2320 ieee transactions on power electronics, · pdf file2320 ieee transactions on power...

2320 IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 9, SEPTEMBER 2010

PI State Space Current Control of Grid-ConnectedPWM Converters With LCL Filters

Joerg Dannehl, Student Member, IEEE, Friedrich Wilhelm Fuchs, Senior Member, IEEE,and Paul Bach Thøgersen, Senior Member, IEEE

Abstract—Design and analysis of PI state space control for grid-connected pulsewidth modulation (PWM) converters with LCL fil-ters based on pole placement approach is addressed. State spacecontrol offers almost full controllability of system dynamic. How-ever, pole placement design is difficult and usually requires muchexperience. In this paper, a suitable pole placement strategy is pro-posed, which ensures fulfilling the requirements, which are com-monly specified with respect to rise time, overshoot, and properresonance damping. Controller parameter expressions are derivedin terms of system parameters and specified poles and zeros.Hence, straightforward controller tuning for a particular systemsetting is possible. Performance is analyzed by means of trans-fer function-based calculations, simulations with MATLAB, andexperimental tests. Dynamic performance and robustness againstgrid impedance variations are addressed as well as harmonic re-jection capability and other practical issues.

Index Terms—Active resonance damping, current control, grid-connected pulsewidth modulation (PWM) converters, LCL filters,state space control.

I. INTRODUCTION

GRID-CONNECTED pulsewidth modulation (PWM) con-verters offer bidirectional power transfer with low-

harmonic current distortion. Typical applications are in thegrowing field of distributed generation systems (DGS), regener-ative motor drives, active filters [1]–[3]. Standards set stringentlimits on the grid current harmonic distortion, which can be di-vided into two parts. The first issue is related to low frequencyharmonics due to grid voltage background distortion. This isconnected to the disturbance rejection capability of the cur-rent controller [4]. High control bandwidth is required for goodharmonic rejection. The second issue of grid current harmonicdistortion is related to harmonics at the PWM frequency of theconverter. The LCL grid filters are increasingly used for miti-gation of PWM harmonics due to their good filtering character-

Manuscript received July 22, 2009; revised November 5, 2009 and January13, 2010; accepted March 16, 2010. Date of current version September 17, 2010.Recommended for publication by Associate Editor Jos R. Espinoza. This workwas supported by the German Research Foundation (DFG) and industry. Thispaper was presented at the Conference on IEEE Energy Conversion Congressand Exposition, San Jose, CA, Sep. 20–24, 2009.

J. Dannehl and F. W. Fuchs are with the Institute for Power Electronics andElectrical Drives, Christian-Albrechts-University of Kiel, 24143 Kiel, Germany(e-mail: [email protected]; [email protected]).

P. B. Thøgersen is with KK-Electronic A/S, 9220 Aalborg, Denmark (e-mail:[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.2010.2047408

istic above their resonance frequency, especially in high powerapplications when the switching frequency is very limited bylosses [5]–[7]. On the other hand, resonance may trigger unde-sired oscillations or even instability. Active resonance dampingis usually preferred as it is more flexible and efficient comparedwith passive damping. However, with high demands on switch-ing harmonic attenuation, the LCL resonance frequency typi-cally becomes lower. In this case, the LCL resonance dynamicsget closer to the desired control bandwidth and it becomes moredifficult to achieve LCL resonance damping and high bandwidthat the same time with reasonable robustness.

Manifold active damping approaches have been proposed inthe literature [4], [8]–[18]. The choice of active damping methoddepends on the sensor signals available. However, fulfilling therequirements of fast rise time, small overshoot and proper reso-nance damping as well as being robust is a challenge, even morewhen the ratio of LCL resonance to control frequency is low.Many approaches with partial state feedback for the purpose ofresonance damping or sensorless solutions are not suitable [19],[20]. Complete state feedback offers almost full controllabil-ity [10], [21]. Hence, grid-connected PWM converters with lowratio of LCL resonance to control frequency can be controlledas well. Even though state space control is a powerful approach,from an industrial point of view there are drawbacks: many sen-sors and difficult controller design. A large number of sensorsincreases the costs and reduces reliability. For this reason, esti-mators or observers could be used in order to reduce the numberof sensors [9], [22]. In this paper, the state space controllerdesign issue is emphasized.

In [21], state space current control is proposed for improvedtransient performance. However, converter delay has not beentaken into account directly and the controller parameters aretuned manually. Linear quadratic optimal control or pole place-ment are more systematic ways of designing state space con-trollers. The LQ controller minimizes the quadratic integral cri-teria [10], [23]. For this purpose, weighting matrices need to bedesigned. A robust controller with guaranteed gain and phasemargins is obtained. However, the control design results in it-erative tuning of weighting matrices. In [10], empirical guide-lines for their selection can be found. The parameter calcu-lation requires numerical tools for solving the mathematicalproblem, e.g., solution of a Riccati equation. Therefore, con-troller tuning and transfer between system settings are difficult.Automatic controller tuning without manual fine tuning for aparticular system setting is often important in industrial appli-cations. Pole placement allows the derivation of control parame-ter expressions as a function of system parameters and specified

0885-8993/$26.00 © 2010 IEEE

DANNEHL et al.: PI STATE SPACE CURRENT CONTROL OF GRID-CONNECTED PWM CONVERTERS WITH LCL FILTERS 2321

Fig. 1. Grid-connected PWM converter with LCL filter.

closed-loop performance characteristics. The idea of pole place-ment is to specify the closed-loop performance in terms of poleand zero locations. The control parameters are tuned in orderto achieve the specified locations. The parameter calculation isstraightforward, even if quite complex for systems of higherorder. However, proper specification of closed-loop poles andzeros is required.

