1. Introduction
The electric vehicle (EV) was conceived in the middle of the previous century. EV’s offer the most promising solutions to reduce vehicular emissions. EV’s constitute the only commonly known group of automobiles that qualify as zero-emission vehicles. These vehicles use an electric motor for propulsion, batteries as electrical-energy storage devices and associated with power electronics, microelectronics, and microprocessor control of motor drives.
The doubly fed induction motor (DFIM) is a wound rotor asynchronous machine supplied by the stator and the rotor from two external source voltages. This machine is very attractive for the variable speed applications such as the electric vehicle and the electrical energy production. Consequently, it covers all power ranges. Obviously, the requested variable speed domain and the desired performances depend of the application kinds (Vicatos & Tegopoulos, 2003, Akagi & Sato, 1999, Debiprasad et al., 2001, Leonhard, 1997, Wang & Ding, 1993, Morel et al., 1998 and Hopfensperger et al., 1999).
The use of DFIM offers the opportunity to modulate power flow into and out of the rotor winding in order to have, at the same time, a variable speed in the characterized super–synchronous or sub–synchronous modes in motor or in generator regimes. Two modes can be associated to slip power recovery: sub–synchronous motoring and super–synchronous generating operations. In general, while the rotor is fed through a cycloconverter, the power range can attain the MW order which presents the size power often reserved to the synchronous machine (Vicatos & Tegopoulos, 2003, Akagi & Sato, 1999, Debiprasad et al., 2001, Leonhard, 1997, Wang & Ding, 1993, Morel et al., 1998, Hopfensperger et al, 1999 a, 1999b , Metwally et al., 2002, Hirofumi & Hikaru, 2002 and Djurovic et al., 1995). The DFIM has some distinct advantages compared to the conventional squirrel-cage machine. The DFIM can be controlled from the stator or rotor by various possible combinations. The disadvantage of two used converters for stator and rotor supplying can be compensated by the best control performances of the powered systems (Debiprasad et al., 2001). Indeed, the input–commands are done by means of four precise degrees of control freedom relatively to the squirrel cage induction machine where its control appears quite simple. The flux orientation strategy can transform the non linear and coupled DFIM-mathematical model into a linear model leading to one attractive solution for generating or motoring operations (Sergei, 2003).
It is known that the motor driven systems account for approximately 65% of the electricity consumed in the world. Implementing high efficiency motor driven systems, or improving existing ones, could save over 200 billion kWh of electricity per year. This issue has become very important especially following the economic crisis due to the oil prices raising, the new energy saving technologies are appearing and developing rapidly in this century (Leonhard, 1997, Longya & Wei, 1995, Wang & Cheng, 2004, Zang & Hasan, 1999, David, 1988 and Rodriguez et al., 2002). In this framework, the DFIM continues to find great interest since the birth of the idea of the double flux orientation (Drid et al., 2005 a, 2005b). The philosophy of this idea is to get a simpler machine model expression (ideal machine) (Drid et al., 2005 a). Consequently, in the same time, we can solve a non linear problem presented by the DFIM control and step up from many digital simulations toward the experimental test by the use of the system dSPACE-1103. This method gives entire satisfaction and consolidates our theory, especially using the Torque Optimization Factor TOF strategy (Drid et al., 2005 b). Always the search for a solution has more optimal, us nap leans towards the minimization of the copper losses in the DFIM.
In this chapter we developed an optimization factor Torque Copper Losses Optimization TCLO. The chapter will be organized as follows. The DFIM mathematical model is presented in section 3. In section 4, the robust nonlinear feedback control is exposed. Section 5 concerns the two energy torque optimization strategies TOF and TCLO. In the section 6, simulation results are exposed and comparative illustration shows the performances in energy saving between TOF and TCLO.
2. The DFIM model
Its dynamic model expressed in the synchronous reference frame is given by
Voltage equations:
Flux equations:
From (Eq. 1) and (Eq. 2), the state-all-flux model is written like:
The electromagnetic torque is done as
The copper losses are giving as:
The motion equation is:
In DFIM operations, the stator and rotor mmf’s (magneto motive forces) rotations are directly imposed by the two external voltage source frequencies. Hence, the rotor speed becomes depending toward the linear combination of theses frequencies, and it will be constant if they are too constants for any load torque, given of course in the machine stability domain. In DFIM modes, the synchronization between both mmf’s is mainly required in order to guarantee machine stability. This is the similar situation of the synchronous machine stability problem where without the recourse to the strict control of the DFIM mmf’s relative position, the machine instability risk or brake down mode become imminent.
