Li-2015-Robust Speed Control

Robust Speed Control of Induction Motor Drives Using First-Order Auto-DisturbanceRejection ControllersJie Li,Hai-Peng Ren,Member,IEEE,and Yan-Ru ZhongAbstract—A novel robust control scheme employing three first-order auto-disturbance rejection controllers(ADRCs)is pre-sented for the speed control of induction motor pared with the existing high-order ADRC-based speed control struc-tures,the proposed method does not need to estimate the rotor flux.As a result,the implementation of the proposed scheme on a digital signal processor(DSP)is easier,and the runtime of the proposed ADRC control algorithm is shorter.The simulation results show that the proposed control scheme can cope with the internal disturbance and external disturbance,such as the motor’s parameter variations,the load disturbances,etc.A TMS320F2812 DSP-based prototype using the proposed control scheme was developed.The comparative experimental results show that the robustness of the proposed ADRC system is obviously better than the conventional proportional–integral system when various disturbances occur,and the scheme is feasible and effective.Index Terms—Auto-disturbance rejection controller(ADRC), induction motor,robust control,vector control.I.I NTRODUCTIONI NDUCTION motors can be controlled similarly as dc mo-tors using thefield-oriented control(FOC)(also called vector control)approach,and the performances of the con-trolled induction motors with FOC are comparable to those of the dc motors.With the development of the power elec-tronics elements and the high-performance microprocessor, induction motor drives employing FOC and conventional proportional–integral(PI)regulators have been commercial-ized.However,the performances of the PI regulator-based FOC suffers from the induction motor parameters’mismatch orManuscript received June22,2013;revised December1,2013and February27,2014;accepted May7,2014.Date of publication June10,2014; date of current version January16,2015.Paper2013-IACC-539.R2,pre-sented at the2012IEEE Industry Applications Society Annual Meeting, Las Vegas,NV,USA,October7–11,and approved for publication in the IEEE T RANSACTIONS ON I NDUSTRY A PPLICATIONS by the Industrial Automation and Control Committee of the IEEE Industry Applications Society.This work was supported in part by the National Natural Science Foundation of China under Grant50907054,in part by the Specialized Research Fund for the Doctoral Program under Grant20126118110008,in part by the Innovation Research Team of Shaanxi Province under Grant2013KCT-04,in part by the Natural Science Research Project of Educational Affairs Committee of Shaanxi Province under Grant2013JK0995,in part by the Scientific Research of Inter-national Cooperation Project of Shaanxi Province under Grant2013KW05-02, and in part by the Scientific Fund of Key Discipline of Shaanxi Province under Grant105-5X1301.The authors are with the School of Automation and Information Engineer-ing,Xi’an University of Technology,Xi’an710048,China(e-mail:lijie@ ;renhaipeng@;zhongyr@).Digital Object Identifier10.1109/TIA.2014.2330062variation with time[1].In addition,when the load disturbances are present,the PI controller scheme has a long recovery period. Generally speaking,the requirements of the motor speed control system are the accuracy,the rapidity,the robustness, and so on.To satisfy these performance requirements of the industry drive system,electrical engineers use two ways:one way is using the online parameter identification to estimate the corresponding parameters in the FOC;and the other way is us-ing the robust controller to replace the conventional PI regulator to get better performance.During the past few decades,many online parameter identification methods have been proposed for FOC[2].However,the parameters of the system are inherently dependent on each other.As a result,more parameters are needed to be identified,which increases the complexity of the algorithm and the difficulty of real-time applications.In order to directly aim at the original goal,i.e.,to improve the control per-formances of the drive systems,researchers attempted various kinds of robust controllers to cope with the parameter uncer-tainty,and at the same time,not to degrade the accuracy and the rapidity of the speed control.For example,an adaptive robust control strategy was proposed in[3]to increase robustness to parameter variations and load disturbance for sensorlessfield-oriented controlled induction motor drives.A back-stepping controller based on the exact model of the induction motors was proposed in[4].Meanwhile,the extended state observer(ESO) was employed to improve the robustness to the uncertainty of the system.The neural network-based robust control schemes were proposed to