The objective of this paper is the development of a PI statespace current control system that can be tuned straightforwardlyby supplying the most important system parameters, namely,LCL filter elements and control frequency. For this purpose, thepole placement approach in the discrete domain is addressed.Controller parameter expressions are derived in terms of systemparameters and specified closed-loop characteristics. The latterare closed-loop poles and zeros that need to be specified in orderto meet the requirements of rise time, overshoot, and proper res-onance damping. The proposed pole placement strategy for theLCL filter-based system makes effective use of the knowledgeof pole and zero locations in a PI-controlled L-filter system.These are well known from standard PI design rules, e.g., sym-metrical or technical optimum [24], [25]. Robustness againstgrid impedance variations is addressed as well as harmonic re-jection capability and other practical issues. With the proposedcontroller, high-dynamic performance with properly dampedresonance can be obtained even for low resonance frequencies,making it most suitable for all applications with high-dynamicand power quality requirements such as DGS or active filters.

The system is described in Section II. Control design is ex-tensively studied in Section IV, using the discrete system modelderived in Section III. Section V provides a theoretical analysisbased on transfer function calculations. The results of simula-tion studies and experimental tests are presented in Sections VIand VII, respectively.

II. SYSTEM DESCRIPTION

A grid-connected PWM converter with LCL filter is shownin Fig. 1. In addition to the dc-link voltage, all the LCL systemstates are measured. The line voltages are measured for thepurpose of synchronizing the control with the line voltage bymeans of a phase-locked loop in the synchronous referenceframe [1]. The three-phase values are linearly dependent, andone phase could be calculated from the two others. However,for better accuracy, all three phases are sensed. Moreover, forprotection against faulty earth currents, sensors on all the threephases are required. For the test setup, control is implemented

Fig. 2. Converter control structure.

on a dSpace board. The converter is loaded by an inverter-fedinduction machine.

The basic control structure is shown in Fig. 2. The dc-linkvoltage is controlled with a PI controller. Active and reactiveconverter currents are controlled in the synchronous referenceframe, employing the proposed state space approach that isdesigned and analyzed in this paper.

III. MODELING

For control design and performance analysis, a system modelis derived. Because the outer dc-link voltage is beyond the scopeof this paper, its dynamics and control are not discussed in thispaper. The reader is referred to [13]. Current control and itsdynamics are emphasized in this paper. Applying Kirchhoff’slaws and transforming into the rotating reference frame yieldthe continuous model of the LCL filter in the dq frame [13]

d

dti→ L,dq

=1

Lf g

(u→ L,dq

− u→ C f,dq

)− jωi

→ L,dq

d

dti→ C,dq

=1

Lf c

(u→ C f,dq

− u→ C,dq

)− jωi

→ C,dq

d

dtu→ C f,dq

=1

Cf

(i→ L,dq

− i→ C,dq

)− jωu

→ C f,dq. (1)

The resonance frequency of the LCL filter can be calculated [6]as follows:

ωRes =

√Lf g + Lf c

Cf Lf g Lf c= 2πfRes . (2)

For the control design, the zero frequency is important as well

ω0 =1√

Lf g Cf

= 2πf0 . (3)

For the control design, the coupling terms in (1) are neglected.Hence, the control can be designed for one axis and used forboth axes. Taking both axes into account leads to more complexcontroller design due to the higher system order. In addition,parasitic resistances are neglected for control design. The cal-culation effort during the design is reduced considerably. It isof more importance to incorporate the delay due to the sam-pling, calculation, and PWM. With the state vector defined asxLC L = [iC , iC f , uC f ]T the simplified continuous state spacerepresentation of the LCL filter for one axis can be written as(indices d or q omitted)

x· LC L (t) = ALC LxLC L (t) + bLC LuC (k) (4)


with the system and input matrices

ALC L =

0 01

Lf c

0 0 −ω2Res Cf

01

Cf0

bLC L =

− 1Lf c

1Lf c

0

.

For control purposes, only sampled data are available and thecontrol implementation is discrete as well. Therefore, the controldesign is performed in the discrete time domain. Hence, thecontinuous model is discretized.

xLC L (k + 1) = ALC Ld xLC L (k) + bLC L

d uC (k) (5)

whereas the discrete system matrices can be calculated with

ALC Ld = e(AL C L Tc ) (6)

bLC Ld =

(ALC L

d − I) (

ALC L)−1

bLC L (7)

where I equals the unity matrix of proper dimension. The ex-ponential matrix is called the fundamental matrix and it is theinverse Laplace transform of (s I − ALC L )−1 . Its calculationyields as follows:

Ad =

11 − cos (ωRes Tc)

Lf c Cf ω2Res

sin (ωRes Tc)Lf c ωRes

0 cos (ωRes Tc) Cf ωRes sin (ωRes Tc)

0sin (ωRes Tc)

Cf ωRescos (ωRes Tc)

(8)

bd =−1

ωRes Lf c

ω2Res Tc −

Tc

Lf c Cf+

sin (ωRes Tc)Lf c Cf ωRes

−ωRessin (ωRes Tc)

1 − cos (ωRes Tc)Cf

. (9)

There is a delay between the reference voltage u∗C and the

converter output voltage uC [26]:

uC (k + 1) = uC,Ref (k). (10)

Hence, the following state space representation of the wholesystem, taking into account the filter dynamics and delay isused for control design

iC (k + 1)

iC f (k + 1)

uC f (k + 1)

uC (k + 1)

︸︷︷︸x(k + 1)

= Asysd x(k) +

0

0

0

1

︸︷︷︸bsysd

·uC,Ref (k) (11)

whereas

Asysd =

a11 a12 a13 b1

a21 a22 a13 b2

a31 a32 a13 b3

0 0 0 0

(12)

Fig. 3. PI state space current control structure.

with aij and bi (i, j ∈ {1, 2, 3}) being entries in Ad and bd . Theconverter current iC is selected as output:

y = [1 0 0 0] · x = cT · x (13)

The performance analysis takes the parasitic resistances intoaccount.

