An LQR-Based Controller Design for an LCL-Filtered Grid ... - MDPI

energies Article

An LQR-Based Controller Design for an LCL-Filtered Grid-Connected Inverter in Discrete-Time State-Space under Distorted Grid Environment Thuy Vi Tran, Seung-Jin Yoon and Kyeong-Hwa Kim *

ID

Department of Electrical and Information Engineering, Seoul National University of Science and Technology, 232 Gongneung-ro, Nowon-gu, Seoul 01811, Korea; [email protected] (T.V.T.); [email protected] (S.-J.Y.) * Correspondence: [email protected]; Tel.: +82-2-970-6406; Fax: +82-2-978-2754 Received: 18 July 2018; Accepted: 6 August 2018; Published: 8 August 2018







Abstract: In order to alleviate the negative impacts of harmonically distorted grid conditions on inverters, this paper presents a linear quadratic regulator (LQR)-based current control design for an inductive-capacitive-inductive (LCL)-filtered grid-connected inverter. The proposed control scheme is constructed based on the internal model (IM) principle in which a full-state feedback controller is used for the purpose of stabilization and the integral terms as well as resonant terms are augmented into a control structure for the reference tracking and harmonic compensation, respectively. Additionally, the proposed scheme is implemented in the synchronous reference frame (SRF) to take advantage of the simultaneous compensation for both the negative and positive sequence harmonics by one resonant term. Since this leads to the decrease of necessary resonant terms by half, the computation effort of the controller can be reduced. With regard to the full-state feedback control approach for the LCL-filtered grid connected inverter, additional sensing devices are normally required to measure all of the system state variables. However, this causes a complexity in hardware and high implementation cost for measurement devices. To overcome this challenge, this paper presents a discrete-time current full-state observer that uses only the information from the control input, grid-side current sensor, and grid voltage sensor to estimate all of the system state variables with a high precision. Finally, an optimal linear quadratic control approach is introduced for the purpose of choosing optimal feedback gains, systematically, for both the controller and full-state observer. The simulation and experimental results are presented to prove the effectiveness and validity of the proposed control scheme. Keywords: distorted grid; digital signal processor (DSP) TMS320F28335; grid-connected inverter; internal model; linear quadratic regulator; LCL filter

1. Introduction The increasing interest in grid-connected voltage source inverters (VSI) for renewable energy conversion systems poses a challenge to the current control design of inverter systems. In particular, the current control scheme is responsible for a high quality of injected current to meet the power quality standard of distributed generation such as the IEEE-519 in USA or the IEC 61000-3-2 in Europe [1] even under harmonically distorted grid voltages. Additionally, the filter connected between the utility grid and VSI plays an essential role to attenuate the current in high switching frequency from the pulse width modulated inverter. In general, LCL filters are regarded as being satisfactory for three-phase voltage source grid-connected inverters because they provide a better grid-side current quality with lower costs and a smaller physical size when compared to the conventional L filters. Nevertheless, the disadvantages of using LCL filters include a high-order system and the resonance behavior. As a

Energies 2018, 11, 2062; doi:10.3390/en11082062

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result, the current control strategies of LCL-filtered inverters are more difficult and complex to stabilize the system. There are two methods to damp the resonance frequency of the LCL filter: passive damping using additional physical components on LCL circuits, and active damping implemented by modifying the control algorithm. A large number of studies in literature address the controller design in both ways. In [2], a passive resistor is added in series with the filter capacitance with the aim of attenuating the peak of LCL filter resonance. However, the main drawback of this method is that it causes extra losses through heat dissipation and overall reduction of the system efficiency. On the other hand, the active damping approaches are generally preferable and used quite commonly due to the fact that they stabilize the system without increasing the losses. An active damping realized by virtual resistance based on the capacitance current feedback is presented in [3–5]. In particular, Jia. Y et al. [3] presents the capacitance current feedback active damping implemented via a proportional gain of the feedback signal. The stability enhancement and robustness against distorted grid voltages are also discussed in this work. Similarly, the capacitor current feedback loop of the LCL filter is implemented to improve both the damping characteristic and inner-loop stability of a hierarchical control structure [4]. Furthermore, an H-infinity repetitive controller in [5] demonstrates a better performance and efficiency of the inverter by introducing the feedback of capacitor current to damp the resonance. Even though the stabilization can be achieved, those schemes increase the complexity and cost in hardware caused by extra sensors to obtain capacitor currents. As in other approaches, the studies in [6–8] present a state-space control scheme which provides a convenient and straightforward way for resonance damping. In order to avoid extra sensing devices, a full-state observer is also presented in these works, whereupon the number of sensors used in the controller is compatible with the design of the conventional L filter case. Aside from the resonance of the LCL filter, the issue of grid voltage distortion should be taken into account in a current controller design for a grid-connected inverter. Thus, the adoption of a proportional-resonant (PR) controller in the control strategy was studied widely in both classical and modern control approaches to improve the power quality. Conventionally, the proportional-integral (PI) controllers in rotating frame and the resonant controller in the stationary frame have been studied in detail in [9], which demonstrates that an equivalent control performance can be achieved by these controllers. The research work in [10] uses multiple PI controllers that are implemented in respective reference frames rotating with the fundamental and harmonic frequencies to achieve control objectives such as reference tracking and harmonic compensation. Another approach uses a PR scheme and harmonic compensation control performing at particular frequencies in the stationary frame to restrain the disturbance caused by the distorted grid voltage [3,11–15]. However, since these approaches require two regulators to compensate both the negative and positive sequences, several regulators might be necessary in the control scheme to meet the required total harmonic distortion (THD) performance, which often leads to a significantly heavy computational burden. In order to reduce the complexity of the digital implementation, the PI and resonant (PI-RES) current control scheme constructed in the synchronous reference frame (SRF) has been studied in [16–18] to achieve multiple harmonic compensation with the number of resonant controllers reduced by half. In particular, PI-RES control schemes in the SRF are proposed for active power filters [18] or a three-phase grid converter system [16,17] for the purpose of compensating multiple harmonics. Aside from the resonant control scheme, an H-infinity repetitive control approach was studied in [6] which presented the robustness against the system parameter variations of the LCL-type grid-connected inverter. In addition, Fu. X et al. investigated a neural network (NN)-based vector control approach for single-phase grid-connected converters to achieve an improved control performance without any damping method for LCL filter [19]. In addition to the typical control structure based on the transfer function design [3,13–15], the internal model (IM) principle proposed by Francis and Woham [20] has been applied to design controllers such as the PI and PR in the state-space. In this regard, several studies considered the IM approach to integrate control terms into the current control structure [17,21–23]. However, such

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a multivariable design approach also poses a challenge to an appropriate selection of controller gains to stabilize the system as well as to ensure both the desired steady-state and transient-state performances. The current controller design using the direct pole placement method in the state-space has been accomplished in the continuous-time domain [7], as well as in the discrete-time domain [8,21]. Although the controller gains can be chosen based on the open-loop poles and the desired dynamics of the closed-loop system, the pole placement method is not an attractive way in a complex system due to the laborious process to select a large number of state-feedback and controller gains. On the other hand, the studies in [12,17,22] solve the linear matric inequalities derived from the stability condition in the Lyapunov sense to obtain the controller gains systematically. In the same vein, an optimal solution based on the linear quadratic regulator (LQR) has been presented in [23], which optimizes the cost function to calculate the optimal gains of the system. In regard to the solution to reduce the number of needed sensors while still meeting the requirement on the availability of system state variables for full-state feedback controller, many types of observer have been studied to estimate the system state variables by using only the information from the system input and output signals. In particular, the research works in [7,8,17,21] employ the prediction-type full-state observer, while the study in [24] presents the reduced-order observer. However, there are not many studies regarding the current full-state observer in the discrete-time domain and its performance applied to three-phase LCL-filtered grid-connected inverters, even though it is known to have the advantages that the estimated value is based on the current measurement in comparison with the prediction-type observer and the impact of possible noise from the system output signals can be avoided. This paper presents a control design methodology for a grid-connected inverter with an LCL filter in the discrete-time state-space, where the current control design is accomplished by a full-state feedback control after incorporating the integral and resonant terms into control structure. In this proposed scheme, the controller is implemented in the SRF in order that the integral control on the DC quantities can ensure zero steady-state current error. Furthermore, four harmonic components in phase currents at the 5th, 7th, 11th and 13th order can be effectively compensated at the same time with only two resonant terms at 6th and 12th order. With an aim of reducing the total number of sensors required for the control of LCL-filtered grid-connected inverters, a current full-state observer is presented in the discrete-time domain with excellent estimation capability. The augmentation of the resonant terms as well as the integral term into an inverter system model causes an increase in the number of feedback gains to be selected. To choose the feedback gains in a systematic way, the optimal linear quadratic control approach is adopted in this paper. By minimizing the cost function to satisfy the stability and robustness requirements of the system, the overall system can be designed in an effective and straightforward way. As a result, both the reference tracking and harmonic compensation capability can be achieved in an LCL-filtered grid-connected inverter with an LQR approach by using only the grid-side current sensor and grid voltage sensor. To demonstrate the effectiveness and validity of the proposed control scheme, the PSIM software-based simulation (9.1, Powersim, Rockville, MD, USA) and experiments have been carried out comprehensively by using a three-phase 2 kVA prototype grid-connected inverter under adverse grid conditions. 2. State-Space Description of a Grid-Connected Inverter with LCL Filter 2.1. Modeling of a Grid-Connected Inverter with LCL Filter In the SRF, three-phase variables “abc” are transformed into two orthogonal DC phasor quantities “dq” by means of the Park’s transformation as follows: 

  cos(θ ) fq   2  f d  =  sin(θ ) 3 1/2 f0

cos(θ − 2π/3) sin(θ − 2π/3) 1/2

  cos(θ + 2π/3) fa   sin(θ + 2π/3)  f b  1/2 fc

(1)

