Cost-Effective Deadbeat Current Control For

IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 2, MARCH/APRIL 2014 1185

Cost-Effective Deadbeat Current Control forWind-Energy Inverter Application With LCL Filter

Katsumi Nishida, Member, IEEE, Tarek Ahmed, Member, IEEE, and Mutsuo Nakaoka, Member, IEEE

AbstractThis paper proposes appropriate analytical expres-sions for a deadbeat control system to be implemented in theoutput inductorcapacitorinductor (LCL) filter of a grid-connected inverter. Unlike in a conventional analysis, the proposedcontrol algorithm with a settling time of three sampling periodsis derived from the system transfer function in the discretizedtime domain instead of the Laplace s-domain. Furthermore, whenintroducing the deadbeat control, the independent control of theintegrated active power and that of the reactive power are alsomade possible. The experimental results of the proposed deadbeatcontrol system indicate that the feedbacks from both grid andinverter currents have similarly high capabilities in attenuatingswitching frequency components and damping resonance. Suchproficiency is brought about because the remaining two unde-tected control variables of the LCL filter are taken as estimates inthe DSP when introduced to the state observer. However, feedbackfrom the grid currents can be seen to significantly reduce the totalharmonic distortion of the actual grid currents themselves.

Index TermsActive damping methods, deadbeat control andadaptive predictor, grid and inverter current feedbacks, grid-connected inverter, LCL filter.

I. INTRODUCTION

POWER regeneration, adjustable power factor, and lowerswitching frequency components in the line current arethe most important advantages of the grid-connected inverterwhen employing the inductorcapacitorinductor (LCL-type)filter instead of the conventional inductor (L-type) filter or theinductorcapacitor (LC-type) filter [1][5]. However, the majordrawback of the LCL-type filter is the resonance phenomenon[6][10]. Several methods for damping the resonance havebeen proposed in order to overcome the stability problem[11][16]. The resonance is attenuated with either passive oractive damping methods. In the passive methods, additionalresistors are connected in series with the filter capacitors, butthis actually creates additional power losses [17][21]. Onthe other hand, the active damping methods are realized by

Manuscript received November 16, 2012; revised February 19, 2013 andJune 22, 2013; accepted July 28, 2013. Date of publication August 28, 2013;date of current version March 17, 2014. Paper 2012-IPCC-667.R2, presentedat the 2012 IEEE Energy Conversion Congress and Exposition, Raleigh, NC,USA, September 1520, and approved for publication in the IEEE TRANS-ACTIONS ON INDUSTRY APPLICATIONS by the Industrial Power ConverterCommittee of the IEEE Industry Applications Society.

K. Nishida is with Ube National College of Technology, Ube 755-8555,Japan (e-mail: [email protected]).

T. Ahmed is with Assiut University, 71516 Assiut, Egypt (e-mail:[email protected]).

M. Nakaoka is with the Electric Energy Saving Research Center, KyungnamUniversity, Masan 631-701, Korea (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/TIA.2013.2279900

modifying the control algorithm in order to take the resonanceinto account in the control strategy [22][24]. Several activedamping methods, proposed in the literature, require additionalsensors [25] or complex control strategies [26] to maintain thesystem stability through controlling the grid-side current andthe capacitor current where this is the main function of theactive damping method.

Although a wide stability margin of the grid-connected in-verter with the LCL filter is very important, it is also desiredto have a minimum number of sensors and, therefore, a simplecontrol structure [12], [22], [27][30]. Some conventional feed-back controls can provide a rather quick and steady response,although only one control variable is actually detected [22].However, the settling time is longer than three sampling peri-ods, which is the shortest settling time to be realized in a 3-Dsystem and can be brought about by the deadbeat control. Onthe other hand, if a model predictive control is introduced as thecurrent control of the LCL filter [12], the implementation of themodel predictive control algorithm is found to be difficult. Thisis mainly due to the necessity of predicting the discrete-timestate evolution for four sampling intervals in advance and thendetermining the desired inverter output voltage to minimize thecost function.

The deadbeat current control algorithm for a grid-connectedinverter with a third-order LCL filter previously derived bythe authors from the basis of the discretized equations was inthe form of the state feedback [13]. As the manipulating value(inverter output voltage) is simply determined by substitutingthe control error of the 3-D state variable to the linear com-bined state feedback equation, the actual implementation of thedeadbeat control scheme is easy.

Furthermore, the proposed deadbeat control system can bemade more cost-effective if the state identification algorithm isadopted in order to achieve the state feedback by detecting onlyone control variable (= element) of the state variable.

