Silicon Steel Department,Baoshan Iron & Steel Co.,Ltd.,Shanghai 200941,China

Abstract: Due to the special secondary rolling and annealing processes,the physical properties of strip ductility and stress relief differ from those of the general products.The proportional integral differential (PID) algorithm for the annealing furnace tension control and the conventional control method that only controls the total tension of the furnace entrance and exit cannot meet production continuity.Aiming at the tension in the annealing furnace,the interroll tension model of the tension of the steel strip between the adjacent furnace rolls is established.By combining the control principle of the annealing furnace roll transmission mechanism,the tension control involves the motor and frequency converter equipment.The coupling relationship between the stator current and the motor’s output torque was determined through the asynchronous equivalent motor circuit.Also,the direct influence of the load torque on the calculated value of the motor’s output tension was obtained through the motor vector control equation.Furthermore,the frequency converter’s voltage control model was analyzed to control the motor’s output tension.Finally,the adjacent furnace roll tension and the horizontal annealing furnace tension were calculated through the interroll tension model.

Key words: asynchronous motor; vector control; tension model

1 Introduction

During the production of non-oriented silicon steel products of Toyota Motor Corporation’s pure elec-tric cars,due to the special secondary rolling and annealing processes,the physical properties of strip ductility and stress relief are different from those of the general products.The proportional integral dif-ferential (PID) algorithm for annealing furnace tension control and the conventional control method that only controls the total tension of the furnace entrance and exit cannot meet production continuity.

Therefore,herein,a tension control scheme for a horizontal annealing furnace that adapts to the secon-dary annealing process was novelly designed.The tension distribution model in the annealing furnace and the interroll tension model between the adjacent furnace rolls were designed as the control basis.Fur-thermore,an artificial intelligence controller based on genetic algorithms was novelly developed to replace the traditional proportional and integral algorithms to control the annealing furnace tension.This paper only introduces the interroll tension model.

The furnace rolls of the horizontal furnace are in the form of idlers to establish the tension of the steel strip between the adjacent furnace rolls.By com-bining the control principle of the annealing furnace and the furnace roll’s transmission mechanism,the tension control involves the motor and frequency con-verter equipment.The coupling relationship between the stator current and output torque of the motor is determined through the asynchronous equivalent motor circuit.Also,the direct influence of the load torque on the calculated value of the motor’s output tension is obtained through the motor’s vector con-trol equation.Furthermore,the voltage control model of the frequency converter is analyzed to achieve the control of the motor’s output tension.Finally,the tension in the horizontal annealing furnace and the adjacent furnace roll is calculated through the interroll tension model.

2 Vector control model of asynchronous motor

2.1 Analysis of motor equivalent circuit

In the study of asynchronous motors,the equi-valent circuit is usually used to identify the par-ameters of the motor.Through the vector control principle,various rotational parameters of the motor in the working state are obtained.In industrial auto-mation control,the ultimate purpose of any tension adjustment system is to convert the motor’s speed control into the torque change and adjust the amount of tension change in this form.The rotational speed is regulated by controlling the motor’s current,so it is necessary to analyze the correlation between the motor’s speed and torque and study the control model of the interaction between the two and the current[1].The rotational inertia equation of the motor is given as follows:

(1)

where,Telecrepresents the motor’s electromagnet torque;Tloadrepresents the motor’s load torque;Jrepresents the motor’s rotational inertia;andωis the motor’s rotational angular velocity.

From Equation (1),the alternating current (AC) motor generates electromagnetic torque from the change in the motor’s rotational speed[2].The principal basis for establishing the vector control theory is to transform the rotational state of an asynchronous AC motor into a motion coordinate so that the asynchronous AC motor is equivalent to a direct current (DC) motor.After excitation,the size of the output current is adjusted to change the motor’s speed,and then the electromagnetic torque output is controlled[3].

By ignoring the effects of the radius of curvature,the magnetic saturation of the stator and rotor core,and the influence of slip on the rotor impedance[4],the mathematical model of the motor in a static state is transformed into a physical modeling form,and Fig.1 can be obtained as the steady-state equivalent circuit of an asynchronous motor.

Fig.1 Equivalent circuit diagram of the asynchronous motor

The steady-state torque,Telec,is given as:

(2)

where,prepresents the number of pole pairs of the motor;andωSrepresents the stator’s rotational angular velocity.SinceEScan be expressed asES=ωS·ψS,Teleccan be written as:

Telec=pψSIS

(3)

where,ψSrepresents the stator flux linkage.The torque size is only related to the stator currentISifψSis constant in the steady state,which can be expressed as:

(4)

where,srepresents the motor slip.On substituting Equation (4) into (3),Telecis given as:

(5)

Equation (5) shows that whenψSof the three-phase asynchronous motor is constant,Telecis equivalent to the DC motor’s output torque,so to keepψSconstant,the flux linkage of the motor is calculated.

