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TRANSCRIPT
Variable Structure Control in Natural Frame forThree-Phase Grid-Connected Inverters with LCL
FilterRamon Guzman, Luis Garcia de Vicuna, Miguel Castilla, Jaume Miret, Member, IEEE and Helena
Martın
Abstract—This paper presents a variable structure con-trol in natural frame for a three-phase gird-connectedvoltage source inverter with LCL filter. The proposedcontrol method is based on modifying the convertermodel in natural reference frame, preserving the lowfrequency state space variables dynamics. Using this modelin a Kalman filter, the system state-space variables areestimated allowing to design three independent currentsliding-mode controllers. The main closed-loop featuresof the proposed method are: 1) robustness against gridinductance variations because the proposed model is in-dependent of the grid inductance, 2) the power lossesare reduced since physical damping resistors are avoided,3) the control bandwidth can be increased due to thecombination of a variable hysteresis comparator with theKalman filter, and 4) the grid-side current is directlycontrolled providing high robustness against harmonics inthe grid. To complete the control scheme, a theoretical sta-bility analysis is developed. Finally, selected experimentalresults validate the proposed control strategy and permitillustrating all its appealing features.
Index Terms—Grid current control, LCL filter, Slidingmode control, Kalman filter, Voltage sensorless.
I. INTRODUCTION
This work has been supported by the European Union ProjectELAC2014/ESE0034 and its linked to Spanish National ProjectPCIN-2015- 001 and also supported by the Ministry of Economyand Competitiveness of Spain under project ENE2015-64087-C2-1-R. Recommended for publication by Associate Editor xxx.
Ramon Guzman is with the Department of Automatic Control,Technical University of Catalonia, 08800 Vilanova i la Geltru, Spain(e-mail: ramon. [email protected]).
Luis Garcia de Vicuna, Miguel Castilla, and Jaume Miret arewith the Department of Electronic Engineering, Technical Univer-sity of Catalonia, 08800 Vilanova i la Geltru, Spain (e-mail: [email protected])
Helena Martn is with the Department of Electric Engineering,Technical University of Catalonia, Barcelona East School of Engi-neering, 08019 Barcelona, Spain (e-mail: [email protected]).
Color versions of one or more of the figures in this paper areavailable online at http://ieeexplore.ieee.org.
Digital Object Identifier xxxxx
POWER converters are commonly used in manypower applications such as uninterruptible power
supplies, unity power factor rectifiers and voltage sourceinverters (VSI’s). The controller design is a difficulttask when the converter is equipped with an LCL filterdue to its inherent resonance problem, specially in gridconnected inverters used in distributed generation wherea large set of grid impedance values may affect thesystem stability [1].
Kalman filter (KF) based-control has been widelyused in power electronics [2], where accurate powerconverter models are considered to estimate the statevariables. Unlike these control methods, the model-basedcontrol presented in this paper uses a modified state-space model which preserves the low frequency state-space variables dynamics, and allows to design a robustsliding mode control (SMC) in natural reference frame.The use of the SMC technique improves the trackingbehaviour and dynamic performances, providing fastdynamic response with high robustness against systemparameters variations [3], [4].
The SMC technique was also introduced in a single-phase grid-connected LCL-filtered VSI with interestingproperties as fast dynamic response, robustness, and si-nusoidal grid currents with low total harmonic distortion[5], [6]. Besides, the use of SMC has been applied tothree-phase active power filters in stationary and rotatingreference frames [7]. Other relevant works regarding theSMC technique can be found in [8]–[13].
Traditionally, VSIs with LCL filter have been con-trolled by means of the inverter-side current. Recentreferences can be found regarding to the control ofthe grid-side current for single-phase systems [14], [15]or for three-phase systems [16]–[18]. In most of theseproposals the use of digital filters is needed in other toimplement the control algorithm, and as a consequencethe robustness against system parameters variations maybe compromised.
Conversely, and as per authors knowledge, there areno references in the literature about the use of the SMC
in abc frame applied to control the grid-side current.Besides, with a direct control of the grid-side currenthigh robustness against a distorted grid is achieved, andas a consequence, a reduction of the total harmonicdistortion (THD) of the currents injected to the grid.Then, with this method, digital filters in the closed loopsystem are not required.
This paper proposes a sliding-mode controller basedon a KF in order to perform a direct control of the currentinjected to the grid by imposing a desired dynamics.The proposed controller scheme estimates the state-spacevariables including the voltages at the point of commoncoupling (PCC). The SMC parameters are obtained byanalyzing the system stability taking into account theinfluence of the KF. The estimated PCC voltages areused in a hysteresis current control (HCC) reducing theswitching noise and improving the switching spectrum.The main advantage of this method is that it is notnecessary to compensate the grid harmonics in orderto achieve sinusoidal grid currents with a low THD[19]. Besides, since the control method is based oncontrolling the grid-side current instead of the inverter-side current some problems are avoided such as: 1) thesensitivity to parameter uncertainties specifically withthe grid inductance variations, and 2) the phase-shiftbetween the current injected to the grid and its reference,which is more important for low values of active power.
