Disturbance Observer-Based Adaptive Current Control With Self-Learning Ability to Improvethe Grid-Injected Current for LCL-Filtered Grid-Connected Inverter

Liu, Jiahao; Wu, Weimin; Chung, Henry Shu-Hung; Blaabjerg, Frede

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Published: 01/01/2019

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Published version (DOI):10.1109/ACCESS.2019.2931734

Publication details:Liu, J., Wu, W., Chung, H. S-H., & Blaabjerg, F. (2019). Disturbance Observer-Based Adaptive Current ControlWith Self-Learning Ability to Improve the Grid-Injected Current for LCL-Filtered Grid-Connected Inverter. IEEEAccess, 7, 105376-105390. https://doi.org/10.1109/ACCESS.2019.2931734

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Received July 1, 2019, accepted July 18, 2019, date of publication July 29, 2019, date of current version August 14, 2019.

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Disturbance Observer-Based Adaptive CurrentControl With Self-Learning Ability to Improvethe Grid-Injected Current for LCL-FilteredGrid-Connected InverterJIAHAO LIU1, WEIMIN WU 1, (Member, IEEE), HENRY SHU-HUNG CHUNG 2, (Fellow, IEEE),AND FREDE BLAABJERG 3, (Fellow, IEEE)1Department of Electronic Engineering, Shanghai Maritime University, Shanghai 201306, China2Department of Electronic Engineering, City University of Hong Kong, Hong Kong3Department of Energy Technology, Aalborg University, DK-9220 Aalborg, Denmark

Corresponding author: Weimin Wu (wmwu@shmtu.edu.cn)

This work was supported in part by the National Natural Science Foundation of China (NSFC) under Grant 51577114, in part by the NSFCunder Grant 51877130, in part by the Shanghai Municipal Education Committee under Grant 14SG43, and in part by the Shanghai Scienceand Technology Commission under Grant 17040501500.

ABSTRACT During the design of the conventional current controller for a grid-connected inverter with LCLfilter, the parameter mismatches and disturbances are generally neglected, which may seriously affect thecontrol performance, even result in instability. In order to improve the ability of disturbance rejection andensure a desired control performance, this paper proposes an Adaptive PID (APID) controller with the self-learning ability based on the Disturbance Observer (DOB). First, the full state-feedback and state observerare utilized to achieve active damping and eliminate the effect of computational delay. Then, aiming toestimate and compensate the lumped disturbance, a DOB is designed. Beneficial from DOB, the steady-stateperformance is almost not affected by model uncertainties and unmodeled dynamics, however, the transientperformance is still deteriorated inevitably due to the limitation of DOB. Thus, an online adaptive methodusing APID is finally proposed to further improve grid-injected current dynamics. The control parameterscan be automatically adjusted in real time by adaptive learning rule, which significantly improves thesystem robustness and the control performance. Simulation and experiments are provided to demonstratethe effectiveness of the proposed strategy.

INDEX TERMS Adaptive PID controller, disturbance observer, full state-feedback, LCL filter, stateobserver.

I. INTRODUCTIONThe LCL filter has been widely adopted in grid-connectedinverters since they can provide improved attenuation onthe pulse width modulation switching harmonics comparedwith the L filter [1], which allows the use of smaller induc-tance to satisfy the stringent harmonic requirements [2], [3].In order to achieve excellent performances and addressglobal stability, various control strategies have been pro-posed, such as PI-based control [4], [5], Repetitive Con-trol (RC) [6], [7], Sliding Mode Control (SMC) [8], [9],

The associate editor coordinating the review of this manuscript andapproving it for publication was Snehal Gawande.

Deadbeat Control (DBC) [10], [11], Model PredictiveControl (MPC) [12], [13], etc. Most of the aforementionedmethods are model-based control scheme, thus the grid-injected current is sensitive to the uncertainties, whichmeans the unmodeled dynamics and disturbances of invert-ers, including nonlinear load, parameter uncertainties, gridinductance variation [14], dead time, on-state voltage dropof switching devices and diodes, would inevitably causeperformance deterioration.

To eliminate the influence of disturbances, severalDisturbance/Uncertainty Estimation andAttenuation (DUEA)schemes had been proposed [15]. Recently, DUEA schemeson the grid-connected inverters, such as Uncertainty

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J. Liu et al.: Disturbance Observer-Based Adaptive Current Control

and Disturbance Estimator (UDE) [16], Extended StateObserver (ESO) [17], Active Disturbance Rejection Con-trol (ADRC) [18] and DOB [19], have become a researchhotspot. In [16], a UDE-based grid-injected current con-trol strategy was proposed. This method allows decouplingbetween disturbance attenuation and reference tracking,however, the introduced differential feedforward may besensitive to noise. Wang et al. [17] separated observationdynamics and parameter mismatches error by utilizing ESO,which can guarantee robust adaption for parameter variation.Nevertheless, the method is only applicable for constant orslowly varying disturbance estimation. Benrabah et al. [18]presented a simplified robust control method based onADRC [20] using Padé approximation. In this strategy,the first order linear ADRC controller is applied to deal withparameter variation of LCL filter, which greatly simplifiesthe controller design, yet brings adverse effects into theclosed-loop control performances. In [19], DOB-based feed-forward scheme was proposed by introducing the estimationof fluctuations into the reference of the inner loop, which canremarkably attenuate disturbance effects both in structure andexternal power.

Among these DUEA strategies above, owing to its rela-tively simple structure and prominent performance [21]–[23],DOB, combining with other controller, is increasinglyadopted in the applications of inverters to improve the abil-ity of disturbance rejection. In [24], PI controller based onfeedback linearization technique and DOB, was proposed toprovide good steady-state performance and improve robust-ness. However, its assumption that the lumped disturbanceapproaches to constant value may be not satisfied some-times, especially when the background harmonic voltageexists. In [25], an internal model-based DOB was proposedto enhance the control performance. This controller fusesthe merits of DOB and RC. Nevertheless, the tracking char-acteristic and stability are affected by the time delay ele-ment introduced by RC. Nguyen and Jung [26] investigateda DOB-based MPC, achieving fast dynamic response underthe uncertain parameters. However, the unconstrained modeand constrained mode bring high computational effort.

It should be pointed out that although the steady-stateperformance is almost not be affected by model uncer-tainties or unmodeled dynamics under the contributionof DOB, the transient performance is still deterioratedinevitably. Because the model uncertainties and external dis-turbances cannot be completely eliminated due to limitationof DOB [25], and they will produce a transient componentto the grid-injected current, which will degrade transientperformance. However, few studies on DOB-based controlof the grid-connected inverters have considered this issue sofar. Aiming to overcome it and further improve grid-injectedcurrent dynamics, an effective way is to integrate onlineadaptive method into the designed controller [22], [27]. Forexample, in [28], DOB-based fuzzy SMC strategy was pro-posed. The switching gain can be adaptively adjusted, which

achieves strong robustness and fast response. However, it onlycompensates constant or slowly varying disturbance.

