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Operation of induction generator in the magnetic saturation region as a self-excited and as a double output system
Journal of Materials Processing Technology 161 (2005) 263–268
Operation of induction generator in the magnetic saturation region as a self-excited and as a double output system Ioannis M. Katiniotis∗ , Maria G. Ioannides, Peter G. Vernados Laboratory of Electric Drives, Faculty of Electrical and Computer Engineering, National Technical University of Athens, Heroon Polytechniou 9, 15773 Athens, Greece
Abstract This paper presents the development of two dynamic models for simulating the operation of an induction machine with saturation considered, operating either as self-excited or as double output induction generator in autonomous wind systems. Both systems’ steady state as well as transient performance during switching on and off from the bank of capacitors or the power line, respectively, is studied considering different wind speed and excitation conditions. A polynomial approximation of the variation of mutual inductance versus magnetizing current is used for the proper account of the effects of core saturation. The transient voltage and current built-up of the two systems, the harmonic distortion as well as the damping of the transient process are obtained from a laboratory autonomous wind generating system. From a comparison between the outputs generated by the two systems the results show that the double output induction generator can be installed for the better use of the wind energy. © 2004 Elsevier B.V. All rights reserved. Keywords: Induction generator; Magnetic saturation region; Double output system
1. Introduction In the last years, the induction machines receive an increasing attention due to their use as generators in wind power plants. An induction machine-based Wind Energy Conversion System (WECS) can operate either as self-excited or as double output induction generator, SEIG or DOIG, respectively [1–3]. When operating in a variable speed environment such as the wind, power fluctuations appear and generate a variable voltage-variable frequency output. Thus, a WECS with DOIG is a multi input-multi output system having as input variables the rotor excitation voltage, the rotor excitation frequency and the driving wind torque, while as output variables has the generated voltage and the generated frequency at stator terminals [4,5]. On the other hand, a WECS with SEIG is a multi-input–multi-output system too, having the same output variables, but its input variables are the wind speed and the voltage of the bank of capacitors [1,6].
∗
Corresponding author. Tel.: + 30 210 7723 791; fax: +30 210 7722 584. E-mail addresses: [email protected] (I.M. Katiniotis), [email protected] (M.G. Ioannides), [email protected] (P.G. Vernados). 0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.034
In this paper the two WECS configurations, SEIG and DOIG, are studied. Both configurations use an induction generator of the same type and ratings, which operates in the saturated region of its magnetic characteristics. Following to this both system’s properties and dynamic responses are studied at the same operating points.
2. Experimental system The induction generator system is shown in Fig. 1. It consists of an 1 kW, four poles, wound rotor induction generator, which is connected first as DOIG and then as SEIG. The first configuration uses a single phase uncontrolled rectifier and a three phase PWM inverter connected at the rotor side. The rectifier converts the line voltage to dc voltage and, then, the PWM inverter converts the dc voltage to three-phase ac voltage Ur at slip frequency fr , which is supplied to the rotor windings. The stator windings are connected to a threephase network and supply the voltage Us at frequency fs . A dc motor simulates the wind turbine, which is mechanically connected to the generator shaft and supplies to the generator the mechanical energy with a driving torque Tw
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Nomenclature C self-excitation capacitance (F) E RMS air gap emf (V) f rated frequency (Hz) isd , isq , ird , irq instantaneous stator and rotor direct and quadrature axis current (A) im , Im instantaneous and RMS magnetizing current (A) Is , Ir RMS stator and rotor current (A) J mass moment inertia of the rotor (kg m2 ) Ls , Lr per phase stator and rotor (referred to stator) inductance (H) M mutual inductance (H) p differential operator (d/dt) P number of poles Ro load resistance () Rs , Rr per phase stator and rotor (referred to stator) resistance () T electromagnetic torque (Nm) Tm mechanical torque (Nm) vsd , vsq , vrd ,vrq instantaneous stator and rotor direct and quadrature axis voltage (V) Vs , Vr RMS stator and rotor voltage (V) Greek letters ωm speed of a P-pole motor (rad/s) ωo speed of a 2-pole motor (rad/s)
