A Novel Hybrid Approach for Optimal Placement of Non-Dispatchable Distributed Generations in Radial Distribution System

Abstract

The objective of the present paper is to study the optimum installation of Non-dispatchable Distributed Generations (NDG) in the distribution network of given sizes under the given scheme. The uncertainty of various random (uncertain) parameters like load, wind and solar operated DG besides uncertainty of fuel prices has been investigated by the three-point estimate method (3-PEM) and Monte Carlo Simulation (MCS) based methods. Nearly twenty percent of the total number of buses are selected as candidate buses for NDG placement on the basis of system voltage profile to limit the search space. Weibull probability density function (PDF) is considered to address uncertain characteristics of solar radiation and wind speed under different scenarios. Load uncertainty is described by Standard Normal Distribution Function (SNDF). To investigate the solution of optimal probabilistic load flow (OPLF) three-point PEM-based technique was applied. For optimization, Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and GA-PSO hybrid-based Artificial Intelligent (AI) based optimization techniques are employed to achieve the optimum value of the multi-objectives function. The proposed multi-objective function comprises loss and different costs. The proposed methods have been applied to IEEE 33- bus radial distribution network. Simulation results obtained by these techniques are compared based on loss minimization capability, enhancement of system bus voltage profile and reduction of cost and fitness functions. The major findings of the present study are the PEM-based method which provides almost similar results as MCS based method with less computation time and as far as loss minimization capacity, voltage profile improvement etc. is concerned, the hybrid-based optimization methods are compared with GA and PSO based optimization techniques.

1. Introduction

In recent years distributed generations (DGs) are becoming very significant in the area of power engineering because DGs pose enumerable benefits over centralized generating units. Therefore, this research area is becoming more popular for active research. Nowadays, efforts are being made to cut down the line losses and emission of harmful greenhouse gases (GHG). Moreover, DG also improves system voltage profile and also improves system reliability. Additionally, DG units, which are powered by renewable energy sources (RES) such as wind turbine generators (WTG), solar photovoltaic (SPV), biomass, small hydro, and geothermal are extremely useful. Other types of DG technologies that are powered by non-renewable energy sources also exist, such as internal combustion engines, fuel cells etc. Most parameters like electric load demand, power generation of DG, and fuel prices suffer from uncertainty. Therefore, uncertainty is to be investigated thoroughly, which is associated to solar and wind operated DG and load demand. Further, it is established that DGs are universally accepted sustainable power generating units because of their inexhaustible and nonpolluting characteristics. Therefore, optimum placement and sizing of DG in the distribution system is extremely important in order to obtain maximum potential benefits [1,2,3,4,5,6,7,8,9,10,11]. It improves the system voltage profile, reliability, power quality in addition to minimizing system power loss [1,2,3,10,11,12,13,14,15,16]. Moreover, non-optimal placement of DG units leads to network losses and system voltage instability which in turn increases system cost [7]. From the literature survey, uncertainty associated with loading, power generation by SPV and WTG-based DG is addressed by very few researchers [1]. That is why this area needs due attention to address the uncertainty of solar and wind operated DGs along with load demand.

Consequently, in view of these facts, optimum placement of DG (OPDG) is essential. Furthermore, most of the researchers have not addressed the uncertainty of load and power generating units in their research. Hence, OPDG is very much crucial when uncertainties of load, fuel prices, and power generating units like wind turbine generators (WTG) and solar photovoltaic (SPV) are adequately investigated. The MCS-based method is commonly used for addressing the uncertainties of various parameters [1,2,3,4,5,12,13,14,17,18,19,20,21] because this method is considered the benchmark method for the analysis of uncertainty. Additionally, different probability density functions (PDFs) are considered for different uncertain parameters [1,2,20,21,22,23]. But MCS based method requires more computational time when data is large. One more attractive method is available for investigations of uncertain parameters. This method is Point Estimate Method (PEM), because in the case of PEM-based approach, it requires less computational time.

In the present study, load demand is a random variable. The variation of load demand is considered as a Gaussian distribution function. Further, wind speed and solar insolation are considered as Weibull distribution functions. Further, fuel prices are considered as an uncertain parameter and follow the properties of Geometric Brownian Motion (GBM). To address the uncertainty of these parameters, three PEM-based techniques were applied and the obtained results give the MCS-based technique. The MCS-based method is the benchmark method for the assessment of uncertainty. For optimization GA, PSO and GA-PSO hybridized based optimization techniques are applied. The present research paper is dedicated to the investigation of uncertainty of NDG (SPV and WTG) and fueled-based DG is also considered for installation along with NDG. The main motive of the present study is optimal sizing and siting of non-dispatchable DG in distribution network by considering uncertainty of load demand addressed by Standard Gaussian PDF, and uncertainty of power generation by SPV and WTG by Weibull based PDF. Uncertainty associated in these parameters needs further attention to address. Therefore, the main focus of this study is to minimize the system power loss by placement of NDG in distribution system.

2. Literature Review

The power generated by solar-based DG and wind-operated DG is an uncertain parameter and has to be addressed carefully. The uncertainty of solar radiation and wind speed is investigated by PEM-based technique. Many researchers have addressed uncertainty of load and power generated by SPV and WTG-based DG. Evangelopoulos and Georgilakis [1] proposed PEM based technique for probabilistic power flow (PPF) in order to consider the uncertainties of the power output of DG, load demand, fuel cost, and electricity prices. Moreover, for optimization GA based technique is applied for optimum sizing and placement of non-dispatchable DG. Authors of [1,2,3,4,12,13,17,20,24,25,26] have applied Weibull probability distribution function (PDF) to address the uncertainty of wind and solar based DG. Additionally, solution of PPF based on deterministic load flow (DLF), has successfully addressed the uncertainties of various parameters such as load, fuel prices, and power generation of non-dispatchable DG with the help of PEM and MCS based methods.

Authors of [15] have applied a new evolutionary programming-based approach for optimum installation of DG which is powered by renewable energy resources. Further, PSO and MOPSO based methods were applied by [6,7,9,10,11,16,23,27] for single as well as multiple DG unit(s) placement by addressing the uncertainty of load and renewable energy sources with the objective to minimize power and energy loss and system cost by PDF. Furthermore, authors have also investigated the impact of DG unit with respect to minimization of system loss, cost, and enhancement of bus voltage profile. The various authors have investigated latest meta-heuristic based techniques such as Equilibrium optimizer (EO) technique [26], improved sunflower optimization algorithm (ISFOA) [28], fuzzy-based multi-objective (FBMO) formulation [29], and modified marine predators optimization (MMPO) algorithm [30].

