- Email: [email protected]
passive electrical networks
Mathematics and Computers in Simulation 63 (2003) 449–459
Generator operations of asynchronous induction machines connected to ac or dc active/passive electrical networks D. Iannuzzi a , E. Pagano a,∗ , L. Piegari a , O. Veneri b a
Electrical Engineering Department, University of Naples, Via Claudio 21, Naples 80125, Italy b Istituto Motori-CNR, Via G. Marconi 8, Naples 80125, Italy
Abstract The paper analyses generator operations of asynchronous machines when they are connected to a passive electrical network either directly or by an intermediate static power converter. The analysis is performed by taking into account the saturation phenomena of the main magnetic circuits of the machine. Mathematical models of both physical configurations are given and their non-linearities are evidenced. The paper investigates the existence of periodical solutions related to stable operations of the systems. Numerical and experimental results support the theoretical analysis. © 2003 Published by Elsevier B.V. on behalf of IMACS. Keywords: Generator operations; Asynchronous machines; Saturation; Mathematical models
1. Introduction In recent years generator operations of asynchronous machines have been the subject of some of our applied researches. The idea has been suggested by various considerations. The evolution of alternative sources of electrical energy (for example, solar and wind energy) that imply weak distributed and disconnected electrical networks requiring self-adjusting operating conditions, suggests the use of low maintenance, high reliability and low cost machines. The growing use of asynchronous machines in electrical drives for industrial and traction applications implies not only traditional motor operations but also generator and plug braking operations. An analysis of technical literature has shown that motor operations of asynchronous induction machines have been widely studied both when they operate with or without intermediate electronic power converters. By contrast, generator operations have been dealt with less often. This lack of knowledge may, perhaps, also be the cause of the poor practical use of asynchronous generators. For this reason we have prepared a theoretical and experimental research programme, devoted to a thorough investigation of the different aspects of generator behaviour, both on passive and active electrical networks, i.e. ∗
Corresponding author. Fax: +39-81-7683223. E-mail address: [email protected] (E. Pagano). 0378-4754/$30.00 © 2003 Published by Elsevier B.V. on behalf of IMACS. doi:10.1016/S0378-4754(03)00088-0
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Nomenclature i r is I r Is Ltm g Lm p rr rs Ru Tel TR v
space vector of rotor currents, referred to stator windings space vector of stator currents steady state vector of rotor current referred to stator windings steady state vector of stator current leakage inductance with reference to stator phase trigonometric air-gap inductance for saturated machines differential air-gap inductance for saturated machines pole-pair number rotor phase resistance stator phase resistance load resistance electromagnetic torque rated value of Tel space vector of stator voltages
Greek letters ϑ angular rotor position Φm amplitude of air-gap flux space vector ωr angular speed of rotor ωn rated value of ωr Ω armature voltage pulsation
self-excitation, self-sustaining oscillations and steady-state operating frequencies on passive networks; slip regions, air-gap magnetic flux densities and variable frequency operating conditions on active networks. The research programme is now in progress. We are convinced that the results so far obtained will be useful for explaining the most interesting aspects of generator behaviour of asynchronous induction machines and, therefore, for increasing their use in those practical applications where these machines appear to better satisfy users’ requirements related to performances, buying prices and maintenance costs. Basically, we have found that in all cases generator operations imply air-gap magnetic flux densities that are higher than those corresponding to motor operations (for example [1]). This situation implies that when traditionally designed machines are used, saturation may occur in their main magnetic circuits. Therefore, theoretically, analysis of generator operations should be performed by means of suitable mathematical models that take into account saturation phenomena. We have found that it is possible to carry out a model which is very simple but at the same time fully satisfactory for practical purposes. It has been presented in a previous paper [2]. The model gives, moreover, rise to simple feeding algorithms for digital control applications. By means of this mathematical model we have evaluated the necessary conditions for obtaining stable generator operations on passive electrical networks. These operations can, however, be really attained only if the system is capable of self-excitation. Self-excitation occurs only if rest conditions of the system are not all identically nil. The physical system, represented by an asynchronous generator and its passive electrical network, has a mathematical model set-up by non-linear differential equations. On the basis of finite initial conditions, their solutions can lead to limit cycles or focal
D. Iannuzzi et al. / Mathematics and Computers in Simulation 63 (2003) 449–459
451