Another interesting property is that the averaged neu-tral point voltage can be changed by modifying thespace-state model of the converter used in the KFalgorithm. As an example a third harmonic voltage canbe injected in the neutral point in order to increase thecontrol dynamic range. This property is useful in PVapplications.
A methodology to analyze and design a robust con-troller of the grid current in natural reference frame ispresented. This method is based on following steps:
1) A modified state-space model of the VSI is pro-posed to design three independent current con-trollers in natural reference frame.
2) The switching surfaces are designed in order toobtain a closed loop dynamics independent ofthe system parameters, using estimated variablesobtained from a KF.
3) A complete stability analysis is performed takinginto account the effect of the KF, the system dis-cretization and the deviation in the system param-eters. This analysis proves the system robustness.
4) Finally, experimental results are reported to provethe aforementioned properties, including the con-troller response against a distorted grid and voltagesags.
The paper is organized as follows. In Section II alinear model of the VSI with LCL filter is presented.Section III introduces the conventional SMC using theinverter-side current. Section IV presents the proposedcontrol system. A stability analysis is presented in sec-tion VI. Experimental results are reported in section VI.Finally, section VII concludes the paper.
II. M ODELING OF VSI WITH LCL FILTER
A circuit scheme of a grid-connected inverter withLCL filter is depicted in Fig.1. The three-phase systemequations can be written as follows:
L1
di1dt
=Vdc
2u− vc − vnI1 (1)
Cdvc
dt= i1 − i2 (2)
L2
di2dt
= vc − v + (vn − v′n)I1 (3)
where i1 = [i1a i1b i1c]T are the inverter-side cur-
rents, i2 = [i2a i2b i2c]T are the grid-side currents,
vc = [vca vcb vcc]T are the capacitor voltages,
v = [va vb vc]T are the voltages at the PCC,
u = [ua ub uc]T are the control variables,vn and v
′
n
are the voltages at the neutral points andI1 is a columnvector defined as[1 1 1]T .
Assuming that the grid voltages are balanced (i.e.,vga + vgb + vgc = 0 and vca + vcb + vcc = 0), theexpressions for the neutral point voltages can be reducedas follows:
vn = v′n =Vdc
6(ua + ub + uc). (4)
The equations for the VSI model can be rewritten asa discrete space-state model. The discrete equations willbe used in the stability analysis section in order to findthe control parameters. The process and measurementdiscrete equations for each phase-legi are expressed asfollows:
xi(k+1) = Axi(k)+Bui(k)+Dvi(k)−Ts
L1
vn(k) (5)
yi(k) = Cxi(k) (6)
where
xi(k) = [i1i(k) vci(k) i2i(k)]T (7)
is the state space vector. The remaining matrices aredefined as:
A =
1 − Ts
L1
0Ts
C1 −Ts
C
0 Ts
L2
1
(8)
Vdc/2
Vdc/2ua ub uc
ua ub uc
i1aL1 i2a
L2Lg
vga
i1bL1 i2b
L2Lg
vgb
i1cL1 i2c
L2Lg
vgc
v′
n
+
−vca
+
−vcb C
+
−vcc
+ va −
+ vb −
+ vc −
vn
Fig. 1. Circuit diagram of three-phase grid-connedcted inverter with LCL-filter
B = [VdcTs
2L1
0 0]T (9)
C = [0 0 1] (10)
D = [0 0 − Ts
L2
]T (11)
beingTs the sampling time.
III. C ONVENTIONAL SMC USING THE
INVERTER-SIDE CURRENT
As a first approach, a sliding surface vector usedto control the inverter-side currents can be defined asfollows:
S = i∗ − i1 (12)
where the reference current vectori∗ = [i∗a i∗b i∗c ]
T isgiven in [20], and defined as follows:
i∗a =P ∗
|v|2 va +Q∗
√3|v|2
(vb − vc) (13)
i∗b =P ∗
|v|2 vb +Q∗
√3|v|2
(vc − va) (14)
i∗c = −(i∗a + i∗b) (15)
being |v|2 = v2a + v2b + v2c =3V 2
p
2with Vp the peak
voltage value, andP ∗ and Q∗ the active and reactivepower references, respectively.