Motivated by the aforementioned limitations, this paperproposes an Adaptive PID (APID) controller with the self-learning ability based on DOB. Firstly, the full state-feedbackwhich plays a role of the active damping is utilized. It iswell known that this feedback strategy can arbitrarily assignthe position of the closed-loop poles, which can obtain thedesired dynamic response [29]. While, since more sensorsare required, a state observer is also constructed as a statepredictor to reduce the number of sensors and eliminatethe influence of computational delay. Secondly, aiming tocompensate the lumped disturbance and relax the sensitivityissue, a DOB is designed. DOB possesses two-degrees-of-freedom structure, allowing to decouple the tracking perfor-mance and disturbance rejection by the independent design ofnominal controller and DOB. Therefore, the excellent track-ing performance and disturbance rejection can be achievedsimultaneously. Finally, an adaptive PID regulator with theself-learning ability is proposed to prevent transient perfor-mance deterioration. The control parameters can be auto-matically adjusted in real time by adaptive learning rule,which further improves the system robustness and the controlperformance.

The rest of this paper is organized as follows. Themodelingof three-phase grid-connected voltage-source inverter withLCL filter is provided in Section II. Then, in Section III,the theoretical analysis and design of the proposed strategyare addressed. The robust stability under model uncertaintiesis analyzed based on the small-gain theorem in Section IV.The effectiveness of the proposed method is demonstratedby a series of comparative simulations and experiments inSection V. Finally, a conclusion of the paper is drawn inSection VI.

FIGURE 1. The proposed control scheme.

II. MODELING OF GRID-CONNECTED VOLTAGESOURCE INVERTER WITH LCL FILTERA three-phase grid-connected voltage-source inverter withLCLfilter is depicted in Fig. 1. Inverter-side inductor L1, filter

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capacitor C and grid-side inductor L2 give the LCL filter.Lg is the equivalent grid inductor. R1, R2 are the equivalentseries resistances of L1 and L2, respectively. Udc, ui andug represent the DC-link, inverter output and the Point ofCommon Coupling (PCC) voltages, respectively. The statevector is selected as x =

[i1, uc, ig

]T , where uc is the filtercapacitor voltage, i1 and ig are the inverter-side and grid-injected currents, respectively.

Thus, in the synchronous reference frame rotating atthe grid angular frequency ωg, the state-space equation isexpressed as: {

x = Ax + B1ui + B2ugy = Ccx

(1)

where

A =

−R1L1− jωg −

1L1

0

1C

−jωg −1C

01L2

−R2L2− jωg

,

B1 =[

1L1

0 0]T,

B2 =[0 0 −

1L2

]T, and Cc =

[0 0 1

].

Due to the coupling terms between d- and q-axis, it iscomplicated to design a desired controller. In this paper, thecoupling terms are treated as a part of internal disturbances,since they have a negligible influence on the stability ofsystem, if the controller parameters are well tuned. Andtaken the parameter mismatches and unmodeled dynamicsinto consideration, the model is rewritten as:{

xd = Axd + B1uid + B2ugd +8λdyd = Ccxd{xq = Axq + B1uiq + B2ugq +8λqyq = Ccxq

(2)

where

A =

−R1L1

−1L1

0

1C

0 −1C

01L2

−R2L2

, and

8 =

−

1L1

0 0

0 −1C

0

0 0 −1L2

.The subscript d and q are the component of d- and q- axis,

respectively. λd and λq represent the lumped disturbance

caused by the parameter variations, cross-coupling and otherunstructured uncertainties, which can be deduced as:

λd =

1L1i1d − L1ωgi1q −1L1ωgi1q +1R1i1d + ε1d1Cucd − Cωgucq −1Cωgucq + ε2d1L2igd − L2ωgigq −1L2ωgigq +1R2igd + ε3d

λq =

1L1 i1q + L1ωgi1d +1L1ωgi1d +1R1i1q + ε1q1Cucq + Cωgucd +1Cωgucd + ε2q1L2igq + L2ωgigd +1L2ωgigd +1R2igq + ε3q

(3)

where the symbol ‘‘1’’ denotes the deviation from the nom-inal values, εd and εq represent the unmodeled uncertaintiesof d- and q-axis, respectively.

Obviously, the expressions in the d- and q-axis are identi-cal, in essence. Hereafter, for the brevity of notation, the sub-script d and q will be omitted.For digital implementation of the control algorithm,

the state space equation is represented in discrete-timedomain as follows:{x (k + 1) = Gx (k)+ H1ui (k)+ H2ug (k)+ 0λ (k)y (k) = Ccx (k)

(4)

where

G = eATs , H1 =

∫ Ts

0eAτdτB1,

H2 =

∫ Ts

0eAτdτB2, 0 =

∫ Ts

0eAτdτ8,

Ts is sample period and k is the discrete sampling instant.

III. THEORETICAL ANALYSIS AND DESIGN OF THEPROPOSED CONTROL STRATEGYThe structure of the proposed strategy is depicted in Fig. 1,from which it can be observed that the control objectives canbe met by the three cascaded control loops. The inner loopis to achieve active damping via full state-feedback usingpredicted state variables; the middle loop is utilized to com-bine the disturbance compensation and tracking performancetogether; whereas the outer loop is designed to improve thequality of grid-injected current by minimizing grid-injectedcurrent error online both in ideal and perturbed conditions.

A. STATE-FEEDBACK AND STATE OBSERVERThe full state-feedback is utilized in inner loop for the activedamping, as described in Fig. 1. Accordingly, the control lawis produced as:

ui(k) = u∗i (k)− Kvx(k) (5)

where u∗i (k) denotes the output of the designed controller withdisturbance compensation, and Kv =

[k1 k2 k3

]is the state-

feedback gain vector.

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Substituting (5) into (4) yieldsx (k + 1) = (G− H1Kv)x (k)+ H1u∗i (k)+ H2ug (k)

+0λ (k)

y (k) = Ccx (k) .

(6)

Here, the analytical relationship from u∗i (k) to ig(k)is treated as a generalized controlled plant PG(z). Thus,from (6), the transfer function of the generalized plant isderived as

PG(z) = Cc(zI3 − G+ H1Kv)−1H1 (7)

where I3 is a three-dimensional identity matrix.According to the guidelines for selecting the pole

locations [30], [31], let the characteristic polynomial of (7) be

det(zI3 − G+ H1Kv) = (z− α1)(z− α2)(z− α3) (8)

where α1, α2 and α3 are the desired poles of the state-feedback, then the state-feedback gain vector Kv can bederived.

It should be noted that the full state-feedback requiresmore sensors compared with capacitor current feedback,which increases the system cost and vulnerability. Moreover,the computational delay has not been considered. In practice,however, one-step delay associated with the digital imple-mentation exists between the control voltage and inverteroutput voltage. To account for the computational delay, thesystem dynamics is rewritten as follows{x (k + 1)=Gx (k)+H1ui (k − 1)+H2ug (k)+0λ (k)

y (k)=Ccx (k) .(9)

Referring to [16], [32], an effective way to compensatefor this time delay is to make one-step-ahead prediction ofthe state variables and utilize predicted states instead of thecurrent states in determining the control law.

