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Electric Power Systems Research

journal homepage: www.elsevier.com/locate/epsr

Observer-based second-order sliding mode control for grid-connected VSIwith LCL-type filter under weak grid

Bin Guoa, Mei Sua, Hui Wanga,c,⁎, Zhongting Tanga, Yuefeng Liaoa, Lei Zhangb, Shuqi Shia

a School of Automation, Central South University, Changsha 410083, Chinab School of mechanical and electrical engineering, Huainan Normal University, Huainan 232038, ChinacWasion Group Co., Ltd., Changsha 410205, China

A R T I C L E I N F O

Keywords:LCL filterPoles placementWeak gridsSliding mode observerSecond order sliding mode control

A B S T R A C T

LCL-type grid-connected voltage source inverters (VSI) have been widely used in distributed generation systems.However, due to the inherent LCL resonance, it brings a stability challenge to the control system. To address thisissue, a sliding mode observer-based second-order sliding mode control method is proposed to achieve arbitrarypoles placement, which can ensure desired resonance damping and dynamic performance even under weak gridoperation. With the help of sliding mode observer (SMO), only the grid current and grid voltage are sensed torealize system stable operation and grid synchronization, which reduces the cost of system. To improve therobustness and dynamic response of LCL-filtered grid-connected inverters under weak grid operation, a super-twisting algorithm (STA) second order sliding mode control (SOSMC) is proposed. Due to the existence of gridimpedance in weak grid, the sliding surface would drift, which increases the track error of grid current. To solvethis problem, an integral term is added to the sliding surface function. The systematic design approach of SOSMCis presented, and the stability of system under the grid impedance uncertainties as well as external disturbancesis proved by Lyapunov theory. Simulation results are finally presented to validate the effectiveness of the pro-posed control strategy.

1. Introduction

Grid-connected voltage source inverters (VSIs) have been widelyused in power conversion applications, such as distributed generation(DG) systems based on photovoltaic (PV), energy storage, and windturbine etc. [1]. The VSIs inject an ac current with the requirement ofspecified frequency and minimum total harmonics distortion (THD)into the grid via a filter [2,3].

In general, two types of filters (L-filter and LCL-filter) are applied toattenuate the switching harmonics of grid-connected inverter.Compared with L-filter, LCL-filter has the merits of high harmonicsattenuation at switching frequencies, which yields small size and costs[4]. However, the inherent resonance of LCL-filter requires properdamping methods to avoid the possible instability of the system [5,6].

A straightforward way to damp the resonance of the LCL filter is thepassive damping method, which introduces a real resistor in series orparallel with the filter capacitors or filter inductors [7-9]. Although thepassive damping attenuates the resonance peaks effectively, it results inextra power loss.

To avoid the power loss generated by the passive resistor, a lot of

active damping methods have been received more and more attentiondue to its high efficiency. Generally, there are two main types of activedamping solutions, including the multiloop active damping solutionsand singleloop active damping solutions. The multiloop active dampingsolutions include the capacitor current proportional feedback [10-12]or proportional integral feedback [13], the capacitor voltage feedback[14], the dual-current feedback [15], and the LC-trap voltage feedback[16]. Although multiloop active damping solutions are flexible andeffective, they require extra sensors, which will increase the costs ofsystem. To reduce the sensors, a lot of single-loop active damping so-lutions have been proposed in literatures [17,19-22]. They can be re-garded as two types, cascading digital filters based and inherentdamping based. In the cascading digital filters based single-loop activedamping solutions, they mainly consist of notch filter, adaptive notchfilter, biquad filter [17], and lead-lag network. However, these methodsare sensitive to the resonance frequency of LCL-filter. Thus, they mayprone to instability under the weak grid, where the system short circuitratio is less than 3 [18]. The digital time delay induces the inherent-damping effect into the current control loop, while the damping effectdepends on where the current is sensed and how long the delay time is

https://doi.org/10.1016/j.epsr.2020.106270Received 21 November 2019; Received in revised form 13 January 2020; Accepted 10 February 2020

⁎ Corresponding author.E-mail addresses: binguo_mail@csu.edu.cn (B. Guo), sumeicsu@csu.edu.cn (M. Su), wanghuicp9@csu.edu.cn (H. Wang), zta@et.aau.dk (Z. Tang).

Electric Power Systems Research 183 (2020) 106270

Available online 22 February 20200378-7796/ © 2020 Elsevier B.V. All rights reserved.

T

set as [19]. Tang et al. [20] first studied the inherent damping char-acteristic of inverter-side current feedback control. Yin et al. [21,22]explored the damping characteristic of time delay in a grid-side currentfeedback based single-loop control system. Wang et al. [23] in-vestigated the previous research work and summarized the time delayregion which can stabilize the system using only inverter-side current orgrid-side current feedback methods, i.e. ωres < ωs/6 for inverter-sidecurrent feedback, and ωres > ωs/6 for grid-side current feedback, whereωres and ωs are the resonant frequency and sampling frequency. Thoughthe inherent damping based methods require less sensors, it is difficultto independently apply them in a current-controlled inverter when thegrid impedance varies and the multiresonance peaks appear [19].

