1. Introduction
Special steel is the key material of major equipment manufacturers, construction projects and defense advanced weapons, and the quality of special steel has a very important influence on the manufacturing level. Currently, the electromagnetic (EM) stirring technology is a key technical way to improve the quality of special billet [1, 2]. By using different forms of magnetic generators to produce a moved magnetic field, a powerful EM stirring force can be induced in the molten steel. The EM stirring force can be used to control the flow and solidification of the molten steel in the continuous casting process, and then it can expand the equiaxed crystal zone and reduce the center porosity to improve the quality of steel products [3, 4].
Nowadays, the structure of two-phase orthogonal power supply (TPOPS) is widely used in the continuous casting process. TPOPS can output two-phase orthogonal currents to power two coils, and then form a rotating EM stirring force to excite a rotary motion of molten steel [5, 6]. In terms of the topology of TPOPS, it is always composed of three-phase rectifier and two-phase inverter. The former rectifier of TPOPS generally adopts three-phase uncontrolled rectifier [7, 8], thus the content of harmonic currents is high. According to the requirements of continuous casting, alternate rotation is needed to make the EM stirring of molten steel more uniformly. As the output currents of TPOPS will change rapidly in the alternate process, the energy of induction coils will vary dramatically. The energy of dc-link capacitor cannot be quickly released and absorbed by the uncontrolled rectifier, which can lead to significant fluctuations of dc-link voltage and affect the normal control of back inverter. As for the topology of two-phase inverter, four legs were adopted to form two single-phase full-bridge inverters [9], so they can be controlled separately. This structure has high utilization rate of dc-link voltage, but the number of power switches is 8. Therefore, two legs were proposed to form a two-phase inverter in [10], which can be considered as two single-phase half-bridge inverters and they were shared with dc-link capacitors. This structure only needs 4 power switches, however, the utilization rate of dc-link voltage is low and a voltage-balanced control is needed for two dc-link capacitors. Therefore, in order to make up for their defects, a two-phase three-leg inverter (TPTLI) was put forward in [11, 12], in which two legs were independent and the third leg was for public use. According to the analysis above, a TPTLI is adopted for TPOPS in the paper. Meanwhile, in order to improve power quality of system and maintain a stable dc voltage for inverter, a three-phase pulse width modulation (PWM) rectifier is adopted.
As for the control of TPOPS, the references were mainly discussed about the control methods of back inverter. In [11], a set of output filters was adopted by TPTLI for EM stirrer, and a deadbeat control method of TPTLI based on the switching model was derived. In [13-15], a proportional-integral (PI) control or hysteresis control was used by two-phase inverter to achieve current closed-loop control, and then the control system adopted sinusoidal PWM method to generate switching signals to drive power switches. In [16], a synchronous frame current control scheme was proposed for voltage-source inverter to drive a two-phase linear motor. Later, space-vector modulation (SVM) methods were proposed for TPTLI to improve the utilization rate of dc-link voltage [12, 17, 18], which can enhance the output capacity of TPTLI for two-phase motors. However, fast positive and negative alternate rotations are needed to make the EM stirring of molten steel more symmetrically and uniformly, so the output currents of TPOPS have to be changed rapidly in the casting process. According to this requirement, TPOPS is expected to have a fast dynamic tracking performance, while the control method of TPOPS is the key section to meet the satisfactory stirring performance.
In order to improve the performance of power supply, the control method of TPOPS is researched in the paper. Firstly the power model of system is analyzed, and the ripple of dc-link voltage is derived mathematically according to the power analysis in Scetion.II. Then an integrated control method with feedforward control for PWM rectifier is proposed in Scetion.III, and a ripple elimination method for voltage control is presented. According to the simplified equivalent circuit of EM stirrer, a composite current control method is proposed for TPTLI in Scetion.VI, and its closed-loop tracking performance is analyzed. Finally, the prototype of TPOPS is developed, meanwhile, the simulation and experimental results have been carried out to verify the analysis and control method.