IV. STATE SPACE CONTROL DESIGN

Different state space approaches are presented in the litera-ture. Pure proportional state feedback yields steady-state errors.For zero-steady-state error, an integrator can be incorporatedin the loop [10]. Here, further improvement is obtained by re-placing the integrator with a PI controller. This improves thetracking performance due to the additional zero of the PI part,which can be used for slow system pole compensation. Fig. 3shows the control structure. The state feedback vector is definedas kT = [kiC kiC f kuC f kuC ], and the PI control parametersare the proportional gain kp and the integrator time constant TI .

The control design procedure can be divided into two steps.First, specification of the desired poles and zeros is required.This implies the translation of closed-loop requirements intopole and zero locations. The second step is the controller pa-rameter calculation. Basically, a relation between system pa-rameters, pole and zero locations, and controller parameters isrequired.

A. Pole Placement

Nowadays, PI control in the synchronous reference frame rep-resents the state-of-the-art in current control of electrical drivesystems, e.g., grid-connected PWM converters with L filtersor induction motor drives [24]. For these applications, closed-current-loop performance requirements are commonly relatedto rise time and overshoot during steps. Rise time is typicallyspecified as a couple of control periods. The overshoot is oftenlimited by the converter current rating. Especially, in high powerapplications, it is more stringent to limit overshoot. Standardcontroller design rules and criteria, like technical or symmetri-cal optimum [24] [25], are known and can be used for designingPI controllers for L filter systems. For the purpose of generality,a symmetrical optimum is used [25]. Special tuning dedicatedto a particular application is possible as well. In any case, thelocations of closed-loop poles and zeros, which fulfill particularrequirements with respect to overshoot and rise time, are knownfor the L-filter-based system.


Fig. 4. Schematics of pole and zero locations: (a) PI-controlled L-filter systemand (b) PI-state space-controlled LCL filter system with the proposed placementstrategy.

However, resonance damping is an additional demand in LCLfilter applications, in the steady state as well as during transients.The strategy proposed in this paper takes advantage of knowingthe closed-loop poles and zeros for the L-filter-based system.The idea is to place all poles and zeros, except those of reso-nance, as for L filter control. In addition, the impact of resonanceis eliminated by a proper shift of the resonance poles.

1) L Filter System With Pure PI Control: Fig. 4(a) showsschematically typical pole and zero locations of an L filter sys-tem. In the open-loop system, there is a pole close to z = 1, dueto the inductor. The one-sample delay gives an additional poleat the origin. The PI controller has a pole at z = 1 and a zeroon the real axis. The PI zero position is a matter of tuning. Theclosed-loop pole zero map is also shown in Fig. 4(a). On thereal axis, there is a pole close to the zero, it is the compensatedpole. The complex conjugated pole pair is dominant. In general,a complex pole pi is characterized by its damping factor Di andits frequency fi .

pi = e−Dif if c ej

√1−D 2

i

f if c . (14)

Here, real poles are characterized only by their real parts.The precise pole and zero locations of the L filter systemwith PI control tuned with a symmetrical optimum (kp =−L/(aSO Tc), TI = a2

SO Tc, aSO = 3) can be calculated. Thelocations of the PI zero z1L , compensated pole p1L , and domi-nant pole pair (p2L , p3L ) are given as follows:

z1L = 1 −(

1a2

SO

)= 8/9 p1L = 0.88z1L

p2L : D2L = 0.9 f2L = fc/12 p3L = p∗2L . (15)

2) LCL Filter System: Fig. 4(b) shows schematically the lo-cation of poles and zeros of an LCL filter. In comparison with4 (a), the LCL open-loop system has additional poles and zeroson the unit circle due to resonance. The angles are related to itsfrequencies ωRes and ω0 , respectively. Note that parasitic lossesof the filter elements yield slight damping of the resonance.Hence, the resonance poles are inside the unit circle in reality.However, the damping factor is quite small. The L filter pole andzero locations from (15) are used to specify most of the desiredclosed-loop LCL system poles and zeros. In order to make theresonance pole pair nondominant, which is to reduce its impact,a small residual is required. Placing the LCL resonance nearthe LCL zeros yields a small residual, since they compensateeach other. A certain distance should be kept in order to have arobustness margin. Note that system zeros are not affected byfeedback [27].

From previous considerations, the desired locations of the PIzero zd

1 , compensated pole pd1 , dominant pole pair (pd

2 , pd3 ), and

nondominant pole pair (pd4 , p

d5 ) are finally specified as follows:

zd1 = z1L pd

1 = p1L

Dd2 = D2L Dd

4 = 0.2

fd2 = f2L fd

4 = f0 . (16)

Note that (pd2 , pd

3 ) and (pd4 , pd

5 ) are complex conjugated polepairs, i.e., pd

3 = pd∗2 and pd

5 = pd∗4 . See (3) for a definition of f0

and (15) for the L filter poles and zeros.

B. Controller Parameter Calculation

Once the desired poles and zeros have been specified, deriva-tion of appropriate control parameter is necessary. Basically,the closed-loop transfer function is written in terms of systemand control parameters. From a comparison with the transferfunction in terms of the desired poles and zeros, the controlparameters can be calculated.