points [3], according to whether the necessary conditions for the existence of stable operations which are quasi-stationary are satisfied or not. Pseudo-sinusoidal steady-state operations are obviously obtained by system parameters and boundary conditions that imply solutions of the former kind. Theoretical results related to this topic have been experimentally validated. Some sample phase-plane pictures are shown in the paper to give readers reasons for the solutions obtained. The behaviour of machines operating in electrical drives, provided by electronic power converters, can be analysed by referring to the same mathematical model of the machine. It should, however, be considered together with the mathematical model of the converter. The mathematical model of the whole physical system is once again set-up by non-linear differential equations, representing the non-linear behaviour of both machine and converter. Solutions of this set of differential equations are, obviously, depending on the operating frequencies of the converter. We have investigated the relationship between passive network parameters and driving frequencies of the converters, both related to the asynchronous machine shaft speed. As a result we have experimentally verified the validity of the relationships, theoretically found, that relate the above-mentioned quantities to each other. The analysis has been devoted both to constant speed operations and to dynamic behaviour of the system. In the paper we shortly summarise some basic results previously published in other papers [4] and then we give the results of the performance analysis of asynchronous generators operating on passive electrical networks both with or without intermediate converters. It is clear that only in the former case real generator operations can be obtained. In the latter case generator operations are mainly used for braking mechanical loads. 2. Basic preliminary considerations For sake of simplicity we initially refer to an asynchronous machine which is supplied by a traditional sinusoidal voltage with constant maximum amplitude and frequency. In this case it is easily seen that the air-gap magnetic flux density has quite different maximum amplitudes when the machine either operates as a motor or as a generator (for example, Fig. 1). The differences are easily explained if we consider that, due to the different sign of the real parts of the machine impedance, e.m.f. in generator operations are motor generator
B/Bn [ p.u.] 1.15
1.05
0.95
0.85 0
0.05
0.1
0.15
σ
Fig. 1. Sample air-gap magnetic flux density in motor and generator operations.
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greater than those in motor operations. Saturation phenomena of the main magnetic circuits, which can be neglected during motor operations of traditionally designed machines, take on, by contrast, significance during generator operations. We have experimentally verified [2] that saturation does not give rise to noteworthy stator current distortion as, perhaps, traditional theoretical considerations would lead to expect. For this reason we have found that an improved Kovacs’s mathematical model of the machine could be successfully used for practical applications. Therefore, we have introduced two different air-gap inductances, i.e. Ltm =
Φm (I ) I
and
Lgm =
dΦm (I ) dI
and we have carried out and experimentally validated a mathematical model, which has been presented in a previous paper [2]. It is as follows: d d di i g i s + Ltm + (Lm − Ltm ) i ; v = rs + ls dt dt i dt d di i g di dωr 0= rr +lr i sr − jpωr (Lr i sr + Lm i s ) + Ltm + (Lm − Ltm ) ; Θ + Tm − Tel = 0; dt dt i dt dt 3 Tel = pLm Im{i s i r exp(jpϑr )}; i = |i m | = |i sr + i s | = [Re(i sr + i s )]2 + [Im(i sr + i s )]2 . (1) 2 From a practical point of view the mathematical model can be easily used. In effect it requires only a preliminary experimental evaluation of the no-load magnetic characteristic of the machine. The evaluation can be made by measuring the corresponding pairs of armature voltage and current values. The non-linearities of the suggested mathematical model disappear when the machine operates at steady-state. Therefore, also when saturation phenomena occur, a sinusoidal steady-state can be obtained for generator operations on passive, linear electrical networks.
3. Generator operations on passive ac electrical networks It is well known from technical literature that generator operations of asynchronous machines on passive electrical networks can occur only on external networks provided by capacitors. This condition is easily and simply understood by considering that at steady-state no external voltages are impressed on the physical electrical system and, therefore, real and imaginary parts of its internal impedance must be nil. Due to the inductance of asynchronous machines, only capacitors can obtain the compensation of the imaginary part. Real parts are compensated by negative equivalent resistance of the rotor. From a general point of view the mathematical model of the whole system, schematically represented by Fig. 2, is given by Eq. (1) together with dv is v = − . dt C Ru C
(2)
Eqs. (1) and (2) represent a system of non-linear differential equations, where the driving mechanical torque is given and all currents are unknown. The torque is also the one and only forcing cause of the physical system. It is easily verified that if all initial conditions of electrical quantities are nil, also
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453
iS
C
R
Tm G 3
iC
iR
Fig. 2. Schematic electrical configuration of a passive electrical network fed by an asynchronous generator.
