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Formulation and application of quantum-inspired tidal firefly technique for multiple-objective mixed cost-effective emission dispatch

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Abstract

In this manuscript, a new quantum computing-based optimization algorithm is proposed to solve multiple-objective mixed cost-effective emission dispatch (MEED) problem of electrical power system. The MEED problem aims at maintaining proper balance between emission of pollutants and generation of power. The problem has been formulated here using cubic equation to reduce the nonlinearities of the system. It is transformed to single-objective problem by considering max to max penalty factor. The proposed optimization technique is inspired by the concept of quantum mechanics, gravitational force and firefly algorithm (FA) and is termed as quantum-inspired tidal FA (QITFA). The proposed QITFA is tested on IEEE 14-bus and IEEE 30-bus test system for four different load conditions. The obtained results are compared with the results yielded by some other state-of-the-art methods like Lagrangian relaxation method, particle swarm optimization (PSO), simulated annealing, quantum-behaved bat algorithm and quantum PSO. This paper proves the superiority of the proposed QITFA over all these methods. Further, the obtained results also suggest its effective and efficient implementation in MEED problem.

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Abbreviations

A2MO:

Artificial mountain ape optimization

ABC:

Artificial bee colony

COOA:

Competitive optimization algorithm

DBC:

Dance bee colony

FA:

Firefly algorithm

FMOPSO:

Fuzzified many-objective PSO

FSA:

Fractal search algorithm

GA:

Genetic algorithm

GSA:

Gravitational search algorithm

IEEE:

Institute of Electrical and Electronics Engineers

LF-VPSO:

Levy flight-based voltage PSO

LR:

Lagrangian relaxation

MEED:

Multiple-objective mixed cost-effective emission dispatch

MHSA:

Modified harmony search algorithm

PF:

Penalty factor

PSO:

Particle swarm optimization

QBAT:

Quantum-behaved bat algorithm

QITFA:

Quantum-inspired tidal FA

QPSO:

Quantum PSO

SA:

Simulated annealing

SOA:

Spiral optimization algorithm

TDSE:

Time-dependent Schrodinger equation

TFA:

Tidal FA

TLBA:

Teacher-learning-based algorithm

\({\text{CO}}_{2}\) :

Carbon dioxide

\(d_{ij}\) :

Gap between firefly i and j

\(D\) :

Number of dimensions

\(f_{i}\) :

Tidal component of the ith firefly

G :

Exploration factor

G:

Gravitational constant

\(h_{{i( {\max/\max} )}}\) :

Max/max penalty factor of the ith generating unit

\(J_{i} \left( {P_{i} } \right)\) :

Emission function of the ith generator

\(J_{{i,{\text{SO}}_{\text{x}} }} \left( {P_{i} } \right)\) :

Emission function for SOx

\(J_{{i,{\text{NO}}_{\text{x}} }} \left( {P_{i} } \right)\) :

Emission function for NOx

\(J_{{i,{\text{CO}}_{2} }} \left( {P_{i} } \right)\) :

Emission function for CO2

\(m_{1}\) :

Mass of one particle

\(m_{2}\) :

Mass of the other particle

\(n_{g}\) :

Total numbers of generators till last generating unit

\(n_{f}\) :

Number of fireflies

\({\text{NO}}_{\text{x}}\) :

Nitrogen oxide

P :

Generated power

\(P_{\text{D}}\) :

Power demand

\(P_{\text{L}}\) :

Total losses

\(P_{i,mu}\) :

Least power limit of the ith generator

\(P_{i,mx}\) :

Extreme power limit of the ith generator

\({\text{SO}}_{\text{x}}\) :

Sulfur oxide

\(x_{i,m}\) :

The position of the ith firefly in mth dimension

\(x_{j,m}\) :

The position of the jth firefly in mth dimension

\(\alpha\) :

Damping factor

\(\alpha_{0}^{{\prime }}\) :

Random constants in [0, 1]

\(\beta^{{\prime }}\) :

Random constants in [0, 1]

\(\beta\) :

Attraction factor

\(\beta_{0} , \gamma\) :

Random constants lying in between 0 and 1

\(\alpha_{0,i} , \alpha_{1,i} , \alpha_{2,i} ,\alpha_{3,i}\) :

Emission coefficient of the ith generator

\(\omega_{0,i} , \omega_{1,i} , \omega_{2,i} , \omega_{3,i}\) :

Fuel cost coefficient of the ith generator

\(\alpha_{{0\left( {{\text{SO}}_{\text{x}} } \right),i}} , \alpha_{{1\left( {{\text{SO}}_{\text{x}} } \right),i}} ,\alpha_{{2\left( {{\text{SO}}_{\text{x}} } \right),i}} , \alpha_{{3\left( {{\text{SO}}_{\text{x}} } \right),i}}\) :

SOx emission coefficient of the ith generator

\(\alpha_{{0\left( {{\text{CO}}_{2} } \right),i}} , \alpha_{{1\left( {{\text{CO}}_{2} } \right),i}} ,\alpha_{{2\left( {{\text{CO}}_{2} } \right),i}} , \alpha_{{3\left( {{\text{CO}}_{2} } \right),i}}\) :

CO2 emission coefficient of the ith generator

\(\alpha_{{0\left( {{\text{NO}}_{\text{x}} } \right),i}} , \alpha_{{1\left( {{\text{NO}}_{\text{x}} } \right),i}} ,\alpha_{{2\left( {{\text{NO}}_{\text{x}} } \right),i}} , \alpha_{{3\left( {{\text{NO}}_{\text{x}} } \right),i}}\) :

NOx emission coefficient of the ith generator

\(\varPsi \left( {d,t} \right)\) :

Position of the wave

\(\hbar\) :

Planck’s constant

\(u\) :

Random variable ranging from 0 to 1

\(\nabla^{2}\) :

Laplace operator

\(t_{\max }\) :

Maximum number of iterations

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Authors and Affiliations

  1. Department of Electrical Engineering, Indian Institute of Technology (Indian School of Mines), Dhanbad, India

    Kapil Deo Bodha & Vivekananda Mukherjee

  2. Department of Electrical Engineering, Delhi Technological University, New Delhi, India

    Vinod Kumar Yadav

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  1. Kapil Deo Bodha
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Correspondence to Vinod Kumar Yadav.

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Bodha, K.D., Yadav, V.K. & Mukherjee, V. Formulation and application of quantum-inspired tidal firefly technique for multiple-objective mixed cost-effective emission dispatch. Neural Comput & Applic 32, 9217–9232 (2020). https://doi.org/10.1007/s00521-019-04433-0

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