The implementation of PV-DGs into the power system largely effects the power flow, power quality, and dynamic performance of the test system. Since the output of PV-DG is affected by real-time changing of solar radiation values, traditional deterministic power flow methods are insufficient for accurately capturing and assessing these effects [17] . So, approaches for computing probabilistic load flow such as Monte Carlo simulation method are required. The authors in [18] presented the probabilistic density function (PDF) following the integration of PV-DG into the grid. A probabilistic power flow methodology using the stochastic formation approach was proposed in [19] , using the principles of uncertainty quantification concept. The Beta probability distribution functions used to estimate hourly solar irradiance are derived from three years of historical data collected from the study site [20] . This data is then used to create a representative frequency distribution of irradiance and wind speed measurements for a typical day in each season.

The research in [17] explored the integration capacity of large-scale PV systems into the grid, PV connection points, and the output of multiple PV power plants using techniques such as probability density distribution, sensitivity evaluation, and over-limit probability assessment. In summary, the research investigated how large-scale PV systems can be connected to the grid and how their output fluctuates and impacts grid performance using statistical and analytical methods. The researchers of the PV probability model in [21] and [17] stated that the solar radiation follows the Beta distribution and the normal distribution. As a result, the probability density function for solar radiation (G) can be presented as in Equation (11):

$f\left(G\right)=\frac{\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\left(\frac{G}{G_{\max }}\right)}^{\alpha -1}{\left(1-\frac{G}{G_{\max }}\right)}^{\beta -1}$ (11)

where $G_{\max }$ is the maximum solar radiation value, and $\alpha$, $\beta$ are the Beta distribution fitting parameters.

Statistical models are crucial for accurately capturing the variability in PV-DG outputs. These models, represented by PDFs, enable the creation of random variables required for Monte Carlo simulations (MCS). A sequential Monte Carlo simulation was used to assess the distribution system’s probabilistic three-phase power flow. This simulation incorporates statistical data on solar resources, advancing through time in a step-by-step manner to account for the time-dependent characteristics of variables [22] .

Figure 2 shows the complete diagram of the model that combines the PV module with the inverter. The model parameters are specified at an irradiance level of 1 kW/m², making it particularly accurate at higher power output levels. When a given irradiance level is input, the panel’s output is adjusted by a factor that depends on the PV module temperature [22] . The following equation was used in [22] to calculate the output power of the PV-DG based on solar radiation and PV module temperature values.

$P_{pv}=P_{mpp}{f}_t\left(T_c\right)G$. (12)

The core goal is to find the optimal allocation sizing of PV-DG systems in the test system. The losses are the key factor, deeply impacting both economic and technical performance. To achieve the optimal integration of PV-DG units must be strategically positioned to minimize power losses across the system without breaching voltage thresholds, as highlighted in [23] . Consequently, the main objective function to be minimized is defined in Equation (13).

$\text{Minimize}{P}_{Loss}={\displaystyle {\sum }_{i=1}^N{P}_{Li}}$ (13)

where, $P_{Li}$ indicates the $i^{\text{th}}$ line losses, while $N$ signifies the total count of lines in the test system.

To enhance the voltage profile as a secondary objective in the radial power system, the aim is to minimize the total voltage deviations at the load buses. This objective function is typically formulated mathematically and expressed as follows [23] :

$\text{Minimize}{\displaystyle {\sum }_{i=1}^N\left|V_i-V_{ref}\right|}$ (14)

In this context, $V_i$denotes the voltage at load bus $i$, while $V_{ref}$ represents the reference voltage at load bus $i$, typically set to 1.0 per unit (pu).

Strategic deployment of PV-DG units and distribution static compensators (DSTATCOM) in a test improves the voltage profile significantly. This optimized configuration supports efficient provision of real and reactive power, thereby minimizing power losses and strengthening voltage stability. The Total Voltage Variation (TVD) across the network is formulated as [24] :

$TVD=\{\begin{array}{l}0,\text{if}0.95\le V_t\le 1.05\\ {\displaystyle {\sum }_{t=1}^N\left|V_{ref}-V_t\right|},\text{else}\end{array}$ (15)

A multi-objective function combines multiple goals to be optimized simultaneously, all within predefined operating constraints. This multi-objective function, described in [24] , was designed to minimize losses while also enhancing the voltage profile and maximizing the voltage stability index. The mathematical formulation for identifying the optimal allocation of DG and DSTATCOM was as follows:

$\text{Minimize}\left(F\right)=\text{Min}\left({\beta }_1\Delta P_{TL}^{\frac{DG}{DST}}+{\beta }_2\Delta TV{D}^{\frac{DG}{DST}}+{\beta }_3\left(\frac{1}{\Delta VS{I}^{\frac{DG}{DST}}}\right)\right)$ (16)

where, ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$ represent the weighting factors associated with minimizing power loss, minimizing TVD, and maximizing voltage stability index (VSI), respectively.

