High Performance Decoupling Current Control by Linear Extended State Observer for Three-Phase Grid-Connected Inverter With an LCL Filter

Compared with traditional single L-type filters, LCL filters for three-phase grid-connected inverters can suppress the switching voltage and current ripple better, thus reducing harmonic pollution to the power grid. However, the increase in system order also makes it difficult to control the system. To solve the problems of complex model and severe coupling of LCL filter in the d-q coordinate system, the decoupling control of current inner loop by the linear extended state observer (LESO) was proposed based on the analysis of mathematical model of grid-connected inverter with an LCL filter. Moreover, given the relatively low speed of system response in traditional active disturbance rejection control (ADRC), the tracking differentiator (TD) was removed, and the nonlinear control law was replaced with the PI controller. Compared with the conventional feedforward decoupling method, the reasonably designed LESO used in this work can not only realize decoupling control of the original system to improve its performance, but also save the cost of sensor and achieve a better decoupling effect. The effectiveness of the proposed current control strategy was verified by experimental results.


I. INTRODUCTION
Recent years, energy consumption and environmental issues promote the rapid development of renewable energy power generation technologies.With the increasing proportion of renewable energy power generation, grid companies are putting forward higher standards for their power quality.Therefore, it is of great significance to improve the gridconnected current quality of renewable energy sources.
For the PWM inverters, the switching frequency is usually spread from 2 kHz to 15 kHz.High-frequency harmonics near the switching frequency and multiple switching frequencies will inevitably occur in grid-side current, which serious interferes and damages power quality of the grid.Once these harmonics are injected into the grid, they will increase the extra loss of the system, result in protection device malfunction in the power system by causing electromagnetic interference, The associate editor coordinating the review of this manuscript and approving it for publication was Zhixiang Zou .and affect reliability of power supply [1].In general, in application of single-inductance filter, large filtering inductance is required on the AC side for effectively suppressing highfrequency harmonics generated by the PWM inverter.Singleinductance filters, which are of large volume and weight, not only cost a lot but also significantly increase the loss.Given the problems, single-inductance filters are usually replaced with LCL filters in high-power inverters [2]- [5].
For LCL filters, the increase of capacitor branch provides a path for high-frequency harmonics in grid-side current of PWM inverter, so it can better attenuate high-frequency harmonics.However, the increase of system order also brings difficulties to the control of the system [6], [7].Control schemes based on the static coordinate system [8], [9] generally adopts proportional resonance (PR) controllers to directly track the sinusoidal signal.Nevertheless, PR controllers, which are designed complexly, tend to be affected by frequency fluctuation of power grid.In addition, in the coordinate system, most of the existing control schemes that ignore the complex coupling between d-and q-axes directly carry out PI control.As a result, the system can hardly achieve the desired effect.
In view of the complex model and serious coupling of LCL filter system, many researches have been done in this aspect.Based on the feedback linearization theory, the decoupling scheme designed by Khajehoddin et al. [10] and Bao et al. [11] could effectively improve the quality of gridside current.He et al. [12] proposed a reduced order method to solve the problem of grid-side current control in LCL filter.Bao et al. [13] applied feedforward compensation to all coupling terms according to the principle of block diagram equivalent transformation in the control theory.Peng et al. [14] analyzed the input and output transfer function of LCL filter, deduced the decoupling expression, and proposed double closed-loop control of grid-side current outer loop and capacitor current inner loop.
Recent years, active disturbance rejection control (ADRC) has been extensively studied as an effective method to solve the problem of controlling uncertain nonlinear systems.ADRC has been preliminarily applied in various fields, because it has excellent control quality, strong robustness and insensitivity to control parameters even when the controlled object undergoes parameter variation or encounters uncertain disturbance [15], [16].[17] solved the resonance problem of LCL filter in light of ADRC.Gao [18] not only linearized and parameterized the nonlinear form of ESO by introducing the concept of bandwidth, but also presented the configuration method of linear ESO parameters, which greatly reduced complexity of parameter configuration and was more convenient for engineering application.
Combining the advantages of traditional control and ADRC, this paper proposes a novel decoupling control method of current loop by the PI controller together with linear extended state observer (LESO) based on the analysis of mathematical model of grid-connected inverter with an LCL filter.Compared with conventional mathematical models, the mathematical model of the system used for LESO, which is built in this work, is much more accurate.Through the reasonable design of the LESO, the coupling amount and other disturbances of the system can be estimated and compensated, thereby largely decoupling the original system and improving the performance of the control system.Compared with conventional feedforward decoupling methods, it not only saves the cost of the sensor, but also has a better decoupling effect.Compared with traditional ADRC, the performance of LESO is improved by analyzing the known model of LCL system.Meanwhile, the control performance of LESO is enhanced by simplifying its control structure, abandoning the tracking differentiator (TD) and replacing the traditional nonlinear control law.Compared with the traditional linear active disturbance rejection control (LADRC), the response speed of the system is raised by replacing PD linear control law with a PI controller.Furthermore, the proposed method was verified to be correct and effective through experimental results analysis.

