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Dynamic behavior of a three-phase self-excited induction generator for single-phase power generation
Electric Power Systems Research 48 (1998) 37 – 44
Dynamic behavior of a three-phase self-excited induction generator for single-phase power generation L.B. Shilpakar, Bhim Singh * Department of Electrical Engineering, IIT Delhi, New Delhi-110 016, India Received 3 April 1998; accepted 27 April 1998
Abstract The inherent advantages of three-phase self-excited induction generators have resulted in their use in renewable energy applications. A practical use for single-phase power generation from three-phase induction generators is the extention of the flexibility and range of application of such generators. Previous work in this field has been limited to steady-state operating conditions. The main objective of this paper is to investigate the transient behavior of the three-phase self-excited induction generator (SEIG) for single-phase power generation. The dynamic model of asymmetrical excitation configuration of a three-phase SEIG is developed, based on stationary reference frame d –q axes theory. The effect of cross saturation is also incorporated in the model. Results of simulation and experimental recordings are compared for the dynamic conditions such as initiation of self excitation, load perturbation and short circuit. The suitability of the scheme for constant-power application is also discussed. © 1998 Published by Elsevier Science S.A. All rights reserved. Keywords: Self-excited induction generator; Single-phase power generation; Cross-saturation
1. Introduction In the isolated regions where grid extension is not economical, the self-excited induction generator (SEIG) is used to generate power from renewable sources of energy, such as micro-hydro, wind and biomass. Use of the SEIG is encouraged, due to its inherent advantages [1]. In remote and rural areas, the population is sparsely distributed and the electric loads for the purpose of lighting, heating, water pumping, etc. are usually of single-phase types. In such cases, the single-phase power supply is preferred to three-phase one, in order to render the distribution system simple and cost effective. The induction machines, used for power generation from renewable sources of energy, would be relatively small in rating, size and cost (less than 20 kW). However, single-phase induction machines are not easily available in integral kW ratings. Moreover, three-phase induction machines have higher efficiency and lower cost than an equivalent-sized single-phase machine. Due to these reasons, it would be desirable to use a three-phase induction machine as a * Corresponding author: Fax: +91 11 6862037.
three-phase SEIG for single-phase power generation [2–4]. However, the use of three-phase SEIG for supplying single-phase loads is an extreme case of unbalanced operation, causing additional losses in the equipment due to the flow of negative sequence current component. Under these conditions, a three-phase machine would have to be de-rated in order to keep the temperature of the machine within allowable limits. Therefore, it is considered desirable to evaluate a singlephase load that can be placed on the three-phase SEIG while maintaining the phase currents of the generator as balanced as possible. Smith [2] had suggested the balanced operation of a three-phase induction generator connected to a singlephase line using two-phase shifting capacitors. This method requires the use of unit-ratio transformer which makes the system costly. A scheme of three-phase induction generator self-excited by a single excitation capacitor across any two stator terminals and supplying a single-phase load connected across it was reported in [3]. The work reported in [4] deals with the performance of a three-phase induction generator when a single excitation capacitor is connected across one phase or between two lines and loads are connected to the other
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L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
phases or lines. Durham et al. [5] studied the feasibility of operating a grid-connected three-phase induction generator in a single-phase power system using power system balancers. An unbalanced excitation configuration which uses capacitances C and 2C respectively, across two-phases with no capacitor connection across the third phase (referred to as C-2C mode) is proposed in [6], where a three-phase SEIG feeding a single-phase load appears balanced at the maximum allowable power output. However, all the studies carried out in [2 – 6] were limited to steady-state operating conditions only. The knowledge of transient values of voltage, current, torque, etc. during different dynamic conditions, such as initial excitation, load perturbation and shortcircuit are equally important for the proper design of the generator, as well as its protection system. For instance, the transient voltages across the excitation capacitances must be known for the purpose of specifying their voltage ratings. The winding current in one phase may be different from another phase due to unbalanced operation of the machine and thus, their transient values must be examined in order to determine appropriate settings of the protection system. Furthermore, the knowledge of the instantaneous torque developed in the machine is necessary to assess the suitability of the shaft dimension. The transient process of voltage and current, both during self excitation and load perturbations of the SEIG, was reported by Grantham et al. [7], but the effect of cross-saturation was not included in the model. The incorporation of the cross-magnetizing effect in the dynamic model of the SEIG was made by Hallenius et al. and Sridhar et al. [8,9], but the literature dealt with the operation of the three-phase SEIG under balanced conditions. A complex current state-space model is presented by Levi and Rauski [10] to predict the transient process of saturated deep-bar and double-cage SEIG for windelectricity generation. This investigation presents the dynamic behavior of a three-phase SEIG subjected to single-phase loading with C-2C connection of excitation capacitors. The dynamic model of the system developed is based on stationary reference frame d – q axis current components as state space variables. The voltage-current equation described by Hallenius et al. [8] is modified to accommodate the terminal condition of C-2C excitation configuration of the SEIG. The main path saturation is modeled by expressing magnetizing inductance (Lm) as a fourth-order polynomial function of magnetizing current (Lm), using the data obtained through synchronous speed test. The developed model also incorporates the effect of cross-saturation in the magnetic circuit of the machine. The machine performances are computed by solving the non-linear differential equations of the developed dynamic model via a numerical method. The
computed results are compared with the experimental ones to validate the developed model.