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4 of 28 cos(𝜃) cos(𝜃 − 2π⁄3) cos(𝜃 + 2π⁄3) 𝑓 𝑓𝑞 𝑎 2 [𝑓𝑑 ] = sin(𝜃) sin(𝜃 − 2π⁄3) sin(𝜃 + 2π⁄3) [𝑓𝑏 ] (1) 3 𝑓𝑐 angle. 𝑓0 where f denotes the variable being 1⁄ transformed and 1⁄ θ is the rotating 1⁄ phasor [ 2 2 2 ] Figure 1 shows a configuration of a three-phase grid-connected inverter with an LCL filter, the variable being transformed and 𝜃 is the rotating phasor angle. inwhere which𝑓 Vdenotes DC denotes the DC-link voltage, R1 , R2 , L1 , and L2 are the filter resistances and filter Figure 1 shows a configuration of filter a three-phase grid-connected with an LCLmodel filter, of in the inductances, respectively, and C is the capacitance. In the SRF,inverter the mathematical which 𝑉 denotes the DC-link voltage, 𝑅 , 𝑅 , 𝐿 , and 𝐿 are the filter resistances and filter 𝐷𝐶 2 inverter system can be expressed as follows:1 2 1 inductances, respectively, and 𝐶 is the filter capacitance. In the SRF, the mathematical model of the . inverter system can be expressed as R2 q 1 q 1 q follows: i2 = − i2 − ωi2d + vc − eq (2) L L L 𝑅22 𝑞 1 2 𝑞 1 2𝑞 𝑞̇ 𝑑 𝑖2 = − 𝑖2 − 𝜔𝑖2 + 𝑣𝑐 − 𝑒 (2) 𝐿2 𝐿2 𝐿 . R2 d 1 d 21 d q d i2 = −𝑅 i2 + ωi2 +1 vc −1 e (3) 2 𝑞 2 𝑐𝑑 − L𝑒2𝑑 𝑖2𝑑̇ = − L2𝑖2𝑑 + 𝜔𝑖2 + L𝑣 (3) 𝐿2 𝐿2 𝐿 . 1 q 21 q R1 q q d (4) i1𝑞 = −𝑅1 i𝑞1 − ωi1 − 1 v𝑞 c +1 𝑞vi 1 𝑐 + L𝑣1𝑖 𝑖1̇ = − L1𝑖1 − 𝜔𝑖1𝑑 − L𝑣 (4) 𝐿1 𝐿1 𝐿1 . 1 1 R q (5) i1d𝑑̇ = −𝑅11 i𝑑1d + ωi𝑞1 − 1 v𝑑 dc +1 𝑑vid L 𝑖1 = − L1𝑖1 + 𝜔𝑖1 − L𝑣 + 𝑣1𝑖 (5) 1𝑐 𝐿 𝐿1 𝐿1 . 1 q d 11 q 1 1 q v𝑞c = −ωv − 𝑞i + 𝑞i1 (6) (6) 𝑣𝑐̇ = −𝜔𝑣𝑐𝑑c − C𝑖2 2+ C 𝑖1 𝐶 𝐶 . 1 1 q v𝑑̇dc = ωv𝑞c −1 𝑑i2d +1 𝑑i1d (7) (7) 𝑣𝑐 = 𝜔𝑣𝑐 − C𝑖2 + C𝑖1 𝐶 𝐶 where the superscript “q” and “d” denote the q-axis and d-axis variables, respectively, ω is the angular where the superscript “q” and “d” denote the q-axis and d-axis variables, respectively, 𝜔 is the frequency of the grid voltage, i is the inverter-side current, i is the grid-side current, vc iscurrent, the capacitor angular frequency of the grid1voltage, 𝑖1 is the inverter-side2 current, 𝑖2 is the grid-side 𝑣𝑐 voltage, e is the grid voltage, and v is the inverter output voltage. i voltage, and 𝑣 is the inverter output voltage. is the capacitor voltage, 𝑒 is the grid Energies 2018, 11, 2062

𝑖

𝑖1

VDC

+ _

𝑣𝑖𝑎 𝑣𝑖𝑏 𝑣𝑖𝑐

L1

𝑖2 R1

C

L2

R2

Grid e

𝑣𝑐

Figure 1. Configuration of a grid-connected inverter with inductive-capacitive-inductive (LCL) filter. Figure 1. Configuration of a grid-connected inverter with inductive-capacitive-inductive (LCL) filter.

From Equations (2) to (7), the continuous-time representation of inverter system can be From Equations expressed in the SRF(2) as:to (7), the continuous-time representation of inverter system can be expressed in the SRF as: . (8) x𝐱̇ ((𝑡) t) = = 𝐀𝐱(𝑡) Ax(t)++𝐁𝐮(𝑡) Bu(t)++𝐃𝐞(𝑡) De(t) (8) (9) 𝐲(𝑡) = 𝐂𝐱(𝑡) y(t) = Cx(t) (9) 𝑇 𝑇 h 2𝑞 𝑖2𝑑 𝑖1𝑞 𝑖1𝑑 𝑣𝑐𝑞 𝑣𝑐𝑑 i] T is the system state vector, 𝐮 = [𝑣𝑖𝑞 𝑣𝑖𝑑 ] is the system input vector, where 𝐱 = [𝑖 q q q q d T 𝑑 ]𝑇 i i d i i d v c v d where is the system vector, u =𝐀,[v𝐁,i v𝐂, is the system input 𝐞 = [𝑒x𝑞 𝑒= is vector, and thestate system matrices 𝐃 are expressed as:vector, c 2 1grid 1 voltage 2 the i ]and

e = [eq ed ] T is the grid voltage vector, and the system matrices A,1/𝐿 B,2C, and −𝜔 0 0 D are expressed as: 0 −𝑅2 /𝐿 2 −𝑅 /𝐿 0 0 1/𝐿 0 2 2 𝜔 2   0 −𝜔0 −1/𝐿 − R /L −ω 1/L 0 −𝑅10/𝐿1 1 2 0 (10) 𝐀= 2 02 −𝑅1 /𝐿01   𝜔0 0 0 −1/𝐿1/L 1 ω0 − R20/L2 2   0 0 1/𝐶 −1/𝐶   0−1/L −𝜔 0 0 0 − R1 /L1 1/𝐶 −ω   1 −1/𝐶 0 0 ] A= [ 0 (10) 𝜔   0 0 ω − R1 /L1 0 −1/L1     −1/C 0 1/C 0 0 −ω  0 −1/C 0 1/C ω 0

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  0 0    0 0       1/L 0    1 B= , D =     0 1/L 1 Energies  2018, 11, x FOR PEERREVIEW     0 0 

−1/L2 0 0 −1/L2 0 0 0 0 0 0 0 −1/𝐿2 0 0

   "   , C =   

1 0

0 1

0 0

0 0

0 0

# 0 05 of 27

(11)

0 0 0 0 −1/𝐿2 0 1/𝐿1 0 0 , C = [1 0 0 0 0 0 ] 0 𝐁= , 𝐃= (11) 2.2. System Model Discretization 0 0 0 1 0 0 0 0 0 1/𝐿1 0 0 0 0 For a digital implementation, the discretized model [ 0 [ 0 0 ] of inverter system is obtained by using the 0 ]

0

0

zero-order hold with the sampling time Ts as [25]: 2.2. System Model Discretization

x(k + 1) = Ad x(k ) + Bd u(k) + Dd e(k)

For a digital implementation, the discretized model of inverter system is obtained by using the zero-order hold with the sampling time 𝑇𝑠 as [25]:

y ( k ) = Cd x ( k )

𝐱(𝑘 + 1) = 𝐀𝐝 𝐱(𝑘) + 𝐁𝐝 𝐮(𝑘) + 𝐃𝐝 𝐞(𝑘)

(12)

𝐲(𝑘) = 𝐂𝐝 𝐱(𝑘)

(13)

where the matrices Ad , Bd , Cd , and Dd can be calculated as follows: ATs as A2follows: Ts2 where the matrices 𝐀𝐝 , 𝐁𝐝 , 𝐂𝐝 , and 𝐃𝐝ATcan be calculated Ad = e s = I + + +...