and at a rotational speed ω. The generator, the dc motor and the inverter are interfaced to the Measurements and Instrumentation unit, which mainly comprises a digital scope and the measuring instruments and, also, produces outputs to the operator. The second configuration uses a bank of capacitors
connected to the stator side while the rotor windings are shortcircuited. 3. Models of the DOIG and SEIG systems The analysis uses the synchronous reference frame of the two axes theory, with the d–q axes 90o apart in space and rotating at synchronous speed as determined by the electrical angular velocity ω10 of the stator voltages. The voltage equations of the induction machine in the d–q stationary reference frame may be expressed as: 1 vsd = Rs + pLs + pC isd + Mpird vsq = Rs + pLs +
1 pC
isq + Mpirq
(1)
vrd = Mpisd + ωo Misq + (Rr + Lr p)ird + ωo Lr irq vrq = −ωo Misd + Mpisq − ωo Lr ird + (Rr + pLr )irq Eq. (1) represents a set of four differential equations which allow the current response of the induction machine to be determined if the rotor speed ω0 is known. The mechanical part of the system is modeled using additional equations, namely the electromagnetic torque Eq. (2) and the load torque Eq. (3) [7–9]: T =
P Lm (isq ird − isd irq ) 3
dωm 1 = (T − Tm ) dt J
(2) (3)
The relation between ωm for a P-pole motor and ωo for a 2-pole motor is: ωo = (p/2)ωm . The additional impedance 1/pC in the stator winding denotes the d–q axis representation of the excitation capacitors of the SEIG [8].
Fig. 1. (a) Double output induction generator system block diagram, DOIG. (b) Self-excited induction generator block diagram, SEIG.
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In the case of DOIG when the stator is connected to an isolated load the stator voltage vsd and vsq become zero [4,5,7,8]. The d–q axis representation includes the voltage equation given by (4) and the output Eq. (5): 0 = (Rs + Ro )isd + Ls pird + Mpird 0 = (Rs + Ro )isq + Ls pisq + Mpirq vrd = Mpisd + ωo Misq + (Rr + Lr p)ird + ωo Lr irq
(4)
vrq = −ωo Misd + Mpisq − ωo Lr ird + (Rr + Lr p) irq vsd = Ro isd (5)
vsq = Ro isq
In the case of SEIG a resistive load is added in parallel with the self-excitation capacitance and the capacitance term 1/pC becomes R/(1 + RpC). Moreover no external voltage is applied to either the stator or the rotor windings [6]. The voltage equation is given by the system of Eq. (6):
The prediction of both induction generator steady state and transient performance requires proper account of the effect of saturation. In this paper the effects of core saturation are incorporated in simulation using an n-order polynomial approximation of the variation of mutual inductance M versus magnetizing current Im [2]. The procedure is as follows: 1 The magnetization characteristic given in Fig. 2 is measured by using the no load test at synchronous speed. According to this test, the machine is connected to a variable voltage at rated frequency 50 Hz and the magnetizing current is measured as the no load stator current. 2 Vectors [V] and [Im ] containing the measured values of the variable voltage and magnetizing current are set up and fed into the computer program. 3 The corresponding values of the rms air gap emf E and the mutual inductance M at 50 Hz are computed for every set of corresponding values of vector [V] and [Im ]. Thus, for the k-th set of values holds: E(k) = V (k) − (Rs + jXs )Im (k)
R 0 = Rs isd + Ls pisd + isd + Mpird 1 + RpC 0 = Rs isq + Ls pisq +
R isq + Mpirq 1 + RpC
M(k) = (6)
0 = Mpisd + ωo Misq + (Rr + Lr p)ird + ωo Lr irq 0 = −ωo Misd + Mpisq − ωo Lr ird + (Rr + Lr p)irq
(7)
E(k) 2πfIm (k)
(8)
These two vectors [Im ] and [M] containing the numerical values for magnetizing current and mutual inductance can be used by the program. 4 A polynomial fitting routine is used. Given the values of vector [Im ], the routine finds two n-order polynomials p1 and p2 such that p1 ([Im ]) and p2 ([−Im ]) fits the values of vector [M] in a least-squares sense. n n−1 M(k) ∼ (k) + imn−1 Im (k) = p1 (Im (k)) = imn Im
4. Saturation model The topic of this paper concerns only the operation in the saturated region of the two systems. Thus, the experiment was designed in such operating conditions that the induction generator be saturated over all operation region, from no load up to full load, as can be seen from the magnetic characteristics from Fig. 2.