2. Modeling Although there are various models available for the transient analysis of an induction machine, the d–q axes model is shown to be more useful in the case of the SEIG, because the cross-saturation effect can be incorporated easily in this model [8–10]. While either the stationary or the synchronously-rotating reference frames are most frequently used, the former one is preferred for analysis of the SEIG, because the frequency of quantities such as terminal voltage, generator current, capacitor current, etc. are found to vary during transient conditions. The model is based on the assumptions that: (a) the air-gap mmf and the flux density are both sinusoidally distributed in space, so that the phase-to-phase mutual inductance coefficients are simple functions of the rotor position; and (b) the effect of saturation in leakage reactances is negligible. The voltage-current relationship of the three-phase induction machine in the d–q axes stationary reference frame is given in Appendix A. As the three-phase induction generator used in the system shown in Fig. 1 is symmetrical in structure, the voltage-current relationship Eq. (A-2)in Appendix A can be applied along with its proper terminal conditions. The basic terminal equations of the proposed SEIG configuration (Fig. 1) are given as
& &
6sb =
1 2C
6sa =
1 C
ic dt
(1)
i2c dt
(2)
6sa = iLRL
(3)
ic = isc − isa − iL
(4)
i2c = − isb + isc
(5)
Fig. 1. Three-phase SEIG with C-2C excitation feeding power to single-phase load.
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
Using these equations, the stator phase voltages 6sa and 6sb can be modified as
& &
6sa =
1 C
6sb =
1 2C
isc −isa −
6sa dt RL
(6)
(isc −isb ) dt
Lsdpids + Ldqpiqs + LmDpidr +Ldqpiqr idr Lsd Ldqiqs Ldqiqr ids − −LmD − CRL CRL CRL CRL
1 R qds qqs + − − s − 3 2 RL C 2C
im = i 2md + i 2mq
+ (LmD + 3 Ldq )pidr +( 3 LmQ +Ldq )piqr =
3 q C qs
(9)
where pqds =ids
(10)
pqqs =iqs
(11)
and p = (d/dt) is a time derivative operator. As there is no change in rotor terminal conditions, the rotor d– q voltage-current equations given in Appendix A are directly applicable and can be written as
where imd = ids + idr and imq = iqs + iqr are direct and quadrature axes components of the magnetizing current space vector. The magnetizing inductance Lm is calculated from non-linear function of im given in Appendix B. The effect of time derivative of Lm has been included in the dynamic model while deriving the direct and quadrature-axes components of the magnetizing voltage as shown in Appendix A. The torque balance equation of mechanical motion of the system is given as Tshaft = Te + J
LmDpids +Ldqpiqs +Lrdpidr +Ldqpiqr
2 pvr P
(15)
(12)
= − Rridr +vrLmiqs +vrLriqr
where Te is the developed electromagnetic torque which can be expressed as
Ldqpids + LmQpiqs +Ldqpidr +Lrqpiqr (13)
= − Rriqr −vrLmids −vrLridr
Eqs. (8), (9), (12) and (13) can be written in matrix form with currents as the state vector as [L1]p[i ]= [G1] [i ]+ [X1] [q] t
It is important to note that the dynamic equations deal with non-linear magnetizing inductance due to saturation, which must be suitably incorporated. This requires determination of non-linear variation of the magnetizing inductance Lm with exciting current im. Careful and accurate measurements on the machine by conducting synchronous speed test [1] yield this result. At each computation step, the amplitude of the magnetizing current space vector is calculated as
(8)
(Lsd + 3 Ldq )pids + ( 3 Lsq +Ldq )piqs
− Rsids − 3 Rsiqs +
Æ(− 3/2 − R s /RL )/C − 3/(2C) 0 0Ç Ã Ã 0
3/(2C) 0 0 [X1]= Ã Ã 0 0 0 0 Ã Ã È 0 0 0 0É
(7)
Expressing the phase variables in terms of d-q variables and using Eq. (A-2)in Appendix A, Eqs. (6) and (7) reduce to
= − Rs −
39
(14) t
where [i ]= [ids iqs idr iqr ] and [q]= [qds qqs 0 0]
Te =
3 2
2 L (i i − i i ) P m dr qs qr ds
(16)
Eqs. (10), (11), (14) and (15) can be written in the derivative form for computer simulation as
Æ Ç L sd L dq L mD L dq à à L sd + 3 Ldq 3 L sq + Ldq L mD + 3 Ldq 3 L mQ + Ldq [L1 ]= à à L mD L dq L rd L dq Ã Ã È É L dq L mQ L dq L rq Æ −R s −Lsd /(CRL ) −L dq /(CRL ) −L mD /(CRL ) − L dq /(CRL ) Ç Ã Ã −R s − 3 R s 0 0 [G1]= à à 0 v rLm −Rr v rLr Ã Ã È É − v rLm 0 − v rLr −Rr
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
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pqds =ids p[i ] = [L1] − 1{[G1] [i] +[X] [q]} pqqs =iqs pvr =
P (T − Te) 2J shaft
(17)
The previous seven differential equations (Eq. (17)) describe the dynamic model for the prediction of transient performance of the three-phase SEIG self-excited with C-2C excitation configuration for single-phase power generation feeding resistive load. The mathematical model for the case of inductive load is given in Appendix C and consists of nine first order differential equations.
3. Results and discussion In the laboratory test rig, a three-phase squirrel cage induction machine is coupled to a separately-excited converter-fed DC motor which operates as the prime mover. The DC drive is operated by closed-loop converter control to provide constant-speed operation with different loads. The machine parameters and magnetization characteristic of the induction generator obtained by conducting various tests are given in Appendix B. Test results and simulated performance of the SEIG are presented for conditions, such as the initiation of self excitation, load perturbation and short circuit. The results of steady state analysis for single-phase power generation with C-2C excitation configuration of threephase SEIG is shown in Fig. 2. Comparison of these results with the results of transient analysis obtained from the developed mathematical model is also made in this section.
3.1. Self excitation of the generator The induction generator is driven at the speed of 1.01 p.u. (base speed 1500 rpm) and at time t = 0, two single-phase capacitors of values 15 mf and 30 mf are switched into a–b and b – c phases, respectively. Fig. 3 shows the build-up of phase voltage and current during the self-excitation. It is observed that the transient build-up of terminal voltage continues until the magnetic circuit of the machine is saturated and it settles down to a steady state value, depending upon the prime-mover speed and the value of excitation capacitors. As expected, the amplitude of the generated
Fig. 2. Characteristics of the three-phase SEIG with C-2C excitation for single-phase power generation: (a) C=15 mf; (b) C =23 mf.
voltages and currents under steady state condition in all three-phases do not differ from one another at any load and are given in Table 1.
