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Performance comparison of permanent magnet synchronous motor (PMSM) drive with delay compensated predictive controllers
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Performance Comparison of Permanent Magnet Synchronous Motor (PMSM) Drive with Delay Compensated Predictive Controllers K. Thangarajan , A. Soundarrajan PII: DOI: Reference:
S0141-9331(19)30637-4 https://doi.org/10.1016/j.micpro.2020.103081 MICPRO 103081
To appear in:
Microprocessors and Microsystems
Received date: Revised date: Accepted date:
25 November 2019 22 January 2020 3 March 2020
Please cite this article as: K. Thangarajan , A. Soundarrajan , Performance Comparison of Permanent Magnet Synchronous Motor (PMSM) Drive with Delay Compensated Predictive Controllers, Microprocessors and Microsystems (2020), doi: https://doi.org/10.1016/j.micpro.2020.103081
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Performance Comparison of Permanent Magnet Synchronous Motor (PMSM) Drive with Delay Compensated Predictive Controllers K. Thangarajan Associate Professor Department of Electrical & Electronics Engg. RVS College of Engineering &Technology. Coimbatore. [email protected]
Abstract High efficient current limiting controller is mandatory to obtain ripple-free torque and required speed level in the output of the Permanent Magnet Synchronous Motor (PMSM) drives. In this paper, substantial analysis is taken to control the output torque ripple, to minimize the acoustic noise and also to get the required speed by simulating PMSM with three different investigators such as SVPWM, Model Predictive Control (MPC) and Dead-Beat (DB) Predictive Controller. The designed controllers are tested through the numerical simulations in the MATLAB Simulink Platform and also experimental validation is taken in the laboratory. These predictive investigators are implemented to get good transient response, less torque ripple; reduced harmonics in phase currents and also with delay compensation. The comparison between the simulation and experimental results are presented at the end. The DBP control issuitable for the high performance applications and it is easy to implement in the PMSM drives. Keywords: Multi Processor chip, Embedded,
Communication, Circuits, Gates, Torque Ripple Reduction, Space Vector Pulse Width Modulation (SVPWM); Model Predictive Control (MPC); Dead-Beat Predictive Controller (DBPC); Permanent Magnet Synchronous Motor (PMSM). I. INTRODUCTION
A. Soundarrajan Professor Department of Electrical & Electronics Engg. PSG College of Technology. Coimbatore.
[email protected] conventional DTC is widely accepted control strategy and it has hysteresis comparators and switching table [2]. The hysteresis comparators has the current limiting values and it maintains the current value within the given limits. By limiting the current values, torque can be controlled. The performance evaluation of DTC, MPDTC, DDTC methodsof a SPMSM is simulated and performance results are explained in [3].The mathematical model of PMSM based on SVPWM with PI controller is implemented using MATLAB in [4]. The simulation of model predictive control with SVPWM based on PMSM using Simulink block is quiet simple [5]. The Model Predictive Control (MPC) is to maintain a certain degree progress compared with the previously explained ideal flux path technique. The researchers did many experiments in the PMSM motor with MPC controller in a different predicting strategies. Among them, some of the few transactions are listed. At first, a comparative study between the MPC and PID controller is made in the aircraft applications. This study is implemented in the vertical take-off and landing in lab prototype is explained in [7]. The simulation of speed control of PMSMs project is done by using MPC controller is presented and concluded that enables the flexible control because of the presence of cost function estimation and prediction methods [8]. The ANFIS and LMS algorithm also implemented to predict the current control strategy, hybrid set control in the field oriented parameter control is presented in the paper [9] and [10]. The torque ripple compensation of variable speed drives [11] and the PMSM model with field-weakening is implemented in the paper [12].
AC drives are dominating in the variable speed drive market. It also replaces the DC drives in the high performance applications and torque control requirements. In the industrial field such as electrical vehicles, electricalField Weakening Techniques have been developed and research is going on to eliminate the drawbacks and to improve the traction, servo control system, wind power generation, better performance. The decoupling of torque and flux are Permanent Magnet Synchronous Motor (PMSM) is widely used to control the torque of the DC machine. The same used. The PMSM drive has several benefits such as i) high principle is behind the field oriented control techniques. It facilitates a self-sufficient control of field and torque efficiency, ii) high power density, iii) high torque to weight independently to handle the parallel field oriented quantities ratio, iv) excellent dynamic performances. For high in order to perform high dynamic applications. The dynamic performances PMSM drives requires high Limitations of FOCs are i) High sensitivity to machine dynamic control strategies[1]. In the recent days commonly parameters and ii) Large computing cost. The speed finite used variable speed control techniques are broadly set of predictive control fed by matrix converter is a novel classified into two categories. They are (i) Field Oriented way in the MPC controller. This method overtakes the Control (FOC) and (ii) Direct Torque Control (DTC). The conventional cascaded control scheme with single control
logic. It controls both the phase current and speed simultaneously in [13].In FOC, no dead-time compensation was necessary because, if a voltage error produces a current error then it will be corrected by the PI controller. The generalized explicit predictive and DB controller are used to control the motor and robust control practice is implemented in[14], [15], [16] and [17].
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(4)
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(
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(
(
(
(5)
(
II. MATHEMATICAL MODELLING OF PMSM
(
[
The PMSM mathematical model can be designed for continuously operating time mode. In a rotating frame of a PMSM model consists of two dynamics. They are:i) Electrical andii) Mechanical dynamics. The general block diagram of PMSM model in the rotating frame (d-q axis) are shown in the Fig.1.