In fact, all the active damping methods described above can be seenas a partial state-feedback. As is known to all, a partial state-feedbackcannot achieve arbitrary pole placement of the closed-loop system,which limits the possibilities of obtaining the desired damping char-acteristic and dynamic performance in the case of grid impedancevaries. To realize arbitrary pole placement, it is sufficient to implementa full-state feedback [24]. This feedback, however, requires to measureone more current and one more voltage, which increases cost and de-creases reliability of system [25]. To reduce the number of sensors, theadditional feedback can be constructed by using a state observer. Ref[26] proposed a Luenberger observer based state-space current controlmethod to achieve pole placement. However, the performance of suchan observer may not be reliable in noisy environments as it is employedmainly for deterministic systems [27]. An extended state observer(ESO) was proposed in [28], where both the internal states and externaldisturbance can be extracted. However, sensitivity analysis was notconducted against grid-side parameter uncertainties, which is manda-tory with ESO strategy since it is sensitive to parameters variation andmeasurement noise. A Kalman filter-based active damping scheme wasproposed in [29] to improve the quality of the estimated variables innoise environments and model uncertainties. However, the online real-time recursive manner of Kalman filter increases the complexity andcomputation burden of system.

Numerous control methods for grid-connected VSIs have been pro-posed to realize fast and accurate current regulation, such as propor-tional integral (PI) control, proportional resonant (PR) control [30],model predictive control [31], repetitive control [32], hybrid control[33] and so on. Due to the inherent nonlinearity of the LCL-type grid-connected system, the aforementioned control strategies yield variousmerits and drawbacks related to steady-state error, dynamic response,control complexity, robustness, and can only fulfill parts of the per-formance requirements [3]. Alternatively, as a nonlinear controlmethod, the sliding mode control (SMC) has gained special interest dueto its superior merits like strong robustness to parameter variations andexternal disturbances, fast dynamic response and easy implementation.In [34], a SMC scheme via multiresonant sliding surface for single-phase LCL-filtered system is proposed to eliminate grid current trackingerror. In [35], a SMC strategy with PR controller is proposed for three-phase LCL-filtered system, where the PR controller is used to generatecapacitor voltage reference. Although the aforementioned SMCmethods showing satisfactory results, they need extra sensors. And thechattering issue is their major drawback, which may result in instabilityand electromagnetic interference (EMI) noise. To eliminate chatteringeffect, SMC has been implemented with double-band hysteresis control[2]. However, the switching frequency is mitigated but cannot be fixed.Chattering problem can also be overcome by replacing the sign functionwith saturation function [36] or sigmoid function [37]. However, thesemethods suffer from the degradation of tracking performance and ro-bustness. Second order sliding mode control (SOSMC) is an effectivesolution to reduce the chattering phenomenon without degradingsystem performance. However, it at the cost of increasing extra in-formation demand, which limits its widespread application.

In this paper, a sliding mode observer (SMO)-based robust SOSMCmethod is proposed for single-phase grid-connected VSI with LCL-filter.

The SMO is proposed to estimate the inverter-side current and capacitorvoltage by only measuring grid-side current and grid voltage. In apractical implementation, the grid voltage sensor is needed to achievegrid synchronization and island detection. Therefore, compared withmultiloop active damping method, the required sensor is reduced,which reduces cost and increases reliability of system. Moreover, todeal with the chattering problem of traditional SMC, a super-twistingalgorithm (STA) SOSMC is proposed. It does not require extra deriva-tive information of sliding mode variable and maintains the merits ofSMC. In order to realize arbitrary pole placement, the formation ofsliding surface function is based on inverter-side current error, capa-citor voltage error and grid-side current error. Thus, the desired re-sonance damping and dynamic performance can be ensured even inweak grid situation. The systematic design method of SMO-basedSOSMC is presented, and the stability of system under the grid im-pedance uncertainties as well as external disturbances is proved. Theeffectiveness and performance of the proposed method is investigatedusing a 1.5 kW system.

2. System modeling

Fig. 1 shows the configuration of single-phase LCL-filtered grid-connected VSIs, where L1, C1, L2 constitute the LCL filter. The gridimpedance is purely inductive with Lg here to present the worst case.Vdc, vpcc, and vg are dc voltage, grid voltage at the point-of-commoncoupling (PCC), grid voltage, respectively. vab denotes the inverteroutput voltage. i1, vc, and ig are inverter-side current, capacitor voltage,and grid current, respectively. Unipolar sinusoidal pulse-width mod-ulation (SPWM) is employed to obtain gating signals. Neglecting thegrid impedance Lg (which will be discussed in Section Ⅲ), the mathe-matical model of the system is expressed as

= −L didt

uV vdc c11

(1)

= −C dvdt

i icg1 1 (2)

= −Ldidt

v vgc g2 (3)

where u is control input and grid voltage vg = Vmsin(ωt).The transfer function from the grid-side current ig to the inverter

output voltage vab is calculated as

= =+ +

=+

G si sv s L L C s L L s sL

ωs ω

( )( )( )

1( )

1 ·g

ab

r

res1 2 13

1 2 1

2

2 2 (4)

where ωres and ωr are the LCL resonance angular frequency and the anti-resonance angular frequency, respectively. They are expressed as

= + =ω L LL L C

ωL C

, 1res r

1 2

1 2 1 2 1 (5)

From (5), it can be seen that the system is prone to instability due tothe two complex conjugate poles. Therefore, an effective resonancedamping should be proposed.