If the current sensors are built into the inverter to protect it,inverter current feedback is reasonable. Thus, the state feedbackcontrol can be achieved by detecting one control variable,although the LCL filter is a 3-D system.

In this paper, the measured results of the proposed deadbeatcontrol system with the use of the grid current as the feedbacksignal are mainly presented. From the observed results, gridcurrent feedback provides the same attenuation of inverterswitching components and damping resonance as comparedto the inverter current feedback, due to it being possible toestimate two undetected remaining control variables in the DSP.

Furthermore, the grid current feedback significantly reducesthe total harmonic distortion (THD) of the grid currents by

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1186 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 2, MARCH/APRIL 2014

Fig. 1. Experimental setup of run-back power system for grid-connected inverter through 3-D LCL filter.

2.0% more than the inverter current feedback. This is directlydue to the fact that the low-order current harmonic componentsin only the detected current can be effectively eliminated withthe adaptive predictor. The adaptive predictor prevents theharmonic components observed in the grid voltage from distort-ing the detected (grid or converter) currents by automaticallyinserting the harmonic voltage with the opposite phase in theinverter output.

The robustness of the line inductance variation of the proposecontrol method is the next practical issue to address. To confirmthe influence of the line inductance on the system stability,different values of line inductance are connected and tested.The experimental results have demonstrated that the proposedsystem is robust against the inductance variation due to itsinherent ability of detecting the grid voltage deviation.

II. DEADBEAT CURRENT CONTROL

A. Difference Equation of LCL Filter in Stationary Frame

The schematic diagram of the grid-side inverter connected tothe grid via an LCL filter is shown in Fig. 1. The state equationof the LCL filter can be expressed as

d

dt

i1i2vc

=

0 0 1/L10 0 1/L21/C 1/C 0

i1i2vc

+

1/L10

0

v1 +

01/L2

0

v2 (1)

where i1, i2, and vc are the instantaneous space vectors ofthe inverter output current, the grid current, and the filtercapacitor voltage, respectively, and they are described in the stationary reference frame.

The state equation (1) is represented in the Laplaces-domain by

sI1(s) i1(k)sI2(s) i2(k)sV C(s) vC(k)

=

0 0 1/L10 0 1/L21/C 1/C 0

I1(s)I2(s)V C(s)

+

1/L10

0

V1(s) +

01/L2

0

V2(s) (2)

where I1(s), I2(s), and V c(s) are the state variables inthe s-domain. V 1(s) and V 2(s) are the inverter output volt-age and the grid voltage in the s-domain, respectively. Inaddition to this, the sampling point k is set to the initialinstant (t = 0).

Assume that the space vectors v1 and v2 are fixed during thesampling period Ts, where the voltages are simply expressedin the s-domain, V 1(s) = v1(k)/s and V 2(s) = v2(k)/s. Re-arranging the above equation defines the state variable in thes-domain by

I1(s)I2(s)V C(s)

= 1

s (s2 + 2r)

s

2 + 1L2C1

L1C sL1

1L2C

s2 + 1L1CsL2

sC

sC s

2

i1(k) +

v1(k)L1s

i2(k) v2(k)L2svC(k)

. (3)

By applying the inverse Laplace transformation to (3), thediscretized system equation is formulated as

i1(k + 1)i2(k + 1)vc(k + 1)

= P

i1(k)i2(k)vc(k)

+Q v1(k) +D v2(k) (4)

NISHIDA et al.: COST-EFFECTIVE DEADBEAT CURRENT CONTROL FOR WIND-ENERGY INVERTER APPLICATION 1187

Fig. 2. Manipulated value expressed in stationary frame.

where P , Q, and D are defined as

P =

+(1) cos f (1)(1cos f )

1X1

sin f(1 cos f ) 1 + cos f 1X2 sin fXc sin f Xc sin f cos f

Q =

fX1+X2

+(1) sin f(X1+X2)

fsin fX1+X2

(1 )(1 cos f )

, D =

f+sin fX1+X2

fX1+X2 sin f

X2

(1 cos f )

where

= L1/(L1 + L2), Ts is the sampling period,

resonance angular frequency r=

(L1 + L2)/L1L2C,

f = rTs, X1 = rL1, X2 = rL2, and Xc = 1/rC.

