stochastic weight trade-off particle swarm … valve point effects i. introduction optimal power...

Journal of Automation and Control Engineering Vol. 2, No. 1, March 2014

31doi: 10.12720/joace.2.1.31-37©2014 Engineering and Technology Publishing

Stochastic Weight Trade-Off Particle Swarm

Optimization for Optimal Power Flow

Luong Dinh Le and Loc Dac Ho Faculty of Mechanical-Electrical-Electronic, Ho Chi Minh City University of Technology, HCMC, Vietnam

Email: [email protected], [email protected]

Jirawadee Polprasert and Weerakorn Ongsakul Energy Field of Study, School of Environment, Resources and Development, Asian Institute of Technology,

Pathumtnani 12120, Thailand


Dieu Ngoc Vo and Dung Anh Le Department of Power Systems, Ho Chi Minh City University of Technology, HCMC, Vietnam


Abstract—This paper proposes a stochastic weight trade-off

particle swarm optimization (SWT-PSO) method solving

optimal power flow (OPF) problem. The proposed SWT-

PSO is a new improvement of PSO method using a

stochastic weight trade-off for enhancing search its search

ability. The proposed method has been tested on the IEEE

30 bus and 57 bus systems and the obtained results are

compared to those from other methods such as conventional

PSO, genetic algorithm (GA), ant colony optimization

(ACO), evolutionary programming (EP), and differential

evolution (DE) methods. The numerical results have

indicated that the proposed SWT-PSO method is better

than the others in terms of total fuel costs, total loss and

computational times. Therefore, the proposed SWT-PSO

method can be a favorable method for solving OPF

problem.

Index Terms—optimal power flow, particle swarm

optimization, stochastic weight trade-off, quadratic fuel

function, valve point effects

I. INTRODUCTION

Optimal power flow (OPF) problem is the important

fundamental issues in power system operation. In essence,

it is the optimization problem and its main objective is to

reduce the total generation cost of units while satisfying

unit and system constraints. Although the OPF problem

developed long time ago but so far it has been extensively

studied due to its importance in power system operation.

There have been many methods developed to solve OPF

problem from classical methods such as Newton’s

method, gradient search, linear programming (LP),

nonlinear programming, quadratic programming (QP), etc

to methods based on artificial intelligence and

evolutionary based methods such as ant colony

Manuscript received July 1, 2014; revised November 21, 2014.

optimization (ACO), genetic algorithm (GA), improved

evolutionary programming (IEP), tabu search (TS),

simulated annealing (SA), etc. These methods have been

effectively for solving the problem.

In 1995, Eberhart and Kennedy suggested a particle

swarm optimization (PSO) method based on the analogy

of swarm of bird flocking and fish schooling [1]. Due to

its simple concept, easy implementation, and

computational efficiency when compared with

mathematical algorithm and other heuristic optimization

techniques, PSO has attracted many attentions and been

applied in various power system optimization problems

such as economic dispatch [2]-[5], reactive power and

voltage control [6]-[8], transient stability constrained

optimal power flow [9], and many others [10], [11]-[13].

In this paper, a stochastic weight trade-off particle

swarm optimization (SWT-PSO) algorithm is proposed

by improvement of conventional PSO method with new

parameter for better optimal solution and faster

computation. The proposed SWT-PSO method has been

tested on the IEEE 30-bus system with quadratic fuel cost

function and fuel cost function with valve point effects

fuel function and the IEEE 57-bus system. The obtained

results are compared to those from many other methods

in the literature such as genetic algorithm GA [20], ant

colony optimization (ACO) [21], improved evolutionary

programming (IEP) [22], evolutionary programming (EP)

[23], gravitational search algorithm (GSA) [24],

differential evolution (DE–OPF) [25], modified

differential evolution (MDE–OPF) [25], base-case [28],

and Matpower [28].

II. OPTIMAL POWER FLOW PROBLEM

The OPF problem can be described as an optimization

(minimization) problem with nonlinear objective function


32©2014 Engineering and Technology Publishing

and nonlinear constraints. The general OPF problem can

be expressed as follows:

Minimize F (u, x) (1)

Subject to g (u, x) = 0 (2)

h (u, x) 0 (3)

where F (u, x) is the objective function, g (u, x) represents

the equality constraints, h (u, x) represents the inequality

constraints, and u is the vector of the control variables

such as generated active power, generation bus voltage

magnitudes, transformers taps, etc), and x is state

variables such as reactive power, load bus voltage

magnitude, bus voltage angle, etc).