3.2. Load switching and perturbation The self excitation of the generator continues to sustain when a single-phase unity power factor load of 0.33 p.u. is suddenly applied across the line a– b. As indicated in Fig. 4, the voltage regulation due to this applied load is within the limit of 9 6% of rated voltage. The steady-state waveforms of the generator voltage and currents depicted in Fig. 5 show that the generator is operating near a balanced condition at this load. The close agreement between results obtained from transient and steady state analyses can be observed from Table 1. The load at which a balanced operation is obtained depends on the capacitance value. A higher capacitance value enhances the power capability of the machine (Fig. 2(b)), but results in a poor part-load performance. Therefore, it is confirmed from this analysis that this scheme is suited to constantpower application like micro-hydro power generation
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
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Fig. 4. Voltage regulation of three-phase SEIG with C-2C excitation: x-axis, (t) 50 ms div. − 1; y-axis, (VL ) 0.894 p.u. div. − 1, (iL ) 1.124 p.u. div. − 1 Fig. 3. Voltage and current build-up in the three-phase SEIG with C-2C excitation: x-axis, (t) 200 ms div. − 1; y-axis, (Vsa ) 0.894 p.u. div. − 1, (isa ) 1.124 p.u. div. − 1
[6]. The oscillograms of the generator currents are not perfectly sinusoidal due to the presence of harmonics. However, the wave shape of load voltage and current, as shown in Fig. 6, are observed to be perfectly sinusoidal and therefore, the quality of generated power supplied to the load is maintained as in the case of balanced operation. Furthermore, an additional load of 0.15 p.u. is switched-in/out in order to demonstrate the effect of load perturbation and the resulting transients in the load voltage and current are shown in Fig. 7.
3.3. Short circuit The response of the voltage and current when a sudden short circuit occurs at the load terminal is shown in Fig. 8. A current surge of 6 p.u. appears in the generator current and the voltage collapses within three cycles as the machine de-excites. This kind of
inherent protection against overload and short circuit makes the SEIG scheme cost effective. The close agreement between the simulated results and experimental oscillograms validates the effectiveness of the developed model.
4. Conclusions The dynamic behavior of the scheme of single-phase power generation from a three-phase SEIG with C-2C capacitor excitation is studied by developing a dynamic model in the d–q axes stationary reference frame. Based on a close agreement between the simulated and experimental oscillograms drawn at different transient conditions, it can be concluded that the approach made in the development of dynamic model is elegant. The scheme can withstand the effect of load perturbation as long as the load does not exceed the maximum power limit, as given by the steady state load characteristics. The single-phase load at which the balanced operation of a three-phase SEIG takes place depends upon the
Table 1 Comparison of steady state and transient analysis with test results Excitation capacitance (phase) a–b
b –c
15 mf
30 mf
Load (p.u.)
No load
0.33
Method of analysis
Steady state Transient Experimental Steady state Transient Experimental
Quantities at steady state condition (p.u.) Vsa
Vsb
Vsc
Isa
Isb
Isc
1.043 1.038 1.060 0.925 0.934 0.936
1.048 1.024 1.060 0.925 0.985 0.925
0.963 0.963 0.983 0.925 0.910 0.925
0.559 0.559 0.667 0.491 0.522 0.547
0.642 0.588 0.570 0.489 0.455 0.502
0.286 0.244 0.310 0.490 0.436 0.500
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L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
Fig. 5. Balanced operation of three-phase SEIG with C-2C excitation at partial loading: x-axis, (t) 10 ms div. − 1; y-axis, (6sa ) 0.894 p.u. div. − 1, (isa, isb, isc ) 1.124 p.u. div. − 1
value of capacitance. Since balanced operation is maintained only at a fixed load, the scheme can be used for constant-power application, such as micro-hydro power generation. The fairly good agreement achieved between the results obtained from transient analysis and steady state analysis further validates the developed mathematical model.
Appendix A The stationary reference frame d –q transformation of stator phase variables to d – q variables is carried out by the transformation as follows. [Vabc ]= [K] [Vodq ]
Æ L sd L dq L mD L dq Ç Ã Ã L dq L sq L dq L mQ à [L]= à L mD L dq L rd L dq Ã Ã È L dq L mQ L dq L rq É Æ 0 0 0Ç 0 à à 0 0 0 0 [G] = à à 0 Lm Lr 0 Ã Ã È −L m 0 − L r 0 É Lsd = Lls + LmD, Ldq = Lls + LmQ, Lrd = Llr + LmD, Lrq = Llr + LmQ and Lr = Lm + Llr.
