Connected Inverters With Lcl Filters

770 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 26, NO. 3, MARCH 2011

Modeling and Control of N -Paralleled Grid-Connected Inverters With LCL Filter Coupled

Due to Grid Impedance in PV PlantsJuan Luis Agorreta, Mikel Borrega, Jesus Lopez, Member, IEEE, and Luis Marroyo, Member, IEEE

Abstract—Designing adequate control laws for grid-connectedinverters with LCL filters is complicated. The power quality stan-dards and the system resonances burden the task. In order to dealwith resonances, system damping has to be implemented. Activedamping is preferred to passive damping so as to improve the effi-ciency of the conversion. In addition, paralleled grid-connectedinverters in photovoltaic (PV) plants are coupled due to gridimpedance. Generally, this coupling is not taken into account whendesigning the control laws. In consequence, depending on the num-ber of paralleled grid-connected inverters and the grid impedance,the inverters installed in PV plants do not behave as expected.In this paper, the inverters of a PV plant are modeled as a mul-tivariable system. The analysis carried out enables to obtain anequivalent inverter that describes the totality of inverters of a PVplant. The study is validated through simulation and field experi-ments. The coupling effect is described and the control law designof paralleled grid-connected inverters with LCL filters in PV plantsis clarified.

Index Terms—Active damping, grid-connected inverter, LCLfilter, photovoltaic (PV) power systems.

I. INTRODUCTION

D ISTRIBUTED power generation systems based on renew-able energy sources are attracting the market and research

interest as a feasible choice in a sustainable development en-vironment. In this context, grid-connected photovoltaic (PV)plants are becoming a common technology to generate energyand its penetration level is gradually increasing [1]. These plantsconsist of sets of PV generators and power electronic invertersconnected in parallel to the distribution network through a dis-tribution transformer. Fig. 1 shows the scheme of a PV plant inwhich the distribution transformer leakage impedance is inte-grated into the grid impedance Zg .

The inverters installed in PV plants are generally voltagesource converters with an output filter. LCL filters are preferredto L filters because their switching harmonic attenuation withsmaller reactive elements is more effective [2], [3]. Thus, the

Manuscript received June 30, 2010; revised September 20, 2010; acceptedNovember 14, 2010. Date of current version May 13, 2011. This work was sup-ported in part by the Spanish Ministry of Education and Science under GrantDPI2009-14713-C03-01. Recommended for publication by Associate EditorM. Liserre.

The authors are with the Department of Electrical and Electronic En-gineering, Public University of Navarra, Pamplona 31006, Spain (e-mail:[email protected]; [email protected]; [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPEL.2010.2095429

Fig. 1. Typical PV power plant scheme.

cost and the weight of the inverters are reduced. However, dueto the need to damp the resonances, the filter and the currentcontrol design are more complex. Active damping [4]–[14],[27] is preferred to passive damping [2] in order to improve theefficiency of the conversion.

Either the inverter side current [2]–[11], [27], or the grid sidecurrent [12], [13], of the LCL filter can be controlled. Each alter-native has its own advantages and drawbacks [15]. Dependingon the controlled current, specific active damping techniqueshave been proposed.

The quality of the grid injected current is a matter of concern.International standards regulate the connection of PV invertersto the grid and limit the harmonic content of the injected current[16]–[18], [28]. Exceeding the harmonic injection limits mayrequire the inverters to disconnect from the grid. Thus, the LCLfilters are implemented to prevent the grid from being pollutedwith switching harmonics, whereas plenty of control algorithms,including repetitive controllers, integrators in multiple rotatingframes, and resonant integrators [16]–[18], [29], have been pro-posed in order to mitigate the low-order current harmonics.

An added difficulty is that PV inverters are coupled due tothe grid impedance Zg and influence each other as a result. Allinverters in Fig. 1 share the voltage in the point of common cou-pling (PCC) vg and are able to modify this voltage by injectingtheir currents. Furthermore, the current of an inverter is able tocirculate through a paralleled inverter instead of coming back

0885-8993/$26.00 © 2010 IEEE

AGORRETA et al.: MODELING AND CONTROL OF N -PARALLELED GRID-CONNECTED INVERTERS 771

through the grid. Notice that if the grid impedance was ideallyconsidered to be zero, coupling would not exist since the voltagein the PCC would always be eg . Depending on the number ofparalleled inverters N and the grid impedance Zg the invertersinstalled in PV plants might not behave as expected. In fact,resonant behavior of the inverters has been detected in manyPV plants. This situation exceeds the harmonic regulations andcauses the breakage of many devices such as electronic meters.Other authors have already studied the interaction of invertersand the distribution network [19], [20]. These works mentionthat a resonant behavior may occur even if all the PV invertersdo individually satisfy the standards. In addition, it is indicatedthat power quality integration problems are expected as the pen-etration level of PV inverters increases.

Every proposal with regard to active damping strategies andcontrol algorithms reviewed [4]–[18] is very interesting. How-ever, their analyses are always done for single grid-connectedinverters. In consequence, coupling between inverters due togrid impedance is ignored and the stability and performance ofthe inverter in a PV plant might be questioned.

The aim of this paper is to describe the consequences of thiscoupling effect and to propose easy guidelines to design theactive damping strategy and the control algorithm for such PVinverters. The N -paralleled inverters are modeled as a multivari-able system. As often the case, inverters in a PV power plant aremanufactured by the same firm, and have the same rated power.In addition, industry standards constantly demand smaller toler-ances. In consequence, all inverters can reasonably be assumedto be equal. This analysis enables to obtain an equivalent inverterthat models the N inverters of a PV plant. Clear criterions todetermine the stability, the effectiveness of the active dampingstrategy, and the bandwidth of the whole system are exposed.

In this paper, PI controllers are considered, and the dampingtechnique based on a lead-lag element applied to the feedbackof the capacitor voltage proposed in [4] is chosen, but the samemethodology can also be applied to other control algorithms oractive damping techniques leading to the same equivalent in-verter model. Although the considered inverters are controlleddigitally, the analysis is performed in the s-domain instead ofthe z-domain. This paper is organized as follows. First, a digitalcontrol emulator that maintains the s-domain analysis with goodreliability is presented. Second, the modeling and control of asingle grid-connected inverter are described. Third, the model-ing and control of the N -paralleled grid-connected inverters of aPV plant are analyzed. Fourth, the theoretical study is validatedthrough simulation. Fifth, experimental waveforms of inverterswith resonant behavior in a PV plant of 1400 kW are shown.In addition, these experimental waveforms are reproduced insimulations making use of the equivalent inverter. Next, somecontrol design guidelines and practical uses are suggested. Fi-nally, conclusion is discussed.

II. CONTINUOUS MODELING OF THE SAMPLER, THE

ZERO-ORDER HOLD AND THE COMPUTATION DELAY

Emulating digitally controlled systems by s-domain continu-ous transfer functions is a common practice. Digitally controlled

Fig. 2. Continuous model of the one sampling period time delay, the samplerand the zero-order hold.

converters have a time delay of one sampling period due to thefact that the computation time of the DSP microprocessor rela-tive to the sampling period is not negligible. Sampling is inherentto the z-domain analysis, but if the aim is to model a discretesystem as a continuous system, the sampler has to be taken intoaccount. The fundamental spectrum of a sampled continuous-time signal can be represented by the sampler continuous ap-proximation [22], [24]. The zero-order hold equivalent of plantsis used in z-domain analysis of pulse width modulation (PWM)converters [2], [7], [8]. However, the continuous expression ofthe zero-order hold is needed in s-domain analysis [22], [24].The continuous block diagram of these three elements is shownin Fig. 2. Ts refers to the sampling period. In this paper, thesampling and the control updating are done at twice the rate ofthe PWM carrier frequency, fswitch .

The continuous transfer function of the block diagram inFig. 2 is expressed in

Ds =e−Ts ·s ·

(1 − e−Ts ·s

)

Ts · s. (1)

In this paper, the transfer function Ds is called digital controlemulator. Delays, such as the computation delay of one samplingperiod or the time delay present in the zero-order hold expressionof Fig. 2, are difficult to handle because they make transferfunctions nonrational. That is why, in power electronics thedigital control emulator Ds has been traditionally approximatedby the following transfer function [3], [4]:

Ds ≈ Ds1 =1

1, 5 · Ts · s + 1. (2)

This approximation is useful when tuning PI controllers, sincethe crossover frequency of the open-loop gain is generally farenough from half the sampling frequency or Nyquist frequency.However, it fails in describing the behavior of digitally con-trolled systems as frequency comes close to the Nyquist fre-quency. Notice that this issue is essential if an active dampingstrategy has to be implemented, since the resonance frequencyis usually within the range of the crossover frequency and theNyquist frequency. Consequently, the great majority of authorsprefer the z-domain analysis [6]–[8], [11], [15], [16].