3. Nonlinear vector control strategy
3.1. Double flux orientation
It consists in orienting, at the same time, stator flux and rotor flux. Thus, it results the constraints given below by (Eq. 7). Rotor flux is oriented on the d-axis, and the stator flux is oriented on the q-axis. Conventionally, the d-axis remains reserved to magnetizing axis and q-axis to torque axis, so we can write (Drid et al., 2005 a, 2005b)
Using (Eq. 7), the developed torque given by (Eq. 4) can be rewritten as follows:
where,
3.2. Vector control by Lyapunov feedback linearization
Separating the real and the imaginary part of (Eq. 3), we can write:
Where f1, f2, f3 and f4 are done as follows :
With:
Tacking into account of the constraints given by (Eq. 7), one can formulate the Lyapunov function as follows
From (Eq. 11), the first and second quadrate terms concern the fluxes orientation process defined in (Eq. 7) with the third and fourth terms characterizing the fluxes feedback control. Where its derivative function becomes
Substituting (Eq. 9) in (Eq. 12), it results
Let us define the following law control as (Khalil, 1996):
Hence (Eq. 14) replaced in (Eq. 13) gives:
The function (Eq. 15) is negative one. Furthermore, (Eq. 14) introduced into (Eq. 9) leads to a stable convergence process if the gains Ki (i=1, 2,3, 4) are evidently all positive, otherwise:
In (Eq. 16), the first and second equations concern the double flux orientation constraints applied for DFIM-model which are define above by (Eq. 7), while the third and fourth equations define the errors after the feedback fluxes control. This latter offers the possibility to control the main machine magnetizing on the d-axis by rd and the developed torque on the q-axis by sq.
3.3. Robust feedback Lyapunov linearization control
In practice, the nonlinear functions
On,
Where:
The ∆fi can be generated from the whole parameters and variables variations as indicated above. We assume that all the ∆fi are bounded as follows: |∆fi| < i; where are known bounds. The knowledge of i is not difficult since, one can use sufficiently large number to satisfy the constraint|∆fi| < i.
The ∆fi can be generated from the whole parameters and variables variations as indicated above.
Replacing (Eq. 17) in (Eq. 9), we obtain
The following result can be stated.
Proposition: Consider the realistic all fluxes state model (Eq. 18). Then, the double fluxes orientation constraints (Eq. 7) are fulfilled provided that the following control laws are used
where Kii i and Kii > 0 for i=1; 4.
Proof. Let the Lyapunov function related to the fluxes dynamics (Eq. 18) defined by
One has
where
The latter inequalities are satisfied since
Finally, we can write:
Hence, using the Lyapunov theorem (Khalil, 1996), on conclude that
The design of these robust controllers, resulting from (19), is given in the followed figure 2
The indices
4. Energy optimization strategy
In this section we will explain why and what is the optimization strategy used in this work. Fig. 1 illustrates the problem which occurs in the proposed DFIM vector control system when the machine magnetizing excitation is maintained at a constant level.
4.1. Why the energy optimization strategy?
Considering an iso-torque-curve (hyperbole form), drawn from (Eq. 8) for a constant torque in the
On the same graph, we define a second iso-torque-curve CeT=Const in the
Once the machine speed reaches its reference, the inertial torque is cancelled ( = 0), then the developed torque must return immediately to the initial load torque Cro, characterized by the second transitions A’–A and B’–B towards the preceding equilibrium points A and B. One can notice that during the transition B–B’, corresponding to the under excited machine, the stator flux can attain very high values greater than the tolerable limit (
Where,
In the other hand, for the case A (excited machine), if the A–A’ transition remains tolerable, the armature currents can present prohibitory magnitude in the steady state operation due to the orthogonal contribution of stator and rotor fluxes at the moment that the machine is sufficiently excited. The steady state armature currents can be calculated by (Eq. 26), where we can note the amplification effect of the coefficients , and .
4.2. Torque optimization factor (TOF) design
In the previous sub-section, the problem is in the transient torque, especially when the machine is low loaded. So it becomes very important to minimize the torque transition such as (Drid, 2005b):
where,
This condition should be realized respecting the stator flux constraint given by
In this way the rotor and stator fluxes, though orthogonal, their modulus will be related by the so-called TOF strategy which will be designed from the resolution of the differential equations (Eq. 27 - Eq. 28) with constraint (Eq. 29) as follows:
from (Eq. 29) we can write
thus,
the resolution of (Eq. 32) leads to
where C is an arbitrary integration constant, therefore
Since, the main torque input-command in motoring DFIM operation is related to the stator flux, it becomes dependent on the speed rotor sign and thus we can write
with (Eq. 35), (Eq. 34), the rotor flux may be rewritten as follows
The resolution of (Eq. 32) gives place to the arbitrary integration constant C from which the TOF-relationship (Eq. 36) can be easily tuned. This one can be adjusted by a judicious choice of the integration constant, while figure 2 presents TOF effect on armature DFIM currents with C-tuning. Note that this method offers the possibility to reduce substantially the magnitude of the armature currents into the machine and we can notice an increase in energy saving. Hence using TOF strategy, we can avoid the saturation effect and reduce the magnitude of machine currents from which the DFIM efficiency could be clearly enhanced.