2. The Topology of TPOPS and its Power Analysis
The topology of TPOPS for EM stirrer in the billet continuous casting is shown as Fig. 1. It is composed of a PWM rectifier and a TPTLI. The PWM rectifier is adopted to eliminate the harmonic currents, improve the power factor (PF) of TPOPS, and maintain dc-link voltage stably. The PWM rectifier is connected with the three-phase grid by filter inductor Lo. The load EM stirrer is two-phase induction coils installed in the cooling water. In order to filter higher harmonics and prevent over-current when short-circuit fault happens, a set of output filter L is adopted. The operation principle of TPOPS is shown as follows: firstly, the three-phase ac voltage sources are converted into a dc voltage source by PWM rectifier, and then it can be converted into two-phase ac sources to supply EM stirrer by TPTLI. The operation principle of EM stirrer is similar to the two-phase asynchronous induction motor [19, 20]. Shown as Fig. 1, the load two-phase induction coils, which can be considered as the stator, are installed around the crystallizer of molten steel. And the induction coils can generate a rotating magnetic field and motion in the molten steel which is considered as the rotor. The features of rotary motion are directly related with the amplitude and frequency of output currents of TPTLI. According to the attribute requirements of different kinds of steel, the expected output currents of TPTLI are different.
Fig. 1.The topology of TPOPS for EM stirring
2.1 The input power calculation of TPOPS
In order to study the operation of TPOPS, the output and input power of system is analyzed. Assuming the three-phase voltages of power grid are:
Where, U is the amplitude of phase voltage, ωo is the angular frequency of power grid. PWM rectifier can achieve a unit PF rectification by the closed-loop control. Assuming three-phase input currents of PWM rectifier are:
Where, Ip is the amplitude of active current. Assuming the active power of load EM stirrer is Pd. If ignore the power loss of TPOPS, then there is:
As the inductance of input filter inductor is Lo, and its resistance is small and ignored here. The three-phase voltages of filter inductors can be obtained as:
The total input power of PWM rectifier is:
Where, PS is the power generated by three-phase gird, and it is 3UIp/2; PL is the power generated by three-phase filter inductors, and it is 0.
2.2 The output power calculation of TPOPS
The load of TPOPS is the EM stirrer shown as Fig. 1. The operation principle of EM stirrer is similar to asynchronous induction motor. Two-phase induction coils are considered as the stator of motor, while the rotating molten steel can be considered as the rotor. According to the equivalent model of asynchronous induction motor in [19, 20], the equivalent circuit of EM stirrer can be established shown as Fig. 2. Assuming Lδ1 , rδ1 are the equivalent leakage inductance and resistance of induction coils respectively; Lδ2 , rδ2 are the equivalent inductance and resistance of EM stirrer’s rotor respectively.
Fig. 2.The equivalent circuit of EM stirrer
At the rated operation status, the excitation reactance XN meets: ｜XN｜>10*｜ rδ2 / Sn + jωLδ2｜ (ω is output angular frequency; Sn is the slip ratio). Therefore, the excitation reactance can be neglected. Meanwhile, the flow speed of molten steel is slow and the slip ratio Sn is always high according to the engineering practice. In order to facilitate analysis, assuming the slip ratio is equal to 1, so the equivalent resistance rδ2 / Sn can be simplified as rδ2 . The simplified equivalent circuit of EM stirrer can be shown as Fig. 2(b). According to the rated parameters, the equivalent parameters of EM stirrer can be obtained as follows:
Where, PM and QM are the rated output active and reactive power of EM stirrer respectively; IN is the rated output current. The α - and β -phase equivalent impedance for three-leg inverter can be obtained as:
where, there are r1=r2= rδ1 + rδ2 and L1=L2= Lδ1 + Lδ2 . According to Fig. 1 and (7), the output voltages of three-leg inverter can be equivalent as:
where, L, r are the inductance and resistance of output filters respectively shown as Fig. 1. Assuming iα = Im sin ωt , iβ = Im cos ωt , and then there is:
where, Im is the amplitude of output currents, ω is output angular frequency of inverter. Then the output voltages can be expressed as:
where, Lm = L + L1 = L + L2 and rm = r + r1 = r + r2 . So the total output power of TPOPS is:
According to the principle of power balance, ignoring the power loss of TPOPS, the dc active power component of Po should meet the following equation:
Therefore, the dc active power absorbed by PWM rectifier is mainly consumed as the work of the electromagnetic force.
