Electrical Power and Energy Systems 63 (2014) 1008–1014

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Optimal Control of Active Power Filters using Fractional OrderControllers Based on NSGA-II Optimization Method

http://dx.doi.org/10.1016/j.ijepes.2014.06.0630142-0615/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: nikkhah_hoda@yahoo.com (H. Nikkhah Kashani).

Hoda Nikkhah Kashani ⇑, S.M.R. RafieiDepartment of Electrical and Robotic Engineering, Garmsar Branch, Islamic Azad University of Iran, Garmsar, Iran

a r t i c l e i n f o

Article history:Received 24 August 2013Received in revised form 27 June 2014Accepted 28 June 2014Available online 26 July 2014

Keywords:Active Power FilterFractional CalculusFractional Order PI controllerRepetitive ControllerMulti-objective optimization method(NSGA-II)

a b s t r a c t

Elitist Non-Dominated Genetic Algorithm (NSGA-II) optimization method offers optimal solution to mul-tidimensional objective functions. In this paper, this optimization method is used for designing FractionalOrder PI controller that features a better performance than the Integer Order PI controller for improvingthe performance of a Shunt Active Power Filter. Controller synthesis is based on required Total HarmonicDistortion and transient specifications. The characteristic equation is minimized to obtain an optimal setof controller variables. This process is done by using the Integer Order PI controller and Fractional OrderPI controller with the high performance Repetitive Controller. Simulation results are obtained by usingthe Fractional Order PI controller are better than the ones are obtained by using the Integer Order PIcontroller.

� 2014 Elsevier Ltd. All rights reserved.

Introduction

Nowadays, with the wide use of nonlinear loads and electronicequipment in distribution systems, the problem of power quality(PQ) has become increasingly serious. This fact has lead to morestringent requirements regarding PQ which include the searchfor solutions for such problems [5,16]. Active Power Filters aredevices which are designed to improve the power quality in distri-bution networks. In order to reduce the injection of non sinusoidalload currents, Shunt Active Power Filters (APFs) can be connectedin parallel with the nonlinear loads.

In the case of harmonic distortion, the Shunt Active Power Filter(APF) appears as the best dynamic solution for harmonic compen-sation. This paper offers a good way to optimize the performance ofa Shunt Active Power Filter. Good performance means having anappropriate transient response and an appropriate steady-stateresponse. Low settling time and low transient time includes a goodtransient response and a low THD includes good steady-stateresponse. Here, the goal is to reduce settling time, transient timeand THD by using Fractional Order PI controller based onNSGA-II multiobjective optimization method applying the highperformance Repetitive Controller. This paper proposes an opinionof a designing method of the Fractional Order PI (FOPI) controllerfor Shunt Active Power Filter by using the Elitist Non Dominated

Genetic Algorithm (NSGA-II) as a powerful multi-objective optimi-zation approach for the design of a FOPI controller, also this paperpresents the performance of the Repetitive Controller. In thisresearch THD and settling time, THD and Transient time are twopairs objective functions in a Shunt Active Power Filter, settlingtime and transient time can affect on transient response, that eachpair of objective functions have been selected to be minimizedsimultaneously. The result of this optimization is a set of optimalsolutions, that it is called Pareto Optimal Set (POS), that in thispaper, POS for FOPI controller of Repetitive Controller has theparameters of Kp, Ki, Vi or a, Vdc. The obtained different values fromPOS members are known as Pareto Optimal Front (POF) that thePOF is corresponding to the values of the objective functions, andfor the different values of each pair of the objective functions inthis research, values of POS members are different.

Fractional systems

Fractional-Order systems are known by fractional-order equa-tions. The FOPID controller is a Fractional Order System. FOPIDcontroller is an English acronym which means Fractional-OrderProportional-Integral-Derivative controller.

