Analysis of probabilistic optimal power flow in the power system with the presence of microgrid correlation coefficients

Abstract

The uncertainty of microgrid resources in power systems has increased, which implies many challenges for their design and planning. The penetration coefficient of microgrids in power systems, as well as the high uncertainty of these sources, requires an analysis of probabilistic methods. These types of energy sources are inherently uncertain and bring many unknowns to the power system. One of the most important aspects to be analyzed is the distribution of the probabilistic optimal power flow (POPF). This research examines some methods for the distribution of possible loads in power systems, namely the Monte Carlo method (MCM), the two-point estimation method (2PEM), and the three-point estimation method (3PEM). Then, these methods are used to distribute the possible POPF. This work studies the adjusted probability density function (PDF), average, and deviation of losses for each method. Moreover, the appropriate selection of microgrid resources is determined while considering correlated variables. In order to compare the effectiveness of the proposed methods, a 30-bus IEEE standard test system is used in the MATLAB software, showing that 2PEM is more suitable than the others. The results related to microgrids are executed by running P_OPF in buses 5, 7, and 8.According to the results, for bus 8, the possible load distribution value is 52.897 MW, which requires the placement of a wind power plant. For buses 2 and 5, the possible load distribution is less than 10 MW, which is predicted in these buses considering the solar power plant.

Keywords:

• POPF
• microgrid
• Monte Carlo
• two/three-point estimation methods
• losses
• correlated

1. Introduction

1.1. General perspective

In the field of probabilistic optimal power flow (POPF), many works have been carried out in the last decade (Montoya et al., 2019; Peng et al., 2022) which, in addition to the development of new and diverse mathematical methods, shows the importance of this issue (Bahrami et al., 2016). However, this challenge is still open for further research. With the expansion of privatization policies in the electricity industry, microgrid resources have expanded (Alayi et al., 2021; Feng et al., 2022; Lee et al., 2022), and this expansion and widespread use have sparked an interest in the technical and economic issues of these systems. The issues related to the exploitation and distribution of power, which were initially limited to wind turbines (Shi et al., 2011; Ying et al., 2015), were also extended to photovoltaic (PV) systems (Gutierrez-Lagos & Ochoa, 2019; Li et al., 2022; Montoya et al., 2022). The power of these systems is evaluated at the distribution level, which is where most of these resources are used (with a limited number at the super distribution level). Now we have to state the main question of the research. How can optimal possible load distribution be done in a power system with the presence of photovoltaic and wind power plants? Now we have to state the main question of the research. How can optimal possible load distribution be done in a power system with the presence of photovoltaic and wind power plants? Appropriate methods have been used to coordinate wind and photovoltaic systems in the distribution network. We seek to answer the questions of how to do this with PPF and the power uncertainty of these resources at the distribution level based on the point estimation method. The basis of the point estimation method is that it is possible to increase the accuracy of the estimates by considering the risk and uncertainties to reduce such analytical problems in the PPF method.

1.2. Review of recent literature

In (Fan et al., 2012a), some methods based on probability distribution functions (PDF) are reviewed, and POPF analysis is carried out at the distribution level. The Monte Carlo Method (MCM) is discussed in (Constante-Flores & Illindala, 2018). Another analytical method to determine these POPF involves combining the accumulation method with Von Mises functions (Zhou et al., 2020). Although this method provides a better estimate than the Gram-Charlier expansion (Yuan et al., 2011), it is associated with a higher computational cost. In (Ye et al., 2022), probability and solar radiation models, which are based on the relationship between the output performance of PV systems and the PDF, are studied to obtain PV power. In the work by (Fan et al., 2012b), solar radiation is considered to be random and regular, and a probability model of PV production dynamics is presented. Probabilistic methods can be effectively applied to the analysis of optimal load distribution. The point estimation method (PEM) is used to solve probabilistic distribution (Chen et al., 2015; Mohammadi et al., 2013). It is an approximate method that can measure the accuracy of output variable data at different times based on the costs and the occurrence of events (Delgado & Domínguez-Navarro, 2014). Although some heuristic methods are successful in determining optimal, they are still slow in terms of convergence.