265
+ · · · + im1 Im (k) + imo
(9)
n n−1 M(k) ∼ (k) + imn−1 Im (k) = p2 (−Im (k)) = imn Im
+ · · · + im1 Im (k) + imo
(10)
To verify the approximation, the polynomial p(Im ) is evaluated at different values of Im . A good correlation is found for the sixth order polynomial fitting, n = 6. The numerical values for the polynomial coefficients obtained are listed in Table 1. Plots of measured and computed curves by using this procedure are given in Fig. 3. The good approximation obtained is evident. Table 1 The coefficients of the polynomials p1 (Im (k)) and p2 (−Im (k)) p1 (Im (k))
Fig. 2. Magnetic characteristics.
im6 im5 im4 im3 im2 im1 im0
= −0.062 = 0.681 = −2.779 = 5.313 = −4.679 = 1.186 = 0.92
p2 (−Im (k)) im6 im5 im4 im3 im2 im1 im0
= −0.062 = −0.681 = −2.779 = −5.313 = −4.679 = −1.186 = 0.92
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their final steady state values have been investigated and the obtained results are in good agreement with those reported in [6,9].
5. Results
Fig. 3. Magnetizing characteristics computed with the polynomial fitting method. The two branches corresponding to both positive and negative magnetizing current are shown.
The saturation model based on the polynomial fitting method, was used in the transient analysis of the induction machine operating either as SEIG or DOIG and it is shown in Fig. 4. This model performs an internal iteration at each integration step because of the nonlinear relationship between mutual inductance M and magnetizing current im . Given the initial condition from the previous integration step, the magnitude of the magnetizing current im is calculated by the function Fcn as: im = (isd + ird )2 + (isq + irq )2 (11) The other two functions Fcn1 and Fcn2 used in the saturation model express the polynomial approximation of the experimentally determined functional dependency of the mutual inductance M on magnetizing current im . In this case the output M passes through Fcn1 or Fcn2 corresponding to the positive or negative branch of the magnetizing current, respectively, Fig. 4. The simulation model described by Eqs. (1)–(6) of the two systems shown in Fig. 1 was solved using Simulinksoftware package to simulate the operation and investigate the performance of autonomous wind systems equipped with self-excited or double output inductions generators. In case of DOIG, system transients were studied from Simulink simulation during switching in and out of the supply grid and reported in [7]. In case of SEIG, the transient process of voltage and current build-up during self-excitation as well as the
Fig. 4. Saturation model based on polynomial fitting method.
First, the magnetic characteristic of the machine was identified and recorded, Fig. 2, and then the generator was loaded in the saturation region. The experimental results obtained for the SEIG and the DOIG were studied during cut-in and cut-out of the bank of capacitors and of the excitation voltage and frequency, respectively. During all the tests the system load was kept constant and the generated steady state and transient stator voltage and current at constant mechanical power supplied to both systems were obtained. Waveforms of the generated stator voltage and current of the machine operating as DOIG during cut-in of the excitation voltage and frequency at the rotor side are shown in Plate 1. Initially the machine being without excitation is driven at constant speed of 1887 rpm by a dc motor while the stator terminals are connected to a resistive load of Ro = 65 . In this situation no voltage or current are generated to the stator side. At t = 20 ms the rotor windings are directly connected to the excitation voltage Vr = 20 V and frequency fr = 35 Hz. The ac waveforms of the generated stator voltage and current with a frequency equal to that of the excitation voltage, i.e. fr = 35 Hz are distorted due to the existence of harmonics produced by the inverter and the nonlinear magnetic circuit. Under the same speed and load conditions and with the machine operating as SEIG a bank of capacitors is directly connected in parallel to the stator terminals. The transient build-up of the generated stator voltage and current during the selfexcitation process shown in Plate 2 is continued until there is virtually no change in the value of mutual inductance M, with a self-excitation frequency very close to the frequency
Plate 1. DOIG operation at cut in the rotor excitation voltage. Stator voltage (CH1) and stator current (CH2) at 1887 rpm, Ro = 65 , Vr = 20 V, fr = 35 Hz, CH1 probe coefficient = 1, CH2 probe coefficient = 10.