The closed-loop transfer function can be written as

Gc(z) =Ic(z)

Ic,Ref (z)=

GP I (z)GRD (z)GP I (z)GRD (z) + 1

(17)

with

GRD(z) = cT(z I −

[Asys

d − bsysd kT

])(18)

GPI(z) = kpz + (Tc/TI − 1)

z − 1. (19)


See also Fig. 3. The closed-loop transfer function from (17) canbe rewritten as follows:

Gc(z) =(z − z1)(z − z2)(z − z3)

a5z5 + a4z4 + a3z3 + a2z2 + a1z + a0. (20)

The coefficients ai(i ∈ [0 . . . 5]) consist of system and controlparameters.

With the desired closed-loop poles and PI zero specified, thedesired closed-loop transfer function can be expressed as

Gdc (z) =

k0(z − zd

1)(z − z2) (z − z3)(

z − pd1

) (z − pd

2

). . .

(z − pd

5

) . (21)

In order to obtain unit gain for zero frequency (z = 0), the factork0 is used. Thus, no steady-state error occurs. For the purposeof comparison with (20) it is rewritten as follows:

Gdc (z) =

(z − zd

1)(z − z2) (z − z3)

ad5 z5 + ad

4 z4 + ad3 z3 + ad

2 z2 + ad1 z1 + ad

0. (22)

The coefficients adi (i ∈ [0 . . . 5]) consist of specified poles.

Expressions for controller parameters in terms of system pa-rameters and desired pole and zero locations are obtained fromcomparison of (20) and (22). The PI integrator time constantthat determines z1 follows directly from a comparison of nu-merators.

1 − Tc

TI= zd

1 =⇒ TI =Tc

1 − zd1. (23)

Comparison of the denominator and subsequent rearrangementyield all the other control parameters. In the interest of clarity, theresults are shown in the Appendix [see (26)]. Even though theyare quite long, direct relation between the control parametersand system parameters and the specified pole and zero locationsare obtained.

C. Decoupling Network

The couplings between d- and q-state variables have beenneglected in the control design. A simple forward decouplingnetwork known from L-filter-based systems is used in this paperfor decoupling [28]. To the output of the proposed controller adecoupling voltage udec is added.

udec,d = ω(Lf g + Lf c)iq ,ref (24)

udec,q = −ω(Lf g + Lf c)id,ref . (25)

More advanced strategies can further improve decoupling per-formance [16].

V. THEORETICAL ANALYSIS

The proposed pole placement strategy is analyzed by meansof transfer function-based calculations. For this purpose, theclosed-loop transfer function from (17) is evaluated and, if notstated otherwise, the system parameters from Table I and thecontroller parameters obtained from (26) with the specified polesand zeros from (16) are used.

TABLE IDEFAULT SYSTEM PARAMETERS

A. Impact of LCL Resonance Frequency

The LCL resonance frequency has been shown to affect sys-tem stability considerably in other control approaches [13]. Inthis paper, two LCL filter settings with different filter capacitorsare selected for the analysis. A typical range of filter parameterswith rather low resonance frequencies relative to the controlfrequency is covered. These settings are difficult to handle byother active resonance damping approaches when a reasonablebandwidth is required [13], [19].

Fig. 5 shows the pole zero maps of the system with filtercapacitor of 32.6 µF (6.7%) and 97.8 µF (20.2%), respectively.Closed-loop poles differ slightly from the desired locations sinceparasitic resistances are taken into account in the analysis. Notethat they have been neglected for controller parameter calcula-tion. However, deviations are quite small for both filter settings.Fig. 6 shows step responses of the LCL filter as well as L fil-ter system. The inductance of the L filter has been chosen asthe sum of the LCL inductances. The LCL system with higherresonance frequency response is almost like the L filter system.

The LCL system with lower resonance frequency shows thatthe additional oscillation due to resonance decays slower. Thusthe step response is slightly more oscillatory and the referenceis crossed one time step later. Placing the resonance poles closerto the origin yields a faster decay, but as the distances to the res-onance zero increase, the amplitude of the oscillation increases.Placing them closer to the zeros decreases the residual, but thedecay time increases. In addition, a certain distance should bekept for robustness reasons. Note that the resonance frequency isin the same range as the closed-loop bandwidth, as can be seen


Fig. 5. Pole zero map of closed-loop with proposed strategy (×) and de-sired ones (∗) with different LCL resonance frequencies: (a) higher reso-nance frequency (fRes = 1.6 kHz) and (b) lower resonance frequency (fRes =0.95 kHz).

from the frequency of the dominant poles and the resonancepoles in Fig. 5(b).

It can be concluded that rise time and overshoot requirementsas well as resonance damping are fulfilled. Note that the re-quirements are specified indirectly by choosing a symmetricaloptimum in the design of the L-filter PI controller. For lowresonance frequencies, damping is more difficult.

Fig. 6. Calculated closed-loop step responses of PI-controlled L-filter system(---) and PI-state space-controlled LCL-filter system (—) with different LCLresonance frequencies: (a) higher resonance frequency (fRes = 1.6 kHz) and(b) lower resonance frequency (fRes = 0.95 kHz).

B. Grid Impedance Variations

The grid impedance is another important parameter that af-fects system stability [4], [17], [29]. Fig. 7 shows the effect ofgrid inductance variation when the controller is tuned assumingno grid impedance. A variation of Lf g up to ± 50% is consid-ered. Note that negative variations can occur when a nominalnonzero grid inductance is taken into account in the parame-ter Lf g , which may be subject to negative variations. Fig. 7(a)shows the Bode diagram of the open loop, whereas the statefeedback loop (kT ) is closed. It can be seen that the resonanceis well damped for the analyzed grid inductance variation. Theevolution of closed-loop poles and zeros with varying grid in-ductance also indicates stability, see Fig. 7(b). The movementof the dominant pole pair indicates degradation of bandwidthwith higher grid inductances, which is typical as long as the PIgain is not adjusted adaptively.

