Optimal Power Flow Analysis using Two Stage Initialization based Flower Pollination Algorithm

DOI : 10.17577/IJERTCONV4IS07004

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Optimal Power Flow Analysis using Two Stage Initialization based Flower Pollination Algorithm

1M.Rama Mohana Rao*

Assistant Professor,

Department of EEE, DVR & Dr.HS MIC College of Technology, Kanchikacherla,

India – 521 180.

2Akula Venkata Naresh Babu

Professor,

Department of EEE, DVR & Dr.HS MIC College of Technology, Kanchikacherla,

India – 521 180.

3Chintalapudi Venkata Suresh

Full-time Research Scholar,

EEE Dept., University College of Engineering, JNTU Kakinada, India – 533003.

4Sirigiri Sivanagaraju

Professor & Head, EEE Dept., University College of Engineering, JNTU Kakinada, India 5 33003.

Abstract In this paper, a new approach to solve optimal power flow (OPF) problem in power systems is presented. In the proposed algorithm, a two stage initialization process have been adopted and also it gives optimal solution with less number of generations which results in the reduction of the computation time. The feasibility of the proposed algorithm is demonstrated for IEEE 30-bus system with different objective functions. The OPF result of proposed algorithm is compared with existing algorithm. The results reveal better solution and computational efficiency of the proposed algorithm.

Keywords Two stage initialization, optimal power flow; optimization techniques; power system operation.

  1. INTRODUCTION

    the proposed algorithm gives better solution than existing algorithm .

    The remaining portion of the paper is organized as follows: Section II discusses the OPF problem formulation with different objective functions. Section III gives the overview of the proposed algorithm . Section IV demonstrates the effectiveness of proposed algorithm through numerical example and finally, conclusion is drawn in Section V.

  2. OPF PROBLEM FORMULATION

    In its general form, the OPF problem can be mathematically represented as

    In the optimal power flow problem, certain control variables are adjusted to minimize an objective function such as the cost of active power generation or emission or losses while satisfying operating limits on various control and

    Minimize

    f (x, u)

    subjected to

    g(x, u) 0

    (1)

    (2)

    dependent variables. The solution of OPF problem must satisfy the network security constraints. In the past, conventional methods like Newtons method [1-3] and

    where

    hmin

    h(x, u) hmax

    (3)

    Interior point method [4-6] were used to solve the OPF problem. But, in recent years, evolutionary methods are commonly used to solve the OPF problem than conventional methods because of their advantages like simple to implement, reduction in computation time, a fast and near global optimal solution .

    An approach for the optimal power flow problem in a deregulated power market using Benders decomposition has been presented[7-9].The other methods to find the solution for OPF problem have been discussed [10-13]. Based on the review above, it reveals that there is a single stage initialization process. But, in this paper the initialization is done in two stages and also it gives better solution with less

    number of generations which results in the reduction of the

    f (x, u) is the objective function.

    x is the vector of dependent variables

    u is the vector of independent variables g(x, u) represents equality constraints h(x, u) represents inequality constraints

    In this article, minimization of fuel cost, emission and total power loss are considered as an objective functions to examine the performance of the proposed algorithm. The optimal solution must satisfy the equality and inequality constraints. The mathematical expressions for total fuel cost function, emission and total power loss are given in Eqn.(4)-

    (6) respectively.

    ng

    computation time. Numerical results are carried out on a

    F (P ) (a b P c P2 ) $ / h

    (4)

    standard IEEE 30 bus system. The OPF results like bus voltages, active power generation, generation cost ,emission, power loss and computation time has been compared for

    gi i i gi i gi i1

    ng

    TSIFPA and FPA. From the results, it can be observed that

    E(P ) ( P P2 ) ton / h

    (5)

    gi i i gi i gi i1

    nl 2 2 xt 1

    xt

    • L xt g

      (8)

      P g (V

      V 2VV

      cos(

      )) MW

      (6)

      i i i *

      loss k i j i j i j k 1

      where L ~

      sin / 2 1

      where

      xt Solution vector x

      S1+

      at iteration t

      a , b & c

      are cost co-efficient of ith

      generator. i i

      i i i

      g* Current best solution

      , & are emission co-efficient of ith

      generator.

      i i i

      gi

      gi

      P is the generation of the ith generator.

      ng is the number of generator buses.