The essence of the optimal power flow problem resides

in reducing the objective function and simultaneously

satisfying the load flow equations (equality constraints)

without violating the inequality constraints. The fuel cost

of generators in form of quadratic function is given by:

2

1

( ) ( )GN

i i Gi i Gi

i

F x a b P c P

(4)

where, NG is the number of generators including the slack

bus, PG is the generated active power at bus i, ai, bi and ci

are the unit costs curve for i

th generator.

The smooth quadratic fuel cost function without valve

point loadings of the generating units are given by (4),

where the valve-point effects are ignored. The generating

units with multi-valve steam turbines exhibit a greater

variation in the fuel-cost functions. Since the valve point

results in the ripples, a cost function contains higher order

nonlinearity. Therefore, the equation (4) should be

replaced by (5) for considering the valve-point effects.

The sinusoidal functions are thus added to the quadratic

cost functions as follows.

2

,min( ) sin( ( ))i i i i i i i i i i iF P a b P c P e f P P (5)

where ei and fi are the fuel cost coefficients of the ith

unit

with valve point effects. The shape of fuel cost function

with valve loading effects is given tin Fig. 1.

Figure 1. Example cost function with 6 valves [14]

While minimizing the cost function, it is necessary to

make sure that the generation still supplies the load

demands plus losses in transmission lines. Usually the

power flow equations are used as equality constraints

[14].

( , ) ( )0

( , ) ( )

i i

i i

i G Di

i i G D

P V P PP

Q Q V Q Q

(6)

where active and reactive power injection at bus i are

defined in the following equation

1

( , ) cos sinBN

i i j ij ij ij ij

j

P V VV G B

(7)

1

( , ) sin cosBN

i i j ij ij ij ij

j

Q V VV G B

(8)

The inequality constraints of the OPF reflect the limits

on physical devices in power systems as well as the limits

created to ensure system security. The most usual types

of inequality constraints are upper bus voltage limits at

generations and load buses, lower bus voltage limits at

load buses, reactive power limits at generation buses,

maximum active power limits corresponding to lower

limits at some generators, maximum line loading limits,

and limits on tap setting. The inequality constraints on the

problem are as follows:

Generation constraint: Generator voltages, real power

outputs, and reactive power outputs are restricted by their

upper and lower bounds:

,min ,maxGi Gi GiP P P

for i = 1, 2, . . . . . , NG (9)

,min ,maxGi Gi GiQ Q Q for i = 1, 2, . . . . . , NG (10)

,min ,maxGi Gi GiV V V for i = 1, 2, . . . . . , NG (11)

Shunt VAR constraint: Shunt VAR compensations are

restricted by their upper and lower bounds:

,min ,maxCi Ci CiQ Q Q

for i = 1, 2, . . . . . , NC (12)

where NC is the number of shunt compensators.

Tap changer constraint: Transformer tap settings are

restricted by their upper and lower bounds:

,min ,maxi i iT T T

for i = 1, 2, . . . . . , NT (13)

where NT is the number of transformer taps.

Security constraint: Voltage magnitudes at load buses

are restricted by their upper and lower bounds as follows

,min ,maxLi Li LiV V V for i = 1, 2, . . . . . , NL (14)

where NL is the number of load buses.



III. IMPROVEMENT OF PSO

A. Overview of the PSO

The conventional PSO was originally introduced by

Kennedy and Eberhart as an optimization technique

inspired by swarm intelligence such as bird flocking, fish

schooling, and even human social behavior. Particles

representing candidate solutions change their positions

with time through search space. During the flight, each

particle adjusts its position according to its own

experience and the experience of neighboring particles as

a constructive cooperation by making use of the best

positions encountered by itself and its neighbors [1]. The

position mechanism of the particles in the search space is

updated by adding the velocity vector to its position

vector as given in equation (20) and as illustrated in Fig.