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Plate 2. SEIG operation at cut in the bank of capacitors. Stator voltage (CH1) and stator current (CH2) at 1887 rpm, Ro = 65 , C = 3 × 100 F, CH1 probe coefficient = 100, CH2 probe coefficient = 10.
corresponding to the speed of the motor. The final ac waveform of the steady state voltage obtained for about 500 ms is higher than these corresponding to the case of DOIG. Plates 3 and 4 present the dynamic performance of the induction generator in case of DOIG and SEIG respectively, running with the same load Ro = 65 as in the previous experiments. The induction generator operates in a transient state when accelerated from start, and settles to a steady state. In the case of DOIG, the excitation voltage and frequency are at Vr = 125 V, and fr = 23 Hz and the generator is forced to rotate with a shaft speed of 1750 rpm. Waveforms of the generated stator voltage and current are shown in Plate 3. In the case of SEIG the generator is driven at constant speed of 1750 rpm and a decrease in the generated frequency during the transient voltage and current build-up is easily observed in comparison to the higher speed conditions as in the case shown in Plate 2.
Plate 3. DOIG operation at starting with cut in the excitation voltage. Stator voltage (CH1) and stator current (CH2) at 1750 rpm, Ro = 65 , Vr = 125 V, fr = 23 Hz, CH1 probe coefficient = 1, CH2 probe coefficient = 10.
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Plate 4. SEIG operation at starting with cut in the bank of capacitors. Stator voltage (CH1) and stator current (CH2) at 1750 rpm, Ro = 65 , C = 3 × 100 F, CH1 probe coefficient = 100, CH2 probe coefficient = 10.
Plate 5. SEIG operation: capacitors cut in at 0.25 s and cut out at 0.5 s. Stator voltage (CH1) and stator current (CH2) at 1750 rpm, Ro = 130 , C = 3 × 100 F, CH1 probe coefficient = 100, CH2 probe coefficient = 10.
Plate 6. SEIG operation: capacitors cut in at 0.25 s and cut out at 0.75 s. Stator voltage (CH1) and stator current (CH2) at 1750 rpm, Ro = 65 , C = 3 × 100 F, CH1 probe coefficient = 100, CH2 probe coefficient = 10.
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Finally, Plates 5 and 6 show the transient operation of the SEIG during cut in and cut out of the bank of capacitors, as well as the steady generated stator voltage and current waveforms, for the same shaft speed at 1750 rpm and system load at Ro = 130 and Ro = 65 respectively. Again a decrease in self-excitation frequency occurred at higher values of the system load is easily observed as well as a reduced time of the transient processes.
6. Conclusions The analysis, the implementation and the experimental validation for a SEIG and DOIG operating as WECS in the region of magnetic saturation and supplying a variable load were carried out. Then the results obtained were investigated. Both systems’ performances were studied considering different wind speed, excitation voltage and load conditions. A dynamic model has been developed and presented for investigating the steady state, as well as the transient behavior of the induction generator operating as DOIG or SEIG, during switching on and off from the power line or the bank of capacitors respectively. Waveforms of the generated voltage and current have been obtained taking into account the variation of mutual inductance with magnetization current using an n-order polynomial approximation. The obtained results show the transient voltage and current build-up of the two systems, their harmonic distortion as well as the damping time of the transient process. Also, the results indicate that an adequate procedure is the operation of this system as DOIG in order to better utilize the wind energy.
Acknowledgement The authors express their gratitude to Mr. Panayotis Zannis for the technical assistance during the experiments described in this paper.