2.3 The calculation of 2nd ripple in dc-link voltage
As there is a secondary power P2 in output power of TPOPS shown as (11), and there is：
According to the principle of power balance, there would be a secondary ripple in dc-link voltage. Assuming the dc-link voltage is udc = Ud + δ sin(2ωt + ϕ) , Ud is the dc component, and δ sin(2ωt + ϕ) is the voltage ripple. As the instantaneous power of dc-link capacitors is:
where, C is the total capacitance of dc-link capacitors. If neglect the 4th ripple, according to the power balance between dc- and ac-link of TPOPS, it can be obtained as:
So the 2nd ripple in dc-link voltage can be derived as:
According to power analysis above, a 2nd ripple would exist in dc-link voltage, and its frequency is 2-times of output frequency. The amplitude of ripple is proportional to the square of output current, and it is inversely proportional to the capacitance and dc-link voltage.
3. The Integrated Control Method of PWM Rectifier
According to the power analysis above, an integrated control method is proposed for PWM rectifier shown as Fig. 3. It is mainly composed of voltage outer-loop and current inner-loop. According to the analysis in Section.II, there is a 2nd ripple in dc-link voltage. If it is not eliminated in voltage outer-loop, it will generate some additional harmonic instructions for current inner-loop. And then it would produce some harmonic currents in the input of PWM rectifier. The calculation of additional harmonic instruction signals is as follows:
Fig. 3.The integrated control diagram of PWM rectifier
The voltage error Δu is processed by a PI controller, and then its output will be multiplied with the synchronous signal sinωot shown as Fig. 3. Assuming the expression of voltage PI controller is k+ki/s, the additional harmonic instruction signal generated by the 2nd voltage ripple is:
Seen from (17), it would generate two kinds of harmonic instruction signals. The frequencies of harmonic signals are 2ω±ωo , and its amplitude is kδ / 2 . Therefore, a ripple filter method is proposed for voltage outer-loop, which can eliminate the 2nd ripple and realize the normal control of dc-link voltage. The 2nd ripple expressed as (16) can be adopted to subtract from the voltage error Δu , then the pure dc error can be got for voltage controller shown as Fig. 3. So the output instruction Idc of voltage controller is dc component without ac ripples. However, in the status of alternate rotation process, the system parameters will be varied. As the α- and β-phase equivalent impedances of EM stirrer are generally symmetrical, the ripple elimination method also can be used for voltage control at this time.
According to Idc and active power Pd of EM stirrer, the current instruction signals of PWM rectifier can be obtained as:
According to (18), the expected input currents of PWM rectifier can be obtained. As the active currents are ac signals, the traditional PI controller has a limited tracking performance and there would be a control error in the steady-state. Therefore, the proportional resonant (PR) controller was put forward to improve the control performance for ac signals in [21, 22]. The s-domain function of PR controller can be expressed as:
where, kp is the proportional coefficient, ki is the integral coefficient. According to the current tracking errors, the control output of current closed-loop can be got by PR controller. So a closed-loop dynamic adjustment of output current can be achieved in the steady state.
According to the circuit shown as Fig. 1, the output voltages of PWM rectifier can be obtained as:
So the modulation signals of PWM rectifier can be obtained as:
The modulation signals are derived from the switching model of PWM converter, and they can be used as the feedforward instruction signals of PWM rectifier. Associate (18) with (21), the expected feedforward instruction signals of PWM rectifier can be obtained as:
So the control system can fast regulate the output of PWM rectifier according to (22), as well as tracking to the change of instruction signals rapidly. By combining the outputs Δmf of PR controller with the outputs mf of feedforward control shown as Fig. 3, the total modulation signals mf* (mfa*, mfb*, mfc*) of PWM rectifier can be obtained. According to mf*, the switching signals can be generated by PWM link, and then it can be used to drive power switches and output the desired voltage.
