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

Original Paper

## 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

$$P_{{{\text{g}}_{i} }}$$

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

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

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

BSOA

Backtracking search optimization algorithm

LR

Loss reduction

## References

1. 1.
Jurado F, Cano A (2006) Optimal placement of biomass fuelled gas turbines for reduced losses. Energy Convers Manag 47(15–16):2673–2681
2. 2.
Hernández J, Medina A, Jurado F (2007) Optimal allocation and sizing for profitability and voltage enhancement of PV systems on feeders. Renew Energy 32(10):1768–1789
3. 3.
Balamurugan K, Srinivasan D (2011) Review of power flow studies on distribution network with distributed generation. Paper presented at the IEEE conference on power electronics and drive systems (PEDS)Google Scholar
4. 4.
Arrillaga J, Arnold C, Harker BJ (1983) Computer modelling of electrical power systems. WileyGoogle Scholar
5. 5.
Deng X, Zhang P, Jin K, He J, Wang X, Wang Y (2019) Probabilistic load flow method considering large-scale wind power integration. J Mod Power Syst Clean Energy 7(4):813–825
6. 6.
Gruosso G, Maffezzoni P, Zhang Z, Daniel L (2019) Probabilistic load flow methodology for distribution networks including loads uncertainty. Int J Electr Power Energy Syst 106:392–400
7. 7.
Zhang J, Xiong G, Meng K, Yu P, Yao G, Dong Z (2019) An improved probabilistic load flow simulation method considering correlated stochastic variables. Int J Electr Power Energy Syst 111:260–268
8. 8.
Bijwe P, Raju GV (2006) Fuzzy distribution power flow for weakly meshed systems. IEEE Trans Power Syst 21(4):1645–1652
9. 9.
Jiang X, Chen YC, Domínguez-García AD (2013) A set-theoretic framework to assess the impact of variable generation on the power flow. IEEE Trans Power Syst 28(2):855–867
10. 10.
Su C-L (2005) Probabilistic load-flow computation using point estimate method. IEEE Trans Power Syst 20(4):1843–1851
11. 11.
Ruiz-Rodriguez F, Hernandez J, Jurado F (2012) Probabilistic load flow for radial distribution networks with photovoltaic generators. IET Renew Power Gener 6(2):110–121
12. 12.
Vidović PM, Sarić AT (2017) A novel correlated intervals-based algorithm for distribution power flow calculation. Int J Electr Power Energy Syst 90:245–255
13. 13.
Lopes JP, Mendonça Â, Fonseca N, Seca L (2005) Voltage and reactive power control provided by DG units. Paper presented at the CIGRE symposium: power systems with dispersed generation, Athens, GreeceGoogle Scholar
14. 14.
Acharya N, Mahat P, Mithulananthan N (2006) An analytical approach for DG allocation in primary distribution network. Int J Electr Power Energy Syst 28(10):669–678
15. 15.
Atwa Y, El-Saadany E, Salama M, Seethapathy R (2010) Optimal renewable resources mix for distribution system energy loss minimization. IEEE Trans Power Syst 25(1):360–370
16. 16.
Borges CL, Falcao DM (2006) Optimal distributed generation allocation for reliability, losses, and voltage improvement. Int J Electr Power Energy Syst 28(6):413–420
17. 17.
Knowles JD, Corne DW (2000) Approximating the nondominated front using the Pareto archived evolution strategy. Evol Comput 8(2):149–172
18. 18.
Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197
19. 19.
Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the strength pareto evolutionary algorithm. TIK-report, 103Google Scholar
20. 20.
Coello CAC, Pulido GT, Lechuga MS (2004) Handling multiple objectives with particle swarm optimization. IEEE Trans Evol Comput 8(3):256–279
21. 21.
Zhang P, Lee ST (2004) Probabilistic load flow computation using the method of combined cumulants and Gram–Charlier expansion. IEEE Trans Power Syst 19(1):676–682
22. 22.
Allan R, Al-Shakarchi M (1976) Probabilistic ac load flow. Paper presented at the proceedings of the institution of electrical engineersGoogle Scholar
23. 23.
Rajaram R, Kumar KS, Rajasekar N (2015) Power system reconfiguration in a radial distribution network for reducing losses and to improve voltage profile using modified plant growth simulation algorithm with distributed generation (DG). Energy Rep 1:116–122
24. 24.
Baran M, Wu FF (1989) Optimal sizing of capacitors placed on a radial distribution system. IEEE Trans Power Deliv 4(1):735–743
25. 25.
Venkatesh B, Ranjan R (2003) Optimal radial distribution system reconfiguration using fuzzy adaptation of evolutionary programming. Int J Electr Power Energy Syst 25(10):775–780
26. 26.
Hollands K, Crha S (1987) A probability density function for the diffuse fraction, with applications. Sol Energy 38(4):237–245
27. 27.
Hollands KGT, Huget R (1983) A probability density function for the clearness index, with applications. Sol Energy 30(3):195–209
28. 28.
Hung DQ, Mithulananthan N (2013) Multiple distributed generator placement in primary distribution networks for loss reduction. IEEE Trans Ind Electron 60(4):1700–1708
29. 29.
Manafi H, Ghadimi N, Ojaroudi M, Farhadi P (2013) Optimal placement of distributed generations in radial distribution systems using various PSO and DE algorithms. Elektronika ir Elektrotechnika 19(10):53–57
30. 30.
El-Fergany A (2015) Optimal allocation of multi-type distributed generators using backtracking search optimization algorithm. Int J Electr Power Energy Syst 64:1197–1205