In order to obtain rational transfer functions, delays are usu-ally approximated by poles and zeros. One of the most popularapproximations for delays is the Pade approximation [22]. Thefirst-order Pade approximation for a time delay of one sampling


period Ts is

e−Ts ·s ≈ 1 − s · 0.5 · Ts

1 + s · 0.5 · Ts= Pade1 . (3)

In this paper, a different continuous approximation for the digi-tal control emulator Ds is proposed. It is obtained provided thatthe first-order Pade approximation is replaced in both the com-putation delay expression and the zero-order hold expression ofFig. 2. Thus, the digital control emulator Ds is approximated asfollows:

Ds ≈ Ds2 =Pade1 · (1 − Pade1)

Ts · s=

1 − 0, 5 · s · Ts

(1 + 0, 5 · s · Ts)2 .

(4)The transfer function in (4) describes with better accuracy than(2) the behavior of digitally controlled systems such as PVinverters. It attains the right approximation also within the rangeof the crossover frequency and the Nyquist frequency as it will bedepicted in Section III. Hence, the approximation Ds2 maintainsthe s-domain analysis with a fair agreement between simplicityand accuracy.

III. MODELING AND CONTROL OF A SINGLE

GRID-CONNECTED INVERTER

A. System Description

Generally, the control strategy used in PV inverters is acascaded-loop control [1], [4], [7], [10], [14]. The inner loopcontrols the inductor current whereas the outer loop controls thedc-bus voltage. Thereby, the inner loop takes care of the grid-injected current quality whereas the outer loop is responsible forthe maximum power point tracking. Obviously, the inner loopbandwidth has to be much higher in order to maintain the outerloop and the inner loop decoupled.

In this section, the modeling and control of a single grid-connected inverter with an LCL filter are described. Althoughthis issue has already been addressed in the available litera-ture [1]–[18], the aim is to get the reader familiarized with thenomenclature and methodology used along this paper. More tothe point, the simplifications derived in the following sectionswill resemble the single inverter case. Solely, the inner currentloop is presented so as to simplify. Consequently, the dc bus ofthe inverter is supposed to be an ideal constant voltage sourceof value vbus .

The circuit for the current control design of a single grid-connected inverter is shown in Fig. 3(a). The impedances in thisfigure are those of the LCL filter (5). Note that parasitic elementsare neglected in order to simplify

Z1 = L1 · s, Z2 = L2 · s, Z3 =1

C3 · s, Zg = Lg · s. (5)

This paper follows the technique proposed in [4]. Accordingly,the inverter side current i1 is controlled and the capacitor volt-age vZ 3 is measured in order to implement the active dampingtechnique. The control variable is the converter voltage v0 . Thevoltage in the connexion point is vg . The voltage source eg rep-resents the grid and behaves as a disturbance. In this paper, onlythe inductor current tracking reference is discussed, so the dis-

Fig. 3. (a) Circuit of a single grid-connected inverter. (b) Thevenin-equivalentcircuit across the terminals AB.

turbance eg will be considered to be zero when describing themodeling and control of a single grid-connected inverter. Sincethe capacitor voltage is near to the grid voltage, which has tobe measured in PV grid-connected applications, it is possible toavoid the use of new sensors. This is the reason why this activedamping technique was chosen.

B. Modeling

A straightforward way to obtain the inverter side inductorcurrent dynamics i1 in relation to the converter voltage v0 isshown in Fig. 3(b). A simplified circuit in which the only cur-rent is i1 and the only voltage source is v0 is found using theThevenin-equivalent impedance calculated across terminals ABZthAB . Thus, the transfer function between i1 and v0 is

Y1 =i1v0

=1

Z1 + ZthAB

=Z3 + Z2 + Zg

Z3 · (Z2 + Zg ) + Z1 · (Z3 + Z2 + Zg ). (6)

The plant Y1 has a resonance frequency ωr and an antiresonancefrequency ωa which are, respectively [2], [16]

ωr =1

√C3 · L1 · (L2 + Lg )/L1 + L2 + Lg

,

ωa =1

√C3 · (L2 + Lg )

. (7)

The design procedure of the LCL filter is out of the scope ofthis paper. For further research, the literature [2], [3] may beconsulted. The plant Y1 is undamped since no resistors havebeen considered. It is depicted in Fig. 4(a) in which the Set I ofLCL filter values shown Table I are chosen. These are standardvalues for a 10-kW inverter. The per unit values referred to thegrid effective phase voltage eg = 230 V and to the rated power,PN = 10 kW are facilitated.


Fig. 4. (a) Undamped plant Y1 . (b) Actively damped plant Y1AD . (c) Open loop OLAD .

TABLE ILCL FILTER VALUES

Fig. 5. (a) Control loop for the inverter side inductor current with feedback ofthe capacitor voltage for active damping. (b) Corresponding control loop withthe actively damped plant, Y1AD .

C. Control Strategy

The scheme of the implemented inner control loop strategyfor the inverter side current i1 is shown in Fig. 5(a). The indexi1ref is the inverter side current reference, u0 is the output of thecontroller, and d is the duty cycle. Fi is the inverter side currentsensor (a low-pass filter). A simple PI is chosen as a controller,but repetitive controllers, integrators in multiple rotating framesor resonant integrators can be considered as well [11], [12],[16]–[18]. In [4], a lead-lag element is suggested. In this paper,it is included in Fv , which contains the voltage capacitor sensor

(a low-pass filter) in addition to the lead-lag element

Fi =1

τi · s + 1,PI = Kp +

Ki

s, Fv =

Kll · (τz · s+ 1)(τv · s + 1)(τp · s + 1)

.

(8)In this paper, a full-bridge single-phase inverter is supposed,so the gain vbus is included in the converter model. This termbehaves as a variable gain and it is compensated through itsinverse value in order to decouple the converter model seen bythe controller from the operating point as in [21].

As it can be observed in Fig. 5(a), a feedback path of thecapacitor voltage for active damping purposes is implemented.The transfer function between the inverter side current i1 andthe capacitor voltage vZ 3 is easily obtained from Fig. 3(b) if i1is supposed to be an ideal current source

vZ 3

i1= ZthAB =

Z3 · (Z2 + Zg )Z3 · Z2 + Zg

. (9)

Provided that the feedback path of the capacitor voltage isclosed, the control scheme of Fig. 5(b) is derived. Accordingly,Y1AD is the actively damped plant, which is obtained as shownin (10). Its bode diagram is plotted in Fig. 4(b), where the SetI of LCL filter values of Table I and the control parameters ofTable II are used

Y1AD =Y1

1 − Fv · Ds · (Z3 · (Z2 + Zg )/Z3 + Z2 + Zg ) · Y1

=Z2 + Z3 +Zg

(1−Fv · Ds) · Z3 · (Z2 + Zg)+ Z1 · (Z3 + Z2 + Zg).

(10)

D. Stability Analysis

In order to discuss the stability of the system of Fig. 5, theopen-loop transfer function has to be analyzed. Three differ-ent open-loop transfer functions are considered so as to showwhether the digital control emulator approximation proposed in


TABLE IICONTROL PARAMETER VALUES

this paper (4) performs adequately

OLADz = PIT · z−1 · Y1ADz

OLAD2 = PI · Ds2 · Y1AD · Fi

OLAD1 = PI · Ds1 · Y1AD · Fi. (11)

In (11), OLADz is the open-loop transfer function in thez-domain, where PIT is the PI controller discretized accord-ing to the Tustin rule and Y1ADz is the actively damped planttransfer function in the z-domain [6]. OLAD2 is the open-looptransfer function in which the digital control emulator has beenapproximated by the proposed transfer function (4). Finally,OLAD1 is the open-loop transfer function in which the digitalcontrol emulator has been approximated by the traditional trans-fer function (2). Note that the digital control emulator appearsnot only in the open-loop transfer function, but also inside theactively damped plant Y1AD .

