Scheduling of combined heat and generation outputs in power systems using a new hybrid multi-objective optimization algorithm

  • Soheil DolatabadiEmail author
  • Ragab A. El-Sehiemy
  • Saeid GhassemZadeh
Original Article


In this paper, a hybrid optimization algorithm, consisted of weighted vertices-based optimizer (WVO) and particle swarm optimization (PSO) algorithm, is proposed to solve three economic frameworks for scheduling of power sources in order to meet the required power demand in power systems. These frameworks are economic power dispatch, economic emission power dispatch and combined heat and economic power dispatch problems. The basic idea of weighted vertices optimizer (WVO) is given from the bisection root-finding method in mathematics. It uses swarm intelligence and evolutionary strategy to efficiently find the optimum solution. However, the original WVO algorithm has some flaws in complex problems with a high number of variables and constraints. Therefore, this paper presents hybrid WVO–PSO algorithm which solved the mentioned flaws and also improved its speed and accuracy. In this algorithm, varying speed is defined for each vertex by using PSO which helps better exploration through the search space. To evaluate the performance of WVO–PSO, it is applied to some of well-known and complex emission/economic dispatch (EED), combined heat and power economic dispatch (CHPED) and combined heat and power emission/economic dispatch (CHPEED) problems and then the driven results are compared with other recent methods which demonstrates better performance of the proposed method in solving non-convex and constrained EED, CHPED and CHPEED problem in terms of minimizing costs and emissions.


Emission and economic dispatch Combined heat and power unit Multi-objective optimization Weighted vertices optimizer Hybrid optimization algorithm 

List of symbols


Total fuel cost


Total gas emission


Total network losses


Output power


Output thermal


Total load demand


Total thermal demand

am, bm, cm

Coefficients of CHP feasible region

ai, bi, ci, ei, fi

Fuel cost coefficients of ith co-generation unit

\( \alpha_{i} ,\beta_{i} ,\gamma_{i} , \xi_{i} ,\lambda_{i} \)

Fuel cost coefficients of ith conventional unit

gi, hi, ki

Fuel cost coefficients of ith heat-only unit

\( \alpha_{i}^{*} ,\beta_{i}^{*} ,\gamma_{i}^{*} , \xi_{i}^{*} ,\lambda_{i}^{*} ,\tau_{j}^{*} \)

Emission coefficients of ith conventional unit

\( \theta_{j}^{*} , \eta_{j}^{*} ,\psi_{j}^{*} \)

Emission coefficients of ith co-generation unit

\( \sigma_{j}^{*} ,\pi_{j}^{*} ,\rho_{j}^{*} \)

Emission coefficients of ith heat-only unit


Loss coefficient


Number of conventional units


Number of co-generation units


Number of heat-only units


Uniform distribution function


ith objective function


Number of optimization variables


Number of population


Number of vertices


Coefficients of moving forward


Coefficient of moving backward


Coefficients of moving toward the global best


Selected vertices set


Geometrical center of vertices


Nearest solution to global best


Farthest solution to global worst

ν (Nu)

Vertex speed


Vertex speed damp coefficient


Number of iteration


1 × n random numbers between 0 and 1


\( x{ \preccurlyeq }y \)

x weakly dominates y

\( x \prec y \)

x strictly dominates y



Co-generation unit


Heat-only unit


Conventional unit



Dr. Ragab A. El-Sehiemy would like to acknowledge the support provided by STDF-IFE collaboration through the project # 31161 during the period from April 1, 2019, to September 30, 2019.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interest.

Supplementary material

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2019

Authors and Affiliations

  1. 1.Electrical and Computer Engineering DepartmentUniversity of TabrizTabrizIran
  2. 2.Electrical Engineering Department, Faculty of EngineeringKafrelsheikh UniversityKafr El-ShaikhEgypt

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