Generalized Normal Distribution Optimization Algorithm for Economic Dispatch with Renewable Resources Integration

Sadmanul Hoque ¹, Md. Rashidul Islam ¹, Md Shafiullah ^2,*, Saymun Adnan ¹, Md. Samiul Azam ¹

1. Electrical and Electronic Engineering Department, International Islamic University Chittagong, Chittagong, Bangladesh

2. Control & Instrumentation Engineering Department, King Fahd University of Petroleum & Minerals, Saudi Arabia

* Correspondence: Md Shafiullah

Academic Editor: Grigorios L. Kyriakopoulos

Special Issue: Machine Learning and Metaheuristic Optimizations for Power System Management 

Received: June 18, 2023 | Accepted: September 25, 2023 | Published: September 28, 2023

Journal of Energy and Power Technology 2023, Volume 5, Issue 3, doi:10.21926/jept.2303030

Recommended citation: Hoque S, Islam MR, Shafiullah M, Adnan S, Azam MS. Generalized Normal Distribution Optimization Algorithm for Economic Dispatch with Renewable Resources Integration. Journal of Energy and Power Technology 2023; 5(3): 030; doi:10.21926/jept.2303030.

© 2023 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

Optimization challenges are prevalent in various scientific and technological domains, posing difficulties due to the practical nature of the target function or model constraints. The ED is one such significant challenge in power systems. The ED tackles the task of determining the best possible combination of power production from various power plants. The objective is to meet the critical power demand while minimizing operational costs and ensuring adherence to system constraints [1]. Enhancing the solution to the ED problem can lead to substantial cost savings, prompting extensive research in this field. Researchers have employed various methodologies to address this optimization challenge, including mathematical, artificial intelligence, and hybrid approaches, each offering unique perspectives and advantages [2].

Today, electric power systems are required to have a significant percentage of renewable energy sources, and there have been numerous initiatives to integrate renewable energy (RE) resources into ED issues [3,4,5]. Solar, wind, biomass, wave, geothermal, and other energy resources that can be used for power production repeatedly are examples of renewable energy. However, the power generated from RE sources cannot be directly supplied to the utility power grid as most RE sources produce direct current (DC) voltag. In contrast, the electric power transmission system and consumer loads are operated in alternating current (AC) systems [6]. To handle challenges like determining the appropriate operating level for electric power plants to fulfill demand, a variety of conventional and nonconventional optimization techniques are used. Traditional methods, i.e., lambda iteration [7], have failed to solve such a problem. The ED problem has been solved using nonconventional ways, such as artificial bee colony algorithms (ABC) [8], hybrid grey wolf optimizer algorithms (GWO) [9], genetic algorithms (GA) [10], particle swarm optimization (PSO) [11], moth flame optimization, algorithms (MFA) [12], chameleon swarm algorithm (CSA) [13], firefly algorithms (FA) [14]. Also, the woodpecker mating algorithm (WMA) [15], bald eagle search (BES) algorithm [16], whale optimization algorithm (WOA) [17], and bat-inspired algorithm [18] were also explored to solve the ED problems.

As a result, using meta-heuristic optimization methods to resolve the ED problem has become increasingly wideheld in the last two decades, particularly ant colony optimization (ACO), genetic algorithm (GA) [10], and particle swarm optimization (PSO) [11], which have been helpful in a variety of fields of study. Meta-heuristics have become increasingly popular for four reasons: simplicity, adaptability, derivation-free method, and avoidance of local optima [19,20,21,22]. This task requires allocating loads to a plant's power generators to achieve the lowest fuel cost while addressing the power demand and transmission loss constraints. Several other versions of this issue model the same goal functions and constraints in different ways. A brief comparison of the proposed model with the noteworthy prior research in this field is provided in Table 1, highlighting the differences and improvements.

Table 1 Comparative study between previous literature and proposed model.

Table 1 shows that the suggested model fixes most of the constraints with the earlier models. On the other hand, traditional methods become exceedingly complicated when dealing with increasingly complex dispatch problems. Their lack of robustness and efficacy further limits them in several practical applications.

