# A Novel Approach of Congestion Management in Deregulated Power System Using an Advanced and Intelligently Trained Twin Extremity Chaotic Map Adaptive Particle Swarm Optimization Algorithm

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## Abstract

This paper addresses the implementation of an advanced twin extremity chaotic map adaptive particle swarm optimization (TECM-PSO) algorithm to the nonlinear congestion management cost problem in deregulated power system. The goal of proposed approach is twofold: firstly, to identify accurate number of participating generators for rescheduling process using a robust upstream real capacity tracing method requiring less information of generator units, and secondly, to achieve minimum possible rescheduled generation cost function using TECM-PSO algorithm while alleviating all the line overloads. Further to preserve the diversity of the algorithm and to increase its near-global searching capability, the incursion of dynamic constraint handling has also been done in the algorithm to retrieve the feasible solutions in the search space. The objective function is solved for near-global optima by step-by-step execution of the proposed algorithm. Twin extremity chaotic maps have been generated by updating the equations governing the PSO algorithm in order to prevent the particle swarm optimization plugging into local minima with less convergence rate at later stages of iterations. The feasibility of the proposed algorithm is validated on various line outage cases of both the small and large test systems, namely modified IEEE 30-, IEEE 57- and IEEE 118-bus systems. Simulation results show a considerable reduction in net rescheduled generation cost, power losses and rescheduled generation amount, ensuring more secure and reliable operation of power system.

## Keywords

Nonlinear congestion cost management Upstream real capacity tracing method Generator rescheduling Twin extremity chaotic map adaptive particle swarm optimization Rescheduled generation cost function## Abbreviations

- TECM-PSO
Twin extremity chaotic map adaptive particle swarm optimization

- URCT
Upstream real capacity tracing

- RPCF
Real power contribution factor

- CM
Congestion management

- CV
Cumulative violation

- MFE
Main function evaluation

- AF
Auxiliary function

- RGC
Rescheduled generation cost

- NGR
Net generation rescheduled

- RGCF
Rescheduled generation cost function

- CMR
Convergence mobility rate

## Decision Variables

- \(\hbox {TECM}r\)
Twin extremity chaotic mapped sequence

- \(\hbox {TECM}r^{0}\)
Initial value of twin extremity chaotic mapped sequence

- \(C_\mathrm{g}^+ \)
Incremental price bids submitted by generators at which they are willing to adjust their real power outputs in $/MWh

- \(C_\mathrm{g}^- \)
Decremented price bids submitted by generators at which they are willing to adjust their real power outputs in $/MWh

- Npg
Number of participating generators which are decided using upstream real capacity tracing algorithm (described in “Appendix A1”)

- \(\Delta P_\mathrm{GK}\)
Change in the real power adjustment at bus k in MW.

*i*Participating generator

- \(P_{{\mathrm{g}}_{i}}^0 \)
Active power generated by the

*i*th generator as determined by the system operator in MW- \(\Delta P_{{\mathrm{g}_i}}\)
Change in real power by the

*i*th generator in MW- \(P_{{\mathrm{g}_{i}}}^\mathrm{resh}\)
Active power generated by the

*i*th generator after the process of rescheduling in MW*j*Non-participating generator

- \(P_{{\hbox {D}_{m}}}^0\)
Active power consumed by the

*m*th load determined by the system operator in MW- Nd
Number of loads

*m*Individual load at each bus

- \(P_\mathrm{L}\)
Active power loss in MW

- \(V_{{i_\mathrm{D}}} ^{k+1}\)
Updated velocity of the ith particle

- \(V_{{i_\mathrm{D}}} ^{k}\)
Old velocity of the

*i*th particle- \(P_{{i_\mathrm{D}}} ^{k+1}\)
Updated position of the

*i*th particle- \(P_{{i_\mathrm{D}}} ^{k}\)
Old position of the

*i*th particle

## Constant Variables

- \(c_1\)
Cognitive accélération coefficient

- \(c_2 \)
Social accélération coefficient

- \(r_i\)
Random numbers between 0 and 1. \(i=1,2\)

*w*Inertia weight

- \(w_{\max } \)
Maximum inertia weight

- \(w_{\min } \)
Minimum inertia weight

*k*Current iteration

- \(k_{\max } \)
Maximum number of iterations

- \(N_\mathrm{PQ} \)
Number of PQ buses

- \(N_\mathrm{PV} \)
Number of PV buses

- \(S_{\mathrm{L}_{\max }} \)
Maximum line limit of the line in MVA

- \(S_\mathrm{L} \)
MVA power flow in the line

- \(G_{ij} \)
Conductance of the transmission line between

*i*and*j*buses- \(B_{ij}\)
Susceptance of the transmission line between

*i*and*j*buses- \(P_i\)
Real power of the

*i*th bus in MW- \(Q_i\)
Reactive power of the

*i*th bus in MVAR- \(V_i\)
Voltage magnitude at

*i*th bus- \(N_{B-1} \)
Number of buses except slack bus

- \(V_{{\mathrm{G}_{i,\mathrm{min}}} } \) and \(V_{{\mathrm{G}_{i,\mathrm{max}}} } \)
Minimum and maximum voltage limits, respectively, of PV buses

- \(V_{{\hbox {L}_{i}}}\)
Load bus voltage of

*i*th bus- \(V_{{\mathrm{L}_{i, \mathrm{min}}} } \) and \(V_{{\mathrm{L}_{i, \mathrm{max}}} } \)
Minimum and maximum values of voltages of the

*i*th bus, respectively- \(Q_{{\mathrm{G}_{i,\min }}} \) and \(Q_{{\mathrm{G}_{i,\max }}} \)
Minimum and maximum reactive power generation limits in MVAR

- \(P_{{\mathrm{G}_{i,\mathrm{min}}} } \) and \(P_{{\mathrm{G}_{i, \mathrm{max}}} } \)
Minimum and maximum real power generation limits of PV buses in MW

- \(x_{\mathrm{best},d}^k \)
Best individual in the

*k*th iteration for*d*th dimension- \({{\overline{x}}} _{{\mathrm{best},d} }\)
Mean value of the best individuals

## Greek Symbols

- \(\lambda \)
Lyapunov exponent

- \(\theta _{ij} \)
Angle between

*i*and*j*buses- \(\eta \)
Number of iterations for which stopping criterion applies

- \(\in \)
Standard deviation threshold for which stopping criterion applies

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## Notes

### Acknowledgements

We are highly thankful to Thapar University, Patiala, to Grant TEQIP-II (Center of Excellence) financial assistance to carry out this research.

### Compliance with ethical standards

### Conflict of interest

We have no conflict of interest with any of the ethical statements.

## Supplementary material

## References

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