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Control Design of LCL Type Grid-Connected Inverter Based on State Feedback Linearization

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Control Design of LCL Type Grid-Connected Inverter Based on State Feedback Linearization electronics Article Control Design of LCL Type Grid-Connected Inverter Based on State Feedback Linearization Longyue Yang 1,2,*, Chunchun Feng 2 and Jianhua Liu 1,2 1 Jiangsu Province Laboratory of Mining Electric and Automation, China University of Mining and Technology, Xuzhou 221116, China 2 School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221116, China * Correspondence: [email protected]; Tel.: +86-0516-83885605 Received: 11 July 2019; Accepted: 3 August 2019; Published: 7 August 2019 Abstract: Control strategy is the key technology of power electronic converter equipment. In order to solve the problem of controller design, a general design method is presented in this paper, which is more convenient to use computer machine learning and provides design rules for high-order power electronic system. With the higher order system Lie derivative, the nonlinear system is mapped to a controllable standard type, and then classical linear system control method is adopted to design the controller. The simulation and experimental results show that the two controllers have good steady-state control performance and dynamic response performance. Keywords: state feedback linearization; grid-connected inverter; LCL filter 1. Introduction As the main equipment of an AC power system, grid-connected voltage source inverters (VSI) have been widely used in recent years, such as photovoltaic inverters [1], Pulse Width Modulation Pulse Width Modulation (PWM) rectifiers [2], static var generators [3], active power filters [4], etc. The filter is an important part of the inverter, the structure of which directly determines the mathematical model and control mode of the inverter. Nowadays, a Inductance-Capacitance-Inductance (LCL) filter, with the advantages of small size, low cost and high harmonic attenuation for high frequency current, is widely used in voltage source type grid-connected converters. Considering that the LCL filter is the 3-order system, the damping of the system is small, resulting in a resonant peak of the grid-side inductance current. This phenomenon will adversely affect the safe and stable operation of grid-connected system. In order to improve the damping characteristics of the system, additional system damping is required. Active damping control method is commonly used at present stage [5]. With the state variables feedback, the traditional proportional integral controller can be used to achieve the grid-side inductance current control. However, for nonlinear power electronic devices, this design method of the controller is based on small signal modeling [6] and harmonic linearization. As the coupling and high order terms in the system are ignored in the Taylor series calculation, the obtained controller is suitable for working at steady working point with poor performance in other control domains [7]. In order to solve the global control problem, many solutions have been put forward [8,9]. Among them, the state feedback linearization method that develops from differential geometry [10,11] has become an effective way to solve the nonlinear power electronics system control problems. Based on the differential homeomorphism, Lie derivative is used to analyze the numerical relation between the state variables, the input variables, and the output variables. Then, the necessary and sufficient conditions for controllability and observability of nonlinear control systems can be established. Choi et Electronics 2019, 8, 877; doi:10.3390/electronics8080877 www.mdpi.com/journal/electronics 2 of 16 theElectronics permanent2019, 8magnet, 877 synchronous motor [12]. In addition, the drive flux and torque ripple2 of were 15 suppressed. Yang et al. combined feedback linearization and sliding mode variable structure controlal. proposed to complete a feedback the control linearization of three-phase direct torque four-leg control inverter for the permanent [13]. This magnet method synchronous was used to decouplemotor [the12]. torque In addition, and stator the drive flux fluxof the and inductiv torquee ripple motor were by Lascu suppressed. et al. [14]. Yang Yang et al. et combined al. applied thefeedback state feedback linearization control and to sliding the modeModular variable Multil structureevel Converter control to complete(MMC) thesystem control and of three-phaseanalyzed the performancefour-leg inverter characteristics [13]. This of method the system was used [15]. to decouple the torque and stator flux of the inductive motorDifferent by Lascu from et al.the [14 local]. Yang approximate et al. applied lineariz the stateation feedback method, control the to thefeedback Modular decoupling Multilevel is achievedConverter