Fig. 1. Topology of the single-phase LCL-type grid-connected VSIs.

B. Guo, et al. Electric Power Systems Research 183 (2020) 106270

2

3. Observer-based second-order sliding mode control scheme

3.1. Sliding mode observer design

As mentioned previously, to decrease the cost and increase systemreliability, a SMO is proposed to replace the measurement sensors.Owing to the merit of strong robustness to parameter variations andexternal disturbances, SMO is very suitable for the case of weak gridoperation. Before designing the SMO, the observability of the systemshould be analyzed.

3.1.1. Observability analysisBy taking the variables i1, vc, ig as system states, (1)–(3) can be re-

presented in the state-space form, i.e.,

= + +=

X AX B DY CX

u˙(6)

where X= [i1 vc ig]T, B=[Vdc/L1 0 0]T, C=[0 0 1], and

=

⎡

⎣

⎢⎢⎢⎢

−

−

⎤

⎦

⎥⎥⎥⎥

= ⎡⎣⎢

− ⎤⎦⎥

A DvL

0 0

0

0 0

0 0 .L

C C

L

gT

1

1 1

12

1

1 1

2

Remark 1. The observability matrix of the system described by (6) isobtained as follow

⎡

⎣⎢⎢

⎤

⎦⎥⎥

=CCACA

rank 32 (7)

It can be seen that the matrix is full-rank, which means that thesystem is observable with only the grid-side current information.

3.1.2. Construction of the sliding mode observerDenote the observation errors = −e i i1 1 1, = −e v vc c2 ,

= −e i ig g3 . To enlarge the sliding patch of SMO, a linear output errorfeedback term is added to the observer. Thus, the SMO is constructed as

= + + + +

=

X AX B D G G

Y CX

u e v^ ^

^ ^l ny

(8)

where Gl=[l1 l2 l3]T, and Gn = [k1 k2 k3]T are the observer gain vector,= − = =C X X Cee e( ^ )y 3 is output estimation error, and ν is a dis-

continuous sliding mode term, which is given as

=ν ρ esign( )y (9)

where ρ is a positive constant.Subtracting (8) from (6), then the error dynamics is given as

= −e A e G v˙ n0 (10)

where e = [e1 e2 e3]T, A0 = (A-GlC).The design problem is transformed into determining the gain vector

Gl and Gn to ensure the convergence of error system. According toRemark 1, there exists a feedback gain vector Gl such that A0 is a strictlyHurwitz matrix. Thus, the gain Gl can be directly obtained by using poleplacement method. A rule of thumb is to place the observer poles to betwo to six times faster than the poles of the system controller [26], sothat the observer dynamics do not limit the bandwidth of controller.Proposition 1. if the gain vector Gn is properly selected so that the slidingmode motion is induced on the system error states, the error system shown in(10) is globally stable.

Proof. Based on an appropriate Lyapunov equation, there exist asymmetric positive definite matrix P as the solution of Lyapunovequation [38].

+ = −A P PA QT0 0 (11)

where Q is a symmetric positive definite matrix such that the followingconstraint is satisfied by a Lyapunov matrices pair (P, Q) for A0.

=C G PnT (12)

To prove the state estimation errors are asymptotically converged tozero, the following quadratic form is considered as a candidateLyapunov function, which gives

= e PeV 12

T(13)

Since the matrix P is a symmetric positive definite matrix, accordingto the properties of positive definite matrix, for ∀ ≠e 0, there hasV = eTPe > 0. Therefore, the Lyapunov function V is definitely posi-tive.

Taking the first time derivative of V gives

= +

= + − +

= − − +

= − −

= − −

= − −

e Pe e Pe

e PA A P e G Pe e PG

e Qe Ce Ce

e Qe

e Qe

e Qe

V

ν ν

ν

νe

ρ e e

ρ e

˙ 12

( ˙ ˙ )

12

( ) 12

( )

12

12

( ( ) )

1212

sign( )

12

nT

n

T

T T

T T T

T

Ty

Ty y

Ty

0 0

(14)

Since the matrix Q is a symmetric positive definite and ρ is a positiveconstant, thus the first time derivative of Lyapunov function V is ne-gative. Then the Proposition 1 is proven.

3.2. Second-order sliding mode controller design

In the LCL-filtered grid-connected VSIs, robustness against para-meters variation and external disturbances must be a key attribute ofinverter control to ensure the system stability and achieve satisfactoryperformance. This paper proposes a robust SOSMC to achieve effectiveresonance damping by using pole placement, good tracking of thecurrent reference, and strong robustness to parametric variations anddisturbance.