In order to reduce the number of sensors and predict statevariables one-step ahead, a full-order state observer is utilizedin this paper. Since the grid-injected current ig(k), PCC volt-age ug(k) are measured and the inverter output voltage ui(k)is internally known, the remaining state variables can beestimated precisely, i.e.

x(k + 1) = Gx(k)+ H1ui(k − 1)+ H2ug(k)

+L[y(k)− y(k)]

y(k) = Ccx(k)

(10)

where the subscript ‘‘ˆ’’ denotes the estimated value, L =[l1 l2 l3

]T is the observer gain vector. With (9) and (10),the dynamics of the estimation error ex(k) = x(k) − x(k)is

ex(k + 1) = (G− LCc)ex(k)+ 0λ (k) . (11)

According to the pole placement, the observer gain vectorL can be determined, if the characteristic polynomial of the

observer dynamics is selected as:

det(zI3 − G+ LCc) = (z− p1)(z− p2)(z− p3) (12)

where p1, p2, and p3 are the desired poles of the state observer.From (11), the estimation error will asymptotically con-

verge under the elaborately selected poles, as long as thesufficient and necessary condition below is satisfied

ρ(G− LCc) = max1≤i≤3

|λi(G− LCc)| < 1 (13)

where λi(G − LCc), ρ(G − LCc) denote the eigenvalues andspectral radius of G-LCc, respectively.Consequently, when the steady-state is reached, ex(k + 1)

and ex(k) will converge to the same, i.e.

limk→∞|ex(k + 1)| ≈ lim

k→∞|ex(k)|

≈ limk→∞|(I3 − G+ LCc)−10λ (k) |. (14)

It is obvious that there exists estimation error in the pres-ence of lumped disturbance, which will inevitably degradethe control performance. Thus, aiming to improve the abilityof disturbance rejection, a DOB is constructed.

FIGURE 2. (a) Block diagram of DOB. (b) The standard M-1 configurationof DOB for the small-gain theorem.

B. DISTURBANCE REJECTION BYDISTURBANCE OBSERVERAsmentioned above, the disturbance and model uncertaintiesmay result in performance deterioration. For estimating andcompensating this lumped disturbance, a DOB is designed,as depicted in Fig. 2. ur (z), d(z), d(z) and n(z) represent thecontroller output, external disturbance, estimated disturbanceand measurement noise, respectively.Q(z) is a low-pass filterwhich needs to be elaborately designed. PG(z) is the actualgeneralized plant defined by (7) and P−1Gn(z) is the inverseof nominal generalized plant. To account for the mismatchesbetween PG(z) and PGn(z), PG(z) is formed into the outputmultiplicative perturbation description as follows

PG(z) = [1+1(z)]PGn(z)1(z) = W (z)δ(z)‖δ(z)‖∞ ≤ 1

(15)

where 1(z), W (z), and δ(z) represent the modeling error,weighting function and variation, respectively.

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Based on the block diagram in Fig. 2, the estimated distur-bance can be derived as

d(z) =Q(z)1(z)ur (z)+ Q(z) [1+1(z)] d(z)+ Q(z)n(z)

1+ Q(z)1(z)(16)

and the grid-injected current ig(z) is written as follows:

ig(z) = Gri(z)ur (z)+ Gdi(z)d(z)+ Gni(z)n(z) (17)

where Gri(z), Gdi(z), and Gni(z) denote the transfer functionsfrom ur (z), d(z), and n(z) to ig(z), respectively. They aregiven by

Gri(z) =PG(z)PGn(z)

PGn(z)+ Q(z) [PG(z)− PGn(z)]

=[1+1(z)]PGn(z)1+ Q(z)1(z)

Gdi(z) =[1− Q(z)]PG(z)PGn(z)

PGn(z)+ Q(z) [PG(z)− PGn(z)]

=[1− Q(z)] [1+1(z)]PGn(z)

1+ Q(z)1(z)

Gni(z) = −Q(z)PG(z)

PGn(z)+ Q(z) [PG(z)− PGn(z)]

= −Q(z) [1+1(z)]1+ Q(z)1(z)

.

(18)

It is noted that the disturbance d(z) is dominant in thelow frequency segment, while the noise n(z) is dominant inthe high frequency segment. Thus, from (18), in the lowfrequency segment, Gdi(z) ≈ 0 provided that Q(z) ≈ 1,which implies the disturbance can be effectively attenuated.Therefore, within the bandwidth of Q(z), the grid-injectedcurrent ig(z) is deduced as

ig(z) ≈ PGn(z)ur (z). (19)

It can be seen from (19) that even if the actual plant isperturbed, the DOB forces it to maintain the characteristicof the nominal plant, which significantly improves the abilityof disturbance rejection and relaxes the lumped disturbancesensitivity issue.

Then, we transfer the perturbed plant and DOB into thestandard M−1 configuration, as shown in Fig. 2(b). Basedon the small-gain theorem [22], [25], the sufficient conditionfor the system stability can be easily obtained

‖1(z)Q(z)‖∞ < 1. (20)

From (20), an improperQ(z) would deteriorate the stability.Hence, when selecting the parameters of Q(z), including therelative order and bandwidth, attention should be paid. In fact,the higher order of Q(z) is, the faster response of the DOBbecomes, but with the increasing order, phase lag may makethe system unstable. On the other hand, a high bandwidthQ(z)can enhance the anti-disturbance ability, however, it may eas-ily result in system instability and increase the sensitivity tomeasurement noise n(z). Thus, a tradeoff among the stability,

disturbance suppression ability and noise sensitivity, must bemade when designing DOB.Considering that PG(z) is a three-order plant, to make

Q(z)P−1Gn(z) realizable, this paper defines Q(z) as

Q(z) = Z[

1(τ s+ 1)3

](21)

where τ , Z [·] denote the time constant and ZOH-transformation, respectively.

It can be obtained from (21) that Q(z) is slightly less thanunity within its bandwidth, which results in a fact that Gdi(z)is not strictly equal to zero. Therefore, the lumped distur-bance cannot be completely eliminated and it will producea transient component to the grid-injected current, which willdegrade the dynamic performance. To further improve thequality of grid-injected current, an online adaptive controlstrategy with the self-learning ability is finally proposed.

FIGURE 3. The schematic diagram of the APID controller.

C. ADAPTIVE PID CONTROLLER WITH THESELF-LEARNING ABILITYFig. 3 shows the schematic diagram of the APID controller,where e(k) is tracking error of grid-injected current at thek th sampling instant, which is transformed into three internalvariables, denoted χ1, χ2, and χ3, respectively; w1, w2, andw3 are the weight coefficients of the each internal variable;rule means the adaptive learning rule, which is utilized toadjust the weight coefficients online; K is the gain coefficientand ur (k) is the output of the APID controller.The tracking error and internal variables are defined as

follows:e(k) = ig,ref (k)− ig(k)− n(k)χ1(k) = 1e(k) = e(k)− e(k − 1)χ2(k) = e(k)χ3(k) = e(k)− 2e(k − 1)+ e(k − 2).