(A-1)
where the transformation matrix [K] is given by
Æ1 Cos u [K] = Ã1 Cos (u − f) È1 Cos (u + f)
Sin u Ç Sin (u −f)Ã Sin (u +f)É
where f= 120° and u is an angle between a-phase axis and d-axis. Selecting the d-axis to align the a-phase axis, in stationary reference frame u = 0°. Similar transformation can also be applied to current and flux linkage quantities. The voltage-current relationship of a three-phase cage induction machine is expressed as [7,10]: [V] = [R] [i] +[L]p[i] +[G] [i]vr
(A-2)
where [V]= [Vds Vqs 0 0]t, [i ]= [ids iqs idr iqr ]t and [R] = diag [Rs Rs Rr Rr ]
Fig. 6. Voltage and current waveforms at load terminal: x-axis, (t) 10 ms div. − 1; y-axis, (6L ) 0.894 p.u. div. − 1, (iL ) 1.124 p.u. div. − 1
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
Vmq =
43
dcmq d(Lmimq ) dLm dimq = = imq + Lm dt dt dt dt
(A-3)
where cmd and cmq are the magnetizing flux linkages in the d and q axes, respectively and imd = ids +idr and imq = iqs + iqr are direct and quadrature axes components of magnetizing current space vector. Since Lm is dependent on the magnetizing current, the time derivative of Lm can be expanded as follows: dLm dLm d i 2md + i 2mq = dt d Im d(i 2md + i 2mq) =
dLm 1 dimd dimq imd + imq d Im Im dt dt
n
(A-4)
Substituting the expression of Eq. (A-4) into Eq. (A-3) and on simplification, we have Fig. 7. Load voltage and current during load perturbation: x-axis, (t) 200 ms div. − 1; y-axis, (6L ) 0.894 p.u. div. − 1, (iL ) 1.124 p.u.div. − 1
Vmd = LmD Vmd = Ldq
p =(d/dt) is a time differential operator and other notations have their usual meanings [8,9,11]; Ldq represents the cross-coupling between the axes in space quadrature due to saturation; LmDd and LmDq are direct and quadrature axes magnetizing inductances and their expressions are derived as follows. The direct and quadrature-axes component of the magnetizing voltage are defined as
dimd di + Ldq mq dt dt
dimd di + LmQ mq dt dt
(A-5)
where Ldq =
dLm imdimq d Im Im
(A-6)
LmD = Lm + Ldq
imd imq
(A-7)
LmQ = Lm + Ldq
imq imd
(A-8)
dc d(Lmimd ) dLm di Vmd = md = = i + md Lm dt dt dt md dt Appendix B
3.7 kW, 3-phase, 4 pole, 50 Hz, 415 V, 7.6 A, delta connected,1420 rpm, Rs = 0.053 p.u., Rr,= 0.061 p.u., Xls = 0.087, Xlr = 0.087 p.u., Rm = 28.5 p.u., base voltage= 415 V, base current=4.38 A, base power= 3.7 kW. The relationship between Lm and Im is given by Lm = 0.84 for Im 5 0.77 =0. 89−4.0039 Im − 0.1071 I 2m + 0.0245 I 3m − 0.001 2 I 4m for 0.775 Im 5 4.0 Lm = 0 43 for Im ] 4.0
Appendix C For the case of inductive load, the basic terminal equations Eqs. (1), (2), (4) and (5) remain the same, while Eq. (3) is modified as Fig. 8. Transients in voltage and current of the SEIG during the occurence of a short circuit at the load terminal: x-axis, (t) 50 ms div. − 1; y-axis, (6L ) 1.788 p.u. div. − 1, (iL ) 4.496 p.u.div. − 1
Vsa = iLRL + LL
diL dt
(A-9)
44
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
The set of differential equations describing the mathematical model of the C-2C excitation configuration of the SEIG feeding inductive load can be expressed as p[i] = [L2] − 1{[G2] [i]+ [X2] [q]} pqds =ids
Æ3/(2C) 3 (2C) − 1/C Ã 0 − 3/C 0 [X2]= Ã 0 0 0 Ã 0 0 0 È 0 0 0
0 0 0 0 0
0Ç 0Ã 0Ã 0Ã 0É
pqqs =iqs References
pqL =iL P pvr = (Tshaft −Te) 2J
(A-10)
where [i ] =[ids iqs idr iqr iL ]t and [q]=[qds qqs qL 0 0]t [L2]
Æ L sd L dq L mD ÃL sd + 3 Ldq 3 L sq +Ldq L mD + =à L mD L dq à L dq L mQ È L sd L dq L dq 0 Ç
3 L mQ +Ldq 0 Ã 0 Ã
3 LdqLrd L dq 0 Ã Ldq L rq LmD L dq −L L É Æ Rs 0 0 0 0 Ç Ã − R s − 3 R s 0 0 0 Ã [G2] = Ã 0 v rLm −Rr v rLr 0 Ã Ã − v rLm 0 −v rLr −R r 0 Ã È −Rs 0 0 0 RLÉ
.
[1] L. Sridhar, Bhim Singh, C.S. Jha, A step towards improvement in the characteristics of the self-excited induction generator, IEEE Trans. Energy Convers. 8 (1) (1993) 40 – 46. [2] O.J.M. Smith, Three-phase induction generator for single-phase line, IEEE Trans. Energy Convers. EC-2 (3) (1987) 382–387. [3] A.H. Al-Bahrani, N.H. Malik, Steady state analysis and performance characteristics of a three-phase induction generator self excited with a single capacitor, IEEE Trans. Energy Convers. 5 (4) (1990) 725 – 732. [4] Y.H.A. Rahim, Excitation of isolated three-phase induction generator by a single capacitor, IEE Proc. B 140 (1) (1993) 44 – 50. [5] M.O. Durham, R. Ramakumar, Power system balancers for an induction generator, IEEE Trans. Indust. Appl. IA-23 (6) (1987) 1067 – 1072. [6] J.L. Bhattacharya, J.L. Woodward, Excitation balancing of a self-excited induction generator for maximum power output, IEE Proc. C 135 (2) (1988) 88 – 97. [7] C. Grantham, D. Sutanto, B. Mismail, Steady state and transient analysis of self-excited induction generator, IEE Proc. B 136 (2) (1989) 61 – 68. [8] K.E. Hallenius, P. Vas, J.E. Brown, The analysis of a saturated self-excited asynchronous generator, IEEE Trans. Energy Convers. 6 (2) (1991) 336 – 345. [9] L. Sridhar, Bhim Singh, C.S. Jha, Transient performance of series regulated short shunt self excited induction generator, IEEE Trans. Energy Convers. 10 (2) (1995) 261 – 267. [10] E. Levi, D. Rauski, Modelling of deep-bar and double-cage self-excited induction generators for wind-electricity studies, Electr. Power Syst. Res. 27 (1993) 73 – 81. [11] Paul C. Krause, Analysis of Electric Machinery, McGraw-Hill, New York, 1987.