When the system is in sliding regime, the converterdynamics is forced to evolve over the sliding surface,and the new dynamics can be derived according to theinvariance conditions,S = 0 and S = 0 [21]. Byapplying these conditions to (1)-(3), the following grid-side current differential equation can be found:
L2Cd2i2dt2
+ i2 = i∗ − C
dv
dt. (16)
The output current dynamics exhibits an oscillatorybehavior, as it can be deduced from (16) where the reso-nance frequency is given by1/2π
√L2C. The straightfor-
ward solution is to apply passive damping by connecting
a resistor in series with the capacitor filterC. Then, theideal sliding-mode dynamics results in:
i1 = i∗ (17)
Cdvc
dt= i
∗ − i2 (18)
L2
di2dt
= vc − v+Rd(i∗ − i2) (19)
whereRd is the damping resistor. From the last equa-tions, the closed-loop grid-side current differential equa-tion can be derived as follows:
L2Cd2i2dt2
+RdCdi2dt
+ i2 = i∗ +RdC
di∗
dt− C
dv
dt.
(20)
Assuming a grid-connected application, where only ac-tive power is delivered to the grid, the reactive poweris not considered (i.e.Q∗=0). Then, by applying theLaplace transform to 20, and using (13)-(15) withQ∗=0,the transfer function between the grid-side current andits reference is found:
i2(s) =1 + (Rd − 3V 2
p
2P ∗)Cs
L2Cs2 +RdCs+ 1i∗(s) = H(s)i∗(s). (21)
As it can be deduced from (21), a phase shift betweeni2 andi∗ is produced. Note that this phase shift dependsof the reference active power value,P ∗.
IV. PROPOSEDCONTROL SYSTEM
The control scheme for phase-legi is depicted inFig.2. The control block diagram consists of a KF,a reference neutral point voltage calculator, and theswitching surface used together with a variable hysteresisband combined with a switching decision algorithm [20].This combination allows to obtain an improved switch-ing spectrum concentrated around the desired switchingfrequency.
Besides, in this section, a KF algorithm based on amodified space-state model is presented. Finally, in orderto obtain the control parameters a complete stabilityanalysis is performed including the KF effect.
KFphasei
(36)
i2i
Switching surfacei(43)
i1i,vi ,i2i Si
Reference NeutralPoint Voltage Calculator
(33)
P ∗ ωo va vb vc
HysteresisBand Generator
Switchingdecisionalgorithm
ui
V ∗dc
vi
hi vi Ts
v∗n
Fig. 2. Proposed control system for phase-legi of the VSI
A. Kalman Filter
1) Proposed Space-State model of the VSI: The pro-posed state-space model is defined by the followingdifferential equations:
L1
di1dt
=Vdc
2u− vc − v∗nI1 (22)
Cdvc
dt= i1 − i2 (23)
L2
di2dt
= vc − v (24)
dv
dt= ωovq (25)
dvq
dt= −ωov (26)
wherev∗n is the reference neutral point voltage,ωo is theangular grid frequency andv and vq are the voltagesvector at the PCC and its quadrature, respectively. Notethat, usingv∗n in (22) instead ofvn as in (1), a perfectdecoupling between phases is obtained in the proposedmodel. This simple change allows the dynamics of theinverter-side current of each phase relying only on itscorresponding control input. This fact can be clearlyobserved by considering the expression ofvn written in(4).
By using this model, the KF is also applied to extractthe fundamental component and its quadrature of thePCC voltages. This fact allows to generate sinusoidalreference currents even in case of a highly distorted grid[22].
In order to use the aforementioned model in a KF,a discrete model is necessary. The augmented discretespace-state model including the PCC voltages is writtenas follows:
xaugi(k + 1) = Aaugxaugi(k) +Baugui(k)
− Ts
L1
v∗n(k) + ηi(k)(27)
yi(k) = Caugxaugi(k) +wi(k) (28)
wherewi(k) andηi(k) are the process and the measure-ment noise vectors respectively, and
xaugi = [i1i(k) vci(k) i2i(k) vi(k) viq(k)]T (29)
Aaug =
1 − Ts
L1
0 0 0Ts
C1 −Ts
C0 0
0 Ts
L2
1 − Ts
L2
0
0 0 0 1 Tsωo
0 0 0 −Tsωo 1
(30)
Baug = [VdcTs
2L1
0 0 0 0]T (31)
Caug = [0 0 1 0 0]T . (32)
Note that the circumflex symbol denotes estimated vari-ables.
We can take advantage of the reference neutral pointvoltage,v∗n, shown in 22 to impose a desired voltage. Forinstance, a third harmonic can be imposed at the neutralpoint, which can be obtained following the steps in [20],and it is expressed as:
v∗n =Vp
6[(L1 + L2)
P ∗ωo
|v|2 cos(3ωot)
+ (1 + (L1 + L2)Q∗ωo
|v|2 ) sin(3ωot)].