(22)

Correspondingly, ur (k) can be expressed as

ur (k) = ur (k − 1)+ K3∑i=1

wi(k)χi(k). (23)

Referring to optimal control theory, for pursuing excellenttracking performance, the quadratic performance index aboute(k) can be utilized. Thus, we insert a performance indexfunction into the self-learning algorithm of the APID con-troller. The discrete-type quadratic error function is defined as

E(k) =12

[ig,ref (k)− ig(k)− n(k)

]2=

12e(k)2. (24)

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In order to minimize the error function E(k), one canevaluate the following equation:

∂E(k)∂wi(k)

= −e(k)∂ig(k)∂ur (k)

∂ur (k)∂wi(k)

, i = 1, 2, 3. (25)

For the steepest descent algorithm, the changes in weightcoefficients can be deduced as

1w1(k) = w1(k)− w1(k − 1) = −ηP∂E(k)∂w1(k)

= ηPe(k)∂ig(k)∂ur (k)

∂ur (k)∂w1(k)

1w2(k) = w2(k)− w2(k − 1) = −ηI∂E(k)∂w2(k)

= ηI e(k)∂ig(k)∂ur (k)

∂ur (k)∂w2(k)

1w3(k) = w3(k)− w3(k − 1) = −ηD∂E(k)∂w3(k)

= ηDe(k)∂ig(k)∂ur (k)

∂ur (k)∂w3(k)

(26)

where ηP, ηI , ηD are the proportional, the integral and thedifferential learning rates, respectively.

It can be observed from (23) that ∂ur (k)∂wi(k)

(i = 1, 2, 3) is

equal to Kχi(k)(i = 1, 2, 3). However, ∂ig(k)∂ur (k)

in (26) is uncer-tain. According to the engineering experience, when tuningthe controller parameters, the e(k) and 1e(k) are expectedto quickly converge to zero, which indicates the controllerparameters tuning is associated with e(k) and 1e(k), so thispaper utilizes e(k)+1e(k) to replace ∂ig(k)

∂ur (k). The calculation

error can be compensated by the adjustable learning rates.Therefore, the adaptive learning rule is rewritten asw1(k) = w1(k − 1)+ ηPKχ1(k)e(k) [e(k)+1e(k)]w2(k) = w2(k − 1)+ ηIKχ2(k)e(k) [e(k)+1e(k)]w3(k) = w3(k − 1)+ ηDKχ3(k)e(k) [e(k)+1e(k)] .

(27)

Then, in order to ensure the convergence and robustness of(23) and (27), wi(k)(i = 1, 2, 3) is normalized as:

wi(k) =wi(k)

3∑j=1

∣∣wj(k)∣∣ , i = 1, 2, 3. (28)

Accordingly, ur (k) in (23) can be modified as follows:

ur (k) = ur (k − 1)+ K3∑i=1

wi(k)χi(k). (29)

Expanding (29) and comparingwith the incremental digitalPID regulator, one can easily obtain the equivalent propor-tional gain KP, the integral gain KI and the differential gainKD of the APID controller, i.e.

KP = Kw1(k)KI = Kw2(k)KD = Kw3(k).

(30)

From the theoretical analysis above, the equivalent gainsare able to be adjusted online to minimize grid-injected cur-rent error e(k) and ensure robustness by the adaptive learning

rule at each sampling instant. Hence, when the grid-injectedcurrent error occurs, caused by no matter the step-changedreference, parameters variation or disturbances, the perfor-mance index function E(k) will be optimized, which willsignificantly improve the control performance.

In addition, the proposed adaptive strategy is designedwith low computational burden. This strategy minimizes therequirements on the complexity and computational capacity.The relevant control parameters can be obtained by the trial-and-error method. And a rule of thumb to determine thevalues of K , ηP, ηI , and ηD is briefly summarized as follows:

1) Firstly, the values of initial weight coefficient wi(0)(i = 1, 2, 3) can be set arbitrarily.

2) Then, the gain coefficient K needs to be elaboratelyselected.K can be adjusted by the experimental results.The response of system becomes fast as the value of Kincreases, but with the increasing value, the systemmayoscillate even become unstable.

3) Finally, determine the learning rates ηP, ηI , and ηD.They are tuned to balance the overshoot and settlingtime.

IV. ROBUSTNESS ANALYSIS BASED-ON THESMALL-GAIN THEOREM UNDERMODEL UNCERTAINTIESIt should be pointed out that, in essence, the APID controlleris a nonlinear incremental digital PID regulator with the self-learning ability. To evaluate the robustness, we simplify itas a linear one, denoted Ce(z), whose equivalent gains areobtained under the nominal plant. This simplification forthe purpose of analysis is feasible, since once the steady-state is reached, the APID controller will be converted asincremental digital PID regulator.While, during the transient-state, the APID controller has stronger adaptability due tothe adjustable learning rule, which will achieve better per-formance than PID regulator. Hence, conclusions about therobustness drawn based-on PID regulator is also suitable forthe APID controller.

In this section, the potential influence of parameter per-turbation which may result in unexpected dynamics willbe investigated. For simplification, the output multiplicativeperturbation form in (15) is modified as follows:

PG(z) = [1+W (z)δ(z)]PGn(z) ‖δ(z)‖∞ ≤ 1. (31)

Then,W (z) can be derived for all frequency range as

|W (z)| ≥

∣∣∣∣ PG(z)PGn(z)− 1

∣∣∣∣ . (32)

The inequality (32) implies that the frequency responsecurve of W (z) is supposed to lie above the clustering fre-quency response curves of relative perturbation [PG(z)/PGn(z)− 1]. Assuming that the filter parameters L1, L2 candrift±25%, C can drift±5% around the nominal values [33]listed in Table 1, the clustering frequency response curvesof [PG(z)/PGn(z)− 1] can be easily obtained, which are

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TABLE 1. Parameters of the system.

FIGURE 4. The frequency responses of the weighting function andrelative perturbation.

depicted in Fig. 4. Taken the sufficient margins into consider-ation, a stable and minimum phase weighting functionW (z),which lies above the upper-bound frequency response curve,is selected as

W (z) =1.55z3 − 2.1480z2 + 0.6249z− 0.0251z3 − 0.6123z2 + 0.1381z− 0.0033

. (33)

Then, we define the sensitivity transfer function S(z) andcomplementary sensitivity transfer function T (z):{

S(z) =[1+ PGn(z)Keq(z)

]−1T (z) = 1− S(z)

(34)

where Keq(z) is the equivalent controller on the standardfeedback control structure and it is calculated as:

Keq(z) =Ce(z)+ Q(z)PGn(z)−1

1− Q(z). (35)

Therefore, the robustness condition based on the small-gain theorem [22], [25] can be obtained

‖W (z)T (z)‖∞=

∥∥∥∥W (z)PGn(z)Keq(z)

1+ PGn(z)Keq(z)

∥∥∥∥∞

≤ 1. (36)

FIGURE 5. The robustness analysis based on the small-gain theorem.

FIGURE 6. The transient responses of the grid-injected current in d- andq-axis under the step change in the active power and performancecomparison between the APID + DOB and PI + DOB.

For simplicity, (36) can be transformed to

|W (z)| ≤

∣∣∣∣1+ 11+ PGn(z)Keq(z)

∣∣∣∣ . (37)

From (37), the frequency response curve of W (z) isexpected to lie below the curve of the right-hand part. Bothof them are depicted in Fig. 5. It is easily observed that therobustness under model uncertainties can be guaranteed.