Fractional Calculus

Fractional Calculus is a branch of mathematics that dealing withreal number powers of differential or integral operators. It

H. Nikkhah Kashani, S.M.R. Rafiei / Electrical Power and Energy Systems 63 (2014) 1008–1014 1009

generalizes the common concepts of derivative and integral. Thereare several definitions for fractional order derivatives. The defini-tion which has been proposed by Riemann and Liouville is the sim-plest and easiest definition to use [19]. The definition is given byEq. (1):

cD�nx f ðxÞ ¼

Z x

c

ðx� tÞn�1

ðn� 1Þ! f ðtÞdt; n 2 R ð1Þ

The general definition of D is given by Eq. (2):

cDvx f ðxÞ ¼

R xcðx�tÞ�v�1

Cð�vÞ f ðtÞdt; if v < 0

f ðxÞ if v ¼ 0Dn½cDv�n

x f ðxÞ� if v > 0

8>><>>:

n ¼minfK 2 R; K > vg

ð2Þ

And C(�) is the gamma function.

Approximation of fractional order derivative and integral

The approximation method which is considered in this paper isbased on the approximation of the fractional-order system behav-ior in the frequency domain. For the frequency-domain transferfunction C(s) which is given by Eq. (3):

CðsÞ ¼ Ksv v 2 R ð3Þ

One of the best-known approximations is due to Oustaloup andis given by Oustaloup [13] The Crone method, which uses a recur-sive distribution of N poles and N zeros. Crone is a French acronymwhich means ‘robust fractional order control’ [8]. Transfer functionis given by Eq. (4):

CðsÞ ¼ K 0YN

n¼1

1þ sxzn

1þ sxpn

ð4Þ

where K0 is adjusted so that if K is 1 then |C(s)| = 0 dB at 1 rad/s.Zeros and poles are found inside a frequency interval [xl; xh] wherethe approximation is valid, and they are given, for a positive, by Eqs.(5)–(7).

xz1 ¼ xlffiffiffigp ð5Þ

xpn ¼ xz;n�1 / n ¼ 1 . . . N ð6Þxzn ¼ xp;n�1g n ¼ 2 . . . N ð7Þ

where a and g can be obtained by Eqs. (8) and (9).

/¼ xh

xl

� �vN

ð8Þ

g ¼ xh

xl

� �1�vN

ð9Þ

For a negative v the role of zeros and poles is interchanged. Thenumber of poles and zeros is selected at first and the desired per-formance of this approximation depends on the order N. Simpleapproximation can be provided with lower order N, but it cancause ripples in both gain and phase characteristics. When |v| > 1,the approximation is not satisfactory. The fractional order v usuallyis separated as Eq. (10) and only the first term sb needs to beapproximated [8].

sv ¼ sbsn; v ¼ nþ b; n 2 R; b 2 ½0;1� ð10Þ

Fractional-Order Proportional-Integral-Derivative controller

In recent years, researchers found that controllers making useof factional-order derivatives and integrals could achieve betterperformance and robustness than those with conventional

integer-order controllers [7]. The fractional-order PID controlleris more flexible and can provide an opportunity to better adjustthe dynamical characteristics of the control system [9]. This con-troller which first time was proposed by Podlubny in 1999, is theexpansion of the conventional PID controller based on FractionalCalculus [2,4]. The most common form of a Fractional Order PIDcontroller is the PIaDb controller [4], involving an integrator oforder / and a differentiator of order b where / and b can beany real numbers. For this controller, besides selecting Kp, Ki,and Kd, it needs to select a and b which are not necessarily inte-ger numbers [6]. It can be expected that the PI/Db controller mayenhance the systems control performance. One of the mostimportant advantages of the PI/Db controller is the better controlof dynamical systems, which are described by fractional ordermathematical models. Another advantage lies in the fact thatthe PI/Db controllers are less sensitive to changes of parametersof a controlled system [18]. The transfer function of such a con-troller is given by Eq. (11):

GcðsÞ ¼ Kp þ Kis�a þ Kdsb ð11Þ

The calculation of the five parameters that can be done throughoptimization makes the design scenario of an FOPID controllermore challenging than conventional PID controllers. Several meth-ods have been proposed for this design by using optimizationmethods [19,2,6,12]. Since the design of FOPID can be consideredas a parameter-optimization problem and the features of our sys-tem are conflicting, we used a multi-objective optimizationmethod called Non-Dominated Sorting Genetic Algorithm (NSGA-II) for this problem. In this paper, Kd and b are zero.