(33)
where |v| = [va vb vc] is the vector of estimatedvoltages at the PCC.
The last expression does not have any dependencewith the control signalsua,b,c. For this reason, this solu-tion not only increases the controllers dynamic range, butalso achieves a perfect decoupling between controllers.
On the other hand, the system is controlled using onlyone sensor in the grid-side current, as it is shown inFig.2. Then, the system observability and controllabilityshould be ensured. The observability matrix can beobtained from(30) and (32) which yields to
O = [Caug CaugAaug CaugA2aug CaugA
3aug
CaugA4aug]
T(34)
and using(30) and (31) the controllability matrixΓ isgiven by
Γ = [Baug AaugBaug A2augBaug A
3augBaug
A4augBaug]
(35)
Matrix O is of full-rank, (i.e. rankO = 5), andas a consequence the system is observable using onlythe measured currenti2i. However, the controllabilitymatrix is of rank 3 and only a controllable subspacecan be considered. Note that in this particular casethe controllable subspace is given by matrix(8) whichcontains the control variablei2i. Besides, the statesviand viq are stable states with bounded limits, thereforethe controllability is ensured.
2) KF Algorithm: The KF algorithm is widely ex-plained in the literature [23] so only a brief summary ofits equations will be given in this section. The recursiveKalman algorithm computation is divided in two parts:1) time updating and 2) measurement updating. From thewell-known equations of the KF [24], the equation forthe estate estimation can be expressed as follows:
xaugi(k + 1) = xaugi(k) + Li(k)(i2i(k)− i2i(k))(36)
where the Kalman gain is computed as:
Li(k) = Pi(k)CT (CPi(k)H
T +Ri(k))−1 (37)
being Pi the error covariance matrix for phase-legiand Ri the measurement noise covariance matrix. Thealgorithm and some interesting steps to reduce the com-putational time are explained in [24].
B. Derivation of the SMC by dynamic imposition
In this paper, the sliding surfaces are designed byimposing a desired dynamics behaviour with activedamping capability. The sliding surfaces will be used asgrid-current controllers in order to achieve high currenttracking accuracy with stable dynamics. In the switchingsurfaces implementation only estimated variables areused.
The first step in a switching surface design is tocalculate the relative degree associated with the outputvariable. The desired closed-loop output-current dynam-ics can be then specified according to the relative degreeof i2. The relative degree of a state variable is thesmallest number of differentiations with regards to time,so that the control inputu appears explicity [21]. From(22)-(24) it can be easily found that
∂
∂u
(dj i2dtj
)= 0 j = 1, 2 (38)
∂
∂u
(dj i2dtj
)6= 0 j = 3 (39)
then, the relative degree ofi2 is three. From (1)-(3) theopen-loop output-current dynamics can be expressed asfollows:
L2Cd3i2dt3
+di2dt
− di1dt
+ Cd2v
dt2= 0. (40)
Note that in the previous expression, the control actionu is in the time derivative term ofi1.
The choice of the sliding surface vectorS constitutesthe second step. Here, we introduce the use of invari-ance conditions to synthesize proper sliding surfaces toguarantee perfect tracking dynamics,i2 = i
∗.
The desired closed-loop linear dynamics which guar-antees a perfect tracking performance until the thirdderivative term of the output-current error is
3∑
n=0
λn
dn(i2 − i∗)
dtn= 0. (41)
It is worth to mention that, this ideal dynamics doesnot rely on the system parameters, and only dependsof the controller parameters,λ. This fact provides ahight robustness against system parameters deviations.However, since a KF is used, the effect of the KF and thesystem discretization should be analyzed. For this reason,in section V, a complete analysis of system stability androbustness against system parameters deviation will beperformed.
Here, it is important to remark that the order of thespecified dynamics coincides with the relative degreeof the output-current; otherwise, the desired dynamicscannot be ensured by the control actionu [25], [26].
In sliding regime, the converter dynamics are forcedto evolve over the sliding surfaceS, according to theinvariance conditionS = S = 0 [25]. Such propertycan be used to design a controller that guarantees thedesired dynamics (41). In fact, a similar approach waspreviously employed in [27] in a different application,with good static and dynamic properties.
Now, by subtracting (40) from (41) and equalizing theresult to the invariance conditionS = 0 [25] yields
dS
dt=
di1dt
− di2dt
− Cd2v
dt2− L2C
d3i2dt3
+3∑
n=0
λn
dn(i2 − i∗)
dtn
(42)
and consequently
S = i1 − i2 − Cdv
dt− L2C
d2i2dt2
+3∑
n=1
λn
dn−1(i2 − i∗)
dtn−1
+ λ0
∫(i2 − i
∗)dt.