V. SIMULATION AND EXPERIMENTAL VERIFICATIONA. SIMULATION VERIFICATIONIn order to verify the effectiveness of the proposed strategy(APID + DOB), simulation tests on a 110 V/50 Hz/3 kWgrid-connected inverter with LCL filter are carried out inMATLAB/Simulink environment based-on the system andcurrent controller presented in Fig. 1, and the parameters uti-lized in simulations are listed in Table 1. In addition, to prove

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FIGURE 7. Robustness to the parameter uncertainties of the proposed strategy and performance comparison between theAPID + DOB and PI + DOB when the filter parameters vary in a large range. (a) The transient responses under the APID +DOB when L1 varies ±25% from the nominal value. (b) The transient responses under the APID + DOB when C varies ±5%from the nominal value. (c) The transient responses under the APID + DOB when L2 varies ±25% from the nominal value.(d) The performance comparison between the APID + DOB and PI + DOB under the parameter variation of −25% in L1,−5% in C and −25% in L2.

more advantageous performances of the proposed strategy,the comparative studies with the conventional PI controllerbased on DOB (PI + DOB) are also conducted. And for afair comparison, the optimal control parameters have beendesigned for PI + DOB.

1) TRANSIENT PERFORMANCE UNDER THE STEPCHANGE IN THE ACTIVE POWERThe objective of this simulation is to verify the transientperformance and tracking ability of the grid-injected currentunder the proposed strategy. To this end, the q-axis currentreference is set to zero, while the d-axis current reference isstepped up from 0 A to 6.43 A at t = 0 s, and then steppedup to 12.86 A at t = 0.06 s. The dynamic performances ofthe grid-injected current controlled by the APID + DOBand PI + DOB are shown in Fig. 6. It can be observedthat the transient response without any overshoot under theAPID + DOB is faster than that under PI + DOB. This can

be explained as follows: when the reference is suddenlychanged, the tracking error e(k) occurs in the grid-injectedcurrent, and the discrete-type quadratic error function E(k)defined in (24) will be optimized online by the steepestdescent algorithm and self-learning rule. Therefore, the actualgrid-injected current quickly converges to the reference andthe error e(k) asymptotically converges to zero. This confirmsthat the proposed strategy can achieve better transient per-formance and tracking ability compared with the traditionalmethod.

2) ROBUSTNESS TO THE PARAMETER VARIATIONTo evaluate the robustness to parametric variation of theproposed control strategy, a set of simulations under parame-ter mismatches have been carried out. Fig. 7(a)-(c) presentthe transient responses under the proposed strategy whenL1 varies ±25%, C varies ±5% and L2 varies ±25%from the nominal values, respectively. It can be seen that,

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compared with the transient performance under the nomi-nal values, the dynamics almost keep unchanged regardlessof the parameter variation. Thus the parameter variation inLCL filter brings little adverse effect on the grid-injectedcurrent under the proposed strategy, which sufficiently ver-ifies the theoretical analysis in Section IV. In order to fur-ther highlight this advantage, the performance comparisonbetween the APID + DOB and PI + DOB under parametervariation of −25% in L1, −5% in C and −25% in L2 ispresented in Fig. 7(d). Clearly, during the transient state,the ripple is observed in the grid-injected current controlledby PI + DOB. Because the effect of the parameter uncer-tainties cannot be completely eliminated due to limitationof DOB, and it produces a transient component to grid-injected current, which degrades transient performance. Thisfully matches the theoretical analysis in Section III. While,it should be noted that the proposed strategy has overcomethis issue, and it can remain the prominent performance underthe severe parameter variation.

3) ADAPTABILITY FOR THE GRID INDUCTANCE VARIATIONThis test is conducted to verify the ability of the proposedstrategy to adapt the grid inductance variation. Fig. 8(a) andFig. 8(b) depict the transient performances when the gridinductance varies from 0 mH to 4 mH under the PI + DOBand proposed strategy, respectively. In Fig. 8(a), the overshootand oscillation are easily observed in the grid-injected currentwhen the grid inductance increases. However, in Fig. 8(b),the proposed strategy can ensure fast convergence charac-teristics and precise tracking ability regardless of the gridinductance variation. As well known, the control bandwidthdecreases as the grid inductance increases. The proposedstrategy attenuates the influence of the grid inductance onthe performance. This is due to the inherent self-tuning capa-bility, which enables the controller to be redesigned in realtime to achieve optimal performances. Thus the proposedstrategy can achieve fast convergence properties with enoughrobustness and high control bandwidth.

4) ABILITY TO REJECT THE GRID DISTURBANCESIn this scenario, the effectiveness of the proposed strategywillbe verified under the grid disturbances, such as the unbal-anced grid voltage and the background harmonic voltage.

For a weak grid, the unbalanced grid condition commonlyoccurs due to the grid fault, highly unbalanced loads, etc.Fig. 9 depicts the simulated result of the grid-injected currentunder the unbalanced grid voltage which is emulated by 10%grid voltage sag in Phase B and 20% grid voltage sag inPhase C. From Fig. 9, it can be found that the grid-injectedcurrent has a perfect sinusoidal waveform with zero track-ing error and low total harmonic distortion (THD), whichdemonstrates the ability of the disturbance rejection under theunbalanced grid voltage.

In addition, there may exist large amounts of the back-ground harmonic voltage caused by the nonlinear loads inthe grid. In order to further enhance the harmonic rejection

FIGURE 8. The transient performances when the grid inductance variesfrom 0 mH to 4 mH under the different control strategies. (a) PI + DOB.(b) APID + DOB.

FIGURE 9. The simulated waveform of the grid-injected current under theunbalanced grid voltage.

ability, a harmonic compensator (HC regulator) [34] canbe added in parallel with the APID controller to atten-uate the grid-injected current distortions. Considering the

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FIGURE 10. The simulated waveform of the grid-injected current underthe distorted grid voltage.

FIGURE 11. The simulated waveform of the measured grid-injectedcurrent and the estimated grid-injected current.

high bandwidth characteristics of the proposed strategy,an 11th HC regulator is utilized:

Ghc(z) = Z

∑h=2,4,6,8,10

Khss2 + (ωgh)2

=

∑h=2,4,6,8,10

Khsin(ωghTs)

2ωghz2 − 1

z2 − 2z cos(ωghTs)+ 1

(38)

where Kh denote the resonant gains and all of them areselected as 800 in this paper. Fig. 10 shows the simulatedresult of the grid-injected current under the grid voltagedistorted by the 3rd 5th 7th 9th and 11th harmonics, whosemagnitudes with respect to the grid fundamental voltage are

FIGURE 12. The experimental setup.

FIGURE 13. The steady-state experimental results under nominal poweroperation. (a) PI + DOB. (b) APID + DOB.

1.2%, 2.8%, 1.3%, 2.4%, and 1.5%, respectively. It can beseen that the grid-injected current remains sinusoidal wave-form with zero steady-state error and low THD, which indi-cates that the proposed strategy can effectively reject up to11th harmonic under the condition of Lg = 4 mH.

5) ACCURACY OF THE STATE VARIABLE ESTIMATEDBY THE FULL-ORDER STATE OBSERVERAs analyzed in Section III, the accuracy of the state vari-ables estimated by the full-order state observer affects thecontrol performance. Fig. 11 presents the simulated wave-form of the measured grid-injected current and the estimated

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FIGURE 14. The dynamic waveforms of the grid-injected current underthe step change in the active power. (a) PI + DOB. (b) APID + DOB.

grid-injected current under a sudden change in the activepower. In Fig. 11, ig, ig_est and ig_err denote themeasured grid-injected current, the estimated grid-injected current and theestimation error, respectively. It can be observed that the esti-mation is consistent with themeasurement, and the estimationerror is significantly small, which can be neglected. Thus,the high accuracy can be guaranteed. similarly, the remainingstate variables i1 and uc can be precisely estimated, as well,which are omitted due to the length of this paper.