(13)

(14)

1! 𝐀2 𝑇2! 2 𝐀𝑇 𝑠 𝑠 𝐀𝐝 = 𝑒 =𝐈+ + +⋯ 1! 2! −1

(14)

𝐁𝐝 = 𝐀−1 (𝐀𝐝 − 𝐈)𝐁, 𝐂𝐝 = 𝐂

(15)

𝐃𝐝 = 𝐀 (𝐀𝐝 − 𝐈)𝐃

(16)

𝐀𝑇𝑠

Bd = A

(12)

(Ad − I)B, Cd = C

1 Dd = A − −1 (Ad − I)D

(15) (16)

3. Proposed Control Scheme

3. Proposed Control Scheme

Figure 2 represents the proposed control scheme for a three-phase inverter connected with the utility Figure 2 represents the proposed control scheme for a three-phase inverter connected with the grid through an LCL filter. controlled by the proposed currentcurrent controller through the utility grid through an The LCL inverter filter. Theisinverter is controlled by the proposed controller space vector pulse width modulation (PWM). Also, the phase-locked loop (PLL) scheme is implemented through the space vector pulse width modulation (PWM). Also, the phase-locked loop (PLL) scheme to generate the phase ofthe thephase gridangle voltage forgrid thevoltage grid synchronization process. process. The proposed is implemented to angle generate of the for the grid synchronization proposed control scheme by an integral-resonant state feedbackand controller and full-state a controlThe scheme is constructed by is anconstructed integral-resonant state feedback controller a current current full-state observer in the discrete-time domain with only the measurements of the grid-side observer in the discrete-time domain with only the measurements of the grid-side currents and grid currents and grid voltages. Besides, the current full-state observer is also implemented by using voltages. Besides, the current full-state observer is also implemented by using LCL-filter inverter model LCL-filter inverter model to estimate the system state variables 𝐱 from the control input 𝐮 and to estimate the system𝐲.state from are theused control input u andfeedback system outputs estimated system outputs Thosevariables estimatedxstates for the full-state controllery.toThose stabilize states are used for the full-state feedback controller to stabilize the whole system. the whole system. i1 𝑣𝑖𝑎 VDC

+

i2

L1

R1

Grid

R2



𝑣𝑖𝑏



_ 𝑣𝑖𝑐



C

abc qd

𝑣𝑐

abc qd

𝑞

𝑖 2 𝑖 2𝑑

6 Space vector PWM 𝜃

L2

𝑣𝑖𝑞

𝑖 𝑞1 𝑖1𝑞 𝑣𝑖𝑑

Resonantintegral state feedback control

𝑞

𝑞

𝑖 2 𝑖2

𝑣𝑐𝑞 𝑣𝑐𝑑

𝑞

LQR-based full-state observer

𝑖 2 𝑖2𝑑

PLL

𝑒𝑞 𝑒𝑑

2. Block diagram of the proposed integral-resonant state feedback current control scheme FigureFigure 2. Block diagram of the proposed integral-resonant state feedback current control scheme with observer. with observer.

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3.1. Internal Model-Based Current Controller To ensure asymptotic reference tracking as well as disturbance rejection for the harmonics in the orders of 6th and 12th in the SRF, the integral-resonant state feedback control is constructed by augmenting the integral and resonant terms in the discrete-time state-space based on the internal model principle. An integral term in the state-space is expressed as [21,25]:  . " q # " # q q x t ( )  .i  = APc xi (t) + BPc ε (t) xid (t) εd (t) xid (t) 

(17)

h iT q∗ where ε = [εq εd ] T = r − Cd x is the current error vector, r = i2 i2d∗ is the reference current vector, " # " # 0 0 1 0 and APc = , BPc = . 0 0 0 1 Similarly, resonant terms for the q-axis and d-axis in the state-space are expressed as [17,26]:       

.q

δ1i (t) .q

δ2i (t) .d

δ1i (t) .d



       = Arci    

δ2i (t)  where Arci =

   

0 −(iω )2

q



δ1i (t) q δ2i (t) d (t) δ1i d (t) δ2i

   + Brci 

"

εq (t) εd (t)

# for i = 6, 12



1 −2ξ (iω ) 0 −(iω )2

1 −2ξ (iω )



    , Brci =   

0 1 0 0

(18)

0 0 0 1

   , and ξ is a 

damping factor. As the damping ratio ξ is increased, it is well known that the magnitude of the frequency response at the resonant frequency is reduced, and the frequency response is flattened. The purpose of introducing the resonant terms is to effectively compensate the grid-side current harmonics caused by distorted grid voltages with the high gain at selective frequencies. Moreover, the proposed scheme can ensure the tracking performance of grid-side currents by adopting integral terms. Thus, damping ratio ξ is selected as zero in this study, which ensures that the harmonics from distorted voltages can be effectively compensated for by the high gain at selective frequencies. The system states in Equations (17) and (18) are augmented as: .

zc (t) = Ac zc (t) + Bc ε(t)

(19)

where zc = [z0 z6 z12 ] T is the entire state variables for integral and resonant terms with z0 = h i h i q q d δd , z = δq δq δd δd z6 = δ16 δ26 δ16 12 26 112 212 112 212 :   Ac = 



APc Arc6 Arc12

h

i q xi xid ,



 BPc    , and Bc =  Brc6  Brc12

The discrete-time counterparts of Ac and Bc can be obtained as: Acd = eAc Ts = I +

Ac Ts A2 T 2 + c s +... 1! 2!

1 Bcd = A− c (Acd − I)Bc

(20) (21)

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Then, the entire control system can be augmented as follows: "

x( k + 1) zc ( k + 1 )

#

"

=

Ad −Bcd Cd

0 Acd

#"

x(k) zc (k)

y( k ) =

h

#

"

Bd 0

+

Cd

0

i

"

#

" u( k ) +

x( k ) zc (k )

Dd 0

#

" e( k ) +

0 Bcd

# r( k )

(22)

# (23)

Considering the augmented system, the state feedback control is expressed as: " u(k) = −[Kx Kz ]

x( k ) zc ( k )

#

= ux (k) + uz (k)

(24)

where ux (k) = −Kx x(k ) and uz (k) = −Kz zc (k). The augmented system in Equations (22)–(24) can be rewritten in a compact form as: Energies 2018, 11, x FOR PEER REVIEW

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xe (k + 1) = Ae xe (k) + Be u(k) + De e(k) + Bre r(k ) (25) (25) 𝐱 𝒆 (𝑘 + 1) = 𝑨𝒆 𝐱 𝒆 (𝑘) + 𝑩𝒆 𝐮(𝑘) + 𝑫𝒆 𝐞(𝑘) + 𝑩𝒓𝒆 𝐫(𝑘) y ( k ) = Ce xe ( k ) (26) (26) 𝐲(𝑘) = 𝑪𝒆 𝐱 𝒆 (𝑘) u(k) = −Kxe (k) (27) (27) 𝐮(𝑘) = −𝑲𝐱 𝒆 (𝑘) where K = [[𝐊 K K ] is a set of feedback gains and K = [K K6z K].12z ]. The detailed block diagram of where 𝑲 = x𝐱 𝐊 𝐳z] is a set of feedback gains and 𝐊 𝐳 z= [𝐊 𝐏𝐳 Pz 𝐊 𝟔𝐳 𝐊 𝟏𝟐𝐳 The detailed block diagram of the proposed 3, 3, where AP 𝐀 , A,r6𝐀, A,r12 , BP , Br6 , and Br12 denote the proposedcurrent currentcontroller controllerisisdepicted depictedininFigure Figure where 𝐏 𝐫𝟔 𝐀 𝐫𝟏𝟐 , 𝐁𝐏 , 𝐁𝐫𝟔 , and 𝐁𝐫𝟏𝟐 the discrete-time counterparts of A , A , A , B , B , and B , respectively. rc6 , 𝐁 , and rc12 𝐁 Pc of rc6 denote the discrete-time counterparts 𝐀 , 𝐀rc12, 𝐀 Pc , 𝐁 , respectively. 𝐏𝐜

𝐫𝐜𝟔

𝐫𝐜𝟏𝟐

𝐏𝐜

𝐫𝐜𝟔

𝐫𝐜𝟏𝟐

𝐞(𝑘) 𝐃𝐝 𝐫(𝑘)

𝜺(𝑘) 𝐁𝐏

+-

𝐮(𝑘)

𝐳𝟎 (𝑘) + +

−1

𝐊 𝐏𝐳

-

+

𝐁𝐝

++ +

−1

𝐱(𝑘)

𝐂𝐝

𝐲(𝑘)

𝐲(𝑘) 𝐀𝐝

𝐀𝐏 𝐫(𝑘)

𝜺(𝑘)

𝐳𝟔 (𝑘) 𝐁𝐫𝟔

+-

+ +

−1

𝐊 𝟔𝐳

- -

𝐊𝐱

𝐲(𝑘)

LQR-based fullstate observer

𝐀𝐫𝟔 𝐫(𝑘)

𝐳𝟏𝟐 (𝑘)

𝜺(𝑘) +-

𝐁𝐫𝟏𝟐

+ +

−1

𝐊 𝟏𝟐𝐳

𝐲(𝑘) 𝐀𝐫𝟏𝟐 Figure 3. Detailed control block diagram of the proposed current controller. Figure 3. Detailed control block diagram of the proposed current controller.

3.2. Design of an Optimal Feedback Control Using Linear Quadratic Regulator (LQR) Approach The state feedback control input 𝐮(𝑘) will be an optimal control input to ensure the control performance and system stability if the gain matrix 𝑲 in system Equations (25)–(27) are evaluated systematically by minimizing the discrete quadratic cost function as follows [27,28]: ∞

1 𝐉 = ∑ 𝐱 𝒆 𝑇 (𝑘)𝐐𝐱 𝒆 (𝑘) + 𝐮𝑇 (𝑘)𝐑𝐮(𝑘) 2

(28)

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3.2. Design of an Optimal Feedback Control Using Linear Quadratic Regulator (LQR) Approach The state feedback control input u(k) will be an optimal control input to ensure the control performance and system stability if the gain matrix K in system Equations (25)–(27) are evaluated systematically by minimizing the discrete quadratic cost function as follows [27,28]: 1 ∞ T xe (k)Qxe (k) + uT (k)Ru(k ) 2 k∑ =0

J=

(28)

where Q is positive semi-definite matrix and R is positive definite matrix. To obtain the optimal control input vector u(k) in closed loop form, an n × n real symmetric matrix P with n being the number of state variables should be determined as the solution of the discrete Riccati equation as follows:   −1 P = Q + Ae T PAe − Ae T PBe R + Be T PBe Be T PAe .

