/ 10.00 c) 2000 IEEE 278  condition, the rates of change of all state variables in 1)- 6) are identically zero. 0,-0.793 pu, VI = 0.726 pu, 111. THE CONTINUATION POWER-FLOW ANALYSIS w . .. U = 0.793 pu. V, = 0.676 pu, U = 0.793 pu, VI = 0.704 pu, To express the above equations under steady-state power-flow analysis, the system equations with a new added parameter h can be written as below: TL I ,973 Nm -147.726ij 1111.227 -190.419 kj 624.315 -223.682kj 150.819 -52.138kj 177.153 where [y] denotes the state vector of the studied system, and h = 0 represents the base load condition. A nontrivial solution of (11) is called the equilibrium point or the stationary solution. The continuation power-flow method belongs to a general class of methods for solving nonlinear algebraic equations known as the path-following methods. Such method has been utilized as a tool for analyzing power system steady-state voltage stability [9-lo]. The continuation power-flow method uses an iterative process involving a predictor step and a corrector step through a known solution of (1 1 to obtain a series of solution branch. The stable and unstable equilibrium points of the studied system can then be effectively determined. Iluring the predictor step, a linear approximation is employed to estimate the next solution for a change in one of [y] or h Taking the derivatives of both sides of 1 I), with the state variables corresponding to the initial solution, will lead to the following hear equations: T, =2 0 Nm T,= 1 973 Nm -147.7152~ 1111.637 -147.645k~ 1111.290 -190.844 kj 623.211 -191.457 kj 621 337 -228.691 kj 152.113 -233.236Itj 153 961 -49.832 j 177.153 -48.384kj 183.934 Since the parameter h is unknown, an additional equation must be employed to solve (12). By setting one of the components of the tangent vector to be +I or -1. This component is referred to as the continuation parameter. Equation (12) becomes: -48.375 - 5.624 -1.998 (13) -41.846 -36.378 -15.657 -15.724 0.0 2 394 where yk s a row vector with all elements equal to zero except for the k-th element being equal to 1. The continuation parameter is usually selected to be the state variable that has the greatest rate of change near the given solution. The sign of its slope determines the sign of the corrtsponding component of the tangent vector. Once the tangent vector is obtained, the prediction for the next solution is given by where s the step length and it should be properly chosen so that a solution exists under the specified continuous parameter. 0-7803-5935-6/00/ 10.00 c) 2000 IEEE In the corrector procedure, the srcinal equation 1 1) is augmented by one equation that specifies the state variable selected as the continuation parameter. The new augmented equation is written as below. where yk is a state variable selected as the continuation parameter and q equals the predicted value of yk. This set of equations can be solved using a slightly modified Newton- Raphson power-flow method. The introduction of the additional equation specifying y, makes the Jacobian non- singular at the critical operating point. The continuation power-flow analysis can be continued beyond the critical point and determine the unstable region where the dynamic responses can not be effectively simulated. Table I Eigenvalues rads) of the studied system under three different IV RESULTS AND DISCUSSIONS Three different variables, i.e., the load torque T, of the IM load, the excitation capacitance C and no-load speed ago of the SEIG, an be selected as the continuation parameter h depicted in the previous section. Figs. 4-6 respectively illustrate the continuous critical points of the minimum excitation capacitance C,,,, the minimum required reactive power Qc, and the maximum load torque TLmax ersus different values of axe It is found from Fig. 4  that the higher the value of ago, he smaller the value of C,,, is. The large the value of T, the larger the value of C requires. It is seen from Fig. that the higher he value of ago he larger the value of Qc is. The large the value of TL he larger the value of Qc requires. It is discovered from Fig. that the higher the value of a@, the larger the value of TLma, s. The large the value of C, the larger the value of Twill be. Table 1 also lists the associated eigenvalues (dynamic stability) of the studied system under three different equilibrium points. According to the eigenvalue results listed in the last row of Table 1, the studied system is operated fiom a stable point (equilibrium point 1 through the critical operating point (equilibrium point 2) to the unstable point (equilibrium point 3) when the terminal voltage V, is continuously decreased. 