Φ

Abstract— This paper presents a simple Differential Evolution (DE) Optimization algorithm to solving security constrained optimal power flow (OPF) with consideration of shunt FACTS Controllers under contingent operation states. Fuel cost minimization, voltage profile, and system loadability improvement are three objective function used to test the efficiency of the proposed strategy. The proposed approach is examined and tested on the standard IEEE-30Bus test system with different objective functions and under sever loading conditions. In addition, the non smooth cost function due to the effect of valve point loading has been considered within the second practical network test (40 generating units). The simulation results compared with the other recent techniques. From the different case studies, it is observed that the results demonstrate the potential of the proposed approach and show clearly its effectiveness to solve practical OPF under contingency situations.

Key Words— Optimal power flow, Differential Evolution, Multi objective, Valve point effect, FACTS, SVC, System loadability.

I. INTRODUCTION

The optimal power flow (OPF) problem is one of the important problems in operation and control of large modern power systems. The main objective of a practical OPF strategy is to determine the optimal operating state of a power system by optimizing a particular objective while satisfying certain specified physical and security constraints. In its most general formulation, the optimal power flow (OPF) is a nonlinear, non-convex, large-scale, static optimization problem with both continuous and discrete control variables. It becomes even more complex when various types of practical generators constraints are taken in consideration, and with the growth integration of new technologies known as Renewable source and FACTS Controllers. The first category: many conventional optimization techniques have been applied to ΦBelkacem Mahdad is with the Department of Electrical Engineering, University of Biskra Algeria (e-mail: bemahdad@yahoo.fr).

solve the OPF problem, this category includes, linear programming (LP) [1], nonlinear programming (NLP) [2], quadratic programming (QP) [3], and interior point methods [4]. These mathematical optimization techniques fail to deal with real systems having complex objective functions. Author in [5] presents a review of the major contributions in this area. During the last two decades, the interest in applying global optimization methods in power system field has grown rapidly. The second category includes many heuristique and stochastic optimization methods known as Global Optimization Techniques. Author in [6] represents the major contributions in this area. In [7] author presents an improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels. In [8] authors present a novel string structure for solving the economic dispatch through genetic algorithm (GA). To accelerate the search process authors in [9] proposed a multiple tabu search algorithm (MTS) to solve the dynamic economic dispatch (ED) problem with generator constraints, simulation results prove that this approach is able to reduce the computational time compared to the conventional approaches. Authors in [10] present an algorithm based simulated annealing to solve the optimal power flow. Author in [11] present a particle swarm optimization to solving the economic dispatch with consideration of practical generator constraints, the proposed algorithm applied with success to many standard network. Based on experience and simulation results, these classes of methods do not always guarantee global best solutions. Differential evolution (DE) is one of the most prominent new generation EAs, proposed by Storn and Price [13], to exhibit consistent and reliable performance in nonlinear and multimodal environment [14] and proven effective for constrained optimization problems. The main advantages of DE are: simple to program, few control parameters, high convergence characteristics. In power system field DE has received great attention in solving economic power dispatch (EPD) problems with consideration of discontinuous fuel cost functions. The third category includes, a variety of combined methods based conventional (mathematical methods) and global optimization techniques like (GA-QP), artificial techniques

Differential Evolution for Optimal Power Flow Considering Shunt FACTS Under Contingency Situation

Belkacem Mahdad Department of Electrical Engineering,

Biskra University, Algeria Email: bemahdad@mselab.org

Kamel Srairi Department of Electrical Engineering,

Biskra University, Algeria Email: ksrairi@mselab.org

2012 6th International Conference on Sciences of Electronics, Technologies of Information and Telecommunications (SETIT)

978-1-4673-1658-3/12/$31.00 ©2012 IEEE 121

with metaheuristic methods, like ‘Fuzzy-GA’, ‘ANN-GA’, ‘Fuzzy-PSO’. Many modified DE have been proposed to enhance the optimal solution, author in [15] present a hybrid method which combines the differential evolution (DE) and Evolutionary algorithms (EAs), with cultural algorithm (CA) to solve the economic dispatch problems associated with the valve-point effect. To overcome the drawbacks of the conventional methods related to the form of the cost function, and to reduce the computational time related to the large space search required by many metaheuristic methods, like GA, authors in [19] proposed an efficient decomposed GA for the solution of large-scale OPF with consideration of shunt FACTS devices under severe loading conditions, in [18] authors present a parallel PSO based decomposed network to solve the ED with consideration of practical generators constraints. This paper presents a differential evolution (DE) algorithm for the solution of the optimal power flow under contingent operation states.

