4 Qua Control

Four-Quadrant Control of a Switched Reluctance Motor for a Highly Dynamic Actuator Load Syed Hossain

Iqbal Husain

Harald Klode

Department of Electrical Engineering

Delphi E&C Systems

The University of Akron

USA

The use of switched reluctance motors for actuators and motion control applications was assessed and tested in [1]. The capability of SR motor for electric vehicles and hybrid electric vehicles was investigated in [2]. A nonlinear controller for high performance speed regulation control of the SRM for direct drive servo applications was proposed in [3]. Unfortunately, these studies focused exclusively on 1st quadrant operation. The strategies provided are not sufficient for highly dynamic actuators requiring quick motion reversal. Successful motor operation for a highly dynamic load, as shown in Fig. 1, demands four-quadrant operation of SRM with appropriate turn-on and turn-off angles. To achieve fast motion reversals, maximum torque in the second quadrant is essential. This paper shows how to select firing angles for the motoring and regenerating modes of operation, which is required in many actuator type applications. In these applications, the sudden command changes translate to maximizing the motor torque either for accelerated forward or reverse motion, or for fast motion-direction reversal. Therefore, the firing angles are chosen to maximize the average torque. Depending on the control command, the motor is switched to operate between the motoring and braking regions by choosing the appropriate turn-on and turnoff angles.

INTRODUCTION

Switched reluctance motors (SRMs) have advantages in terms of large peak-torque capability on an intermittent basis and wide speed range, which are particularly suitable for electromechanical actuators and traction type loads. For actuators, transient response is highly critical, whereas for traction rapid emergency stops are essential. Energy regeneration helps in achieving quick response and smooth stops and the energy is efficiently utilized instead of being wasted. In the end, operation at full torque and speed levels is necessary in all four quadrants to get the fast response characteristics. Power Supply and Control

II. OPTIMAL TURN ON AND TURN OFF ANGLES The average torque, Tav produced by an SR motor is [4]

Position Position Command Controller

Tav =

Position Sensor

Gear

Hydraulic Load

Piston Returning Spring

Fig. 1 A hydraulic drive system.

0-7803-7404-5/02/$17.00 (c) 2002 IEEE

qN rW 2π

where q = number of phases, Nr = number of rotor poles and W = energy conversion per working stroke. Figs. 2(a) and 2(b) represent the energy conversion loops for an SRM for motoring and regenerating modes of operation, respectively (note the arrow directions). The figures show the magnetization curves for the aligned and unaligned positions along with the one for a given commutation angle θoff in the motoring mode. The primary objective is to maximize the

Piston

Ballscrew

Shelby Twp, MI 48315

USA

Abstract- A four-quadrant controller for Switched Reluctance Motor (SRM) drives with optimal turn-on and turn-off angles for each operating quadrant is presented in this paper. The firing angles are chosen to maximize the average torque produced by the motor, which requires the optimization of the area of the energy conversion loop. The four-quadrant SR controller developed with the chosen optimization criterion gives fast motoring response as well as fast braking response. The underlying contribution of this paper is the development and solution of an optimal control problem to provide the best control strategies. The paper also provides the specific guideline to switch the firing angle position when the motor operation changes from one quadrant to another. The developed fourquadrant controller has been applied to control the linear displacement of a highly dynamic actuator load.

SR Motor

Delphi Automotive Systems

Dayton, OH 45408

USA

Avoki Omekanda

Delphi Research Labs

Dayton Technical Center

Akron, OH 44325-3904

I.

Bruno Lequesne

41

θa

C

Flux linkage, ψ

Flux linkage, ψ

Aligned

θoff

R R W W

R R

θu

θu

Unaligned

O

max

Phase current, iph

O

Phase current iph

(b) Regenerating Operation (a) Motoring operation Fig. 2 Energy conversion loops in the current-controlled mode.

average torque, which requires maximization of the energy conversion loop per stroke W. The control parameters of the optimization problem are turn-on angle, turn-off angle and reference current (θon ,θoff, Iref). For the actuator system operated in the position-controlled mode where fast response time is critical, it is advantageous to maintain a reference current at the peak rated value to obtain a high average torque. The control parameters then reduce to θon and θoff, while the phase current is limited to the peak rated value of the motor. The problem can now be defined as θ on ,θ off

θoff

W

W

=

max

θ on ,θ off

optimal drive performance over a wide speed range. This section presents an outline of the method that optimizes the control parameters through off-line simulation. A geometry based nonlinear analytical model is used to simulate the physical motor [5-6] in order to incorporate the significant nonlinearities of the system. The optimized values of the control parameters are then expressed as a function of speed. Neglecting the core loss of the motor, the cost function (which is the work done per stroke) is calculated from the phase voltage and phase current as

{ ∫ i(ψ ,θ )dψ }.