II. MATHEMATICAL MODEL OF GRID-CONNECTED INVERTER WITH AN LCL FILTER
The topology of grid-connected inverter with an LCL filter is exhibited in Figure 1.
The use of LCL filter may lead to resonance problems.In order to ensure stability of the system, additional damping was usually used.In view of existing damping strategies, this work uses the filter capacitor current feedback control.
As shown in Figure 1, it is assumed that: (a) The three-phase grid voltage is balance; (b) There is no magnetic saturation in the filter inductor; (c) There is no loss in the switch devices.
According to KVL and KCL, the mathematical model of LCL filter in the d-q coordinate system can be obtained: It can be seen from Figure 2 that the controlled variables i d and i q of current inner loop in the d-q coordinate system are DC variables, and the reference value can be simply tracked.However, the system is severely coupled in the d-q coordinate system, so it is difficult to achieve satisfied results by directly using PI control.

III. CURRENT INNER LOOP DISTURBANCE ANALYSIS AND CONTROLLER DESIGN A. CURRENT LOOP DISTURBANCE ANALYSIS AND CONTROL STRATEGY
According to the analysis in Section I, the d-q coordinate system needs to be decoupled for applying PI control to the current inner loop.Taking the d-axis component as an example, the equation of state can be obtained from (1) considering the feedback of capacitive current: where and f(t) is the total disturbance of current inner loop under the condition of ideal three-phase grid voltage and no external disturbance; x 1 is d-axis current i d ; and K c is the feedback coefficient of capacitance current.Equation ( 3) can be obtained through calculation: where According to the principle of ADRC, LESO reduces external and internal disturbances of the system into total disturbances and gives corresponding dynamic compensation.Therefore, the system is more robust and is able to suppress disturbances.As can be known from ( 2) and ( 3), in addition to the voltage correlation term in the grid, the current correlation term on the AC side and the voltage correlation term of filter capacitance are also included in the coupling of current inner loop.Given complex composition of the system, it is important to decouple the inner loop.Based on the above analysis, some ideas of ADRC are introduced to solve the existing problems.
The structures of traditional second-order ADRC and LADRC are given in Figure 3.In Figure 3(a), v is the given value of controlled variable; v 1 is the tracked given value; v 2 is the derivative of the tracked given value; and b 0 is the compensation parameter.TD is a tracking differentiator that provides the transition process and nonlinear state error feedback (NLSEF) generates the initial control value u 0 .ESO serves to estimate total disturbance of the system and correct it through the output y and the input u of the controller.The standard second-order LADRC structure is displayed in Figure 3(b).Compared with traditional ADRC, LADRC is mainly improved in the following aspects: (a) TD in traditional ADRC is removed; (b) ESO in traditional ADRC is replaced with LESO; (c) NLSEF is replaced with linear state error feedback (LSEF).The above analysis shows that the traditional PI controller can hardly track the reference value of inner loop effectively.Traditional ADRC can hardly meet the inner loop's requirement for high speed because of the existence of TD.Similarly, traditional LADRC fails to exhibit ideal static performance, because LSEF generally adopts PD controller.The differential quantity of LSEF is measured by LESO, which will also produce certain delay time.Neither traditional ADRC nor tradition LADRC can fully meet the requirements.Therefore, on the basis of PI controller, this study proposed to separately increase LESO to observe and compensate system disturbance.Benrabah et al. [17] who analyzed the frequency domain concluded that LESO had better performance than nonlinear ESO at high frequency.Hence, this study selects LESO to observe disturbance.That is, the study adopts a PI and LESO control structure.The control block diagram is presented in Figure 4.
The reference current signal is provided by an outer loop DC voltage regulator, while the deviation between the given current signal and the actual feedback current signal is input into the inner loop current regulator.The modulated signal output by the inner loop current regulator and the actual current signal are input as LESO.The LESO estimates the coupling amount and other disturbances of the system and outputs the decoupling compensation amount to the modulated signal outputted by the inner loop current regulator.The compensated result is taken as the final modulated signal  which is then compared with the carrier signal of a fixed frequency.In this way, the PWM modulated signal is obtained to drive the operation of switching devices in the inverter.The output of LESO in the diagram is expressed as The state equation of fourth-order ESO can be obtained in accordance with (2): where z 1 is the estimated value of i d ; y is the actual value of i d ; e is the error between the estimated value and the actual value ; z4 = kf (t) ; and β 01 − β 04 are adjustable parameters, respectively.When the nonlinear function g i (e) equal to e (i = 1, 2, 3, 4), equation ( 5) is simplified to (6): where the estimated value of disturbance z 4 (z 4 = kf (t)) is the output of LESO. the PI output control value u d and the actual value y of d-axis current i d are the input LESO.
If the adjustable parameters adopt the configuration method based on bandwidth, the following conditions shall be met: Then, the bandwidth ω 0 becomes the only adjustable parameter of LESO.The specific state equation of LESO can be determined by selecting an appropriate LESO bandwidth.The influence of LESO bandwidth on the system is analyzed by system modeling.

