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


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.

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Total fuel cost


Total gas emission

P Loss :

Total network losses

P :

Output power

T :

Output thermal

P D :

Total load demand

T D :

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

B ij :

Loss coefficient

N P :

Number of conventional units

N C :

Number of co-generation units

N H :

Number of heat-only units


Uniform distribution function

f i :

ith objective function

N Var :

Number of optimization variables

N Pop :

Number of population

N V :

Number of vertices

C F :

Coefficients of moving forward

C B :

Coefficient of moving backward

C G :

Coefficients of moving toward the global best

S sel :

Selected vertices set

Geoc :

Geometrical center of vertices

d min :

Nearest solution to global best

d max :

Farthest solution to global worst

ν (Nu):

Vertex speed

ν d :

Vertex speed damp coefficient


Number of iteration

R n :

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


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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.

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Correspondence to Soheil Dolatabadi.

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Dolatabadi, S., El-Sehiemy, R.A. & GhassemZadeh, S. Scheduling of combined heat and generation outputs in power systems using a new hybrid multi-objective optimization algorithm. Neural Comput & Applic 32, 10741–10757 (2020).

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