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Swarm based mean-variance mapping optimization for convex and non-convex economic dispatch problems

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Abstract

In power system generation, the economic dispatch (ED) is used to allocate the real power output of thermal generating units to meet the required load demand so as the total cost of thermal generating units is minimized. This paper proposes a swarm based mean-variance mapping optimization \((\hbox {MVMO}^{S})\) for solving the ED problems with convex and nonconvex objective functions. The proposed method is the extension of the original single particle mean-variance mapping optimization by initializing a set of particles. The special feature of the proposed method is a mapping function applied for the mutation based on the mean and variance of n-best population. The proposed \(\hbox {MVMO}^{S}\) is tested on various systems and the obtained results are compared to those from many other optimization methods in the literature. Test results have shown that the proposed method can obtain better solution quality than the other methods. Therefore, the proposed \(\hbox {MVMO}^{S}\) is a potential method for efficiently solving the convex and nonconvex ED problems in power systems.

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Abbreviations

N :

Total number of generating units

F :

Total operation cost

\(a_{i}, b_{i}, c_{i}\) :

Fuel cost coefficients of unit i

\(e_{i}, f_{i}\) :

Fuel cost coefficients of unit i reflecting valve-point effects

\(B_{ij}, B_{0i}, B_{00}\) :

B-matrix coefficients for transmission power loss

\(P_{D}\) :

Total system load demand

\(P_{i}\) :

Power output of generator i

\(P_{i,max}\) :

Maximum power output of generator i

\(P_{i,min}\) :

Minimum power output of generator i

\(P_{s}\) :

Power output of slack unit

\(P_{s,max}\) :

Maximum power output of slack unit

\(P_{ismin}\) :

Minimum power output of slack unit

\(n_{i}\) :

Number of prohibited operating zones of unit i

\(P_{L}\) :

Total transmission loss

\(P^{l}_{ik}\) :

Lower bound for prohibited zone k of generator i

\(P^{u}_{ik}\) :

Upper bound for prohibited zone k of generator i

\(DR_{i}\) :

Ramp down rate limit of unit i

\(UR_{i}\) :

Ramp up rate limit of unit i

\(S_{i}\) :

Spinning reserve from unit i

\(S_{i,max}\) :

Maximum spinning reserve contribution of unit i

\(S_{R}\) :

Total system spinning reserve requirement

n_var :

Number of variable (generators)

n_par :

Number of particles

mode :

Variable selection strategy for offspring creation

archive zize :

n-best individuals to be stored in the table

\(d_{i}\) :

Initial smoothing factor

\(\Delta d_0^{\mathrm{ini}}\) :

Initial smoothing factor increment

\(\Delta d_0^{\mathrm{final}}\) :

Final smoothing factor increment

\(f_{s\_ini}^*\) :

Initial shape scaling factor

\(f_{s\_\mathrm{final}}^*\) :

Final shape scaling factor

\(D_{min}\) :

Minimum distance threshold to the global best solution

n_randomly :

Initial number of variables selected for mutation

n_randomly_min :

Final number of variables selected for mutation

indep.runs :

m steps independently to collect a set of reliable individual solutions

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Acknowledgments

This research work is sponsored by GA scheme of Universiti Teknologi PETRONAS.

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Authors and Affiliations

  1. Department of Fundamental and Applied Sciences, Universiti Teknologi PETRONAS, Tronoh, Malaysia

    T. H. Khoa, P. M. Vasant & M. S. Balbir Singh

  2. Department of Power Systems, HCMC University of Technology, Ho Chi Minh City, Vietnam

    V. N. Dieu

Authors
  1. T. H. Khoa
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  2. P. M. Vasant
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  3. M. S. Balbir Singh
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  4. V. N. Dieu
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Corresponding author

Correspondence to T. H. Khoa.

Appendix

Appendix

Table 15 Power output of 140-unit system with both VPE and POZ by \(\hbox {MVMO}^{S}\)
Full size table

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Khoa, T.H., Vasant, P.M., Singh, M.S.B. et al. Swarm based mean-variance mapping optimization for convex and non-convex economic dispatch problems. Memetic Comp. 9, 91–108 (2017). https://doi.org/10.1007/s12293-016-0186-1

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