It is a well-known fact that uncertain inputs give rise to uncertain outputs. Therefore, the methods which are applied to placement of DG units whose output is controllable are not suitable for placement of DG units whose output is intermittent [1,2,3]. Moreover, the solution of deterministic based load flow for optimum installation and sizing of non-dispatchable DG cannot provide adequate results. Therefore, probabilistic based methods are applied to investigate the randomness of uncertain parameters [1,2,3,4,5,12,13,14,15,17,18,19,21,23,24,25]. Further, MCS based method is used as benchmark method to address uncertainty. In this technique DLF solutions are obtained several times. The expected value of desired uncertain quantity is obtained on the basis of statistical moments.

Moreover, uncertainty of non-dispatchable type DG is a crucial feature which needs to be investigated critically as it is evident from available literature [1,2,3,4,5,12,13,14,15,17,18,19,24,25]. Additionally, most of the authors considered the uncertainty of DG power output [1,2,3,4,5,6,7,8,9,10,12,13,14,15,17,18,19,24,25,27,31] in their investigations but very few authors addressed the load and fuel price uncertainty along with power generated by DG [1,9]. Therefore, the author is motivated by reference [1]. The present study investigates the uncertainty of load demand, power generating units like WTG and SPV, and fuel prices. The uncertainty of load demand follows the property Gaussian distribution function. Weibull PDF is applied for uncertainty of wind speed and solar radiation. Further, GBM based PDF is applied for uncertainty of fuel prices.

Table 1. Brief detail of extensive literature review.