Dynamic behavior of a three-phase self-excited induction generator for single-phase power generation L.B. Shilpakar, Bhim Singh * Department of Electrical Engineering, IIT Delhi, New Delhi-110 016, India Received 3 April 1998; accepted 27 April 1998
Abstract The inherent advantages of three-phase self-excited induction generators have resulted in their use in renewable energy applications. A practical use for single-phase power generation from three-phase induction generators is the extention of the flexibility and range of application of such generators. Previous work in this field has been limited to steady-state operating conditions. The main objective of this paper is to investigate the transient behavior of the three-phase self-excited induction generator (SEIG) for single-phase power generation. The dynamic model of asymmetrical excitation configuration of a three-phase SEIG is developed, based on stationary reference frame d –q axes theory. The effect of cross saturation is also incorporated in the model. Results of simulation and experimental recordings are compared for the dynamic conditions such as initiation of self excitation, load perturbation and short circuit. The suitability of the scheme for constant-power application is also discussed. © 1998 Published by Elsevier Science S.A. All rights reserved. Keywords: Self-excited induction generator; Single-phase power generation; Cross-saturation
1. Introduction In the isolated regions where grid extension is not economical, the self-excited induction generator (SEIG) is used to generate power from renewable sources of energy, such as micro-hydro, wind and biomass. Use of the SEIG is encouraged, due to its inherent advantages [1]. In remote and rural areas, the population is sparsely distributed and the electric loads for the purpose of lighting, heating, water pumping, etc. are usually of single-phase types. In such cases, the single-phase power supply is preferred to three-phase one, in order to render the distribution system simple and cost effective. The induction machines, used for power generation from renewable sources of energy, would be relatively small in rating, size and cost (less than 20 kW). However, single-phase induction machines are not easily available in integral kW ratings. Moreover, three-phase induction machines have higher efficiency and lower cost than an equivalent-sized single-phase machine. Due to these reasons, it would be desirable to use a three-phase induction machine as a * Corresponding author: Fax: +91 11 6862037.
three-phase SEIG for single-phase power generation [2–4]. However, the use of three-phase SEIG for supplying single-phase loads is an extreme case of unbalanced operation, causing additional losses in the equipment due to the flow of negative sequence current component. Under these conditions, a three-phase machine would have to be de-rated in order to keep the temperature of the machine within allowable limits. Therefore, it is considered desirable to evaluate a singlephase load that can be placed on the three-phase SEIG while maintaining the phase currents of the generator as balanced as possible. Smith [2] had suggested the balanced operation of a three-phase induction generator connected to a singlephase line using two-phase shifting capacitors. This method requires the use of unit-ratio transformer which makes the system costly. A scheme of three-phase induction generator self-excited by a single excitation capacitor across any two stator terminals and supplying a single-phase load connected across it was reported in [3]. The work reported in [4] deals with the performance of a three-phase induction generator when a single excitation capacitor is connected across one phase or between two lines and loads are connected to the other
0378-7796/98/$ - see front matter © 1998 Published by Elsevier Science S.A. All rights reserved. PII S0378-7796(98)00081-9
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L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
phases or lines. Durham et al. [5] studied the feasibility of operating a grid-connected three-phase induction generator in a single-phase power system using power system balancers. An unbalanced excitation configuration which uses capacitances C and 2C respectively, across two-phases with no capacitor connection across the third phase (referred to as C-2C mode) is proposed in [6], where a three-phase SEIG feeding a single-phase load appears balanced at the maximum allowable power output. However, all the studies carried out in [2 – 6] were limited to steady-state operating conditions only. The knowledge of transient values of voltage, current, torque, etc. during different dynamic conditions, such as initial excitation, load perturbation and shortcircuit are equally important for the proper design of the generator, as well as its protection system. For instance, the transient voltages across the excitation capacitances must be known for the purpose of specifying their voltage ratings. The winding current in one phase may be different from another phase due to unbalanced operation of the machine and thus, their transient values must be examined in order to determine appropriate settings of the protection system. Furthermore, the knowledge of the instantaneous torque developed in the machine is necessary to assess the suitability of the shaft dimension. The transient process of voltage and current, both during self excitation and load perturbations of the SEIG, was reported by Grantham et al. [7], but the effect of cross-saturation was not included in the model. The incorporation of the cross-magnetizing effect in the dynamic model of the SEIG was made by Hallenius et al. and Sridhar et al. [8,9], but the literature dealt with the operation of the three-phase SEIG under balanced conditions. A complex current state-space model is presented by Levi and Rauski [10] to predict the transient process of saturated deep-bar and double-cage SEIG for windelectricity generation. This investigation presents the dynamic behavior of a three-phase SEIG subjected to single-phase loading with C-2C connection of excitation capacitors. The dynamic model of the system developed is based on stationary reference frame d – q axis current components as state space variables. The voltage-current equation described by Hallenius et al. [8] is modified to accommodate the terminal condition of C-2C excitation configuration of the SEIG. The main path saturation is modeled by expressing magnetizing inductance (Lm) as a fourth-order polynomial function of magnetizing current (Lm), using the data obtained through synchronous speed test. The developed model also incorporates the effect of cross-saturation in the magnetic circuit of the machine. The machine performances are computed by solving the non-linear differential equations of the developed dynamic model via a numerical method. The
computed results are compared with the experimental ones to validate the developed model.
