The flux linkage current relations are: Ψ q r = L r i q r + L mp i q sp − L mc i q sc Ψ d r = L r i d r + L mp i d sp − L mc i d sc (65) 0 q r = R r i q r + dΨ q r dt + ω r Ψ d r (66) 0 q r = R r i d r + L r di d r dt + ω r L r i q r + L mp di d sp dt + ω r L mp i q sp − L mc di d sc dt −ω r L mc i q sc 0 q r = R r i d r +(L r i d r + L mp i d sp − L mc i d sc )s +(L r i q r + L mp i q sp − L mc i q sc )ω r 0 q r = R r i d r + dΨ d r dt + ω r Ψ q r (67) The electrical torque creation in the power machine is governed by the same principles that apply to any induction machine. The general equation of the electrical torque in this case is simply: T e = 3 2  P 2  Ψ m I r (68) In the d-q reference frame, however, the last equation is rearranged to show the torque as a function of certain control parameter. As the power machine is grid connected, it will have a constant voltage. The torque could be: T p = 3 2  P p 2  (Ψ q sp i d sp −Ψ d sp i q sp ) (69) It is clear from the above equation that the only control variables are the d-q components of the stator current, because the power machine stator fluxes are almost constant. Furthermore, when the controller reference frame is aligned with one of the flux components, the number of the control variables is reduced. To derive the electrical torque for the control machine, we can use the same general equation for the electrical torque. This case cannot be simplified because the stator fluxes of the control machine will be variable. The control machine torque must be expressed as a function of the excitation current and the purpose in this research is to provide a flexible power control of the BDFIG. So the next equation is the control machine torque, and it is given in terms of the future control quantities. T c = − 3 2  P c 2  L mc (i d sc i q r −i q sc i d r ) (70) The option of the rotor current as the second variable is clearly shown and that there exists an electric coupling between the two stators of the BDFTIG, which is achieved through the common rotor current. This reflects the behaviour of the inner workings of the BDFTIG. The total electric torque (Te) for the BDFTIG is the sum of the individual electrical torques of both machines: T e = − 3 4  P p (Ψ q sp i d sp −Ψ d sp i q sp )+P c L mc (i d sc i q r −i q sc i d r )  (71) The electric torque equation is defined by the friction and total inertia of the power and control machines: T e = T L +(B p F + B c F )ω m +(j p s + j c s ) dω m dt (72) 129 From Dynamic Modeling to Experimentation of Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control Rearranging the last equation to derive the shaft speed: dω m dt = 1 j p s + j c s  T e − T L −(B p F + B c F )ω m  Hence, the shaft speed is ω m = T e − T L (B p F + B c F )+(j p s + j c s )s (73) 6.3 Simulation of the BDFTIG Model The BDFTIG model was tested to determine if it was a true representation of the actual generator. Using Matlab/Simulink to test the BDFTIG, the main tests consisted of disabling one side of the BDFTIG and applying a constant AC voltage on the opposite side, at the same time as changing the load torque to allow both motoring and generation modes of operation. The short circuit test consisted of shorting the stator side of the control machine, and a natural speed of 900 rpm was recorded, because