Optimal power flow for a deregulated power system using adaptive real coded biogeography-based optimization

Electrical Power and Energy Systems 73 (2015) 393–399

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Optimal power flow for a deregulated power system using adaptive real coded biogeography-based optimization A. Ramesh Kumar a,⇑, L. Premalatha b a b

Department of Electrical and Electronics Engineering, S.A. Engineering College, Chennai 600077, India School of Electrical Engineering, VIT University, Chennai Campus, Chennai 600127, India

a r t i c l e

i n f o

Article history: Received 12 July 2014 Received in revised form 30 April 2015 Accepted 5 May 2015

Keywords: Adaptive mutation Biogeography-based optimization Optimal power flow Fuel cost minimization Voltage profile improvement Voltage stability enhancement

a b s t r a c t The optimization is an important role in the wide geographical distribution of electrical power market, finding the optimum solution for the operation and design of power systems has become a necessity with the increasing cost of raw materials, depleting energy resources and the ever growing demand for electrical energy. Using adaptive real coded biogeography-based optimization (ARCBBO), we present the optimization of various objective functions of an optimal power flow (OPF) problem in a power system. We aimed to determine the optimal settings of control variables for an OPF problem. The proposed approach was tested on a standard IEEE 30-bus system and an IEEE 57-bus system with different objective functions. Simulation results reveal that the proposed ARCBBO approach is effective, robust and more accurate than current methods of power flow optimization in literature. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction In the competitive electrical power market, electrical energy must be offered at a least cost with high quality, which is a very difficult task for the market operator in deregulated power system. Optimal power flow (OPF) is the tool for solving these complicated problems. The main objective of optimal power flow is to obtain the optimal operating schedule for each generator which minimizes the cost of production and satisfies the system equality and inequality constraints [1]. The literature discusses numerous OPF problems [2–4], including those involving reactive power control, voltage control, loss minimization, contingency dispatching, and load shedding, using traditional optimization techniques such as gradient methods, linear programming, nonlinear programming, quadratic programming, newton method, P–Q decomposition, and interior point method. An OPF problem is generally a nonlinear and a multiobjective optimization problem, with more than one local optimum solution. Thus, local optimization techniques are lesser suitable for such complex problems, because they may not be able to provide a global optimum solution. Recently, many of the evolutionary algorithms have been successfully applied in solving OPF problems. These evolutionary ⇑ Corresponding author. Tel.: +91 9841371939. E-mail addresses: [email protected] (A. Ramesh Kumar), premaprak@ yahoo.com (L. Premalatha). http://dx.doi.org/10.1016/j.ijepes.2015.05.011 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

algorithms include evolutionary programming (EP) [5–7], improved evolutionary programming (IEP) [8], genetic algorithm (GA) [9], improved genetic algorithm (IGA) [10], enhanced genetic algorithm (EGA) [11,12], tabu search (TS) [13], simulated annealing (SA) [14], particle swarm optimization (PSO) [15,16], differential evolution (DE) [17,18], modified differential evolution (MDE) [19], gravitational search algorithm (GSA) [20], modified shuffle frog leaping algorithm (MSFLA) [21], harmony search algorithm (HS) [22], biogeography-based optimization (BBO) [23], and artifial bee colony algorithm (ABC) [24]. Using any of the above algorithms, OPF problems with different objective functions, such as fuel cost minimization, emission minimization, system loss minimization, piecewise quadratic cost function, fuel cost with valve point effects, enhancement of voltage stability, voltage profile improvement, fuel cost and emission minimization, and system loss minimization, can be optimized. The solutions to OPF problems have been reported for IEEE 9-bus, IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus systems in literature. The power of adaptive real coded biogeography-based optimization (ARCBBO) to solve the OPF problem is discussed in this paper. Bhattacharya and Chattopadhyay employed biogeography-based optimization (BBO) to solve OPF problems [23]. However, the BBO method was reported to lack exploration ability and poorly supports population diversity. In the ARCBBO approach, therefore, an adaptive Gaussian mutation is integrated into the OPF problem, thereby avoiding premature convergence, improving population diversity, and enhancing the exploration ability.

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(iv) Security constraints: These include the limits on load bus voltage and transmission line flow.