obtain a robust speed control of induction motor drives in[5]and[6].The ADRC was proposed by Han in1998[7]–[9].The ADRC is a nonlinear controller for an uncertain system,it estimates and compensates the external disturbances and pa-rameter variations,and as a consequence,the accurate model of the plant is not required.It means that the design of ADRC is inherently independent of the controlled system model and its parameters.Bogdan M.Wilamowski,former Editor-in-Chief of the IEEE T RANSACTIONS ON I NDUSTRIAL E LECTRONICS, pointed out that“ADRC turns modern control theory on its head.The implication of this change of direction proved to be enormous both in theory and practice”[9].As one of the simple robust control methods to deal with the uncertainty,ADRC attracts much attention in many technical fields[10]–[25],particularly in thefield of the motor control [14]–[25].A second-order ADRCs was proposed for the speed control of the brushless dc motor[14],[15].ADRC was used for0093-9994©2014IEEE.Personal use is permitted,but republication/redistribution requires IEEE permission.See /publications_standards/publications/rights/index.html for more information.direct torque control of permanent-magnet synchronous servo motor[16].Aside from speed control,ADRC was also applied to speed estimation to improve the robustness of the estimation algorithms[17].Many ADRC-based research works have been done to im-prove the robustness of the induction motor speed control[18]–[25].The ADRC-based speed control scheme for the induction motor drives has the advantage of good robustness to parameter uncertainty and external disturbance.Thefirst scheme of speed control of the induction motor based on ADRC was proposed in[18]and applied in[19]–[21].In this scheme,twofirst-order ADRCs were used to control the rotor speed and the quadrature component of the stator current,respectively,and one second-order ADRC was used to regulate the rotorflux.A four-ADRC configuration was proposed for the speed control of the induction motor fed by a matrix converter[22].Two second-order ADRCs were used to regulate the rotor speed and the quadrature component of the stator current,respectively,and one third-order ADRC was used to regulate the rotorflux in [23]to achieve robust speed control of induction motor.Two second-order ADRCs were employed for the speed and rotor flux control of the induction motor[24].Two conventional PI regulators were used for the torque current and magnetizing current control;meanwhile,two second-order ADRCs were used for speed control[25].Most of the aforementioned works on induction motor con-trol are limited to simulation results.All these existing schemes using ADRCs require the estimation of the rotorflux;therefore, the real-time computing cost is increased when the algorithms are implemented on digital signal processors(DSPs).Further-more,the existing works usually use the second-order(or third-order)ADRCs,including the third-order(or fourth-order) dynamical equations for ESO.Consequently,the complexity of the entire control algorithm is dramatically increased.In this paper,a novel control scheme based on three first-order ADRCs is presented.The rotorflux estimation is removed to reduce the runtime of the proposed ADRC control algorithm.Because the order of ADRC is low and theflux estimation is removed,the DSP code implementation of the proposed method is easier,and the code running time is shorter as well.Simulation results show that the proposed simplified ADRC robust speed control scheme provides strong ability to resist the uncertainties,such as external load disturbances and motor parameter variations.Experimental results also show the feasibility and effectiveness of the proposed method.This paper is organized as follows:the robust control scheme using threefirst-order ADRCs is given for induction motor speed control in Section II.The robust performances of the proposed scheme are tested by simulation in Section III.Exper-imental results on experimental bench are given in Section IV to show the effectiveness of the proposed method.Conclusion remarks are given in Section V.II.T HREE F IRST-O RDER ADRCS C LOSED-L OOP S PEEDC ONTROL OF I NDUCTION M OTORSFor the FOC scheme of induction motors,three control loops, including the quadrature axis(q-axis)current loop,the direct axis(d-axis)current loop,and the speed loop,are considered. Fig.1(a)shows an example of the induction motor speed control system using high-order ADRCs,which consists of two second-order ADRCs for the speed control and q-axis current control,respectively,and one third-order ADRC for the d-axis current control.Meanwhile,theflux estimator is needed to get the information aboutflux and rotor angle.Fig.1(b)gives the proposed scheme using threefirst-order