both machines have four poles each, as shown in Figure 9. For the next test, the load torque was decreased at time 2.25s to put the BDFTIG into the generation mode as shown in Figure 10. The system responded as expected by increasing its speed and moving into the super-synchronous mode of operation, the electrical torque changed at the same time as the load demand. In this section, the dynamic model of the generator was developed based on the selected d-q reference frame. The model was implemented and tested in MATLAB/Simulink. The simulation results verified that the model can correctly describe the dynamic behaviour of BDFTIG design. Fig. 9. Speed-Torque Curve of BDFTIG with short circuit test. Fig. 10. Generation Mode of BDFTIG. 130 Electric Machines and Drives In this section, the dynamic model of the generator was developed based on the selected d-q reference frame. The model was implemented and tested in MATLAB/Simulink. The simulation results verified that the model can correctly describe the dynamic behaviour of BDFTIG design. 7. Experimentation In order to validate the new controllers, experiments were conducted on a real system. The following controllers were implemented: PBC, PBC+Proportional action on stator currents, PI controller on stator currents, and a combination of PBC and PI control. The experiments were done in the IRII-UPC (Institute of Robotics and Industrial Informatics - University Polytechnic of Catalonia) where a 200W DFIG interconnected with an IM prototype is available (see Fig. (11)). The setup was controlled using a computer working under RT-Linux operating system. With the PBC, only the position sensors of the Generator and the Induction machine were used for the control. For the Proportional and PI controllers of the electrical subsystem, measurement of the two stator currents were also needed. In order to show the behaviour of the system under different load conditions, a non-measured load torque was applied. sw Uc 1:1 SERVO AMPLIFIER Advanced Motion Control Three Phase Inverter C’ B’ A’ 3 3 3 3 3 3 #SD Promax 1 Promax 2 Promax 3 AD215BY Isolation Amplifier Promax 2 PCI DAS 4020/12 PCI8133 1:13 Protection System (Salicrú) Vbus + Vbus - 6N137 Optocouplers Pentium 4; 1,8 GHz; 512 MB RAM 74HC244 Buffer Non-Inverting AD215BY Isolation Amplifier Brake DFIM Generator Rotor Stator Induction Motor 3ph 1:13,4 AD215BY Isolation Amplifier 2 2 2 2 2 2 1A-250mV 1000rpm 1V DL10050 1000rpm 1V DL10050 Jeulin 188 019 Jeulin 188 016 12 Hall Sensor EH050 Hall Sensor EH050 1A-250mV ADC - 12BNCs DAC Board Channel Signal 0 0 0 0 1 1 1 1 2 2 2 2 0 1 2 3 0 1 2 3 0 1 2 3 I4 I3 I2 I1 V4 V3 V2 V1 DIO Encoder 360 pulses/revol. Encoder 100 pulses/revol. dada1 dada2 dada3 dada4 dada5 dada6 dada7 dada8 dada9 dada10 dada11 dada12 W,A,V 1V 100mV 1A 250mV 1W 10mV #PWME A , B 74HC14 Inverter notA notB A , B 2 2 A , B 74HC14 Inverter notA notB A , B 2 2 2 2 2 80% of 46V 75% of 42V POWERBOX 100V-10A DC Motor PWMs U+ (16) V+ (17) W+(18) U- (34) V- (35) W-(36) nB2(24) nA2(23) A2(5) B2(6) 2 nB1(21) nA1(20) A1(2) B1(3) Bridge Off + 5V PCIDAS4020 Ramp Braking DC Motor I 1 I 3 I 2 V 1 V 2 V 3 V 4 I 4 Select 12 DAQs X X M speed M position G speed G position Fig. 11. Experimental setup Since a load torque sensor was not available for the acquisition, we built an estimator of the resistive torque based on the measurement of the mechanical IM speed. 