2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific) A Comparison of Discrete-time Complex Vector Current Regulators at Low Frequency Ratio Zhen Dong Yong Yu School of Electrical Engineering and Automation Harbin Institute of Technology School of Electrical Engineering and Automation Harbin Institute of Technology Harbin, China [email protected] [email protected] Harbin, China Abstract?This paper focuses on the performance and stability of discrete-time domain complex vector PI (CVPI) current regulator operating at low carrier-wave frequency ratio. A novel and accurate discrete-time induction motor model using Pole-zero Matching method is proposed with consideration of control delay and PWM delay. On the foundation of this, an explicit comparison of discrete-time CVPI using Front-Euler method, Back-Euler method, Tustin method and Direct Design method is made to investigate the limit frequency ratio for different discrete-time CVPI topologies. The simulation results are provided with the same conditions except the adapted current regulator for a relatively objective comparison. Keywords?discrete-time domain; Pole-zero Matching Method; complex vector current regulator ; induction motor design of current regulator [9]. In this paper, an accurate discrete-time induction motor model based on Pole-zero Matching method is proposed. On the foundation of this, the performance and stability of different discrete-time CVPI current regulators are compared and analyzed in the condition of low carrier-wave frequency ratio. II. DISCRETE CVPI CURRENT REGULATOR DESIGN A. CVPI Current Regulator Design Considering that the Back_EMF is a relative tardy variant, if the Back_EMF is compensated well with negligible effect on current-loop, the IM model could be simplified as RL load in stationary reference frame. And the transfer function is (1). I. INTRODUCTION Since the current loop, the inner loop, is the important part for the performance of the vector control system, different kinds of current loop topologies have been proposed in last decades, like hysteresis-band control(HBC) [1], synchronous reference frame PI(SFPI), feed-forward PI(FFPI) [2],feedback PI(FBPI) and predict control[3]. Some of them with simple topologies are wildly used by industrial applications but could not realize totally decoupling of d_axis and q_axis current [2], some of them with high control performance but the algorithm is much more complicated [3][4]. Meanwhile, complex vector PI (CVPI) current regulator proposed on the basis of SFPI has been deeply discussed because of its comparatively feasible structure and superior performance especially at low carrierwave frequency ratio or high speed terms [5-9]. In terms of current regulator discretization, generally the Back-Euler method and the Tustin method could satisfy most of the discrete requirement. However, since the switching frequency is severely limited in some specially applications like high-power or high-speed occasions because of the intrinsic frequency of gate turn-off thyristors [8]. In order to solve the problem caused by low frequency ratio, the compensation of PWM delay and control delay caused by frame rotation is considered [10]. Active damping is used to eliminate the current oscillatory [8]. An improved Back-Euler method is proposed in senseless control for a tradeoff between the high-speed requirements and observer complexity [11]. Also, the step respond method is applied for direct discrete G ps ( s ) = (1) Where: R =Rs + L2m L2r Rr , L=? Ls , Rs Stator resistance, Rr Rotor resistance, Lm Mutual inductance, Lr Rotor selfinductance, Ls Stator self-inductance, ? Total leakage factor ( ? = 1 ? L2m Ls Lr ). The transform between stationary reference frame and synchronous reference frame is