the currents are always identically nil. By contrast, the presence of a residual flux density or an initial capacitor charge gives rise to not nil solutions for unknown electrical currents. From a physical point of view the result is easily understood in both cases. It is also easy to explain the analytical result connected with initial capacitor charges. Residual flux density can be mathematically considered as a g Dirac’s pulse at the initial time when defining Lm = dΦm (0)/di (0) by means of the theory of generalised functions. It is well known that solutions of non-linear differential equations can be of different kinds. Our problem can be set in the following terms. Do the set of Eqs. (1) and (2) admit a periodic solution and, if yes, when? A theoretical answer to this question can be given by considering whether the Fourier’s series expansions of all unknown currents represent a set of solutions to the equations. This theoretical procedure is, indeed, very difficult to apply in practice also because the function Φm (i ) can not be successfully approximated with simple analytical functions in the whole domain of variation of Φm and i . Therefore, a numerical procedure has been followed. It is based on the conditions that if the system of Eqs. (1) and (2) admits a periodic solution all its phase-trajectories are closed. The results of the numerical procedure are, indeed, limited to dominia of investigation assumed for the range of the external parameters, i.e. of R and C and for each given machine. We have applied the numerical procedure to different asynchronous machines. As an example in the following, results referred to the sample machine of Table 1 are given. By using our test bench we have driven the asynchronous machine by applying different driving shaft torque. Fig. 3 refer to torque of 10 and 20 Nm. The value of the load resistance is 67.8 per phase and capacitors have 210 mF per phase. Table 1 Main parameters and rating of the tested asynchronous induction motor Parameter
Rating
PR (kW) fR (Hz) VR (V) p rs () rr () ls (mH) lr (mH) Lm (mH) Θ (kg m2 )
11 50 380 2 0.4 0.4 3.0 3.0 69.7 0.23
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Fig. 3. Theoretical and experimental amplitudes of machine angular speeds and of armature current space vectors for different driving torque (M = 10 and 20 Nm).
As can easily be seen, numerical and experimental results correspond very well. Both give evidence of the high value of magnetising current which reaches values of up to 96% of the load currents. Results confirm the necessity to refer to mathematical models of the machine which takes into account saturation phenomena of main magnetic circuits. Theoretical and experimental results also allow us to better understand physical phenomena, which occur during self-excitation. For this reason phase trajectories of state variables can be drawn and examined. In another paper [4] we have dealt more extensively with this problem. For sake of completeness Fig. 4 shows phase-trajectories of armature currents and machine shaft speed. They are drawn for two different load circuits. The former one implies a steady-state, the latter one gives rise to unstable operations. The results can be explained simply if we analyse the behaviour of the system at sinusoidal steady-state by means of Eqs. (1) and (2). The existence of such a state implies that a particular integral of the system of differential Eqs. (1) and (2) is represented by a set of sinusoidal functions, when the shaft speed is constant, i.e. by: i˜ s = I s ejΩt ;
i˜ r = I r ejΩt ;
v˜ s = V s ejΩt
(3)
If Eq. (3) is put into Eqs. (1) and (2), for each given asynchronous machine, a system of seven algebraic equations is obtained, where the values of Ω, ωr , C, Ru , Is , Ir , V are connected to each another:
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1 CRu
Is ; C
455
9 3 jΩI s + Lt rs + jpωr L2m I s = VLt + Lm (i r − jpωr Lt )I r ; 4 2 3 3 dω r jΩI r + (rr − jpωr Lt )Lt I r = − Lm V + Lm (rs + jpωr Lt )I s ; = Tm − AIm{ I s I r }. (4) 2 2 dt
V = jΩ+
=
The situation implies that, given three values of the above-mentioned quantities, the other four can be univocally evaluated.
Fig. 4. Phase-plane trajectories of current amplitude and speed for steady-state (a and b) and unstable (c and d) operations.
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Fig. 4. (Continued ).
When the system of Fig. 2 is used in practice, the values of Ω, Is and Ru are generally given. The other corresponding ones are obtained by solving the system of Eq. (4). It might be interesting to examine how C and ωr vary in a given range. As an example Fig. 5 shows the function C(Ru ) for Ω = 314 rad s−1 and V = 380 V. As an example Fig. 3 compares theoretical and experimental results achieved by a mass production asynchronous motor, whose rating and main parameters an given in Table 1.
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Fig. 5. Function C(Ru ) for Ω = 314 rad s−1 and V = 380 V obtaining self-sustained steady-state generator operations of the asynchronous machine of Table 1.