In order to improve the performance of PWM rectifier, a feedforward control method is added in the voltage outer-loop and current inner-loop respectively. According to the system’s power analysis, the feedforward signal 2Pd/(3U) for voltage outer-loop can be calculated; similarly, in order to improve current inner-loop, a feedforward control signal shown as (22) is used to inhibit the output oscillations of PR controller, and reduce the dynamic adjustment process of current control. Meanwhile, PR controller can be used to improve current control accuracy of PWM rectifier in the steady-state. In this way, the integrated control method can effectively improve the dynamic performance of system, and maintain a stable dc source for the back inverter.
4. The Current Control Method of Two-phase Inverter
As the former PWM rectifier is mainly used to provide a stable dc voltage source, the back TPTLI can be controlled independently. The current composite control diagram of TPTLI is proposed as Fig. 4(a).
Fig. 4.The control diagram of TPTLI and the performance comparison
Firstly, according to the desired amplitude Im* and frequency f* of output currents which are determined by the continuous casting process of molten steel [25], the desired output currents of TPTLI can be obtained as:
where, ω* = 2π f* . Since the load EM stirring is inductive, the amplitude of current instructions is given linearly by the ramp function during the process of positive and negative alternate rotation. The desired output currents subtract the output currents of inverter respectively, then the current errors can be obtained. And then the dynamic modulation signals Δutα , Δutβ , Δutc of TPTLI can be obtained by PR controller shown as Fig. 4(a). These modulation signals are mainly used to carry out the closed-loop tracking control according to current errors.
According to simplified equivalent model of EM stirrer and (10), the feedforward modulation signals of TPTLI for each switch leg can be gained as:
If the current instructions change, these feedforward modulation signals would change correspondingly. So it can rapidly regulate the output voltages of inverter according to the instruction currents. Similarly with the current control of PWM rectifier, this current composite control method can combine the merits of feedforward control and feedback control. So it can effectively improve the tracking performance of inverter to meet the requirements of fast alternate rotation of EM stirring.
In order to further analyze the tracking characteristic of current composite control method for inverter, the closed-loop control diagram is established shown as Fig. 4(b). In Fig. 4(b), kg is the normalized parameter, equal to 2 / udc ; e−Ts is the output delay link, which means that the output delays a control cycle T, and it can be simplified as (1−0.5Ts) / (1+0.5Ts) by 1st order pade approximation [23]; H(s) is the detection feedback link, which can be considered as 1 / (τs +1) ; kPWM is the gain of PWM link, and it is equal to udc / 2 ; Zd (s) is the equivalent impedance of load, and it is equal to Lms + rm .
Seen from Fig. 4(b), assuming:
Then the open-loop transfer function of inverter can be expressed as:
The error-tracking transfer function of inverter can be expressed as:
In Fig. 4(c), it can be seen that when using the composite control method, the gain of open-loop control characteristic is basically 1 at the high-frequency range. It means that the composite control method can realize a quick response to high-frequency input signal and improve the dynamic performance of inverter. At the same time, when only using the traditional feedback control method, the gain of open-loop control of system is attenuated at the high-frequency range. So the dynamic response of system with the proposed control method will be better than the one with the traditional control method.
Seen from Fig. 4(d), when using the composite control method, the error-tracking characteristic of system at the low-frequency working range is attenuated more seriously than traditional control method. The control error at the low-frequency working range with proposed control method would be smaller, so the control precision can be improved.
5. Simulation and Experimental Results
5.1 Simulation results
To verify the analysis and control method of TPOPS proposed in the paper, the simulation of TPOPS shown as Fig. 1 is built by using software PSIM.9. The three-phase source is 380V/50Hz. This design of TPOPS is used for a production line of 200*200mm billet continuous casting. According to the theory and practice of EM stirring [25], the rated output current is set as 400A and the rated output frequency is set as 10Hz. In the steady-state, the output capacity of TPOPS is 300kVA, and the rated output active power and reactive power are 90kW, 286kVar, respectively. The design of main circuit and control parameters of TPOPS are presented in the appendix. The parameter values of system are shown as Table 1.