expressed as (2)(3): f dqs = f dqe e j?et j?e t (2) j?e t sf = s( f e ) = [( s + j?e ) f ]e (3) Thus, the transfer function in synchronous frame is (4): s dq e dq e dq i e ( s) 1L = (4) e u ( s ) s + j?e + R L It is known that the current loop could be equivalent as a first-order inertia loop, thus, the current loop is (5) G pe ( s ) = e GCC ( s) = i e ( s) ?cb? = i ( s ) s + ?cb? e _ ref (5) Where ?cb? the expected bandwidth. Project 51377032 supported by National Natural Science Foundation of China; National science and technology support program, 2014BAF08B05. 978-1-5386-2894-2/17/$31.00 �17 IEEE 978-1-5386-2894-2/17/$31.00 �17 IEEE i s ( s) 1L = s u (s) s + R L e e GCO ( s ) = GCR ( s ) * G pe = ?cb? s (6) 2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific) TABLE I. Front-Euler Back-Euler z ?1 Ts s= DIESCRET METHODS s= z ?1 zTs ? (k ) Ts 0.5Ts Tustin s= i (k ) u (k ) 2( z ? 1) Ts ( z + 1) ? ( k + 1) i( k +1) u ( k + 1) u (k ) output is ,ref + Kp ? is K iTs j?eTs + + + 1 z ?1 + us Sampling and Update Piont Fig. 1. Block diagram of CVPI using Front-Euler Approximation. Carrier Wave And the open loop is (6), hence, the complex vector current regulator in continuous domain is (7): e ( s) = GCR K p Ki + jK p?e u( s ) ?cb? s + j?e + R L * =K p + (7) = 1L ?i ( s ) s s Where K p = L?cb? ?Ki = R / L . The discrete methods commonly used are Front-Euler, Back-Euler and Tustin Approximation, as shown in Tab. 1. Hence, we get the CVPI with Front-Euler method expression (8), and Fig1 is the block diagram: us ( z ) K p z + K p KiTs + jK p?eTs ? K p = (8) ?is ( z ) z ?1 Similarly, we obtain CVPI with Back-Euler method (9) and Tustin method (10): F ( z) = GCR B GCR ( z) = T GCR ( z) = = us ( z ) ( K p + K p Ki Ts + jK p?eTs ) z ? K p (9) = ?is ( z ) z ?1 us ( z ) ?is ( z ) (2 K p + K p KiTs + jK p?eTs ) z + ( K p Ki Ts + jK p?eTs ? 2 K p ) (10) 2( z ? 1) Using the Pole-zero cancellation, the expression of direct discrete-time CVPI current regulation in [9] is (11): us ( z ) K ( ze j?eTs ? e ? RTs / L ) = ?is ( z ) z ?1 Where K is the coefficient related to the bandwidth. D GCR ( z) = Computation (11) The expression (11) is derived from the discrete-time IM model, and the explicit derivation is in the next part. We can infer from (11) that since the exponential function is added in the current regulator structure, theoretically, it should correspond with that in continuous domain better while the realized complexity on chip is neglected. B. Time Delay Analysis in Digital Control System In digital motor control system, time delay is inevitable mainly because of program execution, PWM update and PWM output while the sampling time, turn-on and turn-off delay of IGBT are short enough to be neglectful. For double-samplingdouble-update, sampling point and PWM update happen in Ideal PWM Signal Ideal Modulating Wave Output PWM Signal Output Modulating Wave Fig. 2. Time delay in digital control system with double-sampling-doubleupdate PWM. is ,ref + Kp ? is K iTs j?eTs + + + z +1 + 2( z ? 1) j?e 1.5 TTsj? ee0.75 s e us Fig. 3. Block diagram of CVPI using Tustin Method with phase advance time delay compensation. each Ts , during which program execution for uoutput is finished. Then, in the next Ts , PWM output happens when the output modulating wave intersects with the carrier wave, which in average needs 0.5Ts . Hence, in sum, there is 1.5Ts delay for control system. In stationary frame, control delay could be ?T s equivalent to e ?1.5Ts s ( e p in accurate, if all the time delays are included.). With Taylor Approximation, always it can be written as 1?1.5Ts s + 1?. However, in synchronous frame, ? sample is related to both uoutput and isample , if ?e is thought to be constant during the delay period, there would be an 0.75Ts?e error