(
(
(
)
(
](6)
III. SPACE VECTOR PULSE WIDTH MODULATION CONTROL
A. Electrical dynamics The differential equations of PMSM model with respect to electrical dynamics part can be expressed as
(1) The electromagnetic torque developed by PMSM rotor can be expressed as [ ( ) ] (2)
a. SVPWM based control The Space Vector Pulse Width Modulation (SVPWM) is the basic technique to take the ideal flux path, when the PMSM is powered by 3ϕ sine wave voltage [6]. By approaching the flux circle with the actual flux made by inverter so as the motor could acquire constant sine magnetic field. By this way it is easy to achieve high performance with reduced torque ripple.In the process of controlling PMSM, the currents in3ϕarmature winding in ia, ibandicare undergone certain transformation to get current components in d-q axis. By Clark’s transformation,the 3ϕ Stationary co-ordinates are transformed into 2ϕ Stationary co-ordinates. These expressions are given inequ.7.
[ √
]
√ *
+[
√
](7)
By Park’s transformation,the2ϕ Stationary co-ordinates are transformed into synchronous rotating co-ordinates. These expressions are given in equ. 8. Fig.1.General Block Diagram of PMSM Model – Rotating Frame
B. Mechanical dynamics The differential equations of PMSM model with respect to mechanical dynamics part can be expressed as
(
(3)
Euler method is used to obtain the discrete time model of PMSM. The discretization techniques are presented in [18] and [19]. The Euler method affords an unpretentious model without introducing any undesirable supplementary nonlinear terms.
(
[
]
[
][
] (8)
The basic space vector diagram is shown in the Fig.2. The sector determination and duty cycle calculation is as follows. The reference speed and the actual speed from position and speed sensing unit is given as inputs to the error signal comparator. The output error signal is generated and it given as the input to proportional plus integral (PI-1) controller. Theschematic diagram of SVPWM control is shown in the Fig. 3. The PI-1 gives the value of isqrefandisq from the park transformation block is given to the comparator and error signal produced and then this signal is given to the PI-2 as input to get the Vsqref. Similarly isdref and isdfrom the park
transformation block is compared and the resultant signal generated is given to the PI-3 to get the Vsdref.
Fig. 2. Space Vector Diagram
The simulation of SVPWM control of PMSM is a closed loop control, so that the feedback taken from the output torque and rotor speed is given as an input after comparing with the reference signal. Here the subsystem block in Fig. 4is used to produce fluxes fαandfβand given to get the response from SVPWM. It generates the six gate pulses to the switches which is present in the VSI inverter. . The 3ϕ inverter output currents are transformed in to two stationary coordinates by Clark’s transformation as in equ. 7. The two stationary coordinates is renovated to the rotating coordinates by Park’s transformation as in equ. 8. The variation of torque, flux, current and its derivatives are given as input to the system to maintain the ideal flux path.
Duty Cycle is calculated as D = 0.5 * (1 + [dx, dy, d0]). The VsqrefandVsdref is given as the input the inverse park transformation which is transformed from d-q axis to αβ components. The Vsαref andVsβrefare agreed to generate the gate pulses in the SVPWM. The six gate pulses is generated from the SVPWM block is set as the input the 3ϕ inverter which is coupled with the 3ϕ PMSM motor. The Clark transformation is involved to produce the two αβ components from the abc voltage produced in the inverter output as per the equ.7. This cycle follows the closed path to produce the continuous sine magnetic field. The same principle is simulated in MATLAB as per the numerical calculation made from the theoretical explanation and simulated diagram and its results are clearly presented in the section III.b. The sector number determination and its conditions to find the duty cycle is tabulated in Table 1.
Fig.4. Simulation Diagram of SVPWM Control based PMSM
From the simulation results, Vdc is given from a dc source of 300v is constant for SVPWM which is shown in the Fig. 5(a). The produced output phase currents with distorted harmonics are shown in Fig. 5(b).Due to the harmonics present in the current waveform, the torque ripples are present in the electromagnetic torque output which implies proportionality between current to the torque and is shown in the Fig. 5(e).
TABLE 1 Sector Number and Its Conditions to Find Duty Cycle
Sector 1 2 3 4 5 6
Conditions arg>=0 and arg=pi/3 and arg<2*pi/3 arg>=2*pi/3 and arg=-pi and arg<-2*pi/3 arg>-2*pi/3 and arg=-pi/3 and arg<0
Values 0.1 to 1.047 1.047 to 2.094 2.094 to 3.14 -3.14 to -2.094 -2.094 to -1.047 -1.047 to 0.1 (a)
Fig.3. Schematic diagram of SVPWM Control
b. Simulation of SVPWM Control
(b)
(c)
(d)
methods.To detention the control objectives, an optimization problem can be displayed as: Minimize objective function(Control Set), subject to predictive model andconstraints like Ld≤ Lq, etc.control set includes all possible voltage vectors. Predictive Model – Predicted currents for every vector in control set are generalized. Control objective is entrenched into the objective function explicitly and some constraints can be enforced as well. The earlier step of the MPC control is to compare the reference speed wm* with the actual speed wm and error signal generates is given as an input the PI controller. The output of the PI controller is separated into the two signals; one is directly taken as torque reference and other signal is fed into the Maximum Torque per Ampere (MTPA) gives the flux reference. These signals are directly given to the MPC controller which consists of cost function minimization, torque and flux prediction and estimation. The voltage source, Vsget from the MPC block and send it to the SVPWM produces pulses for the voltage source inverter to the PMSM motor. These blocks are shown in the Fig. 6and the MPC controller is designed on the basis of the expressions (9)-(16).
b. Simulation of MPC Control The MPC control based on PMSM consist of the fragmentdq-abc, PWM inverter, PMSM, MPC are shown in the Fig.7. The dq-abc gets the id*, iq*, i0and θtaken as feedback from the PMSM motor output. The current iabc* is engaged to PWM inverter is set to PMSM motor. The MPC controller acquires duty cycle and angle output vector simultaneously in order to solve the optimization problem. (e) Fig.5. Simulation Results obtained from SVPWM based PMSM Control (a) Input DC Voltage Vdc, (b) Phase Output Currentsiabc, (c) Rotor Speed ωm, (d) Electromagnetic torque developed, (e) Ripple Content in the output torque.