27 9  V. CONCLUSIONS - 0.5 - - 04 - a U = 0.3 11.2 - 0 1 - 3*0 .5 his paper has presented a novel scheme on steady-state performance of an isolated three-phase self-excited induction generator (SEIG) supplying a loaded induction motor, Eigenvalue analyses based on synchronous reference frame has been employed to determine the critical operating conditions and dynamic stability of the studied machines. Experimental results obtained from a laboratory 1.1 kW induction machine driven by a dc motor and a 300 W induction motor with a dc generator as its shaft load are also performed to confirm the feasibility of the proposed method. LL .- E 125 100 TL=1 Nm I I I I I I I 0 4 0 5 0 6 0 7 0 8 0 9 1 0 1.1 1 2 Speed of SEIG P.u.) Fig 4 ?Characteristics f C,,,,,, versus different values of rotor speed of SEE and TL. 0 8 0.7 1 0.6 i TL=1 Nm TL O Nm 0.0 1 4 0 5 0 6 0 7 0 8 0 9 1.0 1.1 1 2 Speed of SEIG pu) Fig. Characteristics of minimum reactive power versus various values of rotor speed of SEIG and T, 2.0 1.5 1 o OS /+ 0.0 I I I 0.6 0.7 0.8 0.9 Speed of SEIG pu) Fig. 6 Characteristics of maximum load torque versus speed of SEIG and C VI. REFERENCES [l] E. D. Basset and F. M. Potter, “Capacitive excitation of induction generators.” Tram. Americon Institute Electrical Engineering, vol. 54, [2] D. B. Watson. J. Amlaga and T. Densem. “Controllable d. c. power supply from wind-driven self-excited induction machines,” IEE Proceedrngs. vol. 126. no. 12, 1979, pp. 1245-1248. [3] J. B Patton and D. Curtice, “Analysis of utility protection problems associated with small wind turbine interconnections,” IEEE Truns. Power ApparatusondSystems,vol. 101, no 10, 1982, pp. 3957-3966. [4] S. S Murthy, 0 P. Malik and A. K. Tandon, *Analysis of self excited induction generators,” IEE Proceedings, Pt. C. vol. 129, no. 6, 1982. pp, [ ] L. Shndhar, Bhim Singh, C. S. ha and B. P.Singh, “Analysis of self excited induchon generator feeding induction motor,” IEEE Trans. EnergVConversron, vol. 9, no. 2, 1994. pp. 390-396. [a] L. Wang and C. -H. Lee. “Dynamic analyses of parallel operated self- excited induction generators feeding an induction motor load,” Paper PE-337-EC-0-12-1997, EEWPES 1997 Summer Meeting Berlin, Germany, 1997. [7] P. C. Krause, Analysis ojElectric Mochineiy, New York: McGraw-Hill Book Co., 1987. [8] 0 Ojo “Minimum airgap flux linkage requirement for self-excitation in stand-alone induction generators,” IEEE Trans. Energy Conversion. vol. 10. no. 3, 1995, pp. 484492. [9] V. Aljarapu and C. Christy, “The continuation power flow: A tool for steady state voltage stability analysis,” ]E€€ Tmns Power Systems. vol 7. no. 1. 1992, pp. 416423. [IO] P. Kundur, Power system Stabili4 and Control. New York: McGraw- Hill Book Company, 1993. 1935, pp 540-545. 260-265 VII. BIOGRAPHIES Sung-Chun Kuo was bom on July\9, 1957 in Tainan, Taiwan. He obtained his M. Sc. degree from Department of Electrical Engineering, National Cheng Kung University. He is now a Ph. D. candidate at the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan. He is currently pursuing his Ph. D degree at the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan. His interest includes AC electric machines and power electronics Ll Wang (S’87-M‘88) Dr. Wang was bom in Changhua, Taiwan, on December 20, 1963 He received a Ph. D. degree from Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, in June 1988. Since August 1995, he has been a professor at the Department of Electrical Engineering, National Cheng Kung University, Tainan. Taiwan. At present, his interests include the science research of power engineering such as power systems, electric machinery, and power electronics. 0-7803-5935-6/00/ 10.00 c) 2000 IEEE 280
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