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

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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.

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Abbreviations

N :

Number of independent random

X 1 :

Random variables

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

Probability density function of random variables

a1, a2, …, an :

Coefficients

α γ :

γ-Order moment

μ :

Expected mean value

M γ :

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

K T :

Daily clearness index

K d :

Hourly diffuse fraction

G t,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

C 1,λ :

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

K Tu :

Upper limit of the daily clearness index

K dl :

Lower limit of the hourly diffuse fraction

P pv :

PV electrical power

η c :

PV cell’s electrical efficiency

A :

PV generator surface area

F i :

Fitness function

P loss :

Total power losses

z :

Branch number

I z :

Branch current

R z :

Resistance of branch

n_br :

Number of branches

V sep :

Specified voltage

V i :

Voltage magnitude at bus ith

VDi :

Voltage deviation at bus ith

NG:

Numbers of generation units

P d :

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

V max :

Maximum voltage

V min :

Minimum voltage

Q loss :

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

x k :

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

U loss :

Power loss normalized value

U VD :

Voltage deviation normalized value

U w :

Weighting of the Pareto solution normalized value

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

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Authors and Affiliations

  1. Department of Electrical Engineering, Faculty of Engineering, Aswan University, Aswan, 81542, Egypt

    Salah Kamel, Ali Selim & Walaa Ahmed

  2. Department of Electrical Engineering, University of Jaén, 23700, Linares, Jaén, Spain

    Ali Selim & Francisco Jurado

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  1. Salah Kamel
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  2. Ali Selim
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  3. Walaa Ahmed
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  4. Francisco Jurado
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Correspondence to Francisco Jurado.

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Kamel, S., Selim, A., Ahmed, W. et al. Single- and multi-objective optimization for photovoltaic distributed generators implementation in probabilistic power flow algorithm. Electr Eng 102, 331–347 (2020). https://doi.org/10.1007/s00202-019-00878-7

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