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

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## Abstract

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.

## Keywords

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

*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

*i*th co-generation unit- \( \alpha_{i} ,\beta_{i} ,\gamma_{i} , \xi_{i} ,\lambda_{i} \)
Fuel cost coefficients of

*i*th conventional unit*g*_{i},*h*_{i},*k*_{i}Fuel cost coefficients of

*i*th heat-only unit- \( \alpha_{i}^{*} ,\beta_{i}^{*} ,\gamma_{i}^{*} , \xi_{i}^{*} ,\lambda_{i}^{*} ,\tau_{j}^{*} \)
Emission coefficients of

*i*th conventional unit- \( \theta_{j}^{*} , \eta_{j}^{*} ,\psi_{j}^{*} \)
Emission coefficients of

*i*th co-generation unit- \( \sigma_{j}^{*} ,\pi_{j}^{*} ,\rho_{j}^{*} \)
Emission coefficients of

*i*th 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

*U*(.)Uniform distribution function

*f*_{i}*i*th 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

## Symbols

- \( x{ \preccurlyeq }y \)
*x*weakly dominates*y*- \( x \prec y \)
*x*strictly dominates*y*

## Superscripts

- C
Co-generation unit

- H
Heat-only unit

- P
Conventional unit

## Notes

### Acknowledgement

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

## References

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