Also, there is a chance to produce the acoustic noise because of the harmonics current and torque ripple present is 9.375%. The rotor speed and torque waveform is shown in the Fig. (c)and(d). The electromagnetic torque produced in the motor is 5.2Nm.The settling time of thespeed of motor is 0.2 s.
To launch the rotating d-q coordinates and support the rotor fluxΨron the d-axis. The stator voltage equation of PMSM in the rotating d-q coordinates can be expressed as:
dxyVd(x,y) = Rsisd + Ld isd -
Lqisq
dxyVq(x,y) = Rsisq + Lq isq - ωrLdisd +ωrψPM(10) Ld≤Lq and VdandVq are the d-axis and q-axis projections of vector Vxy, which can be expressed as: Vd(x,y) = M(x) cos[θ v(x,y) - θr] (11)
Vq(x,y) = M(x) sin[θv(x,y) - θr] Where θr∊ [0,2π] is the rotor flux angle.
IV. MODEL PREDICTIVE CONTROL a. MPC based control The Model Predictive Control (MPC) is intended on the basis of operation research techniques which is subjected to minimize the time and cost and to maximize the performance of the system. To improve steady state error and current ripple, inverter dead-time and actuation delay compensation are used for the predictive control
(9)
Fig.6. Block diagram of MPC Control
(12)
In a successive of equ.(9-12) can be disseminated via Euler approximation and then the predictive model of PMSM can be expressed as:
isdxy(k+1) = ( isqxy(k+1)
=
(14)esd(k )= -
esq(k )= xy
(c)
)isd(k)+ [dxyVd(xy) - esd(k)](13) (
)isq(k)+ [dxyVq(xy)
( Lqisq(k)
-
esq(k)
(15)
( [Lqisq(k)+ ψPM]
(16)
xy
isd (k+1) andisq (k+1) are the predictions of isdandisqw.r.to vs= dxyVxy at instant (k+1)Tc. isd(k)andisq(k) are the sampling quantities of isd and isq at instantkTc.
(d)
From theMPC simulation output, Vdc is given from a dc source and is kept constant for all the three investigators which is shown in the Fig. 7 (a). The output waveform of the phase currents produced with distorted harmonicsis shown inthe Fig. 7 (b).Due to the harmonics present in the current waveform is moderate, less ripple present in the electromagnetic torque output.
(e) Fig.7. Simulation Results obtained from MPC based PMSM Control (a) Input DC Voltage Vdc, (b) Phase Output Currents iabc, (c) Rotor Speed ωm, (d) Electromagnetic torque developed, (e) Ripple Content in the output torque
(a)
The output torque is proportional to the currentas shown in the Fig.7 (e). There is a possibility to produce the acoustic noise because of the harmonic current and torque ripple present in the output which is of the value6.55%.The rotor speed and torque waveform is shown in the Fig.7 (c) and 7 (d). The electromagnetic torque produced in the shaft is 8.2Nm. The speed settling time of this controller is 0.0066s.
V. DEAD-BEAT PREDICTIVE CONTROL a. DBPC based control
(b)
The main objective of the deadbeat controller with delay compensation is to minimize the torque ripples, sinusoidal phase currents with constant torque output. The reasons for the undesired fluctuations of phase currents, oscillations in the output, acoustic noises are uncertainties, imperfection and harmonics produced in the back emf. The proposed model consists of prediction and correction of the flux produced and maintains the ideal flux path by choosing the vector in SVPWM which is fed by DBPC. The block diagram of DBPC based PMSM is shown in the Fig. 8.
Fig.8. Block diagram of DBPC Control based PMSM
b. Simulation of DBPC Control The Dead-Beat predictive (DBP) controller designed on the basis of the equations with delay compensation techniques is used for quick response. The succeeding Fig.9shows thesimulation of DBPC based PMSM motor control. The current prediction vector is calculated as Ḯ(k+1) = ik + T{ Fkik + dk + L0-1vk – (1/T)(Kηηk) } (17)
(a)
By Ḯk = i0 where Kη∊ R2x2 symmetric gain matrix, and
ηk= ∑
(18)
The prediction error can be defined by,
ek = ik - Ḯk
(19)
(b)
Fig.9. Dead-Beat Predictive Controller
The DBPC blockis composed with the required speed reference ωm*, d-q axis reference current idq*, actual speed ωmas feedback, 3ϕ stationary currents iabc, angular rotational theta θrandinputs voltages to produce vαβfor the SVPWM block. The two level voltage source inverter and dead time aregiven iabcand dc voltage vdc as input.The output response got from the PMSM are presented in the Fig. 10. The output values are tabulated at the end and the comparative results also presented in the section VII. A Constant Vdc is given from a dc source toall the three investigators considered in this paper is shown in the Fig. 10(a).The rise time is obtained precisely as one sampling period and no overshoot.There is no harmonics present in the currentwaveform, so torque ripple is less in the output torque as shown in theFig. 10(b)andFig. 10(e). Some delay is introduced to compensate the computation time, sensors and actuation propagation time. In the conventional scheme, the computation time is considered as zero, so that the voltage is applied just in the moment when the current is sampled, in instant (k).
(c)
(d)
waveform is exactly proportional to the electromagnetic torque produced by the motor.