II. OPTIMAL POWER FLOW FORMULATION The OPF problem is considered as a general minimization problem with constraints, and can be written in the following form:

Min )u,x(f (1) Subject to: 0=)u,x(g (2) 0≤)u,x(h (3)

maxmin xxx ≤≤ (4) maxmin uuu ≤≤ (5)

Where )u,x(f is the objective function, )u,x(g and )u,x(h are respectively the set of equality and inequality constraints. The vector of state and control variables are denoted by x and u respectively. In general, the state vector includes bus voltage angles δ , load bus voltage magnitudes LV , slack bus real power generation slack,gP and generator reactive power gQ .

[ ]Tgslack,gL Q,P,V,x δ= (6) The control variable vector consists of real power generation gP , generator terminal voltage gV , shunt capacitors/reactors shB , shunt dynamic compensators (SVC)

svcB and transformers tap ratio t .

[ ]Tsvcshgg B,B,t,V,Pu = (7)

For optimal active power dispatch, the objective function f is total generation cost as expressed follows:

Min ( )∑=

++=NG

igiigiii PcPbaf

1

2 (8)

where NG is the number of thermal units, giP is the active power generation at unit i and ia , ib and ic are the cost

coefficients of the thi generator.

A. Equality constraints

The equality constraints )x(g are the real and reactive power balance equations, expressed as follows:

( )∑=

δ+δ=−N

jijijijijjidigi sinbcosgVVPP

1

(9)

and;

( )∑=

δ−δ=−N

jijijijijjidigi cosbsingVVQQ

1

(10)

Where N is the number of buses, giP , giQ are the active and

the reactive power generation at bus i; diP , diQ are the real and the reactive power demand at bus i ; iV , jV , the voltage magnitude at bus i , j , respectively; ijδ is the phase angle

difference between buses i and j respectively, ijg and ijb are

the real and imaginary part of the admittance ( ijY ).

B. Inequality constraints

The inequality constraints )u,x(h reflect the security limits, which include the following constraints as mentioned below:

• Generator constraints Upper and lower limits on the active power generations:

NPViPPP gigigi ,,2,1,maxmin K=≤≤ (11)

Upper and lower limits on the reactive power generations: NPViQQQ gigigi ,,2,1,maxmin K=≤≤ (12)

Upper and lower limits on the generator bus voltage magnitude:

NPViVVV gigigi ,,2,1,maxmin K=≤≤ (13)

• Security constraints These include the constraints on voltage at loading buses (PQ buses) transmission line loadings, tap ratio (t) of transformer and FACTS Controllers limits.

NPQiVVV LiLiLi ,,2,1,maxmin K=≤≤ (14)

NPQiSS lili ,,2,1,max K=≤ (15)

NTittt iii ,...,2,1,maxmin =≤≤ (16)

maxmin XXX FACTS ≤≤ (17)

C. Non-smooth cost function with valve-point loading effects

Typically, the fuel cost function of the generating units with consideration of valve point loadings is represented as follows [3]:

122

( ) ( )( )gigiii

NG

igiigiii PPedPcPbaf −+++=∑

=

min

1

2 sin (18)

id and ie are the cost coefficients of the unit with valve point effects.

III. OVERVIEW OF DIFFERENTIAL EVOLUTION TECHNIQUE In 1995 Storn and Price [13-14] introduced a parallel direct search method to the evolutionary algorithm (EA) family named Differential Evolution (DE). DE has proven to be promising candidate to solve real valued optimization problem. The strategy of DE is based on stochastic searches, in which function parameters are encoded as floating point variables. The key idea behind differential evolution approach is a new mechanism introduced for generating trial parameter vectors. Details description about mechanism search, advantages and different variants of DE can be found in [13-15-23].