References [1] M.G. Ioannides, P.G. Vernados, J.A. Tegopoulos, Steady-state characteristics of a wind driven self-excited asynchronous generator in both autonomous and nonautonomous operation, in: Proceedings of The Beijing International Conference of Electrical Machines BICEM’87, Beijing, China, Book II, August 1987, pp. 856–859. [2] M.G. Ioannides, J.A. Tegopoulos, Double output induction generator as voltage and frequency converter, in: Proceedings of International Conference on Electric Machines ICEM’90, vol. 3, M.I.T., Cambridge, MA, USA, 12–15 August 1990, pp. 982–987. [3] P.G. Vernados, I.M. Katiniotis, M.G. Ioannides, Development of an experimental investigation procedure on double fed electric machinebased actuator for wind power systems, Sens. Actuators A 106 (2003) 302–305 (Elsevier Science). [4] M.G. Ioannides, State space formulation and transient stability of the double output asynchronous generator, IEEE Trans. Energy Conversion 8 (4) (1993) 732–738. [5] M.G. Ioannides, I.M. Katiniotis, Development of a simulation model for the dynamic operation of double output induction generator in autonomous wind systems, in: Proceedings of the 6th International Conference on Development and Application Systems, Romania, May 2002, pp. 104–111. [6] C. Grantham, O. Sutanto, B. Mismail, Steady-state and transient analysis of self-excited inductions generators, IEE Proc. 136 (2 Pt. B) (1998) 61–68. [7] I.M. Katiniotis, M.G. Ioannides, Effects of cut in and cut out of double output induction generator in autonomous wind systems, in: CDROM of the 15th International Conference on Electrical Machines ICEM 2002, Bruges, Belgium, 25–28 August 2002, pp. 1–6 (paper 557, pdf). [8] L. Cadirci, M. Ermis, Performance evaluation of a wind driven DOIG using a hybrid model, IEEE Trans. Energy Conversion 13 (2) (1998) 148–155. [9] R.C. Bansal, T.S. Bhatti, D.P. Kothari, Bibliography on the application of Induction generators in nonconventional energy systems, IEEE Trans. Energy Conversion 18 (3) (2003) 433–439.
Operation of induction generator in the magnetic saturation region as a self-excited and as a double output system Ioannis M. Katiniotis∗ , Maria G. Ioannides, Peter G. Vernados Laboratory of Electric Drives, Faculty of Electrical and Computer Engineering, National Technical University of Athens, Heroon Polytechniou 9, 15773 Athens, Greece
Abstract This paper presents the development of two dynamic models for simulating the operation of an induction machine with saturation considered, operating either as self-excited or as double output induction generator in autonomous wind systems. Both systems’ steady state as well as transient performance during switching on and off from the bank of capacitors or the power line, respectively, is studied considering different wind speed and excitation conditions. A polynomial approximation of the variation of mutual inductance versus magnetizing current is used for the proper account of the effects of core saturation. The transient voltage and current built-up of the two systems, the harmonic distortion as well as the damping of the transient process are obtained from a laboratory autonomous wind generating system. From a comparison between the outputs generated by the two systems the results show that the double output induction generator can be installed for the better use of the wind energy. © 2004 Elsevier B.V. All rights reserved. Keywords: Induction generator; Magnetic saturation region; Double output system
1. Introduction In the last years, the induction machines receive an increasing attention due to their use as generators in wind power plants. An induction machine-based Wind Energy Conversion System (WECS) can operate either as self-excited or as double output induction generator, SEIG or DOIG, respectively [1–3]. When operating in a variable speed environment such as the wind, power fluctuations appear and generate a variable voltage-variable frequency output. Thus, a WECS with DOIG is a multi input-multi output system having as input variables the rotor excitation voltage, the rotor excitation frequency and the driving wind torque, while as output variables has the generated voltage and the generated frequency at stator terminals [4,5]. On the other hand, a WECS with SEIG is a multi-input–multi-output system too, having the same output variables, but its input variables are the wind speed and the voltage of the bank of capacitors [1,6].
∗
Corresponding author. Tel.: + 30 210 7723 791; fax: +30 210 7722 584. E-mail addresses: [email protected] (I.M. Katiniotis), [email protected] (M.G. Ioannides), [email protected] (P.G. Vernados). 0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.07.034
In this paper the two WECS configurations, SEIG and DOIG, are studied. Both configurations use an induction generator of the same type and ratings, which operates in the saturated region of its magnetic characteristics. Following to this both system’s properties and dynamic responses are studied at the same operating points.