4.3. Torque-copper losses optimization (TCLO) design
In many applications, it is required to optimize a given parameter and the derivative plays a key role in the solution of such problems. Suppose the quantity to be minimized is given by the function
with :
The figure 3 represents the layout of (Eq. 37) for a constant level of torque and copper losses in the (s, r) plan. These curves present respectively a hyperbole for the iso-torque and ellipse for iso-copper-losses. From (Eq. 37) we can write:
To obtain a real and thus optimal solution, we must have:
The equation (Eq. 39) represents the energy balance in the DFIM for one working DFIM point as shown in fig.3. Then, one can write:
This equation shows the optimal relation between the torque and the copper losses.
4.4. Finding minimum Copper-losses values
The Rolle’s Theorem is the key result behind applications of the derivative to optimization problems. The second derivative test is used to finding minimum point.
We can rewrite (Eq. 37) as:
The computations of the first and second derivatives show that the critical point is given by:
For which:
We can see that the second derivative is positive and conclude that the critical point is a relative minimum.
5. Simulation
Figure 4 illustrates a general block diagram of the suggested DFIM control scheme. Here, we can note the placement of optimization block, the first estimator-block which evaluates torque and the second estimator-block which evaluates firstly the modulus and position fluxes, respectively s, r, s and r, from the measured currents using (Eq. 2) and secondly the feedback functions f1, f2, f3, f4 given by (Eq. 10). Optimization process allows adapting the main flux magnetizing defined by rotor flux to the applied load torque characterized by the stator flux. With the analogical switch we can select the type of the reference rotor flux. The switch position 1, 2 gives respectively TCLO and TOF for optimized operation and the position 3 for a magnetizing constant level.
The Figure 5 shows the speed response versus time according to its desired profile drawn on the same figure. Figure 6 illustrate the fluxes trajectory of the closed–loop system. It moves along manifold toward the equilibrium point. We can notice the stability of the system. Figures 7 and 8 show respectively the stator and the rotor input control voltages versus time during the test. Figure 9 present the copper losses according to the stator flux variations in steady state operation and we can see the contribution of the TCLO compared to the TOF. Finally figure 10 present the dissipated energy versus time from which we can observe clearly the influence of the three switch positions on the copper losses in transient state. We can conclude that the TCLO is the best optimization.
6. Conclusion
In this chapter was presented a vector control intended for doubly fed induction motor (DFIM) mode. The use of the state-all-flux induction machine model with a flux orientation constraint gives place to a simpler control model. The stability of the nonlinear feedback control is proven using the Lyapunov function.
The simulation results of the suggested DFIM system control based on double flux orientation which is achieved by the proposed DFIM control demonstrates clearly the suitable obtained performances required by the references profiles defined above. The speed tracks its desired reference without any effect of the load torque. Therefore the high control performances can be well affirmed. To optimize the machine operation we chose to minimize the copper losses. The proposed TCLO factor performs better than the already designed TOF. Indeed, the energy saving process can be well achieved if the magnetizing flux decreases in the same way as the load torque. It results in an interesting balance between the core losses and the copper losses into the machine, so the machine efficiency may be largely improved. The simulation results confirm largely the effectiveness of the proposed DFIM control system.
7. Appendix
The machine parameters are:
Rs =1.2 ; Ls =0.158 H; Lr =0.156 H; Rr =1.8 ; M =0.15 H; P =2 ;J = 0.07 Kg.m² ; Pn = 4 Kw ; 220/380V ; 50Hz ; 1440tr/min ; 15/8.6 A ; cos = 0.85.
8. Nomenclature
s, rRotor and stator indices.
d, q Direct and quadrate indices for orthogonal components
PNumber of pairs poles
Torque angle
s, rStator and rotor flux absolute positions
Mechanical rotor frequency (rd/s)
Rotor speed (rd/s)
sStator current frequency (rd/s)
rInduced rotor current frequency (rd/s)
J Inertia
dUnknown load torque
CeElectromagnetic torque
~Symbol indicating measured value
^Symbol indicating the estimated value
*Symbol indicating the command value
DFIMDoubly Fed Induction Machine
TOFTorque Optimization Factor
TCLO Torque Copper Losses Optimization
References
- 1.
