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

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.

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