Problem formulation Generally, an OPF problem is a large-scale, highly constrained nonlinear optimization problem. It may be defined as

min f ðx; uÞ

ð1Þ

subject to gðx; uÞ ¼ 0

ð2Þ

hðx; uÞ 6 0

ð3Þ

where f is the objective function to be minimized, x and u are the vectors of dependent and independent control variables, respectively, g is the equality constraint, and h is the operating inequality constraint. The vector of dependent variables can be represented as:

xT ¼ ½PG1; V L1 . . . V LNpq ; Q G1 . . . Q GNg ; SL1 . . . SLNl 

V min 6 V Li 6 V max Li Li

i ¼ 1; 2; . . . ; Npq

ð13Þ

MVAk 6 MVAmax k

ð14Þ

where MVAk is the power flow at kth line; MVAmax is the k power flow capacity of kth transmission line. Finally, the objective function with all constraints combined for the OPF problem is given by 2

min F ¼ f þ kPg ðPG1  Plim G1 Þ þ

X 2 kQg ðQ Gi  Q lim Gi Þ i2Ng

þ

ð4Þ

X

kV ðV Li 

i2Npq

2 V lim Li Þ

þ

X 2 kPf ðMVAi  MVAmax Þ i i2Nl

where P G1 denotes the slack bus power; V L denotes the load bus voltage; Q G denotes the reactive power output of the generator; SL denotes the transmission line flow; Npq is the number of load buses; Ng is the number of voltage-controlled buses and Nl is the number of transmission lines. The vector of independent control variables can be represented as:

where kPg ; kQg ; kV and kPf are the penalty factors.

uT ¼ ½P G2 . . . PGNg ; V G1 . . . V GNpq ; T 1 . . . T Nt ; Q C1 . . . Q CNc 

If

ð5Þ

where P G is the active power output of generators; V G is the voltage at the voltage-controlled bus; T is the tap setting of the tap-changing transformer; and Q C is the output of shunt VAR compensators; Nt and Nc are the number of tap-changing transformers and shunt VAR compensators, respectively. Equality constraints (g)

P Gi  P Di 

Nb X

V i V j ½Gij cosðdi  dj Þ þ Bij sinðdi  dj Þ ¼ 0 i ¼ 1;2;...;Nb

ð6Þ

j¼1

Q Gi  Q Di 

Nb X

V i V j ½Gij sinðdi  dj Þ  Bij cosðdi  dj Þ ¼ 0 i ¼ 1;2;...;Nb ð7Þ

j¼1

where P Gi and Q Gi are the injected active and reactive power at ith bus, respectively; PDi and Q Di are the demanded active and reactive power at ith bus, respectively; V i and V j are the magnitude of voltage at ith and jth bus, respectively; Gij and Bij are the real and imaginary part of the admittance of line connected between ith and jth bus; di and dj are the phase angle of voltage at ith and jth bus, respectively; Nb is the number of buses. Inequality constraints (h) (i) Generator constraints: The generator active and reactive power outputs and voltage are restricted by their upper and lower limits. max Pmin Gi 6 P Gi 6 P Gi

Q min Gi V min Gi

6 6

Q Gi 6 Q max Gi V Gi 6 V max Gi

i ¼ 1; 2; . . . ; Ng

ð8Þ

i ¼ 1; 2; . . . ; Ng

ð9Þ

i ¼ 1; 2; . . . ; Ng

ð10Þ

(ii) Transformer constraints: Tap-changing transformers have minimum and maximum setting limits:

T min i

6 Ti 6

T max Gi

i ¼ 1; 2; . . . ; Nt

ð11Þ

(iii) Switchable VAR sources: These have minimum and maximum limits: max Q min Ci 6 Q Ci 6 Q Ci

i ¼ 1; 2; . . . ; Nc

ð12Þ

If

PG1 > Pmax G1

max then Plim G1 ¼ P G1

elseif

PG1 < Pmin G1 If

Q Gi > Q max Gi

max then Q lim Gi ¼ Q Gi

elseif

Q Gi < Q min Gi V Li > V max Li

max then V lim Li ¼ V Li

min then P lim G1 ¼ P G1

min then Q lim Gi ¼ Q Gi

elseif

V Li < V min Li

min then V lim Li ¼ V Li

Biogeography-based optimization Dan [25] proposed a comprehensive algorithm (BBO) for solving optimization problems based on the study of geographical distribution of species. A BBO algorithm has two main operators: migration operator and mutation operator. Migration operator Migration is a process of probabilistically modifying each individual in the habitat randomly. A geographical area with high habitat suitability index (HSI) tends to have a large number of species, high emigration rate, and low immigration rate. Suitability index variables (SIVs) define the characteristics of a habitat. A habitat with a high HSI tends to be more static in its species distribution. Such an habitat signifies a good solution in terms of an optimization problem. Immigration rate, kk , and emigration rate, lk , are functions of the number of species in a habitat. For a habitat with no species, its immigration rate can be the highest. kk is given by:

  k kK ¼ I 1  n

ð15Þ

where I is the maximum possible immigration rate, k is the number of species of kth individual, and n is the maximum number of species. lk is given by:

  k n

lK ¼ E

ð16Þ

where E is the maximum possible emigration rate. Mutation operator Mutation tends to increase the diversity of a species in a habitat. Due to natural events, the HSI of a habitat can change dramatically, causing the species count to shift away from its equilibrium value. Species count may be a probability value ðP i Þ. If this probability value is very low, an individual solution is thought to have been

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mutated with other solutions. So, the mutation rate of an individual solution can be calculated using species count probability, given by:

M i ¼ M max

  1  Pi Pmax

ð17Þ

where Mi is the mutation rate, Mmax is the maximum mutation rate, which is a user-defined parameter, and Pmax is the maximum probability of species count. In BBO, a mutation characteristic function is given by:

X 0i ¼ X i þ randð0; 1Þ  ðX max  X min Þ i i X max i

ð18Þ X min i

where X i is the decision variable; and are the lower and upper limits of the decision variable, respectively. Adaptive real coded biogeography-based optimization RCBBO was proposed by Gong et al. [26] as an extension to BBO. In RCBBO, individuals are represented by a D-dimensional real parameter vector, and a probabilistically based Gaussian mutation is used. The Gaussian mutation characteristic function is given by:

X 0i ¼ X i þ Nðl; r2i Þ where Nðl; r

2 iÞ

ð19Þ

represents the Gaussian random variable with mean

l and variance r2 . The values of mean and variance are considered 0 and 1, respectively [26]. Generally, a probability-based mutation operation is known to improve the convergence characteristics. Therefore, adaptive Gaussian mutation is applied in the present work to improve the solution of a worst half set of habitats in the population. In Eq. (19), l ¼ 0, and ri is found using the following equation [27]:

ri ¼ b 

where b is the scaling factor or mutation probability, F i is the fitness value of ith individual, and f min is the minimum fitness value of the habitat in the population. Adaptive mutation probability is given by

 n  X Fi  ðX max  X min Þ i i f min i¼1

ð20Þ

b ¼ bmax 

bmax  bmin T T max

ð21Þ

where bmax ¼ 1, bmin ¼ 0:005, T max is the maximum iteration, and T is the current iteration. The use of adaptive mutation can prevent premature convergence, thereby producing a smooth convergence. This method of mutation can be easily used with real-coded variables, which have been widely used in EP, and hence to carry out local as well as global searches. The steps for solving the OPF problem using proposed ARCBBO is as follows:  Step 1: Initialization Habitat modification probability ðP mod Þ, minimum and maximum values of adaptive mutation probability (bmin and bmax Þ, maximum immigration and emigration rates for each island, maximum species count (P), and maximum iterations are initialized.  Step 2: Generate SIVs for the habitat randomly within the feasible region. Individuals (control variables) in the habitats are initialized as:

X ij ¼ X min þ randð0; 1Þ  ðX max  X min Þ j j j

ð22Þ

where i ¼ 1; 2; . . . ; P and j ¼ 1; 2; . . . ; N v ar; N v ar is the number of control variables; X max and X min are the upper and lower limits of j j jth control variable.  Step 3: Perform load flow analysis using Newton–Raphson method and determine the dependent variables (Eq. 4). Compute the fitness value (HSI) for each habitat set.