2 [15]. Let Xi = (xi1,…, xin) and Vi = (vi1,…, vin) be particle

position and its corresponding velocity in a n-dimensional

search space, respectively. The best position achieved by

a particle is recorded and denoted

by i1( ,..., )Pbest Pbest

i inPbest x x . The best particle among all

particles in the population is represented as

i1( ,..., )Gbest Gbest

i inGbest x x . The updated velocity and

position of a particle can be calculated by:

1 1k k ki i iX X V (15)

where Vik+1

is the velocity of individual i at iteration k+1

given by:

11 1 2 2( ) ( )k k k k k k

i i i i iV V c r Pbest X c r Gbest X (16)

Xik position of individual i at iteration k,

Xik+1

position of individual i at iteration k+1,

Vik velocity of individual i at iteration k,

c1 cognitive factors,

c2 social factors,

Pbestik the best position of individual i until

iteration k, Gbest

k the best position of the group until

iteration k,

r1, r2 random numbers between 0 and 1.

Figure 2. Concept of a searching point by PSO [15]

B. Stochastic Weight Trade-off

During the study PSO algorithm we found that the

expression affects the ability of the algorithm

convergence mainly falling into the velocity updating

(16). In this expression, the two components including

the cognitive factor c1 and the social factor c2 are

independent on each other. If both coefficients are too

large or small, there will be effects the convergence of the

algorithm. In the case both coefficients are too large, the

search space is too far beyond the seek region, making

difficulty for the algorithm convergence while both

factors are too small, the search space is too narrow,

leading to inexact optimal results.

To solve these problems, we propose several

improvements to the PSO algorithm. Among the

improved PSO methods, the stochastic weight trade-off

PSO (SWT-PSO) [29] is proposed to solve the optimal

power flow problem due to the goal of balancing between

particle experiences and social relationships as follows:

1) Improvement of r1 and r2 coefficients

r1 and r2 coefficients are also the additional factors (1 -

r1) and (1-r2) as in expression (17). The terms (1-r2) r1

and (1-r1) r2 will improve the algorithm efficiency to

converge faster to the optimal solution. When both r1 and

r2 are too large or too small, the term (1-r2) r1 will lead to

an imbalance for the algorithm. It is similar to the case

for the term (1-r1) r2. This method enables the algorithm

to create a balance between the two components of

personal experiences and learning from the community.

2) Improvement c1 and c2 coefficients

The coefficients c1 and c2 now are not constant as the

original PSO algorithm. We are recommend the time

varying coefficients c1 (k) and c2 (k) as in (18) and (19).

The two expressions will create the value of c1 and c2

factors large at the initialization and decrease them until

the maximum number of iterations reached. When

starting, the algorithm searches for large space to put to

the best possible area. At the algorithm termination, the c1

and c2 factors guide the algorithm converge to the optimal

result.

1

1 2 1 1

1 2 2

(1 ) ( ) ( )

+ (1 ) ( ) ( )

k k k k

i i i i

k k

i

V rV r c k r Pbest X

r c k r Gbest X

(17)

1max 1min

1 1min

max

( )( )

c cc k c k

k

(18)

2max 2min

2 2min

max

( )( )

c cc k c k

k

(19)

r1, r2 random numbers between 0 and 1,

c1min and c1max initial and final cognitive factors,

c2min and c2max initial and final social factors,

kmax maximum iteration number,

k current iteration number.



C. SWT-PSO Procedure of OPF Problem

The implementation of SWT-PSO algorithm to solve

OPF problem can be described as follows:

Step 1:

Choose

the

population

size,

the

number

of

generations and

coefficients

c1min,

c1max,

c2min,

c2max.

Step 2:

Initialize

the

velocity

and

position

of

all

particles

by randomly

setting

their

values

within

the

pre-specified

boundaries. Set

the

value

of

particle

positions

to

Pbest

and the

particle

corresponding

to

the

best

case

to

Gbest.

Step 3:

Set

the

iteration

counter

k

=

1

and

particle

counter i =

1.

Step 4:

For

each

particle,

solve

AC

power

flow

using

Newton –

Raphson’s

method.

Step 5:

Evaluate

the

fitness

function

for

each

particle

according to

the

objective

function.

Step 6:

Compare

particle’s

fitness

evaluation

with

its

Pbesti. If

the

current

value

is

better

than

Pbesti,

set

Pbesti

to

the

current

value.