for frame rotation. Hence, the total time delay above can be expressed as e ?1.5Ts ( s + j?e ) .Transforming to the discrete-time domain, it could be written as z ?1e1.5Ts j?e .Thus, in considering of time delay compensation, e1.5Ts j?e need be added to the expression from (8) to (10). Meanwhile, since the direct discrete-time current regulator has taken the PWM delay into consideration as a zero-order hold, only e0.5Ts j?e is needed for time delay compensation. III. PROPOSED DISCRETE-TIME IM MODEL A. Discrete-time IM Model using Pole-zero Matching Method In order to analyze the performance of discrete-time complex vector current regulator theoretically, the accurate discrete-time IM model under synchronic reference frame is necessary to be built. The commonly known transition approaches from continuous-time domain to discrete-time domain includes Front-Euler, Back-Euler, Tustin transform, Pulse Response Method, Step Response Method and Pole-zero Matching Method. Though each approach has own advantages in the appropriate occasions, none of them could transfer all the information entirely. Since in the part 2, the complex vector current regulator is designed using pole-zero cancellation, 2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific) u e ? RTs e L + 1 2R + ze j?eTs + ? e j?eTs z ?1 1 z ? e ? RTs 0.75 0.5 TTs sj?j? ee L ee ie Fig. 4. Block diagram of IM model using Pole-zero Matching Method with time delay. corresponding to the discrete method, the Pole-zero Matching Method is most appropriate to ensure the pole-zero consistency. Considering of the time delay in part3, the accurate discrete-time IM model with time delay z -1 ? e-0.5Ts j?e is obtained (16): Using Pole-zero Matching Method, which would be expressed as follows: i e ( z) ( z ? e j?eTs + 1)(1 ? e ? RTs L ) = (16) e u ( z ) (2 R ? e j?eTs z ? 2 R ? e ? RTs L ) z ? e0.5Ts j?e From (16), it is clear to see that the IM property is mainly caused by varying poles related to ?e . G( s) = k ? ( s + zi ) sT z =e ??? ? G( z ) = m ? (s + p ) k1 ? ( z -e ? ziT ) m ? ( z-e j n ? p jT ) G pe _ td ( z ) = ( z + 1)n ? m n B. Comparison and Analysis of CVPI Current Regulator In part2, CVPI with four different kinds of discrete structure has been given without considering of time delay compensation. In this part, both the discrete precision and the time delay compensation would be taken into comparison. (12) Where zi : zero, p j : pole, m, n are the order of numerator and denominator. If m < n , it indicates that there is zero at s = ? , corresponding to z = ?1 . And the gain k1 is derived by G ( s ) s =0 = G( z) z =1 Fig.5 gives the eigenvalue migration with varying frequency from 0Hz to 300Hz. In part II, we know that in continuous domain, the excellent pole/zero cancellation can be achieved with the CVPI structure. However, the zeros using Front-Euler Approximation achieve the proper cancellation under the low frequency. As the frequency going up, it becomes improper, which means that the regulator is disabled. Similarly, as shown in Fig.6, though the regulator zeros tend to track the varying close-loop poles, the cancellation result is still not proper especially in high frequency situation. However, it is much better by using BE Approximation compared with the FE Approximation. Therefore, to tradeoff the algorithm validity . The expression (1) transformed as discrete?time IM model in stationary reference frame can be written in (13): k1 ( z + 1) z ? e ? RTs L G ps ( z ) = Where k1 = 1 ? e ? RTs 2KR (13) L . 2 Ri s ( k + 1) ? e ? j? ( k +1) ? e j?eTs ? u s (k + 1)(1 ? e ? RTs L ) ? e ? j? ( k +1) ? e j?eTs Pole-Zero Map = 2 K R e ? RTs Li s ( k ) ? e ? j? ( k ) + u s (k )(1 ? e ? RTs L ) ? e ? j? ( k ) 1 0.6?