Research on flatness-based control of three-phase voltage source PWM rectifier

Yu-zhen DUAN，Hong-xi WANG，Yu ZHANG，Wen-feng REN，Yu WANG

（School of Electrical and Information Engineering，Beihua University，Jilin 132021，China

Abstract：Normally，the control of a three-phase voltage source PWM rectifier（VSR）adopts a cascade PIcontrol strategy composed of outer voltage loop and current internal loop.However，the inaccurate decoupling of inner current loop results in poor rapidity and robustness of DC side voltage for VSR.Therefore，flatness-based control（FBC），a new nonlinear control method，is employed in the present study to improve the control performance of the inner current loop.Compared with PIcontrol，this method can achieve the feed-forward dynamic decoupling of current accurately.In the meanwhile，the FBC control can also improve robustness for parameter changes of the system and external disturbances，and then enhance the dynamic performance of the system significantly.Finally，the correctness and superiority of FBC control are verified by conducting theoretical analysis and simulation tests

Key words：Flatness-based control，PWM rectifier，Dynamic and precise decoupling of feed-forward current

At present，the VSR has been extensively applied in various fields such as power drive and new energy due to the advantages of high-power factor and bidirectional energy transmission.Since VSR is a nonlinear system，the traditional linear control strategies for dual closed-loop PI control is difficult to obtain excellent dynamic and static performance.

Therefore，the nonlinear control theory is required to better the performance of VSR.So far，numerous scholars have studied the nonlinear control of rectifier：Reference［1］proposed a double closed-loop control algorithm of voltage and current based on the synchronous rotating reference frame，improving the dynamic and static performance of the system.Reference［2］proposed a method based on reactive current injection to stabilize direct current link voltage which enhances the reliability of the rectifier.Against the defects of the traditional feed-forward decoupling control in the unbalanced power system.Reference［3］put forward the improved feed-forward decoupling control，which achieved the decoupling control of active and reactive current.Reference［4］proposed a new model predictive direct power control（MPDPC）strategy to improve the dynamic performance of PWM rectifier.Despite the good control effect，most of the methods above need an accurate and linear mathematical model，which can complicate the control algorithm.FBC is applied to the design of current loop controller for PMSM and the MMC-RPC controller，improving the dynamic performance of current loop and the anti-disturbance performance of controller［5-6］.In this paper，the FBC is used to improve the traditional inner current loop of PWM rectifier.The improved current loop can achieve the feed-forward dynamic decou-pling of current accurately.Additionally，the proposed method can also improve robustness and adaptability for parameter changes as well as external disturbances of the system，and then enhance the dynamic performance of the system significantly.

1 Mathematical model of the threephase VSR system

Fig.1 presents the circuit topology of three-phase VSR.The switching device is IGBT，N is the ground point of the system，and O is the neutral point of three-phase power grids.Voltages of the AC side are recorded as ea，eb，and ec；.Currents at the AC side are recorded as ia，ib，and ic；.L is filter inductance at the AC side，which is used for energy transmission and harmonic suppression.R is equivalent resistance at the AC side.C is capacitance at the DC side，which is used to stabilize the DCvoltage.Udcis voltage at the DCside，and Idcis current at the DCside.RLis load at the DC side.

Fig.1 The circuit topology of three-phase VSR

According to Kirchhoff’s laws，the mathematical model of the three-phase VSR in three-phase static coordinate system is provided：

In this model，AC side variables of the VSR are time-varying，which brings some difficulties to the design of the control system.According to the equivalent rotation transformation matrix：

The mathematical model of three-phase VSR in the dq-reference frame is obtained：

Obviously，the model of three-phase VSR is a multiple-input／output，and have a strongly coupled nonlinear system：

Where Sd，Sqare the switch functions.

State and control variables are defined as x＝［x1，x2］T＝［id，iq］Tand u＝［u1，u2］T＝［ud，uq］Trespectively.The power supply voltage is fixed on d-axis，namely，eq＝0.There are：

2 The flatness of the system

2.1 Flatness theory

Flatness theory is a new control method proposed by Fliess for nonlinear system in the 1990s［7］.Suppose a nonlinear system：˙x＝f（x，u），x∈Rn，u∈Rm，if there is a set of output vectors：

Meanwhile，make all state and control variables in the system meet the following formula：

That is to say，a system is considered to be flat if all the state and control variables in it can be expressed by flat output variables（with the same dimensions as control variables）and their finite order derivatives［8-10］.In equation（7）and（8），the h，Φ1，andΦ2indicate smooth functions.Moreover，the positive integerαandβare differential orders.

The structure diagram of the FBC system is shown in Fig.2，mainly consisting of two parts：feed-forward and feedback.The leading idea of flatness control is to drive the system to approach the reference trajectory through feed-forward，while the introduction of feedback is to eliminate the deviation caused by model un-certainty and various interference［5-6］.Feed-forward control variable ufis obtained by the expected output Yrefthrough the feed-forward controller.The feedback controller calculates feedback variable ubaccording to the error between the expected output Yrefand flat output Y.Add ufand ubtogether to obtain control input of the flat system.

Fig.2 Structure diagram of FBC system

2.2 Flat properties of three-phase VSR system

Before the design of the flat controller，it is necessary to prove that the three-phase VSR system is flat.Considered output vector as：y＝［y1，y2］T＝［id，iq］T.Apparently：

According to formula（5），there is：

Namely：

Obviously，state and control variables can be expressed by flat outputs and their derivatives，indicating that the three-phase VSR system is flat.