3.2.1. Sliding surface selection and control law designSince the inverter-side current i1, capacitor voltage vc and grid-side

current ig are measured or observed, the sliding surface function isconstructed as

= + +σ λ x λ x x1 1 2 2 3 (15)

where λ1, λ2 are positive constant. The state variables x1, x2, x3 aredefined as

= − = − = −x i i x v v x i i^ *, ^ *, *c c g g1 1 1 2 3 (16)

where i *1 , v*c and i*g are the reference of inverter-side current, capacitorvoltage and grid-side current, respectively. They are obtained as

=

= + = +

= + = − +

i I ωt

v L ddt

i v L I ω ωt v

i i C ddt

v I ωt C L I ω ωt C v

* sin( )

* * cos( )

* * * sin( ) sin( ) ˙

g m

c g g m g

g c m m g

2 2

1 1 1 22

1 (17)

where Im is the amplitude of grid-side current reference, ω is grid vol-tage angular frequency, which is obtained by a phase- locked loop(PLL).

The behavior of the system is described by the following state-spaceform:

B. Guo, et al. Electric Power Systems Research 183 (2020) 106270

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= + +x Ax B Du˙ 2 (18)

where x = [x1 x2 x3], A and B matrix are the same as (6), D2 is

= ⎡⎣⎢

− − ⎤⎦⎥

DvL

i* ˙ * 0 0 .c

T

21

1

Taking the first time derivative of sliding surface (15) gives

⎜ ⎟ ⎜ ⎟

= + +

= ⎛⎝

− + − − ⎞⎠

+ ⎛⎝

− ⎞⎠

+

σ λ x λ x x

λL

x VL

uvL

i λC

xC

xL

x

˙ ˙ ˙ ˙1 * ˙ * 1 1 1dc c

1 1 2 2 3

11

21 1

1 21

11

32

2

(19)

It can be seen that the first time derivative of sliding surface σcontains the control input u, thus, the relative degree is one. To keep thecontrol input u continuous and chattering free, a STA-based control isemployed.

The global control law u consists of two terms, the equivalentcontrol term ueq and the super-twisting control term ust. The equivalentcontrol term ueq is to deal with the term of known system dynamics, anda super-twisting control term ust is to deal with the uncertainties arisingfrom parameters errors and unmodeled dynamics.

Hence, the global control law u can be defined as

= +u u ueq st (20)

The equivalent control law ueq is obtained by solving the algebraicequation

=σ 0 (21)

Substituting (19) into (21), the ueq is calculated as

⎜ ⎟⎜ ⎟= ⎛⎝

− + ⎛⎝

− ⎞⎠

+ + ⎞⎠

uV

λ Lλ C

x x Lλ L

xvL

i1 ( ) 1* ˙ *eq

dc

c2 1

1 13 1

1

1 22

11

(22)

The expression of super-twisting term ust is given as

= − +

= −

uV

α σ σ υ

υ α σ

1 ( sign( ) )

˙ sign( )

stdc

11/2

2 (23)

where α1, α2 are positive constant.

3.2.2. Stability and robustness analysisSince in weak grid operation, there exist external disturbance and

grid impedance Lg is not zero. Therefore, the stability of system and itsrobustness against to external disturbances and grid impedance varia-tion should be analyzed.

For easy analysis, let's define the grid impedance Lg as

= = −L L L LΔg n2 2 2 (24)

where L2 and L2n are the actual and nominal values of grid-side in-ductance of LCL-filter, respectively. That is, the grid impedance Lg isregarded as the variation value of inductance L2.

Then (17) is rewritten as

= + = −

= + = −

v L ddt

i v v L ddt

i

i i C ddt

v i C L ddt

i

˜* * * *

˜* * ˜* * *

c n g g c g g

g c g g

2

1 1 1 12

2 (25)

where ‘~’ denotes the nominal values thereafter.So the state variables x1 and x2 are rewritten as

= +

= +

x x C L ddt

i

x x L ddt

i

˜ *

˜ *

g g

g g

1 1 12

2

2 2 (26)

Thus, the sliding surface becomes

= + + = +σ λ x λ x x σ ξ˜ ˜ ˜ Δ1 1 2 2 3 (27)

where = +ξ λ C L i λ L iΔ * *gddt g g

ddt g1 1 2

22 , σ is the nominal sliding surface

function, i.e., ideal sliding surface function. σ is the actual sliding sur-face function.

Since the control law shown in (20) is designed based on theknowledge of ideal sliding surface, taking the first time derivative of(27), and substituting (20) into (27), the closed-loop system is derivedas

= − + += −

σ α σ σ υ dυ α σ˜ ˜ sign( ˜)˙ sign( ˜)

11/2

2 (28)

where d is a bounded disturbance term due to measurement error andany other external disturbances.Proposition 2. Assume that the disturbance d of the system (28) is globallybounded as [39]

≤d δ σ 1/2 (29)

where δ is a known positive constant.

And if the gains α1, α2 satisfy Eq. (30), the system is globallyasymptotic stable.

