International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 16 (2015) pp 37143-37150 © Research India Publications. http://www.ripublication.com
Artificial Bee Colony Algorithm to Solve Multi Impartial Optimum Power Flow C.Shilaja School of Electrical Engineering, VIT University,Vellore
[email protected]
Abstract: This paper presents a replacement and economical methodology for finding optimum power flow (OPF) drawback in power systems, within the planned approach, artificial bee colony (ABC) algorithmic rule is used because the main optimizer for optimum changes of the facility system management variables of the OPF drawback. The target functions such as Fuel Cost ($/h), Power Loss (MW), Voltage Stability and Emission (ton/h) area unit chosen for this extremely forced nonlinear non-convex improvement drawback. The validity and effectiveness of the planned methodology is tested with the IEEE 30-bus system, and therefore the take a look at results area unit compared with the results found by alternative heuristic ways reported within the literature recently. The OPF results obtained show that the planned alphabet algorithmic rule provides correct solutions for any style of the target functions. Key words: Optimum power flow, artificial bee Colony
Introduction The optimal power flow is one of important optimization problem in the power system. It was introduced first time in 1968 by Dommel and Tinney [1], and it is currently considered one of the most useful tools for modern power systems operations and planning [2, 3, 4, 5], because it is a backbone of power system. In general, the OPF is a nonlinear programming (NLP) problem that determines the optimal control set points of the system to minimize a given objective function, subject at the same time to equality and inequality constraints imposed by the power system. In other words, is to determine the optimal combination of real power generations, voltage magnitudes, shunt capacitor, and transformer tap settings to minimize a desired objective function. Several conventional optimization methods such as linear programming (LP), interior point method, reduced gradient method and Newton method (Huneault & Galiana, 1991; Momoh, Adapa, & El-Hawary, 1999[6]) have been applied to solve OPF problem assuming convex, differentiable and linear cost function. But unfortunately, these methods face problems in yielding optimal solution in practical systems due to nonlinear and nonconvex characteristic [7] like valve point effects loading in fossil fuel burning plants [8-3]. Hence, it becomes essential to develop optimization algorithms that are capable of overcoming these drawbacks and handling such difficulties. Complex constrained optimization problems have been solved by many populationbased optimization algorithms in the recent years. These techniques have been successfully applied to non-convex, nonsmooth and nondifferentiable optimization problems. Some of the
K. Ravi, School of Electrical Engineering VIT University,Vellore
[email protected]
populationbased optimization methods are genetic algorithm [3], Particle Swarm Optimization [9], Differential Evolution [10] Evolutionary Programming [11]. Recently, a new evolutionary computation algorithm, based on simulating the foraging behavior of honey bee swarm called “Artificial Bee Colony” (ABC), has been developed and introduced by Karaboga in 2005 for real-parameter optimization. Since ABC algorithm is simple in concept, easy to implement, and has fewer control parameters, it has been widely used in many optimization applications and was successfully applied to some practical problems, such as unconstrained numerical optimization [12-15], constrained numerical optimization [16-17], digital filter design [18], aircraft attitude control [19], and made a series of good experimental results. In this paper ABC algorithm has been employed to IEEE 30-bus and IEEE-57 bus test systems having linear/nonlinear operating constraints, smooth / nonsmooth cost curves under different objective functions. The objective functions used in this study are minimization of fuel cost, valve point effect and multi-fuel of generation units. The potential and effectiveness of the proposed algorithm are demonstrated and the results are compared with the existing algorithms in the literature survey.
Artificial Bee Colony OPF ABC is a new swarm intelligence algorithm proposed by Karaboga , which is inspired by the behavior of honey bees. Since the development of ABC, it has been applied to solve different kinds of problems. Artificial bee colony (ABC) algorithm is a recently proposed optimization technique which simulates the intelligent foraging behavior of honey bees. A set of honey bees is called swarm which can successfully accomplish tasks through social cooperation. In the ABC algorithm, there are three types of bees: employed bees, onlooker bees, and scout bees. The employed bees search food around the food source in their memory; meanwhile they share the information of these food sources to the onlooker bees. The onlooker bees tend to select good food sources from those found by the employed bees. The food source that has higher quality (fitness) will have a large chance to be selected by the onlooker bees than the one of lower quality. The scout bees are translated from a few employed bees, which abandon their food sources and search new ones.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 16 (2015) pp 37143-37150 © Research India Publications. http://www.ripublication.com • UNTIL (requirements are not met). In the ABC algorithm, the first half of the swarm consists of employed bees, and the second half constitutes the onlooker bees. Detailed pseudo-code of the ABC algorithm is given below: The number of employed bees or the onlooker bees is equal to the number of solutions in the swarm. The ABC generates a randomly distributed initial population of SN solutions (food sources), where denotes the swarm size. Let X¡={x¡,1,x¡,2,….,x¡,D} represent the ith solution in the swarm, Where D is the dimension size. Each employed bee X¡ generates a new candidate solution Vi in the neighborhood of its present position as equation (6): …… (6)
V¡,|=X¡,| + ϕ ¡,| . (X¡,|-X¡¡,|)
Where X¡¡ is a randomly selected candidate solution (i≠ii), j is a random dimension index selected from the set{1,2,….,D} , and ϕ¡,| is a random number within[-1,1] . Once the new candidate solution Vi is generated, a greedy selection is used. If the fitness value of Vi is better than that of its parent Xi, then update Xi with Vi; otherwise keep Xi unchangeable. After all employed bees complete the search process; they share the information of their food sources with the onlooker bees through waggle dances. An onlooker bee evaluates the nectar information taken from all employed bees and chooses a food source with a probability related to its nectar amount. This probabilistic selection is really a roulette wheel selection mechanism which is described as equation (7): Pi= fit¡ / fit| ……… (7) Where fiti is the fitness value of the ith solution in the swarm. As seen, the better the solution i, the higher the probability of the ith food source selected. If a position cannot be improved over a predefined number (called limit) of cycles, then the food source is abandoned. Assume that the abandoned source is Xi, and then the scout bee discovers a new food source to be replaced with Xi as equation (8):
1: Initialize the population of solutions xi,j , i = 1 ...SN,j = 1 ...D 2: Evaluate the population 3: cycle=1 4: repeat 5: Produce new solutions υi,j for the employed bees by using (2) and evaluate them 6: Apply the greedy selection process 7: Calculate the probability values P i,j for the solutions xi,j by (1) 8: Produce the new solutions υi,j for the onlookers from the solutions xi,j selected depending on P i,j and evaluate them 9: Apply the greedy selection process 10: Determine the abandoned solution for the scout, if exists, and replace it with a new randomly produced solution xi,j by (3) 11: Memorize the best solution achieved so far 12: cycle=cycle+1 13: until cycle=MCN
Power-Flow Problem Formulation The objective of OPF is to minimize the production cost while satisfying all the equality and inequality constraints, and can be written in the following form Minimize
X ¡,|=lb|+rand(0,1).(ub|-lb|) ……….(8) Where rand(0,1) is a random number within[0,1] based on a normal distribution and lb,ub , are lower and upper boundaries of the ith dimension, respectively.
Fx ,u
x, g u
(1)
0
Subject to:
(2) h
x, u
0
The basic step of this algorithm is given below: • Initialization phase. • REPEAT (a) In the Memory, Employed bees are placed on the food sources; (b) Generate new offspring from older offspring after Applying crossover operator. (c) In the memory, onlooker bees are placed on the food sources; (d) For finding new food sources, Send the scout bee to the search space.
where F x ,u :
Objective function;
g x ,u :
Equality constraints;
h x ,u :
Inequality constraints;
x : Vector of dependent variables consisting of slack bus active power, load bus voltages, generators reactive powers and transmission lines.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 16 (2015) pp 37143-37150 © Research India Publications. http://www.ripublication.com u : Vector of independent variables consisting of the generators’ active powers except slack bus, generators’ voltages, transformers’ tap settings and shunt VAR compensators. Hence
Voltage magnitudes at each bus in the network
VminV
x & u can be expressed as:
N
T
T
....P
,V
....V
,Q
......,Q
N
,T ....T
(4)
Smax
S G2
GNG
G1
GNG
Sh 1
S h NC
1
N
The transmission Lines (3)
x P G 1 ,V L ...V LNB , Q G 1 ...Q GNG , S L 1...S LNL u P
Vmax
NT NTL
NTL
where
NB
:
Number of load buses
NG
:
Number of generators;
NTL
:
NT
:
The discrete transformer tap settings min
Number of Transmission
T NT
Lines;
T NT T
max NT
Number of regulating transformers;
Multi Object OPF NC : Number of shunt Volt Amperes Reactive (VAR) compensators.
2.1. Equality Constraints The equality constraint set typically consists of the load flow equations, which are given below:
2.2. Inequality Constraints
The paper considers the total fuel cost of generating units, the emission of atmospheric pollutants, the active power losses and the voltage deviations as the optimization objective functions. The methods for solving the multiobjective optimization problem include the weighting method [20] and fuzzy mathematics method [21]. The weighting coefficients of the weighting method are determined by decision maker preferences or several simulations. The fuzzy mathematics method utilizes membership functions to fuzzify N
Generator constraints: P
Generator voltage reactive power of i
th
magnitudes, active
Gi
and
Bij
j1
(5)
Vmax
V Gi
Gi
N
Gi Q
min
P Gi
max
PGi
PGi
min
max
Q QGi
Gi
min
, V Gi
max
:
Minimumandmaximumgeneratorvoltage ofi generating unit;
(6)
Q Gi
,
max QGi
:
V i V j G ij sin ij
Q Li
Bij
where Voltage ofi
th
th
and j bus respectively;
P ,Q Gi
Gi
Active and Reactive power ofi
th
Active and Reactive power ofi
th
P Q Li
G ij
i generating unit.
, Bij ,
max
:
load bus;
Conductance, Admittance and Phase difference of th
,P Gi
generator;
Minimum andmaximum reactivepowerof th
min
cosij
j1
Li
min
Gi
V i ,V j
QGi
th
P Gi
sinij
bus lies between their upper and lower limits as given below:
Vmin
V Gi
V i V j G ij cosij
PLi
th voltages between i and j bus.
Minimumandmaximumactivepowerof th
i generating unit.
ij
N
Number of buses.
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 16 (2015) pp 37143-37150 © Research India Publications. http://www.ripublication.com the objective functions without weighting coefficient settings. In other words, the fuzzy mathematics method can make optimization results more objective and reasonable compared with the weighting method. We therefore utilize the fuzzy mathematics method to solve the multi-objective OPF problem, and use the following linear membership function so as to modify the candidate objective functions to its fuzzy form:
where fi is ith objective function; µ(fi) is the membership function of fi; λimax is upper limit value of fi and its value is the initial value; λimin is lower limit value of fi and its value is obtained by the single objective optimization; m is the number of objective functions. The curves of membership functions given by the Equation (16) are shown in Figure 1. The fuzzy satisfaction-maximizing method [22,23] is used to convert the multi membership function µ(fi) into the objective function µF of fuzzy multi-objective OPF, and it can be expressed as: µF =max {1-µ (f1) The obtained model of fuzzy multi-objective OPF is described as: Minimize: μF (18) Subject to: g(x) = 0, h(x) ≤ 0 (19) The smaller value µF that is obtained by the fuzzy multi-objective OPF described by Equation (18) represents the higher fuzzy satisfaction value and the better optimal scheme.
Figure 2 Flow Chart of (a) Bees Algorithm
Results & Discussion To evaluate performance of developed algorithms, bench mark test case IEEE 30 bus system shown in Figure 2 is considered. Numerical result of IEEE 30 bus is presented and discussed in this chapter. The system has 6 generators include slack bus, hence 5-real power generation, 6 generator bus voltage magnitude and 4 transformer tap position are considered as control variables. Base MVA of the system is 100MVA.
Figure 1. Single line diagram of IEEE 30 bus system
b) Bees Algorithm-OPF Flowchart
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 16 (2015) pp 37143-37150 © Research India Publications. http://www.ripublication.com Table 1: Cost and emission coefficients of test case IEEE 30 bus system Coefficients [13]
G1
G2
G3
G4
G5
G6
Fuel cost coefficients a
100
120
40
60
40
100
b
200
150
180
100
180
150
c
10
10
30
10
20
10
Emission cost coefficients ϒ
0.06490
0.05638
0.04586
0.03380
0.04586
0.05151
β
-0.5554
-0.06047
-0.05094
-0.03550
-0.05094
-0.05555
α
0.04091
0.02543
0.04258
0.05326
0.04258
0.06131
ξ
0.0002
0.0005
0.000001
0.002
0.000001
0.00001
λ
2.857
3.333
8.00
2.00
8.00
6.667
For the test case, generation cost and emission coefficients are given in the table 1. The system has 6 generators and its corresponding coefficients are listed.
generating cost the generation of all six generators is adjusted to the global optimal value as given in the table 2. The cost of generation for various algorithms is given in the figure 2
In this work FOUR objectives are considered. Table 2 to Table 6 provides single objective results for each FOUR considered objective function. Table 2, compares Artificial Bee Colony Algorithm(ABC) with other Evolutionary Programming (EP), Tabus Search (TS), Multi objective Differential Evolution (MDE), Ant colony optimization (ACO), Particle swarm optimization (PSO) and Improved, Particle swarm optimization (IPSO) algorithms. The objective considered in the table is cost minimization. It shows that ABC has better results as compared to other algorithms. Generating cost of ABC is lowest as compared to other algorithms, as given in the table 2. To achieve this
Figure 3: Comparison of generating cost of various algorithms
Table 2: Cost minimization objective Gen
EP [11]
TS [12]
MDE[16]
SGA [17]
ACO [18]
PSO [13]
IPSO [13]
ABC
Pg1
173.848
176.04
176.009
175.974
181.945
178.4646
177.0431
176.357
Pg2
49.998
48.76
48.801
48.884
47.001
46.274
49.209
49.209
Pg3
21.386
21.56
21.334
21.51
20.553
21.4596
21.5135
21.5135
Pg4
22.63
22.05
22.262
22.24
21.146
21.446
22.648
22.648
Pg5
12.928
12.44
12.46
12.251
10.433
13.207
10.4146
10.4146
Pg6
12
12
12
12
12.173
12.0134
12
12
Cost ($/hr)
802.62
802.29
802.376
803.699
802.578
802.205
801.978
800.930
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 16 (2015) pp 37143-37150 © Research India Publications. http://www.ripublication.com To get global optimal solution, ABC is iterated for 200 iterations and shown in figure 4. From the figure it is clear that, the algorithm converged at 10th iteration. The second objective of loss minimization is focused in Table 3. It compares Loss minimization of ABC algorithm with other algorithms as given below. Loss minimization is important issue, since reduction of loss is equivalent of increase in generation. The cost of power due to real power loss is the economic loss for the firm and it produces global warming. ABC gives minimum loss 4.8960 MW as compared to other algorithms.
Figure 4: Convergence characteristic of ABC for cost minimization objective Table 3: Loss minimization objective Control Base case [19]
SPEA [19]
GA [19]
PSO [13]
IPSO [13]
ABC
Vg1
-
-
-
1.045
1.047
0.927072
Vg2
1.045
1.044
1.03
1.043
1.044
0.960628
Vg3
1.01
1.023
1.00
0.998
0.976
0.959655
Vg4
1.01
1.022
1.00
1.009
1.035
1.00159
Vg5
1.05
1.042
1.02
1.014
0.984
1.07081
Vg6
1.05
1.043
1.04
1.047
1.042
0.969761
T1
0.97
1.09
1.00
1.012
1.029
0.952339
T2
0.96
0.90
1.01
0.971
0.98
1.03402
T3
0.93
1.02
1.00
1.023
1.01
0.959757
T4
0.96
0.96
1.04
1.014
0.97
1.07622
Loss (MW)
5.4356
5.199
5.3513
5.2105
5.0732
4.8960
Variables
Loss minimization of ABC is compared to various algorithms as pictured in figure 5. ABC gives loss as low as 4.8960 MW compared other algorithms. Figure 5 gives convergence curve of loss minimization objective, the problem is converged in 10th iteration but for search of global value, ABC is iterated to the maximum value of 200 iterations.
Figure 5: Comparison of loss minimization with various algorithms
Global warming is the important issue for the social welfare and to leave good nature for our next generation. This global warming is increased due to emission CO and CO2 which are produced after the coal burnt for electric power generation. So, this emission has to be minimized as far as possible. For the test case, emission is calculated for all algorithms and given in table 5. From the table ABC gives
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 10, Number 16 (2015) pp 37143-37150 © Research India Publications. http://www.ripublication.com minimum emission 0.2055ton/hr as compared to other algorithms.
Figure 6: Convergence characteristic of ABC for loss minimization objective Table 5: Emission minimization objective Gen
GA [19]
PSO [13]
IPSO [13]
ABC
Pg1
69.73
67.13
67.04
63.8795
Pg2
67.84
68.94
68.14
68.1400
Pg3
49.73
49.73
50
50.0000
Pg4
34.42
34.42
35
35.0000
Pg5
29.15
29.67
30
30.0000
Pg6
39.29
39.29
40
40.0000
Emission (ton/hr)
0.20723
0.2063
0.2060
0.2055
Figure 8 shows the convergence characteristics of emission objective function of ABC. In the figure it is converged at 10th iteration. As the convergence criterion of maximum of 200 iterations is iterated to get global minimum value.
Figure 8: Convergence characteristic of ABC for emission minimization objective
Conclusion An Artificial Bee Colony algorithm is proposed to solve the OPF problem under different formulations and considering different objectives function. The performance of the proposed ABC was tested on the IEEE 30-bus test systems. The results obtained using the ABC algorithm were compared to other methods previously reported in the literature. The comparison verifies the influentially of the proposed ABC approach over stochastic techniques in terms of solution quality for the OPF problem and confirmed its potential for solving a most nonlinear problems. The results obtained are rather encouraging as it can be noticed that the Bees Algorithm can converge towards the better solution slightly faster than the rest methods. Hence, it can be considered as a promising alternative that is suitable for solving the OPF problem. References
Figure 7 shows the comparison chart for emission of various algorithms. The comparison shows ABC gives minimum of 0.2055 ton/hr.
Figure 7: Comparison of emission minimization with various algorithms
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