A. Differential Evolution Mechanism Search

A brief review of differential evolution (DE) algorithm is first presented, the following steps summarizes the basic mechanism search of DE: Phase 1: Initialization: Initialize the initial population of individuals, initialize the generation’s counter, G=1, the initial population may be initialized as:

( ) ( ) ( )( )Lj

Uj

Ljji xxrandxx −∗+= ]1,0[)0(

, (19)

Where, G : is the generation or iteration

],[rand 10 : denotes a uniformly distributed random value between 0 and 1.

( )Ljx and ( )U

jx are lower and upper bounds of the component

of vector control ijx respectively for .n,...,,j 21=

Phase 2: Mutation operation: The key idea of DE is the introduction of new parameter (trail vector) into the population according to the following equation:

( ) ( ) ( )( )Gj,r

Gj,rm

Gj,r

)G(j,i xxfxv 321

1 −+=+ (20) A trail vector of each individual is generated based on three other randomly selected individuals 1rx , 2rx and 3rx , 0fmfis a real parameter, called mutation factor, which represent the amplification of the difference between two individuals so as to avoid search stagnation, typical value of mf is in the range [0.4,1].

Fig 1. Mechanism search principle of DE method.

The efficiency of DE is dependent on the selected value of the scaling factor (mutation factor), many variants of DE have been proposed in the literature [10-13].

Phase 3: Crossover operation: The crossover operator implements a discrete recombination of the trail vector generated at mutation phase and the parent vector to produce offspring. The DE crossover operator generates the offspring ]u,,u,u[U G

n,iG,i

G,i

)G(i L21=

corresponding to the parent ]x,,x,x[X Gn,i

G,i

G,i

)G(i L21= as

follows: [ ]( )

≤

=otherwisex

CR,randifvu

)G(j,i

)G(j,i)G(

j,i

10(21)

Where CR indicates the crossover rate of DE.

Phase 4: Selection operation: The selection operator chooses the vectors that are going to compose the population in the next generation. These vectors are selected from the current population and the trial population. Each individual of the trial population is compared with its counterpart in the current population.

<

=+

otherwiseX

)X(fitness)U(fitnessifUX

)G(i

)G(i

)G(i

)G(i)G(

i1 (22)

Where (.)fitness is the objective function (the function to be minimized) of DE. Phase 5: Verification of the stopping criterion: Loop to step 3 until a stopping criterion is satisfied, usually a maximum number of iterations, maxG .

IV. SHUNT FACTS MODELLING

A. Static VAR Compensator (SVC) The steady-state model presented in Fig. 2 used to incorporate the SVC on power flow problems. This model based on representing the controller as variable impedance.

XX

X

X

X

XX

X

X

X

X

X

O

Minimum ),3,2( GrXGrXF −

),3,2(,1 GrXGrXFGrX −+

X2

X1

GiX ,

GXr ,3

GXr ,2 GXr ,1

123

Fig 2. Basic SVC Controller: steady-state circuit representation.

V. APPLICATION STUDY The proposed algorithm is developed in the Matlab programming language (6.5 version) using Microsoft Windows XP. All the programs were run on 2.6 GHz Pentium IV processor with 500MB of random access memory. The proposed approach has been tested on two test network (IEEE 30-Bus with smooth cost function, and to the 40 generating units with valve point effects).

A. Test System 1: smooth cost function

The first test system has 6 generating units; 41 branch system, the system data taken from [21]. It has a total of 24 control variables as follows: five units active power outputs, six generator-bus voltage magnitudes, four transformer-tap settings, nine bus shunt FACTS controllers (SVC). Fig 3 shows the global structure of the chromosome based on the control variables.

Fig 3. Vector control structure based DE for optimal power flow.

5.1.1 DE parameters: Initially, general database is generated; several runs are done with different values of DE parameters such as: mutation constant fm , crossover constant ‘ CR ’, size of population ’ NP ’, and maximum number of generations maxG which is used in this study as convergence criteria. The following

values are selected based on the topology of the network test and to the loading condition (at normal or contingency):

95080 ..fm −= ; 8050 ..CR −= ; 3010−=NP ; 100=maxG .

Case 1: Minimization of fuel cost with and without

compensation (Shunt FACTS)

In this case, the fuel cost objective 1J is considered as:

∑=

=NG

iifJ

11 (23)

Where if is the fuel cost of the ith generating unit.