A year’s worth of hourly solar radiation data (in W/m²) measured with a PV generation unit is used to create a dataset for a probabilistic PV solar radiation model. The data is collected using a Solar-Log Base100 system, a sensor box equipped with a reference cell for measuring solar radiation. To ensure consistency, the data was recorded as a time series with 5-minute interval at local standard time for every day of the year. This dataset, referred to as the “reference solar irradiance data,” is displayed in Figure 4 and initially recorded as a time series with 5-minute intervals for the entire year. The analysis begins by importing the data from Excel into MATLAB, where missing values are identified and filled through interpolation. After completing this step, the data is preprocessed to calculate the average and variance of solar irradiance, which is then used to estimate the output of the PV system.

The proposed approach implements a comprehensive approach to analyze solar radiation data using statistical methods and probabilistic modeling. It employs concepts from the Beta distribution, seasonal statistics, and linear modeling of PV output, which are critical in energy management and forecasting in photovoltaic systems.

The mean ( $\mu$) and variance ( ${\sigma }^2$) of the solar radiation data are calculated as:

$\mu =\frac{1}{n}{\displaystyle {\sum }_{i=1}^n{G}_i}$ (17)

${\sigma }^2=\frac{1}{n-1}{\displaystyle {\sum }_{i-1}^n{\left(G_i-\mu \right)}^2}$ (18)

where $G_i$ represents the solar radiation data points.

The Beta distribution is parameterized by shape parameters αalphaα and βbetaβ, which are calculated from the mean and variance of the data as follows:

$\beta =\frac{\left(1-\mu \right)\mu }{{\sigma }^2\left(1+\mu \right)}$ (19)

$\alpha =\frac{\beta \mu }{1-\mu }$ (20)

The probabilistic model for solar irradiance is based on Beta distribution. Which is a continuous range of probabilities outlined by the range [0, 1], making it suitable for modelling proportion such as solar irradiance. The probability density function (PDF) of the Beta distribution is defined as:

$f\left(x\right)=\frac{\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{x}^{\alpha -1}{\left(1-x\right)}^{\beta -1},0\le x\le 1$ (21)

where ΓGammaΓ is the gamma function, which generates the factorial function.

$x$ is the normalized irradiance (between 0 and 1).

$\alpha$ and $\beta$ are the shape parameters of the beta distribution.

Random samples are generated from the Beta distribution using the betarnd function in MATLAB, which follows the Beta distribution:

$X~Beta\left(\alpha ,\beta \right)$ (22)

The PV output is modelled as a linear function of solar radiation:

$P_{PV}=\eta \cdot G$ (23)

where, $P_{PV}$ is the PV output.

$\eta$ is the efficiency of the PV system (15% in the script).

$G$ is the solar irradiance.

The probability of solar radiation exceeding a specified threshold ( $G_{th}$):

$P\left(G>G_{th}\right)=\frac{1}{n}{\displaystyle {\sum }_{i=1}^{n}1\left(G_i>G_{th}\right)}$ (24)

where I, is a reflection symbol that returns one if the condition is true and zero otherwise.

The proposed method generates a normalized irradiance profile over a 24-hour period using the Beta PDF:

$MyIrrad\left(t\right)=f\left(x_t;\alpha ,\beta \right)$ (25)

where $x_t$ is a normalized time point for each hour (from 0 to 1).

The model uses the beta distribution to probabilistically represent the solar irradiance over a day. It adjusts the PV output based on the generated irradiance profile, which reflects varying conditions throughout the day.

When minimizing power losses is the only goal, an overproduction of active power from PV-DG can cause voltage levels to exceed permissible limits. To counteract this, Reactive Losses are introduced as a secondary objective as in Equation (27), and the Cumulative Voltage Deviation (CVD) as presented in Equation (28) is considered as a tertiary objective. To streamline the technique and procedure, a restriction function is incorporated in the approach as a Quadratic Penalty Factor (QPF), as outlined in Equation (29). Minimizing this factor then becomes the fourth objective. In this approach, all variables are calculated for 24 hours based on the probability of solar radiation

$f_1=\min {\displaystyle {\sum }_{t=1}^{24}{\displaystyle {\sum }_{i=1}^N{P}_{Li}}}$ (26)

$f_2={\displaystyle {\sum }_{i=1}^N{Q}_{Li}}$ (27)

$f_3=CVD={\displaystyle {\sum }_{i=1}^N\left|V_i-V_{rated}\right|/N}$ (28)

$f_4=QPF=\{\begin{array}{ll}{\left(V_i-V_{\min }\right)}^2\hfill & V_i\ge V_{\min }\hfill \\ 0\hfill & V_{\min }\le V_i\le V_{\max }\hfill \\ {\left(V_i-V_{\max }\right)}^2\hfill & V_i\le V_{\max }\hfill \end{array}$ (29)

where; $Q_{Li}$ denotes the reactive power loss at the at $i^{\text{th}}$ line, and $N$ represents all lines considered in the test system.