Table 1 Optimal settings of control variables for different cases. Control variable

Case-1

Case-2

Case-3

Case-4

Case-5

Case-6

P G1 (MW) P G2 (MW) P G5 (MW) P G8 (MW) P G11 (MW) P G13 (MW) V G1 (p.u) V G2 (p.u) V G5 (p.u) V G8 (p.u) V G11 (p.u) V G13 (p.u) T 69 T 610 T 412 T 2827 Q C10 (MVAR) Q C12 (MVAR) Q C15 (MVAR) Q C17 (MVAR) Q C20 (MVAR) Q C21 (MVAR) Q C23 (MVAR) Q C24 (MVAR) Q C29 (MVAR) Fuel cost ($/h) Power loss (MW) Emission (ton/h) R Voltage deviation L-index

177.1590 48.5610 21.4289 21.2958 11.9803 12.0004 1.0851 1.0651 1.0331 1.0384 1.1000 1.0408 1.0974 0.9006 0.9663 0.9760 2.2567 4.2353 4.2998 4.9446 3.7381 4.9901 2.6502 5.0000 2.3967 800.5159 9.0255 0.3663 0.8867 0.1385

167.6377 48.7253 22.1058 23.7659 14.5143 16.2740 1.0210 1.0101 1.0163 1.0067 1.0591 1.0044 1.0786 0.9009 0.9732 0.9704 4.9087 2.7117 5.0000 0 5.0000 4.9983 4.9880 4.9841 2.9470 806.3264 9.6231 0.3395 0.0920 0.1490

175.9872 47.6242 20.3515 23.8073 11.8486 13.0102 1.0694 1.0504 1.0274 1.0388 1.0982 1.0999 1.0261 0.9004 1.0616 0.9508 1.0742 4.2973 0.7846 0.7347 4.8193 0.0939 0.1686 0.1320 0.0095 801.8076 9.2290 0.3628 0.8521 0.1369

173.4861 46.8215 22.5493 24.1073 12.9195 13.4558 1.0781 1.0619 1.0285 1.0439 1.0999 1.0743 1.0436 0.9005 1.0123 0.9536 4.1628 1.0137 2.4209 4.0993 4.4666 0.5075 0.0566 0.0071 0.0083 805.4892 9.9395 0.3551 0.8826 0.1383

51.5020 80.0000 49.9993 34.9998 29.9999 39.9999 1.0618 1.0577 1.0381 1.0447 1.0854 1.0523 1.0701 0.9113 0.9971 0.9771 0.0147 4.9670 4.5824 4.9837 3.9249 4.9907 2.9803 5.0000 2.2742 967.6605 3.1009 0.2073 0.8913 0.1386

63.6625 68.0000 50.0000 35.0000 30.0000 40.0000 1.0550 1.0489 1.0285 1.0357 1.0847 1.0568 1.0250 0.9496 0.9981 0.9683 2.1009 4.9636 4.5424 4.9788 3.9532 5.0000 2.8842 4.9874 2.2953 945.1597 3.2624 0.2048 0.8647 0.1387

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 Step 4: Based on the HSI value, elite habitats are identified.  Step 5: Iterative algorithm for optimization: (i) Perform migration operation on SIVs of each non-elite habitat selected for migration. (ii) Calculate immigration and emigration rates for each habitat set, using Eqs. (15) and (16). (iii) Update the habitat set after migration operation. (iv) Recalculate the HSI value of modified habitat set; feasibility of the solution is verified and habitat set sorted based on new HSI value. (iv) Perform mutation operation on the worst half set of population by Gaussian adaptive mutation. In Eq. (19), mean (l = 0) and variance ðr2 Þ are calculated from the following equation:

rij ¼ b 

F ij min  ðPmax i ¼ 2; 3; . . . ; Ng Gi  P Gi Þ f min

ð23aÞ

where F ij is the fuel cost of ith generator (individual) of jth habitat; f min is the minimum fitness value of the habitat in the population; Pmax and P min Gi Gi are the maximum and minimum limits of active power generation of ith generator. Fuel cost minimization is the main objective function for all case studies, fuel cost mainly depends on active power generation; each active power control variable contributes to minimize the fuel cost individually. So, rij for active power control variables is calculated individually from the fuel cost of each active power generation. But other control variables (except active power control variables) are not directly related to fuel cost minimization function; they are used to satisfy the constraints of the OPF problem. So rij for other control variables is calculated using the fitness value ðF j Þ of the jth habitat set (not individuals of habitat).