C. ANALYSIS OF SYSTEM DECOUPLING AND ANTI-DISTURBANCE PERFORMANCE
Considering capacitive current feedback, the matrix form of the mathematical model of LCL inverters can be obtained by (1) as follows: where Equation ( 9) can be obtained from (8): where Similarly, the matrix form of LESO transfer function can also be obtained according to (5): (10) where Equation ( 11) can be given as follows: As shown in Figure 4, the PI+LESO control output is expressed as where , k p and k i are the proportional and integral coefficients of the PI controller, respectively.
From ( 9), ( 10), (11), and (12) the output current matrix i 2dq of the system with the PI+LESO controller is obtained as follows where and As will be seen from ( 13), the output current of system is composed of the magnitude of reference currents and disturbance components caused by grid voltage.As far as the proposed control is concerned, the current coupling and voltage fluctuation in power grid are considered as the total disturbance suppression.From (13), the decoupling effect and anti-disturbance performance of the proposed control are reflected by the secondary diagonal gain of matrix G CLP (s) and the gain of matrix G CLD (s), respectively.And the smaller the gain, the better decoupling effect and anti-disturbance ability.
The LESO bandwidth ω 0 is enlarged from ω to 2 ω and 3 ω, which is aimed to study the influence of LESO bandwidth on control system.Amplitude-frequency curves of main and secondary diagonals transfer function G CLP (s) of the closed-loop are shown in Figure 5, where the ω is 5000 rad/s.The gain of secondary diagonal transfer function G CLP (s) is always less than 0 dB in Figure 5, which indicates that the coupling of d-and q-axes currents is suppressed by the proposed control.Additionally, Figure 5 (b) shows that the suppression of the system on the current coupling, especially in the low frequency (less than 100 Hz), is significantly enhanced with the ω 0 enlarging.In Figure 5 (a), as the ω 0 is enlarged, the gain of the system decreases in the high frequency band.And if the ω 0 continues to increase to a certain extent, the stability of the system will be affected.
Similarly, Figure 6 shows amplitude-frequency curves of main and secondary diagonals transfer function G CLD (s).Because the d-and q-axes currents are controlled by the method of power grid voltage orientation, the q-axis component of grid voltage equals to zero.The power grid voltage influences the d-axis output current by the main diagonals of matrix G CLD (s).And the q-axis output current is identified by the secondary diagonals of matrix G CLD (s).As can be seen from Figure 6, with the increase of LESO bandwidth ω 0 , the suppression effect of the system on the disturbance of the power grid voltage, especially on the low frequency band, is significantly enhanced.
It can be seen from the simulation that the performance of the LESO is related to its bandwidth and sampling frequency.In the same bandwidth, the shorter the sampling time is, the smaller the grid side current harmonic is.The higher the bandwidth is, the better effect is in theory [19].However, due to the problem of observer discretization in practical application, the parameters of the LESO are not arbitrary.The values of parameters are limited by the sampling time, and if the parameters are too large, the LESO will diverge [18].In a word, if the sampling frequency is high enough, higher bandwidth means better results [20]- [23].On the contrary, it is wise to choose lower bandwidth.