131 From Dynamic Modeling to Experimentation of Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control 7.1 Estimation of the load torque The mechanical dynamics of the IM is given by: J M ¨ θ M = τ M −τ LM − B M ˙ θ M (74) Since the asymptotic stability of the electrical subsystem Σ e is proven we can consider that in the steady state τ M → τ d M (exponentially). Then,we have in the steady state the following: J M ¨ θ M = τ d M −τ LM −B M ˙ θ M    τ ML (75) Hence, a linear load torque observer was designed (with l 1 , l 2 are design parameters): ˙ ˆ ω mM =  τ d M − ˆ τ ML  /J M + l 1 ( ˆ ω mM −ω mM ) (76) ˙ ˆ τ ML = l 2 ( ˆ ω mM −ω mM ) (77) 7.2 PBC 184 186 188 190 192 194 196 198 200 202 204 500 1000 1500 (a) t(s) ω mM ref & ω mM (rpm) 184 186 188 190 192 194 196 198 200 202 204 1000 2000 3000 (b) t(s) ω mG (rpm) 193.74 193.76 193.78 193.8 193.82 193.84 193.86 193.88 193.9 193.92 193.94 2 4 6 (c) t(s) θ G & θ M (rad) 184 186 188 190 192 194 196 198 200 202 204 0 0.2 0.4 (d) t(s) τ G (N.m) 184 186 188 190 192 194 196 198 200 202 204 −2 −1 0 1 2 (e) t(s) τ Md (N.m) Fig. 12. PBC-(a) Regulated Motor speed and its reference. (b)Generator speed. (c) DFIG & IM rotor position. (d) Generator torque (e) Motor desired torque. Figure 12 presents the mechanical IM speed and its smooth reference, the mechanical DFIG speed, the DFIG and IM rotor positions, the DFIG torque τ G and the IM desired torque τ Md . The real IM speed tracks the reference very well, i.e. low overshoot and no steady state error are observed. Figure 13 shows the stator currents i sa and i sb , and their references over a suitable period of time. The stator currents do not track exactly their desired values but are bounded. This is because the goal of the PBC is to track the IM speed and to keep internal signals bounded. Figure 14 shows the DFIG rotor currents i rGa and i rGb , and their references over a period of time. Again, these currents are sinusoidal and bounded. Figure 15 presents the DFIG rotor voltages v rGa and v rGb , the IM rotor speed ω mM and its estimation ˆ ω mM , the estimated IM load torque ˆ τ ML , and the estimated IM speed, given by 132 Electric Machines and Drives 193.74 193.76 193.78 193.8 193.82 193.84 193.86 193.88 193.9 193.92 193.94 −20 −10 0 10 20 (a) t(s) i sGa & i sGb (A) 193.74 193.76 193.78 193.8 193.82 193.84 193.86 193.88 193.9 193.92 193.94 −20 −10 0 10 20 (b) t(s) i sGa & i d sGa (A) 193.74 193.76 193.78 193.8 193.82 193.84 193.86 193.88 193.9 193.92 193.94 −20 −10 0 10 20 (c) t(s) i sGb & i d sGb (A) Fig. 13. PBC-(a) i sa , i sb (b) i d sa , i sa (c) i d sb , i sb . 193.74 193.76 193.78 193.8 193.82 193.84 193.86 193.88 193.9 193.92 193.94 −15 −10 −5 0 5 10 15 (a) t(s) i rGa & i rGb (A) 193.75 193.8 193.85 193.9 193.95 194 194.05 −50 0 50 (b) t(s) i rGa & i d rGa (A) 193.75 193.8 193.85 193.9 193.95 194 194.05 −50 0 50 (c) t(s) i rGb & i d rGb (A) Fig. 14. PBC-(a) i rGa , i rGb (b) i d rGa , i rGa (c) i d rGb , i rGb . 