An enhanced sitting–sizing scheme for shunt capacitors in radial distribution systems using improved atom search optimization

Abstract

In this paper, an enhanced sitting–sizing scheme for shunt capacitors (4SCs) in a radial distribution system (RDS) based on an improved atom search optimization (IASO) algorithm is proposed. IASO emulates the model of atomic motion in nature based on interaction forces among atoms. The main goal of the 4SCs problem is to reduce the line losses and minimize the capacitor installation cost by searching for the optimal location and sizing of the capacitors. This leads to improvements in the voltage profile and reliability of the system. The IASO algorithm is introduced to achieve the optimal sitting and sizing of capacitors for RDSs. The proposed IASO algorithm is benchmarked and validated on different radial systems, including the IEEE 33-bus, IEEE 34-bus, IEEE 65-bus, IEEE 85-bus and Marsa Matrouh networks, to demonstrate its performance in real-world applications. The results obtained by the proposed IASO algorithm are compared with the standard ASO, PSO, SCA, GWO and SSA algorithms. Furthermore, the significance of the obtained results is confirmed by performing a nonparametric statistical test, i.e., the Wilcoxon’s rank-sum at the 5% significance level. The comprehensive results demonstrate that the results obtained by the proposed IASO algorithm denominate the results obtained by the other algorithms and that IASO minimizes the operating costs while achieving better voltage profiles.

Appendix: Benchmark optimization problems with their characteristics

This section provides the description of each benchmark function regarding the mathematical model, function nature (i.e., unimodal, multimodal) and graph. The major motivation for this representation is to concentrate on the more challenging and complicated benchmark problems while validating the performance of a certain algorithm to depict a fire validation.

• 1. Schaffer (F1)

$$ f(x_{1} ,x_{2} ) = 0.5 + \frac{{\sin^{2} \left( {x_{1}^{2} + x_{2}^{2} } \right)^{2} - 0.5}}{{\left( {1 + 0.001\left( {x_{1}^{2} + x_{2}^{2} } \right)} \right)^{2} }}. $$

Characteristics: unimodal, nonconvex, continuous, non-separable, differentiable and defined on an n-dimensional space.

Search space: ${\mathbf{x}} = (x_{1} ,x_{2} ) \in \left[ { - 100,\,100} \right]$

Global minimum: $f({\mathbf{x}}) = 0$ at ${\mathbf{x}} = (0,0)$

• 2. Griewank (F2)

$$ f({\mathbf{x}}) = \frac{1}{4000}\sum\limits_{i = 1}^{n} {(x_{i} - 100)^{2} } - \varPi_{i = 1}^{n} \cos \left( {\frac{{x_{i} - 100}}{\sqrt i }} \right) + 1,\;\;{\text{with}}\,n = 100. $$

Characteristics: unimodal, nonconvex, continuous, non-separable and defined on an n-dimensional space.

Search space: ${\mathbf{x}} \in \left[ { - 600,\,600} \right]$

Global minimum: $f({\mathbf{x}}) = 0$ at ${\mathbf{x}} = 0.$

• 3. Rastrigin (F3)

$$ f({\mathbf{x}}) = 10n + \sum\limits_{i = 1}^{n} {\left( {x_{i}^{2} - 10\cos (2\pi x_{i} )} \right)} ,\;\;{\text{where}}\,n = 100. $$

Characteristics: multimodal, differentiable, continuous, separable, convex and defined on an n-dimensional space.

Search space: ${\mathbf{x}} \in \left[ { - 5.12,\,5.12} \right]$

Global minimum: $f({\mathbf{x}}) = 0$ at ${\mathbf{x}} = 0$

• 4. Rosenbrock (F4)

$$ f({\mathbf{x}}) = \sum\limits_{i = 1}^{n} {\left( {100(x_{i + 1}^{{}} - x_{i}^{2} )^{2} + (x_{i}^{{}} - 1)^{2} } \right)} + ,\;\;{\text{where}}\,n = 100. $$

Characteristics: Unimodal, differentiable, continuous, non-separable, convex and defined on an n-dimensional space.

Search space: ${\mathbf{x}} \in \left[ { - 30,\,30} \right]$

Global minimum: $f({\mathbf{x}}) = 0$ at ${\mathbf{x}} = 1.$

• 5. Ackley (F5)

$$ f({\mathbf{x}}) = - 20\,\exp \left( { - 0.2\sqrt {\frac{1}{n}\sum\nolimits_{i = 1}^{n} {x_{i} } } } \right) - \exp \left( {\frac{1}{n}\sum\limits_{i = 1}^{n} {\cos (2\pi x_{i} } )} \right) + 20 + \exp (1),\;\;{\text{where}}\,n = 100. $$

Characteristics: multimodal, nonconvex, continuous, non-separable and defined on an n-dimensional space.

Search space: ${\mathbf{x}} \in \left[ { - 32,\,32} \right]$

Global minimum: $f({\mathbf{x}}) = 0$ at ${\mathbf{x}} = 0.$

• 6. Penalized (F6)

$$ f({\mathbf{x}}) = 0.1\left\{ {\sin^{2} \left( {3\pi x_{1} } \right) + \sum\limits_{i = 1}^{n} {\left( {x_{i} - 1} \right)^{2} \left[ {1 + \sin (3\pi x_{i} + 1)} \right] + \left( {x_{n} - 1} \right)^{2} \left[ {1 + \sin^{2} (2\pi x_{n} )} \right]} } \right\} + \sum\nolimits_{i = 1}^{n} {u\left( {x_{i} ,\,5,100,4} \right)} . $$

Characteristics: multimodal, nonconvex, continuous, non-separable and defined on an n-dimensional space.

Search space: ${\mathbf{x}} \in \left[ { - 50,\,50} \right]$

Global minimum: $f({\mathbf{x}}) = 0$ at ${\mathbf{x}} = 0.$