(29)

Then, the gain matrix K can be calculated in terms of P as follows:   −1 K = R−1 Be T Ae T (P − Q).

(30)

The optimal control law is obtained by substituting Equations (30) to (27) as:   −1 u ( k ) = − R−1 Be T Ae T (P − Q )xe ( k ).

(31)

Then, the whole control system can be re-modeled as:  xe ( k + 1 ) =

 Ae − Be R

−1

Be

T



Ae

T

 −1



(P − Q)

xe (k) + De e(k) + Bre r(k).

(32)

The discrete Riccati Equation (29) can be solved by MATLAB (R2017b, The MathWorks, Inc, Natick, MA, USA) functions “dare” and “dlqr”. It is obvious that all the elements in the feedback gain matrix K rely on the choice of the symmetrical weighting matrices Q and R which determine the relative importance of the state variable performance and expenditure of energy by control input signals. The larger value of Q indicates that the system is stabilized with less change in the states, while the smaller Q implies that the states would be in larger variation. Similarly, the emphasis on R represents the behavior of the system states inputs. With the larger value of R, the system is stabilized with less control input signals, whereas more energy is used to stabilize the whole system with smaller value of R. In the proposed control scheme, the weighting matrices Q and R are selected as: 

10−2 ·I6×6  Q= 02×6 08×6

06×2 6.3 × 108 ·I2×2 08×2

 06×8  02×8 , 6.3 × 108 ·I8×8

" R=

1 0

0 1

# (33)

where In×m and 0n×m are the identity and zero matrices with appropriate dimensions, respectively. To improve the transient responses as well as to achieve the control objectives, a large weighting value of 6.3 × 108 is used for the state variables of the IM components zc , while a quite small value of 10−2 is chosen for six system state variables. As a result, a fast reference tracking of state variables and a good suppression capability for the distorted harmonics on grid voltages can be obtained. The simulation and experimental results are presented in next section to demonstrate the performance of the optimal control scheme.

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3.3. LQR-Based Current Full-State Observer in Discrete-Time To realize a full-state feedback controller in the augmented system in Equations (25)–(27), all the system state variables should be available for feedback purpose. However, in a three-phase LCL-filtered grid-connected inverter, the additional sensing devices usually increase the total cost and hardware complexity. Therefore, in the proposed control scheme, an LQR-based discrete-time current full-state observer is employed to produce the estimated signals for the grid-side current iˆ2 , the inverter-side current iˆ1 , and the capacitor voltages vˆ c . Regarding to the selection of observer type, there are three alternatives which are the prediction-type observer, the current observer, and the reduced-order observer [20]. In the prediction-type observer, the estimated states xˆ (k) are determined based on the past measurement of outputs at (k − 1) T. This means that the control signal ux (k ) = −Kx xˆ (k) does not utilize the most current information on outputs y(k), which leads to the inaccuracy of estimated values and might cause control performance degradation. On the other hand, the reduced-order observer can solve the drawback of the prediction-type observer by using the measured states to estimate remaining unmeasurable states at time kT. However, if the measurement variables are noisy, the imprecise measured states may influence directly the feedback control inputs. For these reasons, the current full-state observer is employed for the three-phase LCL-filtered grid-connected inverter in this paper, which yields a precise estimation capability even under harmonically distorted grid voltage condition. From the discretized model of the inverter system in Equations (12) and (13), a current full-state observer is given as follows: x(k + 1) = Ad xˆ (k ) + Bd u(k) + Dd e(k) Energies 2018, 11, x FOR PEER REVIEW xˆ (k + 1)

(34)

= x(k + 1) + Ke [y(k + 1) − Cd x(k + 1)]

9 of 27 (35)

where thethe symbol “ˆ”“^” denotes the estimated and x𝐱̅((𝑘 k+ where symbol denotes the estimatedvariables, variables,K𝐊e𝐞isisthe theobserver observergain gain matrix, matrix, and +1 1)) is theisfirst estimate of the state at time k + 1 T. In this type of observer, x k + 1 is first calculated from ( ) ( ) the first estimate of the state at time (𝑘 + 1)𝑇. In this type of observer, 𝐱̅(𝑘 + 1) is first calculated thefrom dynamics of system input signal kT, and thisthen estimation is addediswith the with correction the dynamics of and system and inputatsignal at then 𝑘𝑇, and this estimation added the term in Equation when the are signals measured at time (k at + time 1) T. (𝑘 Figure 4 presents correction term (35) in Equation (35)output when signals the output are measured + 1)𝑇. Figure 4 a presents a current discrete-time current full-state with a state-feedback controller, where the discrete-time full-state observer withobserver a state-feedback controller, where the estimated state estimated are used to construct thecontrol state feedback variables arestate usedvariables to construct the state feedback inputs. control inputs.

Plant 𝐱 𝑘 + 1 = 𝐀𝐝 𝐱 𝑘 + 𝐁 𝐝 𝐮 𝑘 + 𝐃 𝐝 𝐞 𝑘 𝐲 𝑘 = 𝐂𝐝 𝐱(𝑘)

𝐲 𝑘

𝐞(𝑘) 𝐃𝐝 𝐮 𝑘 𝐮𝐳 𝑘

++

𝐮𝐱 𝑘

𝐱̅(𝑘 + 1) 𝐁𝐝

𝐊𝐱

+ ++

𝐱(𝑘)

−1

𝐱̅(𝑘)

𝐂𝐝

+ -

𝐀𝐝 ++

𝐊𝐞

Figure 4. State feedback control using current full-state observer in the discrete-time domain. Figure 4. State feedback control using current full-state observer in the discrete-time domain.

In order to determine the observer gain matrix 𝐊 𝐞 , the estimation error 𝐱̃ is defined as: 𝐱̃(𝑘) = 𝐱(𝑘) − 𝐱(𝑘)

(36)

Then, the error dynamics of the observer can be obtained by subtracting Equation (35) from Equation (12) as follows:

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In order to determine the observer gain matrix Ke , the estimation error e x is defined as: e x(k) = x(k) − xˆ (k )

(36)

Then, the error dynamics of the observer can be obtained by subtracting Equation (35) from Equation (12) as follows: e x(k + 1) = (Ad − Ke Cd Ad )e x( k ) (37) To ensure that the observer is stable and the estimated states well track the actual ones, the observer gain matrix should be chosen in order that the matrix (Ad − Ke Cd Ad ) or (AdT − (Cd Ad ) T )KeT ) is stable. By applying a LQR approach similar to the design of a state feedback control, the discrete Riccati equation can be applied for observer design as: Po = Qo + Ad Po Ad T − Ad Po (Cd Ad ) T (Ro + (Cd Ad )Po (Cd Ad ) T )−1 (Cd Ad )Po Ad T

(38)

where Po is the solution of Riccati equation (38), and Qo and Ro are weighting matrices. Hence, the observer gain Ke can be calculated in terms of Po as follows: Ke = Ro −1 Cd Ad (Ad )−1 (Po − Qo ).

(39)

In this paper, the optimal observer gains can be chosen by utilizing the MATLAB function “dlqr”. 4. Simulation Results In order to verify the feasibility and validity of the proposed current control scheme, simulations were carried out for an LCL-filtered three-phase grid-connected inverter based on the PSIM software. The configuration of the inverter system and the proposed control scheme are depicted in Figure 2. The system parameters are listed in Table 1. Table 1. System parameters of a grid-connected inverter. Parameters

Value

Units

DC-link voltage Resistance (load bank) Filter resistance Filter capacitor Inverter-side filter inductance Grid-side filter inductance Grid voltage (line-to line rms) Grid frequency

420 24 0.5 4.5 1.7 0.9 220 60

V Ω Ω µF mH mH V Hz

Figure 5 represents three-phase distorted grid voltages used for the simulations. The abnormal grid voltages contain the harmonic components in the order of the 5th, 7th, 11th, and 13th with the magnitude of 5% with respect to the nominal grid voltages, which yields the THD value of 9.99%.

Grid voltage (line-to line rms) Grid frequency

220 60

V Hz

Figure 5 represents three-phase distorted grid voltages used for the simulations. The abnormal grid voltages the harmonic components in the order of the 5th, 7th, 11th, and 13th with the Energies 2018, 11,contain 2062 11 of 28 magnitude of 5% with respect to the nominal grid voltages, which yields the THD value of 9.99%. ea

eb

ec

𝑒𝑎

200

𝑒𝑏

𝑒𝑐

THD: 9.99%

Voltage (V)

100

0

-100

-200 0.2

0.22

0.24

0.26

0.28

0.3

Time (s)

(a) 30

THD: 9.99% 25

Voltage (V)

20 5th

7th

11th

13th

15 10 5 0 0

200

400

600

800

1000

Frequency (Hz)

(b) Figure Figure5.5.Distorted Distortedgrid gridvoltages. voltages.(a) (a)Three-phase Three-phasedistorted distortedgrid gridvoltages; voltages;(b) (b)Fast FastFourier Fouriertransform transform (FFT) result of a-phase voltage. (FFT) result of a-phase voltage.