ADRCs for the speed control of induction motors,where threefirst-order ADRCs are used for the three control loops of the system.It is shown from Fig.1that the proposed control scheme is simpler in the sense of using lower order ADRCs,which means the lower order of dynamical equation to be solved and that the removal of the flux estimator further reduces the complexity of the control algorithm.The proposed speed control scheme employs three different first-order ADRCs for rotor electrical angular speedωr regu-lation,direct component of stator current i d1regulation,and quadrature component of stator current i q1regulation.It means that the proposed control scheme use open loop control for the rotorflux.The motivation of using open loopflux control is to reduce the complexity of the control algorithm when it is implemented on DSP.As shown in Fig.2,eachfirst-order ADRC is composed of three parts:1)nonlinear differentiator(ND);2)ESO;and 3)nonlinear state error feedback control law(NLSEF).The ESO in the ADRC can be treated as a kind of dynamic feedback linearization mechanism.Its structure and performance are not only determined by the model of the system under control,but only by the range of its variation rate.Therefore,the ESO is robust to the system model uncertainty.The role of ND is to define a desirable transition response for the step input.The ND can smoothen the sudden change of the input signal in order to decrease the overshoot of the output response during the transient state.These factors make ADRC get a good balance between the fast transient response and the small overshoot.On the contrary,for the conventional PID controller,it is hard for tuning parameters to achieve this point.The NLSEF gives the control law u0to drive the state trajectory to track the desired reference.Now,we explain how to design the threefirst-order ADRCs as follows.A.ADRC for Speed RegulationThe mathematical model of the speed control loop is˙ωr=k1ψd2i q1+w1(t)(1) wherek1=P2L m/JL2and w1(t)=−P T L/JΨd2is the d-axis rotorflux linkage,i q1is the q-axis stator current,P is the number of pole pairs of the motor,J is the rotor inertia,L m is the mutual inductance,L2is the rotor inductance, and T L is the load torque.The load torque T L and the coupling term between the speed loop andflux loop can be treated as the“disturbances”when weparison of the scheme using higher order ADRC and our proposed scheme.(a)Example of control systems using high-order ADRCs.(b)Proposed scheme based on three first-orderADRCs.Fig.2.Block diagram of first-order ADRC.design the ADRC for the speed loop.These disturbances can be compensated by the ESO given as ⎧⎪⎨⎪⎩e (k )=z 1(k )−y (k )z 1(k +1)=z 1(k )+h (z 2(k )−β1fal (e (k ),α1,δ1)+bu (k ))z 2(k +1)=z 2(k )−hβ2fal (e (k ),α1,δ1)(2)where function fal (e (k ),α,δ)is a nonlinear function given asfal (e (k ),α,δ)=|e (k )|αsgn (e (k )),|e (k )|>δe (k )/δ1−α,|e (k )|≤δand y (k )is the output signal of the system (plant)to be controlled.For the speed loop,y (k )is the speed ωr (as shown in Fig.1),z 1(k )is the estimation of the system state,z 2(k )is the estimation of the disturbances.ESO has four adjustableparameters:α1,δ1,β1,and β2.The range of α1is from 0to 1;the smaller the α1is,the better the ability of the ESO against the uncertainty of the induction motor model and the disturbances is.δ1is the width of linear area of the nonlinear function.The system dynamic performances are affected tremendously by the β1and β2.β1mainly has effect on the estimation of the system state,and β2mainly affects the estimation of the disturbances.The larger the β1and the β2are,the faster the estimation converges,on the other hand,if the β1and the β2are too large,the estimation might not converge.The ND is given asx 1(k +1)=x 1(k )+hx 2(k )x 2(k +1)=x 2(k )+hfst (x 1(k )−v (k ),x 2(k ),r,h 0)(3)where the nonlinear function can be represented byfst (p 1(k ),p 2(k ),r,h 0)=−ra (k )/d,|a (k )|≤dr sgn (a (k )),|a (k )|>dy td (k )=p 1(k )+h 0p 2(k ),a 0(k )= d 2+8r |y td (k )| 12a (k )=p 2(k )+(a 0(k )−d )/2,|y td (k )|>d 0p 2(k )+y td (k )/h 0,|y td (k )|≤d 0d =rh 0,d 0=dh 0v(k)is the reference signal of the ADRC,for the speed loop, v(k)is the speed referenceω∗r(as shown in Fig.1),x1(k)is the tracking signal of v(k),x2(k)is the derivative of x1(k), also approximately the derivative of v(k),h is the step size. ND is used to arrange transition procedure when tracking the reference signal.There are two adjustable parameters in the ND,i.e.,r and h0.r is the convergence rate coefficient. The larger the r is,the faster the procedure that x1(k)converges toω∗r is.h0is afiltering factor used for the noisefiltering. According to the output of the ND and the ESO,the NLSEF gives the control to the corresponding loop⎧⎨⎩e1(k)=x1(k)−z1(k)u0(k)=β3fal(e1(k),α2,δ3)u(k)=u0(k)−z2(k)/b(4)where u(k)is the control output of the ADRC,for the speed loop,u(k)is the q-axis stator current reference i∗q1(as shown in Fig.1).