Identify

the

particle

with

the

neighborhood with

the

best

success

so

far,

and

assign

its

index to

Gbest.

Step 7:

Update

the

particle

velocity

by

using

the

global

best and

individual

best

of

each

particle

according

to

(17).

Step 8:

Update

particle

position

by

using

(15).

Step 9:

If

i <

total

number

of

particles,

i =

i +

1

and

return to

Step

4.

Step 10:

If

k <

Number

of

iterations,

set

i =

1

and

k =

k +

1, return

to

Step

4.

Step 11:

Stop

the

algorithm.

IV. NUMERICAL RESULTS

The proposed SWT-PSO method is tested on two

systems including the IEEE 30-bus system with quadratic

fuel function and valve point effects and the IEEE 57-bus

system with quadratic fuel function. The algorithm of the

SWT-PSO method is coded in Matlab platform and run

on a 2.5 GHz with 4 GB of RAM PC. The control

parameters of the SWT-PSO method for all test systems

are selected as follows: the cognitive and social

parameters are respectively set to c1(k) and c2(k) with

c1max = c2max

= 2.5, c1min = c2min

= 0.5, the velocity limit

coefficient is set to 0.15 (R = 0.15), the maximum

number of iterations ITmax is set to 200; the number of

particles Np is set to 15 for the IEEE 30-bus system and

25 for the IEEE 57-bus system. All penalty factors in the

fitness function are set to 106. For each system, the

proposed method is run 100 independent trials and the

obtained optimal results are compared to those from other

methods. The obtained results for the systems include

minimum total cost, power losses, and computational

time.

Case 1: The IEEE 30 bus system with quadratic fuel cost

function

TABLE I. OPTIMAL SOLUTION FOR THE IEEE 30-BUS SYSTEM WITH

QUADRATIC FUEL COST FUNCTION

Variable Min Max Optimal Solution

Pg1 (MW) 50 200 177.2529

Pg2 (MW) 20 80 48.3832

Pg5 (MW) 15 50 21.3497

Pg8 (MW) 10 35 21.4238

Pg11 (MW) 10 30 11.7042

Pg13 (MW) 12 40 12.0196

Vg1 (pu) 0.90 1.10 1.1000

Vg2 (pu) 0.90 1.10 1.0850

Vg5 (pu) 0.90 1.10 1.0520

Vg8 (pu) 0.90 1.10 1.0647

Vg11 (pu) 0.90 1.10 1.0857

Vg13 (pu) 0.90 1.10 1.0999

T11 (pu) 0.90 1.10 1.0317

T12 (pu) 0.90 1.10 0.9074

T15 (pu) 0.90 1.10 0.9865

T36 (pu) 0.90 1.10 0.9555

Qc10 (MVAr) 0 19 15.9942

Qc24 (MVAr) 0 4.3 4.2934

Ploss (MW) 8.7335

CPU Time (s) 12.184

Total Cost ($/h) 799.4637

To verify the feasibility of the proposed SWT-PSO

method, the standard IEEE 30-bus system [17] has been

used to test the OPF problem. The system line and bus

data are given in [18]. The system has six generators

located at buses 1, 2, 5, 8, 11, and 13 and four

transformers with off-nominal tap ratio in lines 6-9, 6-10,

4-12, and 28-27. The cost curve coefficients are given in

[19].

Table I shows the optimal dispatches of the generators.

Also note that all outputs of generator are within its

permissible limits. The obtained results of the SWT-PSO

are compared with those of other methods in Table II

including GA [20], ACO [21], IEP [22], and EP [23]. In

Table II, it is observed that SWT-PSO algorithm gives

better total cost than other methods in a fester manner.

These results have shown that the proposed method is

feasible and indeed capable of acquiring better solution.

Fig. 3 shows the convergence characteristic of SWT-PSO.

