Interconnection and Damping AssignmentPassivity-Based Current Control of

Grid-Connected PWM Converter withLCL-Filter

Matthias Bottcher, Jorg Dannehl∗, Friedrich W. Fuchs∗∗Institute for Power Electronics and Electrical Drives / University of Kiel, Kiel, Germany,

email: mabo@tf.uni-kiel.de, ∗jda@tf.uni-kiel.de, ∗∗fwf@tf.uni-kiel.de

Abstract—Interconnection and Damping Assignment Pas-sivity-Based Control of grid-connected PWM converterswith LCL-filter is designed and analyzed. Dynamic perfor-mance, resonance damping and robustness against gridimpedance variations are studied by simulation and showgood results. Experimental tests on a 22 kW induction motordrive prove the simulation results.

Index Terms—IDA-PBC, PWM converter, LCL-filter, ro-bustness against grid impedance.

I. INTRODUCTION

Grid-connected PWM converters offer bi-directionalpower transfer with low harmonic current distortion.Typical applications are in the growing field of distributedgeneration systems (DGS), regenerative motor drives andactive filters [1], [2]. Standards set stringent limits on thegrid current harmonic distortion which can be devidedinto two parts. The first issue is related to low frequencyharmonics due to grid voltage background distortion. It isa matter of disturbance rejection capability of the currentcontroller [3]. Typically harmonic compensators like reso-nant controllers are employed for low frequency harmonicrejection [4]. High control bandwidth is required for goodharmonic rejection. Otherwise stability problems mayoccur [5]. Due to this reason stability problems may alsooccur when current control bandwidth suffers from gridimpedance variations [5].

The second issue of grid current harmonic distortionis related to harmonics at the PWM frequency of theconverter and multiples. LCL grid filters are used formitigation of PWM harmonics due to their good filteringcharacteristic above its resonance frequency, especially inhigh power applications when the switching frequencyis very limited by the losses [6], [7], [8]. On the otherhand its resonance may trigger undesired oscillationsor even instability. Active resonance damping is usuallypreferred as it is more flexibel and efficient comparedwith passive damping. Manifold solutions have beenproposed differing in complexity, number of sensors, andperformance [9], [10], [11], [12], [13], [14], [15]. State-based control approaches have also attracted attention dueto the possibility of superior performance [16], [17], [18],[19], [20], [21], [22], [23], [24]. Other approaches based

on nonlinear control methods entered also into the area ofcontrol of power electronics and drives [25], [26], [27],[28], [29]. Especially the passivity-based control (PBC)is a promising control technique, which offers consider-able advantages in robustness compared to linear controlmethods [30]. The recently developed interconnection anddamping assignment passivity-based control (IDA-PBC)is the most auspicious variant [30]. First applications inthe field of power electronics can be found in literature[31], [32], [33], [34], [35].

In this paper the IDA-PBC is applied to the grid-connected PWM converter with LCL-filter. Robustnessagainst grid disturbance with contemporaneously goodbehaviour regarding dynamic performance and active res-onance damping is shown. The paper is organized asfollows. Section II starts with a survey of PBC with thefocus on IDA-PBC. In section III the system is modeledin equations of port-controlled Hamiltonian systems withdissipation (PCHD) as base for the control design in sec-tion IV. After analyzing the controlled system regardingdynamic behaviour, active damping and robustness bymeans of simulations results in section V and experi-mental results in section VI the conclusion is added insection VII.

II. PBC - SURVEY

The recent past has shown a trend towards nonlinearcontrol techniques [30]. Here the system property ofpassivity [36], which is targeted on an interpretation of theenergy flow between system and ambience, is the key formost of these developments [30]. According to notationof Lyapunov passivity implies asymptotic stability [36].Based on the aim to impose passivity on a closed loopsystem in order to stabilize it the term passivity-basedcontrol (PBC) has been introduced in [37]. The principleis firstly to allocate a potential energy function with astrict local minimum in the desired equilibrium pointto the system in Euler-Lagrange (EL) equations (energyshaping). The second step is to impose an additionaldamping in order to reach asymptotic stability (injectiondamping). This technique, known as classical PBC in lit-erature, has been applied very successfully to mechanicalsystems in EL equations [30]. The application to electrical

14th International Power Electronics and Motion Control Conference, EPE-PEMC 2010

978-1-4244-7855-2/10/$26.00 ©2010 IEEE T3-20

and electromechanical systems has necessitated a newmathematic system description [38].