The work by (Ran & Miao, 2016) presents a probabilistic method to analyze the steady-state operating conditions of a distribution system with wind and PV power plants. A fuzzy optimization (Baghaee et al., 2017; Zadeh, 1965) approach to generate power system timing while considering solar and wind energy systems is presented in (Kamal et al., 2021). Deterministic load flow (DLF) and load distribution are used to analyze and evaluate the planning and commissioning of power systems based on daily operation. DLF uses different power generation and load demand values for a selected network structure in order to calculate the system states and power flow (Chen et al., 2008; Sexauer & Mohagheghi, 2013). Therefore, DLF ignores the uncertainties in power systems, such as generator outage rates, grid configuration changes, and load demand. In addition, today’s power systems, by combining wind turbines and PV systems, generate power fluctuations due to the presence of uncontrollable primary sources. Therefore, the deterministic method is not efficient for analyzing new power systems, and the results obtained from POPF may produce an unrealistic estimate of system performance. Various mathematical methods can be used for analysis, such as the probability method, fuzzy sets, and interval analysis. The probabilistic method has a more integrated mathematical background and can be used for the power systems included in the literature. It can also be used in the field of energy management and environmental energy harvesting techniques for energy supply (Kapeller et al., 2021; Singh, 2020).

1.3. Motivation and structure of the paper

The integration of a significant amount of microgrid resources (e.g., wind and solar generation) into power systems has led to very important operation challenges, which is caused by the uncertainty in power production inherent to these sources. In addition to this uncertainty, the elements of the power system are not 100% reliable. Optimal power flow is one of the methods used to determine the state of a system. With unknown parameters, determining the OPF cannot accurately determine the state of a power system. In this analysis, instead of using absolute values for power flow inputs, PDF are used, and, as a result, the outputs, instead of being absolute numbers, are expressed as probability density distribution or cumulative probability distribution functions.

With all these descriptions, it can be said that tracking and controlling the occurrence of processes is not possible in completely microgrid-based systems. POPF methods and system structure, as well as several methods aimed at coordinating the possible load distribution of microgrids, have been proposed in order to address these challenges. The main issues associated with analytical methods are complex mathematical calculations and a lack of accuracy due to various approximations. In this vein, this study uses the Monte Carlo method (MCM), the two-point estimation method (2PEM), and the three-point estimation method (3PEM) to solve the problem under study. For the MCM, a disadvantage is that it takes a lot of time to execute and iterate. Thus, the 2PEM actually seeks to expand the PPF model of the system; the basis of this method is that the accuracy of the estimations can be increased by considering risks and uncertainties, with the aim to reduce issues in PPF analysis. Sometimes, 2PEM and 3PEM can have similar results. In these methods, to obtain the probability density functions in terms of cost and losses. The structure of this article is as follows: the second section examines the POF methods; the third section describes the possibilities in the sources; the fourth section deals with point estimation while considering correlation variables; the fifth section outlines the studied system; the sixth part presents the simulation results; and the last section presents the conclusions of this work. Considering the correlated variables for microgrids, which has been done less in the works. Also, we want to see the average results and the standard deviation of the losses with the point estimation method.

2. POPF methods

There are many methods for calculating the power flow in power systems that differ in speed, accuracy, and computer storage requirements. Optimization is the process of finding the best solution to a problem. Various methods have been used to this effect, as is the case of the MCM, 2PEM and 3PEM. This section reviews these methods and their corresponding equations.