B. Derivation of Deadbeat Control Scheme

The proposed current control system is designed for realizinga 3-D finite-time settling response with the simple implementa-tion of the control scheme. The inverter output voltage v1(k +1) is set to a manipulated value during the interval lasting froma sampling point k + 1 to the next sampling point k + 2 inthe current control implementation. The authors in accordancewith the well-known superposition theory originally define themanipulated value, as expressed by

v1(k + 1) = v1 (k + 1) + v1(k + 1). (5)

From both (5) and Fig. 2, v1(k + 1) is determined as the sumof the steady-state term to keep each space vector expressedin the dq synchronously rotating reference frame constant andthe adjusting term to make the control error vanish. Although,the current reference of the space vector of the grid-connectedinverter is expressed in the dq synchronously rotating refer-ence frame, the function of the finite-time settling controller foreliminating the control error is effective only when the refer-ence can be set to a constant value in the stationary frame.

Fig. 3. Phasor diagram of the state vectors in steady state.

Therefore, synthesizing the two voltage vectors as depicted inFig. 2 is required to attain the finite-time settling response.

Based on the phasor diagram of the state vectors representedin Fig. 3, the steady-state term v1 (k + 1) is simply calculatedfrom

v1 = vc +jL1 i1=(1 2L1C)v2 +

{3L1L2C (L1 + L2)

} i2q

+ j{3L1L2C + (L1 + L2)

} i2d . (6)

On the other hand, the adjusting term of (5), v1(k + 1), iscalculated on the basis of the following state feedback equation:

v1(k + 1) = K x(k + 1) (7)

where the controlled error matrix x(k + 1) is defined by

x(k + 1)=

i1(k + 1)i2(k + 1)vc(k + 1)

=

i1(k + 1) i1 (k + 1)i2(k + 1) i2 (k + 1)vc(k + 1) vc (k + 1)

(8)where asterisks denote the reference of space vectors taken asthe control variables of the LCL filter.

The state vector of the controlled error reaches the origin(i.e., the control error vanishes) in the shortest sampling periodswhen the state feedback gain matrix K = [K1,K2,K3] istuned as follows.

Defining x(0) as the undefined initial error vector, the errorvector of the next sampling point is defined from (4) as follows:

x(1) = P x(0) +Q v1(0) = [P Q K]x(0).

Then, the x(2) is also expressed by

x(2)=[PQ K]x(1)=[PQK][PQ K]x(0)

=

{P 2 [P QQ]

[K

K(P QK)

]}x(0)

where the inverse matrix of [PQ Q]1 does not exist, sox(2) = [0 0 0]T for the undefined x(0) is impossible.

Finally,

x(3)=[PQK]{P 2[P QQ]

[K

K(PQK)

]}x(0)

=

P3[P 2 Q P Q Q]

KK(PQK)K(PQK)2

x(0).

Moreover, in order to always make x(3) = [0 0 0]T forthe undefined x(0), the state feedback matrix K should be


defined as follows:P 3 [P 2 Q P Q Q]

KK(P QK)K(P QK)2

= 0

KK(P QK)K(P QK)2

= [P 2 Q P Q Q]1 P 3.

The state feedback matrix K is

K = [ 1 0 0 ] [P 2 Q P Q Q ]1 P 3. (9)

From (9), it is proven that the condition of the controllabilityis given by

det [P 2 Q P Q Q ] = 0. (10)

Also, the condition of the controllability that satisfies (10)can be defined by

fr =1

2

L1L2CL1+L2

< 0.5 fs. (11)

Equation (11) expresses that resonance frequency fr shouldbe set less than half the sampling frequency fs to damp theresonance.

C. Identification Algorithm for Undetected Control Variables

For compensating the calculation time delay in the DSP,v1(k + 1) must be calculated at the sampling point (k) sothat i1(k + 1), i2(k + 1), and vc(k + 1) defined in (8)have to be estimated at the sampling point (k) in advance.

In addition, the identification algorithm is introduced for theproposed control scheme to be implemented with the reducednumber of sensors. If only i2 can be measured whereas i1and vc are estimated, the detection matrix C, which relatesthe control variables to the output signal, is

C = [ 0 1 0 ] . (12)

In the 3-D system, the undetected controlled errors at thesampling point k 2, x(k 2), should be first estimated atthe sampling point k using i2(k 2), i2(k 1), i2(k),and the adjusting term in the manipulated values v1(k 2)and v1(k 1) as follows:

i2(k 2) =C x(k 2)i2(k 1) =C x(k 1)

=C {P x(k 2) +Q v1(k 2)}i2(k) =C x(k)

=C {P x(k 1) +Q v1(k 1)}=C {P (Px(k 2) +Q v1(k 2))

+ Q v1(k 1)} .