0 10 20 30 40 50 60 70 80 90 100798

799

800

801

802

803

804

805

806

807

Iteration

Cos

t($/

h)

Case 1: Minimize Fuel Cost

Fig 4 Convergence characteristic: Case1: Minimize fuel cost with shunt FACTS. The objective function of the generation cost is presented as indicated in equation (8). The convergence of total fuel cost is shown in Fig 4. Details of the optimal power generation, fuel cost, and the optimal setting of control variables, obtained by the proposed approach will be given in the next full paper.

Case 2: Minimization of fuel cost and voltage profile

considering shunt FACTS Controllers

Voltage magnitude is one of the important indices of power quality. For a secure operation of the power system, it is important to maintain required level of security margin. In this case in order to minimize the fuel cost and improves the voltage profile. The following multi objective function is proposed and expressed as follow:

∑∑∈=

−+=NLi

iV

NG

ii VfJ 0.1

12 λ (24)

Where Vλ is a weighting factor. The optimal power generation, fuel cost, and the optimal setting of control variables, obtained by the proposed approach will be given in details the next full paper.

Vr

minB

maxB

a) Firing angle Model b) Susceptance Model

Power Flow

Vr

minα

maxα

Length of Vector Control

Min

Max

Max

Max

Min

[ Nsvcsvc

Nttt

NPVgg QQnnVV ,...,,,...,,,..., 111NGPgPg ...2

Min

Active Power Control

Reactive Power Control

124

0 10 20 30 40 50 60 70 80 90 100803

804

805

806

807

808

809

810

Iteration

Cos

t($/

h)

Case 2: Minimze cost & Minimize Voltage deviation

Fig 5 Convergence characteristic: Case 2: Minimize fuel cost & minimize voltage deviation.

0 5 10 15 20 25 300.95

1

1.05

1.1

Bus number

Volta

gepr

ofile

(p.u

)

Max

Min

Minimze Cost

Minimze cost and voltage deviation

Fig. 6 System voltage profile: Case 2: Minimize fuel cost & voltage deviation.

The system voltage profile in this case is compared to that of case1as shown in Fig 6. It is clear that the voltage profile is greatly improved compared to that of case 1.

Case 3: Minimization of fuel cost and power losses under contingency condition considering (Shunt FACTS). To guide the decision making of the power system operator, the OPF solution should take in consideration the critical situation due to severe loading conditions and fault in power system, so it is important to maintain the voltage magnitudes within admissible values at consumer bus under abnormal situation (load increase and contingency). In this case a contingency condition is simulated as outage at different candidate lines. The following objective function is proposed and expressed as follow:

∑∑∈=

ω+=NLi

losspi

NG

ii P.fJ

13 (25)

Where; piω is a penalty factor associated to power loss.

TABLE IOPTIMAL CONTROL VARIABLES UNDER CONTINGENCY CONDITION

CONSIDERING SHUNT FACTS: IEEE 30-BUS.

UNITS LINE (2-6) LINE (10-17)

Cost ($/h) 804.76 801.56 Ploss (MW) 10.050 9.368

svcQ (10) 4.00 5.00

svcQ (12) 8.00 10.00

svcQ (15) 2.40 3.00

svcQ (17) 4.00 5.00

svcQ (20) 3.84 4.80

svcQ (21) 4.92 6.15

svcQ (23) 3.092 3.84

svcQ (24) 3.664 4.58

svcQ (29) 0.400 0.50

Table I shows sample results related to the optimal power flow solution under contingency conditions (line 2-6, line 10-17) with contribution of Static Var Compensators (SVC) installed at a specified buses.

B. Test System 2: with valve point effects

This case study consisted of 40 thermal units of generation with the effects of valve-point loading. The system data taken from [21-22]. The total load demand expected to be satisfied was PD= 10500 MW. Table II compares the results obtained in this paper with those of others studies reported in the literature.

C. Solution Quality

Table II, shows that the minimum cost achieved by the proposed approach (121416.28 $/h), is less than those reported in recent literature [18-21-22]. Observing the comparison results depicted in Table III, the proposed approach is shown to be efficient than other global optimization methods.