      Standard Gamma function

      Due to physical proximity and other factors such as wind, local pollination can have a significant fraction p in the overall pollination activities.

      nl is the number of lines.

      xt 1

      xt

    • xt xt

    (9)

    k

    k

    g is the conductance of k th

    line.

    x

    x

    where

    i i j k

  3. OVERVIEW OF PROPOSED ALGORITHM

    t and xt

    are pollen from different flowers of the same

    k

    k

    j

    j

    plant species. This essentially mimics the flower constancy in

    x

    x

    x

    x

    j

    j

    k

    k

    In this paper, a two stage initialization[14] based flower pollination algorithm (TSIFPA) has been presented. It tries to

    a limited neighborhood. Mathematically if t and t

    approach the target in an optimal manner for finding the optimal solution to any mathematical optimization problem. The major stages of the proposed algorithm are briefly described as follows:

    The population is generated by using the following equation

    come from the same species or selected from the same population, this equivalently becomes a local random walk if

    is drawn from a uniform distribution in [0,1].

  4. RESULTS AND DISCUSSIONS

    x x min rand ( 0, 1) (x max x min ) (7)

    i, j j j j

    In this section, a standard IEEE 30-bus system [17] has

    where

    i 1, 2,.., ps ; j 1, 2,.., ncv.

    been considered to demonstrate the effectiveness and

    ps = population size.

    ncv = number of control variables.

    robustness of proposed TSIFPA. In 30-bus test system, bus 1 is considered as slack bus, while bus 2, 5, 8, 11 and 13 are

    j

    j

    x min & x

    max

    j

    are the lower and upper bounds of

    j th

    taken as generator buses and other buses are load buses. A MATLAB program is implemented for the test system on a

    control variable.

    rand ( 0,1 ) is a uniformly distributed random number between 0 and 1.

    The two stage initialization process provides better probability of detecting an optimal solution to the power flow equations that would globally minimize a given objective function. In the first stage, initial population is generated as a multi-dimensional vector of size (spv × ncv). Evaluate the value of objective function for each string in the population vector and select the best string from the population vector corresponding to minimum function value. Repeat the procedure for number of population vectors (n). In the second stage, combine all the best strings to form multi-dimensinal vector of size (n ×ncv) and this new population is used for evolutionary operations. In flower pollination algorithm (FPA) proposed by Xin She Yang[15,16], each flower changes its position according to its constancy and the previous positions in the problem space. The individual best and global best is calculated during iterative process till the stopping criteria satisfied. The flower constancy is updated using the Eqn.(8)-(9).

    Local pollination and global pollination is controlled by a switch probability p [0, 1].

    In the global pollination step, flower pollen gametes are carried by pollinators such as insects, and pollen can travel over a long distance because insects can often fly and move in a much longer range. The inertia weight is updated by using the following equation

    personal computer with core2duo processor and 2 GB RAM. An analysis has been carried out to study the effect of algorithm parameters on the solution of a power flow problem. Based on the analysis, the input parameters of FPA and TSIFPA for the test system are given in Table I.

    The solution for the OPF problem with different objective functions are obtained using FPA and TSIFPA. Table II summarizes the OPF results of both the methods for cost minimization. The OPF results of both the methods for emission and losses minimization are given in Table III and Table IV respectively. The convergence characteristics comparison of the test system using FPA and TSIFPA for cost, emission and losses minimization are shown in Fig.1- Fig.3 respectively.

    TABLE I. INPUT PARAMETERS FOR IEEE 30 BUS SYSTEM

    Optimization Method

    Parameters

    Quantity

    FPA

    & TSIFPA

    Population size

    50

    Number of iterations

    100

    Probability vector

    0.8

    lambda

    1 3

    From Table II- Table IV, it is observed that the control variables obtained using proposed TSIFPA is superior than FPA. Also, the computing time obtained using proposed algorithm is less than FPA. Further, the iterative process

    begins with minimum function value and the change in function value from initial to final is less which indicates convergence rate is fast for proposed TSIFPA method than FPA as shown in the Fig.1 to Fig.3. This is because of best strings selected during the initialization which is known as two stage initialization adopted in the proposed method and evolutionary operations are performed on these best strings.

    TABLE II. COMPARISION OF OPF SOLUTION FOR COST MINIMIZATION

    Control Variables

    FPA

    TSIFPA

    PG1(MW)

    178.695

    177.87

    PG2(MW)

    52.281

    48.450

    PG5(MW)

    20.109

    21.067

    PG8(MW)

    16.975

    21.925

    PG11(MW)

    12.253

    11.114

    PG13(MW)

    12.599

    12.134

    VG1(p.u.)