Then, the above three equations are represented in the matrixform as follows:i2(k 2)i2(k 1)C Q v1(k 2)i2(k)C P Qv1(k 2)C Qv1(k 1)

=

CC PC P 2

x(k 2)

x(k 2)

=

i1(k 2)i2(k 2)vc(k 2)

=

CC PC P 2

1

i2(k 2)i2(k 1)C Q v1(k 2)i2(k)C P Qv1(k 2)C Qv1(k1)

.

(13)

From (13), the observability condition is

det

CC PC P 2

= 0. (14)

Then, the controlled error in the undetected control variablesi1(k 1) and vc(k 1) at sampling point (k 1) can beestimated by using (4) with k 2 substituted to the samplinginstants and the third term of Dv2(k 2) neglected. Moreover,i1(k) and vc(k) at sampling point (k) can be also estimatedusing (4). Finally, i1(k + 1), i2(k + 1), and vc(k + 1)are estimated from

i1(k + 1)i2(k + 1)vc(k + 1)

= P

i1(k)i2(k)vc(k)

+Qv1(k). (15)

D. Stability Analysis Using z-Transform

A well-designed LCL-type filter for the three-phase inverterproduces better attenuation of inverter switching harmonicsthan the conventional L-type and LC-type filters but maycause both dynamic and steady-state current distortions dueto resonance. The stability of the proposed deadbeat currentcontrol system is analyzed on the basis of the characteristic rootlocations in the z plane.

For the open-loop control scheme, the characteristic equationis defined by (15) with the second term in the right-hand sideneglected. The three roots of the characteristic equation are 1,e+jrTs, and ejrTs and lie just on the unit circle in the z plane.This proves that this system is unstable in the open-loop controlscheme.

For the deadbeat control, the characteristic equation can bederived by substituting (15) into (7), which gives

v1(k + 1) = K

i1(k + 1)i2(k + 1)vc(k + 1)

= K

P

i1(k)i2(k)vc(k)

+Qv1(k)

. (16)


Fig. 4. Finite-time settling control system with adaptive predictor.

Fig. 5. System configuration of adaptive predictor.

By rearranging (16), the characteristic equation is finally de-scribed byi1(k + 1)i2(k + 1)vc(k + 1)v1(k + 1)

=

[P QK P K Q

]i1(k)i2(k)vc(k)v1(k)

. (17)

When the resonance frequency fr is made lower than halfthe switching frequency fs, the four roots of the characteristicequation (17) (i.e., eigenvalues of the 4 4 matrix) lie on theorigin of the unit circle in the z plane. If the condition of thecontrollability, defined by (11), is satisfied, the deadbeat controlscheme is able to bring about high stability and increasedresponsiveness to the grid-connected inverter with the LCLfilter.

E. Introduction of Adaptive Predictor

The deadbeat current controller has similar behavior to aproportional controller as shown in (7). The introduction of theadaptive predictor as an integral controller to decrease the con-trolled error of not only the dc but also the ac ripples brings tothe current control implementation a desired robustness against

grid voltage distortion and the nonlinearity characteristics of theoutput voltage of the inverter together with its command.

The adaptive predictor is one kind of technique of the adap-tive finite-impulse response (FIR) filter [13]. Fig. 4 shows theproposed current control system with the adaptive predictor. Inthis current control system, the adaptive predictor is introducedto predict the control error of four sampling periods aheadbecause both the settling time of three sampling periods andthe calculation time delay of one sampling period should becompensated. Moreover, the accuracy of the current controlsystem can be improved by subtracting the adaptive predictoroutput e(k + 4) as an adjustment term from the referencecurrent i2 based on the theory of inverse modeling [14].

Fig. 5 indicates the complete system configuration of theadaptive predictor. The active and reactive components of theinverter output current error vector in the dq synchronouscoordinate frame are separately predicted by using two similaradaptive predictors described in the DSP program. Moreover,the frequency of the input signal should be set to the samefrequency as to be contained in the error signal. In Fig. 6,the control error of the 5th and 7th harmonics observed onthe grid-integrated line current can be eliminated effectively bysetting the input signal of the sixth sine wave (= sin 6t). In


Fig. 6. Step responses for state variables of the LCL filter. (a) q-axiscomponents of state variables. (b) d-axis components of state variables.

Fig. 5, the adaptation process of the digital filter coefficient isillustrated by blue lines where the adaptation of the FIR filter isimplemented by using the leaky least mean square algorithm asindicated by

hl(k + 1) = hl(k) + 2 e(k) sin 6tkl4 (18)

where

l =0, 1, 2, . . . , 62, 63, = 0.999

0


Fig. 7. Variations of the manipulating values of the dq-axis inverter outputvoltages.

resonance frequency (fr = 1440 Hz) is increased from 28% to50% of fs. In this case, improving the control accuracy of thegrid integration inverter is necessary, although the resonance iscompletely damped as described in Section II.