For testing we have used a bench where the shaft of the asynchronous machine is connected to a dc motor, supplied by variable voltage sources, both for armature and excitation windings. The armature windings of the asynchronous machine are connected to a three-phase RC network, whose resistances can be changed simply, ranging from 10 to 250 W. Capacitors are connected to the machine by means of a two-way switch, so that they can either operate in parallel to the resistances or be charged separately. Previous tests have been made without pre-charging capacitors. We have verified that the decaying time of the residual flux density is very short, so that self-excitation of the system, i.e. generator operation of the machine, occurs only if its armature windings are suddenly switched from the ac feeding active network to the RC passive one. In this case steady-state operations will take place for the pair of values of R and C that have been theoretically predicted (see Fig. 5). When armature windings are kept even for short times disconnected from external networks, selfexcitation does not occur. However, if a small charge is given to capacitance (we have used a dc source of 12 V) self-excitation occurs and higher values of voltages across capacitors are reached at steady-state. This experience shows that the system is self-adjusting and, due to its non-linear behaviour, final stable operating conditions can be reached.
4. Generator operation on dc passive electrical network In some electrical drives that require braking operations, asynchronous machines operate as generators on a dc passive electrical network by means of an intermediate inverter, according to the schematic electrical configuration shown in Fig. 6. The mathematical model of the physical system is now represented by Eq. (1), and by: dVab Vab i dc + =− ; dt CRu C
i dc = Re{i s e−(jπ/3)(6t/T) }.
(4)
The system of Eqs. (1) and (4) is still non-linear, but now non-linearities not only occur for the saturation and for the presence of product between unknown quantities, but also for the non-linear characteristic of the inverter-components.
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Ru A
B
ia
C
M 3 Fig. 6. Schematic electrical configuration of asynchronous machine operating as generator on a dc passive electrical network by means of an intermediate inverter.
A preliminary analysis immediately shows that the circuit configuration does not imply sinusoidal steady-state operations of the machine, because the system of Eqs. (1) and (4) does not have solutions with sinusoidal armature currents and for constant idc current. Periodic solutions can, however, still be found. They imply that: is =
∞
Isk e
jkΩs t
;
∞ vs = Vsk ejkΩs t ;
k=1
k=1
∞ ∞ ϑr = ωrk ejkΩ t dt = αk ejkΩ t . k=0
i r
∞ = Irk ejkΩr t ; k=1
(5)
k=0
For sake of simplicity we put: q = i r ejpϑr ;
Q = i s + q.
By taking into account Eq. (5), the second part of Eq. (1) can be written as: 0 = rr
∞ k=1
qke
jkΩs t
∞ ∞ ∞ d jkΩs t 3 d jkΩs t + lr qke + Lm Qk e − jpωr lr q k ejkΩs t dt k=1 2 dt k=1 k=1
∞ 3 − jpωr Lm Qk ejkΩs t . 2 k=1
(6)
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For non-periodical variation of the load torque, the frequency of the speed variation is only due to the frequency of the electromagnetic torque. In particular, because the torque is Tel = kIm{iˇ s q} the speed frequency is always an exact multiple of the stator current frequency; i.e. it is Ω = mΩs . If we consider that speed can be substituted with its Fourier’s series and that according to Cauchy’s rule for the product of sums, Eq. (3) can be written as: k ∞ (ν+(k−ν)m)Ω t s jkΩs t j jϕk−ν j(ν−(k−ν)m)Ωs t −jϕk−ν 0= Mk e + Nk aν ωr,k−ν e e +e e . (7) k=0
ν=0
From the analysis of Eq. (7) it is clear that the variation of speed produces voltage on armature windings at the same frequency of the harmonic generated by the feeder; for this reason the reaction of the armature to the speed variation can be compensated by the feeder and a steady-state solution for the system can exist as normal practice confirms. 5. Conclusions The paper has analysed the behaviour of asynchronous machines where they operate as generators on passive electrical network, directly or by intermediate static power converters. The analysis has been performed by means of suitable mathematical models of machine and networks. Emphasis has been given to self-excitation phenomena and to the existence of periodical solutions. Theoretical results have been validated by experimental tests. References [1] E. Pagano, L. Piegari, O. Veneri, Generator and plug brake operations of asynchronous machines, in: G.A. Capolino, R. Goyet (Eds.), Jubilè Power Electrical Engineering. [2] E. Pagano, O. Veneri, Generator operations of saturated induction machines, IEE Proc. Electron. Power Appl. (2002). [3] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962. [4] D. Iannuzzi, E. Pagano, L. Piegari, O. Veneri, Braking operations of electrical drives for road vehicles, in: Proceedings of the EDPE2001, Slovak Republic, 3–5 October 2001.