133 From Dynamic Modeling to Experimentation of Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control 193.75 193.8 193.85 193.9 193.95 194 194.05 −60 −40 −20 0 20 40 60 (a) t(s) v rGa & v rGb (V) 184 186 188 190 192 194 196 198 200 202 204 400 600 800 1000 1200 1400 1600 (b) t(s) ω mM & ω  mM (rpm) 184 186 188 190 192 194 196 198 200 202 204 −3 −2 −1 0 1 2 (c) t(s) τ τ LM (N.m) Fig. 15. PBC-(a) v rGa , v rGb (b) ω mM , ˆ ω mM (c) ˆ τ ML . (76)-(77), is tracking the real speed. Hence, a good estimation of the real IM load torque is obtained. It has to be noticed that the IM rated torque is 0.7Nm. It can be concluded that the PBC provides good practical performance even when the applied load torque is twice the magnitude of the nominal load torque of the IM. 7.3 PBC + P 55 60 65 70 75 80 500 1000 1500 (a) t(s) ω mM ref & ω mM (rpm) 55 60 65 70 75 80 1000 2000 3000 (b) t(s) ω mG (rpm) 68.85 68.9 68.95 69 69.05 69.1 2 4 6 (c) t(s) θ G & θ M (rad) 55 60 65 70 75 80 0 0.1 0.2 0.3 (d) t(s) τ G (N.m) 55 60 65 70 75 80 −2 −1 0 1 2 (e) t(s) τ Md (N.m) Fig. 16. PBC+P-(a) Regulated Motor speed and its reference. (b)Generator speed. (c) DFIG & IM rotor position. (d)Generator torque (e) Motor desired torque. As with the PBC alone, the results obtained with the PBC+P are given in figures 16-19. On the whole, the system behaviour is the same as the PBC alone. One difference that is noticeable is 134 Electric Machines and Drives 68.85 68.9 68.95 69 69.05 69.1 −15 −10 −5 0 5 10 15 (a) t(s) i sGa & i sGb (A) 68.85 68.9 68.95 69 69.05 69.1 −15 −10 −5 0 5 10 15 (b) t(s) i sGa & i d sGa (A) 68.85 68.9 68.95 69 69.05 69.1 −15 −10 −5 0 5 10 15 (c) t(s) i sGb & i d sGb (A) Fig. 17. PBC+P-(a) i sGa , i sGb (b) i d sGa , i sGa (c) i d sGb , i sGb . 68.85 68.9 68.95 69 69.05 69.1 −15 −10 −5 0 5 10 15 (a) t(s) i rGa & i rGb (A) 68.85 68.9 68.95 69 69.05 69.1 69.15 69.2 69.25 69.3 −40 −20 0 20 40 60 (b) t(s) i rGa & i d rGa (A) 68.85 68.9 68.95 69 69.05 69.1 69.15 69.2 69.25 69.3 −40 −20 0 20 40 60 (c) t(s) i rGb & i d rGb (A) Fig. 18. PBC+P-(a) i rGa , i rGb (b) i d rGa , i rGa (c) i d rGb , i rGb . the small error between the desired stator currents and the real ones thanks to the proportional controller. The PBC+P controller exhibits good practical performance but not significantly better than those obtained with the PBC alone. 7.4 PBC + PI Again, as for the PBC and the PBC+P controllers, figures 20-23 show the results. It can be seen in figure 21 that the integral actions on the stator currents do not decrease the error significantly between the real and desired values in comparison with the results for the PBC+P 135 From Dynamic Modeling to Experimentation of Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control 68.85 68.9 68.95 69 69.05 69.1 69.15 69.2 69.25 69.3 −60 −40 −20 0 20 40 60 (a) t(s) v rGa & v rGb (V) 55 60 65 70 75 80 400 600 800 1000 1200 1400 1600 (b) t(s) ω mM & ω  mM (rpm) 55 60 65 70 75 80 −3 −2 −1 0 1 2 (c) t(s) τ τ LM (N.m) Fig. 19. PBC+P-(a) v rGa , v rGb (b) ω mM , ˆ ω mM (c) ˆ τ ML . 40 45 50 55 60 65 500 1000 1500 (a) t(s) ω mM ref & ω mM (rpm) 40 45 50 55 60 65 1000 2000 3000 (b) t(s) ω mG (rpm) 52.6 52.65 52.7 52.75 52.8 52.85 2 4 6 (c) t(s) θ G & θ M (rad) 40 45 50 55 60 65 0 0.1 0.2 0.3 (d) t(s) τ G (N.m) 40 45 50 55 60 65 −2 −1 0 1 2 (e) t(s) τ Md (N.m) Fig. 20. PBC+PI-(a) Regulated Motor speed and its reference. (b)Generator speed. (c) DFIG & IM rotor position. (d)Generator torque (e) Motor desired torque. 