The simulations have been done in order to confirm thecorrectness and effectiveness of the proposed deadbeat controlalgorithm expressed by (5)(7).

Fig. 6(a) and (b) show the d- and q-axis components ofcurrent and voltage space vectors defined as the controlledvariables of the third-order LCL filter. The desired value ofthe active grid-integrated current i2d is stepped up by 2 A atthe sampling points 3, 6, 9, 12, and 15, where holding thedesired values for three sampling periods (= 3Ts) is vital toobtain a 3-D finite-time settling response. The superior time-domain performance of the proposed deadbeat control schemewas verified with the simulation results.

At the same conditions of Fig. 6, Fig. 7 shows the variationsin the d- and q-axis components of the three-phase inverteroutput voltage space vector v1 as the manipulated value.

As the active component of inverter currents i1d varies up anddown as illustrated in Fig. 6(b), the control error of i1d can berelated to the manipulated value v1d in Fig. 7, in order to dampthe resonance. From the simulation results, the inverter currentfeedback brings about higher performances to the conventionalcontrolled system with a well-designed PI controller comparedto the grid current feedback [12].

IV. EXPERIMENTAL RESULTS ANDPERFORMANCE EVALUATIONS

A. Inverter Mode of Operation

Fig. 1 demonstrates the experimental setup using a run-backpower system. As the 12-pulse rectifier is used as the formerconverter in the run-back system, the output current reference ofthe inverter as the latter converter can be set arbitrarily, and theperformance of the utility inverter with the new control schemetreated here can be easily and accurately tested.

The utility interactive inverter employs the synchronousspace vector pulsewidth modulation (PWM). When using thePLL circuit, one grid voltage period is divided into 128 PWMperiods, and one PWM period is divided into 256 steps equally.The data sampling timing is synchronous with PWM, and thesampling frequency is 64 f = 3840 Hz, as given in Table I.Fig. 8 indicates the harmonic spectrum of the grid voltage withthe grid voltage THD of 1.9%.

Fig. 8. Harmonic spectrum of the grid voltage.

Fig. 9. Measured dq-axis current components of grid current vector dur-ing the active power step change from 50% to 100% of the rated power(rated current =

3 3 = 5.2 [A]). (a) Open-loop control in inverter operat-

ing mode. (b) Deadbeat control without adaptive predictor. (c) Deadbeat controlwith adaptive predictor.

For the open-loop control system, Fig. 9(a) shows the currentcomponents of the grid line current vector i2 in the dq syn-chronously rotating reference frame. The reference of the activecurrent is changed in step from half to the rated value with the


Fig. 10. Comparisons between current error and adaptive predictor output(rated voltage = 100 [V]). (a) Adaptive predictor output for d-axis currenterror. (b) Adaptive predictor output for q-axis current error.

step width set to the feasible and the shortest settling time ofthree sampling periods (3Ts = 3 1 3840 = 0.781 [ms]).

When applying the deadbeat control to the grid utility in-verter, Fig. 9(b) indicates the dq-axis current components ofthe grid current vector i2. Moreover, Fig. 9(c) shows the samecurrent components when adding the adaptive predictor to thedeadbeat controller.

As shown in Fig. 9(a), the resonance in the LCL filter isnot observed for the open-loop control, although no controlis applied for eliminating the control error. In Fig. 9(b), thedynamic response is faster than that of Fig. 9(a) with a smallcontrolled error. This error is due to the 5th and 7th harmonicsand the negative-sequence component observed in the grid volt-age. In Fig. 9(c), the control error in each current component iseffectively reduced owing to the adaptive predictor.

Fig. 10(a) and (b) proves that the adaptive predictors for ac-tive and reactive components estimate accurately the controlledcurrent error vector in Fig. 9(b) when the adaptive predictor isnot used. Therefore, the proposed adaptive predictor improvesthe control accuracy for the deadbeat control implementationthrough modifying the reference by the estimated controllederror in advance.

The experimental results proved that the proposed deadbeatcontrol, introducing the digital signal processing, improvesthe control accuracy without decreasing the control stability,although the step size for the adaptation process of the digitalfilter coefficient should be appropriately tuned.

In Fig. 10(a) and (b), the variations in dq-axis componentsof the grid voltage vector v2 are also illustrated, respectively,which are brought about by integrating the difference betweenthe measured grid current and its estimated value from (15), asillustrated in Fig. 4 [13].