(e)
Fig.11. Experimental Setup – PMSM Control
The d-q axis phase currents with fluctuations are shown in the Fig.12(a) andFig.12(b). From the analytical point of view, there is an impact of VdcandNm plays a vital role in the PMSM drive. The DBP control results obtained from numericalsimulationsand hardware implementation are quite identical with reduced ripples and harmonics.
Symbol
(f) Fig.10. Simulation Results obtained from DBPC (a) Input DC Voltage Vdc, (b) Phase Output Currents iabc, (c) Rotor Speed ωm, (d) Electromagnetic torque developed, (e) Ripple Content in the output torque, (f) Rotor angle and torque vs time.
Acoustic noise and fluctuation of motor is less and torque ripple present is only 2%.The rotor speed and torque waveform is shown in the Fig. 10(c)and10(d). The electromagnetic torque produced in the motor is 12.2Nm. The speed settling time also less as 0.005secin the DBPController.
VII. HARDWARE IMPLEMENTATION
TABLE 2 Parameters of the PMSM Description Rating
P
Rated Power
3 kW
Rs
Stator Resistance
1.3Ω (ohm)
Ls
Stator Inductance
0.000835H
PM Rotor Magnetic Flux
0.175wb
B
Viscous friction Coefficient
0.001 kg m2 s-1
p
No. of Pole Pairs
4
J
Moment of Inertia
0.0008 kg m2
The parameters employed in the PMSM drive in the Table 2 andquantitative comparison and values are clearly presented in the Table 3. The graphical representation shows the d-q axis phase currents idq,stator output voltages vdq, rotor speed output ωm,and electromagnetic torque Teare shown in the Fig. 12.
a. Experimental Setup From the MATLAB simulation sections, Dead-Beat (DB) controller is tremendous in the reduction of the torque ripple in the output of PMSM motor drive. So acoustic noise production can also be in lower range in this controller. The identical hardware setup is made in laboratory to analyse its performance with the regular atmospheric condition. The investigational arrangement consists of voltage source inverter circuit,deadbeat controller implemented in DSP processor,3ϕ PMSM motor as shown in the Fig. 11.
b. Experimental Results Analysis The output results are got from the cathode ray oscilloscope ,tachometer and are plotted. The d-axis current indicates the amplitude of phase currents and q-axis current
(a)
(b)
(f) Fig.12. Experimental Outputs(a) i sd, (b) isq, (c) vsd, (d)vsq, (e)Rotor speed, (f) Electromagnetic torque at 300Vdc. TABLE 3 Quantitative Comparison of Results Parameters
(c)
Phase Output Current, A Rotor Speed Output, RPM Speed Settling Time, s Electromagnetic Torque Output (Nm) Torque Ripple Acoustic Noise Level
Experimental Simulation SVPWM MPC 11 9
DBPC
DBPC
10
10
1498
1499
1499
1499
0.2
0.006
0.005
0.008
5
8
12
12.2
9%
7%
2%
3%
Low
Low
High
Medium
VIII. CONCLUSION
(d)
A Performance comparative analysis is taken over with PMSM drive under SVPWM, MPC and DBPC with delay compensation is carried out in this paper. The analysis is fully focussed on the reduction of torque ripple, harmonics and acoustic noise production through both simulations and implementation of hardware in the laboratory. Among all the investigators, DBP controller has the fast response, better current harmonic reduction, better torque ripple (≈2%) and provides better performance in the simulation. Reduction of switching losses can be achieved by choosing a zero vector in the inverter. As well as in the hardware testing the DBP controller gives better torque ripple reduction (≈3%), fast settling time and comparatively low acoustic noise with other two control strategies. Conflict of Interest
(e) This paper has not communicated anywhere till this moment, now only it is communicated to your esteemed journal for the publication with the knowledge of all co-authors.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.
Appendix:LIST OF NOMENCLATURE Stator Current Components w.r.to Direct Axis
[6]
Stator Current Components w.r.to Quadrature Axis
[7] Stator Voltage Components w.r.to Direct Axis
Stator Voltage Components w.r.to Quadrature Axis
[8] Stator Inductance Components w.r.to d & q Axis
Number of Pole Pairs
[9]
Mechanical Speed w.r.to Rotor frame
Electromotive Force(EMF) Constant
[10]
Electro-Magnetic Torque developed in PMSM
TL& J &Tc ia,ib,ic id,iq iα, iβ
Load Torque& Moment of Inertia Sampling Period&Control Period Actual armature current components Stationary current components referred to dq axis Two stationary current co-ordinates Align Rotor flux on d-axis
Permanent Magnet Flux Modulus
Rs
isq&isd arg& mag
[11]
[12]
[13]
Stator resistance Rotor flux angular velocity
d-axis & q-axis projections of Stator Current is Argument & Magnitude Values
[14]
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Thangarajan K received B.E. degree in Electrical and Electronics Engineering from Bharathiyar University, Coimbatore, India in 2002 and M.E. degree from PSG College of Technology, Coimbatore, India, in 2007. He is currently an Associate Professor of Electrical and Electronics Engineering Department with RVS College of Engineering and Technology, Coimbatore, India. His main research interests include electrical drives, brushless machines, mathematical models of electrical machines, drive-system control and diagnostics, soft computing, renewable energies, and energy management.