$V_i$ represents the voltage at the $i^{\text{th}}$ bus. While, $V_{rated}$ is the rated voltage for the test system, typically set to 1 pu. $V_{lim}$ refers to the voltage limits, either minimum or maximum, for the system.

Equation (30) represents the comprehensive objective function that must be minimized.

$f=\min {\displaystyle {\sum }_{i=1}^{24}\left({\omega }_1{f}_{1i}+{\omega }_2{f}_{2i}+{\omega }_3{f}_{3i}+{\omega }_4{f}_{4i}\right)}$ (30)

Each factor of $\omega$ is assigned a unique weight, and the summition of all these weights must equal 1.

The GA begins by generating an initial population, which includes the PV-DG locations, active power, and power factor, then examines the objective function using the power flow solver in OpenDSS, considering the full distribution system structure. The fitness function can be adjusted for improved outcomes, depending on whether the goal is to enhance the voltage profile or reduce losses. In this study, various weight combinations for each component of Equation (28) were tested through trial and error to determine the optimal configuration. The weights applied are 0.5 for $P_{Loss}$, 0.1 for $Q_{Loss}$, 0.2 for $CVD$, and 0.2 for the voltage limits.

The constraints of the optimization methods are essential for accurately modeling the active and reactive power flows within the test system, as outlined in Equations (31) and (32) [12] .

$P_{Gi}=P_{Di}+{\displaystyle {\sum }_{j=1}^{nb}\left|V_i\right|\left|V_j\left[G_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij}\right]\right|}$ (31)

$Q_{Gi}=Q_{Di}+{\displaystyle {\sum }_{j=1}^{nb}\left|V_i\right|\left|V_j\left[G_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij}\right]\right|}$ (32)

where $V_i$ and $V_j$ denote the voltage magnitudes at buses, $P_{Gi}$ and refers to the active power generated at bus $i$, while $P_{Di}$ indicates the active power demand at that same bus. Similarly, $Q_{Gi}$represents the reactive power generated at bus $i$, and $Q_{Di}$ represents the reactive power demand. The terms $G_{ij}$ and $B_{ij}$ are the conductance and susceptance values of the line connecting buses $i$ and $j$, which characterize the line’s capacity to conduct and store electrical power, respectively.

The constraints set for the optimization include voltage limits across all buses within the network, along with specific operational limits for the PV-DG units. Equation (33) defines the upper and lower voltage boundaries, indicating the allowable range for voltage levels to ensure stable system operation.

$V_{min}\le V_i\le V_{max}$ (33)

where $V_{\min }$ and $V_{\max }$ denote the minimum and maximum permissible voltage magnitudes, respectively.

Determining the optimal size of PV-DG units involves adhering to specific operational limits, defined by the minimum $PVD{G}_{\left(\text{min}\right)}$ and maximum $PVD{G}_{\left(\text{max}\right)}$ boundaries, as outlined in Equation (24).

The load flow results for the base scenario, where no DG units are integrated, are detailed in Table 1 for the 33-bus test system. Table 1 provides a comprehensive comparison, presenting active and reactive power losses, minimum bus voltage levels, and cumulative active and reactive power metrics from this study as well as from comparable research sources. In this investigation, the active power loss for the system is determined to be 211.5 kW, with the lowest observed voltage being 0.9022 per unit at bus 18.

To validate the proposed algorithm’s performance, its results are benchmarked against those from prior studies. In particular, study [24] reports an active power loss of 210.98 kW, with a minimum voltage of 0.9037 per unit. Additionally, the findings in this research and [28] indicate a base case real power loss of 210.98 kW and a reactive power loss of 143.13 kvar. In contrast, findings from [29] reported slightly lower losses, measuring 201.99 kW for real power and 134.77 kvar for reactive power. The worst or minimum voltage is 0.91337 pu at the 18th bus in the base case. This discrepancy arises because researchers in [24] , [28] used resistance and reactance values of 1.7114 Ω and 1.2351 Ω for the 7th line, while studies in [29] , [26] reported these values as 0.7114 Ω and 0.2351 Ω, as clarified in [28] with more references for this difference.

In the first case, where a constant PV-DG production is considered, A comparative analysis are presented with other research studies using loss sensitivity factor (LSF), Exhaustive load flow (ELF), and GA methods, in case of unity power factor of PV-DG as in Table 2 . Table 2 provides a summary of results, including the optimal DG placement, corresponding optimal DG size, and the total power loss within the test system. The findings indicate that placing multiple DG units of optimal size at the ideal location results in a substantial reduction in power losses.