Multi-objective optimization

Optimization methods

In the optimization problems of control systems, normallysimultaneous optimization of different and often conflicting objec-tives is needed [8]. In this optimization case, there is a set of opti-mal solutions that this set is called Pareto Optimal Set (POS). Eachpoint in this set is optimal in the sense that no improvement can beachieved on one optimization objective that does not lead to deg-radation in at least one of the remaining objectives. In the absenceof any further information, none of these POS members could beconsidered as being better than the others [3]. In this paper, POSfor FOPI controller of Repetitive Controller has the followingparameters:

Kp; Ki; a; Vdc:

The obtained different values from POS members are known asPareto Optimal Front (POF), the POF is corresponding to the valuesof the objective functions.

The general multi-objective optimization problem is posed asfollows:

Where x is called design

Minimizeg ¼ f ðxÞ ¼ ðf1ðxÞ; . . . ; fiðxÞ; . . . ; fkðxÞÞSubject To :

x ¼ ðx1; x2; . . . ; xnÞ 2 X

ð12Þ

where k is the number of objective functions, n is the number ofinequality constraints, and e is the number of equality constraints.x is a vector of design variables, and f(x) is a vector of objective func-tions to be minimized.

1010 H. Nikkhah Kashani, S.M.R. Rafiei / Electrical Power and Energy Systems 63 (2014) 1008–1014

Hence, in this paper for first case study the goal is the following:

Minimizeg ¼ f ðxÞ ¼ ðTHDðup to the 50th harmonicÞ; settling timeÞSubject To :

x ¼ ðKp; Ki; a; VdcÞ 2 X for FOPI controller;x ¼ ðKp; Ki; VdcÞ 2 X for Iteger Order PI controller

ð13Þ

Also, for second case study the goal is the following:

Minimizeg¼ f ðxÞ¼ ðTHDðup to the 50th harmonicÞ; transient timeÞSubject To :

x¼ðKp; Ki; a; VdcÞ 2X for FOPI controller;x¼ðKp; Ki; VdcÞ 2X for Iteger Order PI controller

ð14Þ

Elitist Non-Dominated Genetic Algorithm (NSGA-II)

The non-dominated sorting GA (NSGA) proposed by Srinivasand Deb in 1994 has been applied to various problems [17,11].However as mentioned earlier there have been a number of criti-cisms of the NSGA. In this section, we modify the NSGA approachin order to alleviate all the above difficulties. We begin by present-ing a number of different modules that form part of NSGA-II. In thismethod, initially, a random parent population p0 is created. Thepopulation is sorted based on the non-domination. Each solutionis assigned a fitness equal to its non-domination level (1 is the bestlevel). Thus, minimization of fitness is assumed. Binary tourna-ment selection, recombination, and mutation operators are usedto create a child population Q0 of size N. From the first generationonward, the procedure is different. The elitism procedure for t P 1and for a particular generation is shown in the following:

Rt ¼ Pt [ Qt

Combine parent andchildren population

F ¼ fast� nondominated� sort ðRiÞ

F ¼ ðF1; F2; . . .Þ, allnon-dominated Fronts of Ri

Until jPt þ 1j < N

Till the parentpopulation is filled

Crowding – distance – assignmentðF iÞ S

Calculate crowdingdistance in F i

Ptþ1 ¼ Ptþ1 F i

Include i-th non-dominated front in theparent pop

SortðPtþ1;PnÞ

Sort in descendingorder using Pn

Ptþ1 ¼ Ptþ1 [0:N]

Choose the first Nelements of Ptþ1

Qtþ1 ¼make – new – pop ðPtþ1Þ

Use selection,crossover and mutationto create

t ¼ t þ 1

A new population Qtþ1

First, a combined population Rt ¼ Pt [ Q t is formed. The popula-tion Rt will be of size 2N. Then, the population Rt is sorted accordingto non-domination. The new parent population Pt+1 is formed byadding solutions from the first front till the size exceeds N. There-after, the solutions of the last accepted front are sorted accordingto P n and the first N points are picked. This is how we constructthe population Pt+1 of size N. This population of size N is now usedfor selection, crossover and mutation to create a new populationQt+1 of size N. It is important to note that we use a binary tourna-ment selection operator but the selection criterion is now based onthe niched comparison operator P n [1].