Soundarrajan A received the Electrical and Electronics Engineering B.E. degree from Bharathiyar University of Coimbatore, India, in 1999, the M.E. degree in Applied Electronics from PSG College of Technology, Coimbatore, India, in 2003, and the Ph.D. degree from Anna University, Chennai, India, in 2012. From 2007 to 2017, he was an Associate Professor with the Department of Information Technology, PSG College of Technology, and Coimbatore, India. He is currently a Professor with the Department of Electrical and Electronics Engineering, PSG College of Technology, Coimbatore, India. His research interests include power Intelligent Controllers, Big Data Analytics for efficient demand side Energy Management, Soft Computing, Energy Sources and Utilization and Energy Management System
Performance Comparison of Permanent Magnet Synchronous Motor (PMSM) Drive with Delay Compensated Predictive Controllers K. Thangarajan , A. Soundarrajan PII: DOI: Reference:
S0141-9331(19)30637-4 https://doi.org/10.1016/j.micpro.2020.103081 MICPRO 103081
To appear in:
Microprocessors and Microsystems
Received date: Revised date: Accepted date:
25 November 2019 22 January 2020 3 March 2020
Please cite this article as: K. Thangarajan , A. Soundarrajan , Performance Comparison of Permanent Magnet Synchronous Motor (PMSM) Drive with Delay Compensated Predictive Controllers, Microprocessors and Microsystems (2020), doi: https://doi.org/10.1016/j.micpro.2020.103081
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Performance Comparison of Permanent Magnet Synchronous Motor (PMSM) Drive with Delay Compensated Predictive Controllers K. Thangarajan Associate Professor Department of Electrical & Electronics Engg. RVS College of Engineering &Technology. Coimbatore. [email protected]
Abstract High efficient current limiting controller is mandatory to obtain ripple-free torque and required speed level in the output of the Permanent Magnet Synchronous Motor (PMSM) drives. In this paper, substantial analysis is taken to control the output torque ripple, to minimize the acoustic noise and also to get the required speed by simulating PMSM with three different investigators such as SVPWM, Model Predictive Control (MPC) and Dead-Beat (DB) Predictive Controller. The designed controllers are tested through the numerical simulations in the MATLAB Simulink Platform and also experimental validation is taken in the laboratory. These predictive investigators are implemented to get good transient response, less torque ripple; reduced harmonics in phase currents and also with delay compensation. The comparison between the simulation and experimental results are presented at the end. The DBP control issuitable for the high performance applications and it is easy to implement in the PMSM drives. Keywords: Multi Processor chip, Embedded,
Communication, Circuits, Gates, Torque Ripple Reduction, Space Vector Pulse Width Modulation (SVPWM); Model Predictive Control (MPC); Dead-Beat Predictive Controller (DBPC); Permanent Magnet Synchronous Motor (PMSM). I. INTRODUCTION
A. Soundarrajan Professor Department of Electrical & Electronics Engg. PSG College of Technology. Coimbatore.
[email protected] conventional DTC is widely accepted control strategy and it has hysteresis comparators and switching table [2]. The hysteresis comparators has the current limiting values and it maintains the current value within the given limits. By limiting the current values, torque can be controlled. The performance evaluation of DTC, MPDTC, DDTC methodsof a SPMSM is simulated and performance results are explained in [3].The mathematical model of PMSM based on SVPWM with PI controller is implemented using MATLAB in [4]. The simulation of model predictive control with SVPWM based on PMSM using Simulink block is quiet simple [5]. The Model Predictive Control (MPC) is to maintain a certain degree progress compared with the previously explained ideal flux path technique. The researchers did many experiments in the PMSM motor with MPC controller in a different predicting strategies. Among them, some of the few transactions are listed. At first, a comparative study between the MPC and PID controller is made in the aircraft applications. This study is implemented in the vertical take-off and landing in lab prototype is explained in [7]. The simulation of speed control of PMSMs project is done by using MPC controller is presented and concluded that enables the flexible control because of the presence of cost function estimation and prediction methods [8]. The ANFIS and LMS algorithm also implemented to predict the current control strategy, hybrid set control in the field oriented parameter control is presented in the paper [9] and [10]. The torque ripple compensation of variable speed drives [11] and the PMSM model with field-weakening is implemented in the paper [12].
AC drives are dominating in the variable speed drive market. It also replaces the DC drives in the high performance applications and torque control requirements. In the industrial field such as electrical vehicles, electricalField Weakening Techniques have been developed and research is going on to eliminate the drawbacks and to improve the traction, servo control system, wind power generation, better performance. The decoupling of torque and flux are Permanent Magnet Synchronous Motor (PMSM) is widely used to control the torque of the DC machine. The same used. The PMSM drive has several benefits such as i) high principle is behind the field oriented control techniques. It facilitates a self-sufficient control of field and torque efficiency, ii) high power density, iii) high torque to weight independently to handle the parallel field oriented quantities ratio, iv) excellent dynamic performances. For high in order to perform high dynamic applications. The dynamic performances PMSM drives requires high Limitations of FOCs are i) High sensitivity to machine dynamic control strategies[1]. In the recent days commonly parameters and ii) Large computing cost. The speed finite used variable speed control techniques are broadly set of predictive control fed by matrix converter is a novel classified into two categories. They are (i) Field Oriented way in the MPC controller. This method overtakes the Control (FOC) and (ii) Direct Torque Control (DTC). The conventional cascaded control scheme with single control
logic. It controls both the phase current and speed simultaneously in [13].In FOC, no dead-time compensation was necessary because, if a voltage error produces a current error then it will be corrected by the PI controller. The generalized explicit predictive and DB controller are used to control the motor and robust control practice is implemented in[14], [15], [16] and [17].
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(4)
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(5)
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II. MATHEMATICAL MODELLING OF PMSM
(
[
The PMSM mathematical model can be designed for continuously operating time mode. In a rotating frame of a PMSM model consists of two dynamics. They are:i) Electrical andii) Mechanical dynamics. The general block diagram of PMSM model in the rotating frame (d-q axis) are shown in the Fig.1.