2.1. Mcm

MCM is a method of iterative estimation within an algebraic model that uses a set of random numbers in the input (Binder et al., 2010; Carpinelli et al., 2015). This method is often used when the model is complex and nonlinear or contains more than one pair of uncertain parameters (Xie et al., 2017). Although the MCM can provide accurate results, its calculations are time-consuming, so it is not useful in applications where time is important. First, a random number is determined, and then the probability of an event is compared to the value of the generated random number. For any state where the generated number meets the probability criterion, a process or a set of processes or developments will occur. This procedure can be repeated several times, and, for each repetition, a measurable output is produced. Finally, the series of tests with output results are subjected to statistical processing. The process and the events can be simple or very complex and contain multiple loops and algorithms—and even multiple random generators. In the MCM, the input variables are expressed as a random population, as shown in Equation (1)(1) $z^k=f\left(x_1^k,x_2^k,\dots ,x_n^k\right)$(1) :

(1) $z^k=f\left(x_1^k,x_2^k,\dots ,x_n^k\right)$(1)

The calculation of the mean and variance in the MCM is as described in Equations (2)(2) $M\left(z\right)=\frac{1}{k}\sum _{k=1}^n{z}^k$(2) and (3(3) $VAR\left(z\right)=\left[\frac{1}{K}\sum _{k=1}^n{\left(z^k\right)}^2-{\left(E\left(z\right)\right)}^2\right]$(3) ):

(2) $M\left(z\right)=\frac{1}{k}\sum _{k=1}^n{z}^k$(2)

(3) $VAR\left(z\right)=\left[\frac{1}{K}\sum _{k=1}^n{\left(z^k\right)}^2-{\left(E\left(z\right)\right)}^2\right]$(3)

The MCM provides accurate results, but it can be very time consuming. This issue makes Monte Carlo Simulation unattractive in real-time applications.

2.2. PEM

The purpose of the two-point estimation method is to calculate, a random variable that is a function of random input variables (Verbic & Canizares, 2006). In this method, several sub-problems are used, taking only two algebraic values of each unknown variable, which are placed on both sides of the average value (Che et al., 2020; Li et al., 2013). The OPF is run twice, the first time for the lower value, and the second time for the upper value with regard to the average, while the other variables of the average value are preserved. These two points may or may not be symmetrical around the average value. Thus, each sample set of the selected points is subjected to the nonlinear function, where the OPF obtains the transformed sample points. The key is that the value of uncertain input variables changes in different samples, as well as the optimal solution for each definite OPF problem (Che et al., 2020). It should be emphasized that the number of sample points selected in this method increases along with the number of non-deterministic variables. Given that large-sized systems have many non-deterministic variables, the 2PEM is well suited for small- and medium-sized systems.

The main formulation of the 2PEM with symmetric sampled points is as follows:

Step 1- Determining the number of uncertain variables.

Step 2 - Calculation of equation (4)(4) $E\left(Y\right)=0,E\left(Y^2\right)=0,E\left(Y^3\right)=0$(4) :

(4) $E\left(Y\right)=0,E\left(Y^2\right)=0,E\left(Y^3\right)=0$(4)

Step 3 - Setting $K=1$

Step 4 - Determining the value of K₁ ℰ ،K₂ℰ ،Pk₁ and Pk₂ from Equations (5)(5) $\mathcal{E}{k}_1=\frac{\lambda k_3}{2}+\sqrt{n+({\frac{\lambda k_3}{2}}^2}),k=1,2,\dots n$(5) to (12(12) $\lambda k_3=\frac{M_3\left(x_k\right)}{{{\sigma }_{x,k}}^k}$(12) ):

(5) $\mathcal{E}{k}_1=\frac{\lambda k_3}{2}+\sqrt{n+({\frac{\lambda k_3}{2}}^2}),k=1,2,\dots n$(5)

(6) $\mathcal{E}{k}_2=\frac{\lambda k_3}{2}-\sqrt{n+({\frac{\lambda k_3}{2}}^2}),k=1,2,\dots n$(6)

(7) $\mathrm{P}{k}_1=-\frac{\mathcal{E}{k}_2}{n{\xi }_k}\hspace{0.17em}$(7)

(8) $\mathrm{P}{k}_2=-\frac{\mathcal{E}{k}_2}{n{\xi }_k}\hspace{0.17em}$(8)

(9) ${\xi }_k=2\sqrt{n+({\frac{\lambda k_3}{2}}^2}),k=1,2,\dots n$(9)

(10) $\sum _{k=1}^{n}P{k}_1+P{k}_2=1$(10)

(11) $P{k}_1+P{k}_2=\frac{1}{n}$(11)

(12) $\lambda k_3=\frac{M_3\left(x_k\right)}{{{\sigma }_{x,k}}^k}$(12)

Step 5 - Calculating $x{k}_1$and $x{k}_2$

(13) $x{k}_1={\mu }_{x,k}+{\mathcal{E}}_{k,1}{\sigma }_{x,k}$(13)

(14) $x{k}_2={\mu }_{x,k}+{\mathcal{E}}_{k,2}{\sigma }_{x,k}$(14)

Step 6 – Running the OPF for the variables.