rij ¼ b 

Fj

ðX max i

The bus data and line data [28] and the minimum and maximum limits for control variables [29] are obtained from literature. The optimal control parameters for the algorithm are obtained by number of simulation results. They are: habitat size = 50, habitat modification probability = 1, immigration probability = 1, step size for numerical integration = 1, maximum immigration and emigration rate = 1, mutation probability = 0.005, and maximum number of iterations = 200. The results show the corresponding objective functions for 20 independent trails. In the subsequent paragraphs, we discuss the results obtained by the proposed ARCBBO algorithm with regard to each objective function of the OPF problem. The optimal settings of control parameters are given in Table 1. The bolded values represent the optimum value of respective objective functions. It was found that some published studies seem to have violated the constraint limits, thereby yielding infeasible solutions (denoted by ‘a’). C ase 1: Minimization of fuel cost It represents the quadratic cost function whose objective function is expressed as follows:

f ¼ FC ¼

where FC is the total fuel cost; ai ; bi and ci are fuel cost coefficients of the ith unit. The quadratic cost coefficients are obtained from Alsac et al. [28]. Table 2 Comparison of results for minimization of fuel cost. Methods

a

IEEE 30-bus system A IEEE 30-bus system has six generators, four tap-changing transformers, and nine shunt VAR compensation buses for reactive power control. The system active power demand is 283.4 MW and reactive power demand is 126.2 MVAR at 100 MVA base. The magnitude of voltage limits for generator buses are 0.95–1.1 p.u. and for load buses are 0.95–1.05 p.u. Bus 1 is taken as the slack bus.

Best

Average

Worst

800.5159 800.8703 801.0562 800.6600 798.6751a 799.1116a 798.8000a 800.41a 799.2891a 799.56a 800.64 802.376 802.287

800.6412 802.02 801.7414 800.8715 798.9131 799.1985 NA NA NA NA NA 802.382 802.4138

800.9262 802.9431 802.4174 801.8674 799.0284 799.2042 NA NA NA NA NA 802.404 802.5087

Infeasible solution. 845 BBO RCBBO ARBBO

840 835

Results and discussion

830

Fuel Cost in $/h

The power of the ARCBBO algorithm to solve the OPF problem was tested using a IEEE 30-bus and a IEEE 57-bus systems. All simulations were performed on a personal computer (i3 3.1 GHz Intel Processor and 2 GB RAM running MATLAB 10a). Power flow calculations by Newton–Raphson method were performed using 4.1 MATPOWER package [30].

Fuel cost ($/h)

ARCBBO RCBBO BBO ABC [24] GSA [20] BBO [23] HS [22] PSO [15] DE [18] EGA [12] Parallel PSO [16] MDE [19] MSFLA [21]

ð23bÞ

where F j is the fitness value of jth habitat; X max and X min are the i i maximum and minimum limits of ith individual. (v) Compute the fitness value (HSI) for each habitat set after mutation operation and verify the feasibility of the solution. (vi) Sort the habitat set based on new HSI value. (vii) Stop the iteration counter if the maximum number of iterations is reached.  Step 6: Finally SIVs should satisfy the objective function as well as constraints of the problem.

ð24Þ

i¼1

X min Þ i

  f min i ¼ Ng þ 1; Ng þ 2; . . . ; Nv ar

Ng X ðai þ bi  P Gi þ ci  P2Gi Þ

825 820 815 810 805 800 795

0

20

40

60

80

100

120

140

160

180

200

Iterations Fig. 1. Convergence characteristics of proposed ARCBBO, RCBBO, and BBO for minimization of fuel cost.

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A comparison of results across different optimization techniques is presented in Table 2. The first three rows mentioned in the table are obtained from our own implementation of algorithms; i.e. original BBO [25] and RCBBO [26]. Best fuel cost obtained by the proposed ARCBBO was 800.5159 $/h, and average fuel cost was 800.6412 $/h, which is lesser than the minimum fuel cost obtained using algorithms of [25,26] and solution reported in [16,19,21,24]. Convergence characteristics of optimization methods, considered in this work are depicted in Fig. 1, which indicates premature convergence in BBO and smooth convergence in ARCBBO. The best solution as reported in [12,15,18,20,22,23] are infeasible because the voltage magnitudes at most of the load buses are greater than 1.05 p.u., which violate their limits reported in [29]. Case 2: Voltage profile improvement This objective function minimizes the fuel cost while enhancing the voltage profile by minimizing all the load bus voltage deviation from 1.0 p.u. It can be expressed as:

f ¼

Ng Npq X X ðai þ bi  PGi þ ci  P 2Gi Þ þ k jV i  1:0j i¼1

ð25Þ

i¼1

where k is a weighting factor selected by the user. The sum of voltage deviation in this case is 0.0920, which was 0.8867 in the previous case. Hence there is an improvement from 89.62% in the voltage profile. The results of this objective function across different optimization methods considered are presented in Table 3. The proposed ARCBBO shows a better solution than other methods [17,20,23]. The best solution as reported in [15] is infeasible because the reactive power of the slack bus is 20.1144, in violation of the limits reported in [29]. Case 3: Enhancement of voltage stability Voltage stability is the ability of a power system to maintain acceptable voltages at all buses in the system under normal conditions. A system enters a state of voltage instability when a disturbance, such as an increase in load demand or change in system condition, causes a progressive and an uncontrollable decrease in voltage. Voltage stability is an important parameter in a power system operation. It can be defined via minimizing the voltage stability indicator (L-index) values of each bus of a power system. The L-index of a bus specifies the proximity of the voltage collapse condition of that bus. The L-index of a jth load bus is defined as

  Ng  X V i   Lj ¼ 1  F ij  j ¼ 1; 2; . . . ; Npq  V j i¼1

ð26Þ

F ij ¼ ½Y 1 1 ½Y 2 

ð27Þ

where V i is the voltage of the ith generator bus and V j is the voltage of the jth load bus. Y 1 and Y 2 are the submatrices of the system Y bus obtained after separating the load and generator buses parameter. Lj equal to one represents the voltage collapse condition of jth bus. So, a global power system’s L-index is given as

L ¼ maxðLj Þ j ¼ 1; 2; . . . ; Npq

ARCBBO GSA [20] BBO [23] DE [18] PSO [15] a

Infeasible solution.

f ¼

Ng X ðai þ bi  PGi þ ci  P2Gi Þ þ kL

ð29Þ

i¼1

where k is a weighting factor selected by the user. The results of this objective function across different optimization methods considered are presented in Table 4. The maximum value of L-index in this case is 0.1369, which is better than that reported in [24]. The best solution as reported in [15,18,20,22,23] are infeasible because the voltage magnitudes at most of the load buses are greater than 1.05 p.u., which violate their limits reported in [29]. Enhancement of voltage stability under contingency condition In this case, the objective function of the previous case is followed under contingency condition. A contingency is created by the outage of the transmission line between buses 2 and 6. The results of this objective function are presented in Table 5. The maximum value of L-index obtained by proposed ARCBBO is 0.1383, which is better than that reported in [24]. The best solution as reported in [18,20] are infeasible because the voltage magnitudes at load buses are greater than 1.05 p.u., which violate their limits as reported in [29]. Case 5: Minimization of active power loss Total active power loss in the transmission line can be formulated as

PL ¼

Nl X Nb X V i V j ½Gij cosðdi  dj Þ þ Bij sinðdi  dj Þ

ð30Þ

k¼1 j¼1

where Gij and Bij are the real and imaginary parts of the Ybus matrix. di and dj are the voltage angles of ith and jth buses, respectively. The objective function can be expressed as follows:

f ¼

Ng X ðai þ bi  PGi þ ci  P2Gi Þ þ kP L

ð31Þ

i¼1

where k is a weighting factor selected by the user. Table 4 Comparison of results for enhancement of voltage stability. Method

ARCBBO ABC [24] GSA [20] HS [22] BBO [23] DE [18] PSO [15] a

ð28Þ

Table 3 Comparisons of results for voltage profile improvement. Method

A lower value of L-index represents a more stable system. An objective function combining minimization of fuel cost and enhancement of voltage stability is suggested by minimizing the L-index value. This objective function can be expressed as

L-index Best

Average

Worst

0.1369 0.1379 0.1162a 0.1006a 0.1104a 0.1219a 0.1246a

0.1375 0.1960 0.1205 NA 0.1186 NA NA

0.1387 0.7201 0.1228 NA 0.1214 NA NA

Infeasible solution.

Table 5 Comparison of results for enhancement of voltage stability under contingency condition.

R Voltage deviation (p.u) Best

Average

Worst

0.0920 0.0932 0.1020 0.1357 0.0891a

0.1008 0.0939 0.1105 NA NA

0.1257 0.0941 0.1207 NA NA

Method

ARCBBO ABC [24] GSA [20] DE [18] a

Infeasible solution.