C. Effect of Signal Filtering Delay

The currents and voltages needed for control are distortedby switching harmonics. Thus sampling at twice the switchingfrequency can cause aliasing. It is common practice to avoidaliasing by sampling the currents during zero voltage spacevectors at the beginning and in the middle of the switching


Fig. 7. Effect of grid inductance variation (parameter Lf g changed up to± 50%): (a) Bode diagram and (b) pole zero map.

period. However, this sampling technique cannot avoid aliasingin the filter capacitor voltages. For this reason, filter capacitorvoltages are oversampled at two times the control frequency andthe average over one control period is used. This gives a half-sample delay in the feedback path of the filter capacitor voltage.Fig. 8 shows the closed-loop pole zero location when a half-sample delay modeled as 0.5(1 + z−1) is considered. The filterinserts a pole-zero pair close to the origin. It can be seen that theother poles get shifted somewhat closer to the unit circle. Theclosed-loop bandwidth degrades slightly from 915 to 840 Hz,but the robustness margins remain almost the same.

D. Harmonic Rejection Capability

The grid voltage at the point of common coupling (PCC)is subject to background distortions at low frequency harmon-ics [4]. Hence, it is important to have enough harmonic rejectioncapability. In other words, the impedance seen from the grid atthe PCC is of interest and which is given by the transfer functionfrom the line voltage to the line current. This is shown in Fig. 9for the proposed controller. It can be seen that the impedance atthe low frequency harmonics is rather low. However, it is a com-monly known concept for harmonic rejection to put the resonantcontroller in parallel with the fundamental controller [4]. Therequirement for this type of control is a sufficiently high band-

Fig. 8. Effect of half-sample delay in feedback path of filter capacitor voltageon system dynamics (∗: no filter, ×: with filter).

Fig. 9. Impedance seen from the PCC with the proposed control and improve-ment by resonant controller.

width of the fundamental controller. As the proposed controlleryields a higher bandwidth even for very low LCL resonancefrequencies, this kind of harmonic rejection is possible. Fig. 9shows the impedance improvement when a resonant controllerfor the sixth harmonic is added. Note that the sixth harmonic istypically dominant in the background distortion of three-phasesystems.

VI. SIMULATION RESULTS

The performance is verified in MATLAB/Simulink simula-tions. Switching of insulated gate bipolar transistors is takeninto account by using the piecewise linear electrical circuit sim-ulation toolbox. System and control parameters from the theo-retical analysis are used. The purpose is to clearly demonstratethe current control performance. Hence, the dc-link capacitor isreplaced with a constant dc voltage source, and the outer dc-linkvoltage loop is disabled in the simulation study. Thus, currentreferences can be chosen almost arbitrarily. The active currentis controlled constantly at 30 A, and the reactive current refer-ence is varied in the analysis of transients. Currents are sampledduring zero space vectors at the beginning and in the middle of


Fig. 10. Simulated steady-state spectra of converter phase current: (a) withoutUC f oversampling and (b) with UC f oversampling.

Fig. 11. Simulated step response of dq current components with decouplingnetwork not included (· · ·) and included (—): (a) converter currents and (b) linecurrents.

the switching period. If not stated otherwise, the filter capacitorvoltage is oversampled at two times the control frequency , theaverage is used and a decoupling network is used.

Fig. 10 shows the impact of UC f oversampling on the steady-state converter phase current and total distortion factor (THD) upto the 50th harmonic. The converter is operated with Icd = 30 Aand Icq = 20 A. It can be seen that the fifth and seventh harmoniccomponents caused by aliasing are reduced by the proposedfiltering. The THD is reduced by a factor of more than two inthis case. Note that an ideal line voltage is used in the simulation.

Fig. 11 shows the simulated step response of the proposedcontroller. From a comparison with the response without de-coupling network it becomes clear that the simple decouplingnetwork effectively reduces the coupling to the d path withoutaffecting the step response in the q path. The calculated stepresponse from Fig. 6 is well confirmed. It can be seen that theresonance is well damped in the converter current. The linecurrents are slightly more oscillatory during the step.

Fig. 12. Measured response of (internally computed) dq current componentsduring an icq -reference step from 0 to 30 A (no active power except losses) withresonant controller not included (· · ·) and included (—): (a) converter currentsand (b) line currents.

VII. EXPERIMENTAL RESULTS

Performance is verified experimentally on a self-built22-kW test drive in a laboratory environment. The grid-connected converter is loaded by an inverter-fed induction ma-chine that is field-oriented controlled. Settings and operatingpoints from theoretical studies and simulations are used. Forthis purpose, the load torque on the motor shaft is adjusted inorder to give a constant active power at 1300 r/min. The reactivecurrent reference is varied to test the dynamic performance. Thedc-link voltage is controlled by PI control and a slowly tunedresonant harmonic compensator for the sixth harmonic in the dqframe is used to reduce the fifth and seventh harmonics in thestationary frame, if not stated otherwise. Feedback signals aresampled as described in Section V-C.

The dynamic performance is verified by an icq -reference stepfrom 0 to 30 A. No active power except the losses is transferredin this case in order to have a step in the phase current ampli-tude as well. Fig. 12 shows the measured responses of the dqcurrent components in the proposed controller with and withoutthe resonant harmonic compensator for the sixth harmonic, re-spectively. The dq current signals are obtained from the controlsystem. It can be seen that without the harmonic compensatorthe fundamental response is distorted by low frequency harmon-ics. A slow harmonic compensator reduces them considerablywithout affecting the fundamental response. The LCL resonanceis well damped in both cases. The response in icq is also shownwith higher resolution in Fig. 12. The reference is crossed afterfive samples and 30% overshoot is obtained. Thus the simu-lation result from Fig. 11 is matched very well. Phase currentand voltage waveforms are measured with a Dewetron poweranalyzer with a sampling frequency of 100 kHz. Fig. 13 showsthe three-phase waveforms of the converter currents and voltageat the PCC, and Fig. 14 shows the response of the converter andline current in one phase, both with resonant controller included.