Reference Number and Year of Publication	Type of Application	Main Contribution and Characteristics	Analyzed Method/Remarks	Recommended System/Similarity with Current Approach
[1], March 2014	Optimal placement of DGs (Solar, wind and fueled) based DG in distribution system	Design objective function by combining different costs associated to DGs operation and maintenance and designed objective function is optimized by GA based heuristic technique	Application of 3-point PEM based to address uncertainty of different parameters and results are compared with MCS based technique	The current approach is applied on standard IEEE 33-bus test distribution system and designed objective function is optimized by application of GA, PSO and GA-PSO based hybrid technique
[2], 2014	Extends the application PEM	Main contribution of paper to address the random parameters of DG by PEM based method	In proposed paper, the PEM -based method is highlighted to investigate the uncertainty of unsymmetrical system in presence of DG.	Only probabilistic load flow concept is similar
[3], November 2007	Hong’s PEM based method for PLF	To address the uncertainty associated with PLF, Hong’s PEM-based methods are applied.	The authors considered two scenarios as IEEE 14-bus and IEEE 118-bus distribution network.	Similarity only in this way that PLF by PEM based method
[4], November 2005	New algorithm for PLF	Uncertainties of bus injection and line parameters is investigated by PEM based method	The proposed algorithm is implemented on IEEE test system and deterministic load flow is used to address the PLF	PLF is similar parameter to the current approach
[5], 2006	Applied 2-PEM based method for investigating the uncertain parameters	2-PEM based method for investigation of uncertainties of optimal power flow (OPF)	2-PEM and MCS based methods are applied to address the optimal power flow (OPF) and proposed method is implemented on 129-bus test system	No similarity except PEM and MCS based technique
[6], 2015	Evolutionary or swarm-based search algorithms	Three nature inspired techniques such as IGA, IPSO and ICSO are applied.	The cost function which addressing the benefit of network by optimal sizing and siting of DGs and SCs is optimized by IGA, IPSO and ICSO optimization approaches	No similarity: IGA, IPSO and ICSO are applied in different perspective
[7], 2015	Main focus on optimal power flow (OPF) by considering wind uncertainty	The objective function comprising economic aspect of system is optimized by best guided artificial bee colony optimization algorithm (GABC)	The uncertainty of wind speed is addressed by Weibull density function for optimization GABC technique is applied and the proposed technique is tested on IEEE 30-bus test system	Applied methods are different, optimized function is entirely different test system is different but uncertainty of wind speed is addressed by same Weibull density function
[8], October 2011	The chance constrained programming (CCP) based mathematical model proposed	Authors developed a CCP based mathematical model comprising various costs including reliability cost which is optimized by considering different operating constrains of DGs.	An objective function which includes all the attributes of optimal DG sizing and placement are included is optimized by Monte-Carlo simulation embedded GA is applied in order to optimize the considered objective function and the proposed algorithm is tested on IEEE 37-node test feeder	Considered objective function is different and is optimized by GA based technique, therefore GA is one of the similar technique which is applied but objective function is entirely different
[9], July 2015	Optimum placement and sizing of DG in distribution network by considering a novel method.	Authors formulated a multi-objective functions function which comprising of different costs and optimized by an improved NSGA-II based technique.	The proposed algorithm is tested on IEEE 69-bus test radial distribution system the objective function is optimized by NSGA-II for optimal placement and sizing of DG	No similarity
[10], 2011	The paper proposed a meta-heuristic voltage stability index based method for optimal siting and sizing of DG	A multi-objective based objective function comprising functions which leads towards maximize the benefits by multi-objective particle swarm optimization (MOPSO) based method	The proposed method is tested on 30-bus test system and 41-bus Indian distribution system in order to improve the voltage profile and minimize cost	No similarity
[11], January 2011	A methodology has been proposed for optimally allocating wind-based DG	The authors formulated a multi-objective function which combines all possible operating conditions of load and wind operated DGs with their probabilities.	The objective function is optimized by MINLP based technique for optimal allocation of wind based DG in order to minimize annual energy losses. Further, in present reference uncertainty of load is not considered	Not similarity
[12], 2011	Analysis of probabilistic based load flow by 2-PEM based technique	Comparison of obtained results in terms of uncertainty of wind based DG by proposed method and MCS based method	2-point PEM based method is analyzed for uncertainty with MCS based and tested on IEEE 9-bus and 57-bus test systems	Only probabilistic load flow concept is similar. In proposed method 3- PEM based load flow is used but in [2] only 2-point PEM is applied and results are compared by 2-point PEM based method.
[13], May 2014	Analysis of probabilistic power flow	PEM based method for analysis of uncertainty of wind	To address the uncertainty of wind the PEM and MCS based methods are compared	Only probabilistic load flow concept is similar
[14], 2014	Probabilistic approach for uncertain parameters	The proposed paper addresses the time dependent uncertainty of load by considering PEM based technique	PEM based technique is considered to address the uncertainty of load of different nature and an objective function is constituted by active power, reactive power and voltage profile indices	The objective function is optimized by metaheuristic optimization technique Invasive Weed Optimization (IWO) based technique for size and location of DGs
[15], 2013	Evolutionary programming (EP) based method	Address the uncertainty of wind and solar based DGs by considering probabilistic based method	Uncertainty of wind and solar based DGs by probabilistic method and model the uncertainties of wind speed, solar radiation and load the method is implemented on IEEE 33-bus and 69-bus test distribution system	No similarity
[16], June 2015	Conservation voltage reduction (CVR) and distributed-generation (DG) integration	This paper investigates the interactions between CVR and DG placement to minimize load consumption in distribution networks, while keeping the lowest voltage level within the predefined range	The optimal placement of DG units is formulated as a stochastic optimization problem considering the uncertainty of DG outputs and load consumptions for this purpose A sample average approximation algorithm-based technique is developed and proposed technique is tested on IEEE 37-bus test system	No similarity
[17], October 2013	Novel approach for PLF based PEM	PEM and MCS based techniques are applied to investigate different uncertain parameters	The proposed algorithm is implemented on IEEE 14-bus test system	In this paper only PLF is investigated by considering uncertainty of load, while current approach is applied on IEEE 33-bus test system to address uncertainty parameters like Solar, wind, and load and objective function is constituted by considering different weight
[18], 1974	First paper that address the concept of PLF	In this paper concept of PLF was introduced to address the uncertainty of node data	PLF solution is obtained by application of deterministic load flow	PLF is addressed by author to investigate the uncertainty of node data but in current research paper the uncertainty of many parameters is addressed
[19], 2019	The paper deals Energy storage system (ESS) in active distribution network	The objective of the problem is to minimize the yearly cost ESSs.	The authors investigated the AC power flow based method to address the active and reactive power using PSO based optimization method.	No similarity: only metaheuristic based PSO optimization is applied to optimized for different objective function
[20], January 2015	Probabilistic optimal power flow (POPF) by considering wind speed	To obtain the solution of POPF model for integration of wind operated DG in distribution system two kinds of 2m, 2 m+1 PEM based method are employed	The uncertainty of load is investigated by two kinds of 2m, 2 m+1 PEM and from obtained results it is concluded that 2 m+1 based PEM method provide better solution	Only (2m+1) based PEM is provide better solution it is confirmed by this research paper. In our research article the uncertainty of DGs power, load and fuel costs are investigated by the same method along with objective function is optimized by hybrid GA-PSO, PSO and GA based optimization methods.
[21], November 2002	Rosenbleuth’s based PEM technique is analyzed	Author investigated the Rosenbleuth’s based PEM technique to address the uncertainty of different random parameters	Investigate the uncertainty of different random parameters by MCS and Rosenbleuth’s based PEM technique and observed that PEM based technique reduces the computational burden	2m point based PEM is employed but in our paper (2 m+1) point based PEM is employed and it was confirmed by many articles that (2 m+1) point PEM based technique is more effective in terms of reduction in computational burden and accuracy of results
[22] March 2015	Weibull based PDF is applied to investigate the uncertainty	To assess the wind energy potential, Weibull based probability density functions is employed to investigate the uncertainty of wind speed	In this research article an investigation of uncertainty of wind operated DG Weibull based PDF is employed on real time data of 9 power stations which are located in the United Arab Emirates (UAE).	No similarity
[23], 2014	Monte Carlo Simulation (MCS) based probabilistic load flow is investigated	Uncertain nature of wind operated DG is investigated by MCS based probabilistic load flow is addressed to optimal placement of DG and capacitor to improve system voltage profile	The optimal sizing and siting of wind power DG (WPDG) and capacitor in distribution system is optimized by modified PSO (MPSO) based optimization technique is employed and proposed technique is tested on IEEE 33-bus distribution system	No similarity
[24], 2011	AC probabilistic optimal power flow (P-OPF) is applied to address the uncertainty using MCS.	Applied PEM based method and results are compared with MCS based method for wind speed and line outage factor uncertainty	PEM and MCS based technique to address the wind speed and line outage factor uncertainty and proposed technique is applied on IEEE 30-bus test system	No similarity except PEM and MCS based technique
[25], 2015	In proposed technique integrates the diagonal band Copula and sequential Monte Carlo	The authors Considered the multivariable -based stochastic method to investigate uncertainty of solar based systems.	Objective function is optimized by Big Bang-Big crunch method for optimal placement of DGs and proposed method is applied on IEEE 37-bus test system	MCS based method is only similar but applied in different objective function
[26], December 2020	Optimal allocation of biomass based DGs.	Authors suggested an adaptive equilibrium optimizer technique (EO) to enhance performance of biomass DGs.	EO techniques is applied to IEEE-33-bus and practical large-scale 141-bus system of AES-Venuzuala in metropolitan area of Caracas.	No similarity
[27], 2015	MOPSO-based optimization for optimum sizing and allocation of shunt capacitor banks (SCBs) and DGs.	Multi-objective function comprising the balancing current in different sections along with bus voltage stability and system power loss is optimized by SPEA, NSGA, MODE and combination of ICA/GA.	The uncertainty of loads is modeled by using fuzzy data theory, and objective function is optimized by SPEA, NSGA, MODE and combination of ICA/GA, the proposed algorithm is tested on IEEE 33-bus and an actual realistic 94 bus Portuguese test systems	No similarity: different techniques are applied for uncertainty and optimization of objective function
[28], December 2020	An Improved Sunflower Optimization Algorithm (ISFOA)-based Monte Carlo simulations	Authors Investigated performance enhancement of smart distribution system using ISFOA technique.	The technique is tested on IEEE-33 and real time 84-bus distribution system	No similarity
[29], June 2021	Optimal distributed generation units are correlated with fault current limiter sites.	Different types of DGs are allocated and correlated in a single stage with fault current limiters (FCLs). A fuzzy-based multiobjective (FBMO) formulation is provided for performance improvement both in normal and faulty operating conditions.	The proposed method is studied for 33-bus, 69-bus, and the Egyptian East Delta distribution systems.	No similarity
[30], December 2021	A modified marine predators optimizer (MMPO) is used for simultaneous distribution network reconfiguration (DNR).	Authors have shown the superiority of the proposed MMPO for simultaneous DNR and DG allocation.	This technique is tested on IEEE 33-bus and 69-bus distribution systems.	No Similarity
[31], February 2010	A methodology developed on the basis of probabilistic generation-load model	The authors applied a combination of all working conditions of renewable DGs-based methodology to assess the probabilities by MINLP based technique.	The paper addresses the optimal sizing and siting of all types of DGs (dispatchable and non-dispatchable) with the objective to minimize total annual energy loss of the system	No similarity with proposed method.
[32] 2010, [33], 2009,	Analytical and MCS based technique	Investigation of system reliability when wind based DG is integrated in distribution system	MCS and analytical based methods are employed in order to calculate the system reliability in the form of LOEE and LOLE in presence of wind operated DG	No similarity

represents the extensive literature review.