The three open-loop transfer functions of (11) are depicted inFig. 4(c). OLADz is considered to be the most accurate since itis performed in the z-domain. It can be observed that the open-loop transfer function OLAD2 is close to OLADZ even withinthe range of the crossover frequency and the Nyquist frequency,whereas OLAD1 differs in this range. Consequently, with theproposed approximation Ds2 the z-domain analysis might beavoided with good reliability. The accuracy of the approximationDs2 depends on the proximity of the approximated frequenciesto the Nyquist frequency. If the resonant frequency is close tothe Nyquist frequency, the active damping performed with theapproximation Ds2 is less reliable, but so it is with the z-domainanalysis as well.

IV. MODELING AND CONTROL OF N -PARALLELED

GRID-CONNECTED INVERTERS

A. System Description

Next, a set of N -paralleled grid-connected inverters with anLCL filter is described. The dynamics of these inverters is cou-pled due to the grid impedance. The circuit for current con-trol design of the N -paralleled inverters of Fig. 1 is shown inFig. 6, where Z1i (with i = 1···N) are the inverter side inductorimpedances; Z2i are the inverter grid side inductor impedances;Z3i are the inverter capacitor impedances; and Zg is the gridimpedance. Moreover, i1i are the inverter side currents; i2i arethe grid side currents; i3i are the capacitor currents; ig is thegrid-injected current; vZ 3i are the capacitor voltages; v0i arethe converter voltages; and vg is the voltage in the PCC. In

Fig. 6. Circuit of N -paralleled grid-connected inverters.

mathematical expressions and circuits, n is used instead of N ,but both refer to the number of paralleled inverters.

In this paper, the technique proposed in [4] is chosen to de-scribe the consequences derived from the grid impedance cou-pling. Therefore, the inverter side currents i1i are controlled andthe capacitor voltages vZ 3i are measured in order to implementthe active damping strategy. The control variables are the con-verter voltages v0i . The voltage source eg represents the gridand behaves as a disturbance. In this paper, only the inductorcurrent tracking reference is discussed, so the disturbance eg

will be considered to be zero when describing the modeling andcontrol of the N -paralleled grid-connected inverters.

B. Modeling

The dynamics of the N -paralleled grid-connected invertersof Fig. 6 can be described with the multivariable control theory

i1n = G(s) · v0n ⇔

⎛

⎜⎝

i11i12· · ·i1n

⎞

⎟⎠

=

⎛

⎜⎝

G11 G12 · · · G1n

G21 G22 · · · G2n

· · · · · · · · · · · ·Gn1 Gn2 · · · Gnn

⎞

⎟⎠ ·

⎛

⎜⎝

v01v02· · ·v0n

⎞

⎟⎠ . (12)

Thus, i1n is the output vector that contains the controlled vari-ables i1i ; v0n is the input vector representing the control vari-ables v0i ; and G(s) is the transfer matrix, i.e., the matrix whoseproduct with the input vector v0n yields the output vector i1n .

In most PV plants all installed inverters have been manufac-tured by the same firm and are of the same sort. In addition to


Fig. 7. (a) Auxiliary circuit of the N -paralleled inverters provided that allthe converter voltages v0 i are zero except v01 . (b) Thevenin-equivalent circuitacross the terminals CD.

this, demanding industrial standards constantly require narrowertolerances. Hence, the inverters in a PV plant are assumed to beequal. Based on this assumption, the corresponding impedancesof the LCL filter of each inverter have the same value and arerepresented by Z1 , Z2 , and Z3

Z11 = Z12 = · · · = Z1n = Z1

Z21 = Z22 = · · · = Z12 = Z2

Z31 = Z32 = · · · = Z32 = Z3

⇒

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

Z1 = L1 · sZ2 = L2 · s

Z3 =1

C3 · sZg = Lg · s.

(13)

If the inverters are equal, the system adopts a characteristic sym-metry: all diagonal elements of the transfer matrix G(s) will beidentical since each converter voltage v0i influences its own cur-rent i1i in an identical way. Therefore, all diagonal elements ofG(s) can be replaced by G11 . Likewise, all nondiagonal elementsof the transfer matrix G(s) will be identical since each convertervoltage v0i influences another converter current i1j (with i �=j) in an identical way. Hence, all nondiagonal elements of G(s)can be replaced by G12

Gii = G11

Gij (i �=j ) = G12

}⇒ G(s) =

⎛

⎜⎜⎜⎝

G11 G12 · · · G12

G12 G11 · · · G12

· · · · · · · · · · · ·G12 G12 · · · G11

⎞

⎟⎟⎟⎠

.

(14)The elements G11 and G12 are calculated through the superpo-sition principle and Thevenin equivalent circuits. The diagonalelement G11 might be regarded as the transfer function betweenthe inverter side current i11 and its own converter voltage v01 .Accordingly, G11 can be calculated if all the converter voltagesv0i are supposed to be zero except v01 . Thus, the auxiliary cir-cuit of Fig. 7(a) is derived from Fig. 6. A circuit in which theonly current is i11 and the only voltage source is v01 is shownin Fig. 7(b) where ZthCD is the Thevenin-equivalent impedance

Fig. 8. (a) Auxiliary circuit of N -paralleled inverters provided that all theconverter voltages v0 i are zero except v02 . (b) Thevenin-equivalent circuitacross the terminals EF.

calculated across the terminals CD. The diagonal element G11such as the transfer function between i11 and v01 , is directlyobtained as follows:

G11 =i11

v01=

1Z1 + ZthCD

. (15)

The nondiagonal element G12 might be interpreted as the trans-fer function between the inverter side current i11 and the con-verter voltage of a paralleled inverter v02 . Accordingly, G12 canbe calculated if all the converter voltages v0i are supposed tobe zero except v02 . The auxiliary circuit shown in Fig. 8(a) isderived from Fig. 6. A circuit in which the only current is i11and the only voltage source is v02 . ZEF is shown in Fig. 8(b);the Thevenin-equivalent circuit is calculated across the termi-nals EF. The Thevenin-equivalent impedance is ZthEF . Hence,the nondiagonal element G12 , which for instance might be thetransfer function between i11 and v02 , is directly obtained as

G12 =i11

v02= − ZEF

Z1 + ZthEF. (16)

The expanded expressions of G11 and G12 are rather compli-cated and are shown in the Appendix. G11 and G12 denomina-tors coincide, but not their numerators. This is to be expectedsince the poles of a multivariable system, though not the zeros;have to be the same regardless the input [23].

So far, it has been considered that the correspondingimpedances of each inverter have the same value, since all in-verters installed in a PV plant are generally of the same sort.


Fig. 9. Circuit of the equivalent inverter, which models the N -inverters pro-vided that they are equal.

Going ahead with the assumption that all the installed invertersare equal, not only their impedances, but also their hardware,their software, and their PV generators; it is reasonable to as-sume that they will react all in the same way. In this scenario,the converter voltages of all inverters may be considered equal,i.e., v0i = v0 . Replacing this in (12), we obtain

⎛

⎜⎝

i1i1· · ·i1

⎞

⎟⎠ =

⎛

⎜⎝

G11 G12 · · · G12G12 G11 · · · G12· · · · · · · · · · · ·G12 G12 · · · G11

⎞

⎟⎠ ·

⎛

⎜⎝

v0v0· · ·v0

⎞

⎟⎠ . (17)

Multiplying a G(s) row by vector v0n (17), we derive the sim-plification Y1eq

i1 = (G11 + (n − 1) · G12) · v0 = Y1eq · v0 (18)

Y1eq =i1v0

=(Z3 + Z2 + n · Zg )

Z3 · (Z2 + n · Zg ) + Z1 · (Z3 + Z2 + n · Zg ).

(19)

If the expression of Y1eq (19) is compared with the expandedversion of G11 and G12 (shown in the Appendix), it is observedthat Y1eq is much simpler. The transfer function Y1eq can beidentified with that of the single inverter case (6). The onlydifference between both expressions Y1 and Y1eq is that in Y1eqthe grid impedance is multiplied by n. In fact, if all convertervoltages are equal v0i = v0 an equivalent single inverter whosegrid impedance is N times bigger represents the N inverters.In other words, an inverter in a PV plant sees grid impedanceN times bigger. This simplification is referred to as equivalentinverter and its corresponding circuit is shown in Fig. 9.