    1.1

    1.1

    VG2(p.u.)

    1.081

    1.018

    VG5(p.u.)

    1.055

    1.058

    VG8(p.u.)

    1.049

    1.081

    VG11(p.u.)

    1.019

    1.1

    VG13(p.u.)

    1.022

    0.962

    TAP6-9(p.u.)

    0.920

    0.973

    TAP6-10(p.u.)

    1.051

    0.931

    TAP4-12(p.u.)

    0.944

    0.9

    TAP28-27(p.u.)

    1.001

    0.932

    QC10(MVar)

    15.464

    20.781

    QC24(MVar)

    19.926

    13.856

    Cost($/h)

    801.692

    800.834

    Emission(ton/h)

    0.371

    0.368

    PLoss (MW)

    9.512

    9.160

    Time (Sec)

    21.092

    13.772

    Fig.1 Comparison of convergence characteristics for cost minimisation.

    TABLE III. COMPARISION OF OPF SOLUTION FOR EMISSION MINIMIZATION

    Control Variables

    FPA

    TSIFPA

    PG1(MW)

    69.878

    62.792

    PG2(MW)

    66.09

    70.722

    PG5(MW)

    50

    49.472

    PG8(MW)

    31.366

    34.820

    PG11(MW)

    30

    30

    PG13(MW)

    40

    40

    VG1(p.u.)

    1.018

    0.9687

    VG2(p.u.)

    1.015

    0.9

    VG5(p.u.)

    1.019

    0.942

    VG8(p.u.)

    1.1

    1.052

    VG11(p.u.)

    0.9

    1.045

    VG13(p.u.)

    1.1

    1.098

    TAP6-9(p.u.)

    0.932

    0.978

    TAP6-10(p.u.)

    0.9

    0.948

    TAP4-12(p.u.)

    0.927

    0.922

    TAP28-27(p.u.)

    0.931

    0.908

    QC10(MVar)

    26.403

    22.417

    QC24(MVar)

    5

    22.895

    Cost($/h)

    939.054

    949.876

    Emission(ton/h)

    0.206

    0.205

    PLoss (MW)

    3.933

    4.406

    Time (Sec)

    24.718

    14.872

    PLoss (MW)

    Control Variables

    FPA

    TSIFPA

    PG1(MW)

    53.854

    51.998

    PG2(MW)

    78.483

    79.623

    PG5(MW)

    50

    50

    PG8(MW)

    34.909

    35

    PG11(MW)

    29.516

    30

    PG13(MW)

    39.958

    40

    VG1(p.u.)

    1.092

    1.051

    VG2(p.u.)

    1.021

    1.047

    VG3(p.u.)

    1.062

    1.028

    VG8(p.u.)

    1.056

    1.035

    VG11(p.u.)

    0.921

    1.051

    VG13(p.u.)

    1.086

    1.082

    TAP6-9(p.u.)

    1.005

    0.970

    TAP6-10(p.u.)

    1.057

    0.954

    TAP4-12(p.u.)

    1.02

    0.995

    TAP28-27(p.u.)

    1.024

    0.945

    QC10(MVar)

    15.113

    16.597

    QC24(MVar)

    14.767

    6.678

    Cost($/h)

    963.715

    967.139

    Emission(ton/h)

    0.207

    0.207

    3.322

    3.221

    Time (Sec)

    23.109

    12.399

    Control Variables

    FPA

    TSIFPA

    PG1(MW)

    53.854

    51.998

    PG2(MW)

    78.483

    79.623

    PG5(MW)

    50

    50

    PG8(MW)

    34.909

    35

    PG11(MW)

    29.516

    30

    PG13(MW)

    39.958

    40

    VG1(p.u.)

    1.092

    1.051

    VG2(p.u.)

    1.021

    1.047

    VG3(p.u.)

    1.062

    1.028

    VG8(p.u.)

    1.056

    1.035

    VG11(p.u.)

    0.921

    1.051

    VG13(p.u.)

    1.086

    1.082

    TAP6-9(p.u.)

    1.005

    0.970

    TAP6-10(p.u.)

    1.057

    0.954

    TAP4-12(p.u.)

    1.02

    0.995

    TAP28-27(p.u.)