Table 1.Parameters of TPOPS
Before put into the EM stirring load, the dc-link voltage is stable at 750V by PWM rectification. At 0.4s, the EM stirring load is put into system. In Fig. 5, I1, I2, I3 are the output currents of TPOPS respectively. Since the EM stirring load is inductive, the amplitude of output current instructions is given linearly. From 0.4s to 0.6s, the amplitude of current instructions is increased from 0 to 566A. Seen from Fig. 5(a), the output currents of inverter increase linearly from 0 at 0.4s, and almost reach a stable state at 0.6s. Meanwhile, the active power of load almost increases as the square of output currents. With the proposed control method, PWM rectifier can fast track to the change of output active power. Shown as Fig. 5(b), the input currents of PWM rectifier approximate to increase quadratically, and it almost gets into the steady state at 0.6s. The change of input currents is nearly synchronous with the variation of output currents, so the proposed control method can effectively improve the tracking performance of PWM rectifier to maintain dc-link voltage stably shown as Fig. 5(c). According to the analysis in Section. II, there is a 2nd ripple in dc-link voltage shown as Fig. 5(c), and the amplitude of ripple is almost increasing with the rise of output currents.
Fig. 5.The dynamical waveforms when switching into the EM stirring load
In order to verify the ripple filter method, a simulation comparison is carried out. In Fig. 6(a), ‘Udc’ is the detected dc-link voltage; ‘Vripple’ is the value calculated by (16). In the steady-state the detected ripple in dc-link voltage is almost the same as the calculated ripple. In Fig. 6(a), the amplitude of the ripple is about 14V and its frequency is 20Hz.
Fig. 6.The contrast waveforms for the ripple filter
Fig. 6(b) represents the current waveforms and spectrums without using the ripple filter method respectively. As the dc-link voltage contains a 2nd ripple, after the adjustment of PI controller, the ripple would be multiplied with the synchronous signal. Then it would generate two kinds of additional harmonic instruction signals for current inner-loop, of which the frequencies are 30Hz and 70Hz respectively shown as (17). By the closed-loop control of current controller, the input currents of PWM rectifier would contain these two kinds of harmonic currents shown as Fig. 6(b). The amplitude of these two harmonics are both about 15A shown as the spectrums in Fig. 6(b), so it is basically consistent with the analysis in Section.III. Fig. 6(c) represents the current waveforms and spectrums with using the ripple filter method. The ripple can be calculated as (16), then it can be eliminated from the voltage error and the pure dc error can be obtained for voltage PI controller. So the voltage controller would not produce additional harmonic instructions for current controller, and there is little harmonic current in input currents shown as the spectrums in Fig. 6(c). Therefore, the voltage controller of PWM rectifier can operate normally without adding a lower pass filter to attenuate the ac ripples.
As shown in Fig. 6(c), the amplitude of input currents in the steady-state is 153A. The PF of PWM rectifier is 0.99, and the total harmonic distortion (THD) of current is 3.1%. Therefore, PWM rectifier can effectively improve power quality of TPOPS.
In order to verify the current control method of inverter, the EM stirrer would reversely rotate at 2.0s. The amplitude of α- and β-phase current instructions of TPTLI changes linearly at 1.6s from +566A to 0 within 0.2s, then the inverter would stop working for 0.2s. Finally, the amplitude of α-phase current instruction would increase linearly at 2.0s from 0 to +566A within 0.2s, while the amplitude of β-phase current instruction would increase linearly at 2.0s from 0 to -566A within 0.2s shown as Fig. 7(a). So the reverse rotation of EM stirrer can be achieved.