This has entailed the introduction of the port-controlledHamiltonian systems with dissipation (PCHD) in [38].Each dissipassive system [36] can be described in PCHDform by means of matrices J(x) and G(x), which com-prise the interconnection structure, dissipation matrixR(x) and energy function H(x), which represents aLyapunov function [38]:

d

dtx = [J(x)− R(x)]∇xH(x) + G(x)u (1)

y = GT (x)∇xH(x) (2)

Requirements are [38]:

J(x) = −JT (x) (3)

R(x) = RT (x) ≥ 0 (4)

Based on the PCHD form the interconnection and damp-ing assignment passivity-based control (IDA-PBC) hasbeen developed and introduced in [39]. The basic ideais to impose a desired PCHD form to the closed loopsystem with a control action β(x). The mathematic keyis the so-called matching equation [39], whereas index ddenotes the desired system:

[J(x)− R(x)]∇xH(x) + G(x)β(x)!= [Jd(x)− Rd(x)]∇xHd(x)

(5)

There are different ways of solving the matching equationbeing associated with the design procedure [30].

III. SYSTEM MODELING

A. Differential Equations

The system consists of PWM converter, which is con-nected to the grid by LCL-filter, and control as shownin Fig. 1, whereas system parameters are listed in table Iin the appendix. So the control plant basically containsLCL-filter dynamic, PWM and computation delay. Theequivalent circuit model of the LCL-filter with space vec-tors is depicted in Fig. 2. If the resistances are neglectedthe resonance frequency can be calculated by:

fres =12π

√Lfg+LfcLfgLfcCf

(6)

The system equations in grid voltage oriented dq-co-ordinates with angular grid frequency ωg are:

Lfgddt i

dqg−→

= −Rfgidqg−→− jωgLfgidqg−→

− udqCf−−→+ udqg−→

(7)

Lfcddt i

dqc−→

= −Rfcidqc−→− jωgLfcidqc−→

+ udqCf−−→− udqc−→

(8)

Cfddtu

dqCf−−→

= idqg−→− idqc−→

− jωgCfudqCf−−→(9)

Input vector u and output vector y are defined as follows:

u =[ucd ucq

]T(10)

y =[icd icq

]T(11)

PWM and computation delay can be modeled as a first-order delay. In this paper both is neglected for controldesign and thus for plant modeling.

Fig. 1. System consisting of PWM converter with LCL-filter andcontrol

B. Energy Function

In the first step the total system energy has to beexpressed in terms of the state variables. The followingdefinition of the state vector x = [x1 . . . x6]

T turns out tobe advantageous:

x =[Lfgigd Lfgigq Lfcicd Lfcicq CfuCfd CfuCfq

]T(12)

Then the energy function H(x) can be formulated by:

H(x) = 12

[x21

Lfg+

x22

Lfg+

x23

Lfc+

x24

Lfc+

x25

Cf+

x26

Cf

](13)

Thus the gradient of the energy function yields:

∇xH(x) =[igd igq icd icq uCfd uCfq

]T(14)

C. PCHD Modeling

In the second step the PCHD equations of the systemaccording to (1) and (2) have to be derived. Consideringthe system equations (7), (8) and (9), whereas grid voltageis neglected by treating it as disturbance variable, and tak-ing the requirements (3) and (4) into account expressionsfor R, J and G are obtained:

J =

0 ωgLfg 0 0 −1 0

−ωgLfg 0 0 0 0 −1

0 0 0 ωgLfc 1 0

0 0 −ωgLfc 0 0 1

1 0 −1 0 0 ωgCf

0 1 0 −1 −ωgCf 0

(15)

R =

Rfg 0 0 0 0 0

0 Rfg 0 0 0 0

0 0 Rfc 0 0 0

0 0 0 Rfc 0 0

0 0 0 0 0 0

0 0 0 0 0 0

(16)

G =

[0 0 −1 0 0 0

0 0 0 −1 0 0

]T(17)

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Fig. 2. One-phase equivalent circuit model of the LCL-filter

IV. CONTROLLER DESIGN

In this application parameterized IDA, introduced asa basic design procedure in [30], is most suited forcontrol design. This means that energy function as wellas matrices of the desired PCHD system are describedby undefined parameters at first. The energy function isfixed to a particular class in order to restrict the number ofsolutions of the matching equation. The control action isobtained by evaluation of the matching equation, whereasfixed expressions for some of the undefined parametersare exposed on the one hand. The remaining ones turnout to be degrees of freedom for controller tuning on theother hand.

