Abstract
In this paper, a hybrid optimization algorithm, consisted of weighted verticesbased 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 rootfinding 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 wellknown 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 nonconvex and constrained EED, CHPED and CHPEED problem in terms of minimizing costs and emissions.
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Abbreviations
 C(.):

Total fuel cost
 E(.):

Total gas emission
 P _{Loss} :

Total network losses
 P :

Output power
 T :

Output thermal
 P _{D} :

Total load demand
 T _{D} :

Total thermal demand
 a_{m}, b_{m}, c_{m} :

Coefficients of CHP feasible region
 a_{i}, b_{i}, c_{i}, e_{i}, f_{i} :

Fuel cost coefficients of ith cogeneration unit
 \( \alpha_{i} ,\beta_{i} ,\gamma_{i} , \xi_{i} ,\lambda_{i} \) :

Fuel cost coefficients of ith conventional unit
 g_{i}, h_{i}, k_{i} :

Fuel cost coefficients of ith heatonly 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 cogeneration unit
 \( \sigma_{j}^{*} ,\pi_{j}^{*} ,\rho_{j}^{*} \) :

Emission coefficients of ith heatonly unit
 B _{ ij } :

Loss coefficient
 N _{P} :

Number of conventional units
 N _{C} :

Number of cogeneration units
 N _{H} :

Number of heatonly units
 U(.):

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
 Geo_{c} :

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
 iter:

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
 C:

Cogeneration unit
 H:

Heatonly unit
 P:

Conventional unit
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Acknowledgement
Dr. Ragab A. ElSehiemy would like to acknowledge the support provided by STDFIFE collaboration through the project # 31161 during the period from April 1, 2019, to September 30, 2019.
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Dolatabadi, S., ElSehiemy, R.A. & GhassemZadeh, S. Scheduling of combined heat and generation outputs in power systems using a new hybrid multiobjective optimization algorithm. Neural Comput & Applic 32, 10741–10757 (2020). https://doi.org/10.1007/s00521019046101
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Keywords
 Emission and economic dispatch
 Combined heat and power unit
 Multiobjective optimization
 Weighted vertices optimizer
 Hybrid optimization algorithm