TABLE II ECONOMIC DISPATCH RESULTS FOR 40-GENERATING UNITS USING THE

PROPOSED APPROACH: PD =10500 MW.

iPg (MW) iPg (MW)

[P-PSO] [18]

[DEBBO] [22]

Proposed [DE]

[P-PSO] [18]

[DEBBO] [22]

Proposed [DE]

111.6827 110.7998 110.6081 523.2815 523.2794 523.2845

114.0000 110.7998 110.8064 523.2807 523.2794 523.2797

97.4611 97.3999 97.4002 523.2769 523.2794 523.2862

179.7257 179.7331 179.7334 523.2813 523.2794 523.2845

87.7449 87.9576 87.8061 523.2800 523.2794 523.3007

125

iPg (MW) iPg (MW)

[P-PSO] [18]

[DEBBO] [22]

Proposed [DE]

[P-PSO] [18]

[DEBBO] [22]

Proposed [DE]

139.9498 140.0000 140.0000 523.2801 523.2794 523.3007

259.6000 259.5997 259.6048 10.0 10.0 10.0000

284.6000 284.5997 284.6048 10.0 10.0 10.0000

284.5849 284.5997 284.6048 10.0 10.0 10.0000

130.0000 130.0000 130.0000 88.7611 97.0000 87.8165

168.7834 168.7998 94.0000 190.0000 190.0000 190.0000

168.7984 94.0000 94.0013 190.0000 190.0000 190.0000

214.7588 214.7598 214.7601 189.9987 190.0000 190.0000

394.2837 394.2794 394.2845 164.7862 164.7998 164.8049

304.5187 394.2794 394.2797 164.8112 200.0000 194.4141

394.2787 304.5196 394.2845 164.7916 200.0000 200.0000

489.2949 489.2794 489.2836 110.0000 110.0000 110.0000

489.2951 489.2794 489.2845 110.0000 110.0000 110.0000

511.2655 511.2794 511.3007 110.0000 110.0000 523.2845

511.2649 511.2794 511.2845 511.2800 511.2794 523.2797

PD (MW) 10,500 10,500 10,500

Total Cost ($/h) 121418.98 121420.89 121416.28

TABLE III. COMPARISON OF BEST RESULTS FOR FUEL COSTS:

CASE STUDY: 40 GENERATING UNITS.Methods Minimum Cost($/h)

Particle swarm optimization, PSO [15] 122930.4500 Evolutionary programming, EP [15] 122624.3500 Hybrid evolutionary programming with SQP [15] 122379.6300 Improved genetic algorithm with multiplier updating, IGAMU [15]

121819.2500

Anti-predatory particle swarm optimization, APPSO [15] 121663.5200 Quantum particle swarm optimization, QPSO [15] 121501.1400 Civilized swarm optimization, CPSO [15] 121464.6700 Parallel particle swarm optimization, PPSO [18]. 121418.9800 Hybrid differential evolution with biogeography based optimization, DE/BBO [22]

121420.8963

Improved differential evolution based cultural algorithm and diversity measure IDECD [15]

121423.4013

Best result of this proposed approach 121416.2800

D. Robustness The performances of stochastic search algorithms are to be judged over a number of trails. In this study many trails with different initial population have been carried out to test the consistency of the proposed approach. Due to the limited paper length, details results related to values of control variables (active power generation, voltage magnitudes,

reactive power compensation based SVC Controllers (Qsvc), and loading factor ( λ ) will be given in the next full paper.

VI. CONCLUSION A two stages strategy (active power dispatch, and reactive

power planning) based differential evolution (DE) is proposed to enhance the solution of the optimal power flow (OPF) under contingent operation states with consideration of shunt FACTS Controllers. The performance of the proposed approach in terms of solution quality and computational efficiency has been tested with IEEE 30-Bus with smooth cost function, and with 40 generating units with the valve point effects. The results of the proposed algorithm compared with recent global optimization methods. It is observed that the proposed approach is capable of finding the near global solutions of non-linear and non-differentiable objective functions and obtain a competitive solution.

REFERENCES [1] B. Sttot and J. L. Marinho, “Linear programming for power system

network security applications,” IEEE Trans. Power Apparat. Syst. , vol. PAS-98, pp. 837-848, May/June 1979.

[2] M. Huneault, and F. D. Galiana, “A survey of the optimal power flow literature,” IEEE Trans. Power Systems, vol. 6, no. 2, pp. 762-770, May 1991.

[3] J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, 2nd ed. New York: Wiley, 1984.

