Dieu Ngoc Vo^{1*}, Tung The Tran^{1} and Tuan Trong Nguyen^{2}
^{1}Department of Power Systems, Ho Chi Minh City University of Technology, VNUHMC, Ho Chi Minh City, Vietnam
^{2}Southern Electrical Testing Company, Southern Power Company, Ho Chi Minh City, Vietnam
Received Date: March 20, 2015 Accepted Date: April 07, 2015 Published Date: April 30, 2015
Citation: Vo DN, Tran TT, Nguyen TT (2015) Pseudogradient Based Particle Swarm Optimization with Constriction Factor for Multi Objective Optimal Power Flow. Global J Technol Optim 6:181. doi:10.4172/22298711.1000181
Copyright: © 2015 Vo DN, et al. This is an openaccess article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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This paper proposes a pseudogradient based particle swarm optimization with constriction factor (PGPSOCF) method for solving multiobjective optimal power flow (MOOPF) problem. The proposed PGPSOCF is the conventional particle swarm optimization based on constriction factor based on pseudo gradient to enhance its search ability for optimization problems. The proposed method is to deal with the MOOPF problem by minimizing the total cost and emission from generators while satisfying various constraints of real and reactive power balance, real and reactive power limits, bus voltage limits, shunt capacitor limits and transmission limits. Test results on the IEEE 30bus system have indicated that the proposed method is more efficient than many other methods in the literature. Therefore, the proposed PGPSOCF can be an effectively alternative method for solving the MOOPF problem.
Constriction factor; Multiobjective optimal power flow; Particle swarm optimization; Pseudo gradient
The objective of the optimal power flow (OPF) problem is to optimally determine the combination of control variables in power systems such as real power outputs of generators, voltage magnitude at generation buses, position of transformer tap changers, and reactive power outputs of shunt capacitors so that the total cost of thermal generators is minimized [1,2]. In fact, the OPF problem is a nonlinear and largescale problem since it deals with several variables and nonlinear objective and constraints. Therefore, the OPF problem is always a challenge for solution methods, especially for those with nondifferentiable objective functions which cannot be solved by conventional methods. Moreover, the power generation is also a source to release sulphur oxides (SO_{x}), nitrogen oxides (NO_{x}) and carbon dioxides (CO_{2}) into the atmosphere. The US Clean Air Act amendments of 1990 [3] has forced the utilities to adjust their power generation strategies to guarantee a minimum pollution level. Therefore, the OPF problem should also include the emission in its objective to form a multiobjective OPF (MOOPF) problem. The MOOPF problem is to simultaneously minimize total cost and emission of thermal generators while satisfying all unit and system constraints [4].
There have been several conventional methods proposed for solving the OPF problems such as gradientbased method [5], linear programming (LP) [6], nonlinear programming (NLP) [1], quadratic programming (QP) [7], Newtonbased methods [8], semidefinite programming [9], and interior point method (IPM) [10]. In general, these conventional methods can easily find the optimal solution for a smallscale optimization problem in a very short time. However, the main disadvantage of them is that they suffer difficulty when dealing with nonconvex optimization problems with nondifferentiable objective functions. Moreover, they are also very difficult for dealing with largescale problems due to large search space, leading time consuming or no convergence. The metaheuristic search methods have recently developed shown that they are appropriate for dealing with complicated optimization problems, especially for those with nondifferentiable objective functions. Several metaheuristic search methods have been also widely applied for solving the OPF problem such as genetic algorithm (GA) [11], simulated annealing (SA) [12], tabu search (TS) [13], evolutionary programming (EP) [14,15], differential evolution (DE) [16], improved particle swarm optimisation (IPSO) [17,18], and modified shuffle frog leaping algorithm (MSFLA) [19]. These metaheuristic search algorithms can overcome the main drawback suffered by the conventional methods; that means they can deal with the problems which do not require objective functions to be differentiable. However, these metaheuristic search methods may suffer near optimum solution and the solution quality may not high when dealing with largescale and complex problems. That is the obtained solutions obtained by the methods may be local optima with long computational time. Therefore, the hybrid methods have also developed to overcome the drawback from the single metaheuristic methods such as hybrid TS/SA [20], hybrid GAIPM [21], hybrid differential evolution [22], hybrid of fuzzy and PSO [23], and geneticbased fuzzy mathematical programming technique [24]. The aim of the hybrid methods is to utilize the advantages from each element method to obtain the better optimal solution. Although the hybrid methods can obtain better solution quality than the single methods, they may be suffered slower computational time than the single methods due to combination of many operations. Moreover, the hybrid systems are also usually more complex than the element methods.