(43)
The aforementioned expression can be simplified bytaking λ3 = L2C and considering thatλ3
d2i∗
dt2<< λ1i
∗,as:
S = i1 − i2 − Cdv
dt+ λ2
d(i2 − i∗)
dt+ λ1(i2 − i
∗)
+ λ0
∫(i2 − i
∗)dt.
(44)
The last requirement in the design of SMC is to satisfythe reaching conditions. The most often-used reachingconditions for each phase-legi are given by
SiSi < 0 (45)
which allows us to determine the control law
ui =
1 if Si < 0−1 if Si > 0
(46)
In section V, an analysis of the closed-loop systemstability is performed in order to obtain the controlparameters.
V. STABILITY ANALYSIS
In this section the stability of the proposed controlsystem is analyzed. This analysis takes into accountthe effect of the KF because it modifies the systemdynamics behaviour. According to section IV-A, thecontrollers are decoupled, and each phase-leg can betreated independently in order to perform the stabilityanalysis.
A. Discrete equivalent control deduction
For this analysis, only the state variablesi1i, vci andi2i are used, and the voltages at the PCC are consideredas disturbances. In this case the state-space systemequations for each phase-legi can be defined as follows:
xi(k + 1) = Axi(k) +Bu(k) + f(k) (47)
yi(k) = Cxi(k) (48)
where matrixA, and vectorsB and C are defined in(8) , (9) and (10) respectively, andf(k) is a disturbancevector.
The discrete sliding surface equation can be obtainedfrom (43) yielding:
Si(k) = i1i(k)− i2i(k) + λ2
i2i(k)− i2i(k − 1)
Ts
+ λ1 i2i(k) + λ0ξi(k) + g(v, i∗i ).
(49)
where the integral term is defined as
ξi(k) = ξi(k − 1) + TsCxi(k) (50)
andg(v, i∗i ) is a function with bounded limits dependingof the disturbances. The last expression can be rewrittenas follows:
Si(k) = axi(k) + bxi(k − 1) + λ0ξi(k) + g(v, i∗i ).(51)
where
a = [H+C(λ2
Ts
+ λ1 − 1)] (52)
b = −λ2
Ts
C (53)
H = [1 0 0]. (54)
and
xi = [i1i vci i2i]T . (55)
Vector xi(k+1) is the estimated state vector which canbe obtained using the KF estimation as
xi(k + 1) = Axi(k) +Buieq + LiCe(k) (56)
where A is the system matrix obtained from the pro-posed model which contains theLCL nominal values,A is the real system matrix defined in (8),ei(k) =xi(k) − xi(k) is the estimation error anduieq(k) is theequivalent control of phase-legi, which is the solutionof ∆Si = Si(k + 1)− Si(k)=0, [28].
Taking into account the aforementioned expression,the discrete equivalent control can be found by using(56) in (51) evaluated at timek + 1, yielding
uieq(k) = −(GB)−1(Si(k) + a(GA+ b)xi(k)
+GLiCe(k) + λoξi(k) + g(v, i∗i ))(57)
where the gain matrixG is expressed as:
G = a+ λ0TsC. (58)
B. Closed-loop equations
In the ideal sliding regime, one hasSi(k + 1) =Si(k)=0. Then, (57) can be reduced to:
uieq(k) = K1xi(k) +K2e(k) +K3ξi(k) (59)
where the gainsK1, K2 andK3 are defined as follows:
K1 = −(GB)−1(GA+ b) (60)
K2 = −(GB)−1GLiC (61)
K3 = −(GB)−1λ0. (62)
Note that the disturbances functiong(v, i∗i ) has beenremoved for simplicity, since it has no effect in thestability analysis.
In order to find the closed-loop equations,(59) isreplaced in(47) and in (56). Assumingf(k) in (47)has bounded limits, these function can also be removedfor the stability analysis procedure, leading to
xi(k + 1) = (A+BK1)xi(k) +B(K2 −K1)ei(k)
+BK3ξi(k)(63)
xi(k + 1) = (A+BK1)xi(k) + (B(K2 −K1) + LiC
− A)ei(k) +BK3ξi(k).(64)
Now, by subtracting(64) from (63), the equation for theestimation error at timek + 1 is obtained
e(k + 1) = (A− A)xi(k) + (A− LiC)e(k) (65)
The closed-loop equations are defined by(63) and(65). Then by taking into account that
ξi(k + 1) = ξi(k) + TsCxi(k + 1) (66)
and using (56) in (66), the closed-loop equations inmatrix form can be found:
xi(k + 1)ei(k + 1)ξi(k + 1)
= G
xi(k)ei(k)ξi(k)
(67)
where matrixG can defined as follows
G =
A+BK1 B(K2 −K1) BK3
(A− A) A− LiC 0
TsC(A+BK1) J TsBK3 + 1
(68)
being
J = TsCB((K2 −K1)− A+ LiC) (69)
C. Closed-loop poles
The closed-loop dynamic behaviour will be given bythe eigenvalues of matrixG, which are the solutionof det(zI − G) = 0. In order to ensure the systemstability, the eigenvalues should lie inside the unity circlein the z-plane. In a first stage, we assume that thesystem parameters deviations are zero. In this case itis accomplished thatA = A in the matrixG.