The effect of line voltage harmonic reduction becomes clearalso from the steady-state harmonic spectra. The converter isoperated with 15 kW on the grid side without injecting reac-tive power. The frequency spectrum is also obtained from thepower analyzer. Figs. 15 and 16 show the steady-state voltage


Fig. 13. Measured response of phase converter currents (Iconv 1,2,3) andvoltage at PCC (Uline 1,2,3) during an icq -reference step from 0 to 30 A (noactive power except losses).

Fig. 14. Measured response of converter (Iconv 1) and line current (Iline 1) ofone phase during an icq -reference step from 0 to 30 A (no active power exceptlosses).

Fig. 15. Measured steady-state spectra without resonant control: (a) phasevoltage at PCC, (b) phase converter current, and (c) phase line current.

and current spectra without and with harmonic compensators,respectively. The impedances calculated as the ratio of line volt-age and current components at the corresponding frequenciesimprove from 10 dB to 39.5 dB at 250 Hz and 7.5 dB to 32.2 dB

Fig. 16. Measured steady-state spectra with resonant control: (a) phase voltageat PCC. (b) Phase converter current. (c) Phase line current.

at 350 Hz, as expected from Fig. 9. Note that further improve-ment in power quality can be obtained with higher resonantcontroller gains but dynamic performance suffers as long as thefundamental controller is not redesigned. In this case, the har-monic compensator should be taken directly into account duringthe fundamental control design. This is beyond the scope of thispaper.

VIII. CONCLUSION

A PIstate space current control system for grid-connectedPWM converters with LCL filters was developed and analyzed.Controller parameter expressions have been derived in termsof system parameters and specified closed-loop poles and ze-ros. A pole placement strategy in the discrete time domain wasproposed for the LCL-filter-based system, which uses the poleand zero locations of a PI-controlled L-filter system and suitabledamping of resonance poles. Dynamic performance for differentLCL settings and robustness against grid impedance variationsare addressed as well as harmonic rejection capability and otherpractical issues.

The paper shows that the controller can be tuned straightfor-wardly using the derived expressions. Only the LCL parametersand control frequency are required. Good performance is provenfor different system settings compared to an L filter system. Highbandwidth with good resonance damping, specified overshoot,and good robustness is obtained. Even for LCL filter systemswith low resonance frequencies relative to the control frequency,the requirements are fulfilled, with only slightly more oscilla-tory behavior during transients. Distortions of feedback signalsdue to converter switching can be avoided by oversampling andfiltering of the filter capacitor voltage. Simulation and experi-mental results verify the theoretical results.

APPENDIX

Controller parameter expressions are shown in (26) at the topof the next page (ωR = ωRes , kI = 1/TI ).


kiC f =(

14

)Lf c

Tc sin(wRTc)(w2RCf Lf c cos(wRTc) − cos(wRTc) − w2

RCf Lf c + 1)

×

(w3

RTcCf Lf c − wRTc + sin(wRTc)) (

(pd1p

d2p

d3p

d4 − (−pd

1pd2p

d3 − (pd

1pd2 + (pd

1 + pd2 )p

d3 )p

d4 )p

d5 ) − 1

)+

((−pd

1pd2p

d3 − (pd

1pd2 + (pd

1 + pd2 )p

d3 )p

d4 − (pd

1pd2 (p

d1 + pd

2 )pd3 + (pd

1 + pd2 + pd

3 )pd4 )p

d5 ) + 2 cos(wRTc) + 2

)(w3

RTcCf Lf c − 2wRTc cos(wRTc) + 2w3RTcCf Lf c cos(wRTc) + 3 sin(wRTc) − wRTc

)((pd

1pd2p

d3p

d4p

d5 )(p

d1p

d2 − (−pd

1 − pd2 )p

d3 + (pd

1 + pd2 + pd

3 )pd4 + (pd

1 + pd2 + pd

3 + pd4 )p

d5 ) − 2 − 4 cos(wRTc)

)(4w3

RTcCf Lf c cos(wRTc)2 + 2w3RTcCf Lf c cos(wRTc) − w3

RTcCf Lf c − 4wRTc cos(wRTc)2

−2wRTc cos(wRTc) + wRTc + 5 sin(wRTc))((−pd

1 − pd2 − pd

3 − pd4 − pd

5 ) + 2 cos(wRTc) + 2)

kuC f =(

14

)wRCf Lf c

Tc(−2w2RCf Lf c cos(wRTc) + w2

R cos(wRTc)2Cf Lf c + w2RCf Lf c − 1 + 2 cos(wRTc) − cos(wRTc)2)

×

(−w3RTcCf Lf c + wRTc + sin(wRTc) + 2w3

RTcCf Lf c cos(wRTc) − 2wRTc cos(wRTc))

(pd1p

d2p

d3p

d4p

d5 − ((pd

1pd2 + (pd

1 + pd2 )p

d3 + (pd

1 + pd2 + pd

3 )pd4 + (pd

1 + pd2 + pd

3 + pd4 )p

d5 ) − 2 − 4 cos(wRTc)))

−(4w3RTcCf Lf c cos(wRTc)2 − 2w3

RTcCf Lf c cos(wRTc) − w3RTcCf Lf c − 4wRTc cos(wRTc)2

+2wRTc cos(wRTc) + wRTc + sin(wRTc))(−(pd1 + pd

2 + pd3 + pd

4 + pd5 ) + 2 cos(wRTc) + 2)