In this paper, a pseudogradient based particle swarm optimization with constriction factor (PGPSOCF) method is proposed for solving the MOOPF problem. The proposed PGPSOCF is the conventional particle swarm optimization based on constriction factor based on pseudo gradient to enhance its search ability for optimization problems. The proposed method is to deal with the MOOPF problem by minimizing the total cost and emission from generators while satisfying various constraints of real and reactive power balance, real and reactive power limits, bus voltage limits, shunt capacitor limits and transmission limits. Test results on the IEEE 30bus system have indicated that the proposed method is more efficient than many other methods in the literature.
The remaining organization of this paper is follows. Section 2 addresses the formulation of MOOPF problem. A PGPSOCF implementation for the problem is described in Section 3. Numerical results are presented in Section 4. Finally, the conclusion is given.
The objective of the MOOPF problem is to simultaneously minimize the both total cost and emission while satisfying several equality and inequality constraints. Mathematically, the problem is formulated as follows:
Minimize F_{1}(u,x), F_{2}(u,x) (1)
subject to g(u,x) = 0 (2)
h(u,x) ≤ 0 (3)
where F_{1}(u,x) and F_{2}(u,x) are the objective functions representing total cost and emission, respectively; g(u,x) represents the equality constraints representing power balance at buses; h(u,x) represents the inequality constraints representing upper and lower limits of real power outputs, reactive power outputs, bus voltages, transformer tap changers, shunt capacitors, and power flow in transmission lines; u is the vector of the control variables including active power outputs of generators, magnitudes of generation bus voltage, transformers taps, and shunt capacitors; and x represents state variables including reactive power output, magnitudes of load bus voltage , bus voltage angles, and power flow in transmission lines.
The fuel cost function ($/h) of generators in form of quadratic function is represented by:
(4)
where N_{g} is the number of generators including the slack bus; P_{gi} is the active power output of generator at bus i; a_{i}, b_{i} and c_{i} are the cost coefficients of generator i.
The total emission (ton/h) from generators is represented by:
(5)
where α_{i}, β_{i}, γ_{i}, ξ_{i}, and λ_{i} are emission coefficients of generator i.
The equality and inequality constraints of the problem represented mathematical model as follows:
a) Real and reactive power flow equations at each bus:
(6)
(7)
b) Voltage and reactive power limits at generation buses:
(8)
(9)
c) Capacity limits for switchable shunt capacitor banks:
(10)
d) Transformer tap settings constraint:
(11)
e) Security constraints for voltages at load buses and transmission lines:
(12)
(13)
where Q_{gi} is reactive power outputs of generating unit i; P_{di} and Q_{di} are real and reactive load demand at bus i, respectively; N_{b} is the number of buses; V_{i} and θ_{i} are voltage magnitude and angle at bus i, respectively; G_{ij} and B_{ij} are transfer conductance and susceptance between bus i and bus j, respectively; V_{gi} is voltage at generation bus i; Q_{ci} is reactive power compensation source at bus i; Nc is the number of shunt capacitors; T_{k} is tapsetting of transformer branch k; N_{t} is the number of transformers; V_{l}i is voltage magnitude at load bus i; N_{d} is the number of load buses; P_{l} is power flow in transmission line l connecting between bus i and bus j; and N_{l} is the number of transmission lines.
