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Optimal power flow problem plays a major role in the operation and planning of power systems. It assists in acquiring the optimized solution for the optimal power flow problem. It consist s of several objective functions and constraints. This paper solves the multiobjective optimal power flow problem using a new hybrid technique by combining the particle swarm optimization and ant colony optimization. This hybrid method overcome s the drawback in local search such as stagnation and premature convergence and also enhances the global search with chemical communication signal. The best results are extracted using fuzzy approach from the hybrid algorithm solution. These methods have been examined with the power flow objectives such as cost, loss and voltage stability index by individuals and multiobjective functions. The proposed algorithms applied to IEEE 30 and IEEE 118 - bus test system and the results are analyzed and validated. The proposed algorithm results record the best compromised solution with minimum execution time compared with the particle swarm optimization .

The power system is an interconnected electric network, which has generating plant and loads that are connected through transmission and distribution networks. It is a complicated network and it has many objectives to be solved. The reliable result of the objectives is obtained using optimal power flow. Optimized solution for the objective functions is attained by satisfying the power flow equations and constraints of the entire power system network. Various control variables are also influenced to achieve an optimal solution. Many conventional techniques are applied to solve the optimal power flow problem. They are quadratic programming [

In order to overcome the drawbacks of classical methods, many computational methods are used to solve the optimal power flow problem. Particle Swarm Optimization (PSO) [

Multiobjective optimal power flow problem is solved by differential evolution, for active and reactive power dispatch [

In the weighted sum method [

In the PSO algorithm, if the particles are small, local minimum problem will occur and the outcome could be evaluated in multiple runs. If the particles are too large, the global solution is obtained in each run and it reduces the speed of the algorithm. In order to overcome these difficulties and to increase the efficiency of PSO in this paper, a new hybrid technique technique is proposed by combining the particle swarm optimization and ant colony optimization with chemical communication signal.

In this paper cost, loss and Voltage Stability Index (VSI) objectives are analyzed using enhanced PSO with the multiobjective optimal power flow by pareto optimal me- thod. The multiobjective optimization of these objectives is practiced through three cases. In the first case, all the objective functions are simulated individually. In the second case, two objectives are considered simultaneously for optimization such as cost-loss, cost-VSI, loss-VSI and in the third case three objectives are considered concurrently as cost-loss-VSI. These objectives are optimized using the proposed technique with the IEEE 30 and IEEE 118-bus test system. A short introduction of an optimal power flow system has been presented in this section. This paper is planned as a tag on the section. In Section 2, the optimal power flow problem is formulated and discussed. Section 3 explains the central concepts of PSO, Ant colony Optimization (ACO) and Hybrid Swarm Optimization. Section 4 presents the simulated results with discussions of PSO, ACO and hybrid PSO. The conclusion is dealt in the Section 5.

In this paper the cost, transmission loss, voltage stability index are optimized, which is satisfying equality and voltage, generator, shunt VAR and transformer constraints.

1) Minimization of Fuel Cost:

This objective is to minimize the cost. The total fuel cost can be expressed as

(2)

a_{i}, b_{i}, c_{i} is the cost coefficients of the ith generator, P_{i} is the real power output of the ith generator.

2) Minimization of Transmission Loss:

This objective is to minimize the real power transmission loss. The transmission loss can be stated as

V_{i} is the voltage at the ith line, V_{j} is the voltage at the jth line, g_{k} conductance.

3) Minimization of Voltage Stability Index:

This objective is to minimize the voltage stability index. The voltage level ought to maintain below critical level. If the voltage level greater than the critical limit the entire system becomes unstable. The fast voltage stability index expressed as

1) Voltage Constraint:

The bus voltage maintained the maximum and minimum level. The voltage limit conveyed as

2) Generation Constraints:

The real and reactive power of generation sustained the maximum and minimum limit. The generation level of real and reactive power is expressed as

3) Transformer Constraints:

The transformer tap setting values are maintained within the maximum and minimum level. The transformer tap setting limit is expressed as

4) Shunt VAR Constraints:

The shunt VAR restricted the maximum and minimum level. The shunt VAR limit is stated as

5) Equality Constraints:

The equality constraints are expressed as

n: Number of buses.

P_{Gi}, Q_{Gi} Real, reactive power generation at bus i.