Fig. 7.The current and voltage waveforms during the period of alternate rotation
In Fig. 7(b) and (c), I1, I2, I3 are the output currents of TPTLI respectively, and ie2 represents the tracking error of α-phase output current of inverter. Fig. 7(b) shows the waveforms when only using PR current controller. Since the output current instructions of inverter are changing, the modulation signals of inverter are completely from the output of PR controller. The PR controller is a current feedback control, which has a relatively lag to response, so there is a piece of vibration process in the tracking error. The maximum peak of tracking error is almost 20A shown as Fig. 7(b). When using the proposed current control method, the dynamical waveforms are shown as Fig. 7(c). Since the feedforward signals shown as (24) are obtained from the model of EM stirrer, it can fast control TPTLI to output the corresponding voltages according to the instruction currents. When the current instructions change, the feedforward signals would be varied at the same time. So the current tracking error is almost without vibrations shown as Fig. 7(c). Meanwhile, by using PR feedback control, it can achieve a closed-loop tracking to the current instruction signals and improve the control precision. Therefore, the proposed composite control method can effectively improve the tracking performance of inverter, so as to meet the requirements of fast alternate rotation of EM stirrer.
Fig. 7(d) shows the input currents and dc-link voltage of PWM rectifier with the proposed control method during the period of alternate rotation. The input currents of PWM rectifier almost follow the change of output currents, therefore, PWM rectifier can fast track the change of load power. And with the change of output currents, the 2nd ripple of dc-link voltage is varied correspondingly.
5.2 Experimental results
In order to further verify the analysis and control method, a 300kVA-380V low-frequency TPOPS for EM stirrer is developed, and the physical layout diagram of TPOPS is shown as Fig. 8(a). PWM rectifier and three-leg inverter adopt a TMS320F2812 separately as the digital signal processor (DSP), which is used to sample signals, execute algorithm and generate PWM drive signals for IGBTs. The A-phase voltage is processed by zero-crossing comparison circuit, which is used to generate an interrupt for DSP to initialize the phase angle. The temperature of cooling water in EM stirrer is sampled for protection and two DSPs are communicated with each other by RS232. The system parameters of device for experiments are shown in Table. 1. The control frequency of power switches is 10kHz. Fig. 8(b) and (c) are the physical photos of TPOPS, while Fig. 9 and Fig. 10 are the experimental waveforms.
Fig. 8.The physical implementation pictures of TPOPS
Fig. 9.Experimental waveforms during the period of alternate rotation
Fig. 10.Experimental waveforms in the steady-state
Seen from Fig. 9, it shows the dynamic waveforms when the EM stirrer is in the status of positive and negative alternate rotation. With the proposed control method for TPOPS, the output currents of inverter can change rapidly, and the a-phase input current of PWM rectifier can follow the change correspondingly shown as Fig. 9(b). Seen from Fig. 9(c), there is a 20Hz ripple in dc-link voltage and it is varied with the change of output currents. When the output currents are 0, the ripple is almost 0 and the dc-link voltage is stable at 750V. Fig. 10 is experimental waveforms in the steady-state. The input currents of PWM rectifier are about 155A. The PF of PWM rectifier is 0.98, and the THD of current is 4.3% shown as Fig. 10(a). While the output currents of TPOPS are about 401A, and the THD of current is 3.4% shown as Fig. 10(b).
The TPOPS with the proposed control method has been installed successfully on production line of the billet continuous casting. With the proposed control method, the period of alternate rotation of EM stirring is reduced. So it can enhance the effective work time of EM stirring and it can make the stirring of molten steel more symmetrically and uniformly. The equiaxed crystal rate of special steel billet is increased from 30% to 53%. Finally, the rate of finished products increases by 5%.
6. Conclusion
In order to improve the quality of billet continuous casting, a kind of TPOPS is researched which is composed of PWM rectifier and three-leg inverter. According to the circuit structure of TPOPS, the power model of system is analyzed. According to the power analysis, an integrated control method with feed-forward control for PWM rectifier is proposed, while a current composite control method is proposed for inverter to improve the output tracking performance. The closed-loop control diagram in s-domain is established, and the design of control parameters is presented. With the work of proposed control system, the dynamic performance of TPOPS is effectively improved which can meet the requirements of fast and frequent alternate rotation of EM stirring.
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