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(
(
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(
](6)
III. SPACE VECTOR PULSE WIDTH MODULATION CONTROL
A. Electrical dynamics The differential equations of PMSM model with respect to electrical dynamics part can be expressed as
(1) The electromagnetic torque developed by PMSM rotor can be expressed as [ ( ) ] (2)
a. SVPWM based control The Space Vector Pulse Width Modulation (SVPWM) is the basic technique to take the ideal flux path, when the PMSM is powered by 3ϕ sine wave voltage [6]. By approaching the flux circle with the actual flux made by inverter so as the motor could acquire constant sine magnetic field. By this way it is easy to achieve high performance with reduced torque ripple.In the process of controlling PMSM, the currents in3ϕarmature winding in ia, ibandicare undergone certain transformation to get current components in d-q axis. By Clark’s transformation,the 3ϕ Stationary co-ordinates are transformed into 2ϕ Stationary co-ordinates. These expressions are given inequ.7.
[ √
]
√ *
+[
√
](7)
By Park’s transformation,the2ϕ Stationary co-ordinates are transformed into synchronous rotating co-ordinates. These expressions are given in equ. 8. Fig.1.General Block Diagram of PMSM Model – Rotating Frame
B. Mechanical dynamics The differential equations of PMSM model with respect to mechanical dynamics part can be expressed as
(
(3)
Euler method is used to obtain the discrete time model of PMSM. The discretization techniques are presented in [18] and [19]. The Euler method affords an unpretentious model without introducing any undesirable supplementary nonlinear terms.
(
[
]
[
][
] (8)
The basic space vector diagram is shown in the Fig.2. The sector determination and duty cycle calculation is as follows. The reference speed and the actual speed from position and speed sensing unit is given as inputs to the error signal comparator. The output error signal is generated and it given as the input to proportional plus integral (PI-1) controller. Theschematic diagram of SVPWM control is shown in the Fig. 3. The PI-1 gives the value of isqrefandisq from the park transformation block is given to the comparator and error signal produced and then this signal is given to the PI-2 as input to get the Vsqref. Similarly isdref and isdfrom the park
transformation block is compared and the resultant signal generated is given to the PI-3 to get the Vsdref.
Fig. 2. Space Vector Diagram
The simulation of SVPWM control of PMSM is a closed loop control, so that the feedback taken from the output torque and rotor speed is given as an input after comparing with the reference signal. Here the subsystem block in Fig. 4is used to produce fluxes fαandfβand given to get the response from SVPWM. It generates the six gate pulses to the switches which is present in the VSI inverter. . The 3ϕ inverter output currents are transformed in to two stationary coordinates by Clark’s transformation as in equ. 7. The two stationary coordinates is renovated to the rotating coordinates by Park’s transformation as in equ. 8. The variation of torque, flux, current and its derivatives are given as input to the system to maintain the ideal flux path.
Duty Cycle is calculated as D = 0.5 * (1 + [dx, dy, d0]). The VsqrefandVsdref is given as the input the inverse park transformation which is transformed from d-q axis to αβ components. The Vsαref andVsβrefare agreed to generate the gate pulses in the SVPWM. The six gate pulses is generated from the SVPWM block is set as the input the 3ϕ inverter which is coupled with the 3ϕ PMSM motor. The Clark transformation is involved to produce the two αβ components from the abc voltage produced in the inverter output as per the equ.7. This cycle follows the closed path to produce the continuous sine magnetic field. The same principle is simulated in MATLAB as per the numerical calculation made from the theoretical explanation and simulated diagram and its results are clearly presented in the section III.b. The sector number determination and its conditions to find the duty cycle is tabulated in Table 1.
Fig.4. Simulation Diagram of SVPWM Control based PMSM
From the simulation results, Vdc is given from a dc source of 300v is constant for SVPWM which is shown in the Fig. 5(a). The produced output phase currents with distorted harmonics are shown in Fig. 5(b).Due to the harmonics present in the current waveform, the torque ripples are present in the electromagnetic torque output which implies proportionality between current to the torque and is shown in the Fig. 5(e).
TABLE 1 Sector Number and Its Conditions to Find Duty Cycle
Sector 1 2 3 4 5 6
Conditions arg>=0 and arg
Values 0.1 to 1.047 1.047 to 2.094 2.094 to 3.14 -3.14 to -2.094 -2.094 to -1.047 -1.047 to 0.1 (a)
Fig.3. Schematic diagram of SVPWM Control
b. Simulation of SVPWM Control
(b)
(c)
(d)
methods.To detention the control objectives, an optimization problem can be displayed as: Minimize objective function(Control Set), subject to predictive model andconstraints like Ld≤ Lq, etc.control set includes all possible voltage vectors. Predictive Model – Predicted currents for every vector in control set are generalized. Control objective is entrenched into the objective function explicitly and some constraints can be enforced as well. The earlier step of the MPC control is to compare the reference speed wm* with the actual speed wm and error signal generates is given as an input the PI controller. The output of the PI controller is separated into the two signals; one is directly taken as torque reference and other signal is fed into the Maximum Torque per Ampere (MTPA) gives the flux reference. These signals are directly given to the MPC controller which consists of cost function minimization, torque and flux prediction and estimation. The voltage source, Vsget from the MPC block and send it to the SVPWM produces pulses for the voltage source inverter to the PMSM motor. These blocks are shown in the Fig. 6and the MPC controller is designed on the basis of the expressions (9)-(16).
b. Simulation of MPC Control The MPC control based on PMSM consist of the fragmentdq-abc, PWM inverter, PMSM, MPC are shown in the Fig.7. The dq-abc gets the id*, iq*, i0and θtaken as feedback from the PMSM motor output. The current iabc* is engaged to PWM inverter is set to PMSM motor. The MPC controller acquires duty cycle and angle output vector simultaneously in order to solve the optimization problem. (e) Fig.5. Simulation Results obtained from SVPWM based PMSM Control (a) Input DC Voltage Vdc, (b) Phase Output Currentsiabc, (c) Rotor Speed ωm, (d) Electromagnetic torque developed, (e) Ripple Content in the output torque.