P_{Di}, Q_{Di} Real, reactive load demand at bus i.

V_{i}, V_{j} Voltage magnitudes at bus i and j respectively.

G_{ij}, B_{ij} Transfer conductance and susceptance between bus i and j respectively.

δ_{i}, δ_{j} V oltage angle at bus i and j respectively.

Particle swarm optimization (PSO) is a population based optimization technique which is motivated by group manners of birds or fishes. Each particle has its own experience and neighbor particle experience. According to that particle adjusts its position, within the bounded area. Each particle has its local best (lbest) or a personal best. The best position brings into being by all neighbours particles in the specific solution is global best (gbest). The optimal solution is acquired in the specific solution in the course of its current velocity and the experience.

The formation of individual best is

The pattern of global best is

The velocity and the positions are updated in the each iteration. The updating of velocity is

The updating of position of the particle is

_{1}, c_{2} Acceleration factors, rand_{1}, rand_{2} Random numbers between 0 and 1,

Ant colony optimization (ACO) is an evolutionary and adaptive algorithm inspired by the behavior of real ant colonies. The ant when searching food and find food sources, it deposits a chemical called pheromone as a trail during the return path. Based on the quantity and quality of the food available the quantity of pheromone deposited. The pheromone trails allow the ants to identify the shortest path between the ant nest and the food source. Ant colony optimization algorithm updates pheromone values in order to update solutions during run time [

The pheromone trail updating equation is stated as

α is the pheromone trails evaporation co-efficient, which lies between 0 and 1, ΔT_{ij} is the increment of edge for the period Δt.

To enhance the tempo of search in the optimal power flow problem, the chemical communication between the insects will help. It is the signal of wasp which produces alarm pheromone to the inspiration of anxiety in the colony. This increases the speed of search in the problem area.

The chemical communication signal is expressed as

A is the chemical signal, B is the emission rate, D is the diffusion coefficient, r is the radius of the active space, t is the time from the beginning of emission, Comperr is the complementary error function.

ACO not suitable for large optimization problems because it takes long time to search the result and premature convergence. To overcome of finding the best solution with large search space, combine with another algorithm (PSO) which has a better solution. In the hybrid PSO-ACO, the PSO algorithm is used for pheromone update of ACO. This improves the global exploration capabilities of ACO algorithm. This also improves convergence performance of ACO.

The velocity updating equation is expressed as

A is the chemical communication signal,

In the proposed method, the aim is to achieve a set of solution of the multiobjective optimal power flow problem. In the optimal power flow problem with conflicting objectives the single optimal solution does not determine the real solution. The Pareto optimal approach, it is feasible to find the set of solution instead of the single optimal solution. From the set of solution, the best solution is obtained from the fuzzy approach. The equation for the fuzzy approach for best solution is expressed as

x total number of non-dominated solutions.

Step 1: Read input data include equality and inequality constraints.

Step 2: Initialize the population.

Step 3: Calculate the objective function.

Step 4: Apply Pareto optimal method to get the set of solution.

Step 5: Update the velocity using hybrid velocity updating equation.

Step 6: Update the position using position updating equation.

Step 7: Check for stopping criteria. If the iteration reaches the maximum iteration stop else go to step 5.

Step 8: Apply Pareto method and determine the non-dominated solution.

Step 9: Find the best solution using fuzzy approach.

The proposed system has been tested to IEEE 30-bus test system. The 30-bus IEEE test system has 41 transmission lines, six generators and four transformers (T6-9, T6-10, T4-12 and T27-28). The transformer taps are set between 0.9 to 1.1. The proposed algorithm has the particles of 30.

Case 1: Single objective optimization.

The proposed algorithms analyzed with the various values of acceleration factors (c_{1} and c_{2}) individually. This analysis assists to achieve the suitable value of the acceleration factor for the further process. This optimal power flow problem was simulated with the acceleration factor (c_{1}) varied from 1 to 3, using hybrid algorithm. The best value is obtained at c_{1} > 2.5. _{1} = 2.5 are $801.7/hr, 5.21 MW and 0.14 p.u. respectively. The hybrid algorithm also simulated with the various values of the acceleration factor (c_{2}). The best value is obtained at c_{2} > 2. _{2} = 2 are, $801.72/hr, 5.24 MW and 0.13 p.u. respectively.