Figure 6 shows the simulation results of the proposed current control scheme at steady-state Figure 6 shows the simulation results of the proposed current control scheme at steady-state under the distorted grid condition as in Figure 5. Figure 6a shows the grid-side current responses at under the distorted grid condition as in Figure 5. Figure 6a shows the grid-side current responses at the SRF with the reference currents. As can be observed from Figure 6a, the grid-side currents track the SRF with the reference currents. As can be observed from Figure 6a, the grid-side currents track the reference values well. Figure 6b,c represent the steady-state responses for the inverter-side the reference values well. Figure 6b,c represent the steady-state responses for the inverter-side current current and capacitor voltage, respectively. and capacitor voltage, respectively. To demonstrate the transient performance of the proposed current control scheme, Figure 7 shows the simulation results under the same distorted grid condition when the q-axis reference current has a step change from 4 to 7 A at 0.25 s. Similarly, Figure 7a through Figure 7c represents the grid-side current responses, inverter-side current responses, and capacitor voltage responses at the SRF, respectively. As is shown in Figure 7a, the grid-side currents reach the reference very rapidly, which indicates a sufficiently fast transient response of the proposed control scheme. In addition, the fast transient performance of the proposed control scheme can be also inferred from Figure 7b,c.

Energies 2018, 11, 2062 Energies 2018, 11, x FOR PEER REVIEW V15 V16 _Ide2 𝑖 𝑞2

8

12 of 28 11 of 27

_Iqe2 𝑖 𝑞2

𝑖 𝑑2

𝑖 2𝑑

Current (A)

6 4 2 0 -2 0.2

0.22

0.24

0.26

0.28

0.3

0.26

0.28

0.3

0.26

0.28

0.3

Time (s)

iqe1

ide1

(a)

𝑖1𝑞

8

𝑖1𝑑

Current (A)

6 4 2 0 -2 0.2

0.22

0.24 Time (s)

vc_qc

(b)

vc_dc 𝑣𝑐𝑞

200

𝑣𝑐𝑑

Voltage (V)

150 100 50 0 -50 0.2

0.22

0.24 Time (s)

(c) Figure Simulationresults resultsforfor steady-state responses distorted grid voltage the Figure 6. 6. Simulation steady-state responses underunder distorted grid voltage with thewith proposed proposed (a) current Grid-side current at responses at the synchronous reference (SRF); (b) controller.controller. (a) Grid-side responses the synchronous reference frame (SRF);frame (b) Inverter-side Inverter-side current responses at the SRF; (c) Capacitor voltage responses at the SRF. current responses at the SRF; (c) Capacitor voltage responses at the SRF.

To demonstrate the transient performance of the proposed current control scheme, Figure 7 shows the simulation results under the same distorted grid condition when the q-axis reference current has a step change from 4 to 7 A at 0.25 s. Similarly, Figure 7a through Figure 7c represents the grid-side current responses, inverter-side current responses, and capacitor voltage responses at the SRF, respectively. As is shown in Figure 7a, the grid-side currents reach the reference very rapidly, which indicates a sufficiently fast transient response of the proposed control scheme. In

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addition, the11,fast from Energies 2018, 2062transient performance of the proposed control scheme can be also inferred 13 of 28 Figure 7b,c. V15

V16

_Iqe2

𝑖 𝑞2

8

_Ide2 𝑖 𝑞2

𝑖 𝑑2

𝑖 2𝑑

Current (A)

6 4 2 0 -2 0.2

0.22

0.24

0.26

0.28

0.3

0.26

0.28

0.3

0.26

0.28

0.3

Time (s)

iqe1

ide1

(a)

𝑖1𝑞

8

𝑖1𝑑

Current (A)

6 4 2 0 -2 0.2

0.22

0.24 Time (s)

vc_qc

vc_dc

(b)

𝑣𝑐𝑞

200

𝑣𝑐𝑑

Voltage (V)

150 100 50 0 -50 0.2

0.22

0.24 Time (s)

(c) Figure Figure7.7.Simulation Simulationresults resultsfor fortransient transientresponses responsesunder underdistorted distortedgrid gridvoltage voltagewith withthe theproposed proposed controller. (a) Current references and grid-side current responses at the synchronous reference controller. (a) Current references and grid-side current responses at the synchronous reference frame frame (b) inverter-side responses at the SRF; (c) capacitor voltage responses (SRF);(SRF); (b) inverter-side currentcurrent responses at the SRF; (c) capacitor voltage responses at the SRF.at the SRF.

Figure88 shows shows the the simulation simulationresults resultsfor for the the inverter inverter states statesand and estimated estimatedstates statesusing usingthe the Figure proposed integral-resonant state feedback control scheme with the discrete-time current full-state proposed integral-resonant state feedback control scheme with the discrete-time current full-state observeratatthe theSRF. SRF.The Theoptimal optimalobserver observergains gainsare areobtained obtainedby byusing usingthe theMATLAB MATLAB“dlqr” “dlqr”function function observer with given inverter parameters. As can be clearly observed from Figure 8, the estimated states instantly with given inverter parameters. As can be clearly observed from Figure 8, the estimated states converge to the actual ones even during oscillating transient periods, which confirms a fast and stable operation of the observer.

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instantly Energies 2018,converge 11, 2062 to the actual ones even during oscillating transient periods, which confirms a fast 14 of 28 and stable operation of the observer. x1_e

x2_e

_Iqe2

𝑖 𝑞2

8

_Ide2 𝑖𝑞2

𝑖 𝑑2

𝑖𝑑2

Current (A)

6 4 2 0 -2 0.2

0.22

0.24

0.26

0.28

0.3

0.28

0.3

0.28

0.3

Time (s) x3_e

x4_e

iqe1

𝑞

8

ide1

(a) 𝑞

𝑖1𝑑

𝑖1

𝑖𝑑1

𝑖1

Current (A)

6 4 2 0 -2 0.2

0.22

0.24

0.26 Time (s)

x5_e

x6_e

vc_qc

𝑣𝑐𝑞

200

vc_dc

(b) 𝑣𝑐𝑞

𝑣𝑐𝑑

𝑣𝑐𝑑

Voltage (V)

150 100 50 0 -50 0.2

0.22

0.24

0.26 Time (s)

(c) Figure 8. Simulation results for the proposed control scheme with the discrete-time full-state Figure 8. Simulation results for the proposed control scheme with the discrete-time full-state observer observer under step change in q-axis current reference; states and estimated states. (a) Waveforms of under step change in q-axis current reference; states and estimated states. (a) Waveforms of grid-side grid-side currents and estimated states at the SRF; (b) waveforms of inverter-side currents and currents and estimated states at the SRF; (b) waveforms of inverter-side currents and estimated states estimated states at the SRF; (c) waveforms of capacitor voltages and estimated states at the SRF. at the SRF; (c) waveforms of capacitor voltages and estimated states at the SRF.

Figure 9 represents the simulation results for measured three-phase variables using the Figure integral-resonant 9 represents the simulation results for measured three-phase variables current using the proposed proposed state feedback control scheme with the discrete-time full-state observer when state the q-axis reference a step change. As can be seen fromobserver Figure 9a, integral-resonant feedback controlcurrent schemehas with the discrete-time current full-state when the q-axis reference current has a step change. As can be seen from Figure 9a, three-phase grid-side current waveforms remain relatively sinusoidal with a desired transient performance. In fact, the

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three-phase grid-side current waveforms remain relatively sinusoidal with a desired transient performance. In fact, the grid-side phase currents have the THD level of 3.57% in this case. Also, grid-side phase currents the THD level of 3.57% in this case. Also, 9b,c show actual Figure 9b,c show actualhave three-phase inverter-side current waveforms and Figure three-phase capacitor three-phase inverter-side current waveforms and three-phase capacitor voltage waveforms. voltage waveforms. _i_lg_a

_i_lg_b 𝑖 2𝑎

10

_i_lg_c 𝑖 2𝑐

𝑖 2𝑏

Current (A)

5

0

-5

-10 0.2

0.22

0.24

0.26

0.28

0.3

0.26

0.28

0.3

0.26

0.28

0.3

Time (s) ia

ib

ic

(a)

𝑖 1𝑎

10

𝑖 1𝑐

𝑖 1𝑏

Current (A)

5

0

-5

-10 0.2

0.22

0.24 Time (s)

Vac

Vbc

Vcc

𝑣𝑐𝑎

200

(b) 𝑣𝑐𝑐

𝑣𝑐𝑏

Voltage (V)

100

0

-100

-200 0.2

0.22

0.24 Time (s)

(c) Figure 9. Simulation results for measured three-phase variables with the proposed control scheme Figure 9. Simulation results for measured three-phase variables with the proposed control scheme under step change in q-axis current reference. (a) Three-phase grid-side current waveforms; (b) under step change in q-axis current reference. (a) Three-phase grid-side current waveforms; (b) three-phase inverter-side current waveforms; (c) three-phase capacitor voltage waveforms. three-phase inverter-side current waveforms; (c) three-phase capacitor voltage waveforms.