Fig. 11(a)(c) shows the oscillographs of the inverter currenti1v , the grid current i2v , the dc link voltage Vdc, and the gridline-to-line voltage v2wu for the inverter mode of operation at

Fig. 11. Measured waveforms of inverter current i1v , grid current i2v ,dc link voltage Vdc, and grid voltage v2wv in inverter operating mode whichare corresponding to Fig. 9. (a) Open-loop control in inverter mode (i2 THD =2.6%). (b) Deadbeat control without adaptive predictor (i2 THD = 2.3%).(c) Deadbeat control with adaptive predictor (i2 THD = 1.5%).

the same condition of Fig. 9. The THDs of the grid current forthe open-loop control and the deadbeat control without and withan adaptive predictor are 2.6%, 2.3%, and 1.5%, respectively.The THD is measured for harmonic orders up to the 100th.

On the other hand, Fig. 12 shows the manipulating valueof the inverter output voltage vector v1 expressed in the dq


Fig. 12. Transient responses of the manipulating voltage vector.

synchronously rotating reference frame at the same conditionof Figs. 9(c) and 11(c). From Fig. 12, it can be seen that eachcomponent of the manipulating voltage vector almost agreeswith the previous simulation results in Fig. 7.

B. STATCOM Mode of Operation

The grid interactive inverter can control the instantaneousreactive power as well as the active power. Fig. 13(a)(c)shows the differences in the control performance of the threeschemes (the open-loop control and the deadbeat control withand without an adaptive predictor) when increasing the reactivecurrent in step from 17% to 42% of the rated value.

Fig. 13(a) shows the current response for the open-loopcontrol scheme where the LC resonance of fr = 1061 Hz isobserved after the change of the reference reactive current.

On the other hand, Fig. 13(b) shows the d- and q-axis currentcomponents when the deadbeat control without an adaptivepredictor is applied to the grid interactive inverter. The dynamicresponse is roughly improved, and the resonance is perfectlydamped by the deadbeat controller. Moreover, Fig. 13(b) provesthat the same control error as shown also in Fig. 9(b) remainsin the steady state due to the 5th and 7th harmonics and thenegative-sequence component, which are observed in the gridvoltage as a kind of system disturbance.

In Fig. 13(c), the control error, observed on each currentcomponent, is effectively reduced owing to introducing theadaptive predictor. In the meanwhile, Fig. 14(a)(c) shows theoscillographs of inverter current i1v , grid current i2v , filtercapacitor voltage vCwv , and grid line-to-line voltage v2wv atthe same condition of Fig. 13. The difference of the controlperformances between three types of control methods cannotbe obvious in Fig. 14(a)(c), but these differences can be madeclear when observing the dq-axis current components of thegrid current.

On the other hand, Fig. 15 shows the manipulating value ofinverter output voltage vector v1, expressed in the dq rotatingreference frame, at the same condition of Figs. 13(c) and 14(c).With the reactive reference current step change, not v1d but v1qchanges updownup, which is quite opposite to what is givenin Fig. 12.

Fig. 13. Measured dq-axis current components of grid current with reactivecurrent step change from 17% to 42% of the rated current. (a) Open-loopcontrol in STATCOM operating mode. (b) Deadbeat control without adaptivepredictor. (c) Deadbeat control with adaptive predictor.

C. Rectifier Mode of Operation

In the rectifier operating mode, the proposed deadbeat con-troller can be only applied to the converter with the referenceactive current i2d with negative values set as shown in Fig. 16.In Fig. 16, the input power is increased from 260 to 520 Wthrough decreasing the dc load resistance, connected to the dcside of the converter as shown in Fig. 1. The proportional andintegral (PI) controller in the major loop outputs the referenceof the active current for keeping the dc bus voltage constantat 220 V in a steady-state operation with only small changein the transient state (1.2% of 220 V). On the other hand,the minor loop deadbeat controller calculates the manipulatedvalue in order to make the actual active current follow up to itsreference.

Fig. 17 displays the measured waveforms of the tran-sient responses of the inverter output phase current i1v , grid-side current i2v , dc bus voltage Vdc, and line-to-line grid


Fig. 14. Measured waveforms of inverter current i1v , grid current i2v ,filter capacitor voltage vCwv , and grid voltage v2wv in STATCOMoperating mode which are corresponding to Fig. 13. (a) Open-loopcontrol in STATCOM operating mode [current THD before step change =3.7%; current THD after step change = 8.5%)]. (b) Deadbeat control inSTATCOM operating mode (current THD before step change = 4.0%;current THD after step change = 5.0%). (c) Deadbeat control with adaptivepredictor (current THD before step change = 2.1%; current THD after stepchange = 2.1%).

voltage v2wu in the rectifier mode. In three operating modes,the rectifier mode can record the lowest THD of the gridcurrent.