Acceleration factor | 1 | 1.5 | 2 | 2.5 | 3 |
---|---|---|---|---|---|

Cost ($/hr) | 803.7 | 802.4 | 801.9 | 801.7 | 801.9 |

Loss (MW) | 5.41 | 5.322 | 5.24 | 5.21 | 5.91 |

VSI (p.u.) | 0.151 | 0.15 | 0.149 | 0.14 | 0.153 |

Acceleration factor | 1 | 1.5 | 2 | 2.5 | 3 |
---|---|---|---|---|---|

Cost ($/hr) | 803.89 | 802.75 | 801.72 | 801.76 | 801.78 |

Loss (MW) | 5.319 | 5.27 | 5.24 | 5.25 | 5.254 |

VSI (p.u) | 0.131 | 0.129 | 0.13 | 0.132 | 0.131 |

PSO | ACO | Hybrid | |
---|---|---|---|

P_{g1} (MW) | 177.132 | 176.901 | 176.7 |

P_{g2} (MW) | 48.12 | 49.932 | 48.89 |

P_{g3} (MW) | 21.356 | 21.125 | 21.4719 |

P_{g4} (MW) | 22.878 | 21.468 | 21.6423 |

P_{g5} (MW) | 10.121 | 10.116 | 12.0878 |

P_{g6} (MW) | 12 | 12 | 12.00 |

V_{g1} | 1.05 | 1.045 | 1.045 |

V_{g2} | 1.0442 | 1.043 | 1.043 |

V_{g3} | 1.446 | 0.998 | 0.998 |

V_{g4} | 1.0408 | 1.009 | 1.009 |

V_{g5} | 0.9601 | 1.014 | 1.014 |

V_{g6} | 1.05 | 1.047 | 1.047 |

T_{6-9}, (p.u.) | 1.01 | 1.012 | 1.012 |

T_{6-10}, (p.u.) | 0.99 | 0.971 | 0.971 |

T_{4-12}, (p.u.) | 1.01 | 1.023 | 1.023 |

T_{27-28}, (p.u.) | 1.02 | 1.014 | 1.014 |

Cost ($/hr) | 802.15 | 804.86 | 801.54 |

Loss (MW) | 5.974 | 9.42 | 5.208 |

VSI (p.u) | 0.1307 | 0.142 | 0.13 |

The accuracy of the proposed system is measured using the mean and standard deviation.

ACO | PSO | Hybrid | |
---|---|---|---|

Mean | 0.212 | 0.234 | 0.259 |

STD | 0.042 | 0.036 | 0.024 |

respectively). The convergence time taken for the hybrid technique is less. The accuracy and speed of convergence of hybrid are more than the ACO and PSO. Because the hybrid algorithm overcome the drawback in local search such as stagnation and premature convergence by the local experience of the ant and it processed in parallel. This enhances the chance of locating the finest solution in rapidity. The global search increased by the divergence of the pheromone tracks and also with the chemical communication signal. This leads the particle move the quickly to reach the solution. The global utilization, speed up the search and it support to acquire the best solution.

This optimal power flow problem was worked out using P4, 3 GHz. To evaluate the difficulty of the proposed system, the IEEE 14 bus is chosen to solve this OPF problem. The convergence time for this optimization in IEEE 14 bus system of hybrid method is 0.63 s. The convergence time for the hybrid technique is for IEEE 30 bus test system is 1.05 s The convergence speed of the IEEE 30 bus system is twice the times of IEEE 14 bus test system. This will persuade to carry on the proposed technique in higher buses.

Case 2: Two objective optimization.

In a real world, the problems involve simultaneous optimization of several objective functions. These functions are non commensurable and conflicting objective functions. It gives rise to a set of optimal solutions, instead of one optimal solution. The reason for the optimality of many solutions is that no one can be considered to be better than any other with respect to all objective functions.