The optimal placement of a single PV-DG is at bus 6 with a size of 2600 kW, resulting in a system loss of 111.052 kW compared with 2601 kW and 111.1 kW for size and losses, respectively in [16] . While GA presented 2835 kW for size

Also, for the first case, a comparison of GA with other research studies for two cases of unity and 095 lag power factor of PV-DG are presented in Table 3 . As discussed earlier, placing Type 1 DG compensates for active currents within the distribution lines, while Type 2 DG addresses reactive currents. This setup enables compensation for local active and reactive loads at the bus level, leading to a reduction in real power losses attributed to both the active and reactive components of branch currents. The proposed algorithm efficiently determines both the optimal locations and sizes of DG in a single step, unlike a two-stage approach. Here, installing one DG implies the combined installation of one Type 1 DG and one Type 2 (capacitor) DG. Detailed DG location findings are provided in Table 3 .

At unity pf for 1 PV-DG Best Active Power Losses: 111082.136198 Best Reactive Power Losses: 76589.923719 k Var. At unity pf for 2 PV-DG the Active Power Losses: 87147.304554 Best Reactive Power Losses: 54536.497028. At unity pf for 3 PV-DG the Active Power Losses: 72.7142 kW and the Reactive Power Losses: 45.3521 kvar.

At 0.95 pf for 1 PV-DG the Best Active Power Losses: 78234.756217 Best Reactive Power Losses: 56115.774219. At 0.95 pf for 2 PV-DG the Best Active Power Losses: 45182.094978 Best Reactive Power Losses: 26406.109248. At 0.95 pf for 3 PV-DG the active Power Losses: 28.4468 kW and Best Reactive Power Losses: 15.7403 kvar.

Assuming constant PV-DG generation does not adequately capture the typical behavior of a radial distribution system over time while incorporating variable PV-DG output offers a more realistic representation of actual conditions. Additionally, dynamically adjusting PV-DG output to minimize losses over a 24-hour period provides more accurate insights into the optimal DG placement and the maximum DG capacity required to reduce system power losses. Therefore, this research examines two distinct cases.

A year of 5 min averaging solar irradiance data gathered within a PV energy generation in Cairo, Egypt is utilized as raw data for developing the probabilistic model. In this article, this data is used as a reference for solar radiation data. The annual hourly average of the data collected is presented in Figure 5 . The measured irradiance data (after interpolation and normalization) forms the foundation for the distribution parameters in the Beta distribution. The Alpha and Beta values as shown in Figure 6 , reflect how the irradiance changes throughout every hour of the day as illustrated in Figure 6 . The probability distribution functions (PDF) used to estimate hourly solar irradiance are based on one year of historical data collected from the study site. This data is then employed to generate frequency distribution of irradiance measurements for a typical day.

Figure 6 gives an overview of how the Beta distribution parameters change throughout the day, allowing to assess variability and predictability in solar irradiance. Figure 7 zooms in on the probability of different irradiance values for specific hours of the day (10 to 15), helping in modelling the solar generation more accurately by visualizing the Beta-distributed solar radiation for those peak hours. Figure 7 plots the probability density functions (PDFs) of solar radiation modeled by Beta distributions for each hour between 10 and 15. The normalized solar irradiance (values between 0 and 1), which represents the intensity of the solar radiation relative to the maximum possible value for each hour. The probability density for each irradiance value (i.e., how likely a certain irradiance level is at each hour.

Table 4 compares optimal solutions for DG allocation and sizing in a 33 bus test system across different methods: GA based approach from a prior study [29] , This research and the probabilistic and deterministic approaches developed in the current study, and Artificial Bee Colony algorithm from (ABC) in [26] . Where, in Case 1: Minimization of active power losses under voltage constraint. Case 2: A multi-objective optimization problem aimed at minimizing active and reactive power losses while maximizing voltage profile enhancement, under voltage constraint. In OpenDSS, the PV system element has two operational modes, Mode 1 and Mode 2, which define how the PV system interacts with the grid. These modes govern the control of active and reactive power. Mode 1: Fixed Power Factor Mode, where the PV system operates with a fixed power factor. Mode 2: Voltage Control Mode, where the PV system dynamically adjusts its reactive power (Q) to regulate the voltage at the point of interconnection.

The results show consistency in single PV-DG allocation, where Bus 6 is consistently identified as the optimal location across all methods and approaches. The sizes and placements of the second DG vary widely between methods, likely due to differences in optimization criteria and assumptions about network behavior. Deterministic methods in this research tend to favor larger DG sizes, particularly for the second DG at Bus 30, which might indicate a focus on minimizing losses or improving voltage profiles. Probabilistic methods produce more conservative DG sizes, reflecting consideration of uncertainties in PV generation and load variability.