Figure 10 shows the simulation results for the estimated waveforms of three-phase grid-side Figureinverter-side 10 shows the simulation forvoltages the estimated waveforms of three-phase currents, currents, and results capacitor under the same condition of Figuregrid-side 9. In currents, inverter-side currents, and capacitor voltages under the same condition of Figure 9. In these these figures, the estimated three-phase variables are constructed in a DSP by using the estimated figures, the estimated three-phase variables are constructed in a DSP by using the estimated states at the SRF to demonstrate the estimating performance of the discrete-time current full-state observer.

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states at the SRF to demonstrate the estimating performance of the discrete-time current full-state Obviously, estimated variables are compatible with the actual measured three-phase observer. the Obviously, thethree-phase estimated three-phase variables are compatible with the actual measured waveforms in Figure 9. three-phase waveforms in Figure 9. V15

V16

V17

𝑖𝑎2

10

𝑖𝑐2

𝑖𝑏2

Current (A)

5

0

-5

-10 0.2

0.22

0.24

0.26

0.28

0.3

0.26

0.28

0.3

0.26

0.28

0.3

Time (s)

V15

V16

V17

𝑖𝑎1

10

(a) 𝑖𝑐1

𝑖𝑏1

Current (A)

5

0

-5

-10 0.2

0.22

0.24 Time (s)

V15

V16

V17

𝑣𝑐𝑎

200

(b) 𝑣𝑐𝑐

𝑣𝑐𝑏

Voltage (V)

100

0

-100

-200 0.2

0.22

0.24 Time (s)

(c) Figure 10. Simulation results for estimated three-phase variables with state observer under step Figure 10. Simulation results for estimated three-phase variables with state observer under step change change in q-axis current reference. (a) Estimated three-phase grid-side currents; (b) estimated in q-axis current reference. (a) Estimated three-phase grid-side currents; (b) estimated three-phase three-phase inverter-side currents; (c) estimated three-phase capacitor voltages. inverter-side currents; (c) estimated three-phase capacitor voltages.

To verify the quality of injected grid currents for the proposed integral-resonant controller To verify the quality of injected grid forFFT the result proposed integral-resonant controller under a distorted grid voltage, Figure 11currents shows the for grid-side a-phase current withunder the a distorted grid voltage, Figure the FFT result regulation for grid-side a-phase current thebe harmonic harmonic limits specified by 11 theshows grid interconnection IEEE Std. 1547 [29].with As can seen limits specified by the grid interconnection regulation IEEE Std. 1547 [29]. As can be seen clearly, the

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clearly, the grid-side phase current yields5th, only7th, small 5th, 7th, 11th, and 13th harmonics components. grid-side current yields only small 11th, and 13th harmonics The resultant clearly, phase the grid-side phase current yields only small 5th, 7th, 11th, and 13thcomponents. harmonics components. The resultant THD value is 3.569%, which meets the quality criteria of inverter injected current. THD value is 3.569%, which meets the quality criteria of inverter injected current. The resultant THD value is 3.569%, which meets the quality criteria of inverter injected current. 0.30.3

4% 4%

THD: 3.569% THD: 3.569%

0.25 0.25

Odd harmonics Odd harmonics 2% 2%

Current (A) Current (A)

0.20.2 0.15 0.15 5th 5th

0.10.1

11 11thth

77thth

th th 1313

0.05 0.05

0 0 0

0

200 200

400 400

600 Frequency (Hz)

600

800

800

1000

1000

Frequency (Hz)

Figure 11. FFT result for grid-side a-phase current of the proposed controller under distorted grid Figure11. 11.FFT FFT result grid-side a-phase current of proposed the proposed controller distorted Figure result forfor grid-side a-phase current of the controller underunder distorted grid voltages. grid voltages. voltages.

In order to verify the effectiveness of the proposed current control scheme, the performance of Inproposed order verify of the proposed current control scheme, the of orderto toLQR-based verifythe theeffectiveness effectiveness proposed scheme, theperformance performance theIn current controlofisthe compared to current PR plus control harmonic compensator (PR + HC) of the proposed LQR-based current control is compared to PR plus harmonic compensator (PR + HC) thestructure proposed LQR-based current is compared PRproposed plus harmonic compensator + HC) presented in [3] undercontrol the same parameterstoas in Table 1 and grid (PR voltage structure presented in [3] under the same parameters as proposed in Table 1 and grid voltage structure presented in [3] under the same parameters as proposed in Table 1 and grid voltage condition condition presented in Figure 5a. The transfer function of a PI + HC controller is given in the condition presented in The Figure 5a. The transfer function of a PI +is HC controller is givenframe in the presented in frame Figureas: 5a. transfer function of a PI + HC controller given in the stationary as: stationary

stationary frame as: 𝐾 𝑠 𝐾𝑟5 𝑠 𝐾𝑟7 𝑠 𝐾𝑟11 𝑠 𝐾𝑟13 𝑠 Kr5 s + K 𝑟1s + Kr7 s + Kr11 s + Kr13 s 𝐺= (40) (40) 2 +𝐾(11𝜔) 2 +𝐾 𝑠 22 +𝑠 2 + 𝐾𝑟7 𝑠 22 + G= K P𝐾𝑃++ 2𝑠𝐾2𝑟1r1 𝑟11 𝑠 2 2 𝑠+ 𝑟13 𝑠 2 +𝑠 𝜔22+ 𝑠22 𝐾 +𝑟5(5𝜔) (7𝜔) 𝑠 (13𝜔) 2 2 2 s + ω 𝐺 = 𝐾𝑃 + 2 + s2 + (5ω )2 + 2s + (7ω2) + 2s + (11ω2)+ 2s + (13ω2)2 (40) 2 𝑠 +𝜔 𝑠 + (5𝜔) 𝑠 + (7𝜔) 𝑠 + (11𝜔) 𝑠 + (13𝜔) where 𝐾𝑃 is the proportional gain and 𝐾𝑟𝑖 is the resonant gain with 𝑖 = 1, 5, 7, 11, 13. where 𝐾 K𝑃P is the gain K𝐾ri𝑟𝑖isisthe gain with i= 5,5,7, 13. Figure 12 proportional shows the simulation for the control scheme As can be13. observed from where proportional gain and andresults theresonant resonant gain within 𝑖[3]. =1,1, 7, 11, 11, Figure 12 shows simulation scheme inin [3]. AsAs can be be observed from the theFigure grid-side currentthe responses in results Figure 12a, thecontrol PR + HC current control still can compensate 12 simulation resultsfor forthe the control scheme [3]. can observed from grid-side current responses in Figure 12a, the PR + HC current control still can compensate effectively effectively the harmonics caused by background voltages. However, the THD value of a-phase the grid-side current responses in Figure 12a, the PR + HC current control still can compensate isthe slightly increased to 3.69% in comparison to that obtained fromthe the LCL filter parameters thecurrent harmonics caused by background However, the THD value of a-phase current slightly effectively harmonics caused by voltages. background voltages. However, THD value ofisa-phase in [3] isbecause the filter inductor values are reduced. 12b the forthe increased to 3.69% in comparison to in that obtained from theobtained LCLpresents filter parameters [3]results because current slightly increased to 3.69% comparison toFigure that from thesimulation LCL in filter parameters the reference tracking performance of grid-side currents in the stationary frame. filter inductor values are reduced. Figure 12b presents the simulation results for the reference tracking in [3] because the filter inductor values are reduced. Figure 12b presents the simulation results for _i_lg_a _i_lg_b _i_lg_c performance of grid-side currents inofthe stationary frame. the reference tracking performance grid-side currents in the stationary frame. 𝑖 2𝑎 _i_lg_b

10_i_lg_a

𝑖 2𝑎

10

𝑖 2𝑐

𝑏

𝑖2 _i_lg_c

THD: 3.69%

𝑖 2𝑐

𝑖 2𝑏

THD: 3.69%

Current (A) Current (A)

5

5 0

0 -5

-5 -10 -10

0.2

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0.24

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0.2

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(a) Figure 12. Cont.

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Time (s)

(b) Figure 12. Simulation results for the proportional-resonant plus harmonic compensator (PR + HC) Figure 12. Simulation results for the proportional-resonant plus harmonic compensator (PR + HC) control scheme under step change. (a) Three-phase grid-side current waveforms; (b) current control scheme under step change. (a) Three-phase grid-side current waveforms; (b) current references references and grid-side current responses at the stationary frame. and grid-side current responses at the stationary frame.