System under study and objective functions

In this study, a Shunt Active Power Filter (Fig. 1) operates as acurrent source injecting the harmonic components generated bythe load but phase shifted by 180� this filter can be connected inparallel with the nonlinear load with a high performance Repeti-tive Controller (Fig. 2) and a FOPI controller.

Repetitive Controller

This current-control strategy uses a repetitive-based controlleralong with a PI (or FOPI) controller, as shown in Fig. 2. The Repet-itive Controller uses a discrete Fourier transform (DFT), which hasa frequency response approximately equal to the frequencyresponse in tracking the harmonic reference (Fig. 2) [14]. The dis-crete transfer function of this DFT is given by Eq. (13):

FDFTðZÞ ¼2N

XN�1

i¼0

Xk2Nh

cos2pN

hðiþ NaÞ� �0

@1AZ�i ð15Þ

where N is the number of the coefficients, Nh is the set of selectedharmonic frequencies and Na the number of leading steps whichis necessary to guarantee the system stability [14]. This controllerfeatures a high tracking operation, but its operation is inherentlyslow. Hence, it would be of interest to minimize the transientresponse of the compensator as an objective function. Some objec-tive functions chosen to be minimized are:

- THD (up to the 50th harmonic)The term THD means Total Harmonic Distortion and is a widely

used notion in defining the level of harmonic content in alternatingsignals.

- Transient response (rise time or settling time)

Rise time: the time required for a signal (as on an oscilloscope)to increase from one specified value (as 10%) of its amplitude toanother (as 90%).

Settling time: the time elapsed until the step response enters(without leaving it afterwards) a specified tolerance band, aroundthe final value. This tolerance band is usually defined as a percent-age of the final value, say usually 5% or 2%.

Also, the set of design variables to minimize the objective func-tions has been chosen as follows:

Design variables: DC bus voltage (Vdc) and the PI controllerparameters Kp and Ki in PI controller (Fig. 2) also a in FOPIcontroller.

The DC bus voltage can affect both the transient and steadystate characteristics of the system. Hence, it has been selected asa design variable [15].

Case studies

The following two cases have been considered to be minimizedsimultaneously:

Case 1: simultaneously minimizing THD (up to the 50thharmonic) and settling time.

Case 2: simultaneously minimizing THD (up to the 50thharmonic) and transient time.

Simulation results

The active power filter shown in Fig. 1 has been modeled inMATLAB/SIMULINK environment. The NSGA-II algorithm used inthis study has been run for 15 generations with each generation

Fig. 1. Block diagram of the Active Power Filter [10].

Fig. 2. Repetitive Control scheme [10].

3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

0.04

0.045

0.05

0.055

0.06

0.065

0.07FOPI − Case1

THD (%)

Settl

ing

Tim

e (s

ec.)

1

2

Fig. 3. Pareto Front for Fractional Order PI (optimization of THD (up to the 50thharmonic) and settling time).

Table 1Pareto Set and Front for Fractional Order PI, Optimization of THD (up to the 50thharmonic) and settling time.

No Kp Ki Vi or a Vdc THD (%) Ts (s)

1 0.8461 112.9690 1.2292 785.1798 3.8039 0.06612 1.5691 492.2483 0.7059 778.1098 4.2512 0.03993 0.9757 186.9696 0.8196 786.5418 3.9804 0.05074 1.6451 302.1049 0.9194 763.7058 4.4802 0.03895 1.6931 410.1851 1.9106 688.9375 4.3666 0.03906 2.1345 424.4429 1.7104 768.1365 4.5736 0.0389

H. Nikkhah Kashani, S.M.R. Rafiei / Electrical Power and Energy Systems 63 (2014) 1008–1014 1011

including 30 individuals. For two the cases the rest of parametersfor the Repetitive Controller are: N = 200, Na = 3, and kf = 1 [14].For FOPI controller, the Crone approximation with order 5 and fre-quency range equals [0.01; 1,000,000] rad/s is used.

Case 1. minimizing THD (up to the 50th harmonic) and settlingtime for FOPI controller and Integer Order PI controller thencomparison the results of them.