Therefore, from the above literature review, none of the research articles address the uncertainty of power generated from wind, solar operated DG, load, and fuel cost. In present manuscript uncertainty of wind and solar operated DG, along with load demand fuel prices uncertainty is investigated by 3-points PEM and MCS based technique. The cost function or objective function is designed by considering different costs that pertain to DG operation under different scenarios. Further, optimum locations and optimum value of cost function is obtained by GA, PSO, and GA-PSO-based hybrid technique.

From the literature it is noticed that very few researchers have addressed the uncertainty of three parameters like load demand and power generated by SPV and WTG based DG by PEM and MCS based method. For optimization of objective function GA, PSO, and GA-PSO-based hybridized methods are not considered. The fueled operated DG is also considered as dispatchable DG for placement.

From the obtained result, it is confirmed that the GA-PSO-based hybrid method is more feasible and provides an optimal solution close to actual results. Moreover, 3-point PEM-based method yields almost similar results as MCS based technique which is the only available technique to investigate the uncertainty of different parameters.

In the present article, three different scenarios and types of DGs (wind, solar, and fueled) have been considered. For optimization GA, PSO, and hybrid (combination of GA and PSO) based heuristic techniques have been applied. The PEM and MCS-based methods have been applied to incorporate the stochastic nature of different parameters. The configuration and properties of computer on which MATLAB programs have been executed is Intel (R) Core (TM) i 5-4570 CPU @ 3.20 GHz 4 GB RAM.

The remaining part of present research article is organized as follows: Section 2 discusses modeling of uncertainties of wind and solar operated DG. Section 3 describes the PEM-based method. Section 4 deals with the problem formulation of present study. Section 5 highlights the scheme and different scenarios of DGs which are to be placed along with solution process. Section 6 discusses the algorithm for selection of initial population of optimization technique. Finally, Section 7 describes the results and discussion.

3. Uncertainty Modelling for Solar and Wind Operated DG

As solar radiation and wind speed are uncertain parameters, the uncertainty of wind speed and solar radiation followed the Weibull PDF [1,2,3,4,12,13,17].

3.1. Wind Speed Uncertainty and Power Output of Wind Turbine Generator (WTG)

Weibull probability density function (PDF) is generally applied to represent the stochastic nature of wind speed. Many researchers used the same PDF for random behavior of wind speed [1,8,32].

$$f\left(v\right)=\frac{k}{c^k}{v}^{k-1}\exp \left(-{\left(\frac{v}{c}\right)}^k\right),\hspace{1em}\hspace{1em}0\le v<\infty$$

(1)

The applied symbols have been explained previously. The output power of WTG is as

$$\begin{array}{l}{P}_w=\{\begin{array}{l}0, \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} 0\le v\le v_i\\ P_{wn}\frac{\left(v-v_i\right)}{\left(v_n-v_i\right)}, \hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} v_i\le v\le v_n\\ P_{wn} \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} v_n\le v\le v_{co}\\ 0 \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} v_{co}<v\end{array}\\ \end{array}$$

(2)

Here, symbols have their usual meaning.

3.2. Power Output of Solar Photovoltaic

It is observed that solar radiation also follows the characteristic of Weibull PDF. Therefore, intensity of solar illumination is given by [8,32,33]:

$$f\left(s\right)=\frac{k_s}{c_s^{k_s}}{s}^{\left(k_s-1\right)}\exp \left(-{\left(\raisebox{1ex}{$s$}\!\left/ \!\raisebox{-1ex}{$c_s$}\right.\right)}^{k_s}\right),\hspace{1em}\hspace{1em}0\le s<\infty$$

(3)

$$P_s=\{\begin{array}{l}{P}_{sn}\frac{s}{s_n}\hspace{1em}\hspace{1em}0\le s\le s_n\\ P_{sn}\hspace{1em}\hspace{1em}\hspace{1em}{s}_n\le s\end{array}$$

(4)

The relevant symbols of (3) and (4) have their usual meaning.

3.3. Solution of Probabilistic Load Flow (PLF) by PEM

The MCS-based method is the only method that is used as uncertainty evaluation. Further, PEM based method is also used to evaluate uncertainty of various parameters. It is an approximate method in which only a few points of the distribution function of a random variable are considered for calculations. These points are called concentrations. By using these points, the information of the output can be obtained [1,2,3,4,5,6,12,13,14,15,17,18,19,24,25]. The computation time of the PEM-based method is less because it requires only three values of uncertain parameters, including the mean value. In the present study, the power generated from SPV and WTG and two components of load demand (active and reactive) are considered as uncertain parameters. In PEM based method the concentration of each random variable is calculated on the basis of [1,2,3,4,12,13,14,17,18].

In this paper, (2 m + 1) based PEM method is applied because it provides very accurate results for probabilistic load flow (PLF) solutions by running MATLAB program of deterministic load flow only (2m + 1) times [1,2,3,4,12,13,14,17,18], here m is the number of uncertain variables.

Let x_l is lth random variable, and l = 1, 2…m. There is ‘h’ level of PEM based technique required in order to calculate number of points of random variable x_l is x_l,1, x_1,2 or x_l,h as given in (5)

$$x_{l,k}={\mu }_l+{\xi }_{l,k}{\sigma }_l; \mathrm{For} k=1, 2 \dots h$$

(5)

Here, ${\mu }_l$ and ${\sigma }_l$ are the mean and standard deviation of random variable x_l. The value of k shows the number of points to be evaluated in this study k, considered as three, and three points-based PEM technique is applied. The process for estimating the points of each random variable $x_l$ with their matching weights is described below.

For determination of standard central moments ${\lambda }_{l,i}$

$${\lambda }_{l,i}=\frac{E\left[{\left(x_l-{\mu }_1\right)}^j\right]}{{\sigma }_l^i}, i=3\dots \dots .2m$$

Here, it is to be noted that m is number of variables to be considered in the PEM method.