Vicatos M. S. Tegopoulos J. A. 2003 A Doubly-Fed Induction Machine Differential Drive Model for Automobiles 18 2 June 2003),225 230 0885-8969 - 2.
Akagi H. Sato H. 1999 Control and Performance of a Flywheel Energy Storage System Based on a Doubly-Fed Induction Generator-Motor, Proceedings of the 30 th ,32 39 1 02759306 USA, 27 June-1 July, 1999 - 3.
Debiprasad P. et al. 2001 A Novel Control Strategy for the Rotor Side Control of a Doubly-Fed Induction Machine. ,1695 1702 0-78037-114-3 USA, 30 September- 04 October 2001 - 4.
Leonhard W. 1997 Control Electrical Drives, S pringier verlag,3-54041-820-2 Heidelberg, Germany - 5.
Wang S. Ding Y. 1993 Stability Analysis of Field Oriented doubly Fed induction Machine drive Based on Computed Simulation, ,21 1 11 24 1532-5008 - 6.
Morel L. et al. 1998 Double-fed induction machine: converter optimisation and field oriented control without position sensor 145 4 July 1998),360 368 1350-2352 - 7.
Hopfensperger B. et al. 1999 Stator flux oriented control of a cascaded doubly fed induction machine Electric power applications,146 6 November 1999),597 605 1350-2352 - 8.
Hopfensperger B. et al. 1999 Stator flux oriented control of a cascaded doubly fed induction machine with and without position encoder,147 4 July1999),241 250 1350-2352 - 9.
Metwally H. M. B. et al. 2002 Optimum performance characteristics of doubly fed induction motors using field oriented control 43 1 3 13 0196-8904 - 10.
Hirofumi A. Hikaru S. 2002 Control and Performance of a Doubly fed induction Machine Intended for a Flywheel Energy Storage System, ,17 1 January 2002),109 116 0885-8993 - 11.
Djurovic M. et al. 1995 Double Fed Induction Generator with Two Pair of Poles Conferences of Electrical Machines and Drives,449 452 0-85296-648-2 UK, 11-13 September 1995 - 12.
Leonhard W. 1988 Adjustable-Speed AC Drives, Invited Paper, ,76 4 April 1988),455 471 0018-9219 - 13.
Longya X. Wei C. 1995 Torque and Reactive Power control of a Doubly Fed Induction Machine by Position Position Sensorle ss Scheme.31 3 May/June 1995),636 642 0093-9994 - 14.
Sergei P. Andrea T. Tonielli A. 2003 Indirect Stator Flux-Oriented Output Feedback Control of a Doubly Fed Induction Machine, ,11 6 Nov. 2003),875 888 1063-6536 - 15.
Wang D. H. Cheng K. W. E. 2004 General discussion on energy saving Power Electronics Systems and Applications. ,298 303 9-62367-434-1 Nov. 2004 - 16.
Zang L. Hasan K. H. 1999 Neural Network Aided Energy Efficiency control for a Field Orientation Induction Machine Drive. ,356 360 0537-9989 Canterbury, UK, 1-3 September 1999 - 17.
David E. 1988 A Suggested Energy-Savings Evaluation Method for AC Adjustable-Speed Drive Applications,” ,24 6 Nov/Dec. 1988),1107 1117 0093-9994 - 18.
Rodriguez J. et al. 2002 Optimal Vector Control of Pumping and Ventilation Induction Motor Drives ,49 4 August 2002),889 895 0278-0046 - 19.
Drid S. Nait_ Said. M. S. Tadjine M. 2005 Double flux oriented control for the doubly fed induction motor ,33 10 October 2005),1081 1095 1532-5008 - 20.
Drid S. Tadjine M. Nait_ Said. M. S. 2005 Nonlinear Feedback Control and Torque Optimization of a Doubly Fed Induction Motor. ,56 3-4 57 63 1335-3632 - 21.
Drid S. Nait-Said M. S. Tadjine M. Makouf A. 2008 Nonlinear Control of the Doubly Fed Induction Motor with Copper Losses Minimization for Electrical Vehicle CISA08, , AIP Conf. Proc.,1019 339 345 Annaba, Algeria, June 30-July 02, 2008 - 22.
Khalil H. 1996 Nonlinear systems Prentice Hall,0-13067-389-7