A. Setting up the Matching Equation

In the first step of control design the matching equationaccording to (5) has to be set up. The control actionβ(x) = [βd(x) βq(x)]

T has to be derived by meansof solving this equation with desired energy functionHd(x) and matrices Jd and Rd. The energy function hasto represent a Lyapunov function [38], which can beachieved by complying with the class that the energyfunction of the control plant belongs to. Furthermore itmust have a local minimum in the desired equilibriumpoint x0 = [x01 . . . x06]

T . Especially the requirement of[x03 x04] =

[Lfci

∗cd Lfci

∗cq

]has to be satisfied, whereas

i∗cd and i∗cq are the reference values of the convertercurrent.

In order to achieve this aim x01, x02, x05 and x06 mustcapture explicit values that have to be calculated since thestate variables are physically associated with each otherby the LCL-filter. This would entail large mathematicefforts, especially concerning the solution of the matchingequation, and complexity of the control would increaseconsiderably. In order to simplify the control designprocedure the energy function of the desired PCHD closedloop system is chosen with x01 = x02 = x05 = x06 = 0,whereas γ1, γ2 and γ3 are undefined parameters:

Hd(x) =12

[γ1x

21 + γ1x

22 + γ2(x3 − x03)2

+γ2(x4 − x04)2 + γ3x25 + γ3x

26

] (18)

The closed loop system is not able to reach the equi-librium point of this simplified energy function since itincludes the requirement of zero grid current and zerocapacitor voltage during a desired converter current flow.Nevertheless the system aims for the state of minimaltotal energy according to (18). Thus the control variableis aimed to approach the reference value. Steady state

errors, which can be minimized by tuning of the controlparameters, are eliminated by an integral action.

Taking the requirements (3) and (4) into account thematrices Jd and Rd are chosen with undefined parametersas well:

Jd =

0 J12 J13 J14 J15 J16

−J12 0 J23 J24 J25 J26

−J13 −J23 0 J34 J35 J36

−J14 −J24 −J34 0 J45 J46

−J15 −J25 −J35 −J45 0 J56

−J16 −J26 −J36 −J46 −J56 0

(19)

Rd =

r1 0 0 0 0 0

0 r1 0 0 0 0

0 0 r2 0 0 0

0 0 0 r2 0 0

0 0 0 0 r3 0

0 0 0 0 0 r3

(20)

B. Solving the Matching Equation

Combining (13), (15), (16) and (17) with (18), (19) and(20) the matching equation (5) comprises six equations.The control action can be derived from the equations ofthe third and fourth row since they contain x03 and x04and thus the reference values i∗cd and i∗cq of the convertercurrent. At first the equations of the remaining rows canbe used for determination of undefined parameters bymeans of comparison of coefficients. With redefinitionr2 = r and J34 = J this leads to the following differencebetween interconnection and dissipation matrix of thePCHD closed loop system:

Jd − Rd =

−RfgLfgγ1

ωgγ1

0 0 −1Lfgγ1

0

−ωgγ1−RfgLfgγ1

0 0 0 −1Lfgγ1

0 0 −r J 1Lfcγ2

0

0 0 −J −r 0 1Lfcγ2

1Lfgγ1

0 −1Lfcγ2

0 0ωgCfLfgγ1

0 1Lfgγ1

0 −1Lfcγ2

−ωgCfLfgγ1

0

(21)

Now the control action arises from row three and four ofthe matching equation:

βd =(rγ2Lfc −Rfc) icd − rγ2Lfci∗cd+(1− γ1

γ2

LfgLfc

)uCfd (22)

+ (ωgLfc − Jγ2Lfc) icq + Jγ2Lfci∗cq

βq =(rγ2Lfc −Rfc) icq − rγ2Lfci∗cq+(1− γ1

γ2

LfgLfc

)uCfq (23)

− (ωgLfc − Jγ2Lfc) icd − Jγ2Lfci∗cd

Obviously γ2 can be defined as a fixed expression,whereas γ2 = 1

Lfcturns out to be most beneficial for

simplification. Finally with redefinition of γ1 = γ thisyields the control action for d- and q-component with

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tuning parameters r, γ and J :

βd =− r (i∗cd − icd)−Rfcicd + (1− Lfgγ)uCfd+ J

(i∗cq − icq

)+ ωgLfcicq

(24)

βq =− r(i∗cq − icq

)−Rfcicq + (1− Lfgγ)uCfq

− J (i∗cd − icd)− ωgLfcicd(25)