Find the standard locations ${\xi }_{l,k}$, for 3-PEM these locations are calculated using (6)

$${\xi }_{l,k}=\frac{{\lambda }_{l,3}}{2}+{\left(-1\right)}^{3-k}\times \sqrt{{\lambda }_{l,4}-\frac{3}{4}{\lambda }_{l,3}^2}, \mathrm{where} {\xi }_{l,3}=0,\hspace{1em}\hspace{1em}k=1,2$$

(6)

As ${\xi }_{l,k}$ is standard locations and are calculated on the basis of $x_{l,k}$ from Weibull Distribution. The weighting factors w_l,k for three point-PEM method are computed from the following equations as

$$w_{l,k}=\frac{{\left(-1\right)}^{3-k}}{{\xi }_{l,k}\left({\xi }_{l,1}-{\xi }_{l,2}\right)},\hspace{1em}\hspace{1em}{w}_{l,3}=\frac{1}{n}-\frac{1}{{\lambda }_{\mathrm{l},4}-{\lambda }_{l,3}^2}, \hspace{1em}\hspace{1em}k=1, 2$$

$$\mathrm{and} w_{\mu }=1-{\displaystyle \sum _{l=1}^n{\displaystyle \sum _{k=1}^m{w}_{l,k}}}$$

Finally, output of any random variable is obtained as follows

$$E\left(z\right)={\displaystyle \sum _{l=1}^m{\displaystyle \sum _{k=1}^2{w}_{l,k}Z\left(l,k\right)}+w_{\mu }Z\left(\mu \right)}$$

(7)

$$E\left(z^j\right)={\displaystyle \sum _{l=1}^m{\displaystyle \sum _{k=1}^2{w}_{l,k}Z{\left(l,k\right)}^j}}$$

(8)

Here the process expected output is E (z), and $E\left(z^j\right)$ is explained in Figure 1.

4. Problem Formulation

In the present article, the problem is formulated for optimum placement of NDG units on the basis of unknown variables. These variables are as:

• The candidate buses are selected on the basis of bus voltage profile i.e., the bus having the lowest value of bus voltage given first priority for DG placement and so on. Thus the sequence of buses is identified for DG placement.
• Nearly 20% of total buses are selected for DG placement on bus voltage priority.
• Size of all DG units (in kW).
• The DG units which are to be placed are considered as WTG and SPV based DG as non-dispatchable types, DG and fueled operated DG (such as biomass and microturbines etc.) considered as dispatchable type DG.
• The placement of DG which is energized by renewable sources depends upon factors such as availability of wind speed and solar radiation.
• These resources are uncertain in nature and this uncertainty is modeled by using Weibull PDF.
• The proposed method is investigated under the scheme presented in

  Table 2. Candidate schemes for a different type of DGs placement and their sizing.

  Ranking of Buses for DG Placement	Installed Capacity of DG (in kW)					DG Type *
  18	20	40	60	80	100	1, 2
  17	40	80	120	160	200	1, 2, 3
  16	40	80	120	160	200	1, 2
  33	20	40	60	80	100	1, 2
  32	20	40	60	80	100	1, 2, 3
  15	100	200	300	400	500	1, 2, 3
  31	100	200	300	400	500	1, 2, 3
  14	40	80	120	160	200	1, 2
  13	40	80	120	160	200	1, 2, 3

  * DG type: 1-stands for wind DG, 2-stands for solar photovoltaic DG and 3-stands for fueled DG.

  .
• Constant power load is considered and its uncertainty is defined by the standard normal distribution function.

Table 2 describes the ranking of buses for DGs placement and the types of DGs considered for placement.

Table 3. Technical specifications of DGs for different type of DGs.

S.No.	DG Type	Technical Specifications
1.	Wind turbines	Vci = 4 m/s, cut-in speed
2.		Vco = 25 m/s, cut-out speed
3.		Vn=15 m/s, nominal speed
4.		Power factor = 0.9 lagging
5.	Solar photovoltaic	Sn = 1000 W/m2
6.		Power factor = 1
7.	Fueled DGs	Stable power
8.		Power factor = 0.9 lagging

describes the technical details of wind turbine generator and solar operated DG in respect of wind speeds i.e., cut-in speed, nominal speed, and cut-out speed and solar radiations. Table 3 represents the data considered for different scenarios. In this table, the different scenarios have been created by considering the different values of shape and scale parameters of Weibull PDF. Moreover,

Table 4. Data for wind and solar-based DG for different scenarios.

S.No.	Scenarios	Different Parameters (kv Stands for Shape Parameter and cv Stands for Scale Parameters) of Wind Speed	Different Parameters (ks Stands for Shape Parameter and cs Stands for Scale Parameters) of Solar Radiation	Weight for Different Objective Functions
1.	1	kv = 2.1 cv = 7.5	ks = 1.4 cs = 5.5	b1 = 0.1, b2 = 0.11, b3 = 0.34, b4 = 0.34, b5 = 0.11
2.	2	kv = 1.8 cv = 6.0	ks = 1.8 cs = 6.5	b1 = 0.1, b2 = 0.11, b3 = 0.34, b4 = 0.34, b5 = 0.11
3.	3	kv = 2.1 cv = 7.5	ks = 1.4 cs = 5.5	b1 = 0.34, b2 = 0.11, b3 = 0.34, b4 = 0.11, b5 = 0.10

represents the three different values of scale and shape parameters of Weibull PDF and respective weight factors.

4.1. Objective Function

Optimum placement of DG unit of a given size is codified as a constrained nonlinear optimization problem. Further, the objective is to minimize the system total cost considered as an objective function which includes the investment cost, operating maintenance cost, network loss cost as per [1].

$$Min\hspace{1em}f\left(C\right)=w_1{C}^I+w_2{C}^M+w_3{C}^O+w_4{C}^L+w_5{C}^A$$

(9)

where w₁, w₂, w₃, w₄, and w₅ are the weight factors and their sum is 1 and C^I, C^M, C^O, and C^A are the costs of DG units for different parameters such as investment, maintenance, operation, and capacity adequacy costs in ($) respectively and C^L system power loss cost in ($/kW).

$$Min\hspace{1em}f=w_1{\displaystyle \sum _{k=1}^{N_{type}}{\displaystyle \sum _{i\in N_{DGk}}\left(C_{DGk}^I{P}_{DGki}^{Nype}\right)}}+w_2{\displaystyle \sum _{k=1}^{N_{type}}{\displaystyle \sum _{i\in N_{dgKI}}\left(C_{DGk}^M{T}_{DGki}{P}_{DGki}^N\right)}}$$

(10)

Backward forward sweep method (BFS) based load flow solutions are considered in present study. Following considerations have been assumed:

System under consideration is balanced, radial and fed by a substation.

Voltage of all nodes is close to 1 p.u.

Shunt admittances of lines are negligible.

The base case system power loss are evaluated on the basis of the following formula

$$P_L={\displaystyle \sum _{i=1}^{n-1}{I}_i^2{R}_i}$$

(11)

Here symbols have their usual meanings.

4.2. Modeling Constraints

Basically, two types of modeling constraints have been considered in the present study. These are given as below.

4.2.1. Constraints of Equality

In this section, backward forward sweep (BFS) based method is considered for the solution of load flow equations as equality constraints in order to obtain bus voltage.