Fig. 15. Transient responses of the manipulating voltage vector.

Fig. 16. Measured dq-axis current waveforms in rectifier mode.

Fig. 17. Measured waveforms of converter current i1v , grid current i2v , dclink voltage Vdc, and grid voltage v2wu in rectifier operating mode with powerincrease from 260 to 520 W.

D. Reducing the Filter Size

As mentioned in Section III, the proposed grid interactiveinverter can work with high performance for damping the LCresonance, even if the total inductance L1 + L2 is reducedfrom 8 to 4 mH. Fig. 18(a) and (b) shows the transient re-sponses for the inverter and the STATCOM modes, respec-tively. It can be seen that the switching ripple components ofthe inverter current remain much more than that with 8 mH


Fig. 18. Measured waveforms of inverter current i1v , grid current i2v , filtercapacitor voltage vCwv , and grid voltage v2wv when the total inductanceL1 + L2 is reduced to 4 mH. (a) Inverter mode operation (i2 THD = 2.3%).(Active current step change from 50% to 100% of the rated value, with the unitypower factor operation being kept.) (b) STATCOM mode operation (i2 THD =2.9%). (Reactive current step change from 17% to 42% of the rated value,with the active current being fixed at 0.58 p.u.).

TABLE IICOMPARISON OF THD OBSERVED ON GRID CURRENT

while the resonance is damped due to introducing the pro-posed deadbeat control scheme. However, the THD of the gridcurrent with 4 mH is increased by 0.8% more than that withL1 + L2 = 8 mH.

E. Comparison Between Two Types of Feedback

In this section, a comparison in terms of the measured THDof the grid current is made for the two different types offeedback control, inverter current feedback and grid currentfeedback. Table II shows all the measured THDs for differentoperation modes: inverter, STACOM, and rectifier.

It can be seen from Table II that the grid current feedbacksignificantly reduces the THD by 2.0% as compared to the

Fig. 19. Comparisons of measured waveforms of inverter current i1v , gridcurrent i2v , filter capacitor voltage vCwv , and grid voltage v2wv in invertermode between two types of current feedback. (a) Grid current feedback.(b) Inverter current feedback.

inverter current feedback. This is due to the fact that the low-order current harmonic components of the detected (grid orconverter) current can be effectively eliminated by the adaptivepredictor.

As the adaptive predictor counteracts the harmonic com-ponents in the grid voltage by automatically inserting theharmonic voltage with the opposite phase in the inverter out-put, it is impossible to eliminate the harmonic componentscontained in both grid and inverter currents at the same time.The grid current feedback control is reducing the low-ordercurrent harmonics in the grid current which is the target ofthe adaptive predictor much more than that of the invertercurrent as shown in Fig. 19(a). On the other hand, using theinverter current feedback, the low-order current harmonics inthe inverter current can be significantly reduced as shown inFig. 19(b) while the current harmonics in the grid current couldnot be significantly decreased due to the fact that the invertercurrent is set as the target of the adaptive predictor.

If the LCL filter is symmetrically designed with L1 = L2,the current harmonics in both grid and inverter currents canbe rather effectively reduced at the same time as shown inFig. 18(a) and (b) without depending on the detected current.


Fig. 20. Root arrangement of the proposed deadbeat control system withdifferent values of line inductance.

Fig. 21. Measured waveforms of inverter current i1v , grid current i2v , filtercapacitor voltage vcwv , and grid voltage v2wv in inverter mode of operationwith line inductance Lg = 3 mH.

F. Robustness About the Variation of Line Inductance

The four roots of the characteristic equation, i.e., eigenvaluesof the 4 4 matrix in (17), lie on the origin of the unit circlein the z plane when the line inductance Lg is negligible. Toconfirm the influence of the line inductance on the systemstability, different values of line inductance (1, 2, and 3 mH)are used in investigating the root allocation. Fig. 20 indicatesthat the characteristic equation roots are located inside the unitcircle, while the feedback gain matrix K remained constant.This proves that the deadbeat control can work steadily, evenwhen the line inductance is doubled.