B. EXPERIMENTAL VERIFICATIONIn order to further verify the theoretical analysis, a 110 V/50 Hz/3 kW prototype is constructed based on dSPACEDS1202, Danfoss-FC320, Chroma 61830 and Chroma62150H-600S. The experimental setup is shown in Fig. 12and the experimental parameters coincide with those utilizedin simulations. To emulate the grid inductance, the externalinductors are utilized. In addition, the comparative experi-ments based on PI+DOB are also carried out to highlight thesuperiority of the proposed strategy. And all the experimentalwaveforms are captured from a Yokogawa DL 1640 digitaloscilloscope.

FIGURE 15. The dynamic performances under the parameter variation of−25% in L1, −5% in C and −25% in L2. (a) PI + DOB. (b) APID + DOB.

1) STEADY-STATE PERFORMANCE UNDERNOMINAL POWER OPERATIONIn this scenario, the d-axis current reference is set to 12.86 A,while the q-axis current reference is set to zero. The cor-responding steady-state performances under the PI + DOBand proposed strategy are shown in Fig. 13(a) and Fig. 13(b),respectively. It can be seen that both of them can achieve thezero steady-state error and unity power factor.

2) TRANSIENT RESPONSE UNDER THE STEPCHANGE IN THE ACTIVE POWERThe following experiments are performed to investigate thedisturbance rejection performance and the dynamic responseunder a sudden change in the active power. Fig. 14 showsthe dynamic waveforms of the grid-injected current whenthe q-axis current reference is kept equal to zero, while thed-axis current reference is suddenly stepped up from 6.43 Ato 12.86 A. It can be easily observed from Fig. 14(a) thatthe transient response with settling time about 8 ms underPI + DOB is slower than the proposed strategy in Fig. 14(b).As shown in Fig. 14(b), the grid-injected current referencesare well tracked and the proposed scheme provides excellent

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FIGURE 16. The dynamic performances when the grid inductance is 4 mH.(a) PI + DOB. (b) APID + DOB.

dynamic performance with settling time about 3 ms and with-out overshoot. This also demonstrates the proposed strategyhas stronger disturbance rejection capability under the suddenchange in active power.

3) ROBUSTNESS TO THE PARAMETER VARIATIONThese tests evaluate the robustness of the proposed methodby investigating the sensitivity of the grid-injected currentagainst the parameter uncertainties under a reference step-changed condition. To emulate the parameter uncertainties,the values of L1, L2, C are set with −25%, −25% and−5% variation from the nominal values. The dynamic per-formances with parameter uncertainties are shown in Fig. 15,revealing that the perturbed plant behaves as the same asthe nominal plant during the steady state, regardless of thefiltering deterioration due to the physical change of an LCLfilter. That is because the DOB eliminates the influence of theparameter variation. However, the transient response underPI+DOB (with settling time about 7 ms in Fig. 15(a)) is stillslower than that under the proposed strategy (with settlingtime about 3 ms in Fig. 15(b)). Especially, there are somecurrent ripples in Fig. 15(a) during the transient state, whichcompletely agrees with the simulation result, because the

FIGURE 17. The experimental waveforms of the grid-injected currentcontrolled by the proposed strategy under the unbalanced grid voltage.(a) The steady-state performance. (b) The transient performance.

parameter variation as the internal disturbance defined in (3)results in the transient component superimposed on the grid-injected current. Compared with PI + DOB, the APID fullyremoves the ripple by optimizing the grid-injected currenterror, which is consistent with the theoretical analysis.

This simultaneously verifies the effectiveness and robuststability of the proposed strategy.

4) ADAPTABILITY FOR THE GRID INDUCTANCE VARIATIONIn actual applications, the grid inductance may change in alarge range. Thus, these experiments are performed to high-light the benefits of the proposed method to maintain thestability and fast dynamics in the presence of the grid induc-tance. To emulate the grid inductance, the external inductors(Lg = 4 mH) are adopted. Fig. 16 shows the dynamicperformances when the d-axis current reference is steppeddown from 12.86A to 6.43A. In Fig. 16(a), the oscillation canbe easily observed under PI + DOB. In addition, the settlingtime and overshoot are up to 10 ms and 40%, respectively.The performance evidently deteriorates under the traditionalmethod because the control bandwidth significantly reducesdue to the great value of grid inductance. While, it can be

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FIGURE 18. The experimental waveforms of the grid-injected currentunder the distorted grid voltage and Lg = 4 mH when HC regulator isadopted. (a) PI + DOB. (b) APID + DOB.

obtained from Fig. 16(b) that the fast transient performanceagainst grid inductance variation is achieved with no oscil-lation and settling time about 4 ms. Different from the PIcontroller with fixed parameters, the APID can be automati-cally tuned online. Thus, the proposed strategy can achievestronger adaptability and more prominent transient perfor-mance with high control bandwidth characteristics when thegrid impedance varies widely.

5) ABILITY TO REJECT THE GRID DISTURBANCESIn applications, the grid-connected inverters are expected tonormally operate no matter under the balanced grid conditionor the unbalanced one. Fig. 17 depicts the experimental wave-forms of the grid-injected current controlled by the proposedstrategy under the unbalanced grid voltage. In Fig. 17(a),it is effectively demonstrated the ability to maintain the sinu-soidal grid-injected current with zero tracking error evenunder the severely unbalanced grid condition. Simultane-ously, Fig. 17(b) further reveals the prominent property of thefast performance recovery of the proposed strategy under thestep change in the active power.

In order to further prove the ability of the proposed strategyto reject the grid disturbances, a programmable AC source(Chroma 61830) is utilized to emulate the grid voltage whichis distorted by the 3rd, 5th, 7th, 9th and 11th harmonics. Themagnitudes of harmonics with respect to the grid fundamentalvoltage are 1.2%, 2.8%, 1.3%, 2.4% and 1.5%, respectively.Fig. 18(a) depicts the experimental waveform of the grid-injected current under Lg = 4mH, when an 11th HC regulatoris in parallel with the PI controller. The THD of the grid-injected current is 4.7%. As a contrast, Fig. 18(b) depictsthe experimental waveform of the grid-injected current underLg = 4 mH, when an 11th HC regulator is in parallel with theAPID controller. The THD is 1.8%, which is evidently lowerthan that under the traditional method. It can be seen that thegrid-injected current maintains perfect sinusoidal waveformwith low THD and the 11th harmonic has been successfullyattenuated. These tests further verify that the proposedstrategy can maintain the high-bandwidth characteristicsunder the great value of grid inductance and distorted gridvoltage.

VI. CONCLUSIONIn this paper, based on disturbance observer, an adaptivecurrent control strategy with the self-learning ability is pro-posed to overcome the lumped disturbance sensitivity issueand improve the control performance. The principle of theproposed strategy is deduced in detail and the robust stabilityis analyzed based on the small-gain theorem. By theoreticalanalysis, the following conclusions can be drawn.