J

=

ψ off

∫ +∫

0 0

ψ off

∑ (v

ph

− i ph r ph ) * i ph * ∆ t ,

per stroke

where

The cost function J of the optimization problem is the work done per stroke. The energy conversion loop can be divided into two parts at the transition point (θoff) corresponding to fluxing and defluxing the machine respectively. Therefore, the cost function can be expressed as

Cost function, J

=

∆ t = simulation time step .

A univariate search technique has been used to reduce the total amount of computation. The guiding logic behind univariate search is to change one variable at a time so that the function is maximized in each of the co-ordinate directions. The optimization algorithm is described in the following: 1. Start at some arbitrary starting point x0 (θon ,θoff) within the feasible space of solutions. 2. Find the next point x1 by performing the maximization with respect to the variable θoff, i.e.,

i(ψ fluxing , θ )dψ i (ψ defluxing , θ )dψ .

The optimization problem can be solved by defining the following necessary conditions dJ (θ on , θ off ) dJ (θ on , θ off ) =0 ; =0 . dθ on dθ off An analytical solution of the necessary conditions is only possible if the energy conversion loop can be defined by a smooth function of the control parameters. The challenge in the case of SR machines is its highly nonlinear electromagnetic characteristics. The approach presented in the paper is based on numerical optimization techniques that can accommodate models containing significant nonlinearities.

x1 = x0 + λ1e1

where e1=[0 1]. λ1 is scalar such that f(x0+λ1e1) is maximized. In other words, we solve the problem is the local min f ( x 0 + λ1e1 ). This θoff λ1

optimization of the cost function. 3. Find the point x2 by performing maximization with respect to the variable θon where the locally optimized θoff is used, i.e.,

x2 = x1 + λ2e2 ,

e2 = [1 0]

such that f(x1+λ2e2) is maximized. This θon is the locally optimized turn-on angle. 4. Continue successive iteration of steps 2 and 3 until the quantities |λk| are less than some tolerance value. These iterations give the globally optimized turn-on and turnoff angles.

A. Optimization Approach The control parameters are precalculated through the optimization algorithm to maximize the energy conversion per stroke at different operating points in order to achieve an

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B. Results of Optimization Problem The method outlined above was implemented in MATLAB/SIMULINK using a four-phase, 8/6 SRM as the reference motor. The motor parameters and ratings are given in the Appendix-I. The influence of the turn-on angle and turn-off angles on energy conversion per stroke in the motoring mode at different speeds are shown in Fig. 3(a) and 3(b). Turn-on angle is kept constant in Fig. 3(a), while turnoff angle is kept constant in Fig. 3(b). Observing the two figures, it is obvious that the energy conversion is increasing by advancing the turn-on and turn-off angles at high speed. Also, energy is optimized along two lines, one for turn-on angle and one for turn-off angle, denoted by “optimal line” in the figures. The optimal lines for the motoring mode can be represented by the following general equations:

III. FOUR-QUADRANT CONTROL STRATEGIES In many applications, the SRM operates in the torquecontrolled mode with the command torque set by an outerloop position controller. The SR drive controller in the innerloop functions to maintain the desired command torque. Fast response is critical for highly dynamic loads, where the command torque and the motor speed may reverse quickly from positive to negative and vice versa. The motor operates in all four quadrants of its torque-speed characteristic. The flow diagram of the controller is shown in Fig. 4. The controller switches the motor between the motoring and braking regions according to the control command. The appropriate and different turn-on and turn-off angles are used depending on the operating quadrant.

Optimal turn − on angle = a11 + a12 * ω + a13 *ω 2

Commutation Scheduler

θ

Optimal turn − off angle = a 21 + a 22 * ω + a 23 *ω 2 .

Turn-on and Turnoff angle Calculator

ω

Similar analysis for the regenerating mode shows that

θon θoff

Operating Quadrant Determination

Optimal turn − on angle = a31 + a32 * ω + a33 *ω 2 Optimal turn − off angle = a 41 + a 42 * ω + a 43 *ω 2 .