Also, there is a chance to produce the acoustic noise because of the harmonics current and torque ripple present is 9.375%. The rotor speed and torque waveform is shown in the Fig. (c)and(d). The electromagnetic torque produced in the motor is 5.2Nm.The settling time of thespeed of motor is 0.2 s.
To launch the rotating d-q coordinates and support the rotor fluxΨron the d-axis. The stator voltage equation of PMSM in the rotating d-q coordinates can be expressed as:
dxyVd(x,y) = Rsisd + Ld isd -
Lqisq
dxyVq(x,y) = Rsisq + Lq isq - ωrLdisd +ωrψPM(10) Ld≤Lq and VdandVq are the d-axis and q-axis projections of vector Vxy, which can be expressed as: Vd(x,y) = M(x) cos[θ v(x,y) - θr] (11)
Vq(x,y) = M(x) sin[θv(x,y) - θr] Where θr∊ [0,2π] is the rotor flux angle.
IV. MODEL PREDICTIVE CONTROL a. MPC based control The Model Predictive Control (MPC) is intended on the basis of operation research techniques which is subjected to minimize the time and cost and to maximize the performance of the system. To improve steady state error and current ripple, inverter dead-time and actuation delay compensation are used for the predictive control
(9)
Fig.6. Block diagram of MPC Control
(12)
In a successive of equ.(9-12) can be disseminated via Euler approximation and then the predictive model of PMSM can be expressed as:
isdxy(k+1) = ( isqxy(k+1)
=
(14)esd(k )= -
esq(k )= xy
(c)
)isd(k)+ [dxyVd(xy) - esd(k)](13) (
)isq(k)+ [dxyVq(xy)
( Lqisq(k)
-
esq(k)
(15)
( [Lqisq(k)+ ψPM]
(16)
xy
isd (k+1) andisq (k+1) are the predictions of isdandisqw.r.to vs= dxyVxy at instant (k+1)Tc. isd(k)andisq(k) are the sampling quantities of isd and isq at instantkTc.
(d)
From theMPC simulation output, Vdc is given from a dc source and is kept constant for all the three investigators which is shown in the Fig. 7 (a). The output waveform of the phase currents produced with distorted harmonicsis shown inthe Fig. 7 (b).Due to the harmonics present in the current waveform is moderate, less ripple present in the electromagnetic torque output.
(e) Fig.7. Simulation Results obtained from MPC based PMSM Control (a) Input DC Voltage Vdc, (b) Phase Output Currents iabc, (c) Rotor Speed ωm, (d) Electromagnetic torque developed, (e) Ripple Content in the output torque
(a)
The output torque is proportional to the currentas shown in the Fig.7 (e). There is a possibility to produce the acoustic noise because of the harmonic current and torque ripple present in the output which is of the value6.55%.The rotor speed and torque waveform is shown in the Fig.7 (c) and 7 (d). The electromagnetic torque produced in the shaft is 8.2Nm. The speed settling time of this controller is 0.0066s.
V. DEAD-BEAT PREDICTIVE CONTROL a. DBPC based control
(b)
The main objective of the deadbeat controller with delay compensation is to minimize the torque ripples, sinusoidal phase currents with constant torque output. The reasons for the undesired fluctuations of phase currents, oscillations in the output, acoustic noises are uncertainties, imperfection and harmonics produced in the back emf. The proposed model consists of prediction and correction of the flux produced and maintains the ideal flux path by choosing the vector in SVPWM which is fed by DBPC. The block diagram of DBPC based PMSM is shown in the Fig. 8.
Fig.8. Block diagram of DBPC Control based PMSM
b. Simulation of DBPC Control The Dead-Beat predictive (DBP) controller designed on the basis of the equations with delay compensation techniques is used for quick response. The succeeding Fig.9shows thesimulation of DBPC based PMSM motor control. The current prediction vector is calculated as Ḯ(k+1) = ik + T{ Fkik + dk + L0-1vk – (1/T)(Kηηk) } (17)
(a)
By Ḯk = i0 where Kη∊ R2x2 symmetric gain matrix, and
ηk= ∑
(18)
The prediction error can be defined by,
ek = ik - Ḯk
(19)
(b)
Fig.9. Dead-Beat Predictive Controller
The DBPC blockis composed with the required speed reference ωm*, d-q axis reference current idq*, actual speed ωmas feedback, 3ϕ stationary currents iabc, angular rotational theta θrandinputs voltages to produce vαβfor the SVPWM block. The two level voltage source inverter and dead time aregiven iabcand dc voltage vdc as input.The output response got from the PMSM are presented in the Fig. 10. The output values are tabulated at the end and the comparative results also presented in the section VII. A Constant Vdc is given from a dc source toall the three investigators considered in this paper is shown in the Fig. 10(a).The rise time is obtained precisely as one sampling period and no overshoot.There is no harmonics present in the currentwaveform, so torque ripple is less in the output torque as shown in theFig. 10(b)andFig. 10(e). Some delay is introduced to compensate the computation time, sensors and actuation propagation time. In the conventional scheme, the computation time is considered as zero, so that the voltage is applied just in the moment when the current is sampled, in instant (k).
(c)
(d)
waveform is exactly proportional to the electromagnetic torque produced by the motor.
(e)
Fig.11. Experimental Setup – PMSM Control
The d-q axis phase currents with fluctuations are shown in the Fig.12(a) andFig.12(b). From the analytical point of view, there is an impact of VdcandNm plays a vital role in the PMSM drive. The DBP control results obtained from numericalsimulationsand hardware implementation are quite identical with reduced ripples and harmonics.