⎧⎨⎩

>

> +−

α δ

α α

2α δ δ

α δ

1

2 15 42( 2 )

1 2

1 (30)

Proof. To prove the stability of the system, a Lyapunov function ischosen as

= + + −

= ζ P ζ

V α σ υ α σ σ υ2 ˜ 12

12

( ˜ sign( ˜) )

T

22

11/2 2

2 (31)

where

= = ⎡⎣⎢

+−

− ⎤⎦⎥

ζ Pσ σ υ α αα

α[ ˜ sign( ˜) ], 12

42 .T 1/2

22 1

2

1

1

Its time derivative alone the solution of (23) results as follows

= − +ζ Q ζ q ζVσ

dσ

˙ 1˜ ˜

T T1/2 2 1/2 (32)

where

⎜ ⎟= ⎡⎣⎢

+−

− ⎤⎦⎥

= ⎡

⎣⎢

⎛⎝

+ ⎞⎠

− ⎤⎦⎥

Q qα α αα

α αα α

22

1 , 22 2

.T2

1 2 12

1

12

12

1

Appling the bounds shown in (29), we can get that

≤ − ζ Q ζVσ

˙ 1˜

˜T21/2 (33)

where

=⎡

⎣⎢⎢

+ − +

− +

− + ⎤

⎦⎥⎥

( )Q α α α α δ

α δ

α δ˜2

2

( 2 )

( 2 )1

.α

α21 2 1

2 41

1

12

1

It is obvious that V is negative if >Q 02 , which represents thesystem is globally asymptotically stable if (30) is satisfied.

From the above analysis, it can be concluded that the grid-con-nected inverter system with LCL-filter is stable even works in weak grid.Thus, when sliding mode occurs, =σ 0 and =σ 0 are guaranteed.However, as shown in (27), since the grid impedance Lg is not zero inthe weak grid situation, the actual sliding surface function is drift,which increases the tracking error of grid current.

3.2.3. Integral second order sliding mode controllerTo deal with the aforementioned problem, the integral of grid cur-

rent error x3 is added to the sliding surface function, which is shown as

∫= + + +σ λ x λ x x λ x dτ1 1 2 2 3 3 3 (34)

B. Guo, et al. Electric Power Systems Research 183 (2020) 106270

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Since in weak grid situation, the sliding surface function is drift.Consequently, (34) is rewritten as

∫∫

= + + + = +

= + + + +

σ λ x λ x x λ x dτ σ ξ

λ x λ x x λ x dτ ξ

˜ ˜ ˜ Δ

Δ

1 1 2 2 3 3 3

1 1 2 2 3 3 3 (35)

Transforming the sliding surface function (35) into the frequencydomain, we can get that

= ⎛⎝

+ + + + ⎞⎠

+σ s λ C L s λ L s λ λs

x s ξ s˜ ( ) 1 ( ) Δ ( )1 1 22

2 2 13

3 (36)

When the sliding mode occurs, =σ 0 and =σ 0, thus, it has

⎛⎝

+ + + + ⎞⎠

= −λ C L s λ L s λ λs

x s ξ s1 ( ) Δ ( )1 1 22

2 2 13

3 (37)

Rewriting (37) as

=−

=+ + + +

G s x sξ s

sλ C L s λ L s λ s λ

( ) ( )Δ ( ) ( ( 1) )3

1 1 23

2 22

1 3 (38)

Substituting the parameters listed in Table 1 into (38), the bodediagram of the transfer function G(s) with and without the integral termis shown in Fig. 2. It can be seen that, after adding the integral term tothe sliding surface function, the attenuation gain of low frequency bandis increased significantly, which reduces the tracking error of gridcurrent.

3.2.4. Parameters designIn the proposed SMO, there are six parameters need to be designed.

Regarding the gain vector Gl, it can be directly obtained by using poleplacement method. If the characteristic polynomial of the observerdynamic is selected as

− + = +I A G Cs s ωdet( ) ( )l n3 (39)

where ωn determines the dynamics of the observer. A rule of thumb is toplace ωn to be two to six times faster than the dynamics of the systemcontroller [26], so that the observer dynamics do not limit the band-width of controller.

Based on the system parameters listed in Table 1 and according to(39), the gain vector Gl can be obtained as

= × × ×G [1.57 10 3.92 10 4.71 10 ]lT4 5 4 (40)

According to an algorithm for the design of P [40], and letQ = I3×3, the symmetric positive definite matrix P can be determinedas

= × ⎡

⎣⎢

− −− −− −

⎤

⎦⎥−P 1 10

0.1623 0.0075 0.00280.0075 0.0649 0.00030.0028 0.003 0.0138

3

(41)

Therefore, according to (12), the gain vector Gn can be derived as

= ×G 1 10 [0.1261 0.0519 7.2489]nT4 (42)

There are five parameters related to SOSMC, α1, α2, λ1, λ2 and λ3,which plays a significant role in stability, steady-state and dynamicperformance of the system.

According to (30), the equality must be satisfied to guarantee thesliding variable converges to zero in finite time. Hence, gains α1, α2 canbe easily selected based on the bounded disturbance shown in (29).