−(w3RTcCf Lf c − wRTc + sin(wRTc))(((pd

1pd2p

d3p

d4 − (−pd

1pd2p

d3 − (pd

1pd2 + (pd

1 + pd2 )p

d3 )p

d4 )p

d5 ) − 1)

+((−pd1p

d2p

d3 − (pd

1pd2 + (pd

1 + pd2 )p

d3 )p

d4 − (pd

1pd2 + (pd

1 + pd2 )p

d3 + (pd

1 + pd2 + pd

3 )pd4 )p

d5 ) + 2 cos(wRTc) + 2))

kp =(

12

)L2

f cCf w2R

(−1 + cos(wRTc))(−1 + w2RCf Lf c)T 2

c kI

×

(−(pd1p

d2p

d3p

d4p

d5 ) + (pd

1pd2p

d3p

d4 + pd

5pd1p

d2p

d3 + pd

5pd4p

d1p

d2 + pd

5pd4p

d3p

d1 + pd

5pd4p

d3p

d2 )

−(pd1p

d2p

d3 + pd

4pd1p

d2 + pd

4pd3p

d1 + pd

4pd3p

d2 + pd

5pd1p

d2 + pd

5pd3p

d1 + pd

5pd3p

d2 + pd

5pd4p

d1 + pd

5pd4p

d2 + pd

5pd4p

d3 )

+(pd1p

d2 + pd

3pd1 + pd

3pd2 + pd

4pd1 + pd

4pd2 + pd

4pd3 + pd

5pd1 + pd

5pd2 + pd

5pd3 + pd

5pd4 ) + 1)

−(+pd1 + pd

2 + pd3 + pd

4 + pd5 )

kic =Cf L2

f cw2R

2T 2c kI (w2

RCf Lf c cos(wRTc) − cos(wRTc) − w2RCf Lf c + 1)

×

(1 + TckI )(pd1p

d2p

d3p

d4p

d5 ) − ((pd

1pd2p

d3p

d4 + (+pd

1pd2p

d3 + (pd

1pd2 + (pd

1 + pd2 )p

d3 )p

d4 )p

d5 ) − 1)

−(1 − TckI )[(−pd

1pd2p

d3 − (pd

1pd2 + (pd

1 + pd2 )p

d3 )p

d4 − (pd

1pd2 + (pd

1 + pd2 )p

d3 + (pd

1 + pd2 + pd

3 )pd4 )p

d5 )

+2 cos(wRTc) + 2] + (−1 + 2TckI )((pd1p

d2 + (+pd

1 + pd2 )p

d3 + (pd

1 + pd2 + pd

3 )pd4 + (pd

1 + pd2 + pd

3 + pd4 )p

d5 )

−2 − 4 cos(wRTc)) + (−1 + 3TckI )(−(pd1 + pd

2 + pd3 + pd

4 + pd5 ) + 2cos(wRTc) + 2)

kuC = +2 cos(ωRes Tc) + 2 − pd1 − pd

2 − pd3 − pd

4 − pd5 . (26)

REFERENCES

[1] F. Blaabjerg, R. Teodorescu, M. Liserre, and A. Timbus, “Overview ofcontrol and grid synchronization for distributed power generation sys-tems,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1398–1409, Oct.2006.

[2] J. Rodriguez, J. Dixon, J. Espinoza, J. Pontt, and P. Lezana, “Pwm regen-erative rectifiers: State of the art,” IEEE Trans. Ind. Electron., vol. 52,no. 1, pp. 5–22, Feb. 2005.

[3] C. Lascu, L. Asiminoaei, I. Boldea, and F. Blaabjerg, “High performancecurrent controller for selective harmonic compensation in active powerfilters,” IEEE Trans. Power Electron., vol. 22, no. 5, pp. 1826–1835, Sep.2007.

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

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

[6] 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.

[7] R. Teichmann, M. Malinowski, and S. Bernet, “Evaluation of three-levelrectifiers for low-voltage utility applications,” IEEE Trans. Ind. Electron.,vol. 52, no. 2, pp. 471–481, Apr. 2005.

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

[9] M. Malinowski and S. Bernet, “A simple voltage sensorless activedamping scheme for three-phase PWM converters with an LCL fil-ter,” IEEE Trans. Ind. Electron., vol. 55, no. 4, pp. 1876–1880, Apr.2008.


[10] E. Wu and P. 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.

[11] P. A. Dahono, “A control method to damp oscillation in the input LC filter,”in Proc. IEEE Power Electron. Spec. Conf., 2002, vol. 4, pp. 1630–1635.

[12] E. Twining and D. 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] J. Dannehl, C. Wessels, and F. 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.

[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] S. Y. Park, C. 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.

[16] P.-T. Cheng, J.-M. Chen, and C.-L. Ni, “Design of a state-feedback con-troller for series voltage-sag compensators,” IEEE Trans. Ind. Appl.,vol. 45, no. 1, pp. 260–267, Jan./Feb. 2009.

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

[18] M. Bongiorno and J. Svensson, “Voltage dip mitigation using shunt-connected voltage source converter,” IEEE Trans. Power Electron.,vol. 22, no. 5, pp. 1867–1874, Sep. 2007.

[19] J. Dannehl, F. Fuchs, S. Hansen, and P. Thøgersen, “Investigation of activedamping approaches for PI-based current control of grid-connected PWMconverters with LCL filters,” in Proc. IEEE Energy Convers. Congr. Expo.,2009, pp. 2998–3005.