Generator operations of asynchronous induction machines connected to ac or dc active/passive electrical networks D. Iannuzzi a , E. Pagano a,∗ , L. Piegari a , O. Veneri b a
Electrical Engineering Department, University of Naples, Via Claudio 21, Naples 80125, Italy b Istituto Motori-CNR, Via G. Marconi 8, Naples 80125, Italy
Abstract The paper analyses generator operations of asynchronous machines when they are connected to a passive electrical network either directly or by an intermediate static power converter. The analysis is performed by taking into account the saturation phenomena of the main magnetic circuits of the machine. Mathematical models of both physical configurations are given and their non-linearities are evidenced. The paper investigates the existence of periodical solutions related to stable operations of the systems. Numerical and experimental results support the theoretical analysis. © 2003 Published by Elsevier B.V. on behalf of IMACS. Keywords: Generator operations; Asynchronous machines; Saturation; Mathematical models
1. Introduction In recent years generator operations of asynchronous machines have been the subject of some of our applied researches. The idea has been suggested by various considerations. The evolution of alternative sources of electrical energy (for example, solar and wind energy) that imply weak distributed and disconnected electrical networks requiring self-adjusting operating conditions, suggests the use of low maintenance, high reliability and low cost machines. The growing use of asynchronous machines in electrical drives for industrial and traction applications implies not only traditional motor operations but also generator and plug braking operations. An analysis of technical literature has shown that motor operations of asynchronous induction machines have been widely studied both when they operate with or without intermediate electronic power converters. By contrast, generator operations have been dealt with less often. This lack of knowledge may, perhaps, also be the cause of the poor practical use of asynchronous generators. For this reason we have prepared a theoretical and experimental research programme, devoted to a thorough investigation of the different aspects of generator behaviour, both on passive and active electrical networks, i.e. ∗
Corresponding author. Fax: +39-81-7683223. E-mail address: [email protected] (E. Pagano). 0378-4754/$30.00 © 2003 Published by Elsevier B.V. on behalf of IMACS. doi:10.1016/S0378-4754(03)00088-0
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Nomenclature i r is I r Is Ltm g Lm p rr rs Ru Tel TR v
space vector of rotor currents, referred to stator windings space vector of stator currents steady state vector of rotor current referred to stator windings steady state vector of stator current leakage inductance with reference to stator phase trigonometric air-gap inductance for saturated machines differential air-gap inductance for saturated machines pole-pair number rotor phase resistance stator phase resistance load resistance electromagnetic torque rated value of Tel space vector of stator voltages
Greek letters ϑ angular rotor position Φm amplitude of air-gap flux space vector ωr angular speed of rotor ωn rated value of ωr Ω armature voltage pulsation
self-excitation, self-sustaining oscillations and steady-state operating frequencies on passive networks; slip regions, air-gap magnetic flux densities and variable frequency operating conditions on active networks. The research programme is now in progress. We are convinced that the results so far obtained will be useful for explaining the most interesting aspects of generator behaviour of asynchronous induction machines and, therefore, for increasing their use in those practical applications where these machines appear to better satisfy users’ requirements related to performances, buying prices and maintenance costs. Basically, we have found that in all cases generator operations imply air-gap magnetic flux densities that are higher than those corresponding to motor operations (for example [1]). This situation implies that when traditionally designed machines are used, saturation may occur in their main magnetic circuits. Therefore, theoretically, analysis of generator operations should be performed by means of suitable mathematical models that take into account saturation phenomena. We have found that it is possible to carry out a model which is very simple but at the same time fully satisfactory for practical purposes. It has been presented in a previous paper [2]. The model gives, moreover, rise to simple feeding algorithms for digital control applications. By means of this mathematical model we have evaluated the necessary conditions for obtaining stable generator operations on passive electrical networks. These operations can, however, be really attained only if the system is capable of self-excitation. Self-excitation occurs only if rest conditions of the system are not all identically nil. The physical system, represented by an asynchronous generator and its passive electrical network, has a mathematical model set-up by non-linear differential equations. On the basis of finite initial conditions, their solutions can lead to limit cycles or focal
D. Iannuzzi et al. / Mathematics and Computers in Simulation 63 (2003) 449–459
451