IV. ANALYSIS OF EXPERIMENTAL RESULTS
Table 1 lists parameters of the three-phase VSR.In order to further verify the effectiveness of the proposed control strategy, a three-phase two-level grid-connected inverter experimental platform was used and tested with and without the LESO decoupling control strategy.In the experiment, a TMS320F28335 DSP was used as the controller for executing the algorithm, while an XC6SLX9-2TQG144I FPGA was used to generate the driving signals required by the IGBT and adjust the dead time of IGBT.The parameters of the test system are listed in Table 1.The dead-time of switching interval was set to 3 µs, and the switching frequency was set to 5 kHz.In the experiment, current and voltage state variables of the main circuit were directly measured by the E6N current clamp and voltage differential probe and were output by an oscilloscope.The actual current spectrum was measured using a power quality analyzer.The experimental prototype of grid-connected inverter with an LCL filter is exhibited in Figure 7.

A. EXPERIMENTAL RESULTS UNDER STEADY STATE
Experimental results of both control methods under steady state are shown in Figure 8. Waveforms in Figure 8 represent inverter three-phrase output current and the d-and q-axes current of the inverter in different control algorithms.In the steady-state experiment, reference values of d-axis current and q-axis current were set as 12 A and 0 A, respectively.As can be observed from Figure 6, both control methods, under which total harmonic distortion (THD) values of output current on the inverter side are 3.75% and 2.23%, respectively, can both suppress grid-side current harmonics.Compared with the traditional PI control method, the decoupling control method proposed in this paper can result in a smaller THD value of grid-side current and a better control effect because it establishes a more accurate mathematical model and introduces the idea of ADRC.

B. EXPERIMENTAL RESULTS UNDER VARIED INDUCTANCE PARAMETERS
The experimental results of grid-side output current under varied grid inductance L 2 and inverter side inductance L 1 are listed in Table 2.As can be seen from Table 2, THD values of grid-side current corresponding to the two control methods both increase after the system inductance is halved.Compared with the traditional PI control method, the proposed decoupling control method suppresses current ripple better.The experimental results show that the proposed decoupling control scheme is robust to grid inductance variation.

C. EXPERIMENTAL RESULTS UNDER GRID VOLTAGE DISTORTION
In actual industrial operation conditions, the three-phase grid voltage is not a standard sine wave due to the use of various non-linear loads.The grid voltage can be distorted that contains a variety of higher harmonics.Thus, it is necessary to ensure that the control algorithm can maintain a good operating state under the condition of grid voltage distortion.In this experiment, the inverter was powered by an unfiltered three-phase voltage source in the laboratory power.The source has a single-phase voltage amplitude of 334 V, a total  voltage distortion rate of 5.6% and odd harmonics (3 times, 5 times, 7 times, 9 times, etc.).The experimental results of the two algorithms under grid voltage distortion are exhibited in Figure 9.
The waveforms in Figure 9 are the phase A grid-side current and the phase A grid voltage, respectively.Figure 9 indicates that grid-side currents controlled by the two control algorithms are distorted to varying degrees after the distortion of grid voltage.Specifically, the THD value of grid-side current rises from 3.75% under undistorted grid voltage to 4.73% for adopting the traditional PI control method, whereas it increases from 2.23% to 2.31% for adopting the proposed decoupling control method.The traditional PI control method has a poor harmonic suppression effect under grid voltage distortion, and the grid-side output current is distorted seriously, containing a large number of current ripples.In contrast, the proposed decoupling control algorithm achieves a good harmonic suppression effect and ensures a good sinusoidal degree of grid-side output current under grid voltage distortion.