(15) $X=\left[{\mu }_{X,1},{\mu }_{X,2},\dots x_{k,i},\dots {\mu }_{X,n}\right]i=1,2$(15)

Step 7 - Calculation of equations (16)(16) $E\left(Y\right)\cong \sum _{k=1}^n\sum _{i=1}^2{p}_{k,i}h\left(\left[{\mu }_{X,1},{\mu }_{X,2},\dots x_{k,i},\dots {\mu }_{X,n}\right]\right)$(16) to (18(18) $E\left(Y^3\right)\cong \sum _{k=1}^n\sum _{i=1}^2{p}_{k,i}h\left({(\left[{\mu }_{X,1},{\mu }_{X,2},\dots x_{k,i},\dots {\mu }_{X,n}\right])}^3\right)$(18) )

(16) $E\left(Y\right)\cong \sum _{k=1}^n\sum _{i=1}^2{p}_{k,i}h\left(\left[{\mu }_{X,1},{\mu }_{X,2},\dots x_{k,i},\dots {\mu }_{X,n}\right]\right)$(16)

(17) $E\left(Y^2\right)\cong \sum _{k=1}^n\sum _{i=1}^2{p}_{k,i}h\left({(\left[{\mu }_{X,1},{\mu }_{X,2},\dots x_{k,i},\dots {\mu }_{X,n}\right])}^2\right)$(17)

(18) $E\left(Y^3\right)\cong \sum _{k=1}^n\sum _{i=1}^2{p}_{k,i}h\left({(\left[{\mu }_{X,1},{\mu }_{X,2},\dots x_{k,i},\dots {\mu }_{X,n}\right])}^3\right)$(18)

Step 8 - Calculating the mean, standard deviation, and skewness of the output variables.

(19) ${\mu }_Y=E\left(Y\right)$(19)

(20) ${\sigma }_Y=\sqrt{E\left(Y^2\right)-\left(E{\left(Y\right)}^2\right)}$(20)

(21) ${\lambda }_Y\cong \frac{{\sum }_{k=1}^n{\sum }_{i=1}^2{p}_{k,i}h\left(\left[{\mu }_{X,1},{\mu }_{X,2},\dots x_{k,i},\dots {\mu }_{X,n}\right]\right)-{\mu }_Y{)}^3}{{{\sigma }_Y}^3}$(21)

After the 8th step, this process is repeated until the desired results are obtained from the algorithm.

2.3. PEM

This method is like the 2PEM. In the 3PEM (Das & Malakar, 2018; Gupta, 2016), Equations (22)(22) ${\xi }_{lk}=\frac{{\lambda }_{l,3}}{2}+{\left(-1\right)}^{3-k}\sqrt{m-\left(\frac{{{\lambda }_{l,3}}^2}{2}\right)}k=1,2$(22) and (23(23) $w_{l,k}=\frac{1}{m}{\left(-1\right)}^k\frac{1}{{\xi }_{k,r}\left({\xi }_{k,1}-{\xi }_{k,2}\right)}k=1,2$(23) ) are used as expressed in Equations(24)(24) ${\xi }_{lk}=\frac{{\lambda }_{l,K}}{2}+{\left(-1\right)}^{3-k}\sqrt{{\lambda }_{l,4}-\frac{3}{4}{{\lambda }_{l,3}}^2}k=1,2$(24) and (25(25) $\left\{\begin{array}{c}{w}_{l,k}=\frac{{\left(-1\right)}^{3-k}}{{\xi }_{l,k}\left({\xi }_{l,1}-{\xi }_{l,2}\right)}k=1,2\\ w_{l,3}=\frac{1}{m}-\frac{1}{{\lambda }_{l,4}-{{\lambda }_{1,3}}^2}\end{array}\right.$(25) ) for K = 3:

(22) ${\xi }_{lk}=\frac{{\lambda }_{l,3}}{2}+{\left(-1\right)}^{3-k}\sqrt{m-\left(\frac{{{\lambda }_{l,3}}^2}{2}\right)}k=1,2$(22)

(23) $w_{l,k}=\frac{1}{m}{\left(-1\right)}^k\frac{1}{{\xi }_{k,r}\left({\xi }_{k,1}-{\xi }_{k,2}\right)}k=1,2$(23)

(24) ${\xi }_{lk}=\frac{{\lambda }_{l,K}}{2}+{\left(-1\right)}^{3-k}\sqrt{{\lambda }_{l,4}-\frac{3}{4}{{\lambda }_{l,3}}^2}k=1,2$(24)

(25) $\left\{\begin{array}{c}{w}_{l,k}=\frac{{\left(-1\right)}^{3-k}}{{\xi }_{l,k}\left({\xi }_{l,1}-{\xi }_{l,2}\right)}k=1,2\\ w_{l,3}=\frac{1}{m}-\frac{1}{{\lambda }_{l,4}-{{\lambda }_{1,3}}^2}\end{array}\right.$(25)

The main purpose of this method is to estimate the equations presented as follows:

Step 1 - Average and standard deviation of random load and power generation, obtained using curves:

(26) $min{\sum }^{\left(a_{2i}{P}_{Gi}^2+a_{1i}{P}_{Gi}+a_{0i}\right)}$(26)

Here, the coefficients of $a$ are the active units of the generators in the network, and the active and reactive power is also determined as follows:

(27) $P_{Gi}^{min}\le P_{Gi}\le P_{Gi}^{max}$(27)

(28) $Q_{Gi}^{min}\le Q_{Gi}\le Q_{Gi}^{max}$(28)

Step 2 - In all cases, assuming the normal distribution and standard deviation provided by the first step, random sampling is performed.

Step 3 - Performance indicators are used to show the performance of the proposed method.

3. Possibilities in the sources

Due to the relatively large uncertainties in the generation and load of the power network, probabilistic analysis has overcome its deterministic counterpart (Yu & Rosehart, 2011). Of course, the use of probabilistic analysis, especially POPF, has been popular in power networks for a long time. Since wind and solar power have a completely random and probabilistic nature, the electric power produced during the day (and therefore the whole month) is completely variable (Mansouri et al., 2023). For this reason, if the calculations to determine the capacity of the transmission network are carried out in a similar way to other conventional power plants, i.e., with the maximum capacity of wind and solar power plants, it is possible that these resources are not properly exploited during certain periods, given the lack of radiation intensity and wind speed, which is not economical.

3.1. PDF modeling of wind power

The relation of the power that can be extracted from the wind is expressed according to Equation (29)(29) $P_W=\frac{1}{2}\rho \pi R^{2}V{w}^3{C}_P\left(\lambda ,B\right)$(29) (Böttcher, 2012; Zishan et al., 2023):

(29) $P_W=\frac{1}{2}\rho \pi R^{2}V{w}^3{C}_P\left(\lambda ,B\right)$(29)

where V_W is the wind speed, R is the blade radius, β is the blade angle, ρ is the air density, and CP is the turbine power factor, which is a function of λ and β.