The resonance frequency ωreq and the antiresonance fre-quency ωae of the plant Y1eq depend on the number of invertersN . Their expressions are

ωreq =1

√C3 · L1 · (L2 + n · Lg )/(L1 + L2 + n · Lg )

,

ωaeq =1

√C3 (L2 + n · Lg )

. (20)

The bode diagram of Y1eq with the Set I of LCL filter valuesintroduced in Table I is plotted in Fig. 11(a) for different valuesof N . Notice that, if the grid inductance in a certain point ofthe distribution network is Lg = 0.1 mH, and the number ofinverters is N = 1200 [see Fig. 11(a)], an inverter in a PV plantwill see N·Lg = 120 mH, whereas a single inverter would onlysee Lg = 0.1 mH. In consequence, inverters whose control lawhas been designed ignoring the grid impedance coupling may

Fig. 10. (a) Multivariable control loop for the inverter side inductor currentswith feedback of the capacitor voltages for active damping. (b) Correspondingcontrol loop with the actively damped matrix J(s).

not perform as desired. This is because these inverters will seegrid impedance N times bigger than expected.

The equivalent inverter proves that the number of invertersinstalled in a PV plant has a great impact on the functioning ofthe inverters due to the grid impedance coupling.

C. Control Strategy

Fig. 10(a) shows the multivariable control loop correspondingto the N -paralleled grid-connected inverters with LCL filtercoupled due to the grid impedance in PV plants of Fig. 6 Thisis the multiple input, multiple output (MIMO) version of thesingle input, single output (SISO) control loop of Fig. 5(a).

Some of the elements in Fig. 10(a) have been previouslyintroduced. The unknown elements are listed now: i1n ref isthe vector containing the inverter side current references; u0n

is the vector representing the controllers output; and vZ3 isthe vector that includes the capacitor voltages. Moreover, PI(s)is the matrix that contains the controllers, PI′; Fv (s) is thematrix that contains the voltage sensors, F ′

v ; and A(s) is thematrix representing the transfer relation between the inverterside currents i1n and the capacitor voltages vZ3

i1n ref =

⎛

⎜⎝

i11refi12ref· · ·

i1nref

⎞

⎟⎠ , u0n =

⎛

⎜⎝

u01u02· · ·u0n

⎞

⎟⎠, vZ3n =

⎛

⎜⎝

vZ 31vZ 32· · ·

vZ 3n

⎞

⎟⎠

PI(s) =

⎛

⎜⎝

PI ′ 0 · · · 00 PI ′ · · · 0· · · · · · · · · · · ·0 0 · · · PI ′

⎞

⎟⎠ ,

Fv(s) =

⎛

⎜⎝

F ′v 0 · · · 0

0 F ′v · · · 0

· · · · · · · · · · · ·0 0 · · · F ′

v

⎞

⎟⎠ . (21)


The matrices PI(s) and Fv (s) are diagonals because the invertersare completely independent and there is no communication sig-nal between inverters. Each converter has its own controller inorder to control its own inverter side current, but no decouplingcontrollers are implemented. In the same way, each converteronly has access to its own capacitor voltage. Since the invertersinstalled in a PV plant are expected to be equal, the controllersPI′ and the voltage sensors F ′

v are the same for them all.If the MIMO loop in Fig. 10(a) is compared with the SISO

loop in Fig. 5(a), some differences are revealed. In the MIMOsystem, the variable gain vbus has been suppressed since it iscompensated through its inverse value as in [21]. Apparently, inthe MIMO system, the inverter side current sensor Fi and thedigital control emulator Ds are not included. They should be intwo independent diagonal matrices but they can be integrated inPI(s) and Fv (s) provided that their elements are

PI′ = PI · Ds · Fi, F′v = Fv · Ds. (22)

Based on the assumption that in a PV plant all installed invertersare equal, the transfer matrix A(s), which relates the inverterside current vector i1n and the voltage capacitor vector vZ3

acquires the characteristic symmetry of this application. Hence,its diagonal elements are A11 and its nondiagonal elements areA12

vZ3 = A(s) · i1n ⇔

⎛

⎜⎝

vZ 31vZ 32· · ·

vZ 3n

⎞

⎟⎠

=

⎛

⎜⎝

A11 A12 · · · A12A12 A11 · · · A12· · · · · · · · · · · ·A12 A12 · · · A11

⎞

⎟⎠ ·

⎛

⎜⎝

i11i12· · ·i1n

⎞

⎟⎠ . (23)

The elements A11 and A12 are obtained by the superpositionprinciple and Thevenin equivalent circuits. In the following rea-soning, all the converter voltages v0i are considered to be zero.The diagonal element A11 can be regarded as the transfer func-tion between the capacitor voltage vZ 31 and the inverter sidecurrent i11 . Accordingly, A11 can be obtained if the inverterside currents i1i are supposed to be ideal current sources whosevalue is zero for all of them except for i11 . The correspondingauxiliary circuit has been omitted. The transfer function betweenvZ 31 and i11 is

A11 =vZ 31

i11=

Z3 · (Z2 + Z3 · Zg + Z2 · (Z3 + n · Zg ))(Z2 + Z3) · (Z3 + Z2 + n · Zg )

.

(24)The nondiagonal element A12 might be regarded as the transferfunction between the capacitor voltage vZ 31 and the inverter sidecurrent i12 . A12 can be calculated if the inverter side currents i1i

are supposed to be ideal current sources whose value is zero forall of them except for i12 . The corresponding auxiliary circuithas been omitted. The transfer function between vZ 31 and i12 is

A12 =vZ 31

i12=

Z23 · Zg

(Z2 + Z3) · (Z3 + Z2 + n · Zg ). (25)

Each element of the MIMO loop in Fig. 10(a) has been de-scribed. If the feedback path of the capacitor voltages for activedamping is closed; the control scheme of Fig. 10(b) is derivedfrom Fig. 10(a). Note that Fig. 10(b) is the MIMO version ofFig. 5(b). In this sense, J(s) is the actively damped matrix andis obtained as pointed in

i1n = J(s) · u0n ⇔

⎛

⎜⎝

i11i12· · ·i1n

⎞

⎟⎠

=

⎛

⎜⎝

J11 J12 · · · J12J12 J11 · · · J12· · · · · · · · · · · ·J12 J12 · · · J11

⎞

⎟⎠ ·

⎛

⎜⎝

u01u02· · ·u0n

⎞

⎟⎠ (26)

J(s) = [I − G(s) · Fv(s) · A(s)]−1 · G(s). (27)

The I element in (27) is the identity matrix. Based on the as-sumption that in a PV plant all the installed inverters are equal,all matrices involved in the calculation of J(s) have the character-istic symmetry of this application, i.e., all its diagonal elementsare identical and all nondiagonal elements are also identicalbut different to the diagonal ones. Fortunately, the product andinverse operations of this kind of matrices preserve the charac-teristic symmetry as shown in the Appendix, so the J(s) matrixacquires this symmetry. Hence, its diagonal elements are J11and its nondiagonal elements are J12 . The expressions of J11and J12 are rather complicated and are included in the Appendix.Clearly, their denominators are equal but not their numerators.

Since the inverters are expected to be equal, the output ofevery PI controller may be considered equal, i.e., u0i = u0 .Replacing in (26), we obtain

⎛

⎜⎝

i1i1· · ·i1

⎞

⎟⎠ =

⎛

⎜⎝

J11 J12 · · · J12J12 J11 · · · J12· · · · · · · · · · · ·J12 J12 · · · J11

⎞

⎟⎠ ·

⎛

⎜⎝

u0u0· · ·u0

⎞

⎟⎠ . (28)

Multiplying a J(s) row by vector u0n in (28), we derive thesimplification Y1ADeq

i1 = (J11 + (n − 1) · J12) · u0 = Y1ADeq · u0 (29)

Y1ADeq

=Z2 + Z3 + n · Zg

(1 − F ′v ) · Z3 · (Z2 + n · Zg ) + Z1 · (Z3 + Z2 + n · Zg )

.

(30)

If the expression of Y1ADeq (30) is compared with those of J11and J12 (shown in the Appendix) it is observed that Y1ADeqis much simpler. Again, the transfer function Y1ADeq can beidentified with that of the single inverter case (10). The onlydifference between both expressions Y1ADeq and Y1AD is thatin Y1ADeq the grid impedance appears multiplied by n. In con-sequence, if all the outputs of the controllers are equal u0i = u0an equivalent inverter whose grid impedance is N times bigger(see Fig. 9) models the N inverters to which the active dampingtechnique of [4] is applied. The bode diagram of Y1ADeq with


Fig. 11. Bode diagrams for different values of N : (a) Y1eq ; (b) Y1ADeq ; and (c) OLADeq .

the Set I of LCL filter values compiled in Table I and the con-trol parameters gathered in Table II is plotted in Fig. 11(b) fordifferent values of N.