4.2.2. Inequality Constraints

In the intelligent optimization technique, the inequality constraints should not be violated. Designer of network is intended to form direct relationship with other scientific terms to get feasible solution of power supply and voltage. The inequality constraints are specified as a minimum and maximum value of active and reactive powers produced by DG units. Therefore, inequality is designed in terms of system voltage and output power generated by these DG units.

$$V_{i, min}\le V_i\le V_{i,max}, \mathrm{i}=1, 2, 3,\dots \dots N$$

(12)

$$P_{DGi}\le P_{DGi\max },\hspace{1em}i=1, 2, 3,\dots \dots N_{DG}$$

(13)

$$Q_{DGi}\le Q_{DGi\max },\hspace{1em}i=1, 2, 3,\dots \dots N_{DG}$$

(14)

$${\displaystyle \sum _{i=1}^N{P}_i}+{\displaystyle \sum _{i=1}^{N_{DG}}{P}_{DGi}={\displaystyle \sum _{i=1}^N{P}_{Di}+{\displaystyle \sum _{i=1}^{N_B}{P}_{Li}}}}$$

(15)

$${\displaystyle \sum _{i=1}^N{Q}_i}+{\displaystyle \sum _{i=1}^{N_{DG}}{Q}_{DGi}}={\displaystyle \sum _{i=1}^N{Q}_{Di}+{\displaystyle \sum _{i=1}^{N_B}{Q}_{Li}}}$$

(16)

where, V_min, V_max, are the minimum, maximum value of voltage at ith bus, P_DGi, Q_DGi, component of active and reactive power injected from DG at bus i, Q_Di, P_Di, reactive and active power demand at bus i, Q_Li, P_Li reactive and real power loss in branch i and N_DG number of DG units to be installed, and Q_i, P_i reactive active power flow in ith branch respectively.

5. Solution Process for Optimum Placement of DG

The uncertainties have been addressed by three points PEM and MCS-based approaches. Further, for searching out the different candidate buses for optimal placement, different types of DG units are selected on the basis of increasing order of bus voltage profile to limit the search space. The technical details of wind turbine generator and solar operated DG in respect of wind cut-in speed, nominal speed and cut-out speed and solar radiations and different cost along with scenarios are considered taken from [1].

The PEM-based method considered in the present paper is tested on the IEEE 33-bus radial distribution network. The single-line diagram of this system is depicted in Figure 2. The total load on IEEE 33-bus test radial distribution system is (3.715 + j2.300) MVA.

5.1. Structure of Chromosome

Proper encoding of solution vector is the fundamental tool for efficient application of any meta-heuristic optimization technique. The optimum solution of objective function basically depends upon sizing and siting of DG units. As a result, each solution is represented in the form of chromosomes. The illustration is shown as follows.

The solution vector of different types of DGs are to be allocated as, P = (P^WDG, P^SDG, P^FDG); where P^WDG is a vector of K_w dimension corresponding to size of $P_i^{wdg}$ wind operated DG in every candidate bus for placement of wind DG. Similarly, for P^SDG is a vector of K_s dimension corresponding to size of $P_i^{sdg}$ solar operated DG in each of the candidate bus for installation of wind DG and P^FDG is a vector of K_F dimension corresponding to size of $P_i^{fdg}$ fueled operated DG in each of the candidate bus for installation of wind turbine. Moreover, K_W, K_S and K_F are the number of candidate buses for placement of wind, solar, and fuel operated DGs. Therefore, the dimension of the chromosome is equal to K_W + K_S + K_F.

The vector P is zero if there is no DG and its value is 1 or 2 or ……or N_C if there is DG. Further, P is equal to 1 or 2 or ……or N_C corresponds to first, second, or N_Cth candidate DG size [1]. The scheme adopted in the present article is illustrated as, let us assume that bus number 18 is used as the first candidate bus for installing wind turbine DG with five possible states of DG unit sizes as 20, 40, 60, 80, 100 kW etc. According to this table value P = 1 stands for 20 kW allocated DG size while P = 5 stands for 100 kW size of DG to be installed [1].

5.2. Genetic Algorithm (GA)

GA is a random population-based search technique in which solutions are obtained in the form of chromosomes. It is an effective technique for both optimization and machine learning techniques. GA is easily parallelized and it is capable to work on discrete and continuous type problem. GA based technique is a time-consuming algorithm, which means its computational time is more compared to other nature-inspired techniques.

5.3. Particle Swarm Optimization (PSO)

PSO is a nature-inspired evolutionary optimization technique. The technique works on the basis of social behavior like flocking of birds and schooling of fish. Basically, PSO starts with randomly generated solutions; these solutions are known as particles. Random velocities are assigned to these particles and are bound to keep track of their positions in search space in which they search the best solution called (pbest). Further, the particle swarm optimizer holds overall best solution, with reference to their position it is known as (gbest). The advantages of PSO are its computational time is less than heuristic techniques. It does not converge at local minima and disadvantages are it is not suitable for the combinatorial type of problems because of its continuous nature. Finally, the following two equations provide the velocity and position of each particle,

$$\begin{array}{l}{v}_{i,it}=w\times v_{i,it-1}+c_1\times ran{d}_1\times \left(pbes{t}_{i,it-1}-x_{i,it-1}\right)+c_2\times ran{d}_2\times \left(gbes{t}_{i,it-1}-x_{i,it-1}\right)\times \\ \left[1+\frac{-range}{\max ite}\times \left(Ite-1\right)\right]\\ x_{i,it}=x_{i,it-1}+v_{i,it}\end{array}$$

GA-PSO based hybridized technique mitigates the disadvantage of GA and PSO and yields better results in the formless computational time and local minima. The process of computation is presented in flowchart as shown in Figure 3.

6. Algorithm for Initial Population

Generally, the metaheuristic optimization technique starts with a random set of population of chromosomes known as initial solution. The initial population is created whose dimensions are product of (N_pop × N_par) with zero elements. Here N_pop is number of chromosomes and N_par is number of genes. The procedure for the solution is presented as follows

• Firstly, an integer is randomly selected between 1 and N_par for each possible solution.
• From the vector dimension h with integer elements between 1 and N_par M is randomly selected, i.e., let’s assume N_par = 8 and h = 5 is randomly selected, therefore M is filled with integer {1, 2,…,8} and h stand for Ist, IInd, IIIrd, IVth, and Vth gene respectively.
• Lastly, a random number M is generated between 1 and N_C for every element of candidate scenarios placed in the gene of M randomly. It can be understood as, say M₁ = 1 is first gene of the chromosome and selected randomly between 1 and N_C, as M₂ = 4 stands for fourth gene of the chromosome and picked up from 1 and N_C so on.

Table 5. Simulation results for optimal DGs size and their corresponding optimum location by PEM based method for scenario-1 using different metaheuristic optimization techniques.