/T 0.8 (14) Then, the resulting equation can be transformed into the difference equation in the synchronous frame (14), and finally the discrete-time domain transfer function in synchronous frame is obtained (15): ? ? ? j ? ( k )+?e ( k ) Ts ] Where f dqe (k ) = f??s (k )e ? j? ( k ) , e ? j? ( k +1) = e [ 0.5?/T 0.5 0.4?/T 0.1 0.3?/T 0.2 0.3 0.4 0.2?/T 0.5 0.6 0.7 0.1?/T 0.8 0.9 0.7?/T 0.6 0.8?/T Imaginary Axis 0.4 0.2 0 -0.2 0.9?/T 0 1?/T 1?/T 0.9?/T 0.1?/T -0.5 -0.4 0.8?/T -0.6 0.2?/T 0.7?/T -0.8 0.3?/T 0.6?/T -1 -1 -0.8 -0.6 -0.4 -0.2 0.5?/T 0 -1 0.4?/T 0.2 0.4 0.6 0.8 1 0.6 0.7 0.8 0.9 1 1.1 Real Axis G pe ( z ) = j?eTs ? RTs L e ? RTs L i ( z ) z (1 ? e )?e + (1 ? e = ue ( z) 2 R ? e j?eTs z ? 2 R ? e ? RTs L ) Fig. 5. Eigenvalue migration of current close-loop with FE CVPI. (15) Pole-Zero Map Pole-Zero Map 0.6?/T 0.8 0.5?/T 0.8?/T Imaginary Axis 0.4 0.2 0 -0.2 0.9?/T 0.1?/T -0.6 0.8?/T -0.8 -0.6 -0.4 -0.2 0.5?/T 0 0 -0.2 0.9?/T 1?/T 1?/T 0.9?/T 0.1?/T 0.8?/T 0.2?/T 0.7?/T -0.8 0.4?/T 0.2 0.2 -0.6 -0.8 0.3?/T 0.6?/T 0.6 0.8 1 0.3?/T 0.6?/T -1 0.4 0.1 0.3?/T 0.2 0.3 0.4 0.2?/T 0.5 0.6 0.7 0.1?/T 0.8 0.9 0.8?/T -0.4 0.2?/T 0.7?/T 0.4?/T 0.4 -0.4 0.9?/T 0.5?/T 0.7?/T 0.6 -0.2 1?/T 1?/T -0.8 -1 -1 0.8 0 -0.4 -0.6 0.6?/T 0.2 0.1 0.3?/T 0.2 0.3 0.4 0.2?/T 0.5 0.6 0.7 0.8 0.1?/T 0.9 0.7?/T 0.6 1 0.4?/T Imaginary Axis 1 0.6 0.7 0.8 0.9 1 -1 -1 -0.8 Real Axis (a) Fig. 6. Eigenvalue migration of current close-loop with BE CVPI including time delay compensation -0.6 -0.4 -0.2 0.4?/T 0.5?/T 0 0.2 Real Axis (b) 0.4 0.6 0.8 1 1.2 2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific) Pole-Zero Map 0.5?/T 0.6?/T 0.8 0.6 0.1 0.3?/T 0.2 0.3 0.4 0.2?/T 0.5 0.6 0.7 0.8 0.1?/T 0.9 0.8?/T Imaginary Axis 0.4 0.9?/T 0.2 Pole-Zero Map -0.2 0.6?/T 0.8 0.2 -0.4 0.1?/T 0.1 0.3?/T 0.2 0.3 0.4 0.2?/T 0.5 0.6 0.7 0.1?/T 0.8 0.9 0.8?/T 0.9?/T 0.2 1?/T 1?/T 0 -0.2 0.9?/T 0.1?/T -0.4 -0.4 0.8?/T -0.6 0.7?/T -0.8 -0.6 -0.4 0.5?/T 0 -0.2 -0.8 0.2?/T 0.7?/T -0.8 0.3?/T 0.6?/T 0.4?/T 0.2 0.8?/T -0.6 0.3?/T 0.6?/T -1 -1 -0.6 0.2?/T -0.8 0.4 0.6 0.8 1 0.6 0.8 -1 -1 1 -0.8 -0.6 -0.4 -0.2 Pole-Zero Map 0.6?/T 0.8 0.5?/T 0.6 0.8?/T 0 -0.2 0.9?/T 0.8 0.2 0.9?/T -0.4 0.1?/T -0.6 0.8?/T -1 -1 -0.8 -0.6 -0.4 -0.2 0.5?/T 0 0.4?/T 0.6 0.8 1 0.2 0.4 0.6 0.8 0.4?/T 0.1 0.3?/T 0.2 0.3 0.4 0.2?/T 0.5 0.6 0.7 0.8 0.1?/T 0.9 0.8?/T 0 -0.2 0.9?/T 1?/T 1?/T 0.9?/T 0.1?/T 0.8?/T 0.2?/T 0.7?/T -0.8 -1 Real Axis 0.2 -0.6 -0.8 0.3?/T 0.6?/T 0.4 -0.4 0.2?/T 0.7?/T 0.5?/T 0.4 0 1?/T 1?/T -0.8 0.2 0.7?/T 0.6 -0.2 -0.4 -0.6 0.6?/T 0.4 0.1 0.3?/T 0.2 0.3 0.4 0.2?/T 0.5 0.6 0.7 0.8 0.1?/T 0.9 0.4 0.2 0.4?/T Pole-Zero Map 1 0.4?/T 0.7?/T Imaginary Axis 1 0.5?/T 0 Real Axis Real Axis Imaginary Axis 0.4?/T 0.4 -0.2 0.9?/T 0.5?/T 0.7?/T 0.6 0 1?/T 1?/T 0 1 0.4 0.4?/T 0.7?/T Imaginary Axis 1 0.3?/T 0.6?/T 1 0.7 0.8 0.9 -1 -1 1 -0.8 -0.6 -0.4 -0.2 0.5?/T 0 0.4?/T 0.2 0.4 0.6 0.8 1 Real Axis Fig. 7. Eigenvalue migration of current close-loop with Tustin CVPI and PZM CVPI including time delay compensation. and complexity, BE Approximation is the common choice for most under based speed occasion. Meanwhile, of another particular interest is the poles in the dashed box, which would migrate to or out of the unit circle as the synchronous frequency increases. The system will become oscillated without any compensation. Considering of the delay compensation, phase advance method is used to compensate the e ?1.5Ts j?e . Comparing with Fig.6 (a), the unstable poles can be well cancelled in Fig6 (b). Hence, the whole system will be in stability without any effects of the increasing synchronous frequency. Comparing the two latter methods in Fig.7: Tustin Approximation and Direct Discrete-time Method. It is obvious that both would satisfy most of the high-speed application because of the excellent pole/zero cancellation. From the enlarged part, nuanced matching still could be found. Thereby, Direct Discrete-time Method is more precious than Tustin Approximation especially when the varying frequency up to 200Hz. And the phase advance time delay compensation is needed yet. frequency is set to 400Hz(the frequency ratio is 8). Fig.9 shows that BE Method tends to be unstable, which means that with a TABLE II. PARAMETERS OF THE INDUCTION MOTOR Parameters Value Rated power Pe/kW 3.7 Rated voltage Ue/V 380 Rated current Ie/A Rated frequency fe/Hz 8.9 2 Stator resistance Rs/? 