136 Electric Machines and Drives 52.6 52.65 52.7 52.75 52.8 52.85 −10 0 10 (a) t(s) i sGa & i sGb (A) 52.6 52.65 52.7 52.75 52.8 52.85 −10 0 10 (b) t(s) i sGa & i d sGa (A) 52.6 52.65 52.7 52.75 52.8 52.85 −10 0 10 (c) t(s) i sGb & i d sGb (A) Fig. 21. PBC+PI-(a) i sGa , i sGb (b) i d sGa , i sGa (c) i d sGb , i sGb . 52.6 52.65 52.7 52.75 52.8 52.85 −10 0 10 (a) t(s) i rGa & i rGb (A) 52.6 52.65 52.7 52.75 52.8 52.85 52.9 52.95 53 −50 0 50 (b) t(s) i rGa & i d rGa (A) 52.6 52.65 52.7 52.75 52.8 52.85 52.9 52.95 53 −50 0 50 (c) t(s) i rGb & i d rGb (A) Fig. 22. PBC+PI-(a) i rGa , i rGb (b) i d rGa , i rGa (c) i d rGb , i rGb . controller (see fig. 17). This is due to the fact that the reference values are sinusoidal and that the bandwidth of the PI controllers cannot be increased sufficiently experimentally. It can be concluded that the PI action on the stator currents does not improve significantly the performance obtained with the PBC+P controller. 7.5 PI The PI control law (with K p and K i are proportional and integral gains) is given below: Bv rG = B  K p (i sG −i d sG )+K i (i sG −i d sG )  (78) 137 From Dynamic Modeling to Experimentation of Induction Motor Powered by Doubly-Fed Induction Generator by Passivity-Based Control 52.6 52.65 52.7 52.75 52.8 52.85 52.9 52.95 53 −50 0 50 (a) t(s) v rGa & v rGb (V) 40 45 50 55 60 65 500 1000 1500 (b) t(s) ω mM & ω  mM (rpm) 40 45 50 55 60 65 −2 0 2 (c) t(s) τ τ LM (N.m) Fig. 23. PBC+PI-(a) v rGa , v rGb (b) ω mM , ˆ ω mM (c) ˆ τ ML . 230 235 240 245 250 255 500 1000 1500 t(s) ω mM ref & ω mM (rpm) 230 235 240 245 250 255 500 1000 1500 2000 2500 t(s) ω mG (rpm) 241.35 241.4 241.45 241.5 241.55 2 4 6 t(s) θ G & θ M (rad) 230 235 240 245 250 255 −0.4 −0.2 0 t(s) τ G (N.m) 230 235 240 245 250 255 −2 0 2 t(s) τ Md (N.m) Fig. 24. PI-(a) Regulated Motor speed and its reference. (b)Generator speed. (c) DFIG & IM rotor position. (d) Generator torque (e) Motor desired torque. 138 Electric Machines and Drives [...]... sGa 83 .5 83 .55 83 .6 t(s) 83 .65 83 .7 10 5 0 5 isGb & id (A) sGb 10 10 5 0 5 10 d d Fig 29 PBC -robustness test-(a) i sGa, isGb (b) isGa, isGa (c) isGb, isGb 141 142 Electric Machines and Drives irGa & irGb(A) 20 10 0 10 20 83 .5 83 .55 83 .6 83 .65 83 .7 83 .75 83 .8 83 .85 83 .9 83 .95 83 .75 83 .8 83 .85 83 .9 83 .95 83 .75 83 .8 83 .85 83 .9 83 .95 83 .9 83 .95 t(s) irGa & id (A) rGa 50 0 50 83 .5 83 .55 83 .6 83 .65 83 .7... M 85 t(s) 2500 2000 1500 1000 500 4 G 2 G(N.m) 83 .5 83 .55 83 .6 t(s) 83 .65 83 .7 0.1 0 0.1 Md (N.m) 70 75 80 85 90 95 85 90 95 t(s) 2 0 2 70 75 80 t(s) isGa & isGb(A) Fig 28 PBC-robustness test-(a) Regulated Motor speed and its reference (b)Generator speed (c) DFIG & IM rotor position (d) Generator torque (e) Motor desired torque 10 5 0 5 10 83 .55 83 .6 t(s) 83 .65 83 .7 83 .5 83 .55 83 .6 t(s) 83 .65 83 .7 83 .5... 50 83 .5 83 .55 83 .6 83 .65 83 .7 t(s) irGb & id (A) rGb 50 0 50 83 .5 83 .55 83 .6 83 .65 83 .7 t(s) d d Fig 30 PBC-robustness test-(a) irGa, irGb (b) irGa , irGa (c) irGb, irGb 0 v rGa &v rGb (V) 50 50 83 .5 83 .55 83 .6 83 .65 83 .7 83 .75 83 .8 83 .85 hat 1500 1000 mM & mM (rpm) t(s) 500 70 75 80 85 90 95 85 90 95 Switch on Parameters t(s) 1 0.5 0 70 75 80 t(s) Fig 31 PBC-robustness test-(a) vrGa , vrGb (b) mM... systems, in Communications and Control Engineering Berlin,Germany:Spring-Verlag, 19 98 ă [2] Liu, X., G Verghese, J Lang and M Onder, Generalizing the Blondel-Park Transformation