[4] J. A. Momoh and J. Z. Zhu, “Improved interior point method for OPF problems, ” IEEE Trans. Power Syst. , vol. 14, pp. 1114-1120, Aug. 1999.

[5] M. Huneault, and F. D. Galiana, “A survey of the optimal power flow literature,” IEEE Trans. Power Systems, vol. 6, no. 2, pp. 762-770, May 1991.

[6] R. C. Bansal, “Optimization methods for electric power systems: an overview,” International Journal of Emerging Electric Power Systems,vol. 2, no. 1, pp. 1-23, 2005.

[7] C.-L. Chiang, “Improved genetic algorithm for power economic dispatch of units with valve-point effects and multiple fuels,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1690–1699, Nov. 2005.

[8] C. Chien Kuo, “A novel string structure for economic dispatch problems with practical constraints,” Journal of Energy Conversion and management, Elsevier, vol. 49, pp. 3571-3577, 2008.

[9] S. Pothiya, I. Nagamroo, and W. Kongprawechnon, “Application of multiple tabu search algorithm to solve dynamic economic dispatch considering generator constraints,” Journal of Energy Conversion and management, Elsevier, vol. 49, pp. 506-516, 2008.

[10] C. A. Roa-Sepulveda, B. J. Pavez-Lazo, “ A solution to the optimal power flow using simulated annealing,” Electrical Power & Energy Systems (Elsevier), vol. 25,n°1, pp.47-57, 2003.

[11] Z. L. Gaing, “Particle swarm optimization to solving the economic dispatch considering the generator constraints,” IEEE Trans. Power Systems, vol. 18, no. 3, pp. 1187-1195, 2003.

[12] T. Nikmam, “A new fuzzy adaptive hybrid particle swarm optimization algorithm for non-linear, non-smooth and non-convex economic dispatch,” Journal of Applied Energy, Vol. 87, pp. 327-339, 2010.

[13] R. Storn and K. Price, “Differential Evolution—A Simple and Efficient Adaptive Scheme for Global Optimization Over Continuous Spaces,”

126

International Computer Science Institute,, Berkeley, CA, 1995, Tech. Rep. TR-95–012.

[14] K. Price, R. Storn, and J. Lampinen, Differential Evolution: A Practical Approach to Global Optimization. Berlin, Germany: Springer- Verlag, 2005.

[15] L. S. Coelho, R. C. Thom Souza, and V. Cocco mariani, “Improved differential evolution approach based on cultural algorithm and diversity measure applied to solve economic load dispatch problems,” Journal of Mathematics and Computers in Simulation, Elsevier, 2009.

[16] B. Mahdad, T. Bouktir, K. Srairi, “OPF with Environmental Constraints with SVC Controller using Decomposed Parallel GA: Application to the Algerian Network,” Journal of Electrical Engineering & Technology, Korea, Vol. 4, No.1, pp. 55~65, March 2009.

[17] B. Mahdad, T. Bouktir, K. Srairi, and M. EL. Benbouzid, “Dynamic strategy based fast decomposed GA coordinated with FACTS devices to enhance the optimal power flow,” Journal of Energy Conversion and management, Elsevier, 2010.

[18] B. Mahdad, K. Srairi, T. Bouktir,“Parallel PSO to Solving the Large Economic Dispatch with Consideration of non-smooth Cost Function,” ,Published at IEEE Melecon, 2010.

[19] Gupta, “Economic emission load dispatch using interval differential evolution algorithm,” 4th International Workshop on reliable Engineering Computing (REC 2010).

[20] N. Sinha, R. Chakrabarti, and P. K. Chattopadhyay, “Evolutionary programming techniques for economic load dispatch,” IEEE Trans. Evol.Comput., vol. 7, no. 1, pp. 83–94, Feb. 2003.

[21] J. G. Vlachogiannis, and K. Y. Lee, “Economic dispatch-A comparative study on heuristic optimization techniques with an improved coordinated aggregation-based PSO,” IEEE Trans. Power Systems, vol. 24, no. 2, pp. 991-10001, 2009.

[22] A. Bhattacharya, and P. k, Chattopadhyay “Hybrid differential evolution with Biogeaography-based optimization for solution of economic load dispatch,” IEEE Trans. Power Syst., vol. 25, no. 4, pp. 1955–1964, 2010.

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