1.142 Rotor resistance Rr/? 0.825 Mutual inductance Lm/mH 118.9 Stator, rotor inductance Ls,Lr/mH 124.4 Frequency Ratio 12 10 BE DQ_axis Current ?A? A given commend step speed from 0 to 50Hz, the simulation results are in Fig.8, the first platform (0s-0.4s) acceleration period, and the q-axis current is limited, and the second platform (0.5s-0.8s) is the loading period. The switching frequency is set to 600Hz(the frequency ratio is 12). And the BF CVPI, Tustin CVPI and Pole-zero Matching are included in terms of the operation stability. It can be clear seen that all of them have a nice performance. While the switching 1440 Pole pair pn IV. SIMULATION RESULTS The simulation is based on the MATLAB Simulink circumstance. The IM model is built by Simulink modules and the control system is compiled by M files. The parameters needed in simulation are in Tab.2. 50 Rated velocity ne/(r/min) 0 -10 10 Tustin 0 -10 10 Pole-zero Matching 0 -10 0 0.5 isd ,ref Time (s) isq ,ref 1.0 isd isq Fig. 8. Performance of BE, Tustin and Pole-zero Matching under the frequency ratio of 12. 2017 IEEE Transportation Electrification Conference and Expo, Asia-Pacific (ITEC Asia-Pacific) these above objectively, a novel and accurate discrete-time inductive motor model using Pole-zero Matching Method is proposed with consideration of control delay and PWM delay. The eigenvalue migration is applied for theoretical analysis and the simulation results prove the validity of the Pole-zero Matching Method especially under the low frequency ratio. Frequency Ratio 8 10 DQ_axis Current ?A? BE 0 -10 10 Tustin 0 -10 Pole-zero Matching REFERENCES 10 [1] 0 -10 0 0.5 1.0 Time (s) isd ,ref isq ,ref isd isq Fig. 9. Performance of BE, Tustin and Pole-zero Matching under the frequency ratio of 8. Frequency Ratio 6 50A 10 DQ_axis Current ?A? BE 0 0.46s -10 0.52s 50A 10 Tustin 0 0.46s -10 0.52s 15A 10 Pole-zero 0 Matching -10 0 0.5 isd ,ref Time (s) isq,ref 1.0 -10A 0.8s isd 0.84s isq Fig. 10. Performance of BE, Tustin and Pole-zero Matching under the frequency ratio of 6. sudden reference, the response need an oscillated period to follow the track. Meanwhile, the latter methods are still in stability. Finally, the switching frequency is set to 300Hz (the frequency ratio is 6). Both the BE Method and the Tustin Method are out of control because of the too long control period. However, the Pole-zero Matching is still effective though the control performance is degraded in some certain. V. CONCLUSION This paper focuses on the performance and stability of different discrete methods including FE Approximation, BE Approximation, Tustin Approximation and Pole-zero Matching Method based on CVPI structure. In order to compare all of C. N.-M. Ho, V. Cheung, and H. S.-H. Chung, ?Constant-frequency hysteresis current control of grid-connected VSI without bandwidth control,? IEEE Trans. Power Electron., vol. 24, no. 11, pp. 2484?2495, Nov. 2009 [2] Robert D. Lorenz, Donald B. Lawson, ?Performance of Feedforward Current Regulators for Field-Oriented Induction Machine Controllers,? IEEE Trans. Ind. Appl., vol. 23, no. 4, pp. 597 ? 602, July 1987 [3] J. M. Espi, J. Castello, R. Garc韆-Gil, G. Garcera, and E. 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