The closed-loop current control dynamics and damping character-istics depend on the selection of parameters λ1, λ2 and λ3, which is setthrough direct pole placement. During the sliding mode, (34) is writtenas

+ + + =x n x n x n x¨ ˙ 03 1 3 2 3 3 3 (43)

where

= = + =n λλ C

n λλ C L

n λλ C L

, 1 ,12

1 12

1

1 1 23

3

1 1 2

Taking a Laplace transform of (43), the characteristic polynomial ofclosed-loop system is derived as

+ + + = + + + =s n s n s n s ω s ζω s ω( ) ( 2 ) 031

22 3 0

Dominant dynamics

21 1

2

Resonant dynamics

(44)

where ω0 is the natural frequency of the dominant dynamics, whichdetermines the bandwidth of system. ω1 and ζ are the natural frequencyand damping ratio of the resonant dynamics, respectively. They de-termine the resonant damping of system.

The natural frequency ω0 is related to the dynamics response ofsystem, thus, it can be selected by the desired bandwidth. To keepcontrol effort low, the resonant pole pair should be set near the LCLresonance angular frequency (i.e., ω1 ≈ ωres) with typical damp ratioζ = 0.1……0.4 [41]. When the pole locations are placed, the para-meters λ1, λ2 and λ3 can be determined by solving (44).

4. Simulation results

To verify the effectiveness of the proposed method, a 1.5 kW LCL-filtered single-phase grid-connected VSI simulation model has beenbuilt and tested in Simulink/MATLAB. To be as close to the actualsystem as possible, in the numerical simulation, the sampling noise,dead-time and time delay are taken into account. The key system andcontrol parameters are listed in Table 1 and the control block diagramof the closed-loop system is shown in Fig. 3.

4.1. Test of steady-state performance

To verify the superior performance of the proposed sliding modeobserver (SMO), the conventional Luenberger observer method isadopted here to make a comparison. Figs. 4 and 5 show the waveforms

Table 1System and control parameters.

Parameters Values Parameters Values

Output power: Po 1.5 kW Observer gain 1: l1 1.57 × 104

DC-link voltage: Vdc 200 V Observer gain 2: l2 3.92 × 105

Grid voltage: vg (rms) 110 V Observer gain 3: l3 4.71 × 104

Grid frequency: fg 50 Hz Observer gain 4: k1 1.26 × 103

Switching frequency: fsw 20 kHz Observer gain 5: k2 5.19 × 102

Sampling frequency: fs 20 kHz Observer gain 6: k3 7.2 × 104

Dead time: td 0.8 us SOSMC gain 1: α1 1.1 × 104

Inverter-side inductor: L1 1.3 mH SOSMC gain 2: α2 6 × 104

Filter capacitor: C1 15 uF SOSMC gain 3: λ1 1.48Grid-side inductor: L2 0.6 mH SOSMC gain 4: λ2 0.13Grid impedance: Lg 0/2 mH SOSMC gain 5: λ3 1080

Fig. 2. Bode diagram of G(s) with and without integral term.

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of inverter-side current i1, capacitor voltage vc, grid-side current ig, theirobserved value and observation errors with the two observer methodsin the case of grid current reference step change and filter parametersvariations (L1 reduced by 30%, C1 increased by 20% and 2 mH gridimpedance Lg is existed). As can be seen in Figs. 4 and 5, the steady-state observation errors of i1, vc and ig with the traditional Luenbergerobserver method are 1A, 15 V and 1.3 A, respectively. While, the ob-servation errors of i1, vc and ig with the proposed method are 0.5 A, 5 Vand 0.3 A, respectively. In addition, compared with the traditionalobserver, the transient time of the proposed observer is reduced by37.5%. From the above observations, it is obvious that the proposedstrategy outperforms regarding the steady-state (small observation er-rors) and dynamic performance (small setting time). This is because asaforementioned in the introduction, the Luenberger observer is sensi-tive to system parameters and measurement noise, while the proposedSMO has the merits of fast dynamic response and insensitive to para-metric variations.

Fig. 6 shows the steady-state waveforms of grid voltage and currentin the case of the grid impedance is 0 mH, where the integral term ofsliding surface function is not added to the controller. From Fig. 6, itcan be seen that the grid current is almost sinusoidal with the THD is1.53%. What's more, the maximum tracking error of grid current is lessthan 0.2 A. Therefore, it can be concluded that the proposed controllerwithout the integral term can achieve zero steady-state error tracking

when the parameters of LCL-filter are nominal.

4.2. Test of dynamic performance

Fig. 7 shows the dynamic performance of grid current and voltagefor a step change in the current reference amplitude Im from 10 A to 20A and back to 10 A. From Fig. 7, it can be seen that the system has smallovershoot and very short tracking time with less than 5 ms, whichshows that the proposed method has a very good dynamic performance.

4.3. Test of robustness against to external disturbance and parametersvariations

Fig. 8(a) gives the results of the steady-state waveform of gridcurrent with the proposed method under a distorted grid, where theTHD of grid voltage is 4.35%. Fig. 8(b) shows the steady-state spectrumof current, where THD of grid current is 1.87%. Clearly, the proposedcontrol strategy offers a very good harmonic suppression capability.

Fig. 9(a) shows the simulation results of the grid voltage and currentfor a 10% step change in the grid voltage, i.e., the peak value of gridvoltage step change from 155 V to 171 V. Fig. 9(b) shows the dynamicwaveforms of dc-link voltage, grid voltage, grid current and the dutyratio for a 25% step change in the dc-link voltage. The step change ofgrid voltage and dc-link voltage can be viewed as the perturbation

Fig. 3. Control block diagram of the closed-loop system with the proposed method.