Wind speed is modeled with a suitable PDF such as Weibull. It can be determined by the states of the wind speed and the random events associated with it. The Weibull distribution function is shown in Equation (30)(30) $f\left(v\right)=\frac{Kw}{Cw}{\left(\frac{V}{Cw}\right)}^{Kw-1}\mathrm{e}\mathrm{x}\mathrm{p}{\left[-\left(\frac{v}{Cw}\right)\right]}^2$(30) :

(30) $f\left(v\right)=\frac{Kw}{Cw}{\left(\frac{V}{Cw}\right)}^{Kw-1}\mathrm{e}\mathrm{x}\mathrm{p}{\left[-\left(\frac{v}{Cw}\right)\right]}^2$(30)

where Kw is the scale parameter of the Weibull wind speed, Cw is the formation parameter of the Weibull speed, and v is the wind speed. The output power of the wind turbine corresponding to each state is calculated using a power curve, according to Equation (31)(31) $P_{WTG}=\left\{\begin{array}{ccc}0& if& v⟨Viorv⟩V0\\ P_{rw}\frac{V-Vi}{Vr-Vi}& if& Vi<v<Vr\\ P_{rW}& if& Vr<v<Vo\end{array}\right.$(31) :

(31) $P_{WTG}=\left\{\begin{array}{ccc}0& if& v⟨Viorv⟩V0\\ P_{rw}\frac{V-Vi}{Vr-Vi}& if& Vi<v<Vr\\ P_{rW}& if& Vr<v<Vo\end{array}\right.$(31)

where vi, vo, and vr are the starting, stopping, and rated wind speeds, respectively (it is worth noting that v is the wind speed).

3.2. PDF modeling of solar power

Probability distribution functions are very suitable for showing the possible state of the solar radiation phenomenon. It seems that solar radiation has a high degree of uncertainty and changes as a function of several factors, such as the weather conditions, the time, the day, the month, the season, and the orientation of the solar cell generators (P_SCG). The PDF of solar radiation is modeled with a beta distribution function in the form of Equation (32)(32) $f\left(R:{\alpha }_{\beta },{\beta }_{\beta }\right)=\frac{\mathrm{\Gamma }({\alpha }_{\beta }+{\beta }_{\beta })}{\mathrm{\Gamma }({\alpha }_{\beta })\mathrm{\Gamma }({\beta }_{\beta })}{R}^{{\alpha }_{\beta }-1}(1-R{)}^{{\beta }_{\beta }}$(32) (Alsafasfeh et al., 2019):

(32) $f\left(R:{\alpha }_{\beta },{\beta }_{\beta }\right)=\frac{\mathrm{\Gamma }({\alpha }_{\beta }+{\beta }_{\beta })}{\mathrm{\Gamma }({\alpha }_{\beta })\mathrm{\Gamma }({\beta }_{\beta })}{R}^{{\alpha }_{\beta }-1}(1-R{)}^{{\beta }_{\beta }}$(32)

where αβ and ββ are the coefficients of beta distribution, and PSCG is the output power related to solar radiation (so the output power model requires the solar radiation model). The output power as a function of radiation is shown bin Equation (33)(33) $P_{SCG}=\left\{\begin{array}{ccc}{P}_{rs}& if& v\le R<R_C\\ P_{rs}\frac{R}{R_{STD}{R}_C}& if& RC\le R<RSTD\\ P_{rs}& if& R_{STD}\le R\end{array}\right.$(33) :

(33) $P_{SCG}=\left\{\begin{array}{ccc}{P}_{rs}& if& v\le R<R_C\\ P_{rs}\frac{R}{R_{STD}{R}_C}& if& RC\le R<RSTD\\ P_{rs}& if& R_{STD}\le R\end{array}\right.$(33)

where R is the intensity of the sun’s radiation, R_C is the specific radiation intensity (usually 150 W/m2), RSTD is the radiation intensity under standard conditions (usually 1000 W/m2), P_rs is the power production of the solar cell, and P_SCG is the total output power.

4. Point estimate while considering correlation variables

Random numbers with this distribution are not easily generated, which is why our method can be used to generate multivariate random numbers with any distribution.

