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Electrical Engineering

, Volume 101, Issue 3, pp 699–718 | Cite as

Multi-objective optimal power flow considering wind power cost functions using enhanced PSO with chaotic mutation and stochastic weights

  • Anongpun Man-Im
  • Weerakorn OngsakulEmail author
  • J. G. Singh
  • M. Nimal Madhu
Original Paper
  • 65 Downloads

Abstract

A multi-objective optimal power flow (OPF) solution using an enhanced NSPSO, incorporating chaotic mutation and stochastic weight trade-off features, is proposed here. The objective functions considered embody the different aspects of the power system, viz. financial, reliability and operational efficiency. The proposed OPF algorithm balances the exploration of global and the utilization of local bests with the stochastic weight and dynamic coefficient trade-off techniques, thus enhancing the searching capability. Also, for countering the premature convergence issue and to improve diversity, the feature of chaotic mutation is incorporated. The pareto-optimal front is provided by the combination of crowding distance approach and non-dominated sorting principle. The best solution can be obtained from a dual-stage process, by selecting from the collective of local best compromises using a fuzzy function. To assess the performance, the proposed method is tested on a standard IEEE 30-bus test system and is compared with popular multi-objective algorithms. IEEE 118-bus system is also used for testing applicability to large systems. Based on the obtained results, the proposed method gives a more reliable operating solutions, better optimal front and hence an improved solution providing a better trade-off.

Keywords

Particle swarm optimization Multi-objective optimal power flow Wind power Operation fuel cost Active power loss 

List of symbols

\(C_F\)

Total generator fuel cost ($/h)

\(N_\mathrm{GEN}\)

Number of conventional generators

\(a_i\), \(b_i\), \(c_i\)

Cost efficiency of conventional generator i

\(P_{Gi}\)

Active power output of conventional generator i (MW)

\(W_{si}\), \(W_{ri}\), \(W_{ai}\)

Scheduled, rated and actual wind powers

\(v_i\)

Wind speed

\(v_{\mathrm{in},i}\), \(v_{ri}\), \(v_{\mathrm{out},i}\)

The cut-in, rated and cut-out wind speeds

\(C_i\) , \(K_i\)

Scale and shape factors of Weibull PDF

\(N_W\)

Number of wind farms

\(d_i\),\(d_{Pwi}\), \(d_{Rwi}\)

Direct, penalty and reserve cost coefficients

\(V_{Gi}\)

Voltage magnitude of generator bus i (p.u.)

\(V_{Li}\)

Voltage magnitude of load busi (p.u.)

\(N_L\)

Number of load buses

\(P_\mathrm{loss}\)

Transmission system loss (MW)

\(g_l\)

Conductance of transmission line l

\(V_{i}\), \(V_{j}\), \(\delta _{i}\), \(\delta _j\)

Voltage magnitudes and angles for buses i and j (p.u.)

s, u

Vector of dependent and control variables

\(f_\mathrm{obj}\)

Objective function

g(su)

Equality constraints of vector s and u

h(su)

Inequality constraint of vector s and u

\(P_i\), \(Q_i\)

Active and reactive power injection of bus i (MW)

\(P_{Di}\), \(Q_{Di}\)

Active and reactive power load demand of bus i (MW)

\(G_{ij}\), \(B_{ij}\)

Conductance and susceptance between bus i and j

\(P_{Gi}\), \(Q_{Gi}\), \(V_{Gi}\)

Active power, reactive power and voltage magnitude of generator bus i

\(T_i\)

Tap setting of transformer i

\(Q_{Ci}\)

Shunt VAR compensation of compensator i

\(S_{Li}\)

Line flow of transmission line i

\(P_{Gi}^{\min }\), \(P_{Gi}^{\max }\)

Minimum and maximum active power output of conventional generator i (MW)

\(Q_{Gi}^{\min }\), \(Q_{Gi}^{\max }\)

Minimum and maximum reactive power output of conventional generator i (MW)

\(T_i^{\min }\), \(T_i^{\max }\)

Minimum and maximum tap setting of transformer i

\(Q_{Ci}^{\min }\), \(Q_{Ci}^{\max }\)

Minimum and maximum shunt VAR compensation by compensator i

\(V_{Li}^{\min }\), \(V_{Li}^{\max }\)

Minimum and maximum voltage magnitude of load bus i

\(S_{Li}^{\min }\), \(S_{Li}^{\max }\)

Minimum and maximum line flow of transmission line i

\(N_\mathrm{line}\)

Total number of transmission lines

\(N_{Tr}\)

Number of transformers

\(N_C\)

Number of shunt VAR compensators

\(N_\mathrm{obj}\)

Number of objective functions

\(\lambda \)P, \(\lambda \)V, \(\lambda \)Q, \(\lambda \)S

Penalty coefficients

k, \(k_{\max }\)

Current and maximum number of iterations

n

Number of particles

D

Number of optimized variables

\(v_{id}^k\), \(x_{id}^k\)

Velocity and position of particle i in dimension d of k iteration

\(v_{id,\min }\), \(v_{id,\max }\)

Minimum and maximum values of particle velocity

\(x_{id,\min }\), \(x_{id,\max }\)

Minimum and maximum values of position

\(\omega (k)\)

Inertia weight of iteration k

\(\omega _{\max }\), \(\omega _{\min }\)

Maximum and minimum value of inertia weight

\(C_1(k)\), \(C_s(k)\)

Cognitive and social acceleration coefficients of iteration k

\(C_{1,\min }\), \(C_{1,\max }\)

Minimum and maximum cognitive acceleration coefficients

\(C_{2,\min }\), \(C_{2,\max }\)

Minimum and maximum social acceleration coefficients

\(pbest_{id}^k\)

Local best particle i in dimension d of iteration k

\(pbest_{id}^k\)

Global best particle

\(r_1\),\(r_2\), \(r_3\), \(r_4\)

Random numbers between 0 to 1 with uniform distribution

\(P_{ltg}\), \(P_{fr}\)

Lethargy and freak factors

\(v_{id}^{frk}\)

Normal random velocity of the interval [\(v_{id, \min }\), \(v_{id,\max }\)]

\(f_{\mathrm{obj},i}^{\min }\), \(f_{\mathrm{obj},i}^{\max }\)

Minimum and maximum values of the objective function i

\(f_i\)

Membership function of the objective function i

\(f^{j}\)

Selecting function of non-dominated solution j

m

Number of non-dominated solutions

Notes

Acknowledgements

The authors hereby acknowledge the partial financial support provided by the King HRD Scholarship, Energy Conservation and Promotion Fund, Ministry of Energy of Thailand.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Energy, Environment and Climate Change, School of Environment, Resources and DevelopmentAsian institute of TechnologyPathum ThaniThailand

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