α2andδ3possess the similar meanings as that of α1andδ1in(2).β3governs the speed of the system response, however,ifβ3is too large,a big overshoot would appear.B.ADRC for q-Axis Current LoopThe mathematical model of the q-axis stator current is˙iq1=−k2i q1+w2(t)+u q1/Lσ(5)wherek2=R1L22+R2L2m/LσL22w2(t)=−L mψd2ωr/LσL2−i d1ω1Lσ=L1−L2m/L2and u q1is the q-axis stator voltage.R1and R2are the stator resistance and rotor resistance,respectively.L1is the stator inductance,Lσis leakage inductance.ωr is rotor angular speed.ω1is synchronous angular speed.It can be seen from(5)that the“disturbance”w2(t)contains the coupling terms,i.e.,the product of the rotorfluxΨd2 and the rotor angular speedωr,and the product of the d-axis stator current i d1and synchronous angular speedω1.If a PID controller was used,the performances of the system will be degraded because of this nonlinear coupling terms.However, the ADRC can estimate the disturbance and cope with the effect of this coupling term to get better performance.The ADRC used in the q-axis current loop is similar to that of the speed-loop ADRC,except the input and output shown in Fig.1(b). C.ADRC for d-Axis Current LoopThe mathematical model of the d-axis stator current control loop is˙id1=−k2i d1+w3(t)+u d1/Lσ(6) wherew3(t)=k3ψd2+i q1ω1k3=R2L m/LσL22i d1is the d-axis stator current,and u d1is d-axis stator voltage. Here,w3(t)is treated as disturbance,which contains the coupling term,i.e.,the product of the synchronous angular speedω1and the q-axis stator current i q1.The rotor resistance is also considered in the disturbance,and it is subjected to the variations of operation conditions of induction motors. ADRC can automatically estimate all these disturbances and compensate them to achieve better performance.The ADRC for d-axis current loop is designed using the similar way as that in the speed loop,which is omitted for simplicity.III.S IMULATION R ESULTSTo show the performance of the proposed control scheme, an MATLAB/Simulink model[26]has been established for a 1.1-kW induction machine driven by a voltage source inverter (VSI)using the proposed scheme.Eachfirst-order ADRC is written by an S-function with C code.The parameters of the squirrel-cage induction motor are listed as follows:P N=1.1kWU N=380VI N=2.67Af N=50HzR1=5.27ΩR2=5.07ΩR Fe=1370ΩL1=479mHL2=479mHL m=421mHσ=0.228T N=7.45N·mP=2n N=1410r/min.We have investigated the robustness of the proposed scheme under the following three cases a)load disturbance;b)the motor parameter variations;and c)the model uncertainty.The proposed method is compared with the vector control based on the traditional PI regulators.In the simulation,both the adjustable parameters of the ADRC and those of the PI system have been manually tuned to their desirable values.It can be also done by the method proposed in[15].A.Load Disturbance PerformanceFig.3shows the comparative simulation results when the load torque disturbance occurs,Fig.3(a)shows the speed response when the load torque steps up from no load to the rated load(1pu)at1.5s,and Fig.3(b)shows the speed response when the load torque steps down from the rated load to no load at3s.In Fig.3,the speed reference is1300r/min, The motor parameters are constant during the simulations.As shown in Fig.3,in the steady-state performance aspect,the ADRC system can always settle down to the speed referenceparative simulation result when load torque disturbances occur with speed reference is1300r/min.(a)Load torque steps up from no load to the rated load at1.5s.(b)Load torque steps down from the rated load to no load at3s.value without steady-state error,whereas the steady-state error of the PI system increases when the load is heavier,even up to1.3%under rated load.This shows that the ADRC system has better robustness compared with the PI System from the viewpoint of load disturbances.In the dynamic performance aspect,the ADRC system always has larger overshoot than the PI system in the simulation results,the reason is that their linearization mechanisms are essentially different:the PI regulator depends on thefield oriented to realize the decoupling of the torque control and theflux control,when the rotorflux direction coincides with the d-axis,under this circumstance,the induction motor can be treated as a“linear”system.However,in ADRC control scheme,the ESO,a core component of ADRC, estimate the internal and the external disturbances as the“total disturbance”in real time,then compensate it.As a result,the system is dynamically linearized.In a simulation model, the PI system uses the exact parameters,which means