In this case the following functions have been selected to beminimized simultaneously:

a. THD (up to the 50th harmonic) of the source current.b. Settling time of the current (Ts).

In this case study, it is necessary achieve a low THD. The currentTHD represents the steady-state behavior of the system understudy. Also, a low transient response is very important for compen-sation. In this case the transient response involves the settlingtime. The settling time used in this optimization case has been cal-culated based on the time required for the current THD to reachand remain inside a ± 5% error band around its steady-state value[15]. For the NSGA-II approach, each generation has been consid-ered composed by 15 members. Fig. 3 shows the Pareto Front after

15 generations for Fractional Order PI controller. It confirms that,there are several options to select – some with good THD and somewith good settling time [15].

Table 1 for the FOPI controller shows the values of Pareto Set(Kp, Ki, Vi and Vdc) and Pareto Front (THD and settling time) for thiscase study represented in Fig. 3.

Table 2Pareto Set and Front for Integer Order PI, optimization of THD (up to the 50thharmonic) and settling time.

No Kp Ki Vdc THD (%) Ts (s)

1 1.3490 331.5996 704.6626 4.0351 0.07232 0.7115 426.5628 745.4438 3.9454 0.07723 2.5084 87.5634 789.7689 4.4995 0.03974 0.7885 106.3299 692.7906 4.1059 0.05845 1.7771 292.0801 719.4180 4.3450 0.04326 0.6508 87.2712 730.1335 3.9669 0.0757

3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08

THD (%)

Settl

ing

Tim

e (s

ec.)

PIFOPI

Fig. 5. Comparison between Pareto Front of Integer Order PI and Fractional Order PI(optimization of THD (up to the 50th harmonic) and settling time).

1012 H. Nikkhah Kashani, S.M.R. Rafiei / Electrical Power and Energy Systems 63 (2014) 1008–1014

Some interesting information can be obtained from the front,represented in Fig. 3 and Table 1 for the FOPI controller:

- A compromise can be found between the transient characteris-tics and THD that is two different goals in the best way [15].

- Point 1 in Fig. 3 (row No. 1 of Table 1) shows that in order toachieve a very low THD (near 3.8039%) the settling time (Ts)has been increased up to 0.0661 s.

- Point 2 in Fig. 3 (row No. 6 of Table 1) shows that in order toachieve a very low Ts (near 0.0389 s) the THD has beenincreased up to 4.5736%.

Fig. 4 shows the Pareto Front after 15 generations for IntegerOrder PI controller.

Table 2 shows the numerical data of Pareto Set (Kp, Ki, and Vdc)and Pareto Front (THD and settling time) for this case study repre-sented in Fig. 4.

- Point 3 in Fig. 4 (row No. 2 of Table 2) shows that in order toachieve a very low THD (near 3.9454%) the settling time (Ts)has been increased up to 0.0772 s.

- Point 4 in Fig. 4 (row No. 3 of Table 2) shows that in order toachieve a very low Ts (near 0.0397 s) the THD has beenincreased up to 4.4995%.

Fig. 5 shows a comparison between Pareto Front of IntegerOrder PI and Fractional Order PI (optimization of THD (up to the50th harmonic) and settling time), as it can be seen in Fig. 5. Thevalues of Fractional Order PI controller dominated all the valuesof Integer Order PI controller. FOPI controller has featured betterresults in the first case study.

Case 2: minimizing THD (up to the 50th harmonic) and tran-sient time.

Rise time and settling time can affect on the transient response.Also in this case, it has been regarded to reach a low transient timeand a current THD less than 5%. The objective functions are risetime and THD for up to the 50th harmonic:

a. THD (up to the 50th harmonic) of the source current.b. Transient time (Tr).

3.9 4 4.1 4.2 4.3 4.4 4.5

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

0.08PI − Case1

THD (%)

Settl

ing

Tim

e (s

ec.)

3

4

Fig. 4. Pareto Front for Integer Order PI (optimization of THD (up to the 50thharmonic) and settling time).

Table 3 and Fig. 6 show the optimization results after 15 gener-ations for Fractional Order PI controller.

- Point 1 in Fig. 6 (row No. 1 of Table 3) shows that in order toachieve a very low THD (near 3.8932%) the transient time (Tr)has been increased up to 0.0469 s.