Fig. 4. Simulation waveforms of i1, vc, ig and their observed value with the traditional Luenberger observer method. (a) Waveforms; (b) Observation errors.

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applied to the system. From Fig. 9(a), it can be seen that the grid cur-rent has a little oscillation. This is because the step change of gridvoltage will affect the observed capacitor voltage vc in controller.Nevertheless, the system convergences to stable within half of powercycle. As can be seen in Fig. 9(b), although the dc input voltage Vdc stepchanges from 200 V to 250 V, the grid current can track the changequickly with an overshoot of 12%. From the above observation, it canbe concluded that the proposed method has a strong disturbance re-jection ability as pointed out in Section Ⅲ.

To test the robustness against to parameters uncertainties, first, thevariations of filter parameters L1 and C1 are taken into account in thesimulation system. Since the capacitance tolerances of film capacitors

are usually small than±20%, the change of± 20% from their nominalvalues is taken as an example. What's more, the inductance value ofinverter-side inductor L1 changes by± 30% is considered. Fig. 10(a)shows the simulation results of parameters L1 increased by 30% and C1

reduced by 20%. Fig. 10(b) shows the results of parameters L1 reducedby 30% and C1 increased by 20%. From Fig. 10, it can be seen that,compared with Fig. 6, the system performance has slightly declined.Nevertheless, it still exhibits a good steady-state performance. Similarresults are obtained for 0.7L1n, 0.8C1n and 1.3L1n, 1.2C1n. From theseresults, it can be seen that the proposed method provides robust andstable control satisfactorily to parameter variations.

Second, to test the robustness of the controller in weak grid, a 2 mHinductor Lg is added as the grid impedance to emulate the worst case. Inthis situation, the integral term of sliding surface function is not addedinto the controller. Fig. 11(a) shows the simulation result, as can beseen in Fig. 11(a), the system is stable with the THD of grid current is1.81%. However, there exists a considerable steady-state error in thegrid current (approximately 1A). This is because when in the weak grid,the grid impedance Lg can be regarded as the variation of grid-sideinductor L2. Therefore, as analysis in Section Ⅲ, the actual slidingsurface function is drift, which increases the tracking error of gridcurrent. Fig. 11(b) shows the simulation result when the integral term isactivated in the controller. It can be seen that, after the integral term isenabled at time t2, the current error is reduced to nearly zero (0.17 A)within one power cycle. Thus, it can be concluded that the proposedmethod is robust to parameters variations.

5. Experimental results

A prototype of a 1.5 kW single-phase LCL-type grid connected VSIsystem is developed in the laboratory as shown in Fig. 12. The con-troller is implemented by a DSP (TMS320F28335) and the driver signalsof IGBT are generated by Altera FPGA. A Chroma DC source and pro-grammable AC source are adopted as the source and grid. The systemand control parameters of experiment are the same as those in simu-lation, which can be seen in Table 1.

5.1. Test of steady-state performance

First, the steady-state performance of the system with the proposedcontrol strategy is verified by experiment. Fig. 13 shows the waveformsof grid voltage, grid current and its harmonic spectrum. It can be seenthat the system is stable and the grid current is sinusoidal with the THDis 1.75%, which is almost consistent with the simulation results. It isworth noting that the resonance frequency of the system is less than 1/6of the sampling frequency. However, as mentioned in the introduction,for the conventional grid-side current feedback methods, the system is

Fig. 5. Simulation waveforms of i1, vc, ig and their observed value with the proposed SMO method. (a) Waveforms; (b) Observation errors.

Fig. 6. Steady-state performance of the system with the proposed method in thecase of grid impedance Lg is zero.

Fig. 7. Dynamic response of grid voltage vg and current ig for a step change in Imfrom 10 A to 20 A and back to 10 A.

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stable only under the conditions of the system resonance frequency isgreater than 1/6 of the sampling frequency. Therefore, we can concludethat, with the proposed method, the adaptability of LCL-type VSI systemis significantly improved. This is because as aforementioned, with thehelp of the proposed method, the desired resonance damping and dy-namic performance can be achieved by arbitrary poles placement.

5.2. Test of dynamic performance

Fig. 14(a) and (b) shows the dynamic response of grid voltage andcurrent for a step change in grid current reference amplitude Im form

10 A to 20 A and back to 10 A, respectively.It clearly shows that the system has very short setting time with less

than one power frequency cycle and nearly without overshoot.Therefore, it can be concluded that the system has very good dynamicperformance.

5.3. Test of robustness against to external disturbance and parametersvariations

Fig. 15 gives the experimental results of grid voltage vg and gridcurrent ig under a distorted grid, where the THD of grid voltage is

Fig. 8. Simulation results of grid voltage vg and current ig under a distorted grid. (a) Simulation waveform of vg and ig, (b) Spectrum of current ig.

Fig. 9. Simulation results of the system with the proposed method for the following: (a) Step change of vg from 155 V to 170 V (peak value) and (b)Step change of vdcfrom 200 V to 250 V.