Particle Swarm Optimization (PSO)			Genetic Algorithm (GA)			Hybrid Optimization (GA and PSO)
Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)
156.824654	202.6771	93.3447	238.258334	202.6771	89.0562	372.409068	202.6771	98.6957
DG TYPE			DG TYPE			DG TYPE
Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)
0 (0)	40 (13) 160 (14) 300 (15) 20(32) 20(33)	120 (13) 40 (15) 200 (17) 200 (31)	500 (15)	100 (15)	300 (15) 300 (31) 60 (32)	300 (15) 100 (31) 80 (32) 40 (33)	60 (32) 80 (33)	100 (15) (31)

shows the various is results achieved by PEM based method for scenario-1, and

Table 6. Simulation results for optimal DGs size and their corresponding optimum location by MCS for scenario-1 using different metaheuristic optimization techniques.

Particle Swarm Optimization (PSO)			Genetic Algorithm (GA)			Hybrid optimization (GA and PSO)
Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)
1054.205281	202.6771	89.9687	1615.485610	202.6771	91.4555	2656.724803	202.6771	95.2132
DG TYPE			DG TYPE			DG TYPE
Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)
400 (15)	40 (14) 120 (17) 100 (31) 80 (32)	400 (15) 200 (17) 200 (31) 20 (32)	500 (15)	100 (15)	300 (15) 300 (31) 60 (32)	400 (31)	400 (14) 40 (16) 200 (31)	400 (15) (32)

represents the simulation results obtained by MCS based method for scenario-1. Finally,

Table 7. Comparison of different parameters using PEM and MCS based method for uncertainty for different scenarios of various optimization techniques.

Probabilistic Technique	Different Intelligent Technique	Investment Cost before DG (in USD)	Investment Cost after DG (in USD)	Fitness Value of Function
PEM scenario 1	GA	1.4425 × 106	7.3094 × 105	406.0774
	PSO		7.6799 × 105	426.6635
	HYBRID		7.7463 × 105	430.3498
PEM scenario 2	GA		7.4988 × 105	416.5972
	PSO		7.4988 × 105	429.2726
	HYBRID		7.8558 × 105	436.4327
PEM scenario 3	GA		8.6499 × 105	480.5495
	PSO		9.3932 × 105	521.8421
	HYBRID		1.0225 × 106	568.0348
MCS scenario 1	GA		7.3850 × 105	410.2800
	PSO		7.8040 × 105	433.5537
	HYBRID		7.7596 × 105	431.0876
MCS scenario 2	GA		7.4615 × 105	414.5283
	PSO		7.7363 × 105	429.7956
	HYBRID		7.8163 × 105	434.2403
MCS scenario 3	GA		8.4815 × 105	471.1919
	PSO		1.0075 × 106	559.7419
	HYBRID		9.8414 × 105	546.7418

shows the values of investment cost and fitness function obtained by PEM and MCS based method, respectively, for different scenarios [1]. These results are obtained on the basis of artificial intelligent based optimization techniques like GA, PSO and hybrid (combination of GA and PSO). From these results, it is observed that the investment cost and values of fitness function is almost the same for both PEM and MCS-based methods of uncertainty for all above-mentioned optimization methods. Further, the investment incurred is the maximum ($1.0225 × 10⁶) for PEM based method for scenario 3 for the hybrid-based optimization technique. The minimum value of investment cost ($7.3094 × 10⁵) for GA-based optimization technique for scenario 1 for PEM as well as MCS-based methods. On the basis of these results, it is concluded that investment and fitness values are almost the same for PEM and MCS-based methods. Therefore, PEM and MCS-based methods provide almost the same results for the same scenarios except that MCS based method requires more computation time. According to Table 5, the investment cost is reduced after placement of DG.

Furthermore, it is observed from Table 5 and Table 6 that the execution time of hybrid optimization technique is more as compared to GA and PSO algorithms. Further, it is observed from Table 5 and Table 6 that the system loss reduction capability hybrid technique is more as compared to GA and PSO. Additionally, the execution time of PSO-based optimization is less than that of GA-based optimization technique.

Figure 4 demonstrates the convergence of fitness function and cost function, respectively, for scenario-1. From these figures, it is observed that the GA-PSO hybrid technique converges very fast while GA based technique takes more time to converge. It is evident from Figure 4 that GA-based technique converges after 46 iterations, the PSO-based technique converges after 15 iterations, and hybrid technique converges after 8 iterations. The same pattern is followed for cost function also as is evident from Figure 5. Furthermore, Figure 6 shows the voltage profile of the 33-bus system before and after DG placement under different techniques. From this figure, it is noticed that GA-PSO hybrid technique is the most powerful technique as far as voltage improvement is concerned among all methods i.e., GA and PSO. Moreover, PSO has better capability to improve the system voltage profile.

From Figure 7 it is observed that the convergence rate of PSO is slightly higher than hybrid-based technique and GA further gives same type results, which means its convergence rate is very slow and requires 49 iterations to finally settle down. From this figure, it is also noticed that PSO technique converges after 21 iterations and hybrid technique converges after 26 iterations. As we see from Figure 8 this figure also follows the same pattern as is shown in Figure 7.

Figure 9 represents the system bus voltage profile for scenario-2 by PEM-based technique. In this figure, it is observed that system voltage improves after DG placement. Further, it is noticed that GA-PSO hybrid technique is most capable of improving the system bus voltage followed by PSO and GA, respectively. Thus from this figure, it is concluded that the hybrid-based method is most suitable for enhancing the system voltage profile.

Figure 10 and Figure 11 are evidence of variation of fitness and cost functions by using GA, PSO, and GA-PSO based hybrid techniques respectively, under scenario-3. From these figures, it is observed that the GA based approach stabilizes and obtains optimized value of functions after 49 iterations and PSO provides optimized value after 45 iterations. Finally, GA-PSO based hybrid method provides an optimized value of objective after 34 iterations. Therefore, it is concluded that the GA-PSO-based optimization technique provides quick and better results among GA and PSO-based methods. Finally, Figure 12 shows the system voltage profile before and after DG placement by GA, PSO, and GA-PSO-based optimization technique in this method GA based technique provides better improvement in voltage profile than PSO and hybrid GA-PSO based technique.

The simulation results shown in Table 7,

Table 8. Results for optimal DGs size and their corresponding optimum location using by PEM for scenario-2 using different metaheuristic optimization techniques.

Particle Swarm Optimization (PSO)			Genetic Algorithm (GA)			Hybrid optimization (GA and PSO)
Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)
154. 045316	202.6771	92.7158	235.338242	202.6771	93.6265	396.141804	202.6771	93.9004
DG TYPE			DG TYPE			DG TYPE
Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)
200 (15) 160 (25)	300 (15) 40 (16) 300 (31) 60 (33)	120 (13) 400 (15) 40 (32)	0 (0)	500 (15) 500 (31) 60 (33)	100 (31) 100 (32)	200 (15) 120 (4)	80 (14) 200 (15) 80 (16) 400 (31) 40 (33)	400 (15) 300 (31)

,

Table 9. Results for optimal DG sizes and their corresponding optimum location using by PEM for scenario-3 using different metaheuristic optimization techniques.