L-index Best

Average

Worst

0.1383 0.1474 0.0930a 0.1347a

0.1387 0.1659 0.0965 NA

0.1398 0.2607 0.0998 NA

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Its results are presented in Table 6. The minimum loss obtained by proposed ARCBBO is 3.1009 MW, which is better than that reported in [12,24]. The best solution as reported in [22] is infeasible solution because the voltage magnitudes at all load buses violate the upper limit of 1.05 p.u. Case 6: Minimization of emission Emission generated by each generating unit may be expressed by a quadratic and an exponential function of the generator active power output. The objective function is defined as follows:

f ¼ Em ¼

Ng X

ðai þ bi  PGi þ ci  P2Gi þ ni expðki PGi ÞÞ

ð32Þ

i¼1

where Em is the total emission cost (ton/h); ai ; bi ; ci ; ni and ki are the emission coefficients of the ith unit. Its results are presented in Table 7. The minimum emission obtained by ARCBBO is 0.2048 ton/h, which is better than that reported in [21,24].

Table 8 Optimal settings of control variables for a IEEE 57-bus system. Control variables P G1 (MW) P G2 (MW) P G3 (MW) P G6 (MW) P G8 (MW) P G9 (MW) P G13 (MW) V G1 (p.u) V G2 (p.u) V G3 (p.u) V G6 (p.u) V G8 (p.u) V G9 (p.u) V G13 (p.u) T 418 T 418 T 2120 Fuel cost ($/h) Power loss (MW)

Control variables 142.5804 89.8965 44.6317 75.3757 495.0765 96.2061 358.4099 1.0802 1.0765 1.0643 1.0638 1.0800 1.0543 1.0488 0.9008 1.0988 1.0007

T 2425 T 2425 T 2426 T 729 T 3432 T 1141 T 1545 T 1446 T 1051 T 1349 T 1143 T 4056 T 3957 T 955 Q C1 (MVAR) Q C2 (MVAR) Q C3 (MVAR) 41686 15.3769

1.0722 1.0004 1.0667 1.0000 0.9918 0.9102 0.9926 0.9763 0.9920 0.9551 0.9967 1.0002 0.9937 1.0677 8.7709 13.9948 9.9420

IEEE 57-bus system The system consists of seven generators, fifteen tap-changing transformers, and three shunt VAR compensators. The system data are taken from [31]. The voltage magnitude limits for generator buses are in the range of 0.9–1.1 p.u., and the voltage magnitude limits for load buses are in the range of 0.94–1.06 p.u. The system active power demand is 1250.8 MW and reactive power demand is 336.4 MVAR. Bus 1 is taken as slack bus. The simulation is performed using objective function of minimization of fuel cost, is defined in (24). The optimal control parameters for the algorithm are obtained by number of simulation results. They are: habitat size = 100, habitat modification probability = 1, immigration probability = 1, step size for numerical integration = 1, maximum immigration and emigration rate = 1, mutation probability = 0.005, and maximum number of iterations = 500. The optimal settings of control variables for the OPF is presented in Table 8. Results from 20 independent trails are presented. A comparison of results across different optimization methods is presented in Table 9. The best fuel cost obtained by the proposed ARCBBO is 41686 $/h, which is better than that reported in [20,24,31].

Table 6 Comparison of results for minimization of active power loss Method

ARCBBO ABC [24] HS [22] EGA [12] a

Power loss (MW) Best

Average

Worst

3.1009 3.1078 2.9678a 3.2008

3.1156 NA NA NA

3.1817 NA NA NA

Infeasible solution.

Table 7 Comparison of results for minimization of emission. Method

ARCBBO ABC [24] MSFLA [21] SFLA [21] GA [21] PSO [21]

Emission (ton/h) Best

Average

Worst

0.2048 0.204826 0.2056 0.2063 0.2117 0.2096

0.2054 NA NA NA NA NA

0.2064 NA NA NA NA NA

Table 9 Comparison of results for a IEEE 57-bus system. Method

ARCBBO ABC [24] GSA [20] MATPOWER [31]

Fuel cost ($/h) Best

Average

Worst

41686 41693.9589 41695.8717 41738

41718 NA NA –

41737 NA NA –

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