1) Embedding a DOB into control loop can effectivelysuppress the influence of disturbances and maintain thecharacteristic of the nominal plant, which significantlyenhances the robust stability and ability of the distur-bance rejection.

2) Optimizing the grid-injected current error through thesteepest descent algorithm and self-learning rule, APIDcan achieve high control performance and trackingaccuracy, even when the filter parameters and gridinductance vary widely.

Simulations and experiments on a 110 V/50 Hz/3 kWLCL-filter-based three-phase grid-connected inverter proto-type verify the effectiveness of the proposed strategy.

REFERENCES[1] W. Wu, Y. Liu, Y. He, H. S.-H. Chung, M. Liserre, and F. Blaabjerg,

‘‘Damping methods for resonances caused by LCL-filter-based current-controlled grid-tied power inverters: An overview,’’ IEEE Trans. Ind.Electron., vol. 64, no. 9, pp. 7402–7413, Sep. 2017.

[2] D. Yang, X. Ruan, and H. Wu, ‘‘A real-time computation method withdual sampling mode to improve the current control performance of theLCL-type grid-connected inverter,’’ IEEE Trans. Ind. Electron., vol. 62,no. 7, pp. 4563–4572, Jul. 2015.

[3] W. Wu, Y. He, and F. Blaabjerg, ‘‘An LLCL power filter for single-phase grid-tied inverter,’’ IEEE Trans. Power Electron., vol. 27, no. 2,pp. 782–789, Feb. 2012.

[4] R. Errouissi, A. Al-Durra, and S. M. Muyeen, ‘‘Design and implemen-tation of a nonlinear PI predictive controller for a grid-tied photovoltaicinverter,’’ IEEE Trans. Ind. Electron., vol. 64, no. 2, pp. 1241–1250,Feb. 2017.

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J. Liu et al.: Disturbance Observer-Based Adaptive Current Control

[5] M. B. Saïd-Romdhane, M. W. Naouar, I. Slama-Belkhodja, andE. Monmasson, ‘‘Robust active damping methods for LCL filter-basedgrid-connected converters,’’ IEEE Trans. Power Electron., vol. 32, no. 9,pp. 6739–6750, Sep. 2017.

[6] T. Hornik and Q.-C. Zhong, ‘‘A current-control strategy for voltage-sourceinverters in microgrids based on H∞ and repetitive control,’’ IEEE Trans.Power Electron., vol. 26, no. 3, pp. 943–952, Mar. 2011.

[7] D. Chen, J. Zhang, and Z. Qian, ‘‘An improved repetitive control schemefor grid-connected inverter with frequency-adaptive capability,’’ IEEETrans. Ind. Electron., vol. 60, no. 2, pp. 814–823, Feb. 2013.

[8] X. Hao, X. Yang, T. Liu, L. Huang, and W. Chen, ‘‘A sliding-mode con-troller with multiresonant sliding surface for single-phase grid-connectedVSI with an LCL filter,’’ IEEE Trans. Power Electron., vol. 28, no. 5,pp. 2259–2268, May 2013.

[9] R. P. Vieira, L. T.Martins, J. R.Massing, andM. Stefanello, ‘‘Slidingmodecontroller in a multiloop framework for a grid-connected VSI with LCLfilter,’’ IEEETrans. Ind. Electron., vol. 65, no. 6, pp. 4714–4723, Jun. 2018.

[10] Y. He, H. S.-H. Chung, C. N.-M. Ho, and W. Wu, ‘‘Use of boundarycontrol with second-order switching surface to reduce the system orderfor deadbeat controller in grid-connected inverter,’’ IEEE Trans. PowerElectron., vol. 31, no. 3, pp. 2638–2653, Mar. 2016,

[11] Y. He, H. S.-H. Chung, C. N.-M. Ho, and W. Wu, ‘‘Modified cascadedboundary-deadbeat control for a virtually-grounded three-Phase grid-connected inverter with LCL filter,’’ IEEE Trans. Power Electron., vol. 32,no. 10, pp. 8163–8180, Oct. 2017.

[12] N. Panten, N. Hoffmann, and F. W. Fuchs, ‘‘Finite control set modelpredictive current control for grid-connected voltage-source converterswith LCL filters: A study based on different state feedbacks,’’ IEEE Trans.Power Electron., vol. 31, no. 7, pp. 5189–5200, Jul. 2016.

[13] S. Kwak and S. K. Mun, ‘‘Model predictive control methods to reducecommon-mode voltage for three-phase voltage source inverters,’’ IEEETrans. Power Electron., vol. 30, no. 9, pp. 5019–5035, Sep. 2015.

[14] Y. Liu, W. Wu, Y. He, Z. Lin, F. Blaabjerg, and H. S. H. Chung,‘‘An efficient and robust hybrid damper for LCL- or LLCL-based grid-tied inverter with strong grid-side harmonic voltage effect rejection,’’ IEEETrans. Ind. Electron., vol. 63, no. 2, pp. 926–936, Feb. 2016.

[15] W. Chen, J. Yang, L. Guo, and S. Li, ‘‘Disturbance-observer-based controland related methods-an overview,’’ IEEE Trans. Ind. Electron., vol. 63,no. 2, pp. 1083–1095, Feb. 2016.

[16] Y. Ye and Y. Xiong, ‘‘UDE-based current control strategy for LCCL-typegrid-tied inverters,’’ IEEE Trans. Ind. Electron., vol. 65, no. 5,pp. 4061–4069, May 2018.

[17] B. Wang, Y. Xu, Z. Shen, J. Zou, C. Li, and H. Liu, ‘‘Current control ofgrid-connected inverter with LCL filter based on extended-state observerestimations using single sensor and achieving improved robust observationdynamics,’’ IEEE Trans. Ind. Electron., vol. 64, no. 7, pp. 5428–5439,Jul. 2017.

[18] A. Benrabah, D. Xu, and Z. Gao, ‘‘Active disturbance rejection controlof LCL-filtered grid-connected inverter using Padé approximation,’’ IEEETrans. Ind. Appl., vol. 54, no. 6, pp. 6179–6189, Nov./Dec. 2018.

[19] G. Lou, W. Gu, J. Wang, J. Wang, and B. Gu, ‘‘A unified control schemebased on a disturbance observer for seamless transition operation ofinverter-interfaced distributed generation,’’ IEEE Trans. Smart Grid, vol. 9,no. 5, pp. 5444–5454, Sep. 2018.

[20] J. Han, ‘‘From PID to active disturbance rejection control,’’ IEEE Trans.Ind. Electron., vol. 56, no. 3, pp. 900–906, Mar. 2009.

[21] M. Elkayam, S. Kolesnik, and A. Kuperman, ‘‘Guidelines to classicalfrequency-domain disturbance observer redesign for enhanced rejectionof periodic uncertainties and disturbances,’’ IEEE Trans. Power Electron.,vol. 34, no. 4, pp. 3986–3995, Apr. 2019.

[22] H. Muramatsu and S. Katsura, ‘‘An adaptive periodic-disturbance observerfor periodic-disturbance suppression,’’ IEEE Trans. Ind. Informat., vol. 14,no. 10, pp. 4446–4456, Oct. 2018.