[20] C. Wessels, J. Dannehl, and F. Fuchs, “Active damping of LCL-filterresonance based on virtual resistor for PWM rectifiers—Stability analysiswith different filter parameters,” in Proc. IEEE Power Electron. Spec.Conf., 2008 (PESC 2008), Jun., pp. 3532–3538.

[21] A. Draou, Y. Sato, and T. Kataoka, “A new state feedback based transientcontrol of PWM ac to dc voltage type converters,” IEEE Trans. PowerElectron., vol. 10, no. 6, pp. 716–724, Nov. 1995.

[22] B. Bolsens, K. D. Brabandere, J. V. den Keybus, J. Driesen, andR. Belmans, “Model-based generation of low distortion currents in grid-coupled PWM-inverters using an LCL output filter,” IEEE Trans. PowerElectron., vol. 21, no. 4, pp. 1032–1040, Jul. 2006.

[23] B. Bolsens, K. D. Brabandere, J. V. den Keybus, J. Driesen, andR. Belmans, “Three-phase observer-based low distortion grid current con-troller using an LCL output filter,” in Proc. IEEE Power Electron. Spec.Conf., 2005, CD-ROM paper.

[24] M. P. Kazmierkowski, R. Krishnan, and F. Blaabjerg, Control in PowerElectronics: Selected Problems. Oxford, U.K.: Academic, 2002.

[25] D. Schroder, Elektrische Antriebe 2, Regelung von Antriebssystemen,2nd ed. Berlin, Germany: Springer-Verlag, 2001.

[26] P. W. Lehn and M. R. Irvani, “Discrete time modeling and control ofthe voltage source converter for improved disturbance rejection,” IEEETrans. Power Electron., vol. 14, no. 6, pp. 1028–1036, Nov. 1999.

[27] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Anal-ysis and Design. New York: Wiley, 2005.

[28] J. Dannehl and F. Fuchs, “Flatness-based voltage-oriented control of three-phase PWM rectifiers,” in Proc. 13th Int. Power Electron. Motion ControlConf. 2008, Sep. 1–3, pp. 444–450.

[29] J. Dannehl, F. Fuchs, and S. Hansen, “PWM rectifier with LCL-filter usingdifferent current control structures,” in Proc. Power Electron. Appl., 2007Eur. Conf., Sep., pp. 1–10.

Joerg Dannehl (S’07) was born in Flensburg,Germany, in 1980. He received the Dipl.-Ing. degreefrom Christian Albrechts University, Kiel, Germany,in 2005.

Since 2005, he has been a Research Assistant at theInstitute for Power Electronics and Electrical Drives,Christian Albrechts University. His main research in-terests include control of power converters and drives.

Mr. Dannehl is a Student Member of the IEEEIndustrial Electronics Society, the IEEE Power Elec-tronics Society, and the Verband der Elektrotechnik

Elektronik Informationstechnik (VDE).

Friedrich Wilhelm Fuchs (M’96–SM’01) receivedthe Dipl.-Ing. degree in 1975 and the Ph.D. degree in1982, both from the Rheinisch-Westfalische Technis-che Hochschule University of Technology, Aachen,Germany.

In 1975, he was with the University of Aachen,Aachen, Germany, where he was engaged in researchon ac automotive drives. From 1982 to 1991, he wasa Group Manager and engaged in research on devel-opment of power electronics and electrical drives in amedium-sized company. In 1991, he was a Managing

Director at the Allgemeine Elektrizitats-Gesellschaft (AEG), Berlin, Converterand Drives Division (presently Converteam), where he was engaged in re-search on design and development of drive products, drive systems and highpower supplies, covering the power range from 5 kVA to 50 MVA. In 1996, hejoined as a Full Professor at the Christian-Albrechts-University of Kiel, Kiel,Germany, where he is currently the Head of the Institute for Power Electronicsand Electrical Drives. He is the author or coauthor of more than 150 papers. Hisresearch interests include power semiconductor application, converters topolo-gies, variable speed drives as well as their control. Other research interestsinclude renewable energy conversion, especially wind and solar energy, on non-linear control of drives as well as on diagnosis of drives and fault tolerant drives.Many research projects are carried out with industrial partners. His institute ismember of CEwind eG, registered corporative, competence center of researchin universities in wind energy Schleswig-Holstein, as well as of competencecenter for power electronics in Schleswig-Holstein (KLSH).

Prof. Fuchs is the Chairman of the Supervisory Board of Cewind, theConvenor [Deutsche Kommission Elektrotechnik Elektronik Informationstech-nik (DKE), International Electrotechnical Commission (IEC)], InternationalSpeaker for Standardization of Power Electronics as well as a member of Ver-band der Elektrotechnik Elektronik Informationstechnik (VDE) and EuropeanPower Electronics (EPE) and Drives Association.

Paul Thøgersen (M’92–SM’01) was born in Thy,Denmark, on June 29, 1959. He received the M.Sc. EEdegree (in control engineering) in 1984, and the Ph.D.degree (in power electronics and drives) in 1989, bothfrom Aalborg University, Aalborg, Denmark.

From 1988 to 1991, he was an Assistant Professorat Aalborg University. From 1991 to 2005, he was atDanfoss Drives A/S, first as a Research and Develop-ment Engineer and later as a Manager of technology,and he was involved in drives control technology.Since 2006 he has been the Manager of the Model-

ing and Control Group, R&D Department, the KK Electronic A/S, Aalborg,Denmark. Since 1991, he has been keeping a close relationship to Aalborg Uni-versity, resulting in more than 20 coauthored papers and participation in morethan ten Ph.D. student advisory groups.

Dr. Thøgersen was the recipient of the Angelo Award in 1999 for his contri-butions to the development of industrial drives.

2320 ieee transactions on power electronics, · pdf file2320 ieee transactions on power...

Documents