Now the energy function of the PCHD closed loop systemis given by:

Hd(x) =12

[γx21 + γx22 +

1Lfc

(x3 − x03)2

+ 1Lfc

(x4 − x04)2 + LfgγCf

x25 +LfgγCf

x26

](26)

C. Extension by Integrator

In order to eliminate steady state errors an integratorcan be added without losing the property of passivity [30].Here the control action of (24) and (25) is extended asfollows, whereas k is a further design parameter, whichcan be used for controller tuning [33]:

β′d = βd − k ·∫ t

0

(i∗cd − icd) dt (27)

β′q = βq − k ·∫ t

0

(i∗cq − icq

)dt (28)

D. Controller Tuning

There are four tunable parameters within the extendedcontrol action, r, γ, J and k. In this paper analysis oftheir influence on the closed loop system and tuningis done empirically by means of a simulation modelin MATLAB/Simulink, which just contains a simplifiedcontrol plant dynamic according to (7), (8) and (9) andthe control action (27) and (28).

The most important parameter turns out to be r, sinceit represents the gain in the feedback of the controldeviation. According to requirement (4) it must have apositive algebraic sign. It can be stated that the closedloop system is stable for r > 0 which means thatpassivation is already achieved by positive feedback ofthe control deviation. A compromise between fast trackingresponse and overshoot has lead to a value of r = 4.5.

The feedback of the coupled current components dis-poses the steady state deviations in the actual values of thecurrents, which would appear without this feedback dueto couplings. The bigger parameter J is chosen the fasterthese deviations are compensated but also the higher isthe overshoot. In order to avoid overshoot in the actualvalue of one component when the reference value of theother component changes J is defined as zero. Thus onlythe actual currents are fed back with gain ωgLfc.

Reasonably the term g = (1−Lfgγ) has to be positive.g = 1 turns out to be a stability bound since bigger valueslead to instability. Fig. 3 points out grafically the influenceof this term on the dynamic behaviour. Reference andactual values of the converter currents, whereas a step isgiven for the d-component reference, as well as convertervoltages are depicted with comparison of g = 1 (left) andg = 0.5 (right). Steady state deviations between referenceand actual current at the beginning stem from the grid

Fig. 3. Simulation results with simplified model - Influence of tuningparameter γ; waveforms of reference values ( ) and actual values ( )of d-components (a, b) and q-comp. (c,d) of converter currents and d-comp. ( ) and q-comp. ( ) of converter voltages (e, f); parameters:r = 4.5, J = 0, k = 0, g = 1 (left) and g = 0.5 (right).

current, which represents a disturbance variable. Onlyallocation of g = 1 fully compensates this disturbancetheoretically. The actuating variable oscillates with theresonance frequency of the LCL-filter in a damped en-velope oscillation with the frequency of the grid voltage.For g > 1 these oscillations would increase and thus causeinstability. g = 0.5 is chosen as a reasonable value inorder to have a safety margin regarding model uncertaintyof grid-side inductance.

It becomes clear from Fig. 3 that an integral action isrequired for steady state accuracy. k = 1500 is chosenas a compromise between a rise time as short as possibleand an adequate overshoot.

V. SIMULATION RESULTS

A simulation model of the system as depicted in Fig. 1is built in MATLAB/Simulink, whereas grid, LCL-filterand switching converter are implemented in PLECS. Sim-ulation results are shown in Fig. 4 to Fig. 7. Waveformsof grid and converter current as well as capacitor voltageand reference voltage of the PWM, which represents theactuating variable, are depicted with a step in the d- andq-component of the converter current.

Fig. 4 shows the dynamic behaviour of the closed loopsystem when a switching and control frequency of 5 kHzis used. The step responses of the d- and q-componentof the converter current are quite similar. A rise time ofapproximately five sample periods, that is 1msec, can bestated with an overshoot of about 4.5A, which is 22.5%relating to the step height of 20 A. In the waveformsof the q-components there are slight oscillations with

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a frequency of 150 Hz, which is visible especially inthe capacitor voltage. This is due to aliasing in thecapacitor voltage, since this quantity is not sampled on itsaverage value. This effect is diminished by using a biggerswitching respectively control frequency as becomes clearin Fig 5, whereas a switching frequency of 10 kHz istaken. Regarding the converter current it can be stated alower overshoot due to a shorter delay time, caused bythe PWM, which is not respected for control design.