points [3], according to whether the necessary conditions for the existence of stable operations which are quasi-stationary are satisfied or not. Pseudo-sinusoidal steady-state operations are obviously obtained by system parameters and boundary conditions that imply solutions of the former kind. Theoretical results related to this topic have been experimentally validated. Some sample phase-plane pictures are shown in the paper to give readers reasons for the solutions obtained. The behaviour of machines operating in electrical drives, provided by electronic power converters, can be analysed by referring to the same mathematical model of the machine. It should, however, be considered together with the mathematical model of the converter. The mathematical model of the whole physical system is once again set-up by non-linear differential equations, representing the non-linear behaviour of both machine and converter. Solutions of this set of differential equations are, obviously, depending on the operating frequencies of the converter. We have investigated the relationship between passive network parameters and driving frequencies of the converters, both related to the asynchronous machine shaft speed. As a result we have experimentally verified the validity of the relationships, theoretically found, that relate the above-mentioned quantities to each other. The analysis has been devoted both to constant speed operations and to dynamic behaviour of the system. In the paper we shortly summarise some basic results previously published in other papers [4] and then we give the results of the performance analysis of asynchronous generators operating on passive electrical networks both with or without intermediate converters. It is clear that only in the former case real generator operations can be obtained. In the latter case generator operations are mainly used for braking mechanical loads. 2. Basic preliminary considerations For sake of simplicity we initially refer to an asynchronous machine which is supplied by a traditional sinusoidal voltage with constant maximum amplitude and frequency. In this case it is easily seen that the air-gap magnetic flux density has quite different maximum amplitudes when the machine either operates as a motor or as a generator (for example, Fig. 1). The differences are easily explained if we consider that, due to the different sign of the real parts of the machine impedance, e.m.f. in generator operations are motor generator
B/Bn [ p.u.] 1.15
1.05
0.95
0.85 0
0.05
0.1
0.15
σ
Fig. 1. Sample air-gap magnetic flux density in motor and generator operations.
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D. Iannuzzi et al. / Mathematics and Computers in Simulation 63 (2003) 449–459
greater than those in motor operations. Saturation phenomena of the main magnetic circuits, which can be neglected during motor operations of traditionally designed machines, take on, by contrast, significance during generator operations. We have experimentally verified [2] that saturation does not give rise to noteworthy stator current distortion as, perhaps, traditional theoretical considerations would lead to expect. For this reason we have found that an improved Kovacs’s mathematical model of the machine could be successfully used for practical applications. Therefore, we have introduced two different air-gap inductances, i.e. Ltm =
Φm (I ) I
and
Lgm =
dΦm (I ) dI
and we have carried out and experimentally validated a mathematical model, which has been presented in a previous paper [2]. It is as follows: d d di i g i s + Ltm + (Lm − Ltm ) i ; v = rs + ls dt dt i dt d di i g di dωr 0= rr +lr i sr − jpωr (Lr i sr + Lm i s ) + Ltm + (Lm − Ltm ) ; Θ + Tm − Tel = 0; dt dt i dt dt 3 Tel = pLm Im{i s i r exp(jpϑr )}; i = |i m | = |i sr + i s | = [Re(i sr + i s )]2 + [Im(i sr + i s )]2 . (1) 2 From a practical point of view the mathematical model can be easily used. In effect it requires only a preliminary experimental evaluation of the no-load magnetic characteristic of the machine. The evaluation can be made by measuring the corresponding pairs of armature voltage and current values. The non-linearities of the suggested mathematical model disappear when the machine operates at steady-state. Therefore, also when saturation phenomena occur, a sinusoidal steady-state can be obtained for generator operations on passive, linear electrical networks.
3. Generator operations on passive ac electrical networks It is well known from technical literature that generator operations of asynchronous machines on passive electrical networks can occur only on external networks provided by capacitors. This condition is easily and simply understood by considering that at steady-state no external voltages are impressed on the physical electrical system and, therefore, real and imaginary parts of its internal impedance must be nil. Due to the inductance of asynchronous machines, only capacitors can obtain the compensation of the imaginary part. Real parts are compensated by negative equivalent resistance of the rotor. From a general point of view the mathematical model of the whole system, schematically represented by Fig. 2, is given by Eq. (1) together with dv is v = − . dt C Ru C
(2)
Eqs. (1) and (2) represent a system of non-linear differential equations, where the driving mechanical torque is given and all currents are unknown. The torque is also the one and only forcing cause of the physical system. It is easily verified that if all initial conditions of electrical quantities are nil, also
D. Iannuzzi et al. / Mathematics and Computers in Simulation 63 (2003) 449–459
453
iS
C
R
Tm G 3
iC
iR
Fig. 2. Schematic electrical configuration of a passive electrical network fed by an asynchronous generator.