Tref or ωref

The coefficient a’s are determined through the optimization program. The results found for a 4-phase, 8/6 SRM are given in Appendix-II.

(if speed loop)

Iref

Torque (or Speed Error) to Current Transformation

ω

iph

Hysteresis Controller for Main Phase

On –off Signals

Fig. 4 Schematic representation of the controller.

The turn-on and turn-off angles determined in section II are used for first and fourth quadrant operation. These angles are scheduled as a smooth function of speed, which maximizes the average torque at all operating points. Operation in second and third quadrants is the mirror symmetry of that of the fourth and first quadrants, as shown in Fig. 5. The relationships between the firing angles are

1.2

Energy (Joules)

1.0

Optimal line

0.8 0.6 0.4

θ on , III = −

0.2

0 -2

-4

-6

-8

-10

1000

Turn off angle

800

600

400

2π − θ on , I Nr

θ on , II = − θ on , IV

Speed (rad/sec)

; θ off , III = −

2π − θ off , I Nr

; θ off , II = − θ off , IV

(a) 1.2

Energy (Joules)

ωref Tref

Optimal line

ib

1.0

0.8

θon

0.6

θoff, II

0.4

-ω

0 400

800 1000

-39

-38

-37

-36

-35

Turn on angle

-34

-33

-32

II

I

III

IV

θoff, III θon

Fig 3 Energy vs. speed with (a) turn-off angle and (b) turn-on angle variations.

III

-ωref -Tref

-π/Nr

θoff, I θon, I

ω ib

θon, IV

0

-2π/Nr

(b )

00

Lph

im

600

Speed (rad/sec)

II

π/Nr

-π/Nr

0.2

im -2π/Nr

θoff, IV π/Nr

Fig. 5 Operating points in different quadrants.

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The four-quadrant drive may be required to operate either in the position-controlled loop or in the speed-controlled loop. In the controller, the operating quadrant is determined from the sign of reference torque (or reference speed for speed-controlled drive) and motor speed as shown in Fig. 4.

software package. A cascade control structure is used for the drive system. The controller has two parts: outer loop controller and inner loop controller. The outer loop generates the reference torque or reference speed from the position or speed error. The SRM drive system is in the inner loop. The geometry based analytical model having the significant nonlinear saturation characteristics of an SRM has been used to represent the physical motor in the simulation set-up. The simulation set-up incorporated the practical nonidealities, such as controller delay time, position sensor resolution and errors in current measurements. These measures in simulation provided results that closely matched the experimental results. The simulation results of the closed-loop positioncontrolled system using the proposed controller is shown in Fig. 6. In this simulation, the inner torque loop is followed by the outer position loop. The reference torque generated from the outer position loop is the command input for the SR drive. Figs. 6(b) and 6(c) show the speed response and reference torque respectively under load torque disturbances. The torque-speed profile of Fig. 6(d) shows that operation at full torque and speed levels is necessary in all four quadrants. Extensive simulation studies did prove that the optimal turnon and turn-off angles do indeed provide the fastest response of the actuator system.

IV. SIMULATION RESULTS The simulation of the mechanical subsystem as well as of the optimization method for four-quadrant controller of the SR drive was implemented in the MATLAB/SIMULINK (a) Position vs. Time

0.5

Position (mm)

0.4

Position Command Position Output

0.3

0.2

0.1

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec) (b) Speed vs. Time

6000

5000 4000

Speed (rpm)

3000

V. VALIDATION AND TEST

2000

1000

The prototype set-up to verify the four-quadrant control strategy with optimal control parameters experimentally is shown in Fig. 7. The test set-up was designed for motion control applications that can be operated in variable speedcontrolled or position-controlled loop. The controller algorithm has been implemented within a dSPACE rapid prototyping platform. The classic bridge power converter was used to meet the commutation requirements. Four A/D channels having sampling time of 4 µsec are used to digitize the phase current information for the dSPACE controller. The current control is implemented inside the dSPACE system to avoid the necessity of any additional hardware. The position information is obtained from an incremental encoder.

0

-1000

-2000

-3000

0

2

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time (sec) (c) Torque Command vs. Time

Torque Command (N.m)

1.5

1

0.5 0

-0.5 -1

-1.5

-2

0

2

Rotor Position

Time (sec) (d) Profile of Motor Torque vs. Speed

dSPACE with Interface Hardware

Motor Torque (N.m)

1.5

1

0.5

iph

Converter SRM

Hydraulic Load

Gate Signals

0

-0.5

Vdc

-1

Fig 7 Hardware set-up for SRM drive system.