For the MOOPF problem formulation, the vector of control variables u is represented by:
(14)
where bus 1 is selected as the reference bus and the vector of the state variables x represented by:
(15)
PseudoGradient Based Particle Swarm Optimization with Constriction Factor
Particle swarm optimization with constriction factor
The conventional PSO was developed in 1995 by Kennedy and Eberhart [25]. So far, this method has become one of the most popular metaheuristic search methods implemented in the optimization problems of many fields due to its simplicity in application and efficiency in finding near optimum solution. The principle of PSO for searching the optimal solution for a problem is based on a population of particles which moves in the search space of the problem. The movement of the particles is determined via its location and velocity. During the movement, the position of particles will be updated according to the change of their velocity.
For application of PSO to find the optimal solution of an ndimension problem, a population of NP particles will be used where the position and velocity vectors of particle d are represented by x_{d} = [x_{1d}, x_{2d}, …, x_{nd}] and v_{d} = [v_{1d}, v_{2d}, …, v_{nd}], respectively, where d = 1,…, NP. At each step, the best position of each particle represented by pbest_{d} = [p_{1d}, p_{2d}, …, p_{nd}] (d = 1,…, N_{p}) based on the valuation of the fitness function and the best particle in the population represented by gbest will be stored for the next step. The velocity of each particle in the next iteration (k+1) for fitness function evaluation is calculated by:
(16)
where the constants c_{1} and c_{2} are cognitive and social parameters, respectively and rand_{1} and rand_{2} are random values in [0, 1].
The position of the corresponding particle is updated as follows:
(17)
Generally, the solution quality of the PSO method for optimization problems is sensitive to the calculation of the velocity of particles. Therefore, there have been several improvements on the calculation of velocity of particles to enhance its search ability and solution quality. Clerc and Kennedy have proposed an improvement of velocity calculation for particles with added constriction factor [26] which is to insure the stable convergence of the PSO algorithm. The modified velocity of particles with constriction factor C is calculated as follows:
(18)
(19)
In this improvement, the factor Ï• has an impact on the convergence characteristic of the method and must be greater than 4.0 for convergence stability. In the contrary, if the value of Ï• is high, the constriction C will be small, leading diversification and slower response. Therefore, the best typical value of Ï• suggested by Lim, Montakhab and Nouri [27] is 4.1 (i.e. c_{1}=c_{2}=2.05).
Pseudogradient concept
The pseudogradient is usually used for determining the maximum rate of change direction of nondifferentiable functions where the conventional gradient is not applicable. Therefore, it is appropriate for using in population based search methods to enhance their search ability. This concept has been used in population based methods such as genetic algorithm [28] and evolutionary programming [29].
For a nondifferentiable objective function f(x) where x=[x_{1}, x_{2}, …, x_{n}] in a ndimension optimization problem, a pseudogradient g_{p}(x) for the objective function at a certain point x_{k}=[x_{k1}, x_{k2}, …, x_{kn}] in the search space of the problem moving to another one xl is defined for the two cases as follows [29]:
i) f(x_{1}) < f(x_{k}): the direction from point x_{k} to point x_{1} is defined as the positive direction. The pseudogradient at point x_{1} is determined by:
(20)
where δ(x_{li}) is the direction indicator for element xi moving from point k to point l defined by:
(21)
ii) f(x_{l}) ≥ f(x_{k}): the direction from point xk to point xl is defined as the negative direction. The pseudogradient at point xl is determined by:
(22)
As shown in the definition, if the value of the pseudogradient g_{p}(xl)≠0, a better solution for the problem could be found in the next step based on the direction of the pseudogradient g_{p}(x_{l}) at point l. On the contrary, the search direction at this point may not appropriate due to no improvement can be found for the problem based on this direction.
Pseudogradient based particle swarm optimization
In this paper, the proposed PGPSOCF is the PSO with constriction factor guided by pseudogradient to form a new improved PSO method.
