# Multiobjective Economic Load Dispatch Studies in 2-D and 3-D Space by Particle Swarm Optimization Technique

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## Abstract

In this paper, three important objectives of power systems—cost of generation (FC), system transmission losses (FL) and environmental emissions (FE)—have been considered. The multiobjective economic load dispatch problem has been formulated using weighting method. The noninferior set for IEEE 5 bus, IEEE 14 bus and IEEE 30 bus systems has been generated using particle swarm optimization technique. The noninferior set obtained has been displayed in 3-D space considering all the three objectives and in 2-D space all the combinations of two objectives, viz. for FC & FL, FC & FE and FL & FE for IEEE 5 bus, IEEE 14 bus and IEEE 30 bus systems. The target point or best compromise solution has been obtained by maximizing the minimum relative attainments of all the objectives.

## Keywords

Optimization Maximization of minimum relative attainment Noninferior set Ideal point Target point## Abbreviations

*F*Objective function to be optimized

- FC
Cost of generation, measured in $/h

- FL
Transmission losses, measured in MW

- FE
Cost of environmental emission, measured in kg/h

*a*_{i},*b*_{i},*c*_{i}Cost coefficients of

*i*th generator*d*_{i},*e*_{i},*f*_{i}Emission coefficients of

*i*th generator*P*_{i}Active power generation of the

*i*th generator*P*_{imax}Maximum power generation limit of

*i*th generator*P*_{imin}Minimum power generation limit of

*i*th generator*P*_{D}Total power demand

*Q*Penalty factor

- NG
Total number of generators in the system

*B*_{ij},*B*_{oi},*B*_{oo}Loss coefficients

*P*Number of particles in the swarm

- \(v_{ij}^{k}\)
Velocity of

*j*th particle of*i*th generator at*k*th iteration*W*Inertia weight factor

*C*_{p},*C*_{g}Acceleration coefficients

*r*_{p},*r*_{g}Random numbers between 0 and 1

- \(\it \it x_{ij}^{k}\)
Current position of

*j*th particle of*i*th generator at*k*th iteration- \(x_{{{\text{pbest}}ij}}^{k}\)
Personal best position of

*j*th particle of*i*th generator at*k*th iteration- \(x_{{{\text{gbest}}i}}^{k}\)
Global best position of swarm for

*i*th generator till*k*th iteration- IT
Maximum number of iterations

*k*Current iteration

*W*1Weight attached to cost of generation

*W*2Weight attached to transmission losses

*W*3Weight attached to environmental emissions

- FC
_{max} Maximum value of cost of generation

- FC
_{min} Minimum value of cost of generation

- FL
_{max} Maximum value of transmission losses

- FL
_{min} Minimum value of transmission losses

- FE
_{max} Maximum value of environmental emission

- FE
_{min} Minimum value of environmental emission

- Kount
Function evaluation

*τ*_{C}Minimum relative attainment for cost of generation

*τ*_{L}Minimum relative attainment for transmission losses

*τ*_{E}Minimum relative attainment for environmental emission

## Notes

## References

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