-1.5

-2 -6000

-4000

-2000

0

2000

4000

A. Speed-Controlled Loop The SRM was first connected to an inertia load to verify the variable speed operation. A rectangular bi-directional

6000

Speed (rpm)

Fig. 6 Simulation results in the position-controlled mode.

44

speed command is used for four-quadrant speed control. A fixed set of turn-on and turn-off angles are used in the motoring region. To achieve quick reversal, the optimized operating angles are used in the regenerating mode. A PIregulator is used to generate the current command from the speed error. The closed-loop response for a toggling speed command between 1000 and –1000 rpm is shown in Fig. 8. The motor was thus required to switch operation from

In the speed-controlled mode, the current is regulated in the active phase by chopping at the reference level commanded by the speed regulator. Phase conduction angle control becomes necessary for speed commands above the base speed, since the motor enters the single pulse mode. Fig. 9(a) shows the measured phase currents during motor acceleration. Fig. 9(b) shows the measured phase-A current along with the wrapped rotor position during the deceleration mode. The figure also shows the correct positioning of the phase current with respect to the rotor position in the regeneration mode. The optimized turn-on and turn-off angles produce a small amount of opposite polarity torque. The opposite polarity torque produced by the rising or decaying phase current may appear to be undesirable, since it will adversely affect the machine efficiency and torque ripple. However, the small opposite polarity torque helps to maximize the average torque in each phase cycle of operation, which is required for fast actuation. During regeneration, the kinetic energy of the motor and load is converted into electrical energy. Therefore, the capacitor voltage rise due to regeneration must be considered during the converter design. Fig. 10 shows the significant increase in the dc link bus voltage from the nominal value during regeneration.

Speed command

Speed measured Speed (rpm)

1000

0

-1000

0

5

10

15

20

25

35

30

45

40

50

Time (sec)

Fig. 8 Measured four quadrant closed loop speed response.

forward motoring (first quadrant operation) to reverse motoring (third quadrant operation) through regenerating mode (fourth quadrant operation), and also from reverse motoring to forward motoring through regenerating mode (second quadrant operation).

30

20

Ph -B

Phase Currents (Amps)

P h-A

P h -D

Ph -C

Phase -A voltage

20

Ph -A

16

12

0

- 10

- 20

8

- 30

4

0.8

1

1.2

1.4

1.6

1.8

Time (sec)

Fig. 10 Measured DC bus voltage for four-quadrant operation.

0

2

4

Time (sec)

6

8

B. Position-Controlled Loop In this test, the SRM was driving a linear load via a ballscrew arrangement as shown in Fig. 1. Any position change of the motor is converted into force on the piston that has a restoring spring. The experimental results of the closedloop position-controlled system are shown in Fig. 11. The rotor position information is used to measure the translational displacement of the piston. Figs. 11(a) and 11(b) show the linear displacement and speed responses, respectively, under load torque disturbances. These figures represent fourquadrant response of the SRM for the linear displacement control of the actuator system. Smooth transition is observed when the motor switches from one operating quadrant to another. Table 1 shows the effect of the variations of turn-on and turn-off angles on response time. The response time (rise time) is considered as the time required for translational

25

150

20

120

P h ase-A C u rrent

15

90

10

60

5

30

W r ap p ed ro tor p o sitio n 0

-5

0

A ligned position

A lig ned position

0 .0 3 3

0 .0 34

0 .0 3 5

0 .0 36

Time (sec)

U n a lig ned p o sitio n

0 .0 37

0 .0 3 8

-3 0

0 .0 3 9

Wrapped Rotor Position (Mech. Degree)

(a)

Phase-A Current (Amps)

10

(b) Fig 9 Measured phase currents during (a) acceleration and (b) deceleration.

45

(a): Linear Displacement vs. Time 0 .7

20

Position Command

Ph-B

0 .6

Phase Currents (Amps)

Linear Displacement (mm)

0 .8

Position Output 0 .5

0 .4

0 .3

0 .2

0 .1

Ph -B

Ph -C

Ph -C

16

12

8

4

0

2

2 .5

3

3 .5

4

4 .5

Time (sec)

250

258

254

260

Time (sec)

(b) Motor Speed vs. Time 6000

Fig 12 Measured phase currents during position hold.