In spite of the control performance of the study in [3], it is worth mentioning that the main In spite control performance the study in it is worth mentioning the main drawback of of thethe work in [3] lies in the of requirement for[3], additional current sensingthat devices. As drawback of the work in [3] lies in the requirement for additional current sensing devices. As obviously obviously shown in the control structure, the method in [3] requires the measurement of currents in shown in the as control the method in [3]the requires the measurement of currents in inverter-side inverter-side well structure, as in grid-side to obtain capacitor current for active damping. Generally, as well as in grid-side to obtain the capacitor current for active damping. Generally, since additional since additional sensing devices cause complexity in the hardware and a high implementation cost, sensing devices cause complexity in the hardware and a high implementation cost, the purpose of this the purpose of this study is to implement a desired control performance by utilizing only the study is to implement a desired control performance by utilizing only the grid-side current sensors grid-side current sensors and grid voltage sensors. and Furthermore, grid voltage sensors. the control method in [3] requires 5 resonant controllers including the Furthermore, the control in [3] requires 5 resonant controllers includingfor thethe fundamental fundamental component for method the α-axis and additional 5 resonant controllers β-axis to component for the α-axis and additional 5 resonant controllers for the β-axis to compensate for the the compensate for the harmonic components in the order of the 5th, 7th, 11th, and 13th. On harmonic components in the order of the 5th, 7th, 11th, and 13th. On the contrary, since the proposed contrary, since the proposed scheme is designed in the synchronous reference frame, the total of 4 scheme iscontrollers designed inare the sufficient synchronous frame, the total of 4 resonantthe controllers sufficient resonant for reference the q- and d- axes to compensate 6th andare 12th order for the qand daxes to compensate the 6th and 12th order harmonic components. From the viewpoint harmonic components. From the viewpoint of digital implementation burden, the proposed control of digital implementation burden, the proposed control scheme requires only an acceptable level of scheme requires only an acceptable level of computation and complexity. computation and complexity. 5. Experimental Results 5. Experimental Results In order to verify the feasibility of the proposed control scheme, the control algorithm is In order to verify the feasibility of the proposed control scheme, the control algorithm is implemented on 32-bit floating-point DSP TMS320F28335 (Texas Instruments, Inc, Dallas, TX, USA) implemented on 32-bit floating-point DSP TMS320F28335 (Texas Instruments, Inc, Dallas, TX, USA) to control a 2 kVA prototype grid-connected inverter [30]. The configuration of the entire system is to control a 2 kVA prototype grid-connected inverter [30]. The configuration of the entire system is illustrated in Figure 13a. The sampling period is set to 100 µ s, which results in the switching illustrated in Figure 13a. The sampling period is set to 100 µs, which results in the switching frequency frequency of 10 kHz. Figure 13b depicts the photograph of the experimental test setup. The of 10 kHz. Figure 13b depicts the photograph of the experimental test setup. The experimental setup is experimental setup is composed of a three-phase inverter connected to the grid through an LCL composed of a three-phase inverter connected to the grid through an LCL filter, a magnetic contactor filter, a magnetic contactor for grid connecting operations, an AC power source to emulate for grid connecting operations, an AC power source to emulate three-phase grid voltages in the ideal as three-phase grid voltages in the ideal as well as distorted grid conditions, and current and voltage well as distorted grid conditions, and current and voltage sensors used to measure grid-side currents sensors used to measure grid-side currents and grid voltages, respectively. and grid voltages, respectively. Figure 14a shows three-phase distorted grid voltages used for the experimental evaluation. Figure 14a shows three-phase distorted grid voltages used for the experimental evaluation. Similar Similar to Figure 5 in the simulation, these grid voltages contain the 5th, 7th, 11th, and 13th to Figure 5 in the simulation, these grid voltages contain the 5th, 7th, 11th, and 13th harmonics with harmonics with the magnitude of 5% of the fundamental component. Figure 14b presents the FFT the magnitude of 5% of the fundamental component. Figure 14b presents the FFT results for a-phase results for a-phase grid voltage, which shows each harmonic component similar to Figure 5b. grid voltage, which shows each harmonic component similar to Figure 5b.

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Grid-connected Inverter

Rectifier DC Link Capacitor

Magnetic Circuit Breaker

Single-phase AC Source

Gate Drive 12 A/D Host PC

Load Bank

Three-phase Programmable AC Source

Phase Currents Phase Voltages

DSP TMS320F28335

(a)

Three-phase Programmable AC Power Source

Host PC

DSP-based Controller Grid-connected Inverter

Circuit Breaker

Load Bank

(b) Figure 13. Configuration of the experimental system. (a) Block diagram of the overall system; (b) Figure 13. Configuration of the experimental system. (a) Block diagram of the overall system; photograph of the experimental test setup. (b) photograph of the experimental test setup.

Figure 15 shows the experimental results for the proposed control scheme under the step Figure 15 shows thereference experimental forItthe proposed control scheme under the step change change in q-axis current fromresults 4 to 6 A. can be observed from Figure 15a that the inverter in q-axis current reference from 4 to 6 A. It can be observed from Figure 15a that the inverter output currents can track their references well and instantly reach a new steady-state value, which output currentsa can their references and proposed instantly reach a new steady-state demonstrates fasttrack transient response well of the control scheme. Figurevalue, 15b which shows demonstrates a fast transient of the control Figure three-phase three-phase grid-side currentresponse responses. It isproposed confirmed fromscheme. this figure that15b theshows proposed control grid-side current responses. It is sinusoidal confirmed from this figure the proposed control scheme scheme provides considerable grid-side phasethat currents, which coincides well provides with the considerable sinusoidal grid-side phase currents, which coincides well with the simulation results in simulation results in Figure 9a, verifying a stable and reliable operation of the inverter system. Figure 9a, verifying a stable and reliable operation of the inverter system.

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𝑒𝑏 𝑒𝑏

𝑒𝑐 𝑒𝑐

Voltage : 100 [V/div] Voltage : 100 [V/div]

Time base base: :55[ms/div] [ms/div] Time (a) (a)

Voltage : :55[V/div] [V/div] Voltage

55thth

77thth

11 11thth

13 13thth

Frequency : 100 [Hz/div]

Frequency : 100 [Hz/div]

(b)

(b) Figure 14. Distorted three-phase grid voltages used in the experiments. (a) Three-phase distorted Figure 14. three-phase grid used in the experiments. (a) Three-phase distorted grid Figure 14. Distorted Distorted three-phase gridvoltages voltages used in the experiments. (a) Three-phase distorted grid voltages; (b) FFT result of a-phase grid voltage. voltages; (b) FFT result of a-phase grid voltage. grid voltages; (b) FFT result of a-phase grid voltage. 𝑞 𝑖 𝑞2 𝑖2

𝑖 𝑑2

𝑖 𝑑2

𝑞 𝑞 𝑖2

𝑖2

Current : 2 [A/div]

Current : 2 [A/div]

𝑖 2𝑑

𝑖 2𝑑

Time base : 5 [ms/div] (a)

(a) Figure 15. Cont.

Time base : 5 [ms/div]

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𝑖 𝑏2 𝑖 𝑏2

Current : 2 [A/div] Current : 2 [A/div]

𝑖 𝑐2 𝑖 𝑐2

(b) (b)

𝑞

𝑖 𝑞2 𝑖2

Time base : 5 [ms/div] Time base : 5 [ms/div]

Figure 15. Experimental q-axis current Figure 15. Experimental results results for for the the proposed proposed control control scheme scheme under under step step change change in in q-axis current Figure 15. (a) Experimental resultsresponses for the proposed control scheme under step change in responses. q-axis current reference. Grid-side current at the SRF; (b) three-phase grid-side current reference. (a) Grid-side current responses at the SRF; (b) three-phase grid-side current responses. reference. (a) Grid-side current responses at the SRF; (b) three-phase grid-side current responses.

Figure 16 presents the experimental results for the estimated grid-side currents, estimated Figure 16 16 presents the experimental results for estimated grid-side estimated Figure presents theestimated experimental results for the the grid-side currents, currents, inverter-side currents, and capacitor voltages by estimated using the discrete-time currentestimated full-state inverter-side currents, and estimated capacitor voltages by using the discrete-time current full-state inverter-side currents, and estimated by usingreference. the discrete-time current grid-side full-state observer at the SRF under the step capacitor change involtages q-axis current The estimated observer at under the stepstep change in q-axisq-axis current reference. The estimated grid-sidegrid-side currents observer atthe theSRF SRF change current reference. estimated currents at the SRF inunder Figurethe 16a show similarinbehavior with actual statesThe in Figure 15a. Also, the at the SRFatinthe Figure show similar behavior with actual with statesactual in Figure 15a.inAlso, the experimental currents SRF 16a in Figure 16a show similar behavior states experimental estimated waveforms are very similar to the simulation results in Figure Figure 15a. 8. Also, the estimated waveforms are very similar to the simulation results in Figure 8. experimental estimated waveforms are very similar to the simulation results in Figure 8.

𝑖 2𝑞𝑞 𝑖2

𝑖 𝑞𝑞2 𝑖2

Current : 2 [A/div] Current : 2 [A/div]

𝑖𝑑𝑑2 𝑖2

(a) (a) Figure 16. Cont.

Time base : 5 [ms/div] Time base : 5 [ms/div]

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𝑖𝑞1

𝑖𝑑1

Current : 2 [A/div]

Time base : 5 [ms/div] (b)

𝑣𝑐𝑞

𝑣𝑐𝑑

Time base : 5 [ms/div]

Voltage : 100 [V/div] (c)

Figure 16. 16. Experimental the estimated estimated states states with with the the discrete-time discrete-time full-state full-state observer observer Figure Experimental results results for for the under step change in q-axis current reference. (a) Responses of estimated grid-side currents at the the under step change in q-axis current reference. (a) Responses of estimated grid-side currents at SRF; (b) (b) responses responses of of estimated estimated inverter-side inverter-side currents currents at at the the SRF; SRF; (c) (c) responses responses of of estimated estimated capacitor capacitor SRF; voltages at the SRF. voltages at the SRF.