Moreover, the quick dynamic performance response of theproposed system is not affected as shown in Fig. 21, even whenthe feedback gain matrix K is tuned with Lg neglected. Thisis due to the fact that the influence of the increment of theequivalent grid-side inductance L2 can be compensated usingthe phase-locked loop circuit whereas the induced voltage of theLg is included to the grid voltage. In addition, the synthesizedgrid-side voltage vector can be estimated by the same methodas shown in Fig. 10 because its waveform remains sinusoidal.

V. CONCLUSION

In this paper, a novel deadbeat control algorithm with asettling time of three sampling periods is derived from thediscretized time domain instead of the s-domain in the grid-connected inverter through the LCL-type filter. The effective-ness of using grid current as a feedback signal in the proposedcontrol system was shown to be experimentally validated interms of lowered THD, attenuated inverter switching harmon-ics, and dampened resonance. In fact, the grid current feedbackwas observed to significantly reduce the THD of the gridcurrents by 2% more than that realized when employing an in-verter current feedback. Additionally, analysis of the proposeddeadbeat current control system, based on the locations of thecharacteristic roots in the z plane, has demonstrated it to be bothhighly responsive and exceedingly stable.

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Katsumi Nishida (M05) was born in Yamaguchi,Japan, in 1954. He received the B.S. and M.S. de-grees in electrical engineering from Tokyo Instituteof Technology, Tokyo, Japan, in 1976 and 1978,respectively, and the Ph.D. degree in electrical en-gineering from Yamaguchi University, Ube, Japan,in 2002.

He is currently a Professor with the Departmentof Electrical Engineering, Ube National College ofTechnology, Ube. His research interests are in the de-sign and control of power conditioners with deadbeat

technique and adaptive signal processing technique.Prof. Nishida is a member of the Institute of Electrical Engineers of Japan,

the Power Electronics Society of Japan, and the Institute of Installation Engi-neers of Japan. He was the recipient of best paper awards from the 8th and the10th IEEE International Conference on Power Electronics and Drive Systemsin 2009 and 2013, respectively.

Tarek Ahmed (S03M06) received the Ph.D. de-gree in electrical engineering from Yamaguchi Uni-versity, Ube, Japan, in 2006.

He had conducted research within the RenewableEnergy Research Group at the University of Exeterin the area of wave energy development to create amajor grid-connected project off the north coast ofCornwall, U.K., in 2009. He is currently an Asso-ciate Professor with the Electrical Engineering De-partment, Faculty of Engineering, Assiut University,Assiut, Egypt. He joined the Assiut engineering fac-

ulty in 1995, as an Instructor. In 1998, on the completion of his masters degree,he was made a Teaching Assistant in the Department of Electrical Power andMachines within the same faculty. His research interests are in power electron-ics and electric machine control for grid integration of renewable energy.

Dr. Ahmed was the recipient of best paper awards from the Institute ofElectrical Engineers of Japan in 2003, 2004, and 2005, best paper and studentawards from the 30th Annual Conference of the IEEE Industrial ElectronicsSociety in 2004, a best paper award from the International Conference onElectrical Machines and Systems in 2004, and best paper awards from the 8thand the 10th IEEE International Conference on Power Electronics and DriveSystems in 2009 and 2013, respectively.

Mutsuo Nakaoka (M83) received the Ph.D. de-gree in electrical engineering from Osaka University,Suita, Japan, in 1981.

He joined the Department of Electrical and Elec-tronics Engineering, Kobe University, Kobe, Japan,in 1981. Since 1995, he has been a Professor withthe Graduate School of Science and Engineering,Yamaguchi University, Ube, Japan. He is currentlya Visiting Professor with the Electrical Energy Sav-ing Research Center, Kyungnam University, Masan,Korea. His research interests include state-of-the-art

power electronics circuits and systems engineering.Prof. Nakaoka is a member of the Institute of Electrical Engineers of Japan,

the Institute of Electronics, Information, and Communication Engineers ofJapan, the Institute of Illumination Engineering of Japan, the Power ElectronicsSociety of Japan, and the Institute of Installation Engineers of Japan. From 2001to 2006, he served as a Chairman of the IEEE Industrial Electronics SocietyJapan Chapter. He was the recipient of many distinguished paper awards suchas the 2001 Premium Prize Paper Award from the Institution of ElectricalEngineers, U.K., the 2001/2003 IEEE IECON Best Paper Award, the ThirdPaper Award at the 2000 IEEE PEDS, the 2003 IEEE-IAS James Melcher PrizePaper Award, the Best Paper Award of IATC06, the IEEE-PEDS 2009 and2013 Best Paper Awards, and the IEEE-ISIE 2009 Best Paper Award.

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