3 Design of flatness-based controller

The design of three-phase VSR control system often adopts dual closed-loop control.The outer voltage loop aims to control the DC side voltage of the threephase VSR，and the inner current loop aims to achieve the control of active and reactive current［11］.

The general linear control strategy cannot achieve high control performance since the model of threephase VSR is a multiple-input，multiple-output，and strongly coupled nonlinear system.Based on PI controller，the conventional dual closed-loop control method cannot accurately decouple the inner current loops，which result in poor rapidity and robustness of DC side voltage for VSR［12］，causing poor control performance of DC voltage.In the current work，the conventional dual closed-loop control is improved.Besides，the outer voltage loop is reserved to provide an ideal reference trajectory for the inner current loop.Moreover，the differential flat feed-forward is introduced into the inner current loop so as to improve the dynamic and static performance of the system.

The controller based on the FBC theory consists of two parts：the feed-forward control and the nonlinear error feedback compensation.

Through equation（5），the feed-forward control equation of the inner current loop is obtained：

The errors of current tracking are recorded as：The reference values of these error are.Therefore，the model of error is：

The coupling phases of d、q axes are ignored in reference 5.However，the coupling valuesωLΔiqand ωLΔidare considered as a disturbance acting on the feed-forward channel here，making decoupling more accurate.A PIcontroller is introduced to eliminate theerrors and then the control equation of error feedback is obtained：

Where KiPand KiIare respectively，the proportion and integral coefficients of the current loop.To sum up，the control laws of the three-phase VSR system is obtained：

Thus，the FBC block diagram of three-phase VSR can be acquired，as shown in Fig.3 where the driving signals are generated by SVPWM modulation.Because feed-forward plays an important role in the FBC control，the reference value needs to be smooth enough to use the feed-forward effectively.The i*qis usually controlled to zero，while i*dis obtained by the outer voltage loop.Compared with literature 5，this paper adds a low-pass to the outer loop to smooth the reference trajectory of active current，which makes the decoupling accurately.

4 Simulation results

The simulation model in Fig.4 is established with Matlab／Simulink.

Fig.3 FBC block diagram of three-phase VSR

Fig.4 FBC control system simulation diagram

Table 1 shows the basic parameters of the threephase VSR system.The PI parameters of the current and voltage loops are obtained through the simulation：KiP＝4.3，KiI＝10；Kup＝1，KuI＝5。

Table.1 Parameters of the 3-phase VSR system

4.1 Under steady state operation

Simulation waveforms under steady state are shown in Fig.5（a）illustrates the grid voltage and current waveforms of three-phase VSR controlled by FBC.It can be seen that the current is close to a sine wave，and the system operates in the unity power factor state.Fig.5（b）presents DC-side output waveforms，in which the maximal voltage under PI control is1 073 V while it is only 961V under FBC.Moreover，output through FBC accurately tracks the given voltage（800 V）before 0.4 s，while output under PI reaches the given value after 0.6 s.Obviously，VSR controlled by FBC exhibits shorter response time and lower overshoot than that controlled by PI.

Fig.5 Simulation waveforms under steady state

4.2 When the voltage reference changes suddenly

Fig.6 shows the simulation waveforms when voltage reference in DCside increases from800 V to 900 V at 0.6 s.As presented in Fig.6（a），the output voltage through the PI control reaches a new steady state at 0.7 s，while it at FBC reaches a steady state at 0.64 s without overshoot.Fig.6（b）obviously shows that VSR controlled by FBCstill operates with a unity power factor even when given voltage changes suddenly.

Fig.6 Simulation waveforms when the voltage reference changes suddenly

4.3 When the load changes suddenly

The simulation waveforms are shown in Fig.7 when the resistance in the DCside decreases from53 to 26.5 at 0.9 s.In Fig.7（a），the output voltage through FBC control has the same recovery time（0.5 s）with that through PI control.However，during the transition process，the voltage drop through FBC is 35 V，while the voltage drop through PI control is 41 V.It can be seen that FBC has a better anti-interference performance than PI control.In Fig.7（b），obviously，VSR still operates with a unity power factor even when the load changes suddenly.

Fig.7（c）illustrates the active and reactive current in the AC-side of three-phase VSR controlled by FBC.Obviously，idchanges while iqremains around 0 when the load changes suddenly.It can be seen that FBC can achieve the feed-forward dynamic decoupling of current accurately.

Fig.7 When the load changes suddenly

5 Conclusions

The differential Flatness theory designs control algorithm is an advanced nonlinear control theory according to the nonlinear structure of the system.Based on the mathematical model of three-phase VSR，flat properties of VSR are proved by the definition of flatness.Then，the inner current loop controller of threephase VSR is designed by employing the FBC theory.The outer voltage loop is used to generate an ideal reference trajectory of the active current component.The inner current loop is employed to obtain the control input of three-phase VSR by current feed-forward and error feedback.Finally，the correctness and effectiveness of the FBC are verified by the simulation.Compared with PI control，the proposed method achieves the feed-forward dynamic decoupling of current accurately，improves robustness for parameter changes of the system and external disturbances as well as enhances the dynamic performance of the system significantly.