by (MMC)adopting system this nonlin and analyzedear algorithm the performance without ignoring characteristics the higher-order of the system terms. [15]. However, the commonlyDifferent used from thefeedback local approximate linearization linearization algorithm method, at the thepresent feedback stage decoupling is mostly is used achieved in the by 2- orderadopting or 1-order this nonlinear system. algorithm The rese withoutarch on ignoring the controller the higher-order design of terms. high-order However, or the other commonly complex systemsused feedbackis rare. linearizationIn addition, algorithmwhen the at order the present of the stage controlled is mostly object used is in high, the 2-order whether or 1-orderthe state feedbacksystem. control The research can be on transformed the controller into design a simple of high-order form has or other not been complex analyzed systems by is literatures. rare. In addition, whenIn order the order to achieve of the controlled the control object of high-order is high, whether powe ther electronic state feedback systems, control the can design be transformed of controller into a simple form has not been analyzed by literatures. based on LCL filter type grid-connected inverters is studied in this paper. For the 3-order control In order to achieve the control of high-order power electronic systems, the design of controller system, two controller design methods based on state feedback linearization are proposed in this based on LCL filter type grid-connected inverters is studied in this paper. For the 3-order control paper. Lie derivative vector field is used to solve the system relationship and design the decoupling system, two controller design methods based on state feedback linearization are proposed in this paper. matrix. Through the nonlinear mapping, the nonlinear system can be mapped to a controllable Lie derivative vector field is used to solve the system relationship and design the decoupling matrix. standardThrough form. the nonlinearThen, the mapping,classical linear the nonlinear system systemcontrol can method, be mapped which to can a controllable be designed standard easily, is applicableform. Then, for the classicalcontroller linear design. system Finally, control the method, performance which canof the be designedtwo controllers easily, is is applicable compared for and analyzedthe controller by simulations design. Finally, and experiments. the performance of the two controllers is compared and analyzed by simulations and experiments. 2. Single Closed-Loop Controller Design Based on State Feedback Linearization 2. Single Closed-Loop Controller Design Based on State Feedback Linearization 2.1. Model of the Three-Phase Three-Leg Grid-Connected Inverter 2.1. Model of the Three-Phase Three-Leg Grid-Connected Inverter Figure 1 shows the structure of the three-phase three-leg LCL grid-connected inverter. L1 and Figure1 shows the structure of the three-phase three-leg LCL grid-connected inverter. L1 and L2 L2 denote the inverter side inductance and the grid-side inductance, respectively. denote the inverter side inductance and the grid-side inductance, respectively. L L i 1 2 a e i a + b e U b dc − ic ec C Figure 1. Schematic diagram of the three-phase three-leg grid-connected inverter. Figure 1. Schematic diagram of the three-phase three-leg grid-connected inverter. As the model has been analyzed by many literatures, this paper does not repeat the description. TheAs mathematical the model modelhas been of inverter analyzed in dq coordinateby many axisliteratures, is given directlythis paper as does not repeat the description. The mathematical model of inverter in dq coordinate axis is given directly as 2 3 2 i 3 0 ! 1/L 0 0 0 m /L 2 i 3 6 1d 7 6 ω − 1 d 1 76 1d 7 6 7 6 01000− LmL11d 76 7 6 i 7 ii6 ! 0 0 1/L1 0 0 mq/ L1 76 i 7 6 1q 7 11dd6 −−ω 76 1q 7 6 7 6 − 00100− LmL11q 76 7 6 ucd 7 ii11qq6 1/C 0 0 ! 1/C 0 0 76 ucd 7 6 7 6 ω −− 76 7 d 6 u 7 =6 0100CC 1/C ! 0 100 0 1/C 0 76 u 7 dt 6 cq 7 uucd6 cd 76 cq 7 6 7 6 01CC− −−ω 0 01− 0 76 7 6 i 7d 6 = 0 0 1/L2 0 0 ! 0 76 i 7 6 2d 7 uucq6 cq 76 2d 7 (1) 6 7dt 6 00100L ω 0 76 7 6 i2q 7 6 0 0 02 1/L2 ! 0 0 76 i2q 7 6 7 ii22dd6 76 7 46 57 6 md mq −ω− 746 57 (1) u 4 00010 0L2 0 00 0 05 u dc iiCdc Cdc dc 22qq− − T " md mq # " # 0−− 0 0 0 0000001/L2 0 e uudc+ d dc CCdc dc − 0 0 0 0 1/L 0 0 eq − 2 0000 0− 1L 0T e + 2 d − 0000 1L2 0 0 eq Electronics 2019, 8, 877 3 of 15 h iT where the state variables X = i1d i1q ucd ucq i2d i2q udc represent the inverter side inductance current, the filter capacitor voltage, the grid-side inductance current and the DC voltage in the synchronous reference coordinate system, respectively. ed and eq represent grid voltage in the synchronous reference coordinate system, respectively. md and mq represent the nonlinear pulse width modulation variables, respectively. ! represents the fundamental angular frequency of the system.
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