D. Stability Analysis

In order to discuss the stability of the system, the closed-loop poles have to be analyzed. Hence, the matrix T(s) thatrelates the current reference vector i1n ref and the inverter sidecurrent vector i1n is obtained (31) and (32). Based on the as-sumption that in a PV plant all the installed inverters are equal,all matrices involved in the calculation of T(s) have the char-acteristic symmetry of this application. Since the product andinverse operations of this kind of matrixes preserve the charac-teristic symmetry, the closed-loop transfer matrix T(s) acquiresthis symmetry. Hence, its diagonal elements are T11 and itsnondiagonal elements are T12

i1n = T(s) · i1n ref ⇔

⎛

⎜⎝

i11i12· · ·i1n

⎞

⎟⎠

=

⎛

⎜⎝

T11 T12 · · · T12T12 T11 · · · T12· · · · · · · · · · · ·T12 T12 · · · T11

⎞

⎟⎠ ·

⎛

⎜⎝

i11refi12ref· · ·

i1nref

⎞

⎟⎠ (31)

T(s) = [I + J(s) · PI(s)]−1 · J(s) · PI(s). (32)

From expression (32), the diagonal element T11 and the nondi-agonal element T12 are calculated as

T11 =

PI′ · ((J211 − (n−1) · J2

12) ·PI′ +J11 · (1+(n−2) · J12 ·PI′))(1+(J11 −J12) ·PI′) · (1+(J11 +(n−1) · J12) ·PI′)

(33)

T12 =J12 ·PI′

(1+(J11 −J12) ·PI′) · (1+(J11 +(n−1) · J12) ·PI′).

(34)

These expressions, (33) and (34), lead to too high-order transferfunctions, difficult to handle. As expected, their denominatorscoincide, since the poles of any multivariable system have to bethe same regardless the input, but not their zeros. Notice that theterms in the denominators of T11 and T12 are the closed-looppoles of the system. In this paper, they are classified as externalstability poles and internal stability poles (35). If and only if theexternal and the internal stability poles are all in the left halfs-plane, the system will be stable

(1 + (J11 + (n − 1) · J12) · PI′) ⇒ external stability poles

(1 + (J11 − J12) · PI′) ⇒ internal stability poles. (35)

If the inverters are equal, including their hardware, their soft-ware, and their PV generators, they will react all in the sameway. Thus, their inverter side current references will be equal,and so will be the inverter side currents i1i = i1

⎛

⎜⎝

i1i1· · ·i1

⎞

⎟⎠ =

⎛

⎜⎝

T11 T12 · · · T12T12 T11 · · · T12· · · · · · · · · · · ·T12 T12 · · · T11

⎞

⎟⎠ ·

⎛

⎜⎝

i1refi1ref· · ·i1ref

⎞

⎟⎠ . (36)

Multiplying a T(s) row by vector i1n ref in (36), we derive thesimplification (37)

i1 = (T11 + (n − 1) · T12) · i1ref

=(J11 + (n − 1) · J12) · PI′

1 + (J11 + (n − 1) · J12) · PI′· i1ref . (37)

A simpler transfer function than T11 and T12 is obtained in (37).This is because some elements in the numerators of T11 and(n − 1). T12 combine and cancel terms of their denominators.The remaining term corresponds to the external stability poles,


Fig. 12. System externally and internally stable. (a) Theoretical analysis OLADeq . (b) Simulated waveforms with the Set I of LCL filter values of Table I.(c) Simulated waveforms with a 20% tolerance in LCL filter values.

whereas the canceled term corresponds to the internal stabilitypoles.

Using (29) and replacing in (37), we obtain

i1 =Y1ADeq · PI′

1 + Y1ADeq · PI′· i1ref =

OLADeq

1 + OLADeq· i1ref (38)

OLADeq = Y1ADeq · PI′. (39)

Expression (38) represents the closed-loop transfer function ofan equivalent inverter whose grid impedance is N times bigger(see Fig. 9) and in which the control strategy of [4] is imple-mented. Thus, the external stability poles are related with theequivalent inverter and might be analyzed representing the bodediagram of the open-loop transfer function OLADeq (39) as inFig. 11(c). Note that the open-loop transfer function OLADeqnot only is useful to determine the external stability, but alsoto design the PI controller and to fix the bandwidth and thephase margin of N -coupled inverters. Furthermore, the controllaw design of N -coupled inverters greatly resembles the singlegrid-connected inverter case of Section III. The only differenceis that a grid impedance N times bigger has to be taken intoaccount.

In order to discuss the internal stability, the term of the T11and T12 denominators canceled when all current references areequal, has to be analyzed. Remember that unstable poles cannotbe canceled [22]. Consequently, if the internal stability term haspoles in the right half s-plane, the system will be unstable evenif all current references are equal.

If attention is paid to (35), it can be observed that the internalstability poles can be identified with the closed-loop poles of ahypothetical SISO system H expressed as

H =(J11 − J12) · PI′

1 + (J11 − J12) · PI′=

OLH

1 + OLH. (40)

Thus, representing the open-loop bode of this hypothetical sys-tem OLH it is possible to know whether the internal stabilityterm OLH has right-half s-plane poles. The plant of this hypo-thetical system is shown as follows:

J11 − J12 =Z2 + Z3

(1 − F ′v ) · Z3 · Z2 + Z1 · (Z2 + Z3)

. (41)

Comparing expressions, we observe that fortunately expression(41) is equal to the Y1ADeq expression if N = 0. It is difficultto find a physical sense to this fact, since N = 0 means that anyinverter is placed in the PV plant. However, the stated is true

Y1ADeq |n=0 = (J11 + (n − 1) · J12)|n=0

=Z2 + Z3

(1 − F ′v ) · Z3 · Z2 + Z1 · (Z2 + Z3)

= J11 − J12 . (42)

Consequently, (40) and (42) confirm that the internal stabilitycan easily be determined by means of the open loop of theequivalent inverter OLADeq provided that N = 0 (43). Thus, theinternal stability can be analyzed in Fig. 11(c), in which the firstbode plot corresponds to OLADeq when N = 0

OLH = OLADeq |n=0 . (43)

Note that the internal stability neither depends on the numberof inverters N nor on the grid impedance Zg . Thus, the internalstability problem is the same whether in a certain PV plant thereare many or few inverters, or whether the grid is stiff or weak,whereas the external stability problem is completely different.

V. SIMULATION RESULTS

Next, simulation results are analyzed so as to validate thetheoretical study of previous sections. The simulations are per-formed with the software PSIM. Two different ways of simulat-ing the N inverters are considered. The first one is to implementN independent inverters as in Fig. 6. This is the most detailedsimulation, since any simplification is done. However, it is te-dious if the number of inverters is large. A second way is toimplement the equivalent inverter of Fig. 9. Only N = 3 invert-ers are considered. They are supposed to be full-bridge singlephase inverters of 10 kW with unipolar modulation. All invertersare given in the same current reference value i1ref . A grid effec-tive phase voltage eg = 230 V is supposed in these simulations.All current simulated waveforms presented are in amperes.

First, the Set I of LCL filter values of Table I and the controlparameters of Table II are chosen. In Fig. 12(a), the open-loopbode diagrams are plotted. The N = 3 bode plot shows that


Fig. 13. System externally stable but internally unstable. (a) Theoretical analysis OLADeq . (b) Simulated waveforms with the Set II of LCL filter values ofTable I. (c) Simulated waveforms with a 20% tolerance in LCL filter values.

the system is externally stable. The N = 0 bode plot showsthat the system is internally stable. Consequently, the systemis stable since it is both, externally and internally stable. Thesimulated waveforms of this system are shown in Fig. 12(b).The suffix “eq” is added to the equivalent inverter variables inorder to distinguish them. First, the inverter side current i11 ofone of the N independent inverters and the inverter side currenti1 eq of the equivalent inverter are shown. Second, the sum ofthe inverter side currents of the N independent inverters i11 +i12 + i13 and the inverter side current of the equivalent invertermultiplied by N are shown. Last, the duty cycles of one of theN independent inverters d1 and that of the equivalent inverterd1 eq are shown. In this simulation, a step change occurs inthe effective current reference value at t = 45 ms. At everyinstant, the N independent inverter variables and the equivalentinverter variables are overlapped. This is because the two waysof simulating are similar provided that the system is internallystable and the same current reference value is given.