of Electrical Machines: Necessary and Sufcient Conditions, IEEE Trans Circ Syst., Vol 36, No 8, pp 1 085 -1067, 1 989 [3] M Becherif, R Ortega, E Mendes and S Lee, Passivity-based control of a doubly-fed induction generator interconnected... J ( Nm2 /rad) DFIG 0.365 0.559 0.9 38 0.9 38 12.975 4.3 58 ì 103 IM 0.5 0.2 1.2 1.2 9.00 1.1 ì 103 Table 1 The parameters for DFIG and IM PBC Re f M [rpm] LM [N.m] settling time of Re f M settling time of M G [rpm] e M x 105 e i sGa x 103 Observed magnitude of isGa [A] PavgG [W] Pavg M [W] PBC+P PBC+PI PI 5001000 5001000 5001000 5001000 140 080 0 140 080 0 140 080 0 140 080 0 (1st order lter) (1st order lter)... measured current isq and the reference isq at standstill rotor In this test the auxiliary winding (or axis d) is open while the main winding (or axis q) is identied The dotted box in Fig 2 is detailed in Fig 3 where the RMRAC control law applied to q axis is shown In Fig 3, the reference model and plant are given by Wm (s) = k m Zm ( s ) , Rm (s) (5) 1 48 Electric Machines and Drives * isq RMRAC q vsq... 0.5 0.5 1.15 0.5 0.4s 0.4s 0.4s 0.4s 0.1s 1500 2.9 3.7 0.1s 1500 4 .8 2.65 0.1s 1500 5.7 2. 68 2s 1500 38. 6 0.25 5 4.7 58. 9 7 5.6 74.4 8 5 70.7 10 9 177.9 Table 2 Comparison table of experimental results If we take in account the problem of speed tracking of the IM interconnected to the DFIG and according to the robustness tests and the experimental results presented in Table 2 we can say that the... of the BDFTIG 144 Electric Machines and Drives 9 Acknowledgment The authors would like to express their gratitude to Jordi Riera, Enric Fossas and Miguel Alluộ from IRII-UPC, Barcelona, Spain for their help with the practical experiments 10 References [1] R Ortega, A Loria, P.J Nicklasson, and H Sira-Ramirez, Passivity-based control of Euler-Lagrange systems, in Communications and Control Engineering... with the PBC are reported here Figure 28 presents the mechanical IM speed and its smooth reference, the mechanical DFIG speed, the DFIG and IM rotor positions, the DFIG torque G and the IM desired torque Md The real IM speed tracks very well the reference, i.e low overshoot and no steady state error are observed Figure 29 shows the stator currents i sa and isb , and their references over a period of... inductances, r is the speed rotor, ri is the electromagnetic ux and Ni is the number of turns for auxiliary winding or axis d and for main winding or axis q The stator and rotor inductances are relationship with the leakage and mutual inductance as a L si = L lsi + L mi and Lri = L lri + L mi , respectively From Fig 1 and from (Krause et al., 1 986 ) it is possible to derive the dynamical model of a SPIM . Passivity-Based Control 83 .5 83 .55 83 .6 83 .65 83 .7 83 .75 83 .8 83 .85 83 .9 83 .95 −20 −10 0 10 20 t(s) i rGa & i rGb (A) 83 .5 83 .55 83 .6 83 .65 83 .7 83 .75 83 .8 83 .85 83 .9 83 .95 −50 0 50 t(s) i rGa . 193.76 193. 78 193 .8 193 .82 193 .84 193 .86 193 .88 193.9 193.92 193.94 2 4 6 (c) t(s) θ G & θ M (rad) 184 186 188 190 192 194 196 1 98 200 202 204 0 0.2 0.4 (d) t(s) τ G (N.m) 184 186 188 190 192. desired torque. 83 .5 83 .55 83 .6 83 .65 83 .7 −10 −5 0 5 10 t(s) i sGa & i sGb (A) 83 .5 83 .55 83 .6 83 .65 83 .7 −10 −5 0 5 10 t(s) i sGa & i d sGa (A) 83 .5 83 .55 83 .6 83 .65 83 .7 −10 −5 0 5 10 t(s) i sGb