Cost-Loss | Cost-VSI | VSI-Loss | |||||||
---|---|---|---|---|---|---|---|---|---|

PSO | ACO | Hybrid | PSO | ACO | Hybrid | PSO | ACO | Hybrid | |

Pg1, MW | 136.467 | 132.374 | 134.717 | 178.167 | 179.232 | 178.12 | 177.458 | 172.486 | 169.451 |

Pg2, MW | 49.396 | 51.585 | 48.856 | 46.544 | 47.54 | 45.421 | 45.425 | 49.581 | 46.859 |

Pg3, MW | 32.826 | 27.847 | 31.471 | 21.412 | 21.324 | 21.433 | 23.192 | 21.247 | 22.294 |

Pg4, MW | 21.686 | 32.694 | 30.642 | 16.234 | 17.415 | 16.574 | 17.497 | 19.782 | 18.482 |

Pg5, MW | 18.085 | 28.878 | 19.078 | 12.201 | 13.127 | 12.401 | 12.108 | 13.875 | 12.158 |

Pg6, MW | 26.011 | 27.200 | 26.002 | 14.704 | 12.332 | 12.228 | 14.704 | 13.229 | 12.759 |

Vg1 | 1.035 | 1.0265 | 1.045 | 1.049 | 1.056 | 1.045 | 1.049 | 1.056 | 1.045 |

Vg2 | 1.034 | 1.023 | 1.043 | 1.04 | 1.034 | 1.043 | 1.082 | 1.034 | 1.043 |

Vg3 | 0.998 | 0.948 | 0.998 | 0.890 | 0.998 | 0.998 | 1.089 | 0.998 | 0.998 |

Vg4 | 1.090 | 1.089 | 1.009 | 1.000 | 1.005 | 1.009 | 1.002 | 1.005 | 1.009 |

Vg5 | 1.0154 | 1.014 | 1.014 | 1.022 | 1.034 | 1.014 | 1.212 | 1.034 | 1.014 |

Vg6 | 1.0347 | 1.0477 | 1.047 | 1.057 | 1.064 | 1.047 | 1.057 | 1.064 | 1.047 |

T6-9, pu | 1.0152 | 1.016 | 1.012 | 1.019 | 1.023 | 1.012 | 1.019 | 1.023 | 1.012 |

T6-10, pu | 0.9671 | 0.9598 | 0.971 | 0.980 | 0.992 | 0.971 | 0.980 | 0.992 | 0.971 |

T4-12, pu | 1.0123 | 1.002 | 1.023 | 1.034 | 1.039 | 1.023 | 1.034 | 1.039 | 1.023 |

T27-28, pu | 1.0124 | 1.0133 | 1.014 | 1.034 | 1.045 | 1.014 | 1.034 | 1.045 | 1.014 |

Cost ($/hr) | 825.15 | 831.72 | 821.72 | 809.54 | 815.65 | 802.5 | |||

Loss (MW) | 5.376 | 5.637 | 5.003 | 5.2574 | 5.117 | 5.113 | |||

VSI (p.u) | 0.1094 | 0.1096 | 0.1092 | 0.1047 | 0.1052 | 0.1007 | |||

Execution time (s) | 1.4815 | 1.675 | 1.4167 | 1.598 | 1.679 | 1.5297 | 1.4298 | 1.4972 | 1.4475 |

objective optimization such as cost-loss, cost-VSI and loss-VSI using PSO, ACO and Hybrid algorithm. It is not easy to pinpoint the correct concert solution from a single run for the corresponding objective. So, to get the appropriate solution 30 numbers of pare to solution sets are obtained from 30 generations.

In the cost-loss minimization, objectives contrasts to each other. The cost-loss divergence optimization, the cost and loss for the PSO, ACO and Hybrid techniques are $825.15/hr, $831.72/hr, $821.72/hr and 5.376 MW, 5.375 MW, 5.043 MW respectively. The reduction of cost and loss in hybrid is 1.12% and 11%higher than PSO and ACO techniques.

To avoid the stability problem in the system, the VSI should be maintained below critical limit. In addition to the VSI, cost also to be reduced. The cost-VSI conflict optimization, the cost and VSI for the PSO, ACO and Hybrid techniques are $809.54/hr, $815.65/hr, $802.5/hr and 0.1094 p.u, 0.1096 p.u., 0.1092 p.u, respectively. The reduction of cost and VSI of hybrid technique is 1.63% and 0.36% respectively.

In the loss-VSI case, the loss is minimized with the voltage stability, which is maintained below critical limit. The loss-VSI contrast optimization, the loss and VSI for the PSO, ACO and Hybrid techniques are 5.2574 MW, 5.117 MW, 5.113 MW and 0.1047 p.u, 0.1052 p.u., 0.1007 p.u, respectively. It shows that the reduction of loss in the ACO technique is less than the PSO technique

more than the ACO and PSO. The best compromised solution of loss and VSI are 5.113 MW and 0.1007 p.u, obtained using SPPSO algorithm. This result indicates the hybrid optimized result from the loss-VSI multiobjective optimization is better compared to the PSO and ACO techniques.