Let x1 be a random variable with a uniform distribution U (0, 1)—the CDF of a U (0, 1) is H(x) = x, and let us take n samples from this variable. Let k be the sample. Then, in order to generate random numbers, it is necessary to perform the operation on a given distribution with the invertible cumulative distribution function. If this new variable is transformed, then it has a U (0, 1) distribution. This method starts by generating random numbers in a multivariate normal random variable with a specified correlation matrix that forms the array, where m is the number of variables and n is the sample size. Each element is such that $\mathrm{i}$is a variable and j is an example. These numbers can be easily generated by standard programs. In the second step, a normal transformation of these variables is performed to achieve a uniform multivariate x₂ distribution. In other words, x₂ = F(x₁), where F(x) is a multivariate normal CDF. The third step consists of converting multivariate uniform random numbers into series with the desired marginal distribution G.

5. Test system

To verify the effectiveness of the proposed method regarding POPF with correlated variables, a case study with the proposed methods was carried out. The IEEE 30-bus network was used to test the proposed model, whose information is provided in Tables 1 and 2.

Table 1. Network load specifications

Buses	Load
	MW	MVAR
1	0	0
2	21.7	12.7
3	2.4	1.2
4	7.6	1.6
5	94.2	19
6	0	0
7	22.8	10.9
8	30	30
9	0	0
10	5.8	2
11	0	0
12	11.2	7.5
13	0	0
14	6.2	1.6
15	8.2	2.5
16	3.5	1.8
17	9	5.8
18	3.2	0.9
19	9.5	3.4
20	2.2	0.7
21	17.5	11.2
22	0	0
23	3.2	1.6
24	8.7	6.7
25	0	0
26	3.5	2.3
27	0	0
28	0	0
29	2.4	0.9
30	10.6	1.9

Table 2. Network line specifications

Bus lines connected to each other		line resistance	Line reactance
1	2	0.0192	0.0575
1	3	0.0452	0.1852
2	4	0.057	0.1737
3	4	0.0132	0.0379
2	5	0.0472	0.1983
2	6	0.0581	0.1763
4	6	0.0119	0.0414
5	7	0.046	0.116
6	7	0.0267	0.082
6	8	0.012	0.042
6	9	0	0.208
6	10	0	0.556
9	11	0	0.208
9	10	0	0.11
4	12	0	0.256
12	13	0	0.14
12	14	0.1231	0.2559
12	15	0.0662	0.1304
12	16	0.0945	0.1987
14	15	0.221	0.1997
16	17	0.0824	0.1923
15	18	0.1073	0.2185
18	19	0.0639	0.1292
19	20	0.034	0.068
10	20	0.0936	0.209
10	17	0.0324	0.0845
10	21	0.0348	0.0749
10	22	0.0727	0.1499
21	22	0.0116	0.0236
15	23	0.1	0.202
22	24	0.115	0.179
23	24	0.132	0.27
24	25	0.1885	0.3292
25	26	0.2544	0.38
25	27	0.1093	0.2087
28	27	0	0.396
27	29	0.2198	0.4153
27	30	0.3202	0.6027
29	30	0.2399	0.4533
8	28	0.0636	0.2
6	28	0.0169	0.0599

According to the planning policy, and considering the increase in energy production, microgrids should be installed near buses 5, 7, and 8. Network buses can also be selected for connection. The wind turbine, with a rated power of 2.5 MW, starts working at 2.5 m/s, has a rated speed of 13 m/s, and stops at 25 m/s. The capacity of the solar system is also considered to be 10 MW.

6. Simulation results

The input data includes active and reactive loads in the load bus, as well as real power generation and voltage range in the production buses. As mentioned, OPF is a non-linear optimization. To determine the optimal value of electrical variables in a power network that has an objective function and a number of constraints. To achieve a probability distribution in the bus power injection is P. Even if this statement is not explicitly stated, the assumptions made in the PDF calculations in P refer to a specific transmission rule. A commonly used dispatch rule is one of the specific assumptions that the output and load are balanced on an auxiliary bus. In this case, the reduction of loads and lack of production are caused by production changes in this bus. The variance of the power injected into the auxiliary bus can be very important. It is also important to understand that when the variance of Ps is large, the variance of the load spread on the lines connected to the auxiliary bus will also be very large. If the position of the auxiliary bus is changed, the variances in the load distribution will change because the compensation for uncertainty in the load and generation is done through different schemes in the system.