For implementation of the pseudogradient in PSOCF, the two considered points for calculation of the pseudogradient include the particle’s position at iterations k and k+1 those are x^{(k)} and x^{(k+1)}, respectively. Therefore, the updated position for particles in (17) can be rewritten as:
(23)
As observed in (23), if the value of the pseudogradient is nonzero, the particle is moving on the right direction to the optimal solution in the search space of the problem with the enhanced velocity. Otherwise, the particle’s position is normally updated as in (17). With the implementation of the pseudogradient in PSOCF, the new improved PGPSOCF can be more effective than the conventional PSO in solving optimization problems due to the enhanced search ability.
Implementation of PGPSOCF for the MOOPF
For implementation of the proposed PGPSOCF to the MOOPF problem, each particle position representing a vector of control variables is defined as follows:
(24)
The upper and lower boundaries of the position of particles xd are also the upper and lower limits of the variables contained in the vector. The upper and lower limits for the velocity of each particle are determined based on their lower and upper bounds of position:
(25)
(26)
where R is the limit factor for velocity of particles.
The positions and velocities of particles are randomly initialized within their limits as follows:
(27)
(28)
where rand_{3} and rand_{4} are random values in [0, 1].
During the iterative process, the positions and velocities of particles are always adjusted satisfying their limits after each iteration as follows:
(29)
(30)
The fitness function of the problem is defined based on the problem objective functions and the dependent variables including real power output at reference bus, reactive power outputs at generation buses, load bus voltages, and power flow in transmission lines. The fitness function of the problem is represented as follows:
(31)
where ω is the weight factor for objectives; K_{p}, K_{q}, K_{v}, and K_{s} are penalty factors for real power at reference bus, reactive power at generation buses, load bus voltages, and power flow in transmission lines, respectively.
The limits of the state variables in (31) are determined based on their calculated values as follows:
(32)
where x and x^{lim} respectively represent the calculated values and limits of P_{g1}, Q_{gi}, V_{li}, or P_{l,max}
The overall procedure of the proposed PGPSOCF for solving the OPF problem is addressed as follows:
Step 1: Select the controlling parameters for PGPSOCF including number of particles N_{P}, maximum number of iterations It_{max}, cognitive and social acceleration factors c_{1} and c_{2}, limit factor for maximum velocity R, and penalty factors for constraints in fitness function (31). Set the pseudogradient to zeros.
Step 2: Initialize the initial position x_{id} and velocity v_{id} of N_{p} particles within in their limits.
Step 3: For each particle, calculate value of the state variables based on the power flow solution using NewtonRaphson and evaluate the fitness function F_{pbestd} in (31). Determine the best particle with the lowest value of fitness function F_{gbest}=min(F_{pbestd}, d=1,…, N_{P}).
Step 4: Set the best particle’s position of each particle pbestid to x_{id}, d=1,…, N_{P} and the best particle in the population gbesti to the position of the particle corresponding to F_{pbestd} in Step 3. Set iteration counter k=1.
Step 5: Calculate new velocity v^{(k)}_{ id} using (18) and update position x^{(k)}_{ id} using (23) for each particle. Note that the obtained position and velocity of particles should be satisfied their lower and upper bounds given by (29) and (30).
Step 6: Solve power flow problem using NewtonRaphson based on the newly obtained position of particles.
Step 7: Evaluate fitness function FT_{d} in (31) for each particle with the newly obtained power flow solution. Compare the calculated values of FT_{d} to the previous best F^{(k1)}_{pbestd} for each particle to obtain the best fitness function up to the current iteration F(k) pbestd.
Step 8: Select the best position pbest(k)_{id} corresponding to F^{(k)}_{pbestd} for each particle and determine the new global best fitness function F^{(k)}_{pbestd} and the corresponding position gbest^{(k)}_{ i}.
Step 9: Calculate the value of the pseudogradient indictors at the current point.
Step 10: If k<It_{max}, k=k+1 and return to Step 5. Otherwise, stop.