the currents are always identically nil. By contrast, the presence of a residual flux density or an initial capacitor charge gives rise to not nil solutions for unknown electrical currents. From a physical point of view the result is easily understood in both cases. It is also easy to explain the analytical result connected with initial capacitor charges. Residual flux density can be mathematically considered as a g Dirac’s pulse at the initial time when defining Lm = dΦm (0)/di (0) by means of the theory of generalised functions. It is well known that solutions of non-linear differential equations can be of different kinds. Our problem can be set in the following terms. Do the set of Eqs. (1) and (2) admit a periodic solution and, if yes, when? A theoretical answer to this question can be given by considering whether the Fourier’s series expansions of all unknown currents represent a set of solutions to the equations. This theoretical procedure is, indeed, very difficult to apply in practice also because the function Φm (i ) can not be successfully approximated with simple analytical functions in the whole domain of variation of Φm and i . Therefore, a numerical procedure has been followed. It is based on the conditions that if the system of Eqs. (1) and (2) admits a periodic solution all its phase-trajectories are closed. The results of the numerical procedure are, indeed, limited to dominia of investigation assumed for the range of the external parameters, i.e. of R and C and for each given machine. We have applied the numerical procedure to different asynchronous machines. As an example in the following, results referred to the sample machine of Table 1 are given. By using our test bench we have driven the asynchronous machine by applying different driving shaft torque. Fig. 3 refer to torque of 10 and 20 Nm. The value of the load resistance is 67.8 per phase and capacitors have 210 mF per phase. Table 1 Main parameters and rating of the tested asynchronous induction motor Parameter
Rating
PR (kW) fR (Hz) VR (V) p rs () rr () ls (mH) lr (mH) Lm (mH) Θ (kg m2 )
11 50 380 2 0.4 0.4 3.0 3.0 69.7 0.23
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Fig. 3. Theoretical and experimental amplitudes of machine angular speeds and of armature current space vectors for different driving torque (M = 10 and 20 Nm).
As can easily be seen, numerical and experimental results correspond very well. Both give evidence of the high value of magnetising current which reaches values of up to 96% of the load currents. Results confirm the necessity to refer to mathematical models of the machine which takes into account saturation phenomena of main magnetic circuits. Theoretical and experimental results also allow us to better understand physical phenomena, which occur during self-excitation. For this reason phase trajectories of state variables can be drawn and examined. In another paper [4] we have dealt more extensively with this problem. For sake of completeness Fig. 4 shows phase-trajectories of armature currents and machine shaft speed. They are drawn for two different load circuits. The former one implies a steady-state, the latter one gives rise to unstable operations. The results can be explained simply if we analyse the behaviour of the system at sinusoidal steady-state by means of Eqs. (1) and (2). The existence of such a state implies that a particular integral of the system of differential Eqs. (1) and (2) is represented by a set of sinusoidal functions, when the shaft speed is constant, i.e. by: i˜ s = I s ejΩt ;
i˜ r = I r ejΩt ;
v˜ s = V s ejΩt
(3)
If Eq. (3) is put into Eqs. (1) and (2), for each given asynchronous machine, a system of seven algebraic equations is obtained, where the values of Ω, ωr , C, Ru , Is , Ir , V are connected to each another:
D. Iannuzzi et al. / Mathematics and Computers in Simulation 63 (2003) 449–459
1 CRu
Is ; C
455
9 3 jΩI s + Lt rs + jpωr L2m I s = VLt + Lm (i r − jpωr Lt )I r ; 4 2 3 3 dω r jΩI r + (rr − jpωr Lt )Lt I r = − Lm V + Lm (rs + jpωr Lt )I s ; = Tm − AIm{ I s I r }. (4) 2 2 dt
V = jΩ+
=
The situation implies that, given three values of the above-mentioned quantities, the other four can be univocally evaluated.
Fig. 4. Phase-plane trajectories of current amplitude and speed for steady-state (a and b) and unstable (c and d) operations.
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Fig. 4. (Continued ).
When the system of Fig. 2 is used in practice, the values of Ω, Is and Ru are generally given. The other corresponding ones are obtained by solving the system of Eq. (4). It might be interesting to examine how C and ωr vary in a given range. As an example Fig. 5 shows the function C(Ru ) for Ω = 314 rad s−1 and V = 380 V. As an example Fig. 3 compares theoretical and experimental results achieved by a mass production asynchronous motor, whose rating and main parameters an given in Table 1.
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457
Fig. 5. Function C(Ru ) for Ω = 314 rad s−1 and V = 380 V obtaining self-sustained steady-state generator operations of the asynchronous machine of Table 1.