The converter open-loop poles in each phase-legi arethree (5), but the observer introduces three more poles(56). When the system is in closed-loop operation, onemore pole is added due to the presence of the integratorin the sliding surface, being seven the total number ofpoles.
However, it is found that while the system is in sliding-regime, one pole is fixed at the origin (z=0) due to thesliding equation, which forces the grid-side current totrack its reference (i.e.i2 = i
∗), and as a consequence,the system order is reduced in one unit. The position ofthe remaining poles can be adjusted by tuning the controlparameters.
The pole maps for two different cases are presentedin Fig.3. As it can be seen, the three poles provided bythe KF lie in the same position in both figures since theKalman gain is fixed. The position of the other threepoles can be adjusted by changingλ parameters. InFig.3(a) the control parameters are selected for an os-cillating behaviour and these poles are outside the unity
0.8 0.85 0.9 0.95 1-0.5
0
0.5
0.40.50.60.70.80.9
0.1 /T
0.1 /T
0.10.20.3
Pole-Zero Map
Real Axis
Imag
inar
y A
xis
(a)
0.8 0.85 0.9 0.95 1-0.5
0
0.5
0.40.50.60.70.80.9
0.1 /T
0.1 /T
0.10.20.3
Pole-Zero Map
Real Axis
Imag
inar
y A
xis
(b)
Fig. 3. System poles (a) in an oscillation case (λ3 = 34 · 10−9,
λ2=λ0=0 andλ1=1), (b) used in the proposed controller (λ3 = 34 ·
10−9, λ2 = 136 · 10
−6, λ1=1.136 andλ0=1000.
circle in the complexz plane. In Fig.3(b) the controlparameters are changed, and the poles are attracted insidethe unity circle providing stable dynamics.
D. System parameters robustness analysis
In the previous section the stability of the system hasbeen analyzed. However, the deviations of the LCL filterparameters should be studied. Here, we assume thatA 6=A, which means that the reactive components do notcoincide with their nominal values. Note that accordingto the matrixG, the overall system stability does not relyon the grid inductance variations. This is an importantfeature of the proposed control algorithm that will beshown in the experimental results section.
Figs.4(b)-4(c) show the roots locus of the closedloop system in the case of parameter deviations in theLCL filter. According to [29], deviations of±30% onthe rated values have been taken to verify the system
TABLE ISYSTEM PARAMETERS
Symbol Description ValueL1 Filter input inductance 7 mHC Filter Capacitor 6.8 uFL2 Filter output inductance 5 mHLg Grid inductance 0.8 mH-5 mHVdc DC-link Voltage 450 Vfs Sampling frequency 40 kHz
fgrid Grid frequency 60 Hzfsw Switching frequency 6 kHzVgrid Grid Voltage 110 VP ∗ Active Power 1.5 kWQ∗ Reactive Power 0 VAr
robustness. In each case, variations of only one parameterin the system matrixA are applied, while the rest ofthe parameters remains unchanged. It must be noticedthat, for this analysis the matrixA should contain theLCL nominal values. From the figures, it can be seenthat the eigenvalues have small variation for a largevariations of the LCL parameters. This fact verifies thehigh robustness of the proposed control method againstdeviations in the system parameters. In order to concludethe robustness analysis, different tests with harmonicsin the grid and voltage sags will be shown in theexperimental results section.
VI. EXPERIMENTAL VALIDATION
An experimental three-phase three-wire inverter proto-type was built using a 4.5-kVA SEMIKRON full-bridgeas the power converter and a TMS320F28M35 floating-point digital signal processor (DSP) as the control plat-form with a sampling frequency of 40 kHz. The gridand the DC-Link voltages have been generated using aPACIFIC 360-AMX and an AMREL SPS1000-10-K0E3sources respectively. A photograph of the experimentalsetup is shown in Fig.5. Table I lists the system param-eters used in the experimental prototype. Some resultsare imported to Matlab by means of a script whichcommunicates the computer with the DSP.
Fig.6 shows the equivalent control for phase-lega,obtained by low-pass filtering the control signalua. Theequivalent control is affected by the reference neutralpoint voltage which injects a third harmonic in thecontrol signal as desired. As a consequence, the dynamiccontrol range is extended by simply including the suit-able reference neutral point voltage in the KF algorithm.
