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Single- and multi-objective optimization for photovoltaic distributed generators implementation in probabilistic power flow algorithm

  • Salah Kamel
  • Ali Selim
  • Walaa Ahmed
  • Francisco JuradoEmail author
Original Paper
  • 18 Downloads

Abstract

In this study, probabilistic power flow (PPF) for radial distribution systems (RDSs) integrated with photovoltaic (PV) distributed generators (DGs) is presented. The PPF is carried out using a combined approach of cumulants generating function and Gram–Charlier expansion. To express the intermittent nature of the PV power generation and demand powers, the random probabilities for solar irradiance and load demand are considered and modeled in the PPF. The benefits of PVDGs integration into RDS can be accomplished by their optimal placement and sizing. Hence, two optimization approaches are implemented to allocate the PVDG in the RDS. The first optimization approach utilizes a single-objective function based on particle swarm optimization (PSO) to minimize the total power losses in RDSs, while the second approach uses the multi-objective PSO (MOPSO) to minimize the total power losses and voltage deviation. However, in case of MOPSO, a fuzzy logic decision making is developed to adopt a suitable solution from the optimal Pareto set according to the decision-maker preference. The developed algorithm is verified using two standard IEEE radial distribution systems: IEEE 33-bus and 69-bus. The obtained results prove the ability of the developed algorithm in solving the PPF considering the optimal PVDG allocation with low computational time.

Keywords

Probabilistic power flow Distributed generation Radial distribution systems Single- and multi-objective optimization 

List of symbols

N

Number of independent random

X1

Random variables

\(f_{x1} ( {X_{1} }) \)

Probability density function of random variables

a1, a2, …, an

Coefficients

αγ

γ-Order moment

μ

Expected mean value

Mγ

Central moment

Φ(x), φ(x)

CDF and PDF of a normal distribution

σ

Standard deviation

\( c_{\upsilon } \)

Constant coefficients, υ = 1, 2, 3, …

\( H_{\upsilon } \left( x \right) \)

Hermite polynomial

KT

Daily clearness index

Kd

Hourly diffuse fraction

Gt,B

Time averaged hourly total irradiance on a surface sloped at angle B to the horizontal

\( p_{k} \left( {K_{\text{T}} ,\bar{K}_{\text{T}} } \right) \)

PDF of random variable daily clearness index KT for a set of daily events having mean clearness index \( \bar{K}_{\text{T}} \)

\( p_{k} \left( {K_{\text{d}} ,K_{t} } \right) \)

PDF of random variable hourly diffuse fraction kd for a set of hourly events having mean diffuse fraction Kt

C1,λ

Parameters are functions of KTu and \( \bar{K}_{\text{T}} \)

KTu

Upper limit of the daily clearness index

Kdl

Lower limit of the hourly diffuse fraction

Ppv

PV electrical power

ηc

PV cell’s electrical efficiency

A

PV generator surface area

Fi

Fitness function

Ploss

Total power losses

z

Branch number

Iz

Branch current

Rz

Resistance of branch

n_br

Number of branches

Vsep

Specified voltage

Vi

Voltage magnitude at bus ith

VDi

Voltage deviation at bus ith

NG

Numbers of generation units

Pd

Active load demand

\( P_{{{\text{g}}_{i} }} \)

PVDG power generation

\( P_{{{\text{g}}_{i} }}^{{\text{min}} } , P_{{{\text{g}}_{i} }}^{{\text{max} }} \)

Min and max power limits of the PVDG

Vmax

Maximum voltage

Vmin

Minimum voltage

Qloss

Reactive power loss

VDmax

Maximum voltage deviation

POP

Population size

k

Counter refers to particles number

p[k]

Position of particle k

v[k]

Velocity of particle k

xk

Non-dominated vectors

REP[k]

Repository

w

Inertia weight

r1, r2

Random numbers between 0 and 1

Pbest[k]

Best position for particle k

\( F_{i}^{{\text{max} }} \)

Maximum limit of the objective function i

\( F_{i}^{{\text{min}} } \)

Minimum limit of the objective function i

\( U_{i}^{n} \)

Normlized vlaue of objective function i at n non-dominated

Uloss

Power loss normalized value

UVD

Voltage deviation normalized value

Uw

Weighting of the Pareto solution normalized value

Acronyms

PPF

Probabilistic power flow

RDSs

Radial distribution systems

PV

Photovoltaic

DGs

Distributed generators

PVDGs

Photovoltaic distributed generators

GCE

Gram–Charlier expansion

PSO

Particle swarm optimization

MOPSO

Multi-objective PSO

VD

Voltage deviation

CO2

Carbon dioxide

DLF

Deterministic load flow

MCS

Monte Carlo simulation

MOOP

Multi-objective optimization problems

PAES

Pareto archived evolution strategy

NSGA-II

Non-dominated sorting genetic algorithm

SPEA

Strength Pareto evolutionary algorithm

PDFs

Probabilistic density functions

CDF

Cumulative distribution function

SD

Standard deviation

IA

Improved analytical

DAPSO

Dynamic adaptation of PSO

BSOA

Backtracking search optimization algorithm

LR

Loss reduction

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Electrical Engineering, Faculty of EngineeringAswan UniversityAswanEgypt
  2. 2.Department of Electrical EngineeringUniversity of JaénLinaresSpain

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