Second, the Set II of LCL filter values of Table I and thecontrol parameters of Table II are chosen. In Fig. 13(a), theopen-loop bode diagrams are plotted. The N = 3 bode plotshows that the system is externally stable. However, the N =0 bode plot shows that the system is internally unstable sinceinsufficient damping of the resonance frequency is obtained.Consequently, the system is unstable. The simulated waveformsof this system are shown in Fig. 13(b). The depicted variablesare the same as in Fig. 12(b). The information of the internalstability poles is lost in the equivalent inverter simplification, butnot in the N independent inverter simulation. Since the system isinternally unstable, the variables of the N independent inverterswill diverge and the duty cycles will saturate. The behavior ofthe inverters in saturation is nonlinear and, as a result, the bodediagrams are not valid. In the simulation of Fig. 13(b), the PWMmodulators have been replaced by ideal voltage sources. Thus,the switching effect is not taken into account but saturation isavoided and the internal instability can be depicted clearer. Ifthe inverter side current of one of the N inverters i11 and theinverter side currents of the equivalent inverter i1 eq are com-pared, it can be observed that they coincide initially, but at theend, current i11 diverges at a resonant frequency due to internal

instability. This resonant frequency is that of the N = 0 bodeplot of Fig. 13(a). Although not shown in the figure, the currentsi12 and i13 also diverge at this resonant frequency. In contrast,the sum of the currents of the N independent inverters, i11 +i12 + i13 , and the equivalent inverter current multiplied by N ,n·i1 eq coincide at every instant. This is because the currents i11 ,i12 , and i13 have two components. The first component, equalfor the N independent inverters, is at 50 Hz. This componentappears in the sum i11 + i12 + i13 and is injected in the grid as aresult. The second component is at the resonance frequency be-tween inverters that diverges due to internal instability. This lastcomponent goes from the reactive elements of one inverter tothe reactive elements of another inverter and it does not circulatethrough the grid. That is why it does not appear in the sum i11 +i12 + i13 . In Fig. 13(b), the variables of one of the N indepen-dent inverters and the equivalent inverter coincide initially onlybecause the same reference value is given. If different currentreference values were given to the N independent inverters inthis simulation, the internal stability poles would not tend tocancel and the internal instability would become apparent fromthe first instant.

These simulations validate the theoretical results obtainedin previous sections. The equivalent inverter describes the Ninverters of a PV plant. Only the internal stability poles infor-mation is lost but it can be studied theoretically in the bodediagram. Note that the internal stability neither depends on thenumber of inverters N nor on the grid impedance Zg . In conse-quence, only two-paralleled inverters have to be programed inorder to simulate the internal stability.

In order to obtain the equivalent inverter, it is assumed thatall the inverters are equal, including their hardware, their soft-ware, and their PV generators (17), (28), and (36). Differencesin irradiance, partial shadings, and mismatches in PV generatorswill affect the current reference value of each inverter, but donot change the poles of the system. Active anti-islanding meth-ods such as impedance measuring method and frequency shiftmethod [26] are based on giving a different current referencevalue. They involve different frequencies, but neither changethe poles of the system. If the inverters are of the same sort,the software is the same for them all and the programed values


Fig. 14. Experimental measures of the voltage in the PCC vg and the grid side current i2 i of an inverter in a PV plant. (a) Distorted waveforms due to resonantbehavior. (b) Half grid period detail. (c) Frequency spectrum fast Fourier transform (FFT).

will not change. In consequence, the validity of the equiva-lent inverter approach is only affected by the variations on thehardware. A Gaussian distribution of parameters is expected inindustry. The effect of parameter variations on the equivalentinverter approximation is depicted in Figs. 12(c) and 13(c). TheSet I of LCL filter values of Table I is used in Fig. 12(c), andthe Set II is used in Fig. 13(c), but in these simulations, theparameters of the N independent inverters have been variedrandomly a 20% up and down its nominal value, whereas theparameters of the equivalent inverter remain unchanged. Eachinverter side current of the N independent inverters i1i is com-pared with that of the equivalent inverter i1 eq . The same currentreference value is given to all inverters. Fig. 12(c) shows that allinverter side currents greatly resemble the equivalent invertercurrent despite the parameter variations. Moreover, Fig. 13(c)shows that the internal instability becomes apparent in the Nindependent inverters. Thus, it is concluded that the equivalentinverter describes accurately the N independent inverters evenif a tolerance in parameters of 20% exists.

VI. EXPERIMENTAL RESULTS

The experimental results shown in Fig. 14 correspond to realwaveforms of PV inverters with resonant behavior in a PV plantof 1400 kW. The figure depicts well the voltage in the PCC vg

and the grid side current of an inverter i2i . The oscillation occursat a resonant frequency of 2650 Hz.

These inverters were 5-kW full-bridge single phase IngeconSun inverters with unipolar modulation. They satisfied individ-ually all the standards and performed correctly in other PVplants, but had been designed ignoring the coupling effect dueto the grid impedance. They had an isolation transformer, whoseleakage impedances can be neglected, and an LC output filterinstead of an LCL filter. Considering L2 = 0, we can adapt thestudy carried out in this paper to the LC filter case. Note that ifL2 = 0 the voltage in the PCC vg and the capacitors voltagesvZ 3i coincide. The filter values are shown in Table III. The perunit values referred to the grid effective phase voltage eg =230 V and to the rated power of the Ingecon Sun inverterPNI = 5 kW are facilitated in Table III.

TABLE IIIPV POWER PLANT CHARACTERISTICS

The total number of installed inverters in the PV plant was270. The inverters were connected to a three-phase 13-kV net-work. These single-phase inverters were distributed in threeequal phases. In consequence, N = 270/3 = 90 has to be con-sidered in the equivalent inverter model. The grid impedancewas the leakage impedance of the 1500-kW distribution trans-former. It is calculated according to (44) using the transformerparameters shown in Table III

Lg =vcc

2 · π · 50· v2

1L

PNT=

0.042 · π · 50

· 4002

1500= 0.013 mH.

(44)The cable impedances of the PV plant and most parasiticimpedances were neglected. These parasitic impedances mainlyaffect the high-frequency range and are difficult to estimate sincethey lead to too high order transfer functions [2], [25]. More-over, the resonance frequencies associated to these parasiticimpedances, higher than those of the LCL filter, are correctlydamped due to the skin effect and the magnetic effect [2], [25].

After studying the coupling effect, the resonance problem ofthe PV plant shown in Fig. 14 was identified as an externalstability problem. It can be observed, in Fig. 16(a), that thewhole system was externally stable but near instability. Therewas insufficient damping of the resonance and changes in thegrid voltage eg and in the reference current i1 ref excited theresonance frequency. This exceeded the harmonic regulationsand caused the breakage of many devices such as the electronicmeters. No internal stability problems occurred since LCL filter


Fig. 15. Simulated waveforms of the voltage in the PCC vg eq and the grid side current i2 eq of the equivalent Inverter representing the PV plant. (a) Distortedwaveforms due to resonant behavior. (b) Half grid period detail. (c) Frequency spectrum FFT.

Fig. 16. PV plant with resonant behavior. (a) Theoretical analysis OLADeq .(b) Simulated waveforms.

resonance does not exist provided that L2 = 0 and N = 0.Fig. 16(b) shows the corresponding waveforms of the equivalentinverter simulated according to the parameters of Table III. First,the inverter side current i1 eq and its reference value i1ref eq are

Fig. 17. Curves whose points have the same loop characteristics since theirproducts coincide.

shown. Second, the grid side current i2 eq and the voltage in thePCC vg eq are shown. The current i2 eq is scaled by a factor often in order to make it visible. Finally, the duty cycle d1 eq isdepicted.

In order to better compare experimental results and simu-lations results, Fig. 15 contains the simulated waveforms ofFig. 16(a) at a similar instant than the experimental results ofFig. 14. The resonance oscillation occurs at 2650 Hz. It corre-sponds to the resonance frequency ωreq of the equivalent inverterand is easily obtained by replacing the LC filter values of theIngecon Sun inverter (see Table III), the number of inverters N ,and the grid impedance value (44) in following expression:

ωreq =1

√C3 · L1 · (L2 + n · Lg )/(L1 + L2 + n · Lg )

= 16224 rad/s → 2615 Hz. (45)

The equivalent inverter permitted to describe the resonanceproblem and to design a suitable solution for the single phaseIngecon Sun inverters.