For testing we have used a bench where the shaft of the asynchronous machine is connected to a dc motor, supplied by variable voltage sources, both for armature and excitation windings. The armature windings of the asynchronous machine are connected to a three-phase RC network, whose resistances can be changed simply, ranging from 10 to 250 W. Capacitors are connected to the machine by means of a two-way switch, so that they can either operate in parallel to the resistances or be charged separately. Previous tests have been made without pre-charging capacitors. We have verified that the decaying time of the residual flux density is very short, so that self-excitation of the system, i.e. generator operation of the machine, occurs only if its armature windings are suddenly switched from the ac feeding active network to the RC passive one. In this case steady-state operations will take place for the pair of values of R and C that have been theoretically predicted (see Fig. 5). When armature windings are kept even for short times disconnected from external networks, selfexcitation does not occur. However, if a small charge is given to capacitance (we have used a dc source of 12 V) self-excitation occurs and higher values of voltages across capacitors are reached at steady-state. This experience shows that the system is self-adjusting and, due to its non-linear behaviour, final stable operating conditions can be reached.
4. Generator operation on dc passive electrical network In some electrical drives that require braking operations, asynchronous machines operate as generators on a dc passive electrical network by means of an intermediate inverter, according to the schematic electrical configuration shown in Fig. 6. The mathematical model of the physical system is now represented by Eq. (1), and by: dVab Vab i dc + =− ; dt CRu C
i dc = Re{i s e−(jπ/3)(6t/T) }.
(4)
The system of Eqs. (1) and (4) is still non-linear, but now non-linearities not only occur for the saturation and for the presence of product between unknown quantities, but also for the non-linear characteristic of the inverter-components.
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Ru A
B
ia
C
M 3 Fig. 6. Schematic electrical configuration of asynchronous machine operating as generator on a dc passive electrical network by means of an intermediate inverter.
A preliminary analysis immediately shows that the circuit configuration does not imply sinusoidal steady-state operations of the machine, because the system of Eqs. (1) and (4) does not have solutions with sinusoidal armature currents and for constant idc current. Periodic solutions can, however, still be found. They imply that: is =
∞
Isk e
jkΩs t
;
∞ vs = Vsk ejkΩs t ;
k=1
k=1
∞ ∞ ϑr = ωrk ejkΩ t dt = αk ejkΩ t . k=0
i r
∞ = Irk ejkΩr t ; k=1
(5)
k=0
For sake of simplicity we put: q = i r ejpϑr ;
Q = i s + q.
By taking into account Eq. (5), the second part of Eq. (1) can be written as: 0 = rr
∞ k=1
qke
jkΩs t
∞ ∞ ∞ d jkΩs t 3 d jkΩs t + lr qke + Lm Qk e − jpωr lr q k ejkΩs t dt k=1 2 dt k=1 k=1
∞ 3 − jpωr Lm Qk ejkΩs t . 2 k=1
(6)
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For non-periodical variation of the load torque, the frequency of the speed variation is only due to the frequency of the electromagnetic torque. In particular, because the torque is Tel = kIm{iˇ s q} the speed frequency is always an exact multiple of the stator current frequency; i.e. it is Ω = mΩs . If we consider that speed can be substituted with its Fourier’s series and that according to Cauchy’s rule for the product of sums, Eq. (3) can be written as: k ∞ (ν+(k−ν)m)Ω t s jkΩs t j jϕk−ν j(ν−(k−ν)m)Ωs t −jϕk−ν 0= Mk e + Nk aν ωr,k−ν e e +e e . (7) k=0
ν=0
From the analysis of Eq. (7) it is clear that the variation of speed produces voltage on armature windings at the same frequency of the harmonic generated by the feeder; for this reason the reaction of the armature to the speed variation can be compensated by the feeder and a steady-state solution for the system can exist as normal practice confirms. 5. Conclusions The paper has analysed the behaviour of asynchronous machines where they operate as generators on passive electrical network, directly or by intermediate static power converters. The analysis has been performed by means of suitable mathematical models of machine and networks. Emphasis has been given to self-excitation phenomena and to the existence of periodical solutions. Theoretical results have been validated by experimental tests. References [1] E. Pagano, L. Piegari, O. Veneri, Generator and plug brake operations of asynchronous machines, in: G.A. Capolino, R. Goyet (Eds.), Jubilè Power Electrical Engineering. [2] E. Pagano, O. Veneri, Generator operations of saturated induction machines, IEE Proc. Electron. Power Appl. (2002). [3] H.T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, 1962. [4] D. Iannuzzi, E. Pagano, L. Piegari, O. Veneri, Braking operations of electrical drives for road vehicles, in: Proceedings of the EDPE2001, Slovak Republic, 3–5 October 2001.