2. Experimental system The induction generator system is shown in Fig. 1. It consists of an 1 kW, four poles, wound rotor induction generator, which is connected first as DOIG and then as SEIG. The first configuration uses a single phase uncontrolled rectifier and a three phase PWM inverter connected at the rotor side. The rectifier converts the line voltage to dc voltage and, then, the PWM inverter converts the dc voltage to three-phase ac voltage Ur at slip frequency fr , which is supplied to the rotor windings. The stator windings are connected to a threephase network and supply the voltage Us at frequency fs . A dc motor simulates the wind turbine, which is mechanically connected to the generator shaft and supplies to the generator the mechanical energy with a driving torque Tw
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Nomenclature C self-excitation capacitance (F) E RMS air gap emf (V) f rated frequency (Hz) isd , isq , ird , irq instantaneous stator and rotor direct and quadrature axis current (A) im , Im instantaneous and RMS magnetizing current (A) Is , Ir RMS stator and rotor current (A) J mass moment inertia of the rotor (kg m2 ) Ls , Lr per phase stator and rotor (referred to stator) inductance (H) M mutual inductance (H) p differential operator (d/dt) P number of poles Ro load resistance () Rs , Rr per phase stator and rotor (referred to stator) resistance () T electromagnetic torque (Nm) Tm mechanical torque (Nm) vsd , vsq , vrd ,vrq instantaneous stator and rotor direct and quadrature axis voltage (V) Vs , Vr RMS stator and rotor voltage (V) Greek letters ωm speed of a P-pole motor (rad/s) ωo speed of a 2-pole motor (rad/s)
and at a rotational speed ω. The generator, the dc motor and the inverter are interfaced to the Measurements and Instrumentation unit, which mainly comprises a digital scope and the measuring instruments and, also, produces outputs to the operator. The second configuration uses a bank of capacitors
connected to the stator side while the rotor windings are shortcircuited. 3. Models of the DOIG and SEIG systems The analysis uses the synchronous reference frame of the two axes theory, with the d–q axes 90o apart in space and rotating at synchronous speed as determined by the electrical angular velocity ω10 of the stator voltages. The voltage equations of the induction machine in the d–q stationary reference frame may be expressed as: 1 vsd = Rs + pLs + pC isd + Mpird vsq = Rs + pLs +
1 pC
isq + Mpirq
(1)
vrd = Mpisd + ωo Misq + (Rr + Lr p)ird + ωo Lr irq vrq = −ωo Misd + Mpisq − ωo Lr ird + (Rr + pLr )irq Eq. (1) represents a set of four differential equations which allow the current response of the induction machine to be determined if the rotor speed ω0 is known. The mechanical part of the system is modeled using additional equations, namely the electromagnetic torque Eq. (2) and the load torque Eq. (3) [7–9]: T =
P Lm (isq ird − isd irq ) 3
dωm 1 = (T − Tm ) dt J
(2) (3)
The relation between ωm for a P-pole motor and ωo for a 2-pole motor is: ωo = (p/2)ωm . The additional impedance 1/pC in the stator winding denotes the d–q axis representation of the excitation capacitors of the SEIG [8].
Fig. 1. (a) Double output induction generator system block diagram, DOIG. (b) Self-excited induction generator block diagram, SEIG.
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In the case of DOIG when the stator is connected to an isolated load the stator voltage vsd and vsq become zero [4,5,7,8]. The d–q axis representation includes the voltage equation given by (4) and the output Eq. (5): 0 = (Rs + Ro )isd + Ls pird + Mpird 0 = (Rs + Ro )isq + Ls pisq + Mpirq vrd = Mpisd + ωo Misq + (Rr + Lr p)ird + ωo Lr irq
(4)
vrq = −ωo Misd + Mpisq − ωo Lr ird + (Rr + Lr p) irq vsd = Ro isd (5)
vsq = Ro isq
In the case of SEIG a resistive load is added in parallel with the self-excitation capacitance and the capacitance term 1/pC becomes R/(1 + RpC). Moreover no external voltage is applied to either the stator or the rotor windings [6]. The voltage equation is given by the system of Eq. (6):
The prediction of both induction generator steady state and transient performance requires proper account of the effect of saturation. In this paper the effects of core saturation are incorporated in simulation using an n-order polynomial approximation of the variation of mutual inductance M versus magnetizing current Im [2]. The procedure is as follows: 1 The magnetization characteristic given in Fig. 2 is measured by using the no load test at synchronous speed. According to this test, the machine is connected to a variable voltage at rated frequency 50 Hz and the magnetizing current is measured as the no load stator current. 2 Vectors [V] and [Im ] containing the measured values of the variable voltage and magnetizing current are set up and fed into the computer program. 3 The corresponding values of the rms air gap emf E and the mutual inductance M at 50 Hz are computed for every set of corresponding values of vector [V] and [Im ]. Thus, for the k-th set of values holds: E(k) = V (k) − (Rs + jXs )Im (k)
R 0 = Rs isd + Ls pisd + isd + Mpird 1 + RpC 0 = Rs isq + Ls pisq +
R isq + Mpirq 1 + RpC
M(k) = (6)
0 = Mpisd + ωo Misq + (Rr + Lr p)ird + ωo Lr irq 0 = −ωo Misd + Mpisq − ωo Lr ird + (Rr + Lr p)irq
(7)
E(k) 2πfIm (k)
(8)
These two vectors [Im ] and [M] containing the numerical values for magnetizing current and mutual inductance can be used by the program. 4 A polynomial fitting routine is used. Given the values of vector [Im ], the routine finds two n-order polynomials p1 and p2 such that p1 ([Im ]) and p2 ([−Im ]) fits the values of vector [M] in a least-squares sense. n n−1 M(k) ∼ (k) + imn−1 Im (k) = p1 (Im (k)) = imn Im
4. Saturation model The topic of this paper concerns only the operation in the saturated region of the two systems. Thus, the experiment was designed in such operating conditions that the induction generator be saturated over all operation region, from no load up to full load, as can be seen from the magnetic characteristics from Fig. 2.