that the parameters used in the vector control scheme match very well with the motor;thus,the dynamic performance could be perfect if the parameters of the PI regulators are optimized.On the contrary,the dynamic performance of the ADRC depends on the dynamic performance of its ESO,when the load suddenly increases or decreases,the disturbances in(1),(5),and(6)of the three ADRCs will also suddenly change,the ESOs will undergo a transient state to estimate the disturbance and to track reference.Therefore,even if the ADRC parameters are good,the simulation results will inevitably overshoot when load torque steps up or down heavily in high speed range.In practice,the experimental results in Section IV show that when load torque steps up or down,the PI system alsoparative simulation result when rotor resistance steps up and down.shoot because of inexact parameters used in the PI controller, i.e.,the parameter mismatch.B.Parameter Variation PerformanceTo evaluate the rapidity and the accuracy of control algo-rithms,the step response of reference variation or the step response of disturbance is usually observed,because the step signal contains the most abundant frequency components.For induction motor drive systems,if wound-rotor induction motors were selected,the rotor resistance may suddenly increase or decrease.Although,the rapid parameter variation is not the gen-eral case,the step variation of the rotor resistance is still chosen here to do the comparative simulation,in order to evaluate the proposed ADRC scheme under the worst operation condition.A simulation motor model[26]with a varying rotor re-sistance is used to simulate the performance of the proposed scheme.Fig.4shows the comparative simulation results when the motor parameter varies.The solid line,i.e.,the smoother one, is the speed response of the proposed ADRC scheme,and the dash line,i.e.,the oscillating one,is the speed response of the conventional PI controller.The operation condition is the same for both controllers,i.e.,the speed reference is1pu,and the load torque is1pu.The motor parameters are given as the foregoing parameter list.The rotor resistance steps up from1to 1.5pu at4s and down to1pu again at6s.In fact,it is almost impossible in the practical situation that the rotor resistances suddenly increase or decrease for the squirrel-cage induction motor.However,for the purpose of evaluating the robustness, step changes of the rotor resistance is chosen in simulation.As shown in Fig.4,when the rotor resistance suddenly changes, the proposed scheme is more robust than the conventional PI control.The ADRC system is able to quickly complete the adjustment process and stabilize at the reference speed again, whereas the PI control demonstrates a speed oscillation when the motor parameter parative simulations under the conditions that other motor parameters change have been done.The conclusion is that the ADRC is more robust than the PI control when the motor parameters change.C.Model Uncertainty PerformanceThe motor model considering the iron loss[26]is used to simulate the model uncertainty[19].Both PI controller and theparative simulation result when modeling error exists. proposed ADRC(s)are designed according to the ideal motor model(i.e.,without considering the impact of the nonideal factors such as magnetic saturation and iron loss).In fact,an actual motor cannot behave as we expect;thus,the electrical engineers have to use more robust control methods to get the desirable performance under the undesirable condition.Fig.5shows the comparative simulation results when the iron loss exists in the model,whereas the controller design procedure did not take into account.The solid line is the response of the proposed scheme,and the dashed line is the PI system.The operation condition is same for both schemes,i.e., the speed reference steps up from0.4to0.8pu at5s,and the load torque is full load throughout the simulation.The motor parameters,including the iron loss equivalent resistance,are set as the parameters listed aforementioned.If theflux control and the torque control of the induction motor are completely decoupled,the q-axis component rotorflux should be zero at the steady state.As shown in Fig.5,both the ADRC and the PI control do not make the q-axis component rotorflux to be zero, but the steady-state error of the q-axis rotorflux of the ADRC is obviously less than that of the PI control.It means that the decoupling degree of the proposed scheme is better than that of PI control.IV.E XPERIMENTAL R ESULTSThe experimental bench(see Fig.6)is built based on the core of TMS320F2812DSP,the actual motor has the same nominal parameters as that used in the simulations.The ADRC algorithm was written in mixed C language and assembly language in order