A new optimization approach has recently been proposed to address the photovoltaic (PV) module parameter estimations. Since normal distribution theory served as the foundation for this algorithm, it was named the generalized normal distribution optimization algorithm (GNDO) [27]. The efficiency of GNDO in avoiding local minima was demonstrated by its ability to estimate the parameters by minimizing the sum of squared error between the measured and calculated current and voltage values. The most notable quality of GNDO is that it doesn't require any work to fine-tune initial parameters, which makes it superior to most other metaheuristic algorithms now in use. Three photovoltaic model parameters are extracted using GNDO to evaluate its performance. According to experimental findings, GNDO performs better in accuracy and efficiency than the comparison algorithms [27]. Recently, GNDO has been used to solve many recent problems like permutation flow shop scheduling [28], solar PV integration in monopolar DC networks [29], and the medical feature selection Approach [30]. However it has not yet been utilized to solve ED problems. Considering its better performance in solving complex, diverse problems, GNDO is considered in this research to solve ED problems in renewable source integrated power system networks.

The main contributions of this paper are listed as follows:

• Development of a generic mathematical formulation of the ED problem of electric power system networks considering renewable energy sources.
• Deployment of a recent effective meta-heuristic algorithm, the GNDO, to find optimal solutions for four test power system networks.
• Comparison of the outcomes of the GNDO algorithm with the literature-reported strategies to demonstrate the effectiveness and superiority of the employed algorithm.

The remaining sections of this paper are organized as follows: the problem formulation is explained in Section 2, and the generalized normal distribution optimization technique is summarized in Section 3. The results and findings are discussed in Section 4, and the conclusions are provided in Section 5.

2. Problem Formulation

The economic dispatch strategy aims to reduce fuel costs while following several equity and inequality criteria. As a result, the problem is stated as follows:

2.1 Objective Function

The quadratic fuel charge equation of the thermal generating units is the cost function of the ED problem, and it is expressed as follows:

\[ Min\left(\sum\nolimits_{i=1}^NF_i(p_i)\right)=Min\left(\sum\nolimits_{i=1}^Na_i+b_iP_i+c_iP_i^2\right) \tag{1} \]

Here, a_i, b_i, c_i are i^th generating unit's cost coefficients, p_i is the generated power of the unit, and N is the number of generating units.

2.2 Power Balance Constraint

\[ P_{D}+P_{L}-\sum\nolimits_{i=1}^{Ng}P_{i}=0 \tag{2} \]

Equation (2) is the power balance constraints equation. Here, P_L is Transmission losses and P_D is total load demand. The transmission loss, P_L may be depicted by (3):

\[ P_L=\sum\nolimits_{i=1}^N\sum\nolimits_{j=1}^NP_iB_{ij}P_j+\sum\nolimits_{i=1}^NB_{0i}P_i+B_{00} \tag{3} \]

Here, N is the number of generating units and B_ij is transmission loss coefficient.

2.3 Power Output Limits

Each generating unit has minimum and maximum limits, i.e.

\[ P_i^{min}\leq P_i\leq P_i^{max} \tag{4} \]

3. Generalized Normal Distribution Optimization

This section describes the GNDO algorithm's specifics [27], divided into sub-sections. The first subsection focuses on the motivations for the GNDO algorithm. The outline of GNDO is then presented in another sub-section. Lastly, the application of GNDO for optimization is explained.

3.1 Description

Based on population location data, a generalized normal distribution approach is used to keep the individual's position current. The normal distribution concept influenced GNDO. The normal distribution, often called the Gaussian distribution, is a helpful tool for characterizing common phenomena. The following is an explanation of a normal distribution. Here, x is assumed as an arbitrary variable that follows a possible arrangement with location and scale parameters, and its probability solidity function can be formulated as below.

\[ f(x)=\frac{1}{\sqrt{2\pi\delta}}exp\left(-\frac{(x-\mu)^2}{2\delta^2}\right) \tag{5} \]

The normal distribution has two variables, according to equation (5): the scale parameter δ and the location parameter μ. The mean value and standard variance of random variables are expressed using the location and scale parameters.