[23] J. Yang, W.-H. Chen, S. Li, L. Guo, and Y. Yan, ‘‘Disturbance/uncertaintyestimation and attenuation techniques in PMSM drives—A survey,’’ IEEETrans. Ind. Electron., vol. 64, no. 4, pp. 3273–3285, Apr. 2017.

[24] R. Errouissi and A. Al-Durra, ‘‘Design of PI controller together with activedamping for grid-tied LCL-filter systems using disturbance-observer-based control approach,’’ IEEE Trans. Ind. Appl., vol. 54, no. 4,pp. 3820–3831, Jul./Aug. 2018.

[25] Y. Wu and Y. Ye, ‘‘Internal model-based disturbance observer with appli-cation to CVCF PWM inverter,’’ IEEE Trans. Ind. Electron., vol. 65, no. 7,pp. 5743–5753, Jul. 2018.

[26] H. T. Nguyen and J.-W. Jung, ‘‘Disturbance-rejection-based model pre-dictive control: Flexible-mode design with a modulator for three-phaseinverters,’’ IEEE Trans. Ind. Electron., vol. 65, no. 4, pp. 2893–2903,Apr. 2018.

[27] H. Yang, Y. Zhang, J. Liang, J. Liu, N. Zhang, and P. D. Walker,‘‘Robust deadbeat predictive power control with a discrete-time distur-bance observer for PWM rectifiers under unbalanced grid conditions,’’IEEE Trans. Power Electron., vol. 34, no. 1, pp. 287–300, Jan. 2019.

[28] Y. Zhu and J. Fei, ‘‘Disturbance observer based fuzzy sliding mode controlof PV grid connected inverter,’’ IEEE Access, vol. 6, pp. 21202–21211,2018.

[29] C. A. Busada, S. G. Jorge, and J. A. Solsona, ‘‘Full-state feedback Equiva-lent controller for active damping in LCL-filtered grid-connected invertersusing a reduced number of sensors,’’ IEEE Trans. Ind. Electron., vol. 62,no. 10, pp. 5993–6002, Oct. 2015.

[30] J. Kukkola and M. Hinkkanen, ‘‘Observer-based state-space current con-trol for a three-phase grid-connected converter equipped with an LCL fil-ter,’’ IEEE Trans. Ind. Appl., vol. 50, no. 4, pp. 2700–2709, Jul./Aug. 2014.

[31] J. Kukkola, M. Hinkkanen, and K. Zenger, ‘‘Observer-based state-spacecurrent controller for a grid converter equipped with an LCL filter: Ana-lytical method for direct discrete-time design,’’ IEEE Trans. Ind. Appl.,vol. 51, no. 5, pp. 4079–4090, Sep./Oct. 2015.

[32] J. S. Lim, C. Park, J. Han, and Y. I. Lee, ‘‘Robust tracking control ofa three-phase DC-AC inverter for UPS applications,’’ IEEE Trans. Ind.Electron., vol. 61, no. 8, pp. 4142–4151, Aug. 2014.

[33] Z. Zhang, W. Wu, Z. Shuai, X. Wang, A. Luo, H. S.-H. Chung, andF. Blaabjerg ‘‘Principle and robust impedance-based design of grid-tiedinverter with LLCL-filter under wide variation of grid-reactance,’’ IEEETrans. Power Electron., vol. 34, no. 5, pp. 4362–4374, May 2019.

[34] Y. Tang, P. C. Loh, P. Wang, F. H. Choo, and F. Gao, ‘‘Exploring inher-ent damping characteristic of LCL-filters for three-phase grid-connectedvoltage source inverters,’’ IEEE Trans. Power Electron., vol. 27, no. 3,pp. 1433–1443, Mar. 2012.

JIAHAO LIU was born in Henan, China, in 1993.He received the B.S. degree in electrical engi-neering from the Henan University of Scienceand Technology, Henan, China, in 2017. He iscurrently pursuing the M.S. degree in electricalengineering from Shanghai Maritime University,Shanghai, China.

His current research interests include digitalcontrol techniques of power converters and renew-able energy generation system.

WEIMIN WU (M’16) received the Ph.D. degreein electrical engineering from the College ofElectrical Engineering, Zhejiang University,Hangzhou, China, in 2005.

He was a Research Engineer with the DeltaPower Electronic Center (DPEC), Shanghai, fromJuly 2005 to June 2006. Since July 2006, he hasbeen a FacultyMember at ShanghaiMaritimeUni-versity, where he is currently a Full Professor withthe Department of Electrical Engineering. He was

a Visiting Professor with the Center for Power Electronics Systems (CPES),Virginia Polytechnic Institute and State University, Blacksburg, VA, USA,fromSeptember 2008 toMarch 2009. FromNovember 2011 to January 2014,he was also a Visiting Professor with the Department of Energy Technology,Aalborg University, Demark, working at the Center of Reliable PowerElectronics (CORPE). He has coauthored over 100 papers and holds eightpatents. His areas of interests include power converters for renewable energysystems, power quality, smart grid, and energy storage technology. He servesas an Associate Editor for the IEEE TRANSACTIONS ON INDUSTRY ELECTRONICS.

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HENRY SHU-HUNG CHUNG (M’95–SM’03–F’16) received the B.Eng. and Ph.D. degrees inelectrical engineering from the Hong Kong Poly-technic University, Hong Kong, in 1991 and 1994,respectively. Since 1995, he has been with the CityUniversity of Hong Kong, Hong Kong, where he iscurrently a Professor with the Department of Elec-tronic Engineering, and the Director of the Cen-ter for Smart Energy Conversion and UtilizationResearch. His current research interests include

renewable energy conversion technologies, lighting technologies, smart gridtechnologies, and computational intelligence for power electronic systems.He has edited 1 book, authored 8 research book chapters, and over 400 tech-nical papers, including 180 refereed journal papers in his research areas,and holds 47 patents. He was the Chair of the Technical Committee of theHigh-Performance and Emerging Technologies, the IEEE Power ElectronicsSociety (2010–2014). He has received numerous industrial awards for hisinvented energy-saving technologies. He is currently the Editor-in-Chief ofthe IEEE POWER ELECTRONICS LETTERS and an Associate Editor of the IEEETRANSACTIONS ON POWER ELECTRONICS and the IEEE JOURNAL OF EMERGING AND

SELECTED TOPICS IN POWER ELECTRONICS.

FREDE BLAABJERG (S’86–M’88–SM’97–F’03)was with ABB-Scandia, Randers, Denmark, from1987 to 1988. From 1988 to 1992, he was a Ph.D.student in electrical engineering with AalborgUniversity, Aalborg, Denmark. He became anAssistant Professor (1992), an Associate Profes-sor (1996), and a Full Professor of power electron-ics and drives (1998). Since 2017, he has been aVillum Investigator. His current research interestsinclude power electronics and its applications such

as in wind turbines, PV systems, reliability, harmonics, and adjustable speeddrives. He has published more than 450 journal papers in the fields of powerelectronics and its applications. He is the coauthor of two monographs andthe editor of six books in power electronics and its applications. He wasthe Editor-in-Chief of the IEEE TRANSACTIONS ON POWER ELECTRONICS, from2006 to 2012.

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