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Modified grasshopper optimization framework for optimal power flow solution

Abstract

This paper proposes a modified grasshopper optimization algorithm (MGOA) to solve the optimal power flow (OPF) problem. The conventional GOA is a recent optimization technique that is conceptualized from the natural lifestyle of grasshopper including their movement and migration. The MGOA is based on modifying the mutation process in the conventional GOA in order to avoid trapping into local optima. Different single- and multi-objective functions are solved using the proposed optimization technique. These objective functions consist of quadratic fuel cost minimization, emission cost minimization, active power loss minimization, quadratic fuel cost and active power loss minimization, quadratic fuel cost minimization and voltage profile improvement, quadratic fuel cost minimization and voltage stability improvement, quadratic fuel cost minimization and emission minimization, quadratic fuel cost and power loss minimization, voltage profile and voltage stability improvement. The proposed technique is validated using standard IEEE 30-bus, IEEE 57-bus, and IEEE 118-bus test systems with thirteen case studies. Simulation results reveal the better performance and superiority of the proposed technique to solve various OPF problems compared with well-recognized evolutionary optimization techniques stated in the literature review.

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Abbreviations

A i :

The wind effects

a i, b i, c i :

Cost coefficient of the i-th generator

C :

Factor to reduce repulsion, comfort zone, and attraction zone

c 1 and c 2 :

Social behavior coefficients

d ij :

Distance between the i-th and j-th grasshoppers

\( \widehat{{d_{ij}}} \) :

Unit vector between the i-th and j-th grasshoppers

\( \widehat{{e_{\text{g}}}} \) :

Unit vector in the direction of earth center

\( \widehat{{e_{\text{w}}}} \) :

Unit vector in the direction of wind

f :

Attraction strength

g i(x, u):

Equality constraints

g :

The gravitational constant

G i :

The gravitational force

G ij, B ij :

Conductance and susceptance of the admittance matrix

h i(x, u):

Inequality constraints

j :

Number of subjects

i :

Teaching–learning cycle

k :

The population size for TLBO

K P, K Q, K V, K S :

Penalty factors of active, reactive power, voltage, and apparent power

K w :

Variable bandwidth

l :

Length of attraction

lbd, ubd :

Lower and upper limit in D-th dimension

L j :

Voltage stability local indicator of bus j

M j,i :

The average result of learner

N :

Population size (the number of individuals)

NB:

Number of buses

NC:

Number of shunt compensators

NG:

Number of generators

NPV:

Number of generation buses

NB:

Number of buses

NC:

Number of shunt compensators

NG:

Number of generators

NPV:

Number of generation buses

N par :

Dimension of a problem (GA problem)

NPQ:

Number of load buses

NTL:

Number of transmission lines

NT:

The number of regulating transformers

o :

Constant drift

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

Active power generation of slack bus

\( P_{{G_{1}}}^{\min},P_{{G_{1}}}^{\max} \) :

Active power generation limits of slack bus

\( P_{{G_{i}}}^{\min},P_{{{\text{G}}_{i}}}^{\max} \) :

Active power generation limits of bus i

P G, P D :

Active power generation and load demand, respectively

P loss, Q loss :

Active and reactive power transmission losses

\( {\text{PAR}}_{ {\rm max} } \), \( {\text{PAR}}_{ {\rm min} } \) :

Limits of pitch adjusting rate

p gd :

The complete outstanding position

\( p_{{N_{\text{par}}}} \) :

Chromosome encoded

p id :

The outstanding position of particle i

P(C):

The probability of the selected chromosome

Q C :

Shunt VAR compensation

Q G, Q D :

Reactive power generation and load demand

\( Q_{{{\text{G}}_{i}}}^{\min},Q_{{{\text{G}}_{i}}}^{\max} \) :

Reactive power generation limits of the shunt VAR

r 1, r 2, and r 3 :

Random numbers lie in [0,1]

S i :

The social relationship interaction

\( S_{{{\text{l}}_{i}}},S_{ {\rm max} } \) :

Apparent power flow of ith line and its maximum

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

Upper and lower limits of regulating transformer i

T F :

The teaching factor

u :

Vector of the control variables

\( v_{id} \left({t + 1} \right) \) :

The existing position of particle i

\( V_{{{\text{L}}_{i}}}^{ {\rm min} },V_{{{\text{L}}_{i}}}^{ {\rm max} } \) :

Upper and lower limits of voltage magnitude load bus i

\( V_{{{\text{L}}_{i}}} \) :

Voltage magnitude at load bus i

VD:

Load bus voltage deviation

\( V_{{{\text{G}}_{i}}} \) :

Voltage magnitude at PV buses

w :

The inertia constant

x :

Vector of dependent variables or state variables

X i :

Position of i-th grasshopper

\( x_{id} (t) \) :

The existing position of particle i

\( X_{{i\_{\text{new}}}} \) :

Mutation vector

x max, x min :

State variable limits

Y ij :

Admittance matrix between bus i and bus j

γ i, β i, α i, ζ i, and λ i :

Coefficients of the i-th generator emission

Y ij :

Admittance matrix between bus i and bus j

γ i, β i, α i, ζ i, and λ i :

Coefficients of the i-th generator emission

θ :

Polar angle

δ ij :

Phase angle difference between buses i and j

\( \lambda_{{L_{ {\rm max} }}} \) :

Weighting factor of the Lmax with cost term

λ i :

Weighting factor of the emission with the cost term

λ VD :

Weighting factor of the VD term with the cost term

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Appendix

Appendix

See Tables 2, 3, 4, 5, 6, 7, 8, 9, 10 and 11.

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Taher, M.A., Kamel, S., Jurado, F. et al. Modified grasshopper optimization framework for optimal power flow solution. Electr Eng 101, 121–148 (2019). https://doi.org/10.1007/s00202-019-00762-4

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Keywords

  • Modified grasshopper optimization technique
  • Optimal power flow
  • Power system optimization
  • Metaheuristic