Generally, there are three stages to the search procedure for population-based optimization approaches. Firstly, the dispersed distribution contains all initialized persons. Previously, individuals started moving toward the best overall solution while supervised by the planned exploration and exploitation strategies. Finally, everyone gathers around the best solution that has been found. In reality, numerous normal distributions can construct this search procedure.

To put it another way, all people's positions can be considered random variables with a normal distribution. The average position and the ideal location are more comprehensive in the first stage. The ranking standard deviation across all individuals is significantly more significant. The second step steadily reduces the disparity between the mean and optimum positions. The average change in all people's positions is getting smaller and smaller. In addition to the standard change of all individual positions, the space between the mean and optimal positions can be reduced to zero in the final stage.

3.2 The Structure of the Proposed Method

The proposed information exchange process in GNDO is local exploitation and global exploration, and it has a straightforward structure. The present mean and ideal positions guide local exploitation and are built on the generalized normal distribution model. In addition, three randomly chosen persons are linked to global exploration. A detailed explanation of both learning strategies is provided as follows.

3.2.1 Local Exploitation

Local exploitation is locating improved results within a search space that includes all persons' existing places. A GNDO model for optimization can be constructed for the association between a normal distribution and the distribution of people in the population.

\[ v_{i}^{t}=\mu_{i}+\delta_{i}\times\eta,i=1,2,3,\ldots,N \tag{6} \]

Here, δ_i is generalized standard alteration, v^t_i is the trailing vector of the i^th individual at time t, μ_i is the generalized mean position of the i^th individual, and η is the penalty factor. In addition, μ_i, δ_i, and η can be defined as in (7), (8), and (9).

\[ \mu_{i}=\frac{1}{3}\left(x_{i}^{t}+x_{Best}^{t}+M\right) \tag{7} \]

\[ \delta_{i}=\sqrt{\frac13[(x_{i}^{t}-\mu)^{2}+(x_{Best}^{t}-\mu)^{2}+(M-\mu)^{2}]} \tag{8} \]

\[ \left.\eta=\left\{\begin{matrix}\sqrt{-log(\lambda_1)}\times\cos(2\pi\lambda_2),if \ a\leq b\\\sqrt{-log(\lambda_1)}\times cos(2\pi\lambda_2+\pi),otherwise\end{matrix}\right.\right. \tag{9} \]

Here, a, b, λ_1, and λ₂ are unsystematic numbers between 0 and 1, M (10) is the mean position, and x^t_Best is the present top position of the existing population. In total, calculating M,

\[ M=\frac{\sum_{i=1}^{N}x_{i}^{t}}{N} \tag{10} \]

3.2.2 Global Exploration

Global exploration is searching a speech space around the globe for promising places. In GNDO, the global exploration is established on three persons chosen at random, which can be stated as:

\[ v_{i}^{t}=x_{i}^{t}+\beta\times(|\lambda_{3}|\times v_{1})+(1-\beta)\times(|\lambda_{4}|\times v_{2}) \tag{11} \]

In equation (11), β × (|λ₃| × v₁) is local information sharing and (1-β) × (|λ₄| × v₂) is global information sharing.

Here, λ₃ and λ₄ are arbitrary figures dependent on the standard normal distribution. Modify parameter (β) is a random quantity between 0 and 1, and v₁ and v₂ are two trail vectors. Furthermore, v₁ and v₂ can be determined by:

\[ \left.v_1=\left\{\begin{matrix}x_i^t-x_{p1}^t,if \ f(x_i^t)< f\big(x_{p1}^t\big)\\x_{p1}^t-x_i^t,otherwise\end{matrix}\right.\right. \tag{12} \]

\[ \left.v_2=\left\{\begin{matrix}x_{p2}^t-x_{p3}^t,if \ f\big(x_{p2}^t\big) < f\big(x_{p3}^t\big)\\x_{p3}^t-x_{p2}^t,otherwise\end{matrix}\right.\right. \tag{13} \]

Here, p1, p2, and p3 are three different numbers ranging from 1 to N, which come across i ≠ p1 ≠ p2 ≠ p3. Equations (12) and (13), the subsequent term on the right of (11) can be termed local knowing term, which signifies that the result p1 has information in common with the resolution i; global information sharing is the third term on the right of (11) that defines the individual i is specified information by those p2 and p3. Furthermore, λ₃ and λ₄ are casual numbers with the usual normal distribution, which can sort GNDO has more excellent search space in acting out the global exploration. The adjustment parameter (β) equates the two information distribution approaches. The total sign in (11) is to stay steady with the showing mechanism in (12) and (13).