to improve the execution speed,the execution time of each ADRC is about50μs.The sampling periods of the current loops and the speed loop are250μs,so that the code for ADRC can be executed in real time.Compared with the conventional PI regulators,the ADRCs have more parameters to be tuned.However,once these param-eters are set appropriately,it is effective all over the speed range and load condition.For the conventional PI systems,it always need several proportion coefficient and integral coefficient sets to obtain the desired speed control performance in different speed ranges for the same induction motor.From this point of view,the ADRC also has its superiority.On the other hand,some parameters in the ADRCs have their empirical values,e.g.,α1,α2,h0etc.;thus,the tuning procedure of the ADRC parameters is easier than our imagination.Using Fig.6.Photograph of the experimentalbench.parative experimental results when load torque disturbance occurs (speed reference is1300r/min,load torque steps down from rated load to no load,(a)ADRC system and(b)PI system).the trial and error method,the ADRCs parameters are set as follows.1)Speed ADRC controllerTD:r=60,h0=0.01ESO:α1=0.5,δ1=0.2,β1=20,β2=400,b0=11 NLSEF:β3=10,δ2=0.2,α2=0.75.2)q-axis current loop ADRC controllerTD:r=400,h0=0.01ESO:α1=0.5,δ1=10,β1=10,β2=600,b0=300 NLSEF:β3=10,δ2=10,α2=0.75.Fig.8.Controlled experimental result when load torque disturbance occurs (speed reference is1000r/min,and load torque steps up from no load to0.8-pu load,(a)the proposed scheme and(b)conventional PI control).3)d-axis current loop ADRC controllerTD:r=200,h0=0.01ESO:α1=0.5,δ1=0.1,β1=30,β2=600,b0=940 NLSEF:β3=1,δ2=0.1,α2=0.75.Fig.7shows the comparative experimental results of the proposed scheme and the conventional PI control.The speed reference is1300r/min,load torque steps down from the rated load to no load at the marked places in thefigures.In Figs.7–10, CH1is the measured rotor speed curve and CH2is the measured stator current curve.It is shown from Fig.7that because the speed is relatively high,and the load torque perturbation is relatively strong,as mentioned in Section III,the ADRC system will inevitable overshoot when load torque steps up or down heavily in high speed range.On the other hand,different from the simulation results,the PI control overshoots obviously even more than the proposed scheme.The reason is that in the simulation model,the parameters of the plant and the controller are matched well,but in practice,it is impossible.Under this practical case,the proposed method has better performance.In the middle speed range and the low speed range,the ADRC system shows very attractive performances,it is much better than the PI system.When load torque disturbances occur, the PI system still needs relatively longer settling time,and the speed variation is obvious during the transient state,on the contrary,the speed of the ADRC system almost does not vary when load is stepped up ordown.Fig.9.Controlled experimental result when load torque disturbance occurs (speed reference is700r/min,and load torque steps up from no load to0.6-pu load,(a)the proposed scheme and(b)conventional PI control).Fig.8gives the comparative experimental results under the condition that the speed reference is1000r/min,and load torque steps up from no load to0.8-pu load.The speed of the ADRC system seems to befluctuated to a very small extent, but,the PI system spends more than3s to settle down and has about80r/min transient speed decreasing.The similar situation can be observed in the Figs.9and10.Fig.9shows the comparative experimental results under the condition that the speed reference is700r/min,and load torque steps up from no load to0.6pu load.Fig.10shows the comparative exper-imental results under the condition that the speed reference is700r/min,and load torque steps down from rated load to no load.It can be seen that,when the speed reference is in the middle speed range or in the low speed range,the ADRC system has a very short settling downtime and far smaller speedfluctuation amplitude than those of the conventional PI control.Therefore,the conclusion can be drawn that the performance of the control system has been improved via the state observation,the real-time“disturbance”estimation and compensation,the ADRC controller indeed plays a role of “anti-disturbance.”As mentioned in Section III,practically,it is unlikely that the rotor resistance can suddenly increase or decrease for squirrel-cage induction motors.In order to experimentally evaluate the robustness against the parameter variation,there are two choices:i)replace the squirrel-cage induction motor used in the experiments with a wound-rotor induction motor,then,the rotor resistance can be changed by connecting a three-phase。