265
+ · · · + im1 Im (k) + imo
(9)
n n−1 M(k) ∼ (k) + imn−1 Im (k) = p2 (−Im (k)) = imn Im
+ · · · + im1 Im (k) + imo
(10)
To verify the approximation, the polynomial p(Im ) is evaluated at different values of Im . A good correlation is found for the sixth order polynomial fitting, n = 6. The numerical values for the polynomial coefficients obtained are listed in Table 1. Plots of measured and computed curves by using this procedure are given in Fig. 3. The good approximation obtained is evident. Table 1 The coefficients of the polynomials p1 (Im (k)) and p2 (−Im (k)) p1 (Im (k))
Fig. 2. Magnetic characteristics.
im6 im5 im4 im3 im2 im1 im0
= −0.062 = 0.681 = −2.779 = 5.313 = −4.679 = 1.186 = 0.92
p2 (−Im (k)) im6 im5 im4 im3 im2 im1 im0
= −0.062 = −0.681 = −2.779 = −5.313 = −4.679 = −1.186 = 0.92
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their final steady state values have been investigated and the obtained results are in good agreement with those reported in [6,9].
5. Results
Fig. 3. Magnetizing characteristics computed with the polynomial fitting method. The two branches corresponding to both positive and negative magnetizing current are shown.
The saturation model based on the polynomial fitting method, was used in the transient analysis of the induction machine operating either as SEIG or DOIG and it is shown in Fig. 4. This model performs an internal iteration at each integration step because of the nonlinear relationship between mutual inductance M and magnetizing current im . Given the initial condition from the previous integration step, the magnitude of the magnetizing current im is calculated by the function Fcn as: im = (isd + ird )2 + (isq + irq )2 (11) The other two functions Fcn1 and Fcn2 used in the saturation model express the polynomial approximation of the experimentally determined functional dependency of the mutual inductance M on magnetizing current im . In this case the output M passes through Fcn1 or Fcn2 corresponding to the positive or negative branch of the magnetizing current, respectively, Fig. 4. The simulation model described by Eqs. (1)–(6) of the two systems shown in Fig. 1 was solved using Simulinksoftware package to simulate the operation and investigate the performance of autonomous wind systems equipped with self-excited or double output inductions generators. In case of DOIG, system transients were studied from Simulink simulation during switching in and out of the supply grid and reported in [7]. In case of SEIG, the transient process of voltage and current build-up during self-excitation as well as the
Fig. 4. Saturation model based on polynomial fitting method.
First, the magnetic characteristic of the machine was identified and recorded, Fig. 2, and then the generator was loaded in the saturation region. The experimental results obtained for the SEIG and the DOIG were studied during cut-in and cut-out of the bank of capacitors and of the excitation voltage and frequency, respectively. During all the tests the system load was kept constant and the generated steady state and transient stator voltage and current at constant mechanical power supplied to both systems were obtained. Waveforms of the generated stator voltage and current of the machine operating as DOIG during cut-in of the excitation voltage and frequency at the rotor side are shown in Plate 1. Initially the machine being without excitation is driven at constant speed of 1887 rpm by a dc motor while the stator terminals are connected to a resistive load of Ro = 65 . In this situation no voltage or current are generated to the stator side. At t = 20 ms the rotor windings are directly connected to the excitation voltage Vr = 20 V and frequency fr = 35 Hz. The ac waveforms of the generated stator voltage and current with a frequency equal to that of the excitation voltage, i.e. fr = 35 Hz are distorted due to the existence of harmonics produced by the inverter and the nonlinear magnetic circuit. Under the same speed and load conditions and with the machine operating as SEIG a bank of capacitors is directly connected in parallel to the stator terminals. The transient build-up of the generated stator voltage and current during the selfexcitation process shown in Plate 2 is continued until there is virtually no change in the value of mutual inductance M, with a self-excitation frequency very close to the frequency
Plate 1. DOIG operation at cut in the rotor excitation voltage. Stator voltage (CH1) and stator current (CH2) at 1887 rpm, Ro = 65 , Vr = 20 V, fr = 35 Hz, CH1 probe coefficient = 1, CH2 probe coefficient = 10.