3.2.3 The GNDO Implementation for Optimization

The operation of GNDO is discussed in this section. The suggested GNDO is founded on local exploitation and global exploration tactics that have been established. The two strategies are equally crucial to GNDO and have the same chance of being chosen. Furthermore, like with other population-based optimization techniques, GNDO's population is initialized by

\[ x_{i,j}^t=I_j+(u_j-I_j)\times\lambda_5 \tag{14} \]

Here, i is 1, 2, 3, 4, …, N, and j is 1, 2, 3, 4, …, D.

Where λ₅ is an arbitrary digit between 0 and 1, D is the number of design variables, the upper and lower limits of the j^th design variables are u_j and I_j, respectively. Reminder that the i^th specific may not discover a good result through global exploration or local exploitation strategies. To carry the improved solution into the following generation population, a showing mechanism (15) is considered; it can be denoted as:

\[ \left.x_i^{t+1}=\left\{\begin{matrix}v_i^t,if \ f(v_i^t)< f(x_i^t)\\x_i^t,otherwise\end{matrix}\right.\right. \tag{15} \]

3.2.4 The Computation Difficulty of the Proposed Algorithm

Computation difficulty is an essential statistic for estimating the execution time of an algorithm. The computational complexity of GNDO comprises the period spent comparing and updating positions, which depends on the number of participants, iterations, and variables. N individuals must update their locations in each cycle, and N comparisons must be made. As a result, GNDO's overall computing complexity can be expressed as O (NDT_max + NT_max). The proposed GNDO algorithm's flowchart is shown in Figure 1. A flow chart illustrating the simulation process approach is displayed in Figure 2.

Figure 1 GNDO algorithm flowchart.

Figure 2 Methodology for the simulation process.

4. Results and Discussion

The proposed approach has undergone verification in four distinct power systems, encompassing three thermal units, six thermal units, ten thermal units, and three thermal units with two renewable resources power systems. In each of these setups, the cost function for the system's units is represented by a quadratic function. To address the economic dispatch problems, various techniques, including GNDO, GA, WOA, PSO, and other methods, are employed, and their outcomes are thoroughly analyzed. To conduct a comparative study, all algorithms are implemented using MATLAB to tackle ED problems effectively. The primary objective is to use GNDO to optimize power generation costs.

4.1 First Case: Three Thermal-Unit System

Table 2 provides the fuel cost coefficients for three thermal-unit systems. The results acquired for the system employing the GNDO are shown in Table 3 and compared with GA [31] ,BES [16], and WOA [17]. Figure 3 shows the convergence curve of three thermal fuel costs using the GNDO algorithm for 1000 iterations. It gives the optimum value within 40 iterations. Table 3 shows the fuel costs for the three-unit system at 400MW using different algorithms, where the GNDO algorithm provides minimum fuel cost compared to GA, BES, and WOA. Also, compared to GA, BES, and WOA, power loss is reduced while utilizing the GNDO algorithm. In this case, the computation time for 1000 iterations is 9.44 seconds.

Table 2 Coefficients of fuel cost for three thermal-unit systems [32].

Figure 3 Cost convergence characteristics of GNDO for three thermal-unit systems.

Table 3 Comparison of three thermal-unit systems at a power demand of 400MW.

4.2 Second Case: Six Thermal-Unit System

Table 4 for the six thermal-unit systems shows the fuel cost coefficients. The results acquired for the six thermal-unit systems by the GNDO are shown in Table 5 and are compared with GA and WOA.