Fuzzy based mechanism for best compromise solution
In the multiobjective optimization problems, there is always a conflict and tradeoff among the objectives which provides decision maker (DM) several options for decision making. One of the methods to find the best compromise solution from the Paretooptimal front of a multiobjective optimization problem is fuzzy satisfying method [30]. This method determines the distance from the value of each objective in the obtained solutions to its maximum value using a linear membership function. A solution is considered the best if the sum of the distances from all objectives in that solution is greater than the sums of the distances from any other solutions.
The fuzzy goal is represented in linear membership function as follows [31]:
(33)
where μ_{j} is membership value of objective j, and Fj max and F_{j}^{ min} are maximum and minimum values of objective j, respectively.
For each nondominated solution, the membership function is normalized as follows [32]:
(34)
where μ^{k} is membership function of nondominated solution k; N^{obj} is the number of objective functions; and N^{P} is the number of Paretooptimal solutions.
The solution with maximum membership function μ^{k} can be chosen as the best compromise solution for the problem.
The proposed PGPSOCF has been tested on the IEEE 30bus with two objectives including total operation cost and emission. The test system has 41 transmission lines, six generators at buses 2, 5, 8, 11, and 13, and four transformers at lines 69, 610, 412 and 2728. The total load demand of the system is 283.4 MW and 126.2 MVar. The data for the system can be found in [1,33]. The data for total cost, emission and transmission line limits is given in Table 1 and power flow limits of transmission lines are given in Table 2.
G_{1} (bus 1) 
G_{2} (bus 2) 
G_{3} (bus 5) 
G_{4} (bus 8) 
G_{5} (bus 11) 
G_{6} (bus 13) 


Cost coefficients  
a_{i} ($/h)  0  0  0  0  0  0 
b_{i} ($/MWh)  2  1.75  1  3.25  3  3 
c_{i} ($/MW^{2}h)  0.00375  0.0175  0.0625  0.00834  0.025  0.025 
Emission coefficients  
α_{i} (ton/h)  0.04091  0.02543  0.04258  0.05326  0.04258  0.06131 
β_{i} (ton/MWh)  20.05554  0.06047  0.05094  0.03550  0.05094  0.05555 
γ_{i} (ton/MW2h)  0.06490  0.05638  0.04586  0.03380  0.04586  0.05151 
ξ_{i} (ton/h)  0.0002  0.0005  0.000001  0.002  0.000001  0.00001 
λ_{i} (1/MW)  2.857  3.333  8.000  2.000  8.000  6.667 
Table 1: Cost and emission coefficients for generators.
Line  12  13  24  34  25  26  46  57  67  68  69 
P_{l,max }MW)  130  130  65  130  130  65  90  70  130  32  65 
Line  610  910  911  412  1213  1214  1215  1216  1415  1518  1617 
P_{l,max }MW)  32  65  65  65  65  32  32  32  16  16  16 
Line  1819  1920  1020  1017  1021  1022  2122  1523  2224  2324  2425 
P_{l,max} MW)  16  32  32  32  32  32  32  16  16  16  16 
Line  2526  2527  2827  2729  2730  2930  828  628  
P_{l,max} MW)  16  16  65  16  16  16  32  32 
Table 2: Limits of transmission lines.
For obtaining the power flow solution of the system, the Matpower toolbox [34] is used. Since the bus voltage limits have a great effect on the final results. Therefore, in this research two kinds of bus voltage limit at buses are considered in the range [0.95, 1.05] and [0.95, 1.10]. The tap changer limit of transformers is set to [0.9, 1.1] for all cases. The two capacitor banks are installed at buses 10 and 24.
The proposed PGPSOCF is coded in the Matlab platform and run on a 3.2 GHz PC. The control parameters of the proposed PGPSOCF method for all cases of the test system are simply selected as follows: N_{P}=10, c_{1}=c_{2}=2.05, R=0.15, It_{max}=200. For each test case, the proposed method is performed 20 independent runs.