Symbol
(f) Fig.10. Simulation Results obtained from DBPC (a) Input DC Voltage Vdc, (b) Phase Output Currents iabc, (c) Rotor Speed ωm, (d) Electromagnetic torque developed, (e) Ripple Content in the output torque, (f) Rotor angle and torque vs time.
Acoustic noise and fluctuation of motor is less and torque ripple present is only 2%.The rotor speed and torque waveform is shown in the Fig. 10(c)and10(d). The electromagnetic torque produced in the motor is 12.2Nm. The speed settling time also less as 0.005secin the DBPController.
VII. HARDWARE IMPLEMENTATION
TABLE 2 Parameters of the PMSM Description Rating
P
Rated Power
3 kW
Rs
Stator Resistance
1.3Ω (ohm)
Ls
Stator Inductance
0.000835H
PM Rotor Magnetic Flux
0.175wb
B
Viscous friction Coefficient
0.001 kg m2 s-1
p
No. of Pole Pairs
4
J
Moment of Inertia
0.0008 kg m2
The parameters employed in the PMSM drive in the Table 2 andquantitative comparison and values are clearly presented in the Table 3. The graphical representation shows the d-q axis phase currents idq,stator output voltages vdq, rotor speed output ωm,and electromagnetic torque Teare shown in the Fig. 12.
a. Experimental Setup From the MATLAB simulation sections, Dead-Beat (DB) controller is tremendous in the reduction of the torque ripple in the output of PMSM motor drive. So acoustic noise production can also be in lower range in this controller. The identical hardware setup is made in laboratory to analyse its performance with the regular atmospheric condition. The investigational arrangement consists of voltage source inverter circuit,deadbeat controller implemented in DSP processor,3ϕ PMSM motor as shown in the Fig. 11.
b. Experimental Results Analysis The output results are got from the cathode ray oscilloscope ,tachometer and are plotted. The d-axis current indicates the amplitude of phase currents and q-axis current
(a)
(b)
(f) Fig.12. Experimental Outputs(a) i sd, (b) isq, (c) vsd, (d)vsq, (e)Rotor speed, (f) Electromagnetic torque at 300Vdc. TABLE 3 Quantitative Comparison of Results Parameters
(c)
Phase Output Current, A Rotor Speed Output, RPM Speed Settling Time, s Electromagnetic Torque Output (Nm) Torque Ripple Acoustic Noise Level
Experimental Simulation SVPWM MPC 11 9
DBPC
DBPC
10
10
1498
1499
1499
1499
0.2
0.006
0.005
0.008
5
8
12
12.2
9%
7%
2%
3%
Low
Low
High
Medium
VIII. CONCLUSION
(d)
A Performance comparative analysis is taken over with PMSM drive under SVPWM, MPC and DBPC with delay compensation is carried out in this paper. The analysis is fully focussed on the reduction of torque ripple, harmonics and acoustic noise production through both simulations and implementation of hardware in the laboratory. Among all the investigators, DBP controller has the fast response, better current harmonic reduction, better torque ripple (≈2%) and provides better performance in the simulation. Reduction of switching losses can be achieved by choosing a zero vector in the inverter. As well as in the hardware testing the DBP controller gives better torque ripple reduction (≈3%), fast settling time and comparatively low acoustic noise with other two control strategies. Conflict of Interest
(e) This paper has not communicated anywhere till this moment, now only it is communicated to your esteemed journal for the publication with the knowledge of all co-authors.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.
Appendix:LIST OF NOMENCLATURE Stator Current Components w.r.to Direct Axis
[6]
Stator Current Components w.r.to Quadrature Axis
[7] Stator Voltage Components w.r.to Direct Axis
Stator Voltage Components w.r.to Quadrature Axis
[8] Stator Inductance Components w.r.to d & q Axis
Number of Pole Pairs
[9]
Mechanical Speed w.r.to Rotor frame
Electromotive Force(EMF) Constant
[10]
Electro-Magnetic Torque developed in PMSM
TL& J &Tc ia,ib,ic id,iq iα, iβ
Load Torque& Moment of Inertia Sampling Period&Control Period Actual armature current components Stationary current components referred to dq axis Two stationary current co-ordinates Align Rotor flux on d-axis
Permanent Magnet Flux Modulus
Rs
isq&isd arg& mag
[11]
[12]
[13]
Stator resistance Rotor flux angular velocity
d-axis & q-axis projections of Stator Current is Argument & Magnitude Values
[14]
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Thangarajan K received B.E. degree in Electrical and Electronics Engineering from Bharathiyar University, Coimbatore, India in 2002 and M.E. degree from PSG College of Technology, Coimbatore, India, in 2007. He is currently an Associate Professor of Electrical and Electronics Engineering Department with RVS College of Engineering and Technology, Coimbatore, India. His main research interests include electrical drives, brushless machines, mathematical models of electrical machines, drive-system control and diagnostics, soft computing, renewable energies, and energy management.
Soundarrajan A received the Electrical and Electronics Engineering B.E. degree from Bharathiyar University of Coimbatore, India, in 1999, the M.E. degree in Applied Electronics from PSG College of Technology, Coimbatore, India, in 2003, and the Ph.D. degree from Anna University, Chennai, India, in 2012. From 2007 to 2017, he was an Associate Professor with the Department of Information Technology, PSG College of Technology, and Coimbatore, India. He is currently a Professor with the Department of Electrical and Electronics Engineering, PSG College of Technology, Coimbatore, India. His research interests include power Intelligent Controllers, Big Data Analytics for efficient demand side Energy Management, Soft Computing, Energy Sources and Utilization and Energy Management System