Particle Swarm Optimization (PSO)			Genetic Algorithm (GA)			Hybrid optimization (GA and PSO)
Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)
168.890842	202.6771	94.5629	248.739512	202.6771	93.7272	384.948469	202.6771	93.4615
DG TYPE			DG TYPE			DG TYPE
Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)
160 (16)	300 (15) 120 (16) 160 (17) 300 (31) 40 (32)	500 (15) 400 (31)	0 (0)	500 (15) 100 (31) 100 (32)	300 (15) 500 (31)	300 (15) 40 (32)	120 (14) 200 (16) 160 (17)	200 (15) 400 (31)

and

Table 10. Results for optimal DGs size and their corresponding optimum location using by MCS for scenario-2 using different metaheuristic optimization techniques.

Particle Swarm Optimization (PSO)			Genetic Algorithm (GA)			Hybrid optimization (GA and PSO)
Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)
1047. 508148	202.6771	92.7158	1647.367694	202.6771	93.7272	2733.378447	202.6771	94.7006
DG TYPE			DG TYPE			DG TYPE
Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)
300 (15) 200 (31)	0 (0)	500 (15) 120 (17) 300 (31) 40 (32)	0 (0)	40 (32)	100 (31)	200 (14) 200 (16) 200 (31)	80 (13) 300 (31)	300 (15) 160 (17) 400 (31)

for PEM and MCS-based method for different optimization techniques for scenario-2 and scenario-3.

In Figure 6, Figure 9, and Figure 12 represent the system voltage profile (in p.u.) at different buses with and without DG using PEM-based method by incorporating three optimization technique such as GA, PSO, and hybrid (combination of GA and PSO). Further, it is observed from these figures that the hybrid-based optimization technique is more capable of enhancing the system voltage profile as compared to the other two techniques for scenario-1 and 2. Moreover, the same is applicable for MCS based method with and without DG units.

7. Results and Discussion

The proposed algorithm was applied on the IEEE 33-bus radial distribution system with the base power of 1 MVA and the base voltage of 12.66 kV. The total load of the distribution system is 3.715 MW and 2.3 MVAr. The GA-PEM, PSO-PEM, and hybrid (combination of GA and PSO) -PEM along with GA-MCS, PSO-MCS, and hybrid (combination of GA and PSO)-MCS-based method were applied to address uncertainty. The probabilistic load solutions were obtained on the basis of MCS and PEM-based techniques. To implement programs for PLF and GA, PSO and hybrid-based optimization techniques were developed in MATLAB environment. The load is considered as an uncertain parameter and varies according to the standard normal distribution function at each bus. Further, the uncertainties of wind speed and solar radiation are considered in the present study as following the properties of Weibull PDF. The candidate buses for DG placement are selected on the basis of voltage profiles of different buses and are put in ascending order. Further DG sizes and respective candidate buses are shown in Table 2. The fuel prices are also considered an uncertain parameter and follow the characteristic of geometric Brownian motion (GBM). The magnitude of voltage profile at each bus does not exceed ±5% of the nominal grid voltage. Further, it is assumed that the maximum limit of power in distribution lines is 4 MVA. The details of simulated results obtained in this method are presented in Table 5, Table 6, Table 7, Table 8, Table 9, Table 10 and

Table 11. Results for optimal DG sizes and their corresponding optimum location using MCS for scenario-3 using different metaheuristic optimization techniques.

Particle Swarm Optimization (PSO)			Genetic Algorithm (GA)			Hybrid Optimization (GA and PSO)
Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)	Execution time for algorithm (in seconds)	Power loss without DG (in kW)	Power loss with DG (in kW)
1063.508148	202.6771	103.7904	1716.338242	202.6771	93.6265	2773.036281	202.6771	100.2172
DG TYPE			DG TYPE			DG TYPE
Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)	Wind DG size (in kW) and location (bus)	Solar DG size (in kW) and location (bus)	Fueled DG size (in kW) and location (bus)
200 (14) 300 (15)	120 (16) 40 (17) 200 (31) 40 (32)	400 (15) 400 (31)	0 (0)	100 (31)	100 (15) 200 (17) 500 (31) 20 (12)	80 (16) 300 (31)	160 (14) 200 (17) 500 (31) 40 (33)	200 (15) 300 (31) 60 (32)

, for PEM and MCS-based methods for different scenarios. From these results, it has been observed that the proposed method is competent enough to reduce the system loss by DG installation. Moreover, the voltage profile of the system with and without DG has been shown in Figure 6, Figure 9, and Figure 12. From these figures, it was observed that system voltage was enhanced significantly by placement of DG. Moreover, it is also observed that the hybrid-based optimization technique is more capable of enhancing the system voltage profile as compared to GA and PSO-based optimization techniques. Additionally, the convergence of fitness function is depicted in Figure 4, Figure 7, and Figure 10, for PEM-based method embedded with GA, PSO, and hybrid-based optimization techniques for three scenarios. The cost functions are shown in Figure 5, Figure 8, and Figure 11 for the PEM-based method embedded with GA, PSO and hybrid-based optimization techniques for scenario-3. Different tables show the simulation results are obtained by the above-mentioned method. It is concluded that the loss reduction capacity of different methods for different scenarios is almost the same for GA, PSO, and hybrid-based optimization techniques.

8. Conclusions

In this paper (2m + 1) PEM method is presented, which addresses the uncertainty of load, power generated by solar and wind operated DG. The probabilistic load flow (PLF) solution is carried out to investigate the uncertainty of these parameters. The PEM and MCS-based methods were applied to achieve solutions of PLF. Further, for optimization of fitness function and minimization of investment cost with DG placement, GA, PSO, and Hybrid (combination of GA and PSO) based methods were applied with PEM and MCS. The simulation results obtained by the PEM method were compared with the MCS method, which is used as a benchmark method for uncertainty evaluation. Moreover, as it is evident from the results obtained by PEM-GA, PEM-PSO and PEM-Hybrid based methods are almost similar to MCS-GA, MCS-PSO, and PEM-Hybrid method for three different scenarios. On the basis of obtained results, it is noticed that the simulation time of MCS based method is longer, as compared to PEM based method. Further, it is observed that for the same scenario, the hybrid-based optimization technique provides a better improvement in system voltage profile. Finally, it is concluded that PEM based method provides similar results as MCS with a lesser computational burden. For scenario-3, the system voltage profile is enhanced significantly by GA.