Figure 17a shows the experimental results for three-phase grid-side current waveforms at Figure 17a shows the experimental results for three-phase grid-side current waveforms at steady-state with the proposed current control scheme under harmonically distorted grid conditions steady-state with the proposed current control scheme under harmonically distorted grid conditions as in Figure 14. Generally, the harmonic distortion on grid voltages directly influence on the as in Figure 14. Generally, the harmonic distortion on grid voltages directly influence on the grid-side grid-side current control performance, reducing the power quality of distributed generation system. current control performance, reducing the power quality of distributed generation system. However, However, the three-phase grid-side current waveform of the proposed scheme shows quite the three-phase grid-side current waveform of the proposed scheme shows quite sinusoidal phase sinusoidal phase currents in spite of such a severe harmonic distortion on grid voltages. As is shown currents in spite of such a severe harmonic distortion on grid voltages. As is shown in Figure 17b, the in Figure 17b, the FFT result for a-phase current shows negligibly small harmonic components in FFT result for a-phase current shows negligibly small harmonic components in output current, which output current, which successfully meets the requirements for the harmonic limits specified by IEEE successfully meets the requirements for the harmonic limits specified by IEEE Standard 519-1992. Standard 519-1992.

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𝑖 2𝑎

𝑖 𝑐2

𝑖 𝑏2

𝑖 𝑞2

Time base : 5 [ms/div]

Current : 2 [A/div] (a)

Current : 100 [mA/div]

5th

11th

7th

13th

Frequency : 100Hz/div (b) Figure 17. Experimental results for the proposed control scheme at steady-state. (a) Three-phase Figure 17. Experimental results for the proposed control scheme at steady-state. (a) Three-phase grid-side current waveforms; (b) FFT result for a-phase current. grid-side current waveforms; (b) FFT result for a-phase current.

Figure 18 represents the experimental results for the estimating performance of the Figure 18current represents the experimental for the discrete-time full-state observer. Toresults evaluate the estimating estimatingperformance performanceofofthe thediscrete-time observer by current full-state observer. To evaluate the estimating performance of the observer by currents comparison, comparison, Figure 18a,b show the estimating performance for grid-side three-phase and Figure 18a,b show the estimating performance for grid-side three-phase currents and the comparison the comparison of these estimated signals with measured grid-side currents. In these figures, the of these estimated signals with measured grid-side In these figures, the estimated three-phase estimated three-phase variables 𝑖2𝑎 , 𝑖2𝑏 , and 𝑖2𝑐 are currents. constructed in a DSP by using the estimated states q a b c d 𝑞 variables 𝑑 iˆ2 , iˆ2 , and iˆ2 are constructed in a DSP by using the estimated states iˆ2 and iˆ2 at the SRF. 𝑖2 and 𝑖2 at the SRF. The experimental results are well matched with the simulation results in Figure The experimental results are and wellreliability matched of with simulation Figurefull-state 10a andobserver. validate 10a and validate the stability the the estimated statesresults by thein current the stability and reliability of the estimated states by the current full-state observer. Similarly, Similarly, Figure 18c,d show the estimating performance for inverter-side three-phase currents, and Figure the estimating performance for inverter-side three-phase and three-phase Figure 18e,f Figure 18c,d 18e,fshow for three-phase capacitor voltages, respectively. Also, thecurrents, estimated ˆa , iˆb , and iˆc 𝑞 for three-phase voltages, respectively. estimated three-phase 1 the 1 variables 𝑖1𝑎 , 𝑖1𝑏 ,capacitor and 𝑖1𝑐 are calculated in a DSPAlso, fromthe the estimated currents 𝑖variables 𝑖1𝑑 iat SRF,1 1 and q d a b c 𝑞 the SRF, are in a𝑣 DSP from the estimated currentsvoltages iˆ1 and iˆ1𝑣at 𝑐 𝑑 and vˆ c , vˆ c , and vˆ c from the andcalculated 𝑣𝑐𝑎 , 𝑣𝑐𝑏 , and from the estimated capacitor and 𝑣 at the SRF, respectively. It 𝑐 q d at the SRF, respectively. 𝑐It can be𝑐 confirmed from these figures ˆ ˆ estimated capacitor voltages v and v c c can be confirmed from these figures that all the estimated three-phase variables converge to actual that all thethree-phase estimated three-phase variables converge to actual measured three-phase variables well. measured variables well.

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𝑖𝑏 2

𝑖𝑐2

Current : 2 [A/div]

𝑖𝑞 2

Time base : 5 [ms/div] (a)

𝑖 2𝑎

𝑖𝑏 2

𝑖𝑎 2

𝑖𝑏 2

Time base : 10 [ms/div]

Current : 2 [A/div] (b)

𝑖𝑎 1

𝑖𝑏 1

𝑖𝑐1

Time base : 5 [ms/div]

Voltage : 2 [A/div] (c) Figure 18. Cont.

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𝑖 1𝑎

𝑖1𝑏

Time base : 5 [ms/div]

Current : 2 [A/div] (d)

𝑣𝑐𝑎

𝑣𝑐𝑏

𝑣𝑐𝑐

Voltage : 100 [V/div]

Time base : 5 [ms/div] (e)

𝑣𝑐𝑎

𝑣𝑐𝑎

Time base : 5 [ms/div]

Voltage : 100 [V/div] (f)

Figure 18. Experimental results for the estimating performance with the discrete-time full-state Figure 18. Experimental results for the estimating performance with the discrete-time full-state observer. observer. (a) Estimating performance for grid-side three-phase currents; (b) comparison of the (a) Estimating performance for grid-side three-phase currents; (b) comparison of the estimated grid-side estimated grid-side currents and measured grid-side currents; (c) estimating performance for currents and measured grid-side currents; (c) estimating performance for inverter-side three-phase inverter-side three-phase currents; (d) comparison of the estimated inverter-side currents and currents; (d) comparison of the estimated inverter-side currents and measured inverter-side currents; measured inverter-side currents; (e) estimating performance for three-phase capacitor voltages; (f) (e) estimating performance for three-phase capacitor voltages; (f) comparison of the estimated capacitor comparison of the estimated capacitor voltage and measured capacitor voltage. voltage and measured capacitor voltage.

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6. Conclusions This paper has presented an LQR-based current control design for an LCL-filtered grid-connected inverter. The proposed control scheme has been constructed by using the IM principle in the SRF to augment integral and resonant terms into a state feedback control. As a result, the control scheme successfully achieves control objectives such as an asymptotic reference tracking and disturbance rejection, which significantly reduces the impact of grid voltage distortion on the output current. Moreover, since the proposed scheme is implemented in the SRF, both the negative and positive sequence harmonics can be effectively compensated for by only one resonant term, which leads to a decrease in the number of regulators. This feature is usually preferred because the required THD performance can be met with further reduction in computation efforts. Furthermore, with the aim of avoiding the increase of sensing devices in an LCL-filtered grid-connected inverter system in comparison with L-filtered counterpart, a current full-state observer in the discrete-time domain has been discussed in detail to estimate all the state variables. On account of the augmentation of the resonant terms as well as the integral term into the inverter model, an increased number of feedback gains should be selected. To deal with such a limitation, an optimal LQR approach is adopted as a way of choosing the feedback gains systematically, in which the discrete cost function is minimized to satisfy the stability and robustness requirements of the system. As a result, both the system feedback and observer gains can be selected based on the LQR method in an effective and straightforward way. In addition to that, the control objectives such as the reference tracking and harmonic compensation capability can be effectively achieved without increasing the required number of sensing devices. In order to evaluate the feasibility and validity of the proposed control scheme, the whole control algorithm is implemented on 32-bit floating-point DSP TMS320F28335 to control 2 kVA prototype grid-connected inverter. Comprehensive simulation and experimental results are presented under distorted grid voltage conditions to demonstrate the usefulness of the proposed current control scheme. Author Contributions: T.V.T., S.-J.Y., and K.-H.K. conceived the main concept of the control structure and developed the entire system. T.V.T. and S.-J.Y. carried out the research and analyzed the numerical data with guidance from K.-H.K. T.V.T., S.-J.Y., and K.-H.K. collaborated in the preparation of the manuscript. Funding: This work was also supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), grant funded by the Korea government Ministry of Trade, Industry and Energy (NO. 20174030201840). Acknowledgments: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B03930975). Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3. 4.

5. 6.

Trinh, Q.N.; Lee, H.H. An advanced current control strategy for three-phase shunt active power filters. IEEE Trans. Indus. Electr. 2013, 60, 5400–5410. [CrossRef] Peña-Alzola, R.; Liserre, M.; Blaabjerg, F.; Sebastián, R.; Dannehl, J.; Fuchs, F. Analysis of the passive damping losses in LCL-filter-based grid converters. IEEE Trans. Power Electr. 2013, 28, 2642–2646. [CrossRef] Jia, Y.; Zhao, J.; Fu, X. Direct grid current control of LCL-filtered grid-connected inverter mitigating grid voltage disturbance. IEEE Trans. Power Electr. 2014, 29, 1532–1541. Han, Y.; Shen, P.; Zhao, X.; Guerrero, J. Control strategies for islanded microgrid using enhanced hierarchical control structure with multiple current-loop damping schemes. IEEE Trans. Smart Grid 2017, 8, 1139–1153. [CrossRef] Jin, W.; Li, Y.; Sun, G.; Bu, L. H∞ repetitive control based on active damping with reduced computation delay for LCL-type grid-connected inverters. Energies 2017, 10, 586. [CrossRef] Bolsens, B.; Brabandere, K.D.; Den Keybus, J.V.; Driesen, J.; Belmans, R. Model-based generation of low distortion currents in grid-coupled PWM-inverters using an LCL output filter. IEEE Trans. Power Electr. 2006, 21, 1032–1040. [CrossRef]

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9. 10. 11.

12. 13.

14. 15.

16.

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19. 20. 21. 22. 23.

24.

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