From the two objective optimization the reduction of cost is more in cost-VSI than the cost-loss optimization. The loss is optimized more in cost-loss multiobjective than the loss-VSI. Optimizing two objectives will give the dissimilar result for the same objective function. It initiates the objectives which are considered for the optimal power flow should be optimized simultaneously.

Case 3: Three objective optimization.

In order to confirm the performance of the proposed algorithm in large systems, hybrid algorithm applied in IEEE 118 test bus system. The particles are increased to 50. To get the appropriate solution, 30 numbers of pareto solution sets are obtained from 30 generations.

Variables | PSO | ACO | Hybrid |
---|---|---|---|

P_{g1}, MW | 168.163 | 167.937 | 105.71 |

P_{g2}, MW | 46.232 | 45.827 | 74.445 |

P_{g3}, MW | 23.957 | 23.211 | 38.329 |

P_{g4}, MW | 15.328 | 18.185 | 35.517 |

P_{g5}, MW | 18.541 | 17.248 | 19.278 |

P_{g6}, MW | 23.789 | 25.157 | 12.248 |

V_{g1} | 1.065 | 1.074 | 1.045 |

V_{g2} | 1.054 | 1.063 | 1.043 |

V_{g3} | 0.9002 | 0.9098 | 0.998 |

V_{g4} | 1.100 | 1.104 | 1.009 |

V_{g5} | 1.017 | 1.019 | 1.014 |

V_{g6} | 1.066 | 1.078 | 1.047 |

T_{6-9}, pu | 1.015 | 1.019 | 1.012 |

T_{6-10}, pu | 0.980 | 0.998 | 0.971 |

T_{4-12}, pu | 1.034 | 1.045 | 1.023 |

T_{27-28}, pu | 1.016 | 1.017 | 1.014 |

Cost ($/hr) | 814.76 | 815.6 | 805.4 |

Loss (MW) | 9.924 | 9.98 | 6.321 |

VSI (p.u) | 0.1407 | 0.1482 | 0.1127 |

Execution time (s) | 1.4724 | 1.6245 | 1.5348 |

Objectives | Single objective | Two objectives | Three objectives | ||||
---|---|---|---|---|---|---|---|

Cost | Loss | VSI | Cost-loss | Cost-VSI | Loss-VSI | Cost-loss-VSI | |

Cost ($/hr) | 128945 | 152014 | 149156 | 131992 | 131129 | 131119 | |

Loss (MW) | 139.5 | 123.3 | 138.6 | 115.4 | 125.12 | 143.5 | |

VSI (p.u.) | 0.47 | 0.479 | 0.412 | 0.431 | 0.429 | 0.442 | |

Execution time (s) | 35.487 | 35.897 | 35.127 | 48.087 |

algorithm for IEEE 118 test bus system. It indicates the values of cost, loss and VSI of the single objective optimization are less when compared to the multiobjective optimization. It states that while considering more objective function, the optimized value and execution time are increased concurrently. Because of nonlinearity in large system the front is less consistent comparing to the smaller system.

In this paper optimal power flow problem has three objectives with constraints, and it works out in the multiobjective optimization approach. This multiobjective problem is solved by single objective, two-objective optimization such as cost-loss, cost-voltage stability index, loss-voltage stability index, and three-objective optimization of cost-loss- voltage stability index using PSO, ACO and hybrid techniques. Hybrid swarm intelligence is used to avoid the stagnation of global and local search and get the best solution by fuzzy approach. The simulation results show that the hybrid swarm optimization provides better results and faster convergence compared to the ant colony optimization and particle swarm optimization. The proposed technique gives better results with less convergence time and it will help to choose the sensible solution to the multiobjective optimization problem in IEEE 30 & IEEE 118 test bus system.

Rajalashmi, K. and Prabha, S.U. (2016) Hybrid Swarm Algori- thm for Multiobjective Optimal Power Flow Problem. Circuits and Systems, 7, 3589- 3603. http://dx.doi.org/10.4236/cs.2016.711304