# 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.

## Keywords

Probabilistic power flow Distributed generation Radial distribution systems Single- and multi-objective optimization## List of symbols

*N*Number of independent random

*X*_{1}Random variables

- \(f_{x1} ( {X_{1} }) \)
Probability density function of random variables

*a*_{1},*a*_{2}, …,*a*_{n}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

*K*_{T}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

*k*_{d}for a set of hourly events having mean diffuse fraction*K*_{t}*C*_{1,λ}Parameters are functions of

*K*_{Tu}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

*i*th- VD
_{i} Voltage deviation at bus

*i*th- 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

- VD
_{max} 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

*r*_{1},*r*_{2}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

## 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

- CO
_{2} 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

## References

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