Fig.7 compares the tracking performances of the pro-posed control method with the conventional SMC whenonly active power is injected to the grid. In Fig7(a)and Fig.7(b) conventional SMC expressed in (12) with aphysical damping resistor of 68Ω is used. The damping
0.8 0.85 0.9 0.95 1-0.5
0
0.5
0.1 /T
0.1 /T
0.10.20.30.40.50.60.70.80.9
Pole-Zero Map
Real Axis
Imag
inar
y A
xis
(a)
Pole-Zero Map
Real Axis
Imag
inar
y A
xis
0.8 0.85 0.9 0.95 1-0.5
0
0.5
0.2
0.1 /T
0.9
0.1
0.1 /T
0.30.40.50.60.70.8
(b)
Pole-Zero Map
Real Axis
Imag
inar
y A
xis
0.8 0.85 0.9 0.95 1-0.5
0
0.5
0.1 /T
0.1 /T
0.10.20.30.40.50.60.70.80.9
(c)
Fig. 4. Root locus diagrams when the filter parameters vary. a) L1
varies±30%, b) L2 varies±30%, and c) C varies±30%.
Fig. 5. Photograph of the experimental setup
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1
-0.5
0
0.5
1
Fig. 6. Equivalent control with a common-mode third harmonicinjected.
resistor is selected in accordance with guidelines re-ported [30]. Fig.7(a) shows a phase-shift around7 whenthe reference active power is 750 W. This phase-shift isreduced when the reference active power is increasedup to 1500 W as shows Fig.7(b). It can be seen fromboth figures that using the conventional SMC the phaseshift has a dependence on the power injected to the gridaccording to (21).
If the proposed controller (43) is used the grid-sidecurrent tracks its reference without error regardless ofP ∗, as it is shown in Fig.7(c) and Fig.7(d).
A. Test of the VSI against sudden changes in the activepower reference
Fig.8(a) and Fig.8(b) show the grid-side currents andthe active and reactive power, respectively. Three casesare considered: oscillation behavior, stable behaviour anda power step change, as described as follows:
1) In the oscillation case, the control parametersλ0=λ2=0 and λ1=1 are chosen and the system is os-cillating, as predicted in Fig.3(a). In this case the SMC
is used to control the inverter-side current as in sectionIII (see eq.(12)).
2) In the stable case, the control parametersλ0 andλ2 are selected 1000 and 136·10−6 respectively. In thiscase, grid currents and powers track their respectivereferences.
3) Finally, in the case of an active power step change,from 750 W to 1500 W, a stable operation with fasttransient response is achieved due to the use of the SMC,as shown in Fig.8(b).
B. Test of robustness against grid inductance variations
Fig.9(a) shows the grid-side current of phase-lega forthree different values of the grid inductance,0.8mH,2mH and 5mH, when the conventional SMC is used.An oscillatory behaviour in the grid-side currents appearswhen the grid inductance is more than3mH. When theproposed control is used this oscillation disappears, as itcan be seen in Fig.9(b), which makes the system robustagainst grid inductance variations.
C. Test of VSI under a distorted grid
Fig.10 compares the three-phase grid currents usingthe conventional SMC (12) and using the proposedcontrol method in the case of a distorted grid. Thisfigure shows the distorted PCC voltages, which THD isaround 16%, and the grid-side currents. From the figure,when the conventional SMC is used, a distortion in thegrid currents appears. The reason is that the methodfor generating the reference currents uses distorted PCCvoltages. In contrast, when the proposed control methodis used, the grid currents are sinusoidal since the refer-ence current is generated by using only the fundamentalcomponent of the PCC voltage, which is obtained fromthe KF. Note that with this proposal, the use of extrafilters for the grid voltages is not necessary.
D. Test of the VSI under voltage SAGS
The proposed controller can also operate in case ofvoltage sags. Fig.11 shows the VSI performance undergrid voltage sags. For this test, a voltage sag character-ized by a positive and negative sequence ofV + = 0.7 p.u.andV − = 0.3 p.u. respectively, and with a phase anglebetween sequences ofφ = −π/6 has been provoked.The reference currents are obtained using the positivesequence of the PCC voltage, usingi∗a,b,c = P ∗
|v|2 v+
a,b,c.The positive sequence is computed from the estimatedPCC voltages and their quadratures obtained from theKF, following similar steps as shown in [31]. Note thatthe use of a specific PLL algorithm for extracting the
(a) (b)
(c) (d)
Fig. 7. Grid-side current (red line) tracking its reference(blue line) in phase-lega (2A/div) using: a) conventional SMC withP ∗=750 W,b) conventional SMC withP ∗=1500 W, c) proposed controller withP ∗=750 W, and d) proposed controller withP ∗=1500 W.