Table 4 Coefficients of fuel cost for six thermal-unit systems [33].

Table 5 Comparison of a six thermal-unit system at P_D = 1263 MW.

Table 4 shows the fuel costs for six thermal-unit systems at 1263MW using different algorithms, where the GNDO algorithm provides the lowest fuel cost and less power loss compared to the WOA algorithm. And nearly identical fuel costs and power loss compared to GA [33], a well-established methodology. Figure 4 indicates the cost convergence curve of six thermal-unit systems using the GNDO algorithm for 1000 iterations. It provides optimum value within 60 iterations. In this instance, the process takes 9.56 seconds to complete 1000 iterations.

Figure 4 Cost convergence characteristics of GNDO for six thermal-unit system.

4.3 Third Case: Ten Thermal-Unit System

The results acquired for the ten generator systems for ten thermal-unit coefficients [32] by the GNDO are given in Table 6 and compared with other algorithms.

Table 6 Comparison of ten thermal-unit systems at P_D = 2000MW.

Figure 5 illustrates the cost convergence curve of ten thermal units using the GNDO algorithm for 1000 iterations. It offers optimum value within 110 iterations. Table 6 shows the fuel costs for ten thermal-unit systems at 2000 MW using different algorithms, where the GNDO algorithm provides less fuel cost than the WOA algorithms and almost similar fuel cost compared to a well-established algorithm, FPA. In this case, 1000 iterations are completed in 9.48 seconds.

Figure 5 Cost convergence characteristics of GNDO for ten thermal-unit systems.

4.4 Fourth Case: Three Thermal-Unit with Two Renewable-Unit System

In this case, solar and wind generation data are considered renewable energy, as given in Table 7, where Table 8 provides the fuel price coefficients. The GNDO's three thermal-unit and two renewable-unit system results are presented in Table 9 and distinguished with WOA. The wind and solar generation values in Table 7 are plotted in Figure 6. These wind and solar power are added to the load. In this case, total fuel cost is reduced as they are renewable energy.

Table 7 The data of solar and wind generation (24-hour) [34].

Table 8 Fuel cost coefficients for three thermal-unit and two renewable-unit system [34].

Figure 6 Wind and solar power generation for 24 hours.

Figure 7 shows the changes in fuel cost concerning time, considering the power demand variations over 24 hours. Table 9 shows the fuel costs for three thermal-unit and two renewable-unit systems using WOA and GNDO algorithms, where the GNDO algorithm provides the lowest fuel cost compared to the WOA algorithms. One thousand iterations are completed in this case in 14.98 seconds.

Figure 7 Hourly operation cost with GNDO for three thermal-unit and two renewable-unit systems.

Table 9 Data of fuel cost of three thermal-unit and two renewable-unit systems for 24 hours.

5. Conclusions

This paper employed GNDO to solve the ED problem for four electric power system networks: three thermal units, six thermal units, ten thermal units, and three thermal units with two renewable resources under load and generator limit constraints. The placement of each individual is rearranged using the GNDO. The key feature of the GNDO, connected to the most recent metaheuristic algorithms, is that it does not require a contest for suitable initial parameter adjustment. The employed algorithm significantly outperformed WOA in all scenarios. Considering the availability of solar and wind power demonstrated that the proposed method could boost convergence and pursue the ideal global solution.

In most cases, the simulation results have shown that the proposed algorithm outperforms other selected algorithms concerning determining the minimum generation cost and power losses. Besides, the employed strategy exhibited competitive performances for the remaining test cases. However, it is worth noting that this article does not consider network constraints, such as transmission congestion. Including these constraints couldcan be considered regarded for future research and analysis.

Author Contributions

Conceptualization, S.H., M.S., M.R.I.; methodology, S.H., S.A., M.S.A.; software, S.A., M.S.; investigation, S.A., M.S.,A., M.R.I.; data curation, M.R.I., M.S.; writing—original draft preparation, S.H., S.A., M.S.A.; writing—review and editing, M.R.I., M.S.

Competing Interests

The authors have declared that no competing interests exist.

Abbreviations