D. EXPERIMENTAL RESULTS UNDER ONE-PHASE VOLTAGE DROP
In practical industrial applications, the occurrence of a shortcircuit fault or the switch of a high-power non-linear load in the power system may lead to temporary fall of one phase.These faults can be handled by the relay protection system within a short time and can be restored to normal state quickly.Hence, it is of great practical value to ensure stable and normal operation of the algorithm and prevent the inverter from exiting the system within the short period of one-phase voltage drop.
Experimental waveforms obtained by the two algorithms when the A-phase voltage drops by 25% are illustrated in Figure 10, where the three-phase grid voltage and the threephase grid-side output current under the two control methods are given from top to bottom, respectively.
The A-phase voltage drops by 25% at t = 5 s, and it does not recover until five cycles later.Figure 8 demonstrates that after the grid voltage drops, the grid-side current is largely distorted under the PI control algorithm, whereas it remains almost unchanged under the proposed decoupling control algorithm.Under the unbalance state of 100 ms grid voltage, the proposed decoupling control algorithm can ensure the normal operation state of the system and endow the system with excellent fault ride-through performance and good reliability.

V. CONCLUSION
In this paper, based on the proposed accurate mathematical model of grid-connected inverter with an LCL filter in the synchronous rotational coordinate system, the current inner loop was decoupled by observing the disturbance through a LESO in light of the of ADRC.Meanwhile, the proposed decoupling control method was verified by a real test bench.The simulation and experimental results show that the proposed decoupling control method has the advantages of saving sensor cost, insensitivity to control parameters and strong robustness.Moreover, it can effectively improve the quality of grid-side current.

FIGURE 1 .
FIGURE 1. Topology of grid-connected inverter with an LCL filter.

FIGURE 3 .
FIGURE 3. The structures of traditional ADRC and LADRC structures.(a) Standard second-order active disturbance rejection control (ADRC).(b) Standard second-order linear active disturbance rejection control (LADRC).

FIGURE 4 .
FIGURE 4. Control block diagram of grid-connected inverter with an LCL filter.

FIGURE 5 .
FIGURE 5.The influence of the LESO bandwidth ω 0 on amplitude-frequency curves of main and secondary diagonals transfer function G CLP (s).(a) The amplitude-frequency curves of secondary diagonal transfer function.(b) The amplitude-frequency curves of main diagonal transfer function.

FIGURE 6 .TABLE 1 .
FIGURE 6.The influence of the LESO bandwidth ω 0 on amplitude-frequency curves of main and secondary diagonals transfer function G CLD (s) (a) The amplitude-frequency curves of secondary diagonal transfer function.(b) The amplitude-frequency curves of main diagonal transfer function.

FIGURE 7 .
FIGURE 7. Experimental prototype of two-level grid-connected inverter with an LCL filter.

FIGURE 8 .
FIGURE 8. Experimental results of the two control methods under grid voltage distortion.(a) i a,b,c in PI control with feedforward decoupling method.(b) i d ,q in PI control with feedforward decoupling method.(c) i a,b,c in proposed decoupling control method.(d) i d ,q in proposed decoupling control method.

FIGURE 9 .
FIGURE 9. Experimental results of the two control methods under grid voltage distortion.(a) PI control with feedforward decoupling method.(b) Proposed decoupling control method.

FIGURE 10 .
FIGURE 10.Experimental results of the two control methods under one-phase voltage drop.(a) Three-phase grid voltage (under a-phase voltage drop).(b) PI control with feedforward decoupling method.(c) Proposed decoupling control method.

TABLE 2 .
Experimental results under variable inductance parameters.