    1.024

    0.945

    QC10(MVar)

    15.113

    16.597

    QC24(MVar)

    14.767

    6.678

    Cost($/h)

    963.715

    967.139

    Emission(ton/h)

    0.207

    0.207

    PLoss (MW)

    3.322

    3.221

    Time (Sec)

    23.109

    12.399

    TABLE IV. COMPARISION OF OPF SOLUTION FOR POWER LOSS MINIMIZATION

    REFERENCES

    Fig.2 Comparison of convergence characteristics for emission minimisation.

    Fig.3 Comparison of convergence characteristics for loss minimization.

  5. CONCLUSION

In this paper, a two stage initialization based flower pollination algorithm has been proposed to solve optimal power flow problem with different objective functions. The OPF results obtained for test system using the proposed TSIFPA and existing FPA are compared. The observations reveal that the control variables obtained using proposed TSIFPA is superior than FPA. Because of two stage initialization, the computing time obtained using proposed algorithm is less than FPA. Further, the iterative process begins with minimum function value and the change in function value from initial to final is less which indicates convergence rate is fast for proposed TSIFPA method than FPA.

  1. A.Santos and G.R.M.da Costa, Optimal-power-flow solution by Newtons method applied to an augmented Lagrangian function, IEEE proc.-Gener. Transm, Distrib, vol.142, no.1, Jan. 1995.

  2. G.R.M.da Costa and C.E.U.Costa, Improved Newton method for optimal power flow problem, Electric Power Energy Syst., vol. 22,pp. 459-462,2000.

  3. G.Radman and J.Shultz, A new derivation for newton based optimal power flow solution, Electric Power Components and Systems, vol.33, pp.673-684,2005.

  4. J.A.Momoh and J.Z.Zhu, Improved interior method for opf problems, IEEE Trans. Power Syst., vol.14,no.3, Aug.1999.

  5. Wang Min and Liu Shengsong, A trust region interior point algorithm for optimal power flow problems, Electric. Power Energy Syst, vol. 27, pp.293-300,2005.

  6. Florin Capitanescu, Mevludin Glavic and Louic Wehenkel, Interior- Point based algorithms for the solution of optimal power flow problems, Electric Power Syst. Research, vol.77, pp. 508-517, 2007.

  7. N.Alguacil and A.J.Conejo, Multi-period optimal power flow using Benders decomposition, IEEE Trans. Power Syst., vol.15, no.1, Feb.2000.

  8. H.Y.Yamin, Kamel Al-Tallaq and S.M.Shahidehpour, A novel approach for optimal power flow using Benders decomposition in a deregulated power market, Electric Power Components and Sytems,vol.31, pp.1179-1192,2003.

  9. H.Y.Yamin, Kamel Al-Tallaq and S.M.Shahidehpour, New approach for dynamic optimal power flow using Benders decomposition in deregulated power market, Electrical Power Systems Research, vol. 65 , pp. 101-107, 2003.

  10. Xiaoqing Bai, Hua Wei, Katsuki Fujisawa and Yong Wang, Semi definite programming for optimal power flow problems, Electric. Power Energy Syst., vol. 30, pp. 383-392, 2008.

  11. Lin Liu, Xifan Wang, Xiaoying Ding and Haoyong Chen, A robust approach to optimal power flow with discrete variables, IEEE Trans. Power Syst., vol.24, no.3,Aug.2009.

  12. A.M.de Souza,V.A.de Souza and G.R.M.da Costa, Studies of cases in power systems by sensitivity analysis oriented by OPF, Electric. Power Energy Syst., vol. 32, pp.969-974, 2010.

  13. M.Varadarajan and K.S.Swarup, Solving multi-objective optimal power flow using differential evolution, IET Gener. Transm. Distrib., vol. 2, no.5, pp.720-730, March 2008.

  14. A.V.Naresh Babu, T.Ramana and S.Sivanagaraju, Analysis of optimal power flow problem based on two stage initialization algorithm, Electric. Power and Energy Syst., vol. 55, pp. 91-99, 2014.

  15. Yang XS. Flower pollination algorithm for global optimization, Unconventional computation and natural computation, Lecture notes in computer science., vol. 7445, pp. 240-249, 2012.

  16. Yang XS and Karamangolu M, He XS. Multi objective flower algorithm for optimization, Procedia Comput Sci vol.18, No.1, pp.861- 868, 2013.

  17. IEEE 30-bus system data available at http://www.ee.washington.edu/research/pstca.

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