TABLE II. RESULT COMPARISON FOR THE IEEE 30 BUS SYSTEM WITH

QUADRATIC FUEL COST FUNCTION

Variable GA [20] ACO [21] IEP [22] EP [23] SWT-PSO

Pg1 (MW) 179.367 177.8635 176.2358 173.848 177.2529

Pg2 (MW) 44.24 43.8366 49.0093 49.998 48.3832

Pg5 (MW) 24.61 20.8930 21.5023 21.386 21.3497

Pg8 (MW) 19.90 23.1231 21.8115 22.630 21.4238

Pg11 (MW) 10.71 14.0255 12.3387 12.928 11.7042

Pg13 (MW) 14.09 13.1199 12.0129 12.000 12.0196

Ploss (MW) 9.5177 9.4616 9.5105 9.3700 8.7335

CPU Time (s) 315 20 99.013

(minutes) 51.4 12.184

Total Cost ($/h) 803.699 803.123 802.465 802.62 799.4637

Case 2: The IEEE 30-bus system with valve point effects

fuel function.

In this case, the generating units of buses 1 and 2 are considered to have the valve-point effects on their fuel cost characteristics. The fuel cost coefficients of these generators are taken from [28]. The fuel cost coefficients



of the remaining generators have the same values as of

the test Case 1.

Figure 3. Convergence characteristic with quadratic fuel function for the IEEE 30-bus system

Table III shows the generation outputs of the best

solution. We have also observed that the solution

obtained by SWT-PSO always satisfies the equality and

inequality constraints. The result comparison in Table IV

has indicated that the SWT-PSO algorithm gives better

results than other methods with the percentage as follow

IEP [22] 3.3%, GSA [24] 0.82%, DE–OPF [25] 0.96%,

MDE–OPF [25] 0.93%. Therefore, the proposed SWT-

PSO is very effective for solving the OPF problem with

valve point loading effects.

Case 3: The IEEE 57 bus system

To evaluate the effectiveness and efficiency of the

proposed SWT-PSO approach in solving larger power

system, a standard IEEE 57-bus test system is considered.

The IEEE 57-bus system consists of 7 generation buses,

50 load buses, and 80 branches. The generators are

located at buses 1, 2, 3, 6, 8, 9, and 12 and 15

transformers are located at branches 19, 20, 31, 37, 41, 46,

54, 58, 59, 65, 66, 71, 73, 76, and 80. The system has also

3 switchable capacitor banks installed at buses 18, 25,

and 53. For dealing with this system, there 31 control

variables to be handled including real power output of 6

generators except the generator at the slack bus, voltage

at 7 generation buses, tap changer of 15 transformers, and

reactive power output of 3 switchable capacitor banks.

The total load demand of system is 1250.8 MW and

336.4 MVAR. The bus data, line data, cost coefficients,

and minimum and maximum limits of real power

generations are taken from [26], [27]. The maximum and

minimum values for voltages of all generator buses and

transformer tap settings are considered to be 1.1 and 0.9

in p.u. The maximum and minimum values for voltages

of all load buses are 1.06 and 0.94 in p.u [24].

Table V shows the optimal solution of the problem by

the conventional PSO and SWT-PSO methods. The

minimum cost obtained by this algorithm is compared

with BASE-CASE [28], MATPOWER [28], and

conventional PSO as presented in Table VI. The result

comparison has demonstrated that the proposed method

can give the lowest production cost within reasonable

time. The convergence characteristic of the best fuel cost

result obtained from the SWT-PSO approach is shown in

Fig. 4.

TABLE III. OPTIMAL SOLUTION FOR THE IEEE 30-BUS SYSTEM

WITH VALVE POINT EFFECTS

Variable Min Max Optimal Solution

Pg1 (MW) 50 200 191.7845

Pg2 (MW) 20 80 51.0558

Pg5 (MW) 15 50 15.3289

Pg8 (MW) 10 35 10.0758

Pg11 (MW) 10 30 14.2326

Pg13 (MW) 12 40 12.0021

Vg1 (pu) 0.90 1.10 1.0660

Vg2 (pu) 0.90 1.10 1.0534

Vg5 (pu) 0.90 1.10 1.0226

Vg8 (pu) 0.90 1.10 1.0057

Vg11 (pu) 0.90 1.10 1.0848

Vg13 (pu) 0.90 1.10 1.0347

T11 (pu) 0.90 1.10 0.9969

T12 (pu) 0.90 1.10 0.9603

T15 (pu) 0.90 1.10 0.9794

T36 (pu) 0.90 1.10 0.9480

Qc10 (MVAr) 0 19 12.4459

Qc24 (MVAr) 0 4.3 2.9992

Ploss (MW) 11.0797

CPU Time (s) 9.766

Total Cost ($/h) 922.1029

TABLE IV. RESULTS FOR THE VALVE POINT EFFECTS FOR IEEE 30-BUS SYSTEM WITH DIFFERENT METHOD

Variable IEP [22] GSA [24] DE–OPF

[25]