VII. CONTROL DESIGN GUIDELINES AND PRACTICAL USES

OF THE EQUIVALENT INVERTER

In this paper, PI controllers are considered, and the damp-ing technique proposed in [4] is chosen, but the methodologyfollowed in this paper can also be applied to other control algo-rithms or active damping techniques. The authors have essayedseveral of the active damping proposals, such as [4]–[14], tocontrol the inverter side current i1 or the grid side current i2 andthe same equivalent inverter model has been derived. In rela-tion to the external stability and the internal stability problem,analogous results are obtained regardless of the control strat-egy: the open-loop transfer function of the equivalent inverterdetermines the external stability. The internal stability is studiedreplacing N = 0 in the open-loop transfer function expression ofthe equivalent inverter. In consequence, simplifications derivedin this paper depend on the symmetry of the system, but not onthe control strategy.

In this study, the number of inverters N has been variedto depict the coupling effect (see Fig. 11). However, the loopcharacteristics in Fig. 11 do not truly depend on the numberof inverters N itself. The really important thing is the productN·Lg , since all the different values of N and Lg whose productis the same have the same loop characteristics. In other words,the loop characteristics are the same whether a PV plant has thevalues Lg = 0.1 mH, N = 1200, as plotted in Fig. 11, or thevalues Lg = 0.3 mH, N = 400, since their products coincide.Fig. 17 displays several curves of constant N·Lg . Points whosebode diagram is plotted in Fig. 11 are marked, but points of thesame curve have the same bode plots.

The stiffer the grid is the more paralleled inverters can beplaced. In a weak grid, less paralleled inverters have to be placedin order to obtain the same loop characteristics. Thus, the controllaw has to be designed to perform in a certain range of N·Lg .For the external stability requirement, there is a maximum valuefor N·Lg = NLmax . In order to satisfy the internal stabilityrequirement, the product value N·Lg = 0 has to be taken intoaccount. Due to the parameter uncertainty, the most sensible isto design to be stable within the whole range N·Lg = [0, NLmax ].

The resonance frequencies of the equivalent inverter (20) de-pend on the product value N·Lg . If an active strategy is required,it is essential to be able to estimate the resonance frequencyωreq . If the product N·Lg increases, ωreq decreases down to alimited value. As a result, the resonance frequency will alwaysbe between its minimum (when N·Lg = ∞) and its maximumvalue (when N·Lg = 0) as expressed in (46). Thus, the range ofvariation of the resonance ωreq is well defined. This eases theimplementation of the damping strategy

1√C3 · L1

≤ ωreq ≤ 1√

C3 · L1 · L2/(L1 + L2). (46)

On the contrary, the antiresonance ωaeq decreases unlimitedly ifthe product value N·Lg increases. Therefore, the antiresonancefrequency will always be between 0 Hz (when N·Lg = ∞) andits maximum value (when N·Lg = 0) as expressed in (47). Thislimits the bandwidth achievable since the crossover frequency isreduced as N·Lg increases [see Fig. 11(c)]. Note that, if resonantcontrollers or similar were used and the coupling effect was not

taken into account, the resonant frequencies of these controllerscould surpass the bandwidth of the system. In consequence, thesystem would be unstable as pointed by [16]

0 ≤ ωaeq ≤ 1√C3 · L2

. (47)

Different control algorithms and active damping techniquescan be tested in simulations without programing the total Ninverters of the plant thanks to the equivalent inverter. Further-more, the equivalent inverter can be subjected to many exper-imental uses. The behavior of a certain control strategy in areal PV inverter can be predicted without having to install theN inverters in a PV plant. Simply, a single real inverter hasto be connected to the grid through an auxiliary inductor Lauxwhose value has to be varied within the range of the productN·Lg , under consideration i.e., Laux = [0, NLmax ]. The internalstability of a real inverter can be tested if Laux = 0, but thismay be difficult to meet since in every connexion point a gridimpedance always exists. However, note that only two invertersinstalled in the same PCC are needed in order to test the internalstability. This is because the internal stability neither dependson the number of inverters N nor in the grid impedance Zg .

VIII. CONCLUSION

This paper analyses the modeling and control of N -paralleledgrid-connected inverters with the LCL filter in PV plants. Anequivalent inverter that models the N inverters of a PV plantis obtained. The internal stability poles information is lost butit can be studied theoretically through the bode diagram andit can be simulated provided that only two-paralleled invertersare programed. Thus, the coupling effect due to grid impedanceis described and easy control design guidelines are suggested.Although a specific control strategy is analyzed, based on PIcontrol and the active damping technique proposed in [4], theequivalent inverter simplification can be obtained regardless ofthe control strategy. This study is validated by simulated andexperimental results.

APPENDIX

See (15), (16), and (48)–(50) at the top of the next page.

REFERENCES

[1] E. Figueres, G. Garcera, J. Sandia, F. Gonzalez-Espin, and J. C. Rubio,“Sensitivity study of the dynamics of three-phase photovoltaic inverterswith an LCL grid filter,” IEEE Trans. Ind. Electron., vol. 56, no. 3,pp. 706–717, Mar. 2009.

[2] M. Liserre, F. Blaabjerg, and S. Hansen, “Design and control of an LCL-filter-based three-phase active rectifier,” IEEE Trans. Ind. Appl., vol. 41,no. 5, pp. 1281–1291, Sep./Oct. 2005.

[3] K. Jalili and S. Bernet, “Design of LCL filters of active-front-end two-level voltage-source converters,” IEEE Trans. Ind. Electron., vol. 56,no. 5, pp. 1674–1689, May 2009.

[4] V. Blasko and V. Kaura, “A novel control to actively damp resonance ininput LC filter of a three-phase voltage source converter,” IEEE Trans.Ind. Appl., vol. 33, no. 2, pp. 542–550, Mar./Apr. 1997.

[5] P. A. Dahono, “A control method to damp oscillation in the input LC filterof AC-DC PWM converters,” in Proc. – IEEE Annu. Power Electron.Specialists Conf. (PESC) Record, 2002, vol. 4, pp. 1630–1635.

[6] M. Liserre, A. Dell’Aquila, and F. Blaabjerg, “Stability improvementsof an LCL-filter based three-phase active rectifier,” in Proc. IEEE 33rdAnnu. IEEE Power Electron. Spec. Conf. Proc. (Cat. No. 02CH37289),2002, vol. 3, pp. 1195–1201.


G11 =Z1 · (Z2 + Z3) · (Z2 + Z3 + n · Zg ) + Z3 ·

(Z2

2 + (n − 1) · Z3 · Zg + Z2 · (Z3 + n · Zg ))

(Z2 · Z3 + Z1 · (Z2 + Z3)) · (Z3 · (Z2 + n · Zg ) + Z1 · (Z3 + Z2 + n · Zg ))(15)

G12 = − Z23 · Zg

(Z2 · Z3 + Z1 · (Z2 + Z3)) · (Z3 · (Z2 + n · Zg ) + Z1 · (Z3 + Z2 + n · Zg ))(16)

J11 =Z1 · (Z2 + Z3) · (Z2 + Z3 + n · Zg ) + (1 − Fv ) · Z3 ·

(Z2

2 + (n − 1) · Z3 · Zg + Z2 · (Z3 + n · Zg ))

((1 − Fv ) · Z2 · Z3 + Z1 · (Z2 + Z3)) · ((1 − Fv ) · Z3 · (Z2 + n · Zg ) + Z1 · (Z3 + Z2 + n · Zg ))(48)

J12 = − (1 − Fv ) · Z23 · Zg

((1 − Fv ) · Z2 · Z3 + Z1 · (Z2 + Z3)) · ((1 − Fv ) · Z3 · (Z2 + n · Zg ) + Z1 · (Z3 + Z2 + n · Zg ))(49)

⎛

⎜⎜⎜⎝

a b · · · b

b a · · · b

· · · · · · · · · · · ·b b · · · a

⎞

⎟⎟⎟⎠

·

⎛

⎜⎜⎜⎝

e f · · · f

f e · · · f

· · · · · · · · · · · ·f f · · · e

⎞

⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎝

g h · · · h

h g · · · h

· · · · · · · · · · · ·h h · · · g

⎞

⎟⎟⎟⎠

⇔{

g = a · e + (n − 1) · b · fh = a · f + b · e + (n − 2) · b · f

⎛

⎜⎜⎜⎝

a b · · · b

b a · · · b

· · · · · · · · · · · ·b b · · · a

⎞

⎟⎟⎟⎠

−1

=

⎛

⎜⎜⎜⎝

c d · · · d

d c · · · d

· · · · · · · · · · · ·d d · · · c

⎞

⎟⎟⎟⎠

⇔

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

c =a + (n − 2) · b

a2 + (n − 2) · a · b − (n − 1) · b2

d = − b

a2 + (n − 2) · a · b − (n − 1) · b2

(50)

[7] M. Liserre, A. Dell’Aquila, and F. Blaabjerg, “Genetic algorithm-baseddesign of the active damping for an LCL-filter three-phase active rectifier,”IEEE Trans. Power Electron., vol. 19, no. 1, pp. 76–86, Jan. 2004.