(a)
0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.22000
-1000
-500
0
500
1000
1500
2000
2500
PQ
Power step
Conventional SMC
750 W
1500 W
(b)
Fig. 8. a) Grid-side currents (2 A/div) and b) active and reactivepower, in an oscillation case, stable case and with a sudden stepchange in the active power reference.
positive and negative sequence grid voltage componentsis not necessary in this case. Note that the currentsamplitude is increased in order to maintain the desiredactive power due to the sag.
0.08 0.082 0.084 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1−10
−5
0
5
10
15
Time(ms)
Current(A
)
i2a
, Lg=5mH i
2a,L
g=3mH i
2a,L
g=0.8mH
(a)
0.08 0.082 0.084 0.086 0.088 0.09 0.092 0.094 0.096 0.098 0.1−10
−5
0
5
10
15
Time(ms)
Current(A
)
i2a
, Lg=5mH i
2a , L
g= 3mH i
2a, L
g=0.8mH
(b)
Fig. 9. Grid-side current of phase-lega for different grid inductancevalues (a) using conventional SMC, and (b) using the proposedcontroller.
Fig. 10. Distorted PCC voltages (25V/div) with THD = 16%, andgrid-side currents (5 A/div)
Fig. 11. PCC voltages (50 V/div) and grid-side currents (2A/div)under voltage sag.
0 2 4 6 8 10 12 14 16 18 200
0.10.20.30.40.50.60.70.80.9
1
frequency(kHz)
Normalized
Amplitude
Fig. 12. Switching signal spectrum
E. Switching spectrum
In Fig.12 the control signal spectrum is shown. Notethat the spectrum is concentrated around 6 kHz, as de-sired by using the switching decision algorithm presentedin [24].
VII. C ONCLUSIONS
In this paper a sliding-mode observer-based controlfor grid-connected LCL-filtered three-phase inverter isproposed. The control algorithm is based on SMC incombination with a KF. This proposal allows to obtainthree decoupled controllers which, provides a desireddynamics to the grid-side current. The proposed controltechnique also improves the tracking performance of thereference and also increases the robustness against gridinductance variation. Theoretical and experimental re-sults show that with this controller no physical dampingresistors are needed reducing the control losses. In addi-tion a sinusoidal third harmonic voltage can be injectedat the neutral point increasing the control dynamic range.The system stability analysis has been also performedallowing to find the main control parameters.
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Ramon Guzman received the B.S., the M.S.and the Ph.D. degrees in telecommunicationsengineering from the Technical University ofCatalonia, Barcelona, Spain, in 1999, 2004and 2016, respectively. He is currently anAssociate Professor with the Department ofAutomatic Control in the Technical Universityof Catalonia. His research interests includenonlinear and adaptive control for three-phase
power converters.
Luis Garcia de Vicuna a received the M.S.and Ph.D. degrees in telecommunication en-gineering from the Technical University ofCatalonia, Barcelona, Spain, in 1980 and 1990,respectively, and the Ph.D. degree in electricalengineering from the Paul Sabatier Univer-sity, Toulouse, France, in 1992. From 1980to 1982, he was an Engineer with a controlapplications company in Spain. He is currently
a Full Professor with the Department of Electronic Engineering,Technical University of Catalonia, where he teaches courses onpower electronics. His research interests include power electronicsmodeling, simulation and control, active power filtering, and high-power-factor ac/dc conversion.
Miguel Castilla received the B.S., M.S., andPh.D. degrees in telecommunication engineer-ing from the Technical University of Catalonia,Barcelona, Spain, in 1988, 1995, and 1998,respectively. Since 2002, he has been an Asso-ciate Professor with the Department of Elec-tronic Engineering, Technical University ofCatalonia, where he teaches courses on analogcircuits and power electronics. His research
interests include power electronics, nonlinear control, and renewableenergy systems.
Jaume Miret (M98) received the B.S. de-gree in telecommunications, M.S. degree inelectronics, and Ph.D. degree in electronicsfrom the Universitat Politecnica de Catalunya,Barcelona, Spain, in 1992, 1999, and 2005,respectively. From 1993 to 2011, he was anAssistant Professor in the Department of Elec-tronic Engineering, Universitat Politecnica deCatalunya, Spain. Since 2011 he has been an
Associate Professor in the Universitat Politecnica de Catalunya. Hisresearch interests include dc-to-ac converters, active power filters, anddigital control.
Helena Martın received the B.Sc. degree inElectronic Engineering in 1994, M.Sc. in Elec-trical Engineering in 1997, Ph.D in IndustrialEngineering in 2007. She is currently an Asso-ciate Professor with the Department of ElectricEngineering in the Technical University ofCatalonia Her research interests are optimalmanagement and energy policy for microgidswith renewable energy generation.