2. Modeling Although there are various models available for the transient analysis of an induction machine, the d–q axes model is shown to be more useful in the case of the SEIG, because the cross-saturation effect can be incorporated easily in this model [8–10]. While either the stationary or the synchronously-rotating reference frames are most frequently used, the former one is preferred for analysis of the SEIG, because the frequency of quantities such as terminal voltage, generator current, capacitor current, etc. are found to vary during transient conditions. The model is based on the assumptions that: (a) the air-gap mmf and the flux density are both sinusoidally distributed in space, so that the phase-to-phase mutual inductance coefficients are simple functions of the rotor position; and (b) the effect of saturation in leakage reactances is negligible. The voltage-current relationship of the three-phase induction machine in the d–q axes stationary reference frame is given in Appendix A. As the three-phase induction generator used in the system shown in Fig. 1 is symmetrical in structure, the voltage-current relationship Eq. (A-2)in Appendix A can be applied along with its proper terminal conditions. The basic terminal equations of the proposed SEIG configuration (Fig. 1) are given as
& &
6sb =
1 2C
6sa =
1 C
ic dt
(1)
i2c dt
(2)
6sa = iLRL
(3)
ic = isc − isa − iL
(4)
i2c = − isb + isc
(5)
Fig. 1. Three-phase SEIG with C-2C excitation feeding power to single-phase load.
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
Using these equations, the stator phase voltages 6sa and 6sb can be modified as
& &
6sa =
1 C
6sb =
1 2C
isc −isa −
6sa dt RL
(6)
(isc −isb ) dt
Lsdpids + Ldqpiqs + LmDpidr +Ldqpiqr idr Lsd Ldqiqs Ldqiqr ids − −LmD − CRL CRL CRL CRL
1 R qds qqs + − − s − 3 2 RL C 2C
im = i 2md + i 2mq
+ (LmD + 3 Ldq )pidr +( 3 LmQ +Ldq )piqr =
3 q C qs
(9)
where pqds =ids
(10)
pqqs =iqs
(11)
and p = (d/dt) is a time derivative operator. As there is no change in rotor terminal conditions, the rotor d– q voltage-current equations given in Appendix A are directly applicable and can be written as
where imd = ids + idr and imq = iqs + iqr are direct and quadrature axes components of the magnetizing current space vector. The magnetizing inductance Lm is calculated from non-linear function of im given in Appendix B. The effect of time derivative of Lm has been included in the dynamic model while deriving the direct and quadrature-axes components of the magnetizing voltage as shown in Appendix A. The torque balance equation of mechanical motion of the system is given as Tshaft = Te + J
LmDpids +Ldqpiqs +Lrdpidr +Ldqpiqr
2 pvr P
(15)
(12)
= − Rridr +vrLmiqs +vrLriqr
where Te is the developed electromagnetic torque which can be expressed as
Ldqpids + LmQpiqs +Ldqpidr +Lrqpiqr (13)
= − Rriqr −vrLmids −vrLridr
Eqs. (8), (9), (12) and (13) can be written in matrix form with currents as the state vector as [L1]p[i ]= [G1] [i ]+ [X1] [q] t
It is important to note that the dynamic equations deal with non-linear magnetizing inductance due to saturation, which must be suitably incorporated. This requires determination of non-linear variation of the magnetizing inductance Lm with exciting current im. Careful and accurate measurements on the machine by conducting synchronous speed test [1] yield this result. At each computation step, the amplitude of the magnetizing current space vector is calculated as
(8)
(Lsd + 3 Ldq )pids + ( 3 Lsq +Ldq )piqs
− Rsids − 3 Rsiqs +
Æ(− 3/2 − R s /RL )/C − 3/(2C) 0 0Ç Ã Ã 0
3/(2C) 0 0 [X1]= Ã Ã 0 0 0 0 Ã Ã È 0 0 0 0É
(7)
Expressing the phase variables in terms of d-q variables and using Eq. (A-2)in Appendix A, Eqs. (6) and (7) reduce to
= − Rs −
39
(14) t
where [i ]= [ids iqs idr iqr ] and [q]= [qds qqs 0 0]
Te =
3 2
2 L (i i − i i ) P m dr qs qr ds
(16)
Eqs. (10), (11), (14) and (15) can be written in the derivative form for computer simulation as
Æ Ç L sd L dq L mD L dq à à L sd + 3 Ldq 3 L sq + Ldq L mD + 3 Ldq 3 L mQ + Ldq [L1 ]= à à L mD L dq L rd L dq Ã Ã È É L dq L mQ L dq L rq Æ −R s −Lsd /(CRL ) −L dq /(CRL ) −L mD /(CRL ) − L dq /(CRL ) Ç Ã Ã −R s − 3 R s 0 0 [G1]= à à 0 v rLm −Rr v rLr Ã Ã È É − v rLm 0 − v rLr −Rr
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
40
pqds =ids p[i ] = [L1] − 1{[G1] [i] +[X] [q]} pqqs =iqs pvr =
P (T − Te) 2J shaft
(17)
The previous seven differential equations (Eq. (17)) describe the dynamic model for the prediction of transient performance of the three-phase SEIG self-excited with C-2C excitation configuration for single-phase power generation feeding resistive load. The mathematical model for the case of inductive load is given in Appendix C and consists of nine first order differential equations.
3. Results and discussion In the laboratory test rig, a three-phase squirrel cage induction machine is coupled to a separately-excited converter-fed DC motor which operates as the prime mover. The DC drive is operated by closed-loop converter control to provide constant-speed operation with different loads. The machine parameters and magnetization characteristic of the induction generator obtained by conducting various tests are given in Appendix B. Test results and simulated performance of the SEIG are presented for conditions, such as the initiation of self excitation, load perturbation and short circuit. The results of steady state analysis for single-phase power generation with C-2C excitation configuration of threephase SEIG is shown in Fig. 2. Comparison of these results with the results of transient analysis obtained from the developed mathematical model is also made in this section.
3.1. Self excitation of the generator The induction generator is driven at the speed of 1.01 p.u. (base speed 1500 rpm) and at time t = 0, two single-phase capacitors of values 15 mf and 30 mf are switched into a–b and b – c phases, respectively. Fig. 3 shows the build-up of phase voltage and current during the self-excitation. It is observed that the transient build-up of terminal voltage continues until the magnetic circuit of the machine is saturated and it settles down to a steady state value, depending upon the prime-mover speed and the value of excitation capacitors. As expected, the amplitude of the generated
Fig. 2. Characteristics of the three-phase SEIG with C-2C excitation for single-phase power generation: (a) C=15 mf; (b) C =23 mf.
voltages and currents under steady state condition in all three-phases do not differ from one another at any load and are given in Table 1.