I.M. Katiniotis et al. / Journal of Materials Processing Technology 161 (2005) 263–268
Plate 2. SEIG operation at cut in the bank of capacitors. Stator voltage (CH1) and stator current (CH2) at 1887 rpm, Ro = 65 , C = 3 × 100 F, CH1 probe coefficient = 100, CH2 probe coefficient = 10.
corresponding to the speed of the motor. The final ac waveform of the steady state voltage obtained for about 500 ms is higher than these corresponding to the case of DOIG. Plates 3 and 4 present the dynamic performance of the induction generator in case of DOIG and SEIG respectively, running with the same load Ro = 65 as in the previous experiments. The induction generator operates in a transient state when accelerated from start, and settles to a steady state. In the case of DOIG, the excitation voltage and frequency are at Vr = 125 V, and fr = 23 Hz and the generator is forced to rotate with a shaft speed of 1750 rpm. Waveforms of the generated stator voltage and current are shown in Plate 3. In the case of SEIG the generator is driven at constant speed of 1750 rpm and a decrease in the generated frequency during the transient voltage and current build-up is easily observed in comparison to the higher speed conditions as in the case shown in Plate 2.
Plate 3. DOIG operation at starting with cut in the excitation voltage. Stator voltage (CH1) and stator current (CH2) at 1750 rpm, Ro = 65 , Vr = 125 V, fr = 23 Hz, CH1 probe coefficient = 1, CH2 probe coefficient = 10.
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Plate 4. SEIG operation at starting with cut in the bank of capacitors. Stator voltage (CH1) and stator current (CH2) at 1750 rpm, Ro = 65 , C = 3 × 100 F, CH1 probe coefficient = 100, CH2 probe coefficient = 10.
Plate 5. SEIG operation: capacitors cut in at 0.25 s and cut out at 0.5 s. Stator voltage (CH1) and stator current (CH2) at 1750 rpm, Ro = 130 , C = 3 × 100 F, CH1 probe coefficient = 100, CH2 probe coefficient = 10.
Plate 6. SEIG operation: capacitors cut in at 0.25 s and cut out at 0.75 s. Stator voltage (CH1) and stator current (CH2) at 1750 rpm, Ro = 65 , C = 3 × 100 F, CH1 probe coefficient = 100, CH2 probe coefficient = 10.
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Finally, Plates 5 and 6 show the transient operation of the SEIG during cut in and cut out of the bank of capacitors, as well as the steady generated stator voltage and current waveforms, for the same shaft speed at 1750 rpm and system load at Ro = 130 and Ro = 65 respectively. Again a decrease in self-excitation frequency occurred at higher values of the system load is easily observed as well as a reduced time of the transient processes.
6. Conclusions The analysis, the implementation and the experimental validation for a SEIG and DOIG operating as WECS in the region of magnetic saturation and supplying a variable load were carried out. Then the results obtained were investigated. Both systems’ performances were studied considering different wind speed, excitation voltage and load conditions. A dynamic model has been developed and presented for investigating the steady state, as well as the transient behavior of the induction generator operating as DOIG or SEIG, during switching on and off from the power line or the bank of capacitors respectively. Waveforms of the generated voltage and current have been obtained taking into account the variation of mutual inductance with magnetization current using an n-order polynomial approximation. The obtained results show the transient voltage and current build-up of the two systems, their harmonic distortion as well as the damping time of the transient process. Also, the results indicate that an adequate procedure is the operation of this system as DOIG in order to better utilize the wind energy.
Acknowledgement The authors express their gratitude to Mr. Panayotis Zannis for the technical assistance during the experiments described in this paper.
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