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Optimal power flow solution using a learning-based sine–cosine algorithm

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Abstract

The Sine–Cosine algorithm (SCA) is efficient but faces challenges in exploitative abilities, slow convergence, and exploration–exploitation balance. This study proposes a novel optimization method, the learning-based sine–cosine algorithm (L-SCA), to solve the optimal power flow (OPF) problem. The basic SCA has been modified with a learning phase operator inspired by TLBO. The SCA handles global exploration, while the learner phase of teaching–learning based optimization (TLBO) offers strong local search capabilities, which can be utilized to enhance the solution neighborhood space provided by the SCA technique. The L-SCA and original SCA algorithms address OPF in IEEE 57-bus, Algerian 59-bus, and IEEE 118-bus power systems, considering twelve cases with a focus on cost savings, voltage stability, voltage profile, emissions, and power losses. The comparative study shows that the proposed L-SCA consistently outperforms standard SCA and other reported methods in all cases for varied-scale standard test systems as well as for a practical power system, within reasonable execution times. For instance, L-SCA in the Algerian 59-bus system cut fuel costs by around 13.13% compared to initial case, equating to annual savings of $2.2 million, while in the IEEE-118 bus system, power loss is significantly reduced to 17.881 MW, marking an 86.5% reduction compared to the base case.

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Data availability

The data supporting the findings of this study are available upon request from the corresponding author. The IEEE 57-bus, Algerian 59-bus, and IEEE 118-bus test system data can be found in references [75, 83], and [88, 89], respectively.

Abbreviations

\(Z_{\min } (x,u)\) :

Objective function

\(x,u\) :

State variables, control variables

\(g,h\) :

Equality and inequality constraints respectively

\(P_{{{\text{losses}}}} ,Q_{{{\text{losses}}}}\) :

Total active and reactive power losses respectively

\(P_{{G_{i} }} ,Q_{{G_{i} }}\) :

Active and reactive power of ith generator

\(P_{{D_{i} }} ,Q_{{D_{i} }}\) :

Active and reactive power demands at ith bus

\(S_{line} ,S_{{line_{i} }}^{\max }\) :

Line flow limit in MVA, upper limit of ith line

NB:

Number of busses

NG, NPQ:

Number of generators, Load busses

NL:

Number of transmission lines

\(P_{{G_{1} }}\) :

Power generated at slack bus

\(P_{{G_{i} }}^{\min } ,P_{{G_{i} }}^{\max } ,Q_{{G_{i} }}^{\min } ,Q_{{G_{i} }}^{\max }\) :

Limits of active and reactive powers of ith generator respectively

\(V_{{G_{i} }}^{\min } ,V_{{G_{i} }}^{\max }\) :

ith Generator voltage limits

\(V_{{L_{i} }}^{\min } ,V_{{L_{i} }}^{\max }\) :

Load bus voltage limits at ith bus

\(T,NT\) :

Regulating transformer tap setting, count of \(T\)

\(T_{i}^{\min } ,T_{i}^{\max }\) :

Limits for discrete tap settings

\(Q_{c} ,{\text{NC}}\) :

Shunt VAR compensation, Count of \(Q_{c}\) devices

\(Q_{{\mathop c\nolimits_{i} }}^{\min } ,Q_{{\mathop c\nolimits_{i} }}^{\max }\) :

Limits on output of shunt VAR compensators at ith bus

\(a_{i} ,b_{i} ,c_{i}\) :

Fuel cost coefficients for the ith generator

\(\alpha_{i} ,\beta_{i} ,\gamma_{i} ,\omega_{i} ,\mu_{i}\) :

Coefficients used to model the emissions produced by the ith generator unit

\(L_{j}\) :

L-index of any jth load bus

\(Y_{ji}\) :

Mutual admittance of line between jth and ith bus

\(Y_{jj}\) :

Sum of admittances connected to jth load bus

\(W\) :

Matrix obtained by partial inversion of bus admittance matrix

\(G_{{L_{i - j} }}\) :

Conductance of line \(L\) between ith and the jth bus

\(\delta_{ij}\) :

Difference in phase angle between ith and the jth bus voltages

\(x_{i,j}^{t} ,P_{{{\text{best}}_{i,j} }}^{t}\) :

Position of current solution and best solution respectively for \(i{\text{th}}\) search agent at \(t{\text{th}}\) iteration in \(j{\text{th}}\) dimension

\(t,t_{\max }\) :

Current iteration number and maximum number of iterations respectively

\(a\), User defined constant:

\(a\) = 2

SCA:

Sine–cosine algorithm

TLBO:

Teaching–learning based optimization

MGOA:

Modified grasshopper optimization algorithm

DSA:

Differential search algorithm

MOMICA:

Multi-objective modified imperialist competitive algorithm

NKEA:

Neighborhood knowledge-based evolutionary algorithm

ESDE-MC:

Enhanced self-adaptive differential evolution with mixed crossover

BHBO:

Black-hole based optimization

ACO:

Ant colony optimization

LCA:

League championship algorithm

PGA:

Parallel GA

FSLP:

Fast successive linear programming

GWO:

Grey wolf optimizer

FPA:

Flower pollination algorithm

ICBO:

Improved colliding bodies optimization

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Funding

This research study was conducted using internal resources and support from the Electrical Engineering Department of Delhi Technological University. No external funding was received for this study.

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Authors and Affiliations

  1. Department of Electrical Engineering, Delhi Technological University, Delhi, 110042, India

    Udit Mittal, Uma Nangia, Narender Kumar Jain & Saket Gupta

  2. Department of Instrumentation and Control Engineering, Bharati Vidyapeeth’s College of Engineering, New Delhi, 110063, India

    Saket Gupta

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  1. Udit Mittal
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  2. Uma Nangia
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  3. Narender Kumar Jain
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  4. Saket Gupta
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Contributions

UM and SG contributed to conceptualization, methodology, development of algorithm and analysis of simulation results. UM contributed to the writing of the original draft, reviewing and editing. UN and NKJ contributed to conceptualization, validation, reviewing and editing and supervision of the work. All authors read and approved the final manuscript.

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Correspondence to Udit Mittal or Saket Gupta.

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Mittal, U., Nangia, U., Jain, N.K. et al. Optimal power flow solution using a learning-based sine–cosine algorithm. J Supercomput (2024). https://doi.org/10.1007/s11227-024-06043-7

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