The rotor and stator flux linkage equations are respectively given as:

(6)

and

(7)

TheψSequation of the asynchronous motor can be derived from the equivalent circuit as:

ψS=LmISM

(8)

If the phase component of the stator currentISMis constant,the flux linkage and the motor’s torque are controllable[6].Due to the coupling relationship between theψSandψrequations,controlling the output torque of the motor of the furnace rotor requires continuously studying the interaction relation-ship between the stator and rotor current values through the vector space equation.

2.2 Vector equations of motor stator and rotor

Fig.2 shows the resistance model of a three-phase asynchronous motor in a static state.A,B,andCare the stator resistance coordinate axes,anda,b,andcare the rotor resistance coordinate axes.ωrepresents the rotational angular velocity of the motor’s rotor.The equation of the state of the motor under the stationary frame system is described in a matrix form.

Fig.2 Asynchronous motor resistance model

The voltage equation is given as:

u=R·i+ψ

(9)

the voltage vector is given as:

(10)

the current vector is given as:

(11)

the flux linkage vector is given as:

(12)

and the resistance matrix is given as:

(13)

where,urepresents the instantaneous voltage values of the stator and the rotor;irepresents the instan-taneous current values of the stator and the rotor;ψrepresents the stator and rotor flux linkages;andRrepresents the resistance.

Theψequation is given as:

ψ=L·i

(14)

Letψ=[ψS,ψr]T,i=[iS,ir]T,theψequation is converted to a matrix form:

(15)

(16)

where,LiSandLirrepresent the stator and rotor leakage inductance values,respectively;LmSandLmrrepresent the stator and rotor mutual inductance values,respectively,if the mutual inductance and the magnetic resistance of the stator and the rotor are equal,thenLmr=LmS;andθrepresents the space angular displacement variables of the stator and rotor resistance values,respectively,matricesLSrandLrSare transposed to each other.

According to the principles of energy conser-vation and electric energy conversion,Teleccan be expressed as an electromagnetic torque equation:

(17)

We bring Equation (17) into Equation (1),after spatial coordinate conversion,the torque equations in different space coordinates are obtained as follows.

αβspace:

(18)

dqspace:

(19)

andmtspace:

(20)

where,npis the pole number of the motor.

Inmtspace,the stator current components areiSmandiSt.A first-order inertia link transfer function exists betweeniSmandψr,which is similar to the excitation inertia link of the DC motor excitation resistance.The change inψrdepends oniSm,and the simultaneous change in the amount ofiSmandTelecis proportional to the decoupling.TheiSmandiStvalues can calculate the value of the function between the current and the motor’s output torque to control the motor’s output torque[7].Combined with the con-trol algorithm of the torque vector[8],the output torque of the furnace roll motor is adjusted,and the furnace roll transmission is controlled for the annealing furnace tension.

The motor’s torque system equation in theαβanddqspace coordinates has a coupling relationship,and it is difficult to obtain the relationship between the torque and the current through conventional calculations.However,the two spatial coordinate systems both show the influence ofTloadon the motor’s output torque,and the magnitude of the change is directly related to the control perfor-mance decrement[9].Optimizing the accuracy of the calculatedTloadvalue can effectively reduce the calculation error of the annealing furnace tension.

3 Voltage control model of the inverter unit

The inverter unit of the hearth adopts a vector control method to convert the DC bus voltage output of the rectifier into a variable frequency AC voltage.Based on the pulse width modulation (PWM) technology,the power element insulated gate bipolar transistor (IGBT) in the inverter circuit is controlled,and the voltage pulse of equal amplitude is output,which controls the speed and torque of the motor’s output[10].Its form includes multiple voltage pulses in the output cycle that form a sine wave target waveform.The equivalent voltage of the pulse in the sine wave is stable and does not contain low-order harmonics,and the width and frequency of the pulse are controlled to adjust the output voltage and frequency values of the inverter circuit[11].However,the AC motor speed regulation aims to obtain the corresponding electromagnetic torque,which can only be obtained by relying on the rotating magnetic field.Therefore,the motor and the inverter are combined into the same system,and the state of the system dynamically adjusts the operating parameters of the inverter in the rotating magnetic field.Besides,the equivalent voltage that controls the motor’s operation is the output.Finally,by calculating the inductance,impedance,and other data in the system loop,the working current and excitation flux linkage values of the motor are adjusted[12],and then the motor’s speed and output electromagnetic torque are controlled.

Fig.3 shows that a coupling relationship exists betweenψand the current componentiin theαβspace coordinate system of the motor.However,the current component can be eliminated by theψequa-tion by integrating the electromotive force,and finally the inverter voltage and flux linkage associate,as a controllable model of the inverter unit[13].