MDE–OPF

[25] SWT-PSO

Pg1 (MW) 149.7331 199.5994 196.989 197.426 191.7845

Pg2 (MW) 52.0571 51.9464 51.995 52.037 51.0558

Pg5 (MW) 23.2008 15.0000 15.000 15.000 15.3289

Pg8 (MW) 33.4150 10.0000 10.006 10.000 10.0758

Pg11 (MW) 16.5523 10.0000 10.015 10.001 14.2326

Pg13 (MW) 16.0875 12.0000 12.000 12.000 12.0021

Ploss (MW) 7.6458 15.1458 12.605 13.064 11.0797

CPU Time

(s)

93.583

(minutes) 9.8374 44.96 41.85 9.766 (s)

Total Cost

($/h) 953.573 929.7240 931.085 930.793 922.1029

TABLE V. OPTIMAL SOLUTIONS FOR THE IEEE 57-BUS SYSTEM

Variable PSO SWT-PSO Variable PSO SWT-PSO

Pg1 (MW) 139.1571 146.3227 T31 (pu) 1.00 1.0387

Pg2 (MW) 100.0000 78.5966 T37 (pu) 1.05 0.9734

Pg3 (MW) 75.8451 45.3091 T41 (pu) 0.99 0.9673

Pg6 (MW) 38.4932 71.2516 T46 (pu) 0.92 1.0544

Pg8 (MW) 455.5600 461.2314 T54 (pu) 0.99 1.0155

Pg9 (MW) 100.0000 86.5425 T58 (pu) 0.99 0.9561

Pg12 (MW) 360.2540 377.1499 T59 (pu) 0.95 0.9772

Vg1 (pu) 1.0399 1.0593 T65 (pu) 0.98 0.9674

Vg2 (pu) 1.0319 1.0477 T66 (pu) 1.02 0.9430

Vg3 (pu) 1.0378 1.0379 T71 (pu) 0.90 0.9776

Vg6 (pu) 1.0621 1.0596 T73 (pu) 1.00 0.9654

Vg8 (pu) 1.1000 1.0688 T76 (pu) 1.01 1.0246

Vg9 (pu) 1.0369 1.0402 T80 (pu) 0.97 1.0010

Vg12 (pu) 0.9892 1.0440 Qc18 (MVAr) 3.5 7.7338

T19 (pu) 1.02 0.9993 Qc25 (MVAr) 3.0 3.3608

T20 (pu) 1.04 0.9651 Qc53 (MVAr) 3.3 3.8498



TABLE VI. RESULT COMPARISON FOR THE IEEE 57-BUS SYSTEM

Methods Total cost

($/h) Ploss (MW)

CPU time (s)

BASE-CASE [28] 51347.86 - -

MATPOWER [28] 41737.79 - -

PSO 42109.7231 1786.3245 8.814

SWT-PSO 41733.4425 15.6038 22.511

Figure 4.

Convergence characteristic

for

the

IEEE

57-bus

system

V.

CONCLUTION

In this

paper,

the

stochastic

weight

trade-off

particle

swarm optimization

method

has

been

presented

to

solve

the

OPF

problem.

The

improved

SWT-PSO

has

advantages

such

as

simple

algorithm

and

easy

to

use.

Moreover,

the

algorithm

can

be

implemented

in

the

whole problem

search

space

rather

than

individual

points,

leading faster

position

updating

function

of

particles.

The

proposed method

has

been

tested

on

the

IEEE

30

bus

and

57 bus

systems

and

the

obtained

results

are

compared

to

those

from

many

other

methods

in

the

literature.

The

numerical

results

show

the algorithm’s

flexibility

and

capability in

finding

the

optimal

solution.

Therefore,

the

proposed SWT-PSO

can

be

very

favorable

for

solving

OPF

problem,

especially

for

large

scale

systems

with

nonconvex objective

function.

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R.

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Networks,

Perth,

Australia, vol.

IV,

1995,

pp.

1942-1948.

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N. Sinha,

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