Motor Speed (rpm)

4000

VI. CONCLUSIONS 2000

This paper demonstrated the necessity of a four-quadrant controller and appropriate turn-on and turn-off angles for any highly dynamic actuator load. The response time for the actuator system has been minimized by choosing optimal firing angles based on average torque maximization for the motoring and braking or regeneration modes. Any small variation of the turn-on and turn-off from the optimal values has an adverse effect of the response time. A guideline is presented to switch the firing angle position when the motor operation changes from one quadrant to another. The experimental results show very fast dynamic responses for linear displacement control using the proposed controller with smooth transition from one quadrant to another.

0

-2 0 0 0

-4 0 0 0

-6 0 0 0

2 .0

2 .5

3 .0

3 .5

4 .0

4 .5

Time (sec)

Fig 11 Measured position and speed response in the position controlled mode of operation.

movement of 20% (0.14mm) to 80% (0.55mm) of the position command. The hydraulic drive system was operated in the test bench by varying the turn-on and turn-off angle one-mechanical degree around the optimal angles. The test results verify that the optimal turn-on and turn-off angles give the fastest response.

ACKNOWLEDGMENT This work was performed under a contract between Delphi Automotive Systems and the University of Akron. The authors are grateful for Delphi's support in this regard.

Table 1 Response time comparison for deviation from optimal angles Operating parameters

Rise time (msec)

Optimal turn on and turn off angles

48

Optimal turn on plus 10 and optimal turn off

58

Optimal turn on minus 10 and optimal turn off

51

Optimal turn on and optimal turn off plus 10

55

Optimal turn on and optimal turn off minus 10

52

REFERENCES [1]

[2]

When the position is held at a constant level as seen in Fig. 11, the rotor dithers around zero speed. The effect of this dithering is a small rotor movement of the order of one step angle around the position hold. Therefore, the motor goes through more than one phase during position hold. The dithering around zero speed has been incorporated in the controller to minimize the local overheating of any single phase. Fig. 12 shows the phase currents during position hold. Here the rotor oscillates between phase B and phase C to maintain the constant position. Energizing these two adjacent phases would thus distribute the power loss.

[3]

[4]

[5]

[6]

46

Philip C. Kjaer, Jeremy J. Gribble, and T.J.E. Miller, “Dynamic testing of switched reluctance motors for high bandwidth actuators applications”, IEEE/ASME Trans. on Mechatronics, Vol. 2, No. 2, June 1997. K. M. Rahman, Babak Fahimi, G. Suresh, A.V.Rajarathnam, and M. Ehsani, “Advantages of switched reluctance motor applications to EV and HEV: Design and control issues”, IEEE Trans. Industry Application, vol. 36, No. 1, pp. 111-121, Jan./Feb. 2000. Yang Haiqing, Sanjib K. Panda, Liang Yung Chii, “Performance comparison of feedback linearization control with PI control for fourquadrant operation of switched reluctance”, IEEE-APEC Conference Procedings 1996. pp. 956-962. D. A. Staton, WL Soong, TJE Miller, “Unified theory of torque production in switched reluctance and synchronous reluctance motor”, IEEE IAS Annual Meeting, 1993, Pg. 185-193. A.V. Radun, "Design Considerations for the Switched Reluctance Motor," IEEE Trans. on Industry Applications, Vol. 31, No. 5, pp. 1079-1087, Sept./Oct. 1995. I. Husain, A. Radun and J. Nairus, “Unbalanced Force Calculation in Switched Relcutance Machines,” IEEE Trans. on Magnetics, Vol. 36, No.1, pp.330-338, Jan. 2000.

Appendix-I SRM Ratings and Parameters: Inverter DC bus voltage Power Peak current No. of stator poles No. of rotor poles Stator winding resistance

20V 300W 15A 8; 220 pole arc 6; 220 pole arc 0.179Ω

Appendix-II Optimal Controller Parameters: The coefficients a’s were determined for a 4-phase, 8/6 SR motor. The results were: a11= -34.32; a12= 0.0018; a13= 6.05e-6; a21= -4.29; a22= -0.0021; a23= -2.253e-6; a31= -7.07; a32= 0.0014; a33= -6.1e-6; a41= 26.16; a42= -0.0005; a43= 4.06e-6; where, 0 and –600 are aligned positions, and ±300 are unaligned positions.

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