[8] E. Wu and P. W. Lehn, “Digital current control of a voltage source converterwith active damping of LCL resonance,” IEEE Trans. Power Electron.,vol. 21, no. 5, pp. 1364–1373, Sep. 2006.

[9] L. A. Serpa, S. Ponnaluri, P. M. Barbosa, and J. W. Kolar, “A modifieddirect power control strategy allowing the connection of three-phase in-verters to the grid through LCL filters,” IEEE Trans. Ind. Appl., vol. 43,no. 5, pp. 1388–1400, Sep/Oct. 2007.

[10] M. Malinowski and S. Bernet, “A simple voltage sensorless active damp-ing scheme for three-phase PWM converters with an LCL filter,” IEEETrans. Ind. Electron., vol. 55, no. 4, pp. 1876–1880, Apr. 2008.

[11] I. J. Gabe, V. F. Montagner, and H. Pinheiro, “Design, and implementationof a robust current controller for VSI connected to the grid through an LCLfilter,” IEEE Trans. Power Electron., vol. 24, no. 6, pp. 1444–1452, Jun.2009.

[12] E. Twining and D. G. Holmes, “Grid current regulation of a three-phasevoltage source inverter with an LCL input filter,” IEEE Trans. PowerElectron., vol. 18, no. 3, pp. 888–895, May 2003.

[13] F. Liu, Y. Zhou, S. Duan, J. Yin, B. Liu, and F. Liu, “Parameter design of atwo-current-loop controller used in a grid-connected inverter system withLCL filter,” IEEE Trans. Ind. Electron., vol. 56, no. 11, pp. 4483–4491,Nov. 2009.

[14] G. Shen, D. Xu, L. Cao, and X. Zhu, “An improved control strategy forgrid-connected voltage source inverters with an LCL filter,” IEEE Trans.Power Electron., vol. 23, no. 4, pp. 1899–1906, Jul. 2008.

[15] J. Dannehl, C. Wessels, and F. W. Fuchs, “Limitations of voltage-orientedPI current control of grid-connected PWM rectifiers with LCL filters,”IEEE Trans. Ind. Electron., vol. 56, no. 2, pp. 380–388, Feb. 2009.

[16] M. Liserre, R. Teodorescu, and F. Blaabjerg, “Stability of photovoltaic,and wind turbine grid-connected inverters for a large set of grid impedancevalues,” IEEE Trans. Power Electron., vol. 21, no. 1, pp. 263–272, Jan.2006.

[17] M. Castilla, J. Miret, J. Matas, L. G. de Vicuna, and J. M. Guerrero, “Linearcurrent control scheme with series resonant harmonic compensator forsingle-phase grid-connected photovoltaic inverters,” IEEE Trans. Ind.Electron., vol. 55, no. 7, pp. 2724–2733, Jul. 2008.

[18] M. Castilla, J. Miret, J. Matas, L. G. de Vicuna, and J. M. Guerrero,“Control design guidelines for single-phase grid-connected photovoltaicinverters with damped resonant harmonic compensators,” IEEE Trans.Ind. Electron., vol. 56, no. 11, pp. 4492–4501, Nov. 2009.

[19] J. H. R. Enslin and P. J. M. Heskes, “Harmonic-interaction between a largenumber of distributed power inverters, and the distribution network,” IEEETrans. Power Electron., vol. 19, no. 6, pp. 1586–1593, Nov. 2004.

[20] J. H. R. Enslin, Y. Hu, and R. A. Wakefield, “System considerations, andimpacts of AC cable networks on weak high voltage transmission net-works,” in Proc. 2005/2006 IEEE/PES Transm. Distrib. Conf. Exhibition,May 2006, pp. 1030–1034.

[21] P. Sanchis, A. Ursua, E. Gubia, and L. Marroyo, “Boost DC-AC inverter:A new control strategy,” IEEE Trans. Power Electron., vol. 20, no. 2,pp. 343–353, Mar. 2005.

[22] J. J. D’Azzo, C. H. Houpis, and S. N. Sheldon, Linear Control SystemsAnalysis, and Design With Matlab, 5th ed. New York: Marcel Dekker,2003.

[23] O. N. Gasparyan, Linear, and Nonlinear Multivariable Feedback ControlSystems: A Classical Approach. Chichester, U.K.: Wiley, 2008.

[24] C. H. Houpis, S. J. Rasmussen, and M. Garcıa-Sanz, Quantitative Feed-back Theory. Fundamentals, and applications, 2nd ed. New York: Taylor& Francis, 2006.

[25] E. Gubıa, P. Sanchis, A. Ursua, J. Lopez, and L. Marroyo, “Frequency do-main model of conducted EMI in electrical drives,” IEEE Power Electron.Lett., vol. 3, no. 2, pp. 45–49, Jun. 2005.

[26] P. Sanchis, L. Marroyo, and J. Coloma, “Design methodology for the fre-quency shift method of islanding prevention, and analysis of its detectioncapability,” Prog. Photovoltaics: Res. Appl., vol. 13, no. 5, pp. 409–428,Aug. 2005.

[27] S. Y. Park, Ch. L. Chen, J. S. Lai, and S. R. Moon, “Admittance compen-sation in current loop control for a grid-tie LCL fuel cell inverter,” IEEETrans. Power Electron., vol. 23, no. 4, pp. 1716–1723, Jul. 2008.

[28] T. Kerekes, M. Liserre, R. Teodorescu, C. Klumpner, and M. Sumner,“Evaluation of three-phase transformerless photovoltaic inverter topolo-gies,” IEEE Trans. Power Electron., vol. 24, no. 9, pp. 2202–2211, Sep.2009.

[29] A. Timbus, M. Liserre, R. Teodorescu, P. Rodrıguez, and F. Blaabjerg,“Evaluation of current controllers for distributed power generation sys-tems,” in IEEE Trans. Power Electron., no. 3. vol. 24, Mar. 2009,pp. 654–664.

Juan Luis Agorreta was born in Tudela, Spain, in1984. He received the M.Sc. degree in industrialengineering from the Public University of Navarra,Pamplona, Spain, in 2008, where he is currentlyworking toward the Ph.D. degree in the Departmentof Electrical and Electronic Engineering.

In 2008, he was with the Electrical Engi-neering, Power Electronics, and Renewable Energy(INGEPER) Research Group. He is also involved inresearch projects mainly in co-operation with indus-try. His research interests include power electronics

and renewable energy.


Mikel Borrega received the M.Sc. degree in indus-trial engineering from the University of Mondragon,Mondragon, Spain, in 2003.

He has been engaged in research projects in the PVSolar Energy Department of Ingeteam Energy since2003. He joined the Department of Electrical andElectronic, Public University of Navarra, Pamplona,Spain, in 2008, where he is currently an AssistantProfessor. His research interests include power elec-tronics and renewable energies.

Jesus Lopez (M’05) was born in Pamplona, Spain,in 1975. He received the M.Sc. degree from the Pub-lic University of Navarra, Pamplona, Spain, in 2000and the Ph.D. degree from the Public Universityof Navarra, Spain, in collaboration with LAPLACELaboratory, Toulouse, France, in 2008, both in indus-trial engineering.

In 2001, he joined the Power Electronic Group,Department of Electrical and Electronic, Public Uni-versity of Navarra, where he is currently an AssistantProfessor and is also involved in research projects

mainly in co-operation with industry. His research interests include power elec-tronics, power systems quality and renewable energies, such as wind turbinesand photovoltaic plants.

Luis Marroyo (M’04) received the M.Sc. degree inelectrical engineering from the University of Tolouse,France, in 1993, and the Ph.D. degree in electricalengineering from the Public University of Navarra,Pamplona (UPNA), Spain, in 1997, and from theLEEI-ENSEEIHT INP Toulouse, France, in 1999.

From 1993 to 1998, he was an Assistant Professorin the Department of Electrical and Electronic En-gineering, UPNA, where he has been an AssociateProfessor since 1998. He is currently the Head of theINGEPER Research Group. His research interests in-

clude power electronics, grid quality, and renewable energy.

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Connected Inverters With Lcl Filters

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