Fig.3 Voltage model of rotor flux linkage

The model can be transformed into the following equations:

(21)

The stator’s resistance value in Equation (21) is easily measured,and it can be completed in the static identification process of the inverter unit.The stator and rotor flux linkages can be calculated after the voltage and current values are introduced.According to the derivation in the previous section,the flux linkage is introduced into the equation of motion to obtain the motor’s output torque.The integral part of the model calculation equation is also more suitable for the speed control of the medium and high-speed motors,covering the working parameters of the furnace roll motor.By analyzing the voltage model of the rotor flux linkage,it is possible to understand the principle of the inverter unit in controlling the motor’s output torque,which provides a theoretical basis for implementing the control function.

4 Interroll tension model

The strip tension between the furnace rolls was analyzed according to the laws of theoretical mechanics.Hooke’s law is an important theory of modern physics in the basic laws of mechanics,covering many disciplines such as material and elastic mechanics.Definition of Hooke’s law of elasticity[14]:The elasticity value is equal to the product of the stiffness coefficient and the defor-mation variable.

ΔF=kΔx

(22)

As the generalized Hooke’s law of material mechanics,the solid stressσis proportional to the strainε,and the stress-strain ratio coefficient is the Young’s modulus,E.The equation is given as follows:

σ=Eε

(23)

The strainεis equal to the ratio of the change in the strip’s length in timetto the strip’s lengthL0at the initial time:

ε=ΔL/L0

(24)

Ifσ0between the two rolls is zero,the strip’s initial length between the rolls isL0.Then,whenσchanges,the strip length change is ΔL,and Equations (23) and (24) can be combined to deduce:

σ=EΔL/L0

(25)

Then,the length change is given as:

ΔL=σL0/E

(26)

Thus,the length change of the steel strip after being stressed and the magnitude of the force are related to the strip’s initial length and Young’s modulus.Hence,the strip tension between the hearth rolls should be related to the strip’s initial length between the hearth rolls and the strip’s length sent in and out of the adjacent hearth rolls along the strip’s running direction;that is,the speed difference.The strip length between the rolls is given as follows:

(27)

Thus,between two adjacent rolls,L0is given as:

(28)

The initial distance and stress between the furnace rollsnandn+1 along the strip running direction are given as follows:

(29)

where,Ln0represents the strip’s length of the furnace rollnand the furnace rolln+1 without external force;andLnrepresents the strip’s length when the stress between the two rolls isσn.Then,the strip stressσnbetween the rolls should be:

(30)

(31)

Equation (31) reflects the correlation between the change in stress and the change in the length of the steel strip at the entrance and exit of the roll.dLn=vn+1dtrepresents the change in the exit length ofnroll,and dLn-1=vndtrepresents the change in the entrance length.Equation (31) can be rewritten as:

(32)

On simplification,

(33)

vn-1(t)σn(t)

(34)

Taking the Laplace transform of both sides of Equation (34) gives:

(35)

where,Sis the complex function of Laplace transform,S=d/dt;tnis the inertia time constant,which is equal to the ratio of the length of the steel strip at the exit of the furnace rollnto the exit linear speedvn-1,i.e.,tn=Ln/vn-1.Thus,the greater the speed of the furnace roll,the smaller the time constant value consumed and the stress on the strip at the exit of the furnace roll.

Equation (35) shows that the motor’s output torque of the furnace rolln+1 can control the stress of the steel strip between the furnace rolln.Thus,the exit strip tension of the furnace rollnis the output,which is a first-order inertia function when the speed and entrance tension are known condi-tions,and the system has a steady-state response.

From Equation (35),the unit tension of this section of the steel strip is given as:

(36)

The general equation of the strip tension model between any two rolls is given as:

(37)

where,kS=AE/vn.

Equation (37) expresses the interroll tension model,and the strip tensionTnbetween the adjacent furnace rollsnandn+1 is the input of the first-order inertia between the linear velocity difference of the two rolls (vn+1-vn) and the entrance tensionTn-1of the furnace rolln.

The gain coefficientkSis the ratio of the product of the Young’s modulus multiplied by the unit cross-sectional area of the strip and the linear velocity of the furnace rolln.The linear velocity reflects the speed of the furnace roll.The greater the speed of the furnace rolln,the longer the amount of strip conveyed forward,and the smaller the tension toward the exit of the furnace roll after the strip is relaxed.This model changes the scheme of the individual output control of the rolling equipment.The tension between the rolls is used as the control quantity so that the adjacent furnace rolls are linked to improving the performance of tension adjust-ment.The interroll tension model is shown as Fig.4.

Fig.4 Interroll tension model

5 Conclusions

Aiming at the control and execution mechanism of the electric drive,combined with the tension generation principle,with the help of the equivalent circuit of the AC motor,the current effect on the output electromagnetic torque of the motor was obtained.By studying the stator and rotor vector control models,the coupling relationship between the stator and rotor current values and the load torque influence on the final output torque of the motor were confirmed from the torque equation in the space coordinates.In the interroll tension model,the relationship between the speed difference of the adjacent furnace rolls and the change in tension,the stress-strain law,and the temperature change in the annealing furnace were studied.Hence,this study provides theoretical support for furnace tension control adapted to a secondary annealing process.