3.2. Load switching and perturbation The self excitation of the generator continues to sustain when a single-phase unity power factor load of 0.33 p.u. is suddenly applied across the line a– b. As indicated in Fig. 4, the voltage regulation due to this applied load is within the limit of 9 6% of rated voltage. The steady-state waveforms of the generator voltage and currents depicted in Fig. 5 show that the generator is operating near a balanced condition at this load. The close agreement between results obtained from transient and steady state analyses can be observed from Table 1. The load at which a balanced operation is obtained depends on the capacitance value. A higher capacitance value enhances the power capability of the machine (Fig. 2(b)), but results in a poor part-load performance. Therefore, it is confirmed from this analysis that this scheme is suited to constantpower application like micro-hydro power generation
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
41
Fig. 4. Voltage regulation of three-phase SEIG with C-2C excitation: x-axis, (t) 50 ms div. − 1; y-axis, (VL ) 0.894 p.u. div. − 1, (iL ) 1.124 p.u. div. − 1 Fig. 3. Voltage and current build-up in the three-phase SEIG with C-2C excitation: x-axis, (t) 200 ms div. − 1; y-axis, (Vsa ) 0.894 p.u. div. − 1, (isa ) 1.124 p.u. div. − 1
[6]. The oscillograms of the generator currents are not perfectly sinusoidal due to the presence of harmonics. However, the wave shape of load voltage and current, as shown in Fig. 6, are observed to be perfectly sinusoidal and therefore, the quality of generated power supplied to the load is maintained as in the case of balanced operation. Furthermore, an additional load of 0.15 p.u. is switched-in/out in order to demonstrate the effect of load perturbation and the resulting transients in the load voltage and current are shown in Fig. 7.
3.3. Short circuit The response of the voltage and current when a sudden short circuit occurs at the load terminal is shown in Fig. 8. A current surge of 6 p.u. appears in the generator current and the voltage collapses within three cycles as the machine de-excites. This kind of
inherent protection against overload and short circuit makes the SEIG scheme cost effective. The close agreement between the simulated results and experimental oscillograms validates the effectiveness of the developed model.
4. Conclusions The dynamic behavior of the scheme of single-phase power generation from a three-phase SEIG with C-2C capacitor excitation is studied by developing a dynamic model in the d–q axes stationary reference frame. Based on a close agreement between the simulated and experimental oscillograms drawn at different transient conditions, it can be concluded that the approach made in the development of dynamic model is elegant. The scheme can withstand the effect of load perturbation as long as the load does not exceed the maximum power limit, as given by the steady state load characteristics. The single-phase load at which the balanced operation of a three-phase SEIG takes place depends upon the
Table 1 Comparison of steady state and transient analysis with test results Excitation capacitance (phase) a–b
b –c
15 mf
30 mf
Load (p.u.)
No load
0.33
Method of analysis
Steady state Transient Experimental Steady state Transient Experimental
Quantities at steady state condition (p.u.) Vsa
Vsb
Vsc
Isa
Isb
Isc
1.043 1.038 1.060 0.925 0.934 0.936
1.048 1.024 1.060 0.925 0.985 0.925
0.963 0.963 0.983 0.925 0.910 0.925
0.559 0.559 0.667 0.491 0.522 0.547
0.642 0.588 0.570 0.489 0.455 0.502
0.286 0.244 0.310 0.490 0.436 0.500
42
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
Fig. 5. Balanced operation of three-phase SEIG with C-2C excitation at partial loading: x-axis, (t) 10 ms div. − 1; y-axis, (6sa ) 0.894 p.u. div. − 1, (isa, isb, isc ) 1.124 p.u. div. − 1
value of capacitance. Since balanced operation is maintained only at a fixed load, the scheme can be used for constant-power application, such as micro-hydro power generation. The fairly good agreement achieved between the results obtained from transient analysis and steady state analysis further validates the developed mathematical model.
Appendix A The stationary reference frame d –q transformation of stator phase variables to d – q variables is carried out by the transformation as follows. [Vabc ]= [K] [Vodq ]
Æ L sd L dq L mD L dq Ç Ã Ã L dq L sq L dq L mQ à [L]= à L mD L dq L rd L dq Ã Ã È L dq L mQ L dq L rq É Æ 0 0 0Ç 0 à à 0 0 0 0 [G] = à à 0 Lm Lr 0 Ã Ã È −L m 0 − L r 0 É Lsd = Lls + LmD, Ldq = Lls + LmQ, Lrd = Llr + LmD, Lrq = Llr + LmQ and Lr = Lm + Llr.