The results obtained with regard to microgrids in the network and the implementation of the POPF study with the aforementioned methods are summarized in Table 3, which includes the costs, losses, and power production of the microgrids at buses 5, 7, and 8. Production unit 8 is a marginal unit, which means that it the most expensive production unit. Another reason for this is that the standard deviation of the production in this unit has the highest value in the network. The unit at bus 5 has the largest deviation, due to the fact that more consumers are located near that bus and load changes directly affect its production. According to the results obtained from the optimal possible load distribution, for bus 8, the possible load distribution value is 52.70 MW, which requires a wind power plant. The capacity of each wind turbine is 2.5 MW (i.e., the power obtained from the load distribution). This means 22 wind turbines. For buses 2 and 5, the possible load distribution is less than 10 MW, thus requiring a solar power plant.

Table 3. Results obtained with regard to microgrids in the network and the implementation of the POPF

Parameter	Average			Standard deviation
	2PEM	3PEM	MC	2PEM	3PEM	MC
(Cost10^3$)	29.36	29.37	29.48	4.58	4.59	4.61
Lasses(MW)	7.58	7.58	7.67	1.43	1.43	1.44
PG2(MW)	3.24	3.22	3.22	50	0.51	0.52
PG5(MW)	6.56	6.57	6.56	1.04	1.04	1.06
PG8(MW)	52.77	52.76	52.7	8.19	8.18	8.15

The PDF, in terms of the total implementation costs, is shown in Figures 1 - 3. According to the results, the probability density curves obtained from the 2PEM and 3PEM methods are very close to each other, which confirms the accuracy of these methods. Thus, because the execution cost has a relatively small asymmetry, its PDF is relatively symmetrical and may be approximated with a normal distribution. For the total active losses, since the value of asymmetry (skewness in the statistics) is relatively large, the PDF tends to the right and therefore cannot be approximated by a normal distribution. Sometimes, the data creates a curve that does not have a normal distribution, i.e., its two ends are not symmetrical. When there is no proportion between its two ends, the curve is skewed.

Figure 1. PDF regarding execution costs – MCM.

Figure 2. PDF regarding execution costs − 2PEM.

Figure 3. PDF regarding execution costs − 3PEM.

Now, considering the correlated variables, the average results and the standard deviation of the losses should be examined. All times are correlated with each other by a factor of 1.0. The wind power plant has a correlation coefficient of −2.0, and the solar power plant shows a value of −3.0. The convergence of the results regarding the average value shows the standard deviation of the active losses for the studied system (Figures 4 and 5). The simulation times are also shown in Table 4.

Figure 4. Convergence of the results regarding the average.

Figure 5. Convergence of results regarding the standard deviation.

Table 4. Execution times

Method	Execution time (seconds)
MC	145
3PEM	34
2PEM	26

7. Conclusion

In this article, the POPF was examined with regard to power systems with the presence of microgrids. Some suggested methods for calculating PDF and the power flow in networks with loads and resources such as solar and wind generation (which variable production) were also assessed. In buses with more consumers, these resources are used to provide power and support. Thus, they were placed in the desired buses, and the possible production capacity, losses, and costs were determined. The results regarding uncertainty and correlation between the input variables during the operation of the power system are also presented. Therefore, the development of new methods for probabilistic studies with uncertain correlated variables is of great interest in this field. To evaluate the implementation of these methods, an IEEE 30-bus test system was employed, with favorable results. The optimal load distribution for the system was obtained for the desired buses. As seen, the possible optimal load distribution in the system was obtained for the desired tires. For complex systems, using the 2PEM method has better performance than other methods. The non-deterministic load of the system can lead to significant changes in various parameters of the system. The intensity of these changes depends on the quality of cargo and production at any moment. Therefore, to maintain the security of the system at the optimal level, the rate of penetration of non-deterministic power productions such as renewable energies should be limited in any area.

To continue the work done, the following recommendations are provided for future research in this field.

• Using modern methods and meta-heuristic algorithms to get the lowest cost

• Network reliability

Disclosure statement

No potential conflict of interest was reported by the author(s).