# Enhanced multi-objective crisscross optimization for dynamic economic emission dispatch considering demand response and wind power uncertainty

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## Abstract

In this paper, the dynamic economic emission dispatch problem in electric power system is formulated as a multi-objective optimization problem in a smart grid perspective. Accordingly, two additional subproblems are included in the dynamic economic emission dispatch formulation. Firstly, the wind power generation is penetrated into the system such that the uncertain power varies between a predicted upper and lower bounds. Secondly, a demand response program is implemented at the customer end, to modify the consumption pattern of electricity according to different electricity prices at valley, peak and off-peak periods. A two-level optimization is proposed to determine the optimal schedule of generating units such that the upper level solves for the minimization of cost and emission, whereas the lower level minimizes the wind power output interval reduction. An enhanced multi-objective crisscross optimization using non-dominated sorting approach is proposed as main optimizer to solve upper-level problem, and a linear programming is adopted to solve the lower-level problem. A 10-unit system is taken as a case study for demonstration, and the result shows the effectiveness of proposed formulation in terms of minimizing cost and emission.

## Keywords

Dynamic economic emission dispatch Demand response Wind power Crisscross optimization Smart grid## List of symbols

*a*_{i},*b*_{i},*c*_{i},*e*_{i},*f*_{i}Cost coefficients of generating unit

*i**α*_{i},*β*_{i},*γ*_{i},*η*_{i},*δ*_{i}Emission coefficients of generating unit

*i**P*_{i,min},*P*_{i,max}Minimum and maximum power generating capacity of unit

*i**B*_{ij},*B*_{io},*B*_{00}Transmission loss coefficients

*DR*_{i},*UR*_{i}Down/up ramp rate limit of unit

*i**P*_{i, t}Power output of unit

*i*at time*t**P*_{l, t}Transmission line loss at time

*t**P*_{d, t}Power demand at time

*t*- \( \underline{w}_{t} ,\overline{w}_{t} \)
Lower and upper bounds of predicted wind power

- \( \tilde{w}_{t} \)
Actual wind power at time

*t*- \( \tilde{P}_{t,i} \)
Actual power output of generating unit

*i*at time*t**Δw*_{t}Wind power curtailment at time

*t**ɛ*_{i}Scaling factor

*ρ*_{A},*ρ*_{P},*ρ*_{OP},*ρ*_{V}Initial average, peak, off-peak, valley electricity price

- El
Elasticity

- NP
Net profit

- PR
Profit of customer

*M*Population size

*D*Number of variables

*X*Random population

- MH
Horizontal moderation solutions

- DH
Dominant horizontal solutions

- MV
Vertical moderation solutions

- DV
Dominant vertical solutions

- LF
Load factor

- PV
Peak valley factor

- Peak
_{comp} Peak compensation

## Notes

### Acknowledgements

This study was not funded by any individual/organization.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

### Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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