(A-1)
where the transformation matrix [K] is given by
Æ1 Cos u [K] = Ã1 Cos (u − f) È1 Cos (u + f)
Sin u Ç Sin (u −f)Ã Sin (u +f)É
where f= 120° and u is an angle between a-phase axis and d-axis. Selecting the d-axis to align the a-phase axis, in stationary reference frame u = 0°. Similar transformation can also be applied to current and flux linkage quantities. The voltage-current relationship of a three-phase cage induction machine is expressed as [7,10]: [V] = [R] [i] +[L]p[i] +[G] [i]vr
(A-2)
where [V]= [Vds Vqs 0 0]t, [i ]= [ids iqs idr iqr ]t and [R] = diag [Rs Rs Rr Rr ]
Fig. 6. Voltage and current waveforms at load terminal: x-axis, (t) 10 ms div. − 1; y-axis, (6L ) 0.894 p.u. div. − 1, (iL ) 1.124 p.u. div. − 1
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
Vmq =
43
dcmq d(Lmimq ) dLm dimq = = imq + Lm dt dt dt dt
(A-3)
where cmd and cmq are the magnetizing flux linkages in the d and q axes, respectively and imd = ids +idr and imq = iqs + iqr are direct and quadrature axes components of magnetizing current space vector. Since Lm is dependent on the magnetizing current, the time derivative of Lm can be expanded as follows: dLm dLm d i 2md + i 2mq = dt d Im d(i 2md + i 2mq) =
dLm 1 dimd dimq imd + imq d Im Im dt dt
n
(A-4)
Substituting the expression of Eq. (A-4) into Eq. (A-3) and on simplification, we have Fig. 7. Load voltage and current during load perturbation: x-axis, (t) 200 ms div. − 1; y-axis, (6L ) 0.894 p.u. div. − 1, (iL ) 1.124 p.u.div. − 1
Vmd = LmD Vmd = Ldq
p =(d/dt) is a time differential operator and other notations have their usual meanings [8,9,11]; Ldq represents the cross-coupling between the axes in space quadrature due to saturation; LmDd and LmDq are direct and quadrature axes magnetizing inductances and their expressions are derived as follows. The direct and quadrature-axes component of the magnetizing voltage are defined as
dimd di + Ldq mq dt dt
dimd di + LmQ mq dt dt
(A-5)
where Ldq =
dLm imdimq d Im Im
(A-6)
LmD = Lm + Ldq
imd imq
(A-7)
LmQ = Lm + Ldq
imq imd
(A-8)
dc d(Lmimd ) dLm di Vmd = md = = i + md Lm dt dt dt md dt Appendix B
3.7 kW, 3-phase, 4 pole, 50 Hz, 415 V, 7.6 A, delta connected,1420 rpm, Rs = 0.053 p.u., Rr,= 0.061 p.u., Xls = 0.087, Xlr = 0.087 p.u., Rm = 28.5 p.u., base voltage= 415 V, base current=4.38 A, base power= 3.7 kW. The relationship between Lm and Im is given by Lm = 0.84 for Im 5 0.77 =0. 89−4.0039 Im − 0.1071 I 2m + 0.0245 I 3m − 0.001 2 I 4m for 0.775 Im 5 4.0 Lm = 0 43 for Im ] 4.0
Appendix C For the case of inductive load, the basic terminal equations Eqs. (1), (2), (4) and (5) remain the same, while Eq. (3) is modified as Fig. 8. Transients in voltage and current of the SEIG during the occurence of a short circuit at the load terminal: x-axis, (t) 50 ms div. − 1; y-axis, (6L ) 1.788 p.u. div. − 1, (iL ) 4.496 p.u.div. − 1
Vsa = iLRL + LL
diL dt
(A-9)
44
L.B. Shilpakar, B. Singh / Electric Power Systems Research 48 (1998) 37–44
The set of differential equations describing the mathematical model of the C-2C excitation configuration of the SEIG feeding inductive load can be expressed as p[i] = [L2] − 1{[G2] [i]+ [X2] [q]} pqds =ids
Æ3/(2C) 3 (2C) − 1/C Ã 0 − 3/C 0 [X2]= Ã 0 0 0 Ã 0 0 0 È 0 0 0
0 0 0 0 0
0Ç 0Ã 0Ã 0Ã 0É
pqqs =iqs References
pqL =iL P pvr = (Tshaft −Te) 2J
(A-10)
where [i ] =[ids iqs idr iqr iL ]t and [q]=[qds qqs qL 0 0]t [L2]
Æ L sd L dq L mD ÃL sd + 3 Ldq 3 L sq +Ldq L mD + =à L mD L dq à L dq L mQ È L sd L dq L dq 0 Ç
3 L mQ +Ldq 0 Ã 0 Ã
3 LdqLrd L dq 0 Ã Ldq L rq LmD L dq −L L É Æ Rs 0 0 0 0 Ç Ã − R s − 3 R s 0 0 0 Ã [G2] = Ã 0 v rLm −Rr v rLr 0 Ã Ã − v rLm 0 −v rLr −R r 0 Ã È −Rs 0 0 0 RLÉ
.
[1] L. Sridhar, Bhim Singh, C.S. Jha, A step towards improvement in the characteristics of the self-excited induction generator, IEEE Trans. Energy Convers. 8 (1) (1993) 40 – 46. [2] O.J.M. Smith, Three-phase induction generator for single-phase line, IEEE Trans. Energy Convers. EC-2 (3) (1987) 382–387. [3] A.H. Al-Bahrani, N.H. Malik, Steady state analysis and performance characteristics of a three-phase induction generator self excited with a single capacitor, IEEE Trans. Energy Convers. 5 (4) (1990) 725 – 732. [4] Y.H.A. Rahim, Excitation of isolated three-phase induction generator by a single capacitor, IEE Proc. B 140 (1) (1993) 44 – 50. [5] M.O. Durham, R. Ramakumar, Power system balancers for an induction generator, IEEE Trans. Indust. Appl. IA-23 (6) (1987) 1067 – 1072. [6] J.L. Bhattacharya, J.L. Woodward, Excitation balancing of a self-excited induction generator for maximum power output, IEE Proc. C 135 (2) (1988) 88 – 97. [7] C. Grantham, D. Sutanto, B. Mismail, Steady state and transient analysis of self-excited induction generator, IEE Proc. B 136 (2) (1989) 61 – 68. [8] K.E. Hallenius, P. Vas, J.E. Brown, The analysis of a saturated self-excited asynchronous generator, IEEE Trans. Energy Convers. 6 (2) (1991) 336 – 345. [9] L. Sridhar, Bhim Singh, C.S. Jha, Transient performance of series regulated short shunt self excited induction generator, IEEE Trans. Energy Convers. 10 (2) (1995) 261 – 267. [10] E. Levi, D. Rauski, Modelling of deep-bar and double-cage self-excited induction generators for wind-electricity studies, Electr. Power Syst. Res. 27 (1993) 73 – 81. [11] Paul C. Krause, Analysis of Electric Machinery, McGraw-Hill, New York, 1987.