Oscillations with the resonance frequency of the LCL-filter according to (6) are obvious from the capacitorvoltage during the transient effect of the converter current.This phenomenon is also visible in the waveforms ofgrid current and converter voltage, but with less intensity.Concerning the waveform of the control variable, whichis the converter current, a good active damping of theresonance can be stated.

Fig. 6 shows the analysis of robustness against gridimpedance variations. In comparison with Fig. 4 theonly difference is the present grid-side inductance of thesystem, which is twice as high as the nominal one whichis used for tuning. This means that a grid inductanceequal to the grid-side filter inductance is regarded. Forcontrol design this additional inductance is not respected.Concerning the converter current step the simulationresults show that the rise time is lengthened by one sampleperiod. Overshoot stays approximately the same. Thus thecontrol is very robust against grid impedance variations.

Fig. 4. Simulation results - Dynamic performance with fa = 5 kHz;waveforms of grid currents (a, b), reference values ( ) and actual values( ) of inverter currents (c, d), capacitor voltages (e, f) and referencevalues of inverter voltages (g, h).

Fig. 5. Simulation results - Dynamic performance with fa = 10 kHz;waveforms of grid currents (a, b), reference values ( ) and actual values( ) of inverter currents (c, d), capacitor voltages (e, f) and referencevalues of inverter voltages (g, h).

Fig. 6. Simulation results - Robustness against grid impedance (+200%of Lfg): Dynamic performance with fa = 5 kHz; waveforms of gridcurrents (a, b), reference values ( ) and actual values ( ) of invertercurrents (c, d), capacitor voltages (e, f) and reference values of invertervoltages (g, h).

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As described in subsection IV-D it has to be valid(1 − Lfgγ) < 1 for stability. So if the grid-side induc-tance is assumed too small there is no danger of gettingunstable due to model uncertainties. In the reverse casethis may happen, but prevention is possible by includinga reasonable safety margin.

In order to constitute effects in the experimental resultsdue to grid harmonics Fig. 7 shows the behaviour at 3%of 5th and 7th harmonic, which are the most importantharmonics in the grid. In dq-coordinates 5th and 7th har-monic of same percentage just yield a 6th harmonic in thed-component when they are in phase with the fundamentalcomponent. It can be extracted from the waveforms howthese grid harmonics affect the grid and converter currentas well as capacitor and converter voltage.

VI. EXPERIMENTAL RESULTS

Experimental tests have been accomplished on a 22 kWinduction motor drive. The control has been implementedwith dSPACE. Results with a switching frequency of5 kHz are shown in Fig. 8, whereas the diagrams arearranged according to the simulation results. In contrastto the simulations a step in the converter current isonly made for the q-component. The waveforms aresuperposed by 6th harmonic because of 5th and 7thharmonic in the grid voltage as shown in the simulationresults. Thus comparison of the step response with thetheoretic curve is difficult. Nevertheless it can be statedthat the rise time is quiet close to the rise time ofthe simulations. An additional resonant controller wouldimprove the performance.

Fig. 7. Simulation results - Behaviour at harmonics in grid voltages(3% of 5th and 7th harmonic): Dynamic performance with fa = 5kHz;waveforms of grid voltages (a, b), grid currents (c, d), reference values( ) and actual values ( ) of inverter currents (e, f), capacitor voltages(g, h) and reference values of inverter voltages (i, j).

Fig. 8. Experimental results - Dynamic performance with fa = 5kHz;waveforms of grid currents (a, b), reference values ( ) and actual values( ) of inverter currents (c, d), capacitor voltages (e, f) and referencevalues of inverter voltages (g, h).

VII. CONCLUSION

The basic idea of IDA-PBC with a survey of the fieldof PBC is presented. A possibility to apply the nonlinearcontrol method of IDA-PBC to the grid-connected PWMconverter with LCL-filter is worked out and implemented.Analysis based on simulation and experimental resultshas been carried out. The control offers a very goodrobustness against grid impedance variations with contem-poraneously good dynamic behaviour and active damping.

APPENDIX

TABLE ISYSTEM PARAMETERS

Symbol Quantity Valueug Grid voltage (line-line) 400 Vωg Angular grid frequency 2π50 HzLfg Grid-side filter inductance 0.75 mHRfg Grid-side filter resistance 50 mΩCf Filter capacitance 32 µFLfc Converter-side filter inductance 2 mHRfc Converter-side filter resistance 50 mΩCDC DC link capacitance 2710 µFfres Resonance frequency 1.2 kHzfs = fc Switching frequency = control frequency 5 kHz

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