Fig. 10. Simulation results of the system with the proposed method in the case of filter parameters variations.

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3.46%. It can be seen from Fig. 15 that, although the grid voltage ishighly distorted, the grid current ig is still sinusoidal with small THD.Clearly, the proposed method offers very good harmonic suppressioncapability.

Fig. 16 shows the experimental results under the conditions of gridvoltage amplitude step change from 150 V to 170 V at time t3. Fig. 17shows the dynamic waveforms of dc-link voltage, grid voltage andcurrent for a 25% step change in the dc-link voltage at time t4. The stepchange of grid voltage and dc-link voltage can be viewed as the per-turbation applied to the system. From Figs. 16 and 17, it can be seenthat the grid current is hardly affected. From these results, it can beconcluded that the proposed method has strong robustness to externalrobustness.

To show the robustness of the proposed method to parametersvariations, the system performance is tested under weak grid, where a 2mH inductor Lg is added to emulate the grid impedance. In this op-eration, the integral term is included in the sliding surface function.Fig. 18 shows the experimental results under weak grid operation.Obviously, compared with Fig. 13, the THD of grid current is increased.Nevertheless, the system is still stable with an acceptable power quality.

5.4. Sensitivity analysis

Sensitivity analysis is carried out to study the robustness of theproposed method against the wrong knowledge of the system para-meters. As analysis in SectionⅢ, the filter parameters L1, C1 and L2 playan important role in determining the stability and performance ofsystem. Therefore, the variation of± 20% capacitance value C1

and± 30% inductance value L1 in controller are considered. Fig. 19shows the experimental results of parameters L1 increased by 30% andC1 reduced by 20% in controller. It can be seen that the steady-state anddynamic performance are almost the same as the results shown inFigs. 13 and 14. Similar results are obtained for 0.7L1n, 1.2C1n; 0.7L1n,0.8C1n and 1.3L1n, 1.2C1n. Therefore, these results verified that theproposed method is robust and insensitive to system parameters var-iations.

Fig. 11. Simulation results of the system with the proposed method in weak grid (Lg = 2 mH) (a) The integral term is disabled in controller; (b) The integral term isactivated in controller at the time t2.

Fig. 12. Experimental setup.

Fig. 13. Steady-state performance of the system with the proposed method inthe case of grid impedance Lg is zero. (a) Experimental waveform; (b) Harmonicspectrum of grid current.

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6. Conclusion

This paper proposed a SMO-based SOSMC method for single-phasegrid-connected LCL-type VSI under weak grid operation. A robust SMOis proposed to observe the inverter-side current and capacitor voltageby only sampling the grid-side current and grid voltage, which reducesthe cost of system. Since the inverter-side current and capacitor voltageare observed, the desired resonance damping and dynamic performancecan be achieved by arbitrary poles placement. More importantly, withthe proposed method, the resonance frequency of an LCL filter can be

designed to be less than 1/6 of sampling frequency when only the grid-side current is sampled. Thus, the adaptability of LCL-type inverters inweak grid is significantly enhanced. What's more, to improve the ro-bustness and dynamic response of the system, a chattering free robustSTA-SOSMC method is proposed. It is worth noting that there exists asteady-state error when system operates in weak grid. To solve thisissue, an integral term is added to the sliding surface function. Theinfluences of parameters variations and external disturbances on thesystem are studied, and the stability of the system is proved byLyapunov approach. Finally, various simulation and experiment results

Fig. 14. Experimental response of ig and vg for a step change in Im from 10 A to 20 A and back to 10 A. (a) Im step changes from 10 A to 20 A; (b) Im step changes from20 A to 10 A.

Fig. 15. Experimental results of grid voltage vg and current ig under a distortedgrid.

Fig. 16. Experimental results of grid voltage vg and current ig is the case of gridvoltage amplitude step changes from 150 V to 170 V.

Fig. 17. Experimental results of grid voltage vg and current ig in the case of dc-link voltage Vdc step changes from 250 V to 200 V.

Fig. 18. Experimental results of grid voltage vg and current ig is the case of weakgrid (Lg = 2 mH).

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are presented to verify the effectiveness of the proposed controlmethod.

CRediT authorship contribution statement

Bin Guo: Data curation, Writing - review & editing. Mei Su:Supervision. Hui Wang: Conceptualization, Methodology. ZhongtingTang: Software, Validation. Yuefeng Liao: Software, Validation. LeiZhang: Formal analysis. Shuqi Shi: Writing - review & editing.

Declaration of Competing Interest

All the authors declare that they have no known competing financialinterests or personal relationships that could have appeared to influ-ence the work reported in this paper.

Acknowledgement

This work was supported in part by the National Key R&D Programof China under Grant 2018YFB0606005, in part by the National NaturalScience Foundation of China under Grant 61873289 and 61933011, inpart by Hunan Provincial Key Laboratory of Power ElectronicsEquipment and Grid under Grant 2018TP1001, in part by the MajorProject of Changzhutan Self-Dependent Innovation Demonstration Areaunder Grant 2018XK2002 and in part by the Hunan ProvincialInnovation Foundation for Postgraduate under Grant CX20190122.

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