- Point 2 in Fig. 6 (row No. 5 of Table 3) shows that in order toachieve a very low Tr (near 0.0379 s) the THD has beenincreased up to 4.2548%.

Table 4 and Fig. 7 show the optimization results after 15 gener-ations for Integer Order PI controller.

- Point 3 in Fig. 7 (row No. 3 of Table 4) shows that in order toachieve a very low THD (near 3.9454%) the transient time (Tr)has been increased up to 0.0491 s.

- Point 4 in Fig. 7 (row No. 9 of Table 4) shows that in order toachieve a very low Tr (near 0.0382 s) the THD has beenincreased up to 4.5032%.

Table 3Pareto Set and Front for Fractional Order PI, optimization of THD (up to the 50thharmonic) and transient time.

No Kp Ki Vi or a Vdc THD (%) Tr (s)

1 0.9365 90.4301 0.2550 746.6514 3.8932 0.04692 0.9398 81.9984 0.8491 672.4850 4.0240 0.04533 1.4212 193.0049 0.5425 661.0839 4.1409 0.03874 1.8016 150.5031 1.8718 715.7803 4.1472 0.03875 2.1499 280.8287 1.4114 709.6378 4.2548 0.03796 1.2415 148.6682 0.2348 703.5407 4.0733 0.0389

3.85 3.9 3.95 4 4.05 4.1 4.15 4.2 4.25 4.30.036

0.038

0.04

0.042

0.044

0.046

0.048FOPI − Case2

THD (%)

Tran

sien

t Tim

e (s

ec.)

1

2

Fig. 6. Pareto Front for Fractional Order PI (optimization of THD (up to the 50thharmonic) and transient time).

Table 4Pareto Set and Front for Integer Order PI, optimization of THD (up to the 50thharmonic) and transient time.

No Kp Ki Vdc THD (%) Tr (s)

1 0.9247 377.2732 812.1734 3.9860 0.04892 1.7594 167.9010 740.7735 4.3496 0.03873 0.7115 426.5628 745.4438 3.9454 0.04914 1.4006 168.7666 652.3901 4.2501 0.03885 1.5564 305.5118 714.9233 4.3034 0.03886 1.6214 327.2613 743.5220 4.1128 0.03887 1.0316 170.5051 791.3396 4.0173 0.04548 1.6834 175.1086 695.5721 4.3558 0.03869 2.1272 326.5600 709.0945 4.5032 0.0382

3.8 3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.70.036

0.038

0.04

0.042

0.044

0.046

0.048

0.05

THD (%)

Tran

sien

t Tim

e (s

ec.)

PIFOPI

Fig. 8. Comparison between Pareto Front of Integer Order PI and Fractional Order PI(optimization of THD (up to the 50th harmonic) and transient time).

H. Nikkhah Kashani, S.M.R. Rafiei / Electrical Power and Energy Systems 63 (2014) 1008–1014 1013

Fig. 8 shows a comparison between Pareto Front of IntegerOrder PI and Fractional Order PI (optimization of THD (up to the50th harmonic) and rise time), as it can be seen in Fig. 8. The valuesof Fractional Order PI controller dominated all of the values of Inte-ger Order PI controller. Also, FOPI controller has featured betterresults in second case study.

3.9 4 4.1 4.2 4.3 4.4 4.5 4.6 4.70.038

0.04

0.042

0.044

0.046

0.048

0.05

0.052PI − Case2

THD (%)

Tran

sien

t Tim

e (s

ec.)

3

4

Fig. 7. Pareto Front for Integer Order PI (optimization of THD (up to the 50thharmonic) and transient time).

Conclusion

In this paper, a new method for designing of Fractional Order PIcontroller for Active Power Filter was presented. The obtained simu-lation results using a Fractional Order PI controller are better thanthe obtained results by using an Integer Order PI controller. Thus,The FOPI controller can provide much better Transient response(Ts, Tr) and steady-state response (THD) than the Integer Order PIcontroller. In this paper, using FOPI controller in a Repetitive Control-ler based on NSGA-II of multiobjective optimization approach wasproposed that showed this method is a resultful and useful method.

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