Cost objective function
In this case, there is only the total cost objective function is considered. The results obtained by the PGPSOCF method including min total cost, average total cost, max total cost, standard deviation and average computational time for two kinds of bus voltage limits are given in Table 3. As observed from the table, the total cost for the case with bus voltage limit of 1.05 pu is higher than that for the case with bus voltage limit of 1.1 pu while the total emission for the two cases are nearly the same. For the both cases, the standard deviation is very small which indicates that the proposed method can obtain high quality solution for this case.
V_{busmax} = 1.05 pu  V_{busmax} = 1.10 pu  

Min total cost ($/h)  802.2801  799.1994 
Average total cost ($/h)  802.7527  799.9818 
Max total cost ($/h)  805.4520  804.4023 
Standard deviation ($/h)  0.9124  1.2758 
Average CPU time (s)  15.335  15.248 
Total emission (ton/h)  0.3631  0.3666 
Power losses (MW)  9.4364  8.6699 
Table 3: Results by PGPSOCF for cost dispatch case with different bus voltage limits.
The best results by the proposed PGPSOCF for the two cases have been compared to those from other methods as shown in Tables 4 and 5. For the both cases, the proposed method can obtain better total cost than the others. Therefore, the proposed PGPSO is very effective for solving the OPF problem.
NLP [1]  EP [14]  TS [13]  IEP [15]  MDEOPF [16]  MSLFA [19]  PGPSOCF  

P_{g1} (MW)  176.26  173.848  176.04  176.2358  175.974  179.1929  176.0340 
P_{g2} (MW)  48.84  49.998  48.76  49.0093  48.884  48.9804  48.8786 
P_{g3} (MW)  21.51  21.386  21.56  21.5023  21.51  20.4517  21.5350 
P_{g4} (MW)  22.15  22.63  22.05  21.8115  22.24  20.9264  22.1439 
P_{g5} (MW)  12.14  12.928  12.44  12.3387  12.251  11.5897  12.2448 
P_{g6} (MW)  12  12  12  12.0129  12  11.9579  12.0000 
Cost ($/h)  802.4  802.62  802.29  802.465  802.376  802.287  802.2801 
Table 4: Result comparison for cost dispatch case with bus voltage limit of 1.05 pu.
PSO [17]  IPSO [17]  PGPSOCF  

P_{g1} (MW)  178.4646  177.0431  177.2254 
P_{g2} (MW)  46.274  49.209  48.6302 
P_{g3} (MW)  21.4596  21.5135  21.3220 
P_{g4} (MW)  21.446  22.648  21.0422 
P_{g5} (MW)  13.207  10.4146  11.8500 
P_{g6} (MW)  12.0134  12  12.0000 
Cost ($/h)  802.205  801.978  799.1994 
Table 5: Result comparison for cost dispatch case with bus voltage limit of 1.1 pu.
Emission objective function
In this case, there is only the emission objective function is considered. The obtained results by the proposed method for the two cases of bus voltage limits including min emission, average emission, max emission, standard deviation, and average CPU time are given in Table 6. The total emission for the both cases of bus voltage limits is not different. Moreover, the standard deviation of the proposed method for the both cases is also very small.
V_{busmax} = 1.05 pu  V_{busmax} = 1.10 pu  

Min emission (ton/h)  0.2049  0.2048 
Average emission (ton/h)  0.2092  0.2063 
Max emission (ton/h)  0.2398  0.2195 
Standard deviation ($/h)  0.0087  0.0032 
Average CPU time (s)  15.137  15.241 
Total cost ($/h)  944.7824  943.7578 
Power losses (MW)  3.3514  3.0357 
Table 6: Results by PGPSOCF for emission dispatch case with different bus voltage limits.
The result comparisons from the proposed method and other methods for this case with two bus voltage limits are given in Tables 7 and 8. As shown in the tables, the total emission from the proposed method is less than that from the others. Therefore, the proposed PGPSOCF is also very effective for this case.
GA [19]  PSO [19]  SLFA [19]  MSLFA [19]  PGPSOCF  

P_{g1} (MW)  78.2885  59.8075  64.4840  65.7798  63.9471 
P_{g2} (MW)  68.1602  80  71.3870  68.2688  67.4886 
P_{g3} (MW)  46.7848  50  49.8573  50  50.0000 
P_{g4} (MW)  33.4909  35  35  34.9999  35.0000 
P_{g5} (MW)  30  27.1398  30  29.9982  30.0000 
P_{g6} (MW)  36.3713  40  39.9729  39.9970  40.0000 
Emission (ton/h)  0.21170  2.096  2.063  0.2056  0.2049 
Table 7: Best result comparison for emission dispatch case with bus voltage limit of 1.05 pu.
GA [17]  PSO [17]  IPSO [17]  PGPSOCF  

P_{g1} (MW)  69.7300  67.1300  67.0400  63.9471 
P_{g2} (MW)  67.8400  68.9400  68.1400  67.4886 
P_{g3} (MW)  49.7300  49.8600  50.0000  50.0000 
P_{g4} (MW)  34.4200  34.8900  35.0000  35.0000 
P_{g5} (MW)  29.1500  29.6700  30.0000  30.0000 
P_{g6} (MW)  39.2900  39.9400  40.0000  40.0000 
Emission (ton/h)  0.2072  0.2063  0.2058  2.048 
Table 8: Best result comparison for emission dispatch case with bus voltage limit of 1.1 pu.
Multiobjective function
In this case, both total cost and emission are simultaneously considered in the problem. Since there is not much different total cost and emission between the bus voltage limits, only the case with bus voltage limit of 1.05 pu is considered for the multiobjective function. For obtaining the Pareto front for this case, multiple solutions are determined by changing the value of weight factor ω from 0 to 1. Figure 1 depicts the Pareto front obtained by the proposed method for different bus voltage limits.
Based on the obtained solution for the Pareto front, the fuzzy based mechanism is used for obtaining the best compromise solution for the problem. The best compromise solution obtained by the proposed method is 866.0267 ($/h) and 0.2229 (ton/h) which is better than other methods as shown in Table 9. Therefore, the proposed PGPSOCF is also very effective for the multiobjective case of the problem.
GA [19]  PSO [19]  SLFA [19]  MSLFA [19]  PGPSOCF  

P_{g1} (MW)  96.1251  97.8588  98.9772  97.55027  95.0194 
P_{g2} (MW)  68.5168  61.9419  58.6832  60.42367  61.4059 
P_{g3} (MW)  26.7031  31.1310  35.0661  31.6343  31.9402 
P_{g4} (MW)  35  34.4808  31.7585  35  35.0000 
P_{g5} (MW)  30  29.7100  29.9182  30  30.0000 
P_{g6} (MW)  34.7555  36.0884  35.8174  35.21483  35.1872 
Total cost ($/h)  872.9601  872.8731  872.8533  867.713  866.0267 
Emission (ton/h)  0.2270  0.2253  0.2249  0.2247  0.2229 
Table 9: Best result comparison for multiobjective dispatch case with bus voltage limit of 1.05 pu.
In this paper, the proposed PGPSOCF method has been effectively and efficiently implemented for solving the MOOPF problem. The PGPSOCF is the conventional PSO method with constriction factor guided by pseudogradient for enhancement its search ability and solution quality. The proposed can properly deal with the MOOPF problem using the fuzzy based mechanism for best compromise solution. The test results for the IEEE 30 bus system with different bus voltage limits have indicated that the proposed method can obtain better solution quality than many other methods. Moreover, the proposed method can be also extended for dealing with more complex and larger scale OPF problems. Therefore, the proposed PGPSOCF could be a powerful and favorable method